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 ::