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(Download) CBSE Class-12 Sample Paper (Computer Science) 2014-15

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(Download) CBSE Class-12 Sample Paper (Computer Science) 2014-15

SECTION A

Time allowed : 3 hours

Maximum Marks: 70

Section A (C++)

Q1. a. Differentiate between ordinary function and member functions in C++.

Explain with an example. [2]

b. Write the related library function name based upon the given information in C++.
(i) Get single character using keyboard. This function is available in stdio.h file.
(ii) To check whether given character is alpha numeric character or not. This function is available in ctype.h file. [1]

c. Rewrite the following C++ program after removing all the syntactical errors (if any), underlining each correction. : [2]

include<iostream.h>
#define PI=3.14
void main( )
{ float r;a;
cout<<’enter any radius’;
cin>>r;
a=PI*pow(r,2);
cout<<”Area=”<<a
}

(Download) CBSE Class-12 Sample Paper (Biotechnology) 2014-15

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(Download) CBSE Class-12 Sample Paper (Biotechnology) 2014-15

SECTION A

1. “Golden rice is nutritionally superior to normal rice”. Justify. 1

2. Write the complementary sequence of the following sequence

5’ ATMKGCSWNB 3’ 1

3. What is the specific role of baffles in large scale fermentation process? 1

4. A,B and C are type I, type II and type III enzymes respectively. Which of these is mostly used in recombinant DNA technology and why? 1

5. Curd is used as a pro-biotic. Give reason 1

6. Give the commercial importance of flavr savr variety of tomatoes. 1

SECTION B

7. Both Hind III and Pvu I are unique restriction sites in insert and host DNA. Why is Hind III still a preferred restriction enzyme? 2

8. Detergent manufacturers supplement their products with a protease. Indicate why the enzyme is inactivated by bleach and how can this probelm be overcome? 2

OR

American scientists have developed a formulation based on whey proteins for reducing the viral load in Hepatitis patients. What could be the possible scientific explanation = for this therapeutic effect?

9. Indicate two techniques which can be used for amplifying DNA. 2

10. An interesting property of restriction enzymes is to precisely cut DNA. Restriction enzymes typically recognize a symmetrical sequence of DNA. 2



A. What is this symmetrical sequence of DNA known as?
B. What is the advantage of these overhanging chains?

11. What are data retrieval tools? Which tool would be useful for obtaining comprehensive information on a biological question? 2

12. How can microbes producing novel products be identified using metagenomics?. 2

13. E.coli is not the preferred host for the expression of a protein produced in papaya.Justify. 2

14. (i) Mention the number of primers required in each cycle of polymerase chain 2 reaction (PCR).
(ii) Give the characteristic feature and source organism of the DNA polymerase used in PCR.

SECTION C

15. Protoplasts from two different sources are isolated and allowed to randomly fuse with each other. Name this process and indicate how this fusion can be done and give its agricultural importance? 3

16. Discuss two important medical applications of tissue engineering. 3

17. Study the following enzyme purification table and answer the questions that follow:  3

(Download) CBSE Class-12 Sample Paper (Biology) 2014-15

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(Download) CBSE Class-12 Sample Paper (Biology) 2014-15

Time allowed: 3hrs

 Maximum Marks: 70

General Instructions:

1. There are a total of 26 questions and five sections in the question paper. All questions are compulsory.

2. Section A contains question number 1 to 5, Very Short Answer type questions of one mark each.

3. Section B contains question number 6 to 10, Short Answer type I questions of two marks each.

4. Section C contains question number 11 to 22, Short Answer type II questions of three marks each.

5. Section D contains question number 23, Value Based Question of four marks.

6. Section E contains question number 24 to 26, Long Answer type questions of five marks each.

7. There is no overall choice in the question paper, however, an internal choice is provided in one question of two marks, one question of three marks and all three questions of five marks. An examinee is to attempt any one of the questions out of the two given in the question paper with the same question number.

Section - A

1. A tissue culture experiment has been performed with a plant tissue infected with TMV. Meristematic tissue produces healthy plant. Reason out the possibility of obtaining such result. 1

2. State a method of cellular defence which works in all eukaryotic organisms. 1

3. In case of an infertile couple, the male partner can inseminate normally but the mobility of sperms is below 40 percent. Judge, which kind of ART is suited in this situation to form an embryo in the laboratory, without involving a donor? 1

4. Calculate the length of the DNA of bacteriophage lambda that has 48502 base pairs. 1

5. If two genes are located far apart from each other on a chromosome, how the frequency of recombination will get affected? 1

Section - B

(Download) CBSE Class-12 Sample Paper (Accountancy) 2015

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(Download) CBSE Class-12 Sample Paper (Accountancy) 2015

Part A : Accounting for Partnership Firms and Companies 60 Marks 150 Periods

Unit 1: Accounting for Partnership Firms

  •  Partnership: features, Partnership deed.

  •  Provisions of the Indian Partnership Act 1932 in the absence of partnership deed.

  •  Fixed v/s fluctuating capital accounts. Preparation of Profit & Loss Appropriation account- division of Profit among partners, guarantee of profits.

  •  Past adjustments (relating to interest on capital, interest on drawing, salary and profit sharing ratio).

  •  Goodwill: nature, factors affecting and methods of valuation - average profit, super profit and capitalization.

Scope : Interest on partner’s loan is to be treated as a charge against profits.

Accounting for Partnership firms - Reconstitution and Dissolution.

  •  Change in the Profit Sharing Ratio among the existing partners - sacrificing ratio, gaining ratio. Accounting for revaluation of assets and re-assessment of liabilities and treatment of reserves and Accumulated profits.

  •  Admission of a partner - effect of admission of a partner on change in the profit sharing ratio, treatment of goodwill (as per AS 26), treatment for revaluation of assets and re - assessment of liabilities, treatment of reserves and accumulated profits, adjustment of capital accounts and preparation of balance sheet.

  •  Retirement and death of a partner: Effect of retirement /death of a partner on change in profit sharing Ratio, treatment of goodwill (as per AS 26), treatment for revaluation of assets and re - assessment of Liabilities, adjustment of accumulated profits and reserves, adjustment of capital accounts and preparation of balance sheet. Preparation of loan account of the retiring partner.
    – Calculation of deceased partner's share of profit till the date of death. Preparation of deceased Partner’s capital account, executor's account and preparation of balance sheet.

  •  Dissolution of a partnership firm: types of dissolution of a firm. Settlement of accounts - preparation of

Realization account, and other related accounts: Capital accounts of partners and Cash/Bank A/c (Excluding piecemeal distribution, sale to a company and insolvency of partner(s)).

Note:

(i) If value of asset is not given, its realized value should be taken as nil.
(ii) In case, the realization expenses are borne by a partner, clear indication should be given regarding the payment thereof.

Unit 2: Accounting for Companies

(Download) CBSE Class-10 Sample Paper (Social Science) 2016

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(Download) CBSE Class-10 Sample Paper (Social Science) 2016

Very short answers -VSA (1 mark)

1. What did the inland Emigrating Act of 1859 declare?
2. Name any one fossil fuel used to generate thermal electricity.
3. Give one special feature that distinguishes a pressure group from a political party.
4. Give one reason why multi party system has evolved in India.
5. With the help of an example each compare a single issue movement and a long term movement.
6. What is meant by ‘Fair Globalisation’?
7. Why do banks ask for collateral while giving credit to the borrower?
8. How will you justify that you are an alert consumer while buying a commodity from the market. Give two example/ situations to support you.

(Download) CBSE Class-10 Sample Paper (Science) 2016

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(Download) CBSE Class-10 Sample Paper (Science) 2016

VSA: (1 marks each)

1. State one reason for placing Mg and Ca in the same group of the periodic table 1
2. State two roles of testes in male reproductive system 1
3. Why is re-use of materials better than recycling in saving the environment? 1

SA: (2 marks each)

1. What is a homologous series? Write the name and draw the structure of the second member of the alkene series. 2
2. Give one example of eacha)
a) Metal having valency 2.
b) Non metal having valency 2.
c) Element with completely filled outermost shell.
d) d) Element with three shells, having 4 electrons in the outermost shell.
2
3. To protect the food plants from insects, an insecticide was sprayed in small amounts but it was detected in high concentration in human beings. How did it happen? 2

 SA: II (3 marks each)

1. How do the following traits change in a period from left to right in the periodic table –
a) Atomic size
b) Valency
c) Metallic character.
3
2. What are isomers? Draw all possible isomers of C4 H10 and name them 3
3. Two elements ‗X‘ and ‗Y‘ belong to the second group of the periodic table. ‗X‘ has 2 shells and ‗Y‘ has 3 shells in it –
a) Which of these is more metallic in nature and why?
b) What is the formula of the chloride of ‗X‘ and sulphide of ‗Y‘?
c) Is the valency of ‗X‘ same as that of ‗Y‘ or different? Why?
3
4. a) Why do we see different variety of organisms around us?
b) In which type of reproduction –
i) Off springs are identical?
ii) Exact similar offspring‘s are not produced?
3
5. How do species of two isolated subpopulations become two different species? 3
6. Define –
a) Spore formation
b) Regeneration
c) Multiple fission
3
7. ―Variation is useful for the survival of species overtime but the variants have unequal chances of survival.''
Explain the statement.
3
8. What is ‗Placenta‘? State its function in human female. 3
9. a) State the law of refraction of light that defines the refractive index of one medium with respect to the other.
b) Express it mathematically also.
c) Write the expression relating the refractive index of medium ‗A‘ with respect to the medium ‗B‘ to the speed of light in the two media ‗A‘ & ‗B‘. Name the constant when medium ‗B‘ is vacuum.
3
10. Why does the sky appear blue to an observer from the surface of earth? What will be the colour of the sky for an astronaut in a apace station? Give reason for your answer. 3
11. Name the device (type of lens / mirror) used in the following cases and draw ray diagrams to show the image formation in each case –
a) Object is placed between the device and its focus, the enlarged image is formed behind it
b) The object is placed between infinity and the device, the image is formed behind the device between its pole and focus.
3
11. An object of height 2 cm is placed at a distance of 30 cm from a convex lens of focal length 10 cm. Find the position, nature and height of the objec 3

(Download) CBSE Class-10 Sample Paper (English Language & Literature) 2014-2015

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(Download) CBSE Class-10 Sample Paper (English Language & Literature) 2014-2015

Section A
Reading - 20 Marks

Time: 3hrs.

M.M.70

VSAQ - 1 MARK

Q.1. Read the passage given below:

 8 Marks

Big, bold and beautiful, the Rafflesia arnoldii boasts the title of the largest flower in the world and can grow to massive proportions, with a flower diameter of up to one meter (three feet) and a hefty weight of up to 11 kilograms (24 lbs). It might seem like a great gift for that special someone except that it's nicknamed the corpse flower and smells like rotting meat, so may not be quite as endearing as expected.

Resembling the coiled tentacles of an octopus up close, the stinky flower leaves such a lasting impression that it was once described by Swedish zoologist Eric Mjöberg in 1928 as having “a penetrating smell more repulsive than any buffalo carcass in an advanced stage of decomposition.” Nice. Technically a plant, although it has no leaves, stems or roots that the eye can see, the corpse flower relies on its strong perfume to attract insects that help with pollination. The other not so pleasant qualities of the flower are its parasitic tendencies; by living off the water and nutrients from the hapless Tetrastigma vine, the corpse flower is able to grow as large as it does. And whether it's considered a beauty, beast, or both, the lure of this bewitching flower is hard to resist. However, to be successful in a quest to find it, a few stars need to align. Found only in the dwindling rainforests of Sumatra and Borneo, pollination is rare and the bud death rate is high at 80-90%. The few buds that actually bloom take many months to do so, and when they do they last no more than a few days before dying. The good news is that there are great conservation efforts in place to protect the habitat of the Rafflesia species so future generations can experience the sight and smell of the largest flower on Earth. (314 words)

(Download) CBSE Class-10 Sample Paper (English Communicative) 2016

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(Download) CBSE Class-10 Sample Paper (English Communicative) 2016

Section A
Reading

Q1. VSA (1 mark each)

Read the following passage carefully

First introduced in 1927, The Hardy Boys Mystery Stories are a series of books about the adventures of brothers Frank and Joe Hardy, teenaged detectives who solve one baffling mystery after another. The Hardy Boys were so popular among young boys that in 1930 a similar series was created for girls featuring a sixteen-year-old detective named Nancy Drew. The cover of each volume of The Hardy Boys states that the author of the series is Franklin W. Dixon; the Nancy Drew Mystery Stories are supposedly written by Carolyn Keene. Over the years, though, many fans of both series have been surprised to find out that Franklin W. Dixon and Carolyn Keene are not real people. If Franklin W. Dixon and Carolyn Keene never existed, then who wrote The Hardy Boys and Nancy Drew mysteries?

The Hardy Boys and the Nancy Drew books were written through a process called ghostwriting. A ghostwriter writes a book according to a specific formula. While ghostwriters are paid for writing the books, their authorship is not acknowledged, and their names do not appear on the published books .Ghostwriters can write books for children or adults, the content of which is unspecific. Sometimes they work on book series with a lot of individual titles, such as The Hardy Boys and the Nancy Drew series.

The initial idea for both the Hardy Boys and the Nancy Drew series was developed by a man named Edward Stratemeyer, who owned a publishing company that specialized in children’s books. Stratemeyer noticed the increasing popularity of mysteries among adults, and surmised that children would enjoy reading mysteries about younger detectives with whom they could identify. Stratemeyer first developed each book with an outline describing the plot and setting. Once he completed the outline, Stratemeyer then hired a ghostwriter to convert it into a book of slightly over 200 pages. After the ghostwriter had written a draft of a book, he or she would send it back to Stratemeyer, who would make a list of corrections and mail it back to the ghostwriter. The ghostwriter would revise the book according to Stratemeyer’s instructions and then return it to him. Once Stratemeyer approved the book, it was ready for publication. Because each series ran for so many years, Nancy Drew and The Hardy Boys both had a number of different ghostwriters producing books; however, the first ghostwriter for each series proved to be the most influential. The initial ghostwriter for The Hardy Boys was a Canadian journalist named Leslie McFarlane. A few years later, Mildred A. Wirt, a young writer from Iowa, began writing the Nancy Drew books. Although they were using prepared outlines as guides, both McFarlane and Wirt developed the characters themselves. The personalities of Frank and Joe Hardy and Nancy Drew arose directly from McFarlane’s and Wirt’s imaginations. For example, Mildred Wirt had been a star college athlete and gave Nancy similar athletic abilities. The ghostwriters were also responsible for numerous plot and  setting details. Leslie McFarlane used elements of his small Canadian town to create Bayport, the Hardy Boys fictional hometown.

(Date Sheet) CBSE : Class 10th Board Examination - 2016

GENERAL: 

(Date Sheet) CBSE : Class 10th Board Examination - 2016

DAY - DATE - TIME SUB-CODE SUBJECT NAME

Tuesday, 01 March, 2016 10.30 A.M.

  • 401 DYNAMICS RETAIL(O)
  • 402 INFO TECH (O)
  • 403 SECURITY (O)
  • 404 AUTO TECH (O)
  • 406 INT. TOURISM(O)
  • 461 DYNAMICS RETAIL(C)
  • 462 INFO TECH (C)
  • 463 SECURITY (C)
  • 464 AUTO TECH (C)
  • 466 INT. TOURISM(C)

Wednesday, 02 March, 2016 10.30 A.M.

  • 086 SCIENCE
  • 090 SCIENCE W/O PRAC

Thursday, 03 March, 2016 10.30 A.M.

  • 007 TELUGU
  • 018 FRENCH
  • 076 NATIONAL CADET CORPS
  • 166 INFORMATION & COMM TECH

Saturday, 05 March, 2016 10.30 A.M.

  • 049 PAINTING
  • 096 SPANISH

Tuesday, 08 March, 2016 10.30 A.M.

  • 002 HINDI COURSE A
  • 006 TAMIL
  • 085 HINDI COURSE B

(Date Sheet) CBSE : Class 12th Board Examination - 2016

GENERAL: 

(Date Sheet) CBSE : Class 12th Board Examination - 2016

DAY - DATE - TIME SUB-CODE SUBJECT NAME

Tuesday, 01 March, 2016 10.30 A.M.

  • 001 ENGLISH ELECTIVE
  • 101 ENGLISH ELECTIVE-C
  • 301 ENGLISH CORE

Thursday, 03 March, 2016 10.30 A.M.

  • 054 BUSINESS STUDIES
  • 199 BAHASA MELAYU
  • 624 ELECT APPLIANCES
  • 654 B THERAPY&HAIR DES
  • 745 BEAUTY & HAIR
  • 762 BASIC HORTICULTURE
  • 778 PRINTED TEXTILE
  • 789 OP & MAINT. OF COMM DEV

Saturday, 05 March, 2016 10.30 A.M.

  • 042 PHYSICS
  • 070 HERITAGE CRAFTS
  • 123 PERSIAN
  • 197 KASHMIRI
  • 605 SECT PRAC & ACCNTG
  • 619 CASH MGMT & H-KEEP
  • 623 ELECTRICAL MACHINE
  • 630 FABRICATN.TECH-II
  • 632 AC & REFRGTN-III
  • 642 VEGETABLE CULTURE
  • 655 COSMETIC CHEMISTRY
  • 658 OPTICS
  • 661 CLINICAL BIOCHEMISTRY
  • 663 FUND OF NURSING II
  • 667 RADIOGRAPHY-GENL
  • 684 TEXTILE SCIENCE
  • 699 I T SYSTEM
  • 731 CHILD HEALTH NURSG
  • 751 BAKERY
  • 776 GARMENT CONSTRUCTION
  • 777 TRADITIONAL IND.TXT
  • 787 ELECTRICAL MACHINE
  • 800 SECURITY

(News) CBSE books and learning material will be available online for free

GENERAL: 

CBSE books and learning material will be available online for free


CBSEAll CBSE books and learning material will be made available online free as part of the Centre’s good governance efforts, Union Human Resource Minister Smriti Irani said on Saturday.

At a function in Kendriya Vidyalaya in east Delhi, she said that initiatives would be undertaken to ensure holistic nurturing and improve learning outcomes at these schools.

“We made NCERT books available online free through e-books and mobile applications a month-and-a-half ago. We are similarly going to make CBSE books available online along with additional learning material and videos as part of our good governance efforts,” the Minister said.

She said the Centre would launch Shaala Darpan and Saransh services (for Class I to Class XII students) in Kendriya Vidyalayas in the next academic year.

The Shaala Darpan service is aimed at using SMSs to keep parents informed about their wards’ attendance, time table and marks in exams, while Saransh will help parents compare the subject-wise learning outcomes of their children, with others at the district, State and national levels.

Earlier, addressing the function, Delhi Deputy Chief Minister Manish Sisodia stressed the role of education in the lives of children.

(News) CBSE has launched ‘e-pathshala’ for NCERT text-books and various other learning resources

GENERAL: 

CBSE has launched ‘e-pathshala’ for NCERT text-books and various other learning resources


CBSEAs a part of the Digital India Campaign, the Ministry of HRD has launched ‘e-pathshala’ which is a single point repository of e-resources containing, NCERT text-books and various other learning resources.

CBSE prescribes textbooks published by National Council of Educational Research and Training (NCERT) for classes IX to XII. For classes I to VIII, CBSE although approves syllabus as per pattern of syllabus given by NCERT, it does not prescribe any textbooks for these classes. The National Curriculum Framework, 2005 (NCF) stipulates that region specific books take care of the local context, culture and resources and therefore different books for different regions better relate to the daily lives of the local students. CBSE has directed all its affiliated schools on July 20th, 2015 that prescribing too many textbooks and coercing parents and children to buy them is an unhealthy practice and appropriate action will be taken against the erring schools if any such complaint is received. As regards schools forcing students/parents to buy books from private publishers of their choice, Central Board of Secondary Education (CBSE) has not received any substantiated report in this regard.

NCERT Physics Question Paper (Class - 11)

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NCERT Physics Question Paper (Class - 11)


:: Chapter 1 Physical World ::


Question 1.1 Some of the most profound statements on the nature of science have come from Albert Einstein, one of the greatest scientists of all time. What do you think did Einstein mean when he said : “The most incomprehensible thing about the world is that it is comprehensible”?

Question 1.2 “Every great physical theory starts as a heresy and ends as a dogma”. Give some examples from the history of science of the validity of this incisive remark.

Question 1.3
“Politics is the art of the possible”. Similarly, “Science is the art of the soluble”. Explain this beautiful aphorism on the nature and practice of science.

Question 1.4 Though India now has a large base in science and technology, which is fast expanding, it is still a long way from realising its potential of becoming a world leader in science. Name some important factors, which in your view have hindered the advancement of science in India.

Question 1.5 No physicist has ever “seen” an electron. Yet, all physicists believe in the existence of electrons. An intelligent but superstitious man advances this analogy to argue that ‘ghosts’ exist even though no one has ‘seen’ one. How will you refute his argument ?

Question 1.6
The shells of crabs found around a particular coastal location in Japan seem mostly to resemble the legendary face of a Samurai. Given below are two explanations of this observed fact. Which of these strikes you as a scientific explanation ?

(a) A tragic sea accident several centuries ago drowned a young Samurai. As a tribute to his bravery, nature through its inscrutable ways immortalised his face by imprinting it on the crab shells in that area
(b) After the sea tragedy, fishermen in that area, in a gesture of honour to their dead hero, let free any crab shell caught by them which accidentally had a shape resembling the face of a Samurai. Consequently, the particular

shape of the crab shell survived longer and therefore in course of time the shape was genetically propagated. This is an example of evolution by artificial selection. [Note :This interesting illustration taken from Carl Sagan’s ‘The Cosmos’ highlights the fact that often strange and inexplicable facts which on the first sight appear ‘supernatural’ actually turn out to have simple scientific explanations. Try to think out other examples of this kind].

Question 1.7
The industrial revolution in England and Western Europe more than two centuries ago was triggered by some key scientific and technological advances. What were these advances ?

Question 1.8 It is often said that the world is witnessing now a second industrial revolution, which will transform the society as radically as did the first. List some key contemporary areas of science and technology, which are responsible for this revolution.

Question 1.9 Write in about 1000 words a fiction piece based on your speculation on the science and technology of the twenty-second century.

Question 1.10 Attempt to formulate your ‘moral’ views on the practice of science. Imagine yourself stumbling upon a discovery, which has great academic interest but is certain to have nothing but dangerous consequences for the human society. How, if at all, will you resolve your dilemma ?

Question 1.11 Science, like any knowledge, can be put to good or bad use, depending on the user. Given below are some of the applications of science. Formulate your views on whether the particular application is good, bad or something that cannot be so clearly categorised :

(a) Mass vaccination against small pox to curb and finally eradicate this disease from the population. (This has already been successfully done in India).
(b) Television for eradication of illiteracy and for mass communication of news and ideas.
(c) Prenatal sex determination
(d) Computers for increase in work efficiency
(e) Putting artificial satellites into orbits around the Earth
(f ) Development of nuclear weapons
(g) Development of new and powerful techniques of chemical and biological warfare).
(h) Purification of water for drinking
(i) Plastic surgery
(j ) Cloning

Question 1.12 India has had a long and unbroken tradition of great scholarship — in mathematics, astronomy, linguistics, logic and ethics. Yet, in parallel with this, several superstitious and obscurantistic attitudes and practices flourished in our society and unfortunately continue even today — among many educated people too. How will you use your knowledge of science to develop strategies to counter these attitudes ?

Question 1.13 Though the law gives women equal status in India, many people hold unscientific views on a woman’s innate nature, capacity and intelligence, and in practice give them a secondary status and role. Demolish this view using scientific arguments, and by quoting examples of great women in science and other spheres; and persuade yourself and others that, given equal opportunity, women are on par with men.

Question 1.14 “It is more important to have beauty in the equations of physics than to have them agree with experiments”. The great British physicist P. A. M. Dirac held this view. Criticize this statement. Look out for some equations and results in this book which strike you as beautiful.

Question 1.15 Though the statement quoted above may be disputed, most physicists do have a feeling that the great laws of physics are at once simple and beautiful. Some of the notable physicists, besides Dirac, who have articulated this feeling, are : Einstein, Bohr, Heisenberg, Chandrasekhar and Feynman. You are urged to make special efforts toaccess to the general books and writings by these and other great masters of physics. (See the Bibliography at the end of this book.) Their writings are truly inspiring !

Question 1.16
Textbooks on science may give you a wrong impression that studying science is dry and all too serious and that scientists are absent-minded introverts who never laugh or grin. This image of science and scientists is patently false. Scientists, like any other group of humans, have their share of humorists, and many have led their lives with a great sense of fun and adventure, even as they seriously pursued their scientific work. Two great physicists of this genre are Gamow and Feynman. You will enjoy reading their books listed in the Bibliography.


:: Chapter 2 Units And Measurement ::


Question 2. 1 Fill in the blanks

(a) The volume of a cube of side 1 cm is equal to .....m3
(b) The surface area of a solid cylinder of radius 2. 0 cm and height 10.0 cm is equal to ...(mm)2
(c) A vehicle moving with a speed of 18 km h–1 covers....m in 1 s
(d) The relative density of lead is 11.3. Its density is ....g cm–3 or ....kg m–3.

Question 2. 2 Fill in the blanks by suitable conversion of units

(a) 1 kg m2 s–2 = ....g cm2 s–2
(b) 1 m = ..... ly
(c) 3.0 m s–2 = .... km h–2
(d) G = 6.67 × 10–11 N m2 (kg)–2 = .... (cm)3 s–2 g–1.

Question 2. 3 A calorie is a unit of heat or energy and it equals about 4.2 J where 1J = 1 kg m2 s–2. Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, the unit of time is γ s. Show that a calorie has a magnitude 4.2 α–1 β –2 γ 2 in terms of the new units.

Question 2. 4
Explain this statement clearly : “To call a dimensional quantity ‘large’ or ‘small’ is meaningless without specifying a standard for comparison”. In view of this, reframe the following statements wherever necessary :

(a) atoms are very small objects
(b) a jet plane moves with great speed
(c) the mass of Jupiter is very large
(d) the air inside this room contains a large number of molecules
(e) a proton is much more massive than an electron
(f) the speed of sound is much smaller than the speed of light.

Question 2. 5 A new unit of length is chosen such that the speed of light in vacuum is unity. What is the distance between the Sun and the Earth in terms of the new unit if light takes 8 min and 20 s to cover this distance ?

Question 2. 6
Which of the following is the most precise device for measuring length :

(a) a vernier callipers with 20 divisions on the sliding scale
(b) a screw gauge of pitch 1 mm and 100 divisions on the circular scale
(c) an optical instrument that can measure length to within a wavelength of light ?

Question 2. 7 A student measures the thickness of a human hair by looking at it through a microscope of magnification 100. He makes 20 observations and finds that the average width of the hair in the field of view of the microscope is 3.5 mm. What is the estimate on the thickness of hair ?

Question 2. 8 Answer the following :

(a)You are given a thread and a metre scale. How will you estimate the diameter of the thread ?
(b)A screw gauge has a pitch of 1.0 mm and 200 divisions on the circular scale. Do you think it is possible to increase the accuracy of the screw gauge arbitrarily by increasing the number of divisions on the circular scale ?
(c) The mean diameter of a thin brass rod is to be measured by vernier callipers. Why is a set of 100 measurements of the diameter expected to yield a more reliable estimate than a set of 5 measurements only ?

Question 2. 9 The photograph of a house occupies an area of 1.75 cm2 on a 35 mm slide. The slide is projected on to a screen, and the area of the house on the screen is 1.55 m 2. What is the linear magnification of the projector-screen arrangement. 2.10 State the number of significant figures in the following :

(a) 0.007 m2
(b) 2. 64 × 1024 kg
(c) 0.2370 g cm–3
(d) 6.320 J
(e) 6.032 N m–2
(f) 0.0006032 m2

Question 2.11 The length, breadth and thickness of a rectangular sheet of metal are 4.234 m, 1.005 m, and 2. 01 cm respectively. Give the area and volume of the sheet to correct significant figures.

Question 2.12 The mass of a box measured by a grocer’s balance is 2. 300 kg. Two gold pieces of masses 20.15 g and 20.17 g are added to the box. What is (a) the total mass of the box, (b) the difference in the masses of the pieces to correct significant figures ?

Question 2.13 A physical quantity P is related to four observables a, b, c and d as follows : P = a3b2/ ( c d ) The percentage errors of measurement in a, b, c and d are 1%, 3%, 4% and 2%, respectively. What is the percentage error in the quantity P ? If the value of P calculated using the above relation turns out to be 3.763, to what value should you round off the result ?

Question 2. 14 A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion :

(a) y = a sin 2π t/T
(b) y = a sin vt
(c) y = (a/T) sin t/a
(d) y = (a 2) (sin 2πt / T + cos 2πt / T ) (a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds.

Question 2.15 A famous relation in physics relates ‘moving mass’ m to the ‘rest mass’ mo of a particle in terms of its speed v and the speed of light, c. (This relation first arose as a consequence of special relativity due to Albert Einstein). A boy recalls the relation almost correctly but forgets where to put the constant c. He writes : ( ) m m 1 v = 0 − 2 1/2. Guess where to put the missing c.

Question 2.16 The unit of length convenient on the atomic scale is known as an angstrom and is denoted by Å: 1 Å = 10–10 m. The size of a hydrogen atom is about 0.5 Å. What is the total atomic volume in m3 of a mole of hydrogen atoms ?

Question 2. 17 One mole of an ideal gas at standard temperature and pressure occupies 22.4 L (molar volume). What is the ratio of molar volume to the atomic volume of a mole of hydrogen ? (Take the size of hydrogen molecule to be about 1 Å). Why is this ratio so large ?

Question 2. 18 Explain this common observation clearly :
If you look out of the window of a fast moving train, the nearby trees, houses etc. seem to move rapidly in a direction opposite to the train’s motion, but the distant objects (hill tops, the Moon, the stars etc.) seem to be stationary. (In fact, since you are aware that you are moving, these distant objects seem to move with you).

Question 2. 19 The principle of ‘parallax’ in section 2.3.1 is used in the determination of distances of very distant stars. The baseline AB is the line joining the Earth’s two locations six months apart in its orbit around the Sun. That is, the baseline is about the diameter of the Earth’s orbit ≈ 3 × 1011m. However, even the nearest stars are so distant that with such a long baseline, they show parallax only of the order of 1” (second) of arc or so. A parsec is a convenient unit of length on the astronomical scale. It is the distance of an object that will show a parallax of 1” (second) of arc from opposite ends of a baseline equal to the distance from the Earth to the Sun. How much is a parsec in terms of metres ?

Question 2. 20 The nearest star to our solar system is 4.29 light years away. How much is this distance in terms of parsecs? How much parallax would this star (named Alpha Centauri) show when viewed from two locations of the Earth six months apart in its orbit around the Sun ?

Question 2. 21 Precise measurements of physical quantities are a need of science. For example, to ascertain the speed of an aircraft, one must have an accurate method to find its positions at closely separated instants of time. This was the actual motivation behind the discovery of radar in World War II. Think of different examples in modern science where precise measurements of length, time, mass etc. are needed. Also, wherever you can, give a quantitative idea of the precision needed.

Question 2. 22 Just as precise measurements are necessary in science, it is equally important to be able to make rough estimates of quantities using rudimentary ideas and common observations. Think of ways by which you can estimate the following (where an estimate is difficult to obtain, try to get an upper bound on the quantity) :

(a) the total mass of rain-bearing clouds over India during the Monsoon
(b) the mass of an elephant
(c) the wind speed during a storm
(d) the number of strands of hair on your head
(e) the number of air molecules in your classroom.

Question 2. 23 The Sun is a hot plasma (ionized matter) with its inner core at a temperature exceeding 107 K, and its outer surface at a temperature of about 6000 K. At these high temperatures, no substance remains in a solid or liquid phase. In what range do you expect the mass density of the Sun to be, in the range of densities of solids and liquids or gases ? Check if your guess is correct from the following data : mass of the Sun =2.0 × 1030 kg, radius of the Sun = 7.0 × 108 m.

Question 2. 24 When the planet Jupiter is at a distance of 824.7 million kilometers from the Earth, its angular diameter is measured to be 35.72” of arc. Calculate the diameter of Jupiter. Additional Exercises

Question 2. 25 A man walking briskly in rain with speed v must slant his umbrella forward making an angle θ with the vertical. A student derives the following relation between θ and v : tan θ = v and checks that the relation has a correct limit: as v → 0, θ →0, as expected. (We are assuming there is no strong wind and that the rain falls vertically for a stationary man). Do you think this relation can be correct ? If not, guess the correct relation.

Question 2. 26 It is claimed that two cesium clocks, if allowed to run for 100 years, free from any disturbance, may differ by only about 0.02 s. What does this imply for the accuracy of the standard cesium clock in measuring a time-interval of 1 s ?

Question 2. 27 Estimate the average mass density of a sodium atom assuming its size to be about 2. 5 Å. (Use the known values of Avogadro’s number and the atomic mass of sodium). Compare it with the density of sodium in its crystalline phase :

970 kg m–3. Are the two densities of the same order of magnitude ? If so, why ?

Question 2. 28 The unit of length convenient on the nuclear scale is a fermi : 1 f = 10–15 m. Nuclear sizes obey roughly the following empirical relation : r = r0 A1/3 where r is the radius of the nucleus, A its mass number, and ro is a constant equal to about, 1.2 f. Show that the rule implies that nuclear mass density is nearly constant for different nuclei. Estimate the mass density of sodium nucleus. Compare it with the average mass density of a sodium atom obtained in Exercise. 2. 27.

Question 2. 29 A LASER is a source of very intense, monochromatic, and unidirectional beam of light. These properties of a laser light can be exploited to measure long distances. The distance of the Moon from the Earth has been already determined very precisely using a laser as a source of light. A laser light beamed at the Moon takes 2. 56 s toreturn after reflection at the Moon’s surface. How much is the radius of the lunar orbit around the Earth ?

Question 2. 30 A SONAR (sound navigation and ranging) uses ultrasonic waves to detect and locate objects under water. In a submarine equipped with a SONAR the time delay between generation of a probe wave and the reception of its echo after reflection from an enemy submarine is found to be 77.0 s. What is the distance of the enemy submarine? (Speed of sound in water = 1450 m s–1).

Question 2. 31 The farthest objects in our Universe discovered by modern astronomers are so distant that light emitted by them takes billions of years to reach the Earth. These objects (known as quasars) have many puzzling features, which have not yet been satisfactorily explained. What is the distance in km of a quasar from which light takes 3.0 billion years to reach us ?

Question 2. 32 It is a well known fact that during a total solar eclipse the disk of the moon almost completely covers the disk of the Sun. From this fact and from the information you can gather from examples 2. 3 and 2. 4, determine the approximate diameter of the moon.

Question 2. 33 A great physicist of this century (P.A.M. Dirac) loved playing with numerical values of Fundamental constants of nature. This led him to an interesting observation. Dirac found that from the basic constants of atomic physics (c, e, mass of electron, mass of proton) and the gravitational constant G, he could arrive at a number with the dimension of time. Further, it was a very large number, its magnitude being close to the present estimate on the age of the universe (~15 billion years). From the table of fundamental constants in this book, try to see if you too can construct this number (or any other interesting number you can think of ). If its coincidence with the age of the universe were significant, what would this imply for the constancy of fundamental constants ?


:: Chapter 3 Motion In A Straight Line ::


Question 3.1 In which of the following examples of motion, can the body be considered approximately a point object:

(a) a railway carriage moving without jerks between two stations.
(b) a monkey sitting on top of a man cycling smoothly on a circular track.
(c) a spinning cricket ball that turns sharply on hitting the ground.
(d) a tumbling beaker that has slipped off the edge of a table.

Question 3.2 The position-time (x-t) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig.

Question 3.19 Choose the correct entries in the brackets below ;
(a) (A/B) lives closer to the school than (B/A)
(b) (A/B) starts from the school earlier than (B/A)
(c) (A/B) walks faster than (B/A)
(d) A and B reach home at the (same/different) time
(e) (A/B) overtakes (B/A) on the road (once/twice).

Question 3.3 A woman starts from her home at 9.00 am, walks with a speed of 5 km h–1 on a straight road up to her office 2.5 km away, stays at the office up to 5.00 pm, and returns home by an auto with a speed of 25 km h–1. Choose suitable scales and plot the x-t graph of her motion.

Question 3.4 A drunkard walking in a narrow lane takes 5 steps forward and 3 steps backward, followed again by 5 steps forward and 3 steps backward, and so on. Each step is 1 m long and requires 1 s. Plot the x-t graph of his motion. Determine graphically and otherwise how long the drunkard takes to fall in a pit 13 m away from the start.

Question 3.5 A jet airplane travelling at the speed of 500 km h–1 ejects its products of combustion at the speed of 1500 km h–1 relative to the jet plane. What is the speed of the latter with respect to an observer on the ground ?

Question 3.6 A car moving along a straight highway with speed of 126 km h–1 is brought to a stop within a distance of 200 m. What is the retardation of the car (assumed uniform), and how long does it take for the car to stop ?

Question 3.7 Two trains A and B of length 400 m each are moving on two parallel tracks with a uniform speed of 72 km h–1 in the same direction, with A ahead of B. The driver of B decides to overtake A and accelerates by 1 m s–2. If after 50 s, the guard of B just brushes past the driver of A, what was the original distance between them ?

Question 3.8 On a two-lane road, car A is travelling with a speed of 36 km h–1. Two cars B and C approach car A in opposite directions with a speed of 54 km h–1 each. At a certain instant, when the distance AB is equal to AC, both being 1 km, B decides to overtake A before C does. What minimum acceleration of car B is required to avoid an accident ?

Question 3.9 Two towns A and B are connected by a regular bus service with a bus leaving in either direction every T minutes. A man cycling with a speed of 20 km h–1 in the direction A to B notices that a bus goes past him every 18 min in the direction of his motion, and every 6 min in the opposite direction. What is the period T of the bus service and with what speed (assumed constant) do the buses ply on the road?

Question 3.10 A player throws a ball upwards with an initial speed of 29.4 m s–1.
(a) What is the direction of acceleration during the upward motion of the ball ?
(b) What are the velocity and acceleration of the ball at the highest point of its motion ?
(c) Choose the x = 0 m and t = 0 s to be the location and time of the ball at its highest point, vertically downward direction to be the positive direction of x-axis, and give the signs of position, velocity and acceleration of the ball during its upward, and downward motion.
(d) To what height does the ball rise and after how long does the ball return to the player’s hands ? (Take g = 9.8 m s–2 and neglect air resistance).

Question 3.11 Read each statement below carefully and state with reasons and examples, if it is true or false ; A particle in one-dimensional motion
(a) with zero speed at an instant may have non-zero acceleration at that instant
(b) with zero speed may have non-zero velocity,
(c) with constant speed must have zero acceleration,
(d) with positive value of acceleration must be speeding up.

Question 3.12 A ball is dropped from a height of 90 m on a floor. At each collision with the floor, the ball loses one tenth of its speed. Plot the speed-time graph of its motion between t = 0 to 12 s.

Question 3.13 Explain clearly, with examples, the distinction between :
(a) magnitude of displacement (sometimes called distance) over an interval of time, and the total length of path covered by a particle over the same interval; (b) magnitude of average velocity over an interval of time, and the average speed over the same interval. [Average speed of a particle over an interval of time is defined as the total path length divided by the time interval]. Show in both (a) and (b) that the second quantity is either greater than or equal to the first. When is the equality sign true ? [For simplicity, consider one-dimensional motion only].

Question 3.14 A man walks on a straight road from his home to a market 2.5 km away with a speed of 5 km h–1. Finding the market closed, he instantly turns and walks back home with a speed of 7.5 km h–1. What is the (a) magnitude of average velocity, and (b) average speed of the man over the interval of time

(i) 0 to 30 min,
(ii) 0 to 50 min,
(iii) 0 to 40 min ?

[Note: You will appreciate from this exercise why it is better to define average speed as total path length divided by time, and not as magnitude of average velocity. You would not like to tell the tired man on his return home that his average speed was zero !]

Question 3.15 In Exercises 3.13 and 3.14, we have carefully distinguished between average speed and magnitude of average velocity. No such distinction is necessary when we consider instantaneous speed and magnitude of velocity. The instantaneous speed is always equal to the magnitude of instantaneous velocity. Why ?

Question 3.16 Look at the graphs (a) to (d) (Fig. 3.20) carefully and state, with reasons, which of these cannot possibly represent one-dimensional motion of a particle.

Question 3.17 Figure3.21 shows the x-t plot of one-dimensional motion of a particle. Is it correct to say from the graph that the particle moves in a straight line for t < 0 and on a parabolic path for t >0 ? If not, suggest a suitable physical context for this graph.

Question 3.18 A police van moving on a highway with a speed of 30 km h–1 fires a bullet at a thief’s car speeding away in the same direction with a speed of 192 km h–1. If the muzzle speed of the bullet is 150 m s–1, with what speed does the bullet hit the thief’s car ?

(Note: Obtain that speed which is relevant for damaging the thief’s car).

Question 3.19 Suggest a suitable physical situation for each of the following graphs (Fig 3.22):

Question 3.20 Figure3.23 gives the x-t plot of a particle executing one-dimensional simple harmonic motion. (You will learn about this motion in more detail in Chapter14). Give the signs of position, velocity and acceleration variables of the particle at t = 0.3 s, 1.2 s, – 1.2 s. Fig.

Question 3.21 Figure 3.24 gives the x-t plot of a particle in one-dimensional motion. Three different equal intervals of time are shown. In which interval is the average speed greatest, and in which is it the least ? Give the sign of average velocity for each interval.

Question 3.22 Figure 3.25 gives a speed-time graph of a particle in motion along a constant direction. Three equal intervals of time are shown. In which interval is the average acceleration greatest in magnitude ? In which interval is the average speed greatest ? Choosing the positive direction as the constant direction of motion, give the signs of v and a in the three intervals. What are the accelerations at the points A, B, C and D ?

Question 3.23 A three-wheeler starts from rest, accelerates uniformly with 1 m s–2 on a straight road for 10 s, and then moves with uniform velocity. Plot the distance covered by the vehicle during the nth second (n = 1,2,3….) versus n. What do you expect this plot to be during accelerated motion : a straight line or a parabola ?

Question 3.24 A boy standing on a stationary lift (open from above) throws a ball upwards with the maximum initial speed he can, equal to 49 m s–1. How much time does the ball take to return to his hands? If the lift starts moving up with a uniform speed of 5 m s-1 and the boy again throws the ball up with the maximum speed he can, how long does the ball take to return to his hands ?

Question 3.25 On a long horizontally moving belt (Fig.3.26), a child runs to and fro with a speed 9 km h–1 (with respect to the belt) between his father and mother located 50 m apart on the moving belt. The belt moves with a speed of 4 km h–1. For an observer on a stationary platform outside, what is the

(a) speed of the child running in the direction of motion of the belt ?.
(b) speed of the child running opposite to the direction of motion of the belt ?
(c) time taken by the child in (a) and (b) ? Which of the answers alter if motion is viewed by one of the parents ?

Question 3.26 Two stones are thrown up simultaneously from the edge of a cliff 200 m high with initial speeds of 15 m s–1 and 30 m s–1. Verify that the graph shown in Fig.

Question 3.27 correctly represents the time variation of the relative position of the second stone with respect to the first. Neglect air resistance and assume that the stones do not rebound after hitting the ground. Take g = 10 m s–2. Give the equations for the linear and curved parts of the plot.3.27 The speed-time graph of a particle moving along a fixed direction is shown in Fig. 3.28. Obtain the distance traversed by the particle between

(a) t = 0 s to 10 s, (b) t = 2 s to 6 s. Fig.3.28 What is the average speed of the particle over the intervals in (a) and (b) ?

Question 3.28 The velocity-time graph of a particle in one-dimensional motion is shown in Fig.3..29 : Fig

Question 3.29 Which of the following formulae are correct for describing the motion of the particle over the time-interval t 1 to t 2 :

(a) x(t2 ) = x(t1) + v (t1) (t2 – t1) +(½) a (t2 – t1)2
(b) v(t2 ) = v(t1) + a (t2 – t1)
(c) vaverage = (x(t2) – x(t1))/(t2 – t1)
(d) aaverage = (v(t2) – v(t1))/(t2 – t1)
(e) x(t2 ) = x(t1) + vaverage (t2 – t1) + (½) aaverage (t2 – t1)2 (f) x(t2 ) – x(t1) = area under the v-t curve bounded by the t-axis and the dotted line shown.


:: Chapter 4 Motion In A Plane ::


Question 4.1 State, for each of the following physical quantities, if it is a scalar or a vector : volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.

Question 4.2 Pick out the two scalar quantities in the following list : force, angular momentum, work, current, linear momentum,electric field, average velocity, magnetic moment, relative velocity.

Question 4.3 Pick out the only vector quantity in the following list : Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.

Question 4.4 State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful :
(a) adding any two scalars,
(b) adding a scalar to a vector of the same dimensions ,
(c) multiplying any vector by any scalar,
(d) multiplying any two scalars,
(e) adding any two vectors,
(f) adding a component of a vector to the same vector.

Question 4.5 Read each statement below carefully and state with reasons, if it is true or false :
(a) The magnitude of a vector is always a scalar,
(b) each component of a vector is always a scalar,
(c) the total path length is always equal to the magnitude of the displacement vector of a particle. (
d) the average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time,
(e) Three vectors not lying in a plane can never add up to give a null vector.

Question 4.6 Establish the following vector inequalities geometrically or otherwise :
(a) |a+b| < |a| + |b|
(b) |a+b| > ||a| −|b||
(c) |a−b| < |a| + |b|
(d) |a−b| > ||a| − |b||

When does the equality sign above apply?

Question 4.7 Given a + b + c + d = 0, which of the following statements are correct :
(a) a, b, c, and d must each be a null vector,
(b) The magnitude of (a + c) equals the magnitude of ( b + d),
(c) The magnitude of a can never be greater than the sum of the magnitudes of b, c, and d,
(d) b + c must lie in the plane of a and d if a and d are not collinear, and in the line of a and d, if they are collinear ?

Question 4.8 Three girls skating on a circular ice ground of radius 200 m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in Fig. 4.20. What is the magnitude of the displacement vector for each ? For which girl is this equal to the actual length of path skate ?

Question 4.9 A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (a) net displacement, (b) average velocity, and (c) average speed of the cyclist ?

Question 4.10 On an open ground, a motorist follows a track that turns to his left by an angle of 600 after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case .

Question 4.11 A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (a) the average speed of the taxi, (b) the magnitude of average velocity ? Are the two equal ?

Question 4.12 Rain is falling vertically with a speed of 30 m s-1. A woman rides a bicycle with a speed of 10 m s-1 in the north to south direction. What is the direction in which she should hold her umbrella ?

Question 4.13 A man can swim with a speed of 4.0 km/h in still water. How long does he take to cross a river 1.0 km wide if the river flows steadily at 3.0 km/h and he makes his reaches the other bank ?

Question 4.14 In a harbour, wind is blowing at the speed of 72 km/h and the flag on the mast of a boat anchored in the harbour flutters along the N-E direction. If the boat starts moving at a speed of 51 km/h to the north, what is the direction of the flag on the mast of the boat ?

Question 4.15 The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that a ball thrown with a speed of 40 m s-1 can go without hitting the ceiling of the hall ?

Question 4.16 A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much high above the ground can the cricketer throw the same ball ?

Question 4.17 A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and direction of acceleration of the stone ?

Question 4.18 An aircraft executes a horizontal loop of radius 1.00 km with a steady speed of 900 km/h. Compare its centripetal acceleration with the acceleration due to gravity.

Question 4.19 Read each statement below carefully and state, with reasons, if it is true or false :
(a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre
(b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point
(c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector

Question 4.20 The position of a particle is given by r = 3.0t ˆi − 2.0t 2 ˆj + 4.0 kˆ m where t is in seconds and the coefficients have the proper units for r to be in metres.
(a) Find the v and a of the particle?
(b) What is the magnitude and direction of velocity of the particle at t = 2.0 s ?

Question 4.21 A particle starts from the origin at t = 0 s with a velocity of 10.0 j m/s and moves in the x-y plane with a constant acceleration of + j) m s-2.
(a) At what time is the x- coordinate of the particle 16 m? What is the y-coordinate of the particle at that time?
(b) What is the speed of the particle at the time ?

Question 4.22 j are unit vectors along x- and y- axis respectively. What is the magnitude and direction of the vectors ij ? What are the com along the directions [You may use graphical method]

Question 4.23 For any arbitrary motion in space, which of the following relations are true :
(a) vaverage = (1/2) (v (t1) + v (t2))
(b) v average = [r(t2) - r(t1) ] /(t2 – t1)
(c) v (t) = v (0) + a t
(d) r (t) = r (0) + v (0) t + (1/2) a t2
(e) a average =[ v (t2) - v (t1 )] /( t2 – t1)
(The ‘average’ stands for average of the quantity over the time interval t1 to t2)

Question 4.24 Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that
(a) is conserved in a process
(b) can never take negative values
(c) must be dimensionless
(d) does not vary from one point to another in space
(e) has the same value for observers with different orientations of axes.

Question 4.25 An aircraft is flying at a height of 3400 m above the ground. If the angle subtended at a ground observation point by the aircraft positions 10.0 s apart is 30°, what is the speed of the aircraft ?

Question 4.26 A vector has magnitude and direction. Does it have a location in space ? Can it vary with time ? Will two equal vectors a and b at different locations in space necessarily have identical physical effects ? Give examples in support of your answer.

Question 4.27 A vector has both magnitude and direction. Does it mean that anything that has magnitude and direction is necessarily a vector ? The rotation of a body can be specified by the direction of the axis of rotation, and the angle of rotation about the axis. Does that make any rotation a vector ?

Question 4.28 Can you associate vectors with (a) the length of a wire bent into a loop, (b) a plane area, (c) a sphere ? Explain.

Question 4.29 A bullet fired at an angle of 30° with the horizontal hits the ground 3.0 km away. By adjusting its angle of projection, can one hope to hit a target 5.0 km away ? Assume the muzzle speed to the fixed, and neglect air resistance.

Question 4.30 A fighter plane flying horizontally at an altitude of 1.5 km with speed 720 km/h passes directly overhead an anti-aircraft gun. At what angle from the vertical should the gun be fired for the shell with muzzle speed 600 m s-1 to hit the plane ? At what minimum altitude should the pilot fly the plane to avoid being hit ? (Take g = 10 m s-2 ).

Question 4.31 A cyclist is riding with a speed of 27 km/h. As he approaches a circular turn on the road of radius 80 m, he applies brakes and reduces his speed at the constant rate of 0.50 m/s every second. What is the magnitude and direction of the net acceleration of the cyclist on the circular turn ?

Question 4.32 (a) Show that for a projectile the angle between the velocity and the x-axis as a function of time is given by ( ) − ox 0y v v gt θ t = tan-1 (b) Shows that the projection angle θ0 for a projectile launched from the origin is given by 0 R 4h θ = tan-1 m where the symbols have their usual meaning


:: Chapter 5 Laws Of Motion ::


Question 5.1 Give the magnitude and direction of the net force acting on (a) a drop of rain falling down with a constant speed, (b) a cork of mass 10 g floating on water, (c) a kite skillfully held stationary in the sky, (d) a car moving with a constant velocity of 30 km/h on a rough road, (e) a high-speed electron in space far from all material objects, and free of electric and magnetic fields.

Question 5.2 A pebble of mass 0.05 kg is thrown vertically upwards. Give the direction and magnitude of the net force on the pebble, (a) during its upward motion, (b) during its downward motion, (c) at the highest point where it is momentarily at rest. Do your answers change if the pebble was thrown at an angle of 45° with the horizontal direction? Ignore air resistance.

Question 5.3 Give the magnitude and direction of the net force acting on a stone of mass 0.1 kg,
(a) just after it is dropped from the window of a stationary train
(b) just after it is dropped from the window of a train running at a constant velocity of 36 km/h
(c ) just after it is dropped from the window of a train accelerating with 1 m s-2
(d) lying on the floor of a train which is accelerating with 1 m s-2, the stone being at rest relative to the trainNeglect air resistance throughout.

Question 5.4 One end of a string of length l is connected to a particle of mass m and the other to a small peg on a smooth horizontal table. If the particle moves in a circle with speed v the net force on the particle (directed towards the centre) is :
(i) T
(ii) l mv T 2 −
(iii) l mv T + 2
(iv) 0 T is the tension in the string.
[Choose the correct alternative].

Question 5.5 A constant retarding force of 50 N is applied to a body of mass 20 kg moving initially with a speed of 15 m s-1. How long does the body take to stop ?

Question 5.6 A constant force acting on a body of mass 3.0 kg changes its speed from 2.0 m s-1 to 3.5 m s-1 in 25 s. The direction of the motion of the body remains unchanged. What is the magnitude and direction of the force ?

Question 5.7 A body of mass 5 kg is acted upon by two perpendicular forces 8 N and 6 N. Give the magnitude and direction of the acceleration of the body.

Question 5.8 The driver of a three-wheeler moving with a speed of 36 km/h sees a child standing in the middle of the road and brings his vehicle to rest in 4.0 s just in time to save the child. What is the average retarding force on the vehicle ? The mass of the three-wheeler is 400 kg and the mass of the driver is 65 kg.
Question 5.9 A rocket with a lift-off mass 20,000 kg is blasted upwards with an initial acceleration of5.0 m s-2. Calculate the initial thrust (force) of the blast.

Question 5.10 A body of mass 0.40 kg moving initially with a constant speed of 10 m s-1 to the north is subject to a constant force of 8.0 N directed towards the south for 30 s. Take the instant the force is applied to be t = 0, the position of the body at that time to be x = 0, and predict its position at t = –5 s, 25 s, 100 s.

Question 5.11 A truck starts from rest and accelerates uniformly at 2.0 m s-2. At t = 10 s, a stone is dropped by a person standing on the top of the truck (6 m high from the ground). What are the (a) velocity, and (b) acceleration of the stone at t = 11s ? (Neglect air resistance.)

Question 5.12 A bob of mass 0.1 kg hung from the ceiling of a room by a string 2 m long is set into oscillation. The speed of the bob at its mean position is 1 m s-1. What is the trajectory of the bob if the string is cut when the bob is (a) at one of its extreme positions, (b) at its mean position.

Question 5.13 A man of mass 70 kg stands on a weighing scale in a lift which is moving
(a) upwards with a uniform speed of 10 m s-1
(b) downwards with a uniform acceleration of 5 m s-2
(c) upwards with a uniform acceleration of 5 m s-2. What would be the readings on the scale in each case?
(d) What would be the reading if the lift mechanism failed and it hurtled down freely under gravity ?

Question 5.14 Figure 5.16 shows the position-time graph of a particle of mass 4 kg. What is the (a) force on the particle for t < 0, t > 4 s, 0 < t < 4 s? (b) impulse at t = 0 and t = 4 s ? (Consider one-dimensional motion only).

Question 5.15 Two bodies of masses 10 kg and 20 kg respectively kept on a smooth, horizontal surface are tied to the ends of a light string. a horizontal force F = 600 N is applied to (i) A, (ii) B along the direction of string. What is the tension in thestring in each case?

Question 5.16 Two masses 8 kg and 12 kg are connected at the two ends of a light inextensible string that goes over a frictionless pulley. Find the acceleration of the masses, and the tension in the string when the masses are released.

Question 5.17 A nucleus is at rest in the laboratory frame of reference. Show that if it disintegrates into two smaller nuclei the products must move in opposite directions.

Question 5.18 Two billiard balls each of mass 0.05 kg moving in opposite directions with speed 6 m s-1 collide and rebound with the same speed. What is the impulse imparted to each ball due to the other ?

Question 5.19 A shell of mass 0.020 kg is fired by a gun of mass 100 kg. If the muzzle speed of the shell is 80 m s-1, what is the recoil speed of the gun ?

Question 5.20 A batsman deflects a ball by an angle of 45° without changing its initial speed which is equal to 54 km/h. What is the impulse imparted to the ball ? (Mass of the ball is 0.15 kg.)

Question 5.21 A stone of mass 0.25 kg tied to the end of a string is whirled round in a circle of radius 1.5 m with a speed of 40 rev./min in a horizontal plane. What is the tension in the string ? What is the maximum speed with which the stone can be whirled around if the string can withstand a maximum tension of 200 N ?

Question 5.22 If, in Exercise 5.21, the speed of the stone is increased beyond the maximum permissible value, and the string breaks suddenly, which of the following correctly describes the trajectory of the stone after the string breaks :

a) the stone moves radially outwards
(b) the stone flies off tangentially from the instant the string breaks
(c) the stone flies off at an angle with the tangent whose magnitude depends on the speed of the particle ?

Question 5.23
Explain why (a) a horse cannot pull a cart and run in empty space, (b) passengers are thrown forward from their seats when a speeding bus stops suddenly, (c) it is easier to pull a lawn mower than to push it, (d) a cricketer moves his hands backwards while holding a catch. Additional Exercises

Question 5.24 Figure 5.17 shows the position-time graph of a body of mass 0.04 kg. Suggest a suitable physical context for this motion. What is the time between two consecutive impulses received by the body ? What is the magnitude of each impulse ?

Question 5.25 Figure 5.18 shows a man standing stationary with respect to a horizontal conveyor belt that is accelerating with 1 m s-2. What is the net force on the man? If the coefficient of static friction between the man’s shoes and the belt is 0.2, up to what acceleration of the belt can the man continue to be stationary relative to the belt ? (Mass of the man = 65 kg.)

Question 5.26 A stone of mass m tied to the end of a string revolves in a vertical circle of radius R. The net forces at the lowest and highest points of the circle directed vertically downwards are :[Choose the correct alternative] Lowest Point Highest Point

(a) mg – T1 mg + T2
(b) mg + T1 mg – T2
( c) mg + T1 – (m v 2 1 ) / R mg – T2 + (m v 2 1 ) / R (d) mg – T1 – (m v 2 1 ) / R mg + T2 + (m v 2 1 ) / R T1 and v1 denote the tension and speed at the lowest point. T2 and v2 denote corresponding values at the highest point.

Question 5.27 A helicopter of mass 1000 kg rises with a vertical acceleration of 15 m s-2. The crew and the passengers weigh 300 kg. Give the magnitude and direction of the (a) force on the floor by the crew and passengers, (b) action of the rotor of the helicopter on the surrounding air, (c) force on the helicopter due to the surrounding air.

Question 5.28 A stream of water flowing horizontally with a speed of 15 m s-1 gushes out of a tube of cross-sectional area 10-2 m2, and hits a vertical wall nearby. What is the force exerted on the wall by the impact of water, assuming it does not rebound ?

Question 5.29 Ten one-rupee coins are put on top of each other on a table. Each coin has a mass m. Give the magnitude and direction of (a) the force on the 7th coin (counted from the bottom) due to all the coins on its top, (b) the force on the 7th coin by the eighth coin, (c) the reaction of the 6th coin on the 7th coin.

Question 5.30 An aircraft executes a horizontal loop at a speed of 720 km/h with its wings banked at 15°. What is the radius of the loop ?

Question 5.31 A train runs along an unbanked circular track of radius 30 m at a speed of 54 km/h. The mass of the train is 106 kg. What provides the centripetal force required for this purpose — The engine or the rails ? What is the angle of banking required to prevent wearing out of the rail ?

Question 5.32 A block of mass 25 kg is raised by a 50 kg man in two different ways as shown in Fig. 5.19. What is the action on the floor by the man in the two cases ? If the floor yields to a normal force of 700 N, which mode should the man adopt to lift the block without the floor yielding ?

Question 5.33 A monkey of mass 40 kg climbs on a rope (Fig.5.20) which can stand a maximum tension of 600 N. In which of the following cases will the rope break: the monkey (a) climbs up with an acceleration of 6 m s-2 (b) climbs down with an acceleration of 4 m s-2 (c) climbs up with a uniform speed of 5 m s-1 (d) falls down the rope nearly freely under gravity? (Ignore the mass of the rope).

Question 5.34 Two bodies A and B of masses 5 kg and 10 kg in contact with each other rest on a table against a rigid wall (Fig. 5.21). The coefficient of friction between the bodies and the table is 0.15. A force of 200 N is applied horizontally to A. What are (a) the reaction of the partition (b) the action-reaction forces between A and B ? What happens when the wall is removed? Does the answer to (b) change, when the bodies are in motion? Ignore the difference between μs and μk.

Question 5.35 A block of mass 15 kg is placed on a long trolley. The coefficient of static friction between the block and the trolley is 0.18. The trolley accelerates from rest with 0.5 m s-2 for 20 s and then moves with uniform velocity. Discuss the motion of the block as viewed by (a) a stationary observer on the ground, (b) an observer moving with the trolley.

Question 5.36 The rear side of a truck is open and a box of 40 kg mass is placed 5 m away from the open end as shown in Fig.5.22. The coefficient of friction between the box and the surface below it is 0.15.On a straight road, the truck starts from rest and accelerates with 2 m s-2. At what distance from the starting point does the box fall off the truck? (Ignore the size of the box).

Question 5.37 A disc revolves with a speed of 33 1 3 rev/min, and has a radius of 15 cm. Two coins are placed at 4 cm and 14 cm away from the centre of the record. If the co-efficient of friction between the coins and the record is 0.15, which of the coins will revolve with the record ?

Question 5.38 You may have seen in a circus a motorcyclist driving in vertical loops inside a ‘deathwell’ (a hollow spherical chamber with holes, so the spectators can watch from outside). Explain clearly why the motorcyclist does not drop down when he is at the uppermost point, with no support from below. What is the minimum speed required at the uppermost position to perform a vertical loop if the radius of the chamber is 25 m ?

Question 5.39 A 70 kg man stands in contact against the inner wall of a hollow cylindrical drum of radius 3 m rotating about its vertical axis with 200 rev/min. The coefficient of friction between the wall and his clothing is 0.15.What is the minimum rotational speed of the cylinder to enable the man to remain stuck to the wall (without falling) when the floor is suddenly removed ?

Question 5.40 A thin circular loop of radius R rotates about its vertical diameter with an angular frequency ω. Show that a small bead on the wire loop remains at its lowermost point for ω ≤ g / R . What is the angle made by the radius vector joining the centre to the bead with the vertical downward direction for ω = 2g / R ? Neglect friction.


:: Chapter 6 Work, Energy And Power ::


Question 6.1 The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative:
(a) work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket.
(b) work done by gravitational force in the above case,
(c) work done by friction on a body sliding down an inclined plane,
(d) work done by an applied force on a body moving on a rough horizontal plane with uniform velocity,
(e) work done by the resistive force of air on a vibrating pendulum in bringing it to rest.

Question 6.2 A body of mass 2 kg initially at rest moves under the action of an applied horizontal force of 7 N on a table with coefficient of kinetic friction = 0.1. Compute the (a) work done by the applied force in 10 s, (b) work done by friction in 10 s, (c) work done by the net force on the body in 10 s, (d) change in kinetic energy of the body in 10 s, and interpret your results.

Question 6.3 Given in Fig. 6.11 are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case, specify the regions, if any, in which the particle cannot be found for the given energy. Also, indicate the minimum total energy the particle must have in each case. Think of simple physical contexts for which these potential energy shapes are relevant.

Question 6.4 The potential energy function for a particle executing linear simple harmonic motion is given by V(x) = kx2/2, where k is the force constant of the oscillator. For k = 0.5 N m-1, the graph of V(x) versus x is shown in Fig6.12. Show that a particle of total energy 1 J moving under this potential must ‘turn back’ when it reaches x = ± 2 m.

Question 6.5 Answer the following :

(a) The casing of a rocket in flight burns up due to friction. At whose expense is the heat energy required for burning obtained? The rocket or the atmosphere?
(b) Comets move around the sun in highly elliptical orbits. The gravitational force on the comet due to the sun is not normal to the comet’s velocity in general. Yet the work done by the gravitational force over every complete orbit of the comet is zero. Why ?
(c) An artificial satellite orbiting the earth in very thin atmosphere loses its energy gradually due to dissipation against atmospheric resistance, however small. Why then does its speed increase progressively as it comes closer and closer to the earth ?
(d) In Fig. 6.13(i) the man walks 2 m carrying a mass of 15 kg on his hands. In Fig. 6.13(ii), he walks the same distance pulling the rope behind him. The rope goes over a pulley, and a mass of 15 kg hangs at its other end. In which case is the work done greater ?

Question 6.6 Underline the correct alternative :

(a) When a conservative force does positive work on a body, the potential energy of the body increases/decreases/remains unaltered.
(b) Work done by a body against friction always results in a loss of its kinetic/potential energy.
(c) The rate of change of total momentum of a many-particle system is proportional to the external force/sum of the internal forces on the system.
(d) In an inelastic collision of two bodies, the quantities which do not change after the collision are the total kinetic energy/total linear momentum/total energy of the system of two bodies.

Question 6.7 State if each of the following statements is true or false. Give reasons for your answer.

(a) In an elastic collision of two bodies, the momentum and energy of each body is conserved.
(b) Total energy of a system is always conserved, no matter what internal and external forces on the body are present.
(c) Work done in the motion of a body over a closed loop is zero for every force in nature.
(d) In an inelastic collision, the final kinetic energy is always less than the initial kinetic energy of the system.

Question 6.8 Answer carefully, with reasons :
(a) In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e. when they are in contact) ?
(b) Is the total linear momentum conserved during the short time of an elastic collision of two balls?
(c) What are the answers to (a) and (b) for an inelastic collision ?
(d) If the potential energy of two billiard balls depends only on the separation distance between their centres, is the collision elastic or inelastic ? (Note, we are talking here of potential energy corresponding to the force during collision, not gravitational potential energy).

Question 6.9 A body is initially at rest. It undergoes one-dimensional motion with constant acceleration. The power delivered to it at time t is proportional to (i) t1/2 (ii) t (iii) t3/2 (iv) t2

Question 6.10 A body is moving unidirectionally under the influence of a source of constant power. Its displacement in time t is proportional to (i) t1/2 (ii) t (iii) t3/2 (iv) t2

Question 6.11 A body constrained to move along the z-axis of a coordinate system is subject to a constant force F given by F = −ˆi + 2 ˆj + 3 kˆ N where ˆi, ˆj, kˆ are unit vectors along the x-, y- and z-axis of the system respectively. What is the work done by this force in moving the body a distance of 4 m along the z-axis ?

Question 6.12 An electron and a proton are detected in a cosmic ray experiment, the first with kinetic energy 10 keV, and the second with 100 keV. Which is faster, the electron or the proton ? Obtain the ratio of their speeds. (electron mass = 9.11× 10-31 kg, proton mass = 1.67×10–27 kg, 1 eV = 1.60 ×10–19 J).

Question 6.13 A rain drop of radius 2 mm falls from a height of 500 m above the ground. It falls with decreasing acceleration (due to viscous resistance of the air) until at half its original height, it attains its maximum (terminal) speed, and moves with uniform speed thereafter. What is the work done by the gravitational force on the drop in the first and second half of its journey ? What is the work done by the resistive force in the entire journey if its speed on reaching the ground is 10 m s–1 ?

Question 6.14 A molecule in a gas container hits a horizontal wall with speed 200 m s–1 and angle 30° with the normal, and rebounds with the same speed. Is momentum conserved in the collision ? Is the collision elastic or inelastic ?

Question 6.15 A pump on the ground floor of a building can pump up water to fill a tank of volume 30 m3 in 15 min. If the tank is 40 m above the ground, and the efficiency of the pump is 30%, how much electric power is consumed by the ump ?

Question 6.16 Two identical ball bearings in contact with each other and resting on a frictionless table are hit head-on by another ball bearing of the same mass moving initially with a speed V. If the collision is elastic, which of the following (Fig. 6.14) is a possible result after collision ?

Question 6.17 The bob A of a pendulum released from 30o to the vertical hits another bob B of the same mass at rest on a table as shown in Fig. 6.15. How high does the bob A rise after the collision ? Neglect the size of the bobs and assume the collision to be elastic.

Question 6.18 The bob of a pendulum is released from a horizontal position. If the length of the pendulum is 1.5 m, what is the speed with which the bob arrives at the lowermost point, given that it dissipated 5% of its initial energy against air resistance ?

Question 6.19
A trolley of mass 300 kg carrying a sandbag of 25 kg is moving uniformly with a speed of 27 km/h on a frictionless track. After a while, sand starts leaking out of a hole on the floor of the trolley at the rate of 0.05 kg s–1. What is the speed of the trolley after the entire sand bag is empty ?

Question 6.20
A body of mass 0.5 kg travels in a straight line with velocity v =a x3/2 where a = 5 m–1/2 s–1. What is the work done by the net force during its displacement from x = 0 to x = 2 m ?

Question 6.21
The blades of a windmill sweep out a circle of area A. (a) If the wind flows at a velocity v perpendicular to the circle, what is the mass of the air passing through it in time t ? (b) What is the kinetic energy of the air ? (c) Assume that the windmill converts 25% of the wind’s energy into electrical energy, and that A = 30 m2, v = 36 km/h and the density of air is 1.2 kg m–3. What is the electrical power produced ?

Question 6.22
A person trying to lose weight (dieter) lifts a 10 kg mass, one thousand times, to a height of 0.5 m each time. Assume that the potential energy lost each time she lowers the mass is dissipated. (a) How much work does she do against the gravitational force ? (b) Fat supplies 3.8 × 107J of energy per kilogram which is converted to mechanical energy with a 20% efficiency rate. How much fat will the dieter use up?

Question 6.23 A family uses 8 kW of power. (a) Direct solar energy is incident on the horizontal surface at an average rate of 200 W per square meter. If 20% of this energy can be converted to useful electrical energy, how large an area is needed to supply 8 kW? (b) Compare this area to that of the roof of a typical house. Additional Exercises

Question 6.24 A bullet of mass 0.012 kg and horizontal speed 70 m s–1 strikes a block of wood of mass 0.4 kg and instantly comes to rest with respect to the block. The block is suspended from the ceiling by means of thin wires. Calculate the height to which the block rises. Also, estimate the amount of heat produced in the block.

Question 6.25 Two inclined frictionless tracks, one gradual and the other steep meet at A from where two stones are allowed to slide down from rest, one on each track (Fig. 6.16). Will the stones reach the bottom at the same time? Will they reach there with the same speed? Explain. Given θ1 = 300, θ2 = 600, and h = 10 m, what are the speeds and times taken by the two stones ?

Question 6.26 A 1 kg block situated on a rough incline is connected to a spring of spring constant 100 N m–1 as shown in Fig.6.17. The block is released from rest with the spring in the unstretched position. The block moves 10 cm down the incline before coming to rest. Find the coefficient of friction between the block and the incline. Assume that the spring has a negligible mass and the pulley is frictionless.

Question 6.27 A bolt of mass 0.3 kg falls from the ceiling of an elevator moving down with an uniform speed of 7 m s–1. It hits the floor of the elevator (length of the elevator = 3 m) and does not rebound. What is the heat produced by the impact ? Would your answer be different if the elevator were stationary ?

Question 6.28 A trolley of mass 200 kg moves with a uniform speed of 36 km/h on a frictionless track. A child of mass 20 kg runs on the trolley from one end to the other (10 m away) with a speed of 4 m s–1 relative to the trolley in a direction opposite to the its motion, and jumps out of the trolley. What is the final speed of the trolley ? How much has the trolley moved from the time the child begins to run ?

Question 6.29 Which of the following potential energy curves in Fig. 6.18 cannot possibly describe the elastic collision of two billiard balls ? Here r is the distance between centres of the balls.


:: Chapter 7 Systems of Particles and Rotational Motion ::


Question 7.1 Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body ?

Question 7.2 In the HC1 molecule, the separation between the nuclei of the two atoms is about 1.27 Å (1 Å = 10-10 m). Find the approximate location of the CM of the molecule, given that a chlorine atom is about 35.5 times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.

Question 7.3
A child sits stationary at one end of a long trolley moving uniformly with a speed V on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, what is the speed of the CM of the (trolley + child) system ?

Question 7.4 Show that the area of the triangle contained between the vectors a and b is one half of the magnitude of a × b.

Question 7.5 Show that a.(b × c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors , a, b and c.

Question 7.6 Find the components along the x, y, z axes of the angular momentum l of a particle, whose position vector is r with components x, y, z and momentum is p with components px, py and pz. Show that if the particle moves only in the x-y plane the angular momentum has only a z-component.

Question 7.7 Two particles, each of mass m and speed v, travel in opposite directions along parallel lines separated by a distance d. Show that the vector angular momentum of the two particle system is the same whatever be the point about which the angular momentum is taken.

Question 7.8 A non-uniform bar of weight W is suspended at rest by two strings of negligible weight as shown in Fig.

Question 7.9. The angles made by the strings with the vertical are 36.9° and 53.1° respectively. The bar is 2 m long. Calculate the distance d of the centre of gravity of the bar from its left end.7.9 A car weighs 1800 kg. The distance between its front and back axles is 1.8 m. Its centre of gravity is 1.05 m behind the front axle. Determine the force exerted by the level ground on each front wheel and each back wheel.

Question 7.10 (a) Find the moment of inertia of a sphere about a tangent to the sphere, given the moment of inertia of the sphere about any of its diameters to be 2MR2/5, where M is the mass of the sphere and R is the radius of the sphere.
(b) Given the moment of inertia of a disc of mass M and radius R about any of its diameters to be MR2/4, find its moment of inertia about an axis normal to the disc and passing through a point on its edge.

Question 7.11 Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time.

Question 7.12 A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad s-1. The radius of the cylinder is 0.25 m. What is the kinetic energy associated with the rotation of the cylinder? What is the magnitude of angular momentum of the cylinder about its axis?

Question 7.13 (a) A child stands at the centre of a turntable with his two arms outstretched. The turntable is set rotating with an angular speed of 40 rev/min. How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to 2/5 times the initial value ? Assume that the turntable rotates without friction.
(b) Show that the child’s new kinetic energy of rotation is more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy?

Question 7.14 A rope of negligible mass is wound round a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N ? What is the linear acceleration of the rope ? Assume that there is no slipping.

Question 7.15 To maintain a rotor at a uniform angular speed or 200 rad s-1, an engine needs to transmit a torque of 180 N m. What is the power required by the engine ? (Note: uniform angular velocity in the absence of friction implies zero torque. In practice, applied torque is needed to counter frictional torque). Assume that the engine is 100% efficient.

Question 7.16 From a uniform disk of radius R, a circular hole of radius R/2 is cut out. The centre of the hole is at R/2 from the centre of the original disc. Locate the centre of gravity of the resulting flat body.

Question 7.17 A metre stick is balanced on a knife edge at its centre. When two coins, each of mass 5 g are put one on top of the other at the 12.0 cm mark, the stick is found to be balanced at 45.0 cm. What is the mass of the metre stick?

Question 7.18 A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination.
(a) Will it reach the bottom with the same speed in each case?
(b) Will it take longer to roll down one plane than the other?
(c) If so, which one and why?

Question 7.19 A hoop of radius 2 m weighs 100 kg. It rolls along a horizontal floor so that its centre of mass has a speed of 20 cm/s. How much work has to be done to stop it?

Question 7.20 The oxygen molecule has a mass of 5.30 × 10-26 kg and a moment of inertia of 1.94×10-46 kg m2 about an axis through its centre perpendicular to the lines joining the two atoms. Suppose the mean speed of such a molecule in a gas is 500 m/s and that its kinetic energy of rotation is two thirds of its kinetic energy of translation. Find the average angular velocity of the molecule.

Question 7.21 A solid cylinder rolls up an inclined plane of angle of inclination 30° . At the bottom of the inclined plane the centre of mass of the cylinder has a speed of 5 m/s.
(a) How far will the cylinder go up the plane?
(b) How long will it take to return to the bottom? Additional Exercises

Question 7.22 As shown in Fig.7.40, the two sides of a step ladder BA and CA are 1.6 m long and hinged at A. A rope DE, 0.5 m is tied half way up. A weight 40 kg is suspended from a point F, 1.2 m from B along the ladder BA. Assuming the floor to be frictionless and neglecting the weight of the ladder, find the tension in the rope and forces exerted by the floor on the ladder. (Take g = 9.8 m/s2) (Hint: Consider the equilibrium of each side of the ladder separately.)

Question 7.23 A man stands on a rotating platform, with his arms stretched horizontally holding a 5 kg weight in each hand. The angular speed of the platform is 30 revolutions per minute. The man then brings his arms close to his body with the distance of each weight from the axis changing from 90cm to 20cm. The moment of inertia of the man together with the platform may be taken to be constant and equal to7.6 kg m2.
(a) What is his new angular speed? (Neglect friction.)
(b) Is kinetic energy conserved in the process? If not, from where does the change come about?

Question 7.24 A bullet of mass 10 g and speed 500 m/s is fired into a door and gets embedded exactly at the centre of the door. The door is 1.0 m wide and weighs 12 kg. It is hinged at one end and rotates about a vertical axis practically without friction. Find the angular speed of the door just after the bullet embeds into it. (Hint: The moment of inertia of the door about the vertical axis at one end is ML2/3.)

Question 7.25 Two discs of moments of inertia I1 and I2 about their respective axes (normal to the disc and passing through the centre), and rotating with angular speeds ω1 and ω2 are brought into contact face to face with their axes of rotation coincident.
(a) What is the angular speed of the two-disc system?
(b) Show that the kinetic energy of the combined system is less than the sum of the initial kinetic energies of the two discs. How do you account for this loss in energy? Take ω1 ≠ ω2.

Question 7.26 (a) Prove the theorem of perpendicular axes. (Hint : Square of the distance of a point (x, y) in the x–y plane from an axis through the origin perpendicular to the plane is x2+y2). (b) Prove the theorem of parallel axes. (Hint : If the centre of mass is chosen to be the origin Σmi ri = 0 ).

Question 7.27 Prove the result that the velocity v of translation of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an inclined plane of a height h is given by ( ) 2 2 2 2 1 / gh v k R = + using dynamical consideration (i.e. by consideration of forces and torques). Note k is the radius of gyration of the body about its symmetry axis, and R is the radius of the body. The body starts from rest at the top of the plane.

Question 7.28
A disc rotating about its axis with angular speed ωo is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is R. What are the linear velocities of the points A, B and C on the disc shown in Fig.7.41? Will the disc roll in the direction indicated ?

Question 7.29 Explain why friction is necessary to make the disc in Fig. 7.41 roll in the direction indicated.

(a) Give the direction of frictional force at B, and the sense of frictional torque, before perfect rolling begins.
(b) What is the force of friction after perfect rolling begins ?

Question 7.30 A solid disc and a ring, both of radius 10 cm are placed on a horizontal table simultaneously, with initial angular speed equal to 10 π rad s-1. Which of the two will start to roll earlier ? The co-efficient of kinetic friction is μ k = 0.2.

Question 7.31 A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination 30o. The coefficient of static friction μs = 0.25.
(a) How much is the force of friction acting on the cylinder ?
(b) What is the work done against friction during rolling ?
(c) If the inclination θ of the plane is increased, at what value of θ does the cylinder begin to skid, and not roll perfectly ?

Question 7.32 Read each statement below carefully, and state, with reasons, if it is true or false
(a) During rolling, the force of friction acts in the same direction as the direction of motion of the CM of the body.
(b) The instantaneous speed of the point of contact during rolling is zero.
(c) The instantaneous acceleration of the point of contact during rolling is zero.
(d) For perfect rolling motion, work done against friction is zero.
(e) A wheel moving down a perfectly frictionless inclined plane will undergo slipping (not rolling) motion.

Question 7.33 Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass :

Essay Writing Competition for Students of Classes X and XI (Topic: Shodh Yatri Contest)

GENERAL: 

Essay Writing Competition for Students of Classes X and XI (Topic: Shodh Yatri Contest)

In order to trace India’s historical relations with other Asian countries, the National Book Trust, an autonomous organization under the Ministry of Human Resource Development, Government of India along with the support of the Indian Council of Historical Research (ICHR) is organizing an Essay writing competition under the Shodh Yatri Contest for the students of Classes X and XI.

1. For participation, the eligible participants have to fill an application form and submit an essay of 5000 words in PDF on any one of the following specified themes on www.mygov.in:

a. Shodh Yatri to Sri Lanka (Search for Buddha’s Footprints)
b. Shodh Yatri to Thailand (Search for India’s contribution)
c. Shodh Yatri to Myanmar (Burma)
d. Shodh Yatri to Bangladesh (Search for Martyrs and Freedom Fighters)
e. Shodh Yatri to Cambodia (Angkor Vat)

2. A write up on any one of the identified themes has to be further developed by the participants while submitting the essay. The participant has to explain in 1000 words about his/her own idea on the proposed visit, and how they would utilize the experience gained.

3. The selected entries will be interviewed on skype ,and 10 winning entries will be finalized along with a list of waitlisted candidates.

4. The final winning entries will be awarded with a visit to the concerned Asian country for about a week as a part of a team. After the visit, the candidates have to submit a write-up on his/her Shodh Yatra within a fortnight and submit it to the Director, National Book Trust. The write-up will be published by the National Book Trust with appropriate modifications.

NCERT Mathematics Question Paper (Class - 10)

CBSE Special TX: 
GENERAL: 
Exam: 
Subjects: 

NCERT Mathematics Question Paper (Class - 10)


:: Chapter 1: Number System ::


Exercise 1.1

Question 1. Use Euclid’s division algorithm to find the HCF of

(i) 135 and 225
(ii) 196 and 38220
(iii) 867 and 255

Question 2: Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

Question 3. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns iwhich they can march?

Question 4. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m

Question 5. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

Exercise 1.2

Question 1. Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

Question 2. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. 26 and 91 (ii) 510 and 92 (iii) 336 and 54

Question 3. Find the LCM and HCF of the following integers by applying the prime factorization method.

(i) 12, 15 and 21
(ii) 17, 23 and 29
(iii) 8, 9 and 25

Question 4. Given that HCF (306, 657) = 9, find LCM (306, 657).

Question 5. Check whether 6n can end with the digit 0 for any natural number n.

Question 6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

Question 7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

Exercise 1.3

Question 1. Prove that √5 is irrational.

Question 2. Prove that 3 + 2√5 is irrational.

Question 3. Prove that the following are irrationals: (i) 1/√2 (ii) 7√5 (iii) 6 + √2

Exercise 1.4

Question 1. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

(i)13/3125
(ii)17/8
(iii)64/455
(iv)15/1600
(v)29/343
(vi)23/2³*5²
(vii)129/2²* 57* 75
(viii)6/15
(ix)35/50
(x)77/210

Question 2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

(i)13/3125 = 0.009375
(ii)17/8
(iii)64/455 none- terminating
(iv)15/1600
(v)29/343 it is none – terminating
(vi)23/2³*5² = 23/200
(vii)129/2²* 57*75 it is none terminating
(viii) 6/15 = 2/5 = 0.4
(ix)35/50

Question 3. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form p , q you say about the prime factors of q?


:: Chapter 2: Polynomial ::


Exercise 2.1

Question 1. The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

Exercise 2.2

Question 1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(iii) 4u² + 8u

Question 1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) x² – 2x – 8
(ii) 4s² – 4s + 1
(iii) 6x² – 3 – 7x
(v) t² – 15
(vi)3x² – x – 4

Question 2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

(i)1/4 , -1 (ii) √2 , 1/3 (iii) 0, √5 (iv) 1,1 (v) -1/4 ,1/4 (vi) 4,1

Exercise 2.3

1. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following :

Question 2. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

Question 3. Obtain all other zeroes of 3x4 + 6x3 – 2x2 – 10x – 5, if two of its zeroes are √(5/3) and - √(5/3)

Question 4. On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4, respectively. Find g(x).

Question 5. Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm


:: 3: MATRIX ::


Exercise 3.1

Question 1. Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then .Also, three years from now, I shall be three times as old as you will be.” (Isn’t this interesting?) Represent this situation algebraically and graphically.

Question 2. The coach of a cricket team buys 3 bats and 6 balls for Rs 3900. Later, she buys another bat and 2 more balls of the same kind for Rs 1300. Represent this situation algebraically and geometrically.

Question 3. The cost of 2 kg of apples and 1kg of grapes on a day was found to be Rs 160. After a month, the cost of 4 kg of apples and 2 kg of grapes is Rs 300. Represent the situation algebraically and geometrically.

Exercise 3.2

Question 1 (ii). 5 pencils and 7 pens together cost Rs 50, whereas 7 pencils and 5 pens together cost Rs 46. Find the cost of one pencil and that of one pen

Question 2. On comparing the ratios a1/a2 , b1/b2 and c1/c2, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:

Question 3. On comparing the ratios a1/a2 , b1/b2 and c1/c2 find out whether the following pair of linear equations are consistent, or inconsistent.

(i) 3x + 2y = 5 ; 2x – 3y = 7
(ii) 2x – 3y = 8 ; 4x – 6y = 9
(iii) 3/2x + 5/3 y = 7 ; 9x – 10y = 14
(iv) 5x – 3y = 11 ; – 10x + 6y = –22
(v)4/3x + 2y =8 ; 2x + 3y = 12
(i) 3x + 2y = 5 ; 2x – 3y = 7

Question 4. Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:

Question 5. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

Question 6. Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is:

Question 7. Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.

Exercise 3.3

Question 1. Solve the following pair of linear equations by the substitution method.

(i) x + y = 14 ; x – y = 4
(ii) s – t = 3 ; s/3 + t/2 = 6
(iii) 3x – y = 3 ; 9x – 3y = 9
(iv) 0.2x + 0.3y = 1.3 ; 0.4x + 0.5y = 2.3
(v)√2 x+ √3y = 0 ; √3 x - √8y = 0 (vi)3/2 x - 5/3y = -2 ; x/3 + y/2 = 13/6

Question 2. Solve 2x + 3y = 11 and 2x – 4y = – 24 and hence find the value of ‘m’ for which y =mx + 3.

Question 3. Form the pair of linear equations for the following problems and find their solution by substitution method.

(i) The difference between two numbers is 26 and one number is three times the other. Find them.
(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.
(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, she buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball

Exercise 3.4

Question 1. Solve the following pair of linear equations by the elimination method and the substitution method:

x + y =5 and 2x –3y = 4
3x + 4y = 10 and 2x – 2y = 2
3x – 5y – 4 = 0 and 9x = 2y + 7
x/2 + 2y /3 = - 1 and x – y/3 = 3

Question 2. Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:

(i) If we add 1tothe numerator and subtract 1fromthe denominator, a fractionreduces to 1. It becomes1/2 if we only add 1 to the denominator.What is the raction?
(ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?
(iii) The sum of the digits of a t

Exercise 3.5

Question 1. Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions In case there is a unique solution, find it by using cross multiplication method.

(i)x – 3y – 3 = 0 ; 3x – 9y – 2 =0
(ii)2x + y = 5 ; 3x +2y =8
(iii)3x – 5y = 20 ; 6x – 10y =40
(iv)x – 3y – 7 = 0 ;3x – 3y – 15= 0

Question 2. (i) For which values of a and b does the following pair of linear equations have an infinite number of solutions?

2x +3y =7; (a – b) x +(a +b) y =3a +b –2

Question 3. Solve the following pair of linear equations by the substitution and cross-multiplication methods:

8x +5y =9 …(1)
3x +2y =4 …(2)

Quesiton 4. Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method :

(ii) A fraction becomes 1 3 when 1 is subtracted from the numerator and it becomes 1 4 when 8 is added to its denominator. Find the fraction.

(iii) Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

(v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.

EXERCISE 3.6

Question 1. Solve the following pairs of equations by reducing them to a pair of linear equations:

Question 2. Formulate the following problems as a pair of equations, and hence find their solutions:

(i) Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.
(ii) 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to finish the work, and also that taken by 1 man alone
(iii) Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.

EXERCISE 3.7

Question 1. The ages of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju.

Question 2. One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the amount of their (respective) capital? [From the Bijaganita of Bhaskara II] [Hint : x + 100 = 2(y – 100), y + 10 = 6(x – 10)

Question 3. A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train.

Question 4. The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.

Question 5. In a Δ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

Question 6. Draw the graphs of the equations 5x – y = 5 and 3x – y = 3. Determine the co-ordinates of the vertices of the triangle formed by these lines and the y axis.

Question 7. Solve the following pair of linear equations:


:: Chapter 4 Quadratic Equations ::


EXERCISE 4.1

Question 1. Check whether the following are quadratic equations :

(i) (x + 1)2 = 2(x – 3)
(ii) x2 – 2x = (–2) (3 – x)
(iii) (x – 2)(x + 1) = (x – 1)(x + 3)
(iv) (x – 3)(2x +1) = x(x + 5)
(v) (2x – 1)(x – 3) = (x + 5)(x – 1)
(vi) x2 + 3x + 1 = (x – 2)2
(vii) (x + 2)3 = 2x (x2 – 1)
(viii) x3 – 4x2 – x + 1 = (x – 2)3

Question 2. Represent the following situations in the form of quadratic equations :

(i) The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
(ii) The product of two consecutive positive integers is 306. We need to find the integers.
(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

EXERCISE 4.2

Question 1. Find the roots of the following quadratic equations by factorisation:

(i) x2 – 3x – 10 = 0
(ii) 2x2 + x – 6 = 0
(iii) 2 x2 + 7 x + 5 2 = 0
(iv) 2x2 – x + 1 8 = 0
(v) 100 x2 – 20x + 1 = 0

Question 2. Solve the problems given in Example 1.

Question 3. Find two numbers whose sum is 27 and product is 182.

Question 4. Find two consecutive positive integers, sum of whose squares is 365.

Question 5. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Question 6. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.

EXERCISE 4.3

Question 1. Find the roots of the following quadratic equations, if they exist, by the method of completing the square:

(i) 2x2 – 7x + 3 = 0
(ii) 2x2 + x – 4 = 0
(iii) 4x2 + 4 3x + 3 = 0
(iv) 2x2 + x + 4 = 0

Question 2. Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.

Question 3. The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is 1.3 Find his present age.

Question 4. In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.

Question 5. The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

Question 6. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

Question 7. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Question 8. Two water taps together can fill a tank in 9 3 8 hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Question 9. An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11km/h more than that of the passenger train, find the average speed of the two trains.

Question 10. Sum of the areas of two squares is 468 m2. If the difference of their perimeters is 24 m, find the sides of the two squares.

EXERCISE 4.4

Question 1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x2 – 3x + 5 = 0
(ii) 3x2 – 4 3 x + 4 = 0
(iii) 2x2 – 6x + 3 = 0

Question 2. Find the values of k for each of the following quadratic equations, so that they have two equal roots.

(i) 2x2 + kx + 3 = 0
(ii) kx (x – 2) + 6 = 0

Question 3. Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2? If so, find its length and breadth.

Question 4. Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

Question 5. Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so, find its length and breadth. 


:: Chapter 5 Arithmetic Progressions ::


EXERCISE 5.1

Question 1. In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?

(i) The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each additional km.
(ii) The amount of air present in a cylinder when a vacuum pump removes 1 4 of the air remaining in the cylinder at a time.
(iii) The cost of digging a well after every metre of digging, when it costs Rs 150 for the first metre and rises by Rs 50 for each subsequent metre.
(iv) The amount of money in the account every year, when Rs 10000 is deposited at compound interest at 8 % per annum.]

EXERCISE 5.2

Question 1. Check whether – 150 is a term of the AP : 11, 8, 5, 2 . . .

Question 2. Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.

Question 3. An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.

Question 4. If the 3rd and the 9th terms of an AP are 4 and – 8 respectively, which term of this AP is zero?

Question 5. The 17th term of an AP exceeds its 10th term by 7. Find the common difference.

Question 6. Which term of the AP : 3, 15, 27, 39, . . . will be 132 more than its 54th term?

Question 7. Two APs have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th terms?

Question 8. How many three-digit numbers are divisible by 7?

Question 9. How many multiples of 4 lie between 10 and 250?

Question 10. For what value of n, are the nth terms of two APs: 63, 65, 67, . . . and 3, 10, 17, . . . equal? 16. Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.

EXERCISE 5.3

Question 1. Find the sum of the following APs:

(i) 2, 7, 12, . . ., to 10 terms.
(ii) –37, –33, –29, . . ., to 12 terms.
(iii) 0.6, 1.7, 2.8, . . ., to 100 terms.
(iv) 1 , 1 , 1 15 12 10 , . . ., to 11 terms.

Question 2. Find the sums given below :

(i) 7 + 10 1 2 + 14 + . . . + 84
(ii) 34 + 32 + 30 + . . . + 10
(iii) –5 + (–8) + (–11) + . . . + (–230) 3. In an AP:

(i) given a = 5, d = 3, an = 50, find n and Sn.
(ii) given a = 7, a13 = 35, find d and S13.
(iii) given a12 = 37, d = 3, find a and S12.
(iv) given a3 = 15, S10 = 125, find d and a10.
(v) given d = 5, S9 = 75, find a and a9. (vi) given a = 2, d = 8, Sn = 90, find n and an.
(vii) given a = 8, an = 62, Sn = 210, find n and d.
(viii) given an = 4, d = 2, Sn = –14, find n and a.
(ix) given a = 3, n = 8, S = 192, find d. (x) given l = 28, S = 144, and there are total 9 terms. Find a

Question 4. How many terms of the AP : 9, 17, 25, . . . must be taken to give a sum of 636?

Question 5. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

Question 6. The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?

Question 7. Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.

Question 8. Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.

Question 9. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

Question 10. Show that a1, a2, . . ., an, . . . form an AP where an is defined as below :

(i) an = 3 + 4n
(ii) an = 9 – 5n Also find the sum of the first 15 terms in each case.

Question 11. If the sum of the first n terms of an AP is 4n – n2, what is the first term (that is S1)? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.

Question 12. Find the sum of the first 40 positive integers divisible by 6.

Question 13. Find the sum of the first 15 multiples of 8.

Question 14. Find the sum of the odd numbers between 0 and 50.

Question 15. A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs 200 for the first day, Rs 250 for the second day, Rs 300 for the third day, etc., the penalty for each succeeding day being Rs 50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days?

Question 16. A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs 20 less than its preceding prize, find the value of each of the prizes.

Question 17. In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?

Question 18. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . . as shown in Fig. 5.4. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take π = 22 7 ) 114 MATHEMATICS Fig. 5.4 [Hint : Length of successive semicircles is l1, l2, l3, l4, . . . with centres at A, B, A, B, . . ., respectively.]

Question 19. 200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on (see Fig. 5.5). In how may rows are the 200 logs placed and how many logs are in the top row? Fig. 5.5

Question 20. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato, and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line (see Fig. 5.6). Fig. 5.6 A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run? [Hint : To pick up the first potato and the second potato, the total distance (in metres) run by a competitor is 2 × 5 + 2 × (5 + 3)]

EXERCISE 5.4

Question 1. Which term of the AP : 121, 117, 113, . . ., is its first negative term? [Hint : Find n for an < 0]

Question 2. The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.

Question 3. A ladder has rungs 25 cm apart. (see Fig. 5.7). The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and the bottom rungs are 2 1 2 m apart, what is the length of the wood required for the rungs? [Hint : Number of rungs = 250 25 ]

Question 4. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x. [Hint : Sx – 1 = S49 – Sx]

Question 5. A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of 1 4 m and a tread of 1 2 m. (see Fig. 5.8). Calculate the total volume of concrete required to build the terrace. [Hint : Volume of concrete required to build the first step = 1 1 50 m3 4 2 × × ]


:: Chapter 6 Triangles ::


Theorems

Theorem 6.1: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Theorem 6.8 : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.


:: Chapter 7 Coordinate Geometry ::


EXERCISE 7.1

Question 1. Find the distance between the following pairs of points :

(i) (2, 3), (4, 1)
(ii) (– 5, 7), (– 1, 3)
(iii) (a, b), (– a, – b) 

Question 2. Find the distance between the points (0, 0) and (36, 15). Can you now find the distance between the two towns A and B discussed in Section 7.2.

Question 3. Determine if the points (1, 5), (2, 3) and (– 2, – 11) are collinear.

Question 4. Check whether (5, – 2), (6, 4) and (7, – 2) are the vertices of an isosceles triangle.

Question 5. In a classroom, 4 friends are seated at the points A, B, C and D as shown in Fig. 7.8. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli disagrees. Using distance formula, find which of them is correct.

Question 6. Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:

(i) (– 1, – 2), (1, 0), (– 1, 2), (– 3, 0)
(ii) (–3, 5), (3, 1), (0, 3), (–1, – 4)
(iii) (4, 5), (7, 6), (4, 3), (1, 2)

Question 7. Find the point on the x-axis which is equidistant from (2, –5) and (–2, 9).

Question 8. Find the values of y for which the distance between the points P(2, – 3) and Q(10, y) is 10 units

Question 9. If Q(0, 1) is equidistant from P(5, –3) and R(x, 6), find the values of x. Also find the distances QR and PR.

Question 10. Find a relation between x and y such that the point (x, y) is equidistant from the point (3, 6) and (– 3, 4).

EXERCISE 7.2

Question 1. Find the coordinates of the point which divides the join of (–1, 7) and (4, –3) in the ratio 2 : 3.

Question 2. Find the coordinates of the points of trisection of the line segment joining (4, –1) and (–2, –3).

Question 3. To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD, as shown in Fig. 7.12. Niharika runs 1 4 th the distance AD on the 2nd line and posts a green flag. Preet runs 1 5 th the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?

Question 4. Find the ratio in which the line segment joining the points (– 3, 10) and (6, – 8) is divided by (– 1, 6).

Question 5. Find the ratio in which the line segment joining A(1, – 5) and B(– 4, 5) is divided by the x-axis. Also find the coordinates of the point of division.

Question 6. If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.

Question 7. Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, – 3) and B is (1, 4).

Question 8. If A and B are (– 2, – 2) and (2, – 4), respectively, find the coordinates of P such that AP = 3 AB 7 and P lies on the line segment AB.

Question 9. Find the coordinates of the points which divide the line segment joining A(– 2, 2) and B(2, 8) into four equal parts.

Question 10. Find the area of a rhombus if its vertices are (3, 0), (4, 5), (– 1, 4) and (– 2, – 1) taken in order. [Hint : Area of a rhombus = 1 2 (product of its diagonals)]

EXERCISE 7.3

Question 1. Find the area of the triangle whose vertices are :

(i) (2, 3), (–1, 0), (2, – 4)
(ii) (–5, –1), (3, –5), (5, 2)

Question 2. In each of the following find the value of ‘k’, for which the points are collinear.

(i) (7, –2), (5, 1), (3, k)
(ii) (8, 1), (k, – 4), (2, –5)

Question 3. Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, –1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.

Question 4. Find the area of the quadrilateral whose vertices, taken in order, are (– 4, – 2), (– 3, – 5), (3, – 2) and (2, 3).

Question 5. You have studied in Class IX, (Chapter 9, Example 3), that a median of a triangle divides it into two triangles of equal areas. Verify this result for Δ ABC whose vertices are A(4, – 6), B(3, –2) and C(5, 2).

EXERCISE 7.4

Question 1. Determine the ratio in which the line 2x + y – 4 = 0 divides the line segment joining the points A(2, – 2) and B(3, 7).

Question 2. Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear.

Question 3. Find the centre of a circle passing through the points (6, – 6), (3, – 7) and (3, 3).

Question 4. The two opposite vertices of a square are (–1, 2) and (3, 2). Find the coordinates of the other two vertices.

Question 5. The Class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Sapling of Gulmohar are planted on the boundary at a distance of 1m from each other. There is a triangular grassy lawn in the plot as shown in the Fig. 7.14. The students are to sow seeds of flowering plants on the remaining area of the plot.

(i) Taking A as origin, find the coordinates of the vertices of the triangle.
(ii) What will be the coordinates of the vertices of Δ PQR if C is the origin? Also calculate the areas of the triangles in these cases. What do you observe?

Question 6. The vertices of a Δ ABC are A(4, 6), B(1, 5) and C(7, 2). A line is drawn to intersect sides AB and AC at D and E respectively, such that AD AE 1 AB AC 4 = Calculate the area of the Δ ADE and compare it with the area of Δ ABC. (Recall Theorem 6.2 and Theorem 6.6).

Question 7. Let A (4, 2), B(6, 5) and C(1, 4) be the vertices of Δ ABC.

(i) The median from A meets BC at D. Find the coordinates of the point D.
(ii) Find the coordinates of the point P on AD such that AP : PD = 2 : 1
(iii) Find the coordinates of points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1.
(iv) What do yo observe?

[Note : The point which is common to all the three medians is called the centroid and this point divides each median in the ratio 2 : 1.]


:: Chapter 8 Introduction To Trigonometry ::


EXERCISE 8.1

Question 1. In Δ ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine :

(i) sin A, cos A
(ii) sin C, cos C ]

Question 2. In Fig. 8.13, find tan P – cot R.

Question 3. If sin A = 3 , 4 calculate cos A and tan A.

Question 4. Given 15 cot A = 8, find sin A and sec A.

Question 5. Given sec θ = 13 , 12 calculate all other trigonometric ratios.

Question 6. If ∠ A and ∠ B are acute angles such that cos A = cos B, then show that ∠ A = ∠ B.

Question 7. If cot θ = 7 , 8 evaluate :

(i) (1 sin ) (1 sin ) , (1 cos ) (1 cos ) + θ − θ + θ − θ
(ii) cot2 θ

Question 8. If 3 cot A = 4, check whether 2 2 1 tan A 1 + tan A − = cos2 A – sin2A or not.

Question 9. In triangle ABC, right-angled at B, if tan A = 1 , 3 find the value of: (i) sin A cos C + cos A sin C (ii) cos A cos C – sin A sin C

Question 10. In Δ PQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.

Question 11. State whether the following are true or false. Justify your answer.

(i) The value of tan A is always less than 1.
(ii) sec A = 12 5 for some value of angle A.
(iii) cos A is the abbreviation used for the cosecant of angle A.
(iv) cot A is the product of cot and A.
(v) sin θ = 4 3 for some angle θ.

EXERCISE 8.2

Question 1. Evaluate the following :

(i) sin 60° cos 30° + sin 30° cos 60°
(ii) 2 tan2 45° + cos2 30° – sin2 60° 

Question 3. If tan (A + B) = 3 and tan (A – B) = 1 3 ; 0° < A + B ≤ 90°; A > B, find A and B.

Question 4. State whether the following are true or false. Justify your answer.

(i) sin (A + B) = sin A + sin B.
(ii) The value of sin θ increases as θ increases.
(iii) The value of cos θ increases as θ increases.
(iv) sin θ = cos θ for all values of θ.
(v) cot A is not defined for A = 0°.

EXERCISE 8.3

Question 1. Evaluate :

(i) sin 18 cos 72 ° °
(ii) tan 26 cot 64 ° °
(iii) cos 48° – sin 42°
(iv) cosec 31° – sec 59°

Question 2. Show that :

(i) tan 48° tan 23° tan 42° tan 67° = 1
(ii) cos 38° cos 52° – sin 38° sin 52° = 0

Question 3. If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A.

Question 4. If tan A = cot B, prove that A + B = 90°. 5. If sec 4A = cosec (A – 20°), where 4A is an acute angle, find the value of A.

EXERCISE 8.4

Question 1. Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

Question 2. Write all the other trigonometric ratios of ∠ A in terms of sec A.

Question 3. Evaluate :

(i) 2 2 2 2 sin 63 sin 27 cos 17 cos 73 ° + ° ° + °
(ii) sin 25° cos 65° + cos 25° sin 65°

Question 4. Choose the correct option. Justify your choice.

(i) 9 sec2 A – 9 tan2 A =

(A) 1
(B) 9
(C) 8
(D) 0

(ii) (1 + tan θ + sec θ) (1 + cot θ – cosec θ) =

(A) 0
(B) 1
(C) 2
(D) –1

(iii) (sec A + tan A) (1 – sin A) =

(A) sec A
(B) sin A
(C) cosec A
(D) cos A

(iv) 2 2 1 tan A 1 + cot A + =

(A) sec2 A
(B) –1
(C) cot2 A
(D) tan2 A

Question 5. Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

(i) (cosec θ – cot θ)2 = 1 cos 1 cos − θ + θ
(ii) cos A 1 sin A 2 sec A
(iii) tan cot 1 sec cosec 1 cot 1 tan θ θ + = + θ θ − θ − θ [Hint : Write the expression in terms of sin θ and cos θ]
(iv) 1 sec A sin2 A sec A 1 – cos A + = [Hint : Simplify LHS and RHS separately]
(v) cos A – sin A + 1 cosec A + cot A, cos A + sin A – 1 = using the identity cosec2 A = 1 + cot2 A.
(vi) 1 sinA sec A + tan A 1 – sin A + =
(vii) 3 3 sin 2 sin tan 2 cos cos θ − θ = θ θ − θ
(viii) (sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A
(ix) (cosec A – sin A)(sec A – cos A) 1 tanA + cot A = [Hint : Simplify LHS and RHS separately]
(x) 2 2 2 1 tan A 1 tanA 1 + cot A 1 – cot A = tan2 A


:: Chapter 9 Some Applications of Trigonometry ::


EXERCISE 9.1

Question 1. A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30°

Question 2. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

Question 3. A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m, and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case?

Question 4. The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower.

Question 5. A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string, assuming that there is no slack in the string.

Question 6. A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building.

Question 7. From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower.

Question 8. A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.

Question 9. The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.

Question 10. Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles and the distances of the point from the poles.

Question 11. A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is 60°. From another point 20 m away from this point on the line joing this point to the foot of the tower, the angle of elevation of the top of the tower is 30° (see Fig. 9.12). Find the height of the tower and the width of the canal.

Question 12. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower.

Question 13. As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

Question 14. A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°. After some time, the angle of elevation reduces to 30° (see Fig. 9.13). Find the distance travelled by the balloon during the interval.

Question 15. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.

Question 16. The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is 6 m.


:: Chapter 10 Circles ::

NCERT Biology Question Paper (Class - 11)

Exam: 
GENERAL: 
Subjects: 
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NCERT Biology Question Paper (Class - 11)


(Biology) Chapter 1 The Living World


Question 1: Why are living organisms classified?

Question 2: Why are the classification systems changing every now and then?

(Exam Date Sheet) Council of Indian School Certificate Examination 2016


Council for the Indian School Certificate Examinations


Indian School Certificate Examination, Year 2016

Day & Date Time Subject/Paper Duration
Monday, February 8 9.00 A.M. Physics - Paper 2 (Practical) 3 hrs.
Tuesday, February 9 9.00 A.M. Indian Music - Hindustani Paper 2 (Practical) Indian Music - Carnatic Paper 2 (Practical) Home Science - Paper 2 (Practical) - Planning Session 20 minutes for each candidate 20 minutes for each candidate 1 hr.
Wednesday, February 10 9.00 A.M. Chemistry - Paper 2 (Practical) 3 hrs.
Thursday, February 11 9.00 A.M. Art Paper 1 (Drawing or Painting from Still Life) Biotechnology - Paper 2 (Practical) 3 hrs. 3 hrs.
Monday, February 15 9.00 A.M. Biology - Paper 2 (Practical) 3 hrs.
Tuesday, February 16 9.00 A.M Home Science - Paper 2 (Practical) - Examination Session 3 hrs.
Wednesday, February 17 9.00 A.M. Computer Science - Paper 2 (Practical) Planning Session Examination Session 3 hrs.
Thursday, February 18 2.00 P.M. Psychology 3 hrs.
Saturday, February 20 9.00 A.M. Fashion Designing - Paper 2 (Practical) Western Music - Paper 2 (Practical) 3 hrs. 28 minutes for each candidate
Monday, February 22 2.00 P.M. Computer Science - Paper 1 (Theory) 3 hrs.
Wednesday, February 24 2.00 P.M. Physical Education - Paper 1 (Theory) 3 hrs.
Friday, February 26 2.00 P.M. English - Paper 1 (English Language) 3 hrs.
Saturday, February 27 9.00 A.M. Art Paper 2 (Drawing & Painting from Nature) 3 hrs.
Monday, February 29 2.00 P.M. English - Paper 2 (Literature in English) 3 hrs.
Tuesday, March 01 2.00 P.M. Indian Music - Hindustani - Paper 1 (Theory) Indian Music - Carnatic - Paper 1 (Theory) Western Music - Paper 1 (Theory) Fashion Designing - Paper 1 (Theory) 3 hrs. 3 hrs. 3 hrs. 3 hrs.
Wednesday, March 02 2.00 P.M Sociology 3 hrs.
Friday, March 04 2.00 P.M. Physics - Paper 1 (Theory) 3 hrs.
Saturday, March 05 9.00 A.M. Art Paper 3 (Drawing or Painting of a Living Person) 3 hrs.
Tuesday, March 08 2.00 P.M. Accounts 3 hrs.
Friday, March 11 2.00 P.M. Chemistry - Paper 1 (Theory) 3 hrs.
Saturday, March 12 9.00 A.M. Art Paper 4 (Original Imaginative Composition in Colour) 3 hrs.
Monday, March 14 9.00 A.M. Art Paper 5 (Crafts ‘A’) 3 hrs.
Tuesday, March 15 2.00 P.M. Mathematics 3 hrs.
Wednesday, March 16 2.00 P.M. Home Science - Paper 1 (Theory) Geometrical & Mechanical Drawing Geometrical & Building Drawing 3 hrs. 3 hrs. 3 hrs.
Friday, March 18 2.00 P.M. Biology - Paper 1 (Theory) 3 hrs.
Saturday, March 19 2.00 P.M. Elective English 3 hrs.
Monday, March 21 2.00 P.M. Economics 3 hrs.
Tuesday, March 22 2.00 P.M. Environmental Science - Paper 1(Theory) 3 hrs.
Monday, March 28 2.00 P.M. Indian Languages / Modern Foreign Languages Classical Languages 3 hrs. 3 hrs.
Wednesday, March 30 2.00 P.M. History Biotechnology - Paper 1 (Theory) 3 hrs. 3 hrs.
Friday, April 1 2.00 P.M. Commerce Electricity and Electronics 3 hrs. 3 hrs.
Monday, April 4 2.00 P.M. Political Science 3 hrs.
Wednesday, April 6 2.00 P.M. Business Studies 3 hrs.
Friday, April 8 2.00 P.M. Geography - Paper 1 (Theory) 3 hrs.

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