INDIAN SCHOOL MUSCAT QUESTION BANK 2017-18...

INDIAN SCHOOL MUSCAT QUESTION BANK – 2017-18

DEPARTMENT OF MATHEMATICS – SENIOR SECTION

CLASS X

REAL NUMBERS

SECTION A: (1 MARK)

1. State Euclid’s division lemma.

2. If a divides b and b divides a, what can you say about a and b?

3. Express 0.3 as a fraction in the simplest form.

4. Without actual division find whether the rational number 17

3125 is a terminating or non-

terminating repeating decimal.

5. Write the condition to be satisfied by q so that the rational number p/q is a terminating

decimal.

6. For any integer a and 3, there exists unique integers q and r such that a = 3q + r. Find the

possible values of r.

7. If LCM of two numbers 480 and 672 is 3360, find their HCF.

8. Find the number which when divided by 117 gives 41 as quotient and 23 as remainder.

9. If p and q are positive integers such that p = ab2 and q = a

3b, where a, b are prime numbers,

then find their LCM.

10. If a = (22 x 3

3 x 5

4) and b = (2

3 x 3

2 x 5), then find HCF (a, b).

SECTION B: (2 MARKS)

11. If the HCF of 65 and 117 is of the form (65m – 117), then find the value of m.

12. State whether x = 2.23 + 3

4 is rational or not. Justify your answer.

13. LCM of two numbers is 45 times their HCF. If one of the numbers is 125 and sum of their

HCF and LCM is 1150, find the other number.

14. Using Euclid’s division algorithm, find the HCF of 867 and 255.

15. Show that 21n cannot end with the digits 0, 2, 4, 6, and 8 for any natural number n.

16. By what number should 1365 be divided to get 31 as quotient and 32 as remainder?

17.

Complete the missing entries in the following factor tree.

18. Which of the following has a terminating decimal expansion?

(a) 133

152 (𝑏)

19

80 (𝑐)

27

45 (𝑑)

9

35

19. Which of the following numbers are irrational?

(a) √3 (b) √83

(c) 5.27414141… (d) π

20. Determine the values of p and q so that the prime factorization of 2520 is expressible as

23 x 3

p x q x 7.

SECTION C: (3 MARKS)

21. Using Euclid’s division algorithm, find whether the pair of numbers 847, 2160 are co

primes or not.

22. Using Euclid’s division lemma, show that the square of any positive integer is of the form

2m or 2m +1 for some integer m.

23. Find the largest number of four digits exactly divisible by 12, 15, 18 and 27.

24. Using fundamental theorem of arithmetic, find the LCM and HCF of 816 and 170.

25. Let d be the HCF of 24 and 36. Find two numbers a and b, such that d = 24a + 36b.

26. Find two numbers which on multiplication with √180 gives the smallest rational number.

Are these numbers rational or irrational?

27. The product of three consecutive positive integers is divisible by 6. Is this statement true

or false? Justify your answer.

28. If n is an odd integer, then show that n2 – 1 is divisible by 8.

29. Write whether

2√45+3√20

2√5 on simplification gives a rational or an irrational number.

30. Find the HCF and LCM of 288, 360 and 384 by prime factorization method.

SECTION D: (4 MARKS)

31. A merchant has 120 litres of oil of one kind and 180 litres of another kind and 240 litres of

third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity.

What should be the greatest capacity of such a tin?

32.

In a seminar the number of participants in Mathematics, Physics and Biology are 336, 240

and 96. Find the minimum number of rooms required if in each room same number of

participants is to be seated and all of them being in the same subject.

33. Find the smallest number which when increased by 17 is exactly divisible by 520 and 468.

34. Find the largest number which divides 615 and 963 leaving remainder 6 in each case.

35. Find the HCF and LCM of 306 and 54. Verify that HCF x LCM = Product of two

numbers.

36. Find the LCM of 2.5, 0.5 and 0.175.

37. Prove that

2√3

5 is irrational.

38. Show that the square of any positive integer is of the form 3q or 3q + 1 for some integer q.

39. Prove that 3 + 5√2 is an irrational number.

40. There is a circular path around a sports field. Priya takes 18 mins. to drive one round of the

field, while Ravi takes 12 mins. 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?

POLYNOMIALS

SECTION A: (1 MARK)

1. If α, β , are the zeroes of the polynomial 7x3 + x

2 – 13x +8 then find the value of α β + β + α.

2. If 1 is a zero of the polynomial p(x) = ax2 – 3(a – 1) x – 1, find the value of a.

3. Find the value of “a” for which one zero of p(x) = 3x2 + 12x – a is the reciprocal of the other.

4. Frame a quadratic polynomial having -3 and 2 as their zeroes.

5. If α, β are zeroes of the polynomial p(x) = x

2 – 2x + 1, the find the value of

1+

1

6. Find the degree of the polynomial (x – 1)(x2 + x – x

4 + 2)

7. Find the degree of the polynomial

2

25 1

x

xx

8. The graph of y= f(x) is given in the fig 1, how

many zeroes are there for f(x).

fig 1

9. For what value of k, - 4 is a zero of the polynomial x2

– x – (2k + 2).

10. Find the product of the zeroes of - 2x2

+ kx + 6


11. Write a quadratic polynomial with zeros 5 + √3 & 5 – √3

12. If α and β are the zeros of quadratic polynomial 2x2 – 4x + b, find b if 2α + 3β = 8.

13. If α and 1/α are the zeroes of the polynomial 3x2 + x + (k – 2) find k.

14. If α and β are the roots of the quadratic polynomial p(x) = 3x2 + (k – 2) x + (3k – 2). Find k

if α + β = 2αβ.

15. Form a quadratic polynomial whose one zero is 7 and the product of zeros is given 28.

16. Find the quotient and remainder when x3 + 2x

2 – x – 4 is divided by x + 2.

17. If the sum and product of the roots of the quadratic equation ax2 – 5x + c = 0 are both equal

to 10, then find the values of a and c.

18. Find all other zeroes of the polynomial p(x) = 3x3 -36x +48 of one of its zero is −4

19. Find the zeroes of the quadratic polynomial x2 + 7x + 10 and verify the relationship

between the zeroes and its coefficients.

20. If one root of the quadratic equation x2 – 3x + q = 0 is twice the other, find the value of q.


21. Divide 26x4 – 13x

3 – 74x

2 – 46x – 24 by 2x

2 – 3x – 5 and verify the result by division

algorithm.

22. What must be subtracted from 8x4 + 14x

3 – 2x

2 + 7x – 8 so that the resulting polynomial is

exactly divisible by 4x2 + 3x – 2

23. If α and β are the zeros of the polynomial 20p2 – 19p + 3, find a quadratic polynomial

whose zeros are

24. If α, β and γ are the zeroes of the polynomial x3 – 6x

2 + 11x – 6 find α

−1 + β

−1 + γ

−1.

25. What must be added to the polynomial p(x) = 6x4 + 5x

3 − 11x

2 – 42x + 9 so that the

resulting polynomial is divisible by polynomial q(x) = x2 + 3x + 2.

26. Factorize: x3 – 5x

2 – 2x + 24

27. If α and β are the zeroes of the polynomial x2 + 3x + 2, form a quadratic polynomial whose

zeroes are

28. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time

and the product of its zeroes are 3, −8, −12 respectively.

29. If the zeroes of the polynomial x3 – 3x

2 + x + 1 are p – q, p and p + q, find p and q.

30. What is the remainder when the polynomial x3 – 2x

2 +x – 4 is divided by (x – 2)? What

number is to be added to the given polynomial so that (x – 2) is a factor of it?


31. Find all the zeroes of the polynomial 2x4 – 11x

3 – 16x

2 + 55x + 30 if two of its zeros are

√5, −√5.

32. If one zero of the polynomial p(x) = 2x2 – 4kx + 6x – 7 is the negative of other find the

zeroes of x2 – kx – 1.

33. Find all the zeroes of the polynomial 2x4 + 4x

3 – 22x

2 – 32x + 48 if two of its zeroes are

2√2 and −2√2.

34. Obtain all the zeroes of x4 – 8x

3 + 3 x

2 + 72 x - 108 is two of its zeroes are 2 and 3.

35. Show that 3z + 10 is factor of 9z3 –27z

2 -100z + 300. Also, find the other factors.

36. Factorise 3x4 – 10x

3 + 5x

2 + 10x – 8

37. If the squared difference of the zeroes of the quadratic polynomial f(x) = x2 + kx + 30 is

equal to 169, find the value of k & the zeroes.

38. If the polynomial x4 – 6x

3 + 16x

2 – 25x + 10 is divided by another polynomial x

2 – 2x + k,

the remainder comes out to x – a, find k and a.

39. If two zeroes of the polynomial 2x4 – 12x

3 – 52x

2 + 276x – 70 are 2 ± √3 find the other

zeroes.

40. Frame a quadratic polynomial whose zeroes are

2

24 and

2

24 .

Pairs of Linear Equations in two variables

Section A(1mark)

1. Find the value of k for which the pair of linear equations 4x + 6y – 1 = 0 and 2x + ky - 7 =0

represent parallel lines. (Ans: k = 3)

2. If x = a and y = b is the solution of equations x – y = 2 and x + y = 4, find a and b .

(Ans: a = 3, b =1)

3. The pair of equations y = 0 and y = -7 has how many solutions?

(Ans: no solution)

4. If one equation of a pair of dependent linear equations is - 5x + 7y = 2, give a possible second

equation.

5. Find k if the pair of linear equations kx + 2y = 5 and 3x+ y = 1 has unique solutions.

(Ans: k is any real number except 6)

6. The pair of linear equations 4x + 6y = 12 and 2x +3y = 6 has how many solutions?

(Ans: Infinitely many solutions)

7. What does each solution (x, y) of a linear equation in two variables ax + by + c =0 (a≠0, b≠0)

correspond to?

(Ans: A point in the plane)

8. Give the condition that the system of equations a 1 x+ b 1 y = c 1 and a 2 x+ b 2 y = c 2 has unique

solution. (Ans: 2

1

a

a≠

2

1

b

b )

9. Find the area of the triangle formed by the coordinate axes and line x + y = 6 is :

(Ans: 18 sq units )

10. The equation 2x – y = 12 and the equation 3ax + 2by = 18 represents two parallel lines.Find

the value of a:b

(Ans: -4:3)

Section B(2marks)

11. For which value of p does the pair of equations given below has unique solution?

4x + py+ 8 = 0 and 2x + 2y + 2 = 0.

(Ans: p is any real no.except 4)

12. For what value of k 1 the following system of linear equations have no solution:

3x + y =1 and (2k - 1)x + (k - 1)y = 2k + 1

(Ans: k = 2)

13. Solve for x and y:

37x + 43y =123 ; 43x + 37y = 117

(Ans: x=1; y = 2)

14.

Solve for x and y

4x + andy

156= 3x -

y

4 = 7 (Ans: x = 3, y = 2)

15. For what value of k, 2x + 3y = 4 and (k + 2) x + 6y = 3k + 2 will have infinitely many

solutions? (Ans: k = 2)

16. For what value of k the following equations have unique solution 2x + ky = 1 and 3x – 5y = 7?

(Ans: k is any real number except -3

10)

17. Determine a and b for which the system of linear equation has infinite number of solutions.

(2a -1) x + 3y - 5 = 0 and 3x + (b -1)y – 2 =0 (Ans: a = 4 ; b = 5

11)

18. For what value of p will the following system of equations have no solution

(2p – 1)x + (p -1) y = 2p + 1 and y + 3x – 1 = 0 (Ans: p = 2)

19. Is the system of linear equations 2x + 3y = 9 and 4x + 6y = 18, consistent?

(Ans: consistent and dependent)

20. Without drawing the graph, find out whether the line representing by the following pair of

equations, intersect at a point ,are parallel or are coincident.

9x – 10y = 21 and 2

3x -

3

5 y =

2

7 (Ans: co-incident)

Section C(3 marks)

21. Solve the following equations graphically:

(i) 3x + y + 4 = 0 (ii) 3x - 4y -12 = 0 (iii) x + y =3

6x - 2y + 4= 0 x + 2y – 4= 0 3x+ 3y=9

(iv) 2x+ 7y = 11 ( Ans. (i) x= -1,y= -1 (iii) infinite solutions

5x + 2

35y=25 (ii) x= 4, y= 0 (iv) no solution)

22. For what values of a and b does the following pair of linear equations have infinite number of

solutions? 2x + 3y = 7, a (x + y) – b (x - y) = 3a + b - 2 ( Ans 5; 1)

23. In a bag containing red and white balls, half the number of white balls is equal to one-third the

number of red balls. Thrice the total number of balls exceeds seven times the number of white

balls by 6. How many balls of each colour does the bag contain? (Ans 18; 12)

24. Solve the following pair of linear equations by substitution method:

3x + 2y - 7=0

4x + y - 6=0 (Ans x = 1; y = 2)

25. Solve using cross multiplication method :

5x + 4y – 4 = 0

x – 12y – 20 = 0 (Ans x = 2; y = 2

3- )

26. Solve the following equations:

a) 8x – 9y = 6xy ; 10x + 6y = 19xy (Ans x = 0; y = 0 or x =2

3;

3

2=y )

b) 6(ax + by) = 3a + 2b ; 6(bx + ay) = 3b – 2a (Ans x = 1/2 ; y = 1/3 )

27. The difference between two numbers is 26 and one number is three times the other. Find the

numbers. (Ans 39; 13)

28. A fraction becomes

11

9 if 2 added to both numerator and denominator. If 3 is added to both

numerator and denominator it becomes 6

5. Find the fraction. (Ans 7/9)

29. A two digit number is obtained either by multiplying the sum of digits by 8 and then

subtracting 5 or by multiplying the difference of digits by 16 and adding 3. Find the number.

( Ans 83)

30. The age of the father is twice the sum of the ages of his 2 children. After 20 years his age will

be equal to the sum of the ages of his children. Find the age of the father.

(Ans :40 yrs)

Section D (4 marks)

31. A boat covers 32 km upstream and 36 km downstream in 7 hours. Also, it covers 40 km

upstream and 48 km downstream in 9 hours. Find the speed of the boat in still water and that of

the stream . (Ans: 10km/hr ; 2km/hr)

32. A and B are two points 150 km apart on a highway. Two cars start from A and B at the same

time. If they move in the same direction they meet in 15 hours. But if they move in the

opposite direction, they meet in 1 hour. Find their speeds. (Ans: 80km/hr ; 70km/hr)

33. Draw the graphs of the pair of linear equations:

x+ 2y = 5 and 2x – 3y = - 4

Also find the points where the lines meet the x-axis. (Ans : (5,0) ,(-2,0)

34. Solve the following pair of linear equations graphically:

2x + 3y = 12 and x – y = 1.

Find the area of the region bounded by the two lines representing the above equations and

y-axis. (Ans: 7.5 sq units)

35. Solve the following pair of linear equations graphically:

x+ 3y = 6 , 2x – 3y = 12

Also shade the region bounded by the line 2x - 3y = 12 and both the co-ordinate axes.

(Ans x = 6, y = 0)

36. Raghav scored 70 marks in a test, getting 4 marks for each right answer and losing 1 mark for

each wrong answer. Had 5 marks been awarded for each correct answer and 2 marks been

deducted for each wrong answer, then Raghav would have scored 80 marks. How many

questions were there in the test? Which values would have Raghav violated if he resorted to

unfair means? (Ans : 30 ; honesty)

37. The incomes of two persons A and B are in the ratio 8 : 7 and the ratio of their expenditures is

19 : 16 . If their savings are Rs 2550 per month, find their monthly income.

What is the importance of saving in life?

(Ans : Rs 12240,Rs 10710 ; value)

38. Three lines x + 3y = 6, 2x - 3y= 12 and x = 0 are enclosing a beautiful triangular park. Find the

points of intersection of the lines graphically and the area of the park, if all measurements are

in km. What type of behaviour should be expected by public in these types of parks?

(Ans: 18 km2

; value)

39 A lending library has a fixed charge for the first three days and an additional charge for each

day thereafter. Shristi paid Rs27 for a book kept for seven days, while Rekha paid Rs 21 for the

book she kept for five days. Find the fixed charge and the additional charge paid by them.

What is the importance of library? (Ans : Rs15, Rs 12, Rs 6; importance)

40. In a painting competition of a school a child made Indian national flag whose perimeter was 50

cm. Its area will be decreased by 6 square cm, if length is decreased by 3 cm and breadth is

increased by 2cm, then find the dimensions of the flag? What does the saffron colour in the

flag signify? (Ans : 15cm; 10cm; courage and sacrifice)

QUADRATIC EQUATIONS

SECTION A: (1 MARK) 1. Find the roots of the quadratic equation x

2 + 5x – (α + 1)(α + 6) = 0, where α is a constant.

2. If mx2 + 2x + m = 0 has 2 equal roots, then find the value(s) of m.

3. If 1 is a root of the equations ay2 + ay + 3 = 0 and y

2 + y + b = 0, then find the value of ab.

4. Check the nature of roots of the equation x2 – 4x + 3√2 = 0.

5. Find the roots of the equation ax2 + a = a

2x + x

6. Determine whether -2√3 is a solution of the quadratic equation x2 - 3√3x + 6 = 0.

7. A polygon of n sides has 𝑛(𝑛−3)

2 diagonals. How many sides has a polygon with 54 diagonals?

8. Rewrite the following as a quadratic equation in x

4

𝑥− 3 =

5

2𝑥+3 ; x ≠ 0,

−3

2

9. Check whether the following equations are quadratic equations:

(x + 3)2 = (x + 2)

2 + 1

10. Determine whether √7 y2 – 6y - 13√7 = 0 has roots.

SECTION B :(2 MARKS)

1. If one root of the quadratic equation 2x

2 – 3x + p = 0 is 3, find the other root of the equation .

Also find the value of p.

2. For what value of k does the quadratic equation (k – 5)x2 + 2(k – 5)x + 2 = 0 have equal roots?

3. Solve for x : √3𝑥2 + 𝑥 + 5 = 𝑥 − 3

4. If the equation x2 + 2(k +2)x + 9k = 0 has repeated root, find the values of k.

5. If one root of quadratic equation x2 – 3x + q = 0 is twice the other root, find the value of q.

6. If x = 2

3 𝑎𝑛𝑑 𝑥 = −3 are roots of the quadratic equation ax

2 + 7x + b = 0, find the values of

a and b.

7. Find the set of values for p for which 4x2 – 3px + 9 = 0 has real roots.

8. Find the set of values for k for which x2 + 5kx + 16 = 0 has no real roots.

9. One day, I asked the son of my close friend about his age. The child replied in a different way.

He said, “One year ago, my dad was 8 times as old as me and now his age is equal to square of

my age.” Represent this situation in the form of a quadratic equation.

10. Using quadratic formula solve the following quadratic equation for x:

x2 – 4ax + 4a

2 – b

2 = 0.


1. Solve for x : 2

2x + 3 = 65(2

x – 2) + 122

2. Find the roots of the following quadratic equation : √2𝑥2 + 3𝑥 + √2 = 0

3. Find the roots of the following quadratic equation : 3x2 + 4√3x + 4 = 0

4. If -5 is a root of the quadratic equation 2x2 + px – 15 = 0 and the quadratic equation

p(x2 + x) + k = 0 has equal roots, find the value of k.

5. Three rods have to be fitted together to form a right triangle. If the hypotenuse is to be 4 cm

longer than the base and 8 cm longer than the altitude, find the length of the rods.

6. A piece of cloth costs Rs. 200. If the piece was 5m longer and each meter costed Rs. 2 less, the

cost would remain unchanged. How long is the piece and what is the original rate per meter?

7. Find the roots of the following quadratic equation:

3 (3𝑥 − 1

𝑥 + 2) − 2 (

𝑥 + 2

3𝑥 − 1) = 5; 𝑥 ≠ −2,

1

3

8. Two circles touch internally. Sum of their areas is 116𝜋 cm2 and distance between their centers

is 6 cm. Find the radii of the circles.

9. If the roots of the quadratic equation p(q – r)x2 + q(r – p)x + r(p – q) = 0 be equal, show that

1

𝑝+

1

𝑟=

2

𝑞.

10. Solve for x : x2 + 5x – (a

2 + a – 6) = 0.


1. Solve for x :

2𝑥

𝑥−3+

1

2𝑥+3+

3𝑥+9

(2𝑥+3)(𝑥−3)= 0 (x≠ 3, −

3

2)

2. If the roots of the equation (a – b) x2 + (b – c) x + (c – a) = 0 are equal,

prove that 2a = b + c.

3. A tank can be filled by one pipe in x minutes and emptied by another pipe in (x + 5) minutes.

Both the pipes when opened together can fill the empty tank in 16.8 minutes. Find x.

4. While boarding an aeroplane, a passenger got hurt. The pilot, showing promptness and

concern, made arrangements to hospitalize the injured and so the plane started late by 30

minutes. To reach the destination, 1500 km away, in time, the pilot increased the speed by 100

km/hr. Find the original speed per hour of the plane. Do you appreciate the values shown by

the pilot, namely, promptness in providing help to the injured and his efforts to reach in time?

5. A farmer wishes to start a 100 sq.m ‘rectangular’ vegetable garden. Since he has only 30 m

barbed wire, he fences 3 sides of the rectangular garden letting his house compound wall act as

the fourth side. Find the dimensions of the garden.

6. A path of length 32 m is constructed along a chord PQ of a circular park of radius 20 m. Two

more paths are constructed from an external point T to the park and tangential to it at P and Q.

If the cost of digging the path(taking it to be narrow of negligible width) is ₹120 per meter,

find the total cost incurred.

7. Solve for x: 1

2𝑎+𝑏+2𝑥=

1

2𝑎+

1

𝑏+

1

2𝑥

8. Prove that the tangent at any point of a circle is perpendicular to the radius at the point of

contact.

9. The sum of the squares of two consecutive even numbers is 340. Find the numbers.

10. The numerator of a fraction is 1 less than the denominator. If 3 is added to each of the

numerator and denominator, the fraction is increased by 3/28. Find the fraction.

ARITHMETIC PROGRESSIONS

SECTION A: (1 MARK)

1. Check whether the sequence formed with an =2n2 forms an A.P.

2. Find an A.P., if the nth

term of an A.P is 5 – 2n.

3. If the sum of first n terms of an A.P. is 2n2 + 5n, then find its n

th term.

4. If 4/5, a and 2 are three consecutive terms of an A.P., then find the value of a.

5. If the sum of first p terms of an A.P. is ap2 + bp, find its common difference.

6. Find the next term of an A.P. √8 , √18, √32, …..

7. Find the common difference of the A.P. (-9-4a), (-8-3a), (-7-2a),……

8. For what value of k, (2k+1), 8, 3k form an A.P.?

9. Find the missing terms of the A.P. 23, ____, 69,____.

10. Find the nth

term of the A.P. -2, -5, -8, …..


11. Which term of the A.P 36, 31, 26, 21, … is the first negative term?

12. Is 497 a term of the A.P 56, 63, 70, ….?

13. Find the 6th

term from the end of the A.P. 17, 14, 11, …. – 40.

14. Which term of the A.P 3, 10, 17,…. Will be 84 more than its 13th

term?

15. Show that (a – b), a, (a + b) are in A.P.

16. How many numbers of three digits are divisible by 7?

17. Which term of the A.P. 18, 16, 14, …… is zero?

18. Find the sum of first n odd natural numbers?

19. The sum of n terms of an A.P is n2 + 3n. Find its 20

th term.

20. Find the sum of first 225 natural numbers.


21. Find the sum of the first 25 terms of an A.P. whose nth

term is given by an = 2 – 3n.

22. If m times the mth

term of an A.P. is equal to n times its nth

term, find its (m + n)th

term.

23. The 4th

and 10th

terms of an A.P. are 13 and 25 respectively. Find the A.P.

24. If the sum of the first 14 terms of an A.P is 1050 and its first term is 10, find the 20th

term.

25. Find the three terms of an A.P. whose sum is 15 and their product is 105.

26. If fifth term of an A.P. is zero, show that its 33rd

term is four times its 12th

term.

27. How many terms of the AP 21, 18, 15, ….. must be added to get the sum zero?

28. The 8th

of an AP is equal to three times its third term. If 6th

term is 22, find the AP.

29. Determine the A.P. whose fourth term is 18 and the difference of the 9th

term from the 15th

term is 30.

30. The 9th

term of an AP is 499 and 499th

term is 9, find the term which is equal to zero.


31. In an AP, the first term is 2, the last term is 29 and the sum of all its terms is 155. Find the

common difference of the AP.

32. The angles of a quadrilateral are in AP whose common difference is 10°. Find the angles.

33. The sum of first q terms of an AP is 63 – 3q2. If its p

th term is –60, find the value of p.

Also, find the 11th

term of the AP.

34. The first and the last terms of an AP., are – 4 and 146 respectively and the sum of the AP is

7171. Find the number of terms in AP. And the common difference.

35. Find the sum of all two digit numbers, which leave 1 as remainder, when divided by 3.

36. How many terms of the AP 24, 21, 18, …. must be taken so that their sum is 78?

37. Find the 31st term of an AP whose 11

th term is 38 and 16

th term is 73.

38. Find the sum of first 24 terms of the list of numbers whose nth

term is given by an = 3 + 2n.

39. If pth

, qth

and rth

terms of an AP are a , b and c respectively, then show that

a(q – r) + b(r – p) + c(p – q) = 0.

40. A club consists of members whose ages are in AP., common difference being 3 months.

The youngest member of the club is just 7 years old and the sum of the ages of members is

250 years. Find the number of members in the club.

TRIANGLES

SECTION A: (1 MARK)

1. If sides of two similar triangles are in the ratio of 4 : 9 then Areas of these triangles

are in the ratio___

(a) 2 :3 (b) 4 :9 (c) 81 :16 (d) 16 : 81

2. In a triangle ABC, DE is parallel to BC. If AB = 10 cm, AC = 8 cm and AD = 5 cm,

then find CE.

�

3. In ΔABC, DE II BC intersecting AB at D and AC at E, AD = 1cm, DB = 3cm, AE =

1.5cm, AC =?

(a) 6 cm (b) 10 cm (c) 8 cm (d) None

4. Two triangles are similar but not congruent and the lengths of the sides of the first are

6cm, 11cm and 12cm. The ratio of corresponding sides of first and second triangle is

1 : 2. What is the perimeter of the second triangle:

(a) 29cm (b) 53cm (c) 58cm (d) 56cm

5. If in two triangles ABC and PQR, AB/ PQ =BC/ QR= CA/ PR , then

(a) ∆PQR ~ ∆ CAB (b) ) ∆PQR ~ ∆ABC (c) ) ∆CBA ~ ∆PQR d) ) ∆ BCA ~ ∆

PQR


6. ∆ABC ~ ∆PQR, perimeter of ∆ABC =32cm and perimeter of ∆PQR =48cm

and PR=6cm . Find AC

7. In the given figure , AB ∥ DC, find x if AO= x+5, OC= x+3, OD= x-2, OB= x-1

D C

A B

8. In the ΔABC, ACB = 90˚ and CD is perpendicular to AB, D lies on AB. Prove

that CD2 = BD X AD

9. In a triangle PQR, N is a point on PR such that QN is perpendicular to PR.

If PN X NR = QN2, prove that PQR = 90°

10. Diagonals AC and BD of a trapezium ABCD with AB||DC intersect each other at

the point O. Using a similarity criterion for two triangles, show that OC

OA =

OD

OB


11. If AD and PM are medians of ∆ABC and ∆PQR respectively where ∆ABC ~

∆PQR then

Prove that PQ

AB=

PM

AD

12. Prove that the ratio of the areas of two similar triangles is equal to the squares

of the ratio of their corresponding sides.

13. In the given figure , D, E and F are mid points of sides BC , CA and AB

respectively of ∆ABC Find the ratio of area of ∆ DEF to area of ∆ABC

A

D E

B C

14. Prove that If a straight line divides any two sides of a triangle in the same ratio,

then the line must be parallel to the third side.

15. Prove that If a perpendicular is drawn from the vertex of a right angled triangle

to its hypotenuse, then the triangles on each side of the perpendicular are similar

to the whole triangle.

16. Prove that In a right angled triangle, the square of the hypotenuse is equal to the

sum of the squares of the other two sides.

O

17. In PQR, given that S is a point on PQ such that ST II QR and PS/SQ= 3/5 If PR

= 5.6 cm, then find PT

18. P and Q are the mid points on the sides CA and CB respectively of triangle

ABC right angled at C.

Prove that 4(AQ2 +BP

2) = 5 AB

2

19. In an isosceles triangle ABC if AC = BC and AB2 = 2AC

2, Prove that C is a

right angle Q

20. PQR is a right triangle right angled at P and M is a point on QR such that

PM 𝑖𝑠 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑡𝑜 QR. Show that PM2 = QM X MR.


21. In an equilateral triangle ABC, D is a point on side BC such that BD =1/3 BC.

Prove that 9AD2 = 7AB

2.

22. O is any point inside a rectangle ABCD. Prove that OB2

+ OD2 = OA

2 + OC

2.

23. In the given figure, D and E trisect side BC. Prove that 8AE2 =3AC

2 + 5AD

2

A

B D E C

24. ABC is a right triangle right-angled at C and AC= √3 BC. Prove that

<ABC=60

25. In the given figure, CE and DE are equal chords of a circle with centre O

If AOB=90 find ar(∆CED) = ar(∆ AOB)

26. D and E are points on the sides CA and CB respectively of a right triangle ABC

where C=90 prove that AE

2 + BD

2 = AB

2+ DE

2

27. An equilateral triangle is inscribed in a circle of radius 6cm. Find its side.

28. ABC is a right triangle, right angled at C. If BC=a, CA=b , AB=c and p be the

length of the perpendicular from C on AB then

Prove that i) c p= a b

ii) 1/p2

= 1/a2

+ 1/b2

29. State and prove Basic Proportionality Theorem

30. A farmer has a field in the shape of a right triangle with legs of length 128m and

64m. He wants to leave a space in the form of square of largest size inside the

field for growing wheat and the remaining for growing vegetables. Find the

length of the side of such a square space. What value is indicated from this?

Coordinate Geometry

Section A

1. Find the perpendicular distance of A (5,12) from the y axis. (Ans: 5 units)

2. Find the distance of the point (-4 , -7) from the y axis (Ans : 4 units)

3. State whether the following statement is true or false. Justify your answer.

A circle has its centre at the origin and a point P (5, 0) lies on it. The point Q (6, 8) lies

outside the circle.

(Ans: True: r = 5 and 0Q> 5 units)

4. The point P (-2, 4) lies on a circle of radius 6 and centre C (3, 5). State whether true or false

and justify.

(Ans : False; PC = 26 <6, P lies inside)

5. Find the distance between the points

2,

5

2,2,

5

8QP

(Ans: 2 units)

6. Find the perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0)

(Ans: 12 units)

7. Find the value of a, for which point

2,

3

aP is the midpoint of the line segment joining the

points Q(-5,4) and R (-1,0). (Ans: a = -9)

8. If the centre and radius of circles is (3, 4) and 7 units respectively, then what is the position

of the point A(5, 8) with respect to circle? (Ans: inside the circle)

9. The ordinate of a point A on y axis is 5 and B has co-ordinates (-3,1). Find the length of

AB. (Ans: 5 units)

10. To calculate a point Q on line segment AB such that ABBQ

7

5. What is the ratio of line

segments into which AB is divided? (Ans : 2:5)

Section B

11. The x co-ordinate of a point P is twice its y co-ordinate. If P is equidistance from Q (2, -5)

and R(-3, 6), find the co-ordinates of P.

(Ans : 16,8)

12. If A (5, 2), B (2,-2) and C (-2, t) are the vertices of a right angled triangle with

°=∠ 90B , then find the value of t.

(Ans: t = 1)

13. Find the ratio in which the point

12

5,

4

3P divides the line segment joining the point

2

3,

2

1A and B (2, -5). (Ans: (1:5)

14. The points A 7,4 , B ( )3,p and C ( )3,7 are the vertices of a right triangle , right-angled at B.

Find the value of p.

(Ans: p = 4, p≠7)

15. Find the point on the x-axis which is equidistant from the points (2, -5) and (-2, 9).

(Ans: (-7,0)

16. Find ‘a’ so that ( )a,3 lies on the line represented by 2x - 3y – 5 = 0. Also, find the

co-ordinates of the point where the line cuts the x-axis.

(Ans: a=1/3;(5/2,0))

17. If the vertices of ΔABC are A (5,-1) , B(-3,-2) , C(-1,8) , find the length of median

through A.

(Ans: 65 units)

18. Find the mid-point of side BC of ΔABC, with A (1, - 4) and the mid-points of the sides

through A being (2,-1) and (0,-1).

(Ans: (1,2))

19. Find the relation between x and y, if the points A (x, y), B (- 5, 7) and C (- 4, 5) are

collinear.

(Ans: 2x + y + 3 = 0)

20. For what values of k are the points (8, 1), (3, -2k) and (k, -5) collinear?

(Ans: k = 2,2

11)

Section C

21. Find the co-ordinates of a point P on the line segment joining A(1,2) and B (6,7) such that

AP=5

2AB

(Ans: (3, 4) )

22. If the point C (-1, 2) divides internally the line segment joining the points A (2, 5) and

B (x, y) in the ratio 3 : 4, find the value of 22 yx .

(Ans: 29)

23. The co-ordinates of the vertices of ABCΔ are A (7, 2), B (9,10) and C(1, 4). If E and F are

the midpoints of AB and AC respectively, prove that EF = BC2

1.

24. Find the ratio in which the point (-3, p) divides the line segment joining the points (-5, -4)

and (-2, 3). Hence the value of p.

(Ans: 2:1 ; p = 2/3)

25. Prove that the points (7, 10),(- 2, 5) and (3, - 4) are the vertices of an isosceles right triangle.

26. If (a, b) is the midpoint of the segment joining the points A(10, - 6) and B(k, 4) and

a - 2b = 18, find the value of k and the distance AB.

(Ans: K = 22;AB = 2 61 units)

27. Find the area of the rhombus of vertices (3, 0), (4, 5), (- 1, 4) and (-2, -1) taken in order.

(Ans: 24 sq units)

28. Find the area of a triangle ABC with A (1, -4) and mid points of sides through A being

(2, -1) and (0,-1).

(Ans: 12 sq units)

29. Find the value of p for which the points (p+1, 2p - 2), (p - 1, p) and (p – 3, 2p - 6) are

collinear.

(Ans: p = 4)

30. If A (4, -1), B (5, 3), C (2, y) and D (1, 1) are the vertices of a parallelogram ABCD,

find y.

(Ans: y = 5)

Section D

31. The co-ordinates of vertices of ABCΔ are A (0, 0), B(0, 2) and C(2, 0). Prove that ΔΑΒC is

an isosceles triangle. Also find its area.

(Ans: 2 sq units)

32. Find the values of k so that the area of the triangle with vertices (1, - 1),(- 4, 2k) and (-k, -5)

is 24 sq units.

(Ans: k = 3, -9/2)

33. Find the ratio in which the line 2x + 3y – 5 = 0 divides the line segment joining the points

(8, -9) and (2, 1). Also find the co-ordinates of the point of division.

(Ans: 8:1 ; ( 8/3, 1/9) )

34. If P (9a – 2, -b) divides the segment joining A(3a+ 1, -3) and B (8a, 5) in the ratio 3 : 1, find

the values of a & b.

(Ans: a = 1, b = -3)

35. Find the ratio in which the point P(x, 2) divides the line segment joining the points A (12, 5)

and B (4, -3). Also find x.

(Ans: 3 : 5 , x = 9)

36. The base QR of an equilateral triangle PQR lies on x axis .The co-ordinates of point Q are

(- 4,0) and the origin is the mid-point of the base . Find the co-ordinates of the point P

andR.

(Ans: P (0, 4 3 ) or (0, -4 3 ) ; R (4, 0) )

37. The base BC of an equilateral triangle ABC lies on y axis. The co-ordinates of point C are

(0, -3). The origin is the mid-point of the base. Find the co-ordinates of the point A and B.

Also find the co-ordinates of another point D such that BACD is a rhombus.

(Ans: B(0, 3) , A(3 3 ,0) , D(-3 3 , 0) )

38. Find the ratio in which the line joining points (a + b, b + a) and (a - b, b - a) is divided by

the point (a, b).

(Ans: 1:1)

39. Find the area of the quadrilateral ABCD, the co-ordinates of whose vertices are A(5, -2),

B(-3, -1) , C(2, 1) & D (6, 0)

(Ans: 15 sq units)

40. Prove that the area of a triangle with vertices (t, t - 2), (t + 2, t + 2) and (t + 3, t) is

independent of t.

TRIGONOMETRY

1mark questions:

1. Given A = 300, verify sin 2A = 2cosA sinA.

2. Sin4A = cos (A-200), where 4A is an acute angle, find the value of A.

3. Show that tan4 A + tan

2 A= sec

4 A- sec

2 A for 0

0 ≤ A ≤ 90

0.

4. If tan A = cot (300+A), find the value of A.

5. If cos P = 7/25, find the value of tan P + cot P.

6. In triangle ABC , angle B = 900, AB = 5cm angle ACB = 30

0.Find the length of side AC.

7. Prove tan 100.tan 75

0.tan 15

0.tan 80

0 = 1

8. Write sec A in terms of cot A

9. If sin x +sin 2x = 1, find the value of cos

2x + cos

4x

10. Evaluate: 2tan 300 /1+tan

2 30

0.

2marks questions

11. If tanA = √2 -1 show that

4

2

tan1

tan2

A

A

12. EVALUATE: sin300 cos 45

0 + cos30

0 sin 45

0.

13. If tan A = 1/√3 find the value of sinA cosB +cosA sinB

14. Evaluate:

00000

0000

85tan55tan35tan30tantan

75sin15cos75cos15sin

15. If sec +tan =m and sec - tan =n, find the value of √mn.

16. If sin A = √3/2, find the value of 2cot2A -1

17. If cos +sin =√2cos ,show that cos -sin =√2sin

18. Find the value of x if cos2x = sin600.cos30

0 – cos60

0.sin30

0.

19. Prove that:

22 cottancos.sin

cottan

20. Find A and B if sin (A+B) = 1 and cos(A-B) = 1

3marks questions

21. If a cos -b sin =x and a sin +b cos =y. Prove that a2+b2 =x2+y2

22. ABC is a right angled triangle, D is the midpoint of BC, C = 900, D = ,B = ,

then show that tan /tan = ½.

23. Evaluate without using trigonometric table,

)85tan65tan45tan25tan5(tan7

20cos70(cos4

35sin7

55cos300000

)0.0

0

0 ec

24. If 5sin +3cos = 4, find the value of 3sin -5cos . 25. Show that (sec +cos )(sec -cos ) = tan2 +sin2

26. If sec = x+ 1/4x, prove that sec +tan =2x or 1/2x

27. Evaluate: 02020202 30tan

4

360cos260sin330cot

3

4 ec

28. If sin =cos ,find the value of 2tan2 +sin2 -1

29. If 3cotA = 4 show that AA

A

A 22

2

2

sincostan1

tan1

30. Without using table evaluate:

sin (500+ )-cos (400- )+tan10.tan10.tan200.tan700.tan800.tan890.

4marks questions

31. Determine the value of x such that 2cosec2 300 +xsin2600 -3/4 tan2300 = 10

32. If A+B = 900, prove that

A

BBA

BABA2

2

cos

sinsecsec

cottantantan = tan A

33. Prove that: AA

AA

A

A

Acossin

sincos

sin

tan1

cos 2

34. If xsin3 +ycos3 =sin cos and x sin = y cos , prove that x2+y2=1

35. If 6x =sec and 6/x = tan , find the value of 9(x2- 1/x

2)

36. Show that 3(sin - cos )4 + 6(sin + cos )

2 + 4(sin

6 +cos6 ) is independent of .

37. If 2cos –sin = x and cos -3sin = y, prove that 2x2 + y

2-2xy = 5

38. Prove that: ( 1+cotA + tanA)( sinA –cosA) =

A

ecA

Aec

A22 sec

cos

cos

sec

39. Evaluate:

0200

22020202

30cot60sec30cos

90cos530sec345sin460tan

ec

40. Prove that:

cottan1

tan1

cot

cot1

tan

SOME APPLICATIONS OF TRIGONOMETRY

SECTION A: (1 MARK)

1. Find sun’s altitude at a time when the length of the shadow of a tower is equal to its height.

2. What is the angle of elevation of a 30 m high tower at a point 30 m away from the base?

3. The length of shadow of a tower on the ground is 1/√3 times the height of the tower. Find the angle

of elevation of the sun.

4. A tower of height 40 m casts shadow of 40√3 m when sun’s elevation is θ. Find θ.

5. Find the angle of elevation of a point to the top of a tower if it’s at a distance √3 times the height of

tower.

6. A pole 30 m long rests against a vertical wall at an angle of 30° with the ground. Find the height

upto which the pole reaches.

7. If a ladder 10 m long is touching the top of a wall, making an angle of 60° with the wall, the height

of wall

8. In the given figure1, PQ is the pole of height 20

m. If the suns angles of elevation are 60°, find

the length of shadow QR.

fig 1

9. A kite is flying at a height of 40 cm from the ground with an angle of elevation of 30° find the

length of string from the kite to the ground.

10. A flagstaff 12 m high on the top of a tower casts a shadow of 4√3 along the ground. Find the angle

of elevation of the sun.


11. In the given figure 2, find the angle of depression from A.

12. From the given figure 3, find h1 : h2.

13. Find AB from the given figure 4

Fig 2

.

Fig3

Fig 4

14. Two persons are standing on the opposite sides of a tower. They observe the angles of

elevation of the top of the tower to be 30° and 60°. Find the distance between them if the

height of tower is 60 m.

15. . The angles of elevation and depression of the top and bottom of a light-house from the top

of a 60 m high building are 30° and 60° respectively. Find

(i) the difference between the heights of the light-house and the building.

(ii) the distance between the light-house and the building.

16. From a point 30 m away from the tool of a tower, the angle of elevation of the top of tower

is found to be 30°. Find the height of tower. use √3 = 1.73

17.

fig 5 fig 6

fig 7

18. The length of the shadow of a tower increase by 10√3 m when sun’s altitude changes from 45° to

30° in figure 5. Find height of tower.

19. In the figure6, given below, it is given that AB is perpendicular to BD and is of length X metres.

DC = 40 m, ∠ADB = 30° and ∠ACB = 45°. Find x

20. Find AB from the given figure 7.


21. A person in a helicopter flying at a height of 500 m, observes two objects lying opposite to

each other on either bank of a river. The angles of depression of objects are 30° and 45°.

Find the width of the river. (√3 = 1.732)

22. The angles of elevation of a jet plane from a point A on the ground is 60°. After a flight of

15 seconds, the angle of elevation changes to 30°. If the jet plane is flying at a constant

height of 1500√3, find the speed of the jet plane.

23. From the top of a tree, the angle of depression of an object on the horizontal ground is

found to be 60°. On descending 20 ft from the top of the tree the angle of depression of the

object is found to be 30°. Find the height of the tree.

24. The angles of elevation of a jet plane from a point A on the ground is 60°. After a flight of

15 seconds, the angle of elevation changes to 30°. If the jet plane is flying at a constant

height of 1500√3, find the speed of the jet plane.

25. From the top of a building 60 m high the angles of depression of the top and the bottom of

a tower are observed to be 30° and 60°. Find the height of the tower.

26. From a point, 25 m above the surface of a lake, the angle of elevation of a bird is observed

to 30° and angle depression of its image in the water of the lake is observed to be 60°. Find

the actual h eight of the bird above the surface of the lake.

27. From a window A in figure 8, 10 m above the ground the angle of elevation of the top C of

a tower is x°, where and the angle of depression of the foot D of the tower is y°,

where Calculate the height CD of the tower in metres.

fig 8 Fig 9 fig 10

28. Find the perimeter of the given figure 9.

29. With the reference to the given figure 10, a man stands on the ground at point A, which is

on the same horizontal plane as B, the foot of the vertical pole BC. The height of the pole is

10 m. The man’s eye is 2 m above the ground. He observes the angle of elevation of C, the

top of the pole, as x°, where calculate the distance AB

30. The angle of elevation of a stationary cloud from a point 25 m above a lake is 30° and the

angle of depression of its reflection in the lake is 60°. What is the height of the cloud above

that lake-level?


31. The angles of elevation of the top of a tower from two points at distance of 8 m and 9 m

from the base of tower are in the same straight line which are complementary. Find the

height of tower.

32. The angle of elevation of a plane from a point on the ground is 60°. After a flight of 20

seconds, the angle of elevation changes to 30°. If the plane is at a height of 2400√3 m, find

the speed of the jet plane.

33. From a point on the ground, the angle of elevation of the top of a tower is 30° and that of

flag staff fixed on the top of the tower is 45°. If the length of flag staff is 3 m, find the

height of tower.

34. From the top of a 75 m high tower, the angles of depression of the top & bottom of a pole

are found to be 30° & 45°. What is the height of pole. (use √3 = 1.732)

35. At the foot of a mountain, the angle of elevation of its summit is 45°. After ascending 2 km

towards the mountain up an incline of 30°, the elevation becomes 60°. Find the height of

the mountain.

36. A boy, 1.4 metres tall standing at the edge of a river bank sees the top of a tree on the edge

of the other bank at an elevation of 55°. Standing back by 3 metres, he sees it at elevation

of 45°.

(a) Draw a rough figure showing these facts.

(b) How wide is the river and how tall is the tree?

[ sin 55° = 0.8192, cos 55° = 0.5736, tan 55° = 1.4281]

37. A student sitting in a classroom sees a picture on the black board at a height of 1.5 m from

the horizontal level of sight. The angle of elevation of the picture is 30°. As the picture is

not clear to him, he moves straight towards the black board and sees the picture at an angle

of elevation of 45°. Find the distance moved by the student.

38. Find the height of the chimney, when it is found that one walking towards it for 50 m, in a

horizontal line through its base, the elevation of its top changes from 30° to 60°.

39. At the foot of a mountain, the angle of elevation of its summit is 45°. After ascending 2 km

towards the mountain up an incline of 30°, the elevation becomes 60°. Find the height of

the mountain.

40. A person standing at point P on the bank of a river observes that the angle of elevation of

the top (T) of a tree (BT) standing on the opposite bank is 60°. When he moves 40 m away

from the bank and reaches at the point Q, he finds the angle of elevation to be 30°. Find the

height (BT) of the tree and the width (BP) of the river (Use √3 = 1.732)

CIRCLES

SECTION A: (1 MARK)

1. A line touches a circle of radius 4 cm. Another line is drawn which is tangent to the circle. If the

two lines are parallel then find the distance between them.

2. From a point P, which is at a distance of 13 cm from the centre O of a circle of radius 5 cm, the

pair of tangents PQ and PR are drawn to the circle. Find the area of the quadrilateral PQOR

.

3. In fig 1, if O is centre of a circle, PQ is a chord and the tangent PR at P makes an angle of 50°

with PQ, then find POQ.

4. In the figure 2, QR is a common tangent to given circle which meet at T. Tangent at T meets

QR at P. If QP = 3.8 cm, then find the length of QR.

5. In fig. 3, CP and CQ are tangents to a circle with centre O. ARB is another tangent touching the

circle at R. If CP = 11 cm, BC = 7 cm, then find the length of BR.

Fig. 1

Fig. 2

Fig.3

6. Point P is 26 cm away from the centre O of a circle and the length PT of the tangent drawn from

P to the circle is 24 cm. Then find the radius of the circle.

7. TP and TQ are two tangents to a circle with centre O such that POQ = 110⁰. Then find PTQ.

8. In the figure , QR is a common tangent to

given circle which meet at T. Tangent at T

meets QR at P. If QP = 3.8 cm, then find the

length of QR.

9. If the angle between two radii of a circle is 100⁰, find the angle between the tangents at the ends

of the radii.

10. From a point P which is at a distance of 13 cm from the center of a circle of radius 5 cm, the

pair of tangents PQ and PR to the circle are drawn. Then find the area of the quadrilateral

PQOR.

SECTION B :(2 MARKS)

1. Two concentric circles of radii a and b , where a>b are given. Find the length of chord of the

larger circle which touches the smaller circle.

2. If d1, d2(d2>d1) be diameters of two concentric circles, c be the length of a chord of the circle

which is tangent to the other circle, prove that d22 = c

2 + d1

2.

3. If two tangents inclined at an angle 60⁰ are drawn to a circle of radius 3 cm, find the length of each tangent.

4. PA and PB are two tangents drawn from an external point P to a circle with centre O. If OP is

equal to diameter of the circle, prove that ∆ABP is equilateral.

5. A circle touches the side AC of ∆ABC at

P, touches BC at Q and AB at R. Show

that AC + AR = BC + BR.

6. If two tangents inclined at an angle 60⁰ are drawn to a circle of radius 3 cm, find the length of

each tangent.

7. Two parallel lines touch a circle at points A and B separately. If area of the circle is 25 cm2, then find the length of AB.

8. In figure, if CP = 11 cm and BC = 7 cm,

then find the length of BR.

9. In the given figure , BOA is a diameter of

a circle and the tangent at a point P meets

BA produced at T. If PBO = 30°, what

is the measure of ?PTA

10. In fig , a tangent PT is drawn parallel to a

chord AB of a circle, as shown in given

figure. Prove that APB is an isosceles

triangle.


1. In figure, a triangle ABC is drawn to circumscribe a circle of

radius 10 cm such that the segments BP and PC into which BC

is divided by the point of contact P, are of lengths 15 cm and

20 cm respectively. If the area of ∆ABC is 525 cm2, then find

the lengths of sides AB and AC.

2. If angle between two tangents PT and QT drawn from a point T to a circle of radius r and centre

O is 60⁰, then prove that PT = √3 r.

3. A circle touches the side AC of ∆ABC at P,

touches BC at Q and AB at R. Show that

PC = 1

2 (Perimeter of ∆ABC).

4. In fig.4, AB is diameter of circle with centre O. If AB = 5 cm and AT = 12 cm, find CT.

Fig.4

Fig.5

Fig.6

5. In fig.5, find the value of x.

6. In fig.6, tangents PQ and PR are drawn to circle such that RPQ = 30⁰. A chord RS is drawn

parallel to tangent PQ. Find RQS.

7. OAB is quarter circle. If perimeter of OAB is 360 cm, find the area of the circle.

8. Two tangents RQ and RP are drawn to a circle with center O from an external point R. If QRP

= 120⁰, then prove that OR = PR + PQ.

9. Two tangents PQ and PR are drawn to a circle with center O from an external point P. Prove

that QPR = 2OQR.

10. In figure, find x, if PA and PB are tangents

from P.


1. Prove that the lengths of tangents drawn from an external point to a circle are equal.

2. A quadrilateral is drawn to circumscribe a circle. Prove that the sums of opposite sides are

equal.

3. In fig.9, a tangent PT and a line segment

PBA is drawn to a circle with centre O. If

OL AB, prove that

PA x PB = PT2.

4. In fig.12, QR is tangent at Q. PR‖AQ,

where AQ is a chord through A and P is a

centre, the end point of the diameter AB.

Prove that BR is tangent at B.

5. PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at points P and Q intersect

at point T. Find the length of TP.

6. In fig 8, ABC is a right triangle in which

90B . A circle is inscribed in the

triangle. If AB = 8 cm and BC = 6 cm,

find the radius r of the circle.

7. Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right

angle at the centre.

8. If two circles touch internally, the points of contact lies on the line joining their centres.

9. In a quadrilateral ABCD, D = 90⁰ and a circle is inscribed within it , which touches the sides

AB, BC, CD and DA at P,Q,R and S respectively. If BC = 40 cm, BP = 29 cm and DC = 24 cm,

find the radius of the circle . If AD = 20 cm, then find the length of AB.

10. Two concentric circles with centre O have radii 5 cm and 3 cm. From an external point P,

tangents PA (to bigger circle) and PB (to the smaller circle) are drawn. If AP is 12 cm, find the

length of BP.

GEOMETRICAL CONSTRUCTIONS


1. Draw two tangents to a circle of radius 3.5 cm from a point P at a distance of

5.5 cm from its centre.

2. Draw a circle of radius 4.2 cm. From a point P, 9 cm from the centre of the

Circle, draw a pair of tangents to the circle. Measure the length of each tangent

segment.

3.

Construct a Triangle ABC in which AB=5cm, BC=6cm and AC=7cm.

Construct another triangle whose sides are 3/5 times the corresponding sides of

triangle ABC

4. Draw a circle of radius 3 cm. take two points P and Q on either side of the

extended diameter at a distance of 8 cm and 5 cm respectively from the centre.

Draw tangents to the circle from points P and Q. Write the steps of

construction.

5. Construct a tangent to a circle of radius 2 cm from a point on the concentric

circle of radius 2.6cm and measure its length. Also, verify the measurements by

actual calculations.

6. Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then

another triangle whose sides are 2

11 times the corresponding sides of the

isosceles triangle

7. Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4

cm and 3 cm. Then construct another triangle whose sides are 5/3 times the

corresponding sides of the given triangle

8. Draw a circle of radius 6 cm. From a point 10 cm away from its centre,

construct the pair of tangents to the circle and measure their lengths.

9. Draw a pair of tangents to a circle of radius 4.8 cm that are inclined to each

other at an angle of 90 .

10. Draw line segment AB of length 5.5cm and divide it equally into 6 parts.


11. Construct a triangle ABC in which AB = 5 cm, BC = 6 cm and ABC = 600.

Now construct another triangle whose sides are 5/7 times the corresponding

sides of ΔABC.

12. Construct two circles of radii 3 cm and 5 cm, such that their centre is 12 cm

apart. Join their centre and construct the perpendicular bisector of the line

segment thus obtained. Take a point M which is 3.5 cm away from the mid -

point of the line segment joining the two centre and lying on perpendicular

bisector. From M, construct tangents to the bigger circle. Write the steps of

construction.

13. Draw a triangle ABC with side BC = 7 cm, B = 45°,A= 105°. Then,

construct a triangle whose sides are 4/ 3 times the corresponding sides of

triangle ABC.

14. Draw a pair of tangents to a circle of radius 5 cm which are inclined to each

other at an angle of 60°.

15. Draw a line segment AB of length 4.4cm. Taking A as centre, draw a circle of

radius 2cm and taking B as centre, draw another circle of radius 2.2cm.

Construct tangents to each circle from the centre of the other circle.

16. Draw a pair of tangents to a circle of radius 4.8 cm that are inclined to each

other at an angle of 45 .

17. Draw a circle of radius 7cm. From a point P, 8cm away from its centre,

Construct a pair tangents to the circle. Measure the length of the tangent

segments.

18. Draw a circle of radius 5cm. from a point P, 7cm away from its centre,

construct a pair of tangents to the circle. Measure the length of the tangent

segments.

19. Draw a triangle ABC with side BC = 6 cm, B = 45°, A = 90°. Then,

construct a triangle whose sides are 4/ 3 times the corresponding sides of

triangle ABC.

20. Draw a circle of radius 5cm. from a point P, 7cm away from its centre ,

construct a pair of tangents to the circle.

AREAS RELATED TO CIRCLES

SECTION A: (1 MARK)

1. A road which is 7m wide surrounds a circular park whose circumference is 352

m. Find the area of the road.

(a) 2618 m2 (b) 2518 m

2 (c) 1618 m

2 (d)

none

2. The area of the circle that can be inscribed in a square of side 6cm is____

(a) 36𝜋 sq.cm (b) 18𝜋 sq.cm (c) 12𝜋 sq.cm (d) 9𝜋 sq.cm

3. A steel wire when bent in the form of a square , encloses an area of 121 sq. cm.

The same wire is bent in the form of a circle. Find the area of the circle.

(a) 111 cm2

(b) 184 cm2

(c) 154 cm2

(d) 259 cm2

4. On decreasing the radius of the circle by 30%, its area is decreased by

(a) 30% (b) 60% (c) 45% (d)

none

5. The area of the square is the same as the area of the circle. Their perimeter are

in the ratio

(a) 1 : 1 (b) 𝜋: 2 (c) 2 : 𝜋 (d)

none

6. In making 1000 revolutions, a wheel covers 88 km. The diameter of the wheel

is

(a) 14 m (b) 24 m (c) 28 m (d)

40 m

7. Find the circumference of a circle of diameter 21 cm.

(a) 62 cm (b) 64 cm (c) 66 cm (d) 68

cm

8. A square ABCD is inscribed in a circle of radius ‘r’. Find the area of the square

in sq. units.

(a) 3r2

(b) 2r2

(c) 4r2 (d)

none

9. Find the area of a right-angled triangle, if the radius of its circum circle is 2.5

cm and the altitude drawn to the hypotenuse is 2 cm long.

(a) 5 cm2 (b) 6 cm

2 (c) 7 cm

2 (d)

none

10. The diameter of a wheel is 40 cm. How many revolutions will it make in

covering 176 m?

(a) 140 (b) 150 (c) 160 (d)

166


11. Two circles touch externally. The sum of their areas is 130╥ sq. cm and the

distance between their centre are 14 cm. Find the radii of the circles.

12. A chord of a circle of radius 12cm subtends an angle of 120 at the centre. Find

the area of the segment of the circle. (Use 𝜋 = 3.14 and √3 = 1.73)

13. Find the area of the minor segment of a circle of radius 14 cm, when its central

angle is 600. Also find the area of the corresponding major segment (Use

𝜋=22/7)

14. A paper is in the form of a rectangle ABCD in which AB = 18cm and BC =

14cm. A semicircular portion with BC as diameter is cut off. Find the area of

the remaining paper

15. A racetrack is in the form of a ring whose inner circumference is 352 m and

outer circumference is 396 m. Find the width of the track.

16. The area of a sector is one-twelfth that of the complete circle. Find the angle of

the sector.

17. Find the area of the circle in which a square of area 64 sq. cm is inscribed.

(use 𝜋 = 3.14)

18. The cost of fencing a circular field at the rate of Rs 24 per m is Rs 5280. The

field is to be ploughed at the rate of Rs 0.50 per sq.m. Find the cost of

Ploughing the field.

19. If four circles of radius r each , are drawn such that each touches the other two,

the find the area included between them. (Take 𝜋 = 3.14)

20. The length of an arc subtending an angle of 72 at the centre is 44 cm. Find the

area of the circle.


21. The two sectors of a circle have the central angles as 1200 and 150

0

respectively. Then the find the ratio between the areas of the two sectors.

22. In the figure below, AB and CD are two diameters of a circle (with centre O)

perpendicular to each other and OD is the diameter of the smaller circle. If

OA=14cm, find the area of the shaded region.

23.

In the given figure AB = 3cm and AC = 4 cm and A = 900, Semicircles are

drawn on AB, AC and BC as diameters. Find the area of the shaded region.

24. Find the area of the minor sector of a circle of radius 72 cm, when the angle of

the corresponding sector is 60°.

25.

A circular pond is 21 m in diameter. It is surrounded by 3.5 m wide path. Find

the cost of constructing the path at the rate of Rs. 25 per m2

26. Arcs have been drawn of radius 21 cm each with vertices A, B, C and D of

quadrilateral ABCD as centres and side of the square is 42cm. Find the area

A

B

C D O

A

B

C

of the remaining part.

27. O is the centre of a circlular arc and AOB is a straight line and point C lies on

the semicircle with AC= 12cm and BC=16cm. Find the area of the remaining

region. (use 𝜋 = 3.14)

28. The length of the minute hand of a clock is 5cm . Find the area swept by the

minute hand during the time period 6:05 a.m. and 6:40 a.m.

29. A path of 4m width runs round a semicircular grassy plot whose circumference

is

163 7

3m. Find (i) the area of the path (ii) the cost of gravelling the path at the

rate of Rs. 1.50 per sq. m (iii) the cost of turfing the plot at the rate of 45 paise

per sq. m.

30. A park is in the form of rectangle 120 m by 100 m. At the centre of the park,

there is a circular lawn. The area of the park excluding the lawn is 11384 sq. m.

Find the radius of the circular lawn.


31. On a square cardboard sheet of area 784 sq.cm, four congruent circular plates of

maximum size are placed such that each circular plate touches the other two

plates and each side of the square sheet is tangent to two circular plates. Find

the area of the square sheet not covered by the circular plates.

32. Sides of a triangular field are 15m 16m and 17m With the three corners of the

field a cow, a buffalo and horse are tied separately with the ropes of length 7m

each to graze in the field. Find the area of the field which cannot be grazed by

the three animals.

33. The two sectors of a circle have the central angles as 900 and 150

0 respectively.

Then find the ratio between the areas of the two sectors.

34. A kite in which BCD is the shape of the quadrant of circle of radius 42cm.

ABCD is a SQUARE and CEF is the right angled isosceles triangle whose

equal sides are 6cm long. Find the area of shaded region.

A

5. A park is in the form of rectangle 120 m by 100 m. At the centre of the park,

F E

D

C B

there is a circular lawn. The area of the park excluding the lawn is

11384 sq. m.

Find the radius of the circular lawn.

36. In the given figure three sectors of a circle of radius 7cm , making an angles of

40 , 60

, and 80

at the centre are shaded. Find the area of the shaded

region.

37. In the given figure AB =36cm and M is the midpoint of AB as diameter.

Width of semicircular path is 9cm Find the area

of the shaded region.

A M B

38

Find the area of shaded in the given figure with a radius OC=7.7 cm and AB

and CD are diameters and O as centre.

39.

A chord 20 cm long is drawn in a circle whose radius is 50cm and the angle

subtended

at the centre is 30. Find the area of minor segment.

40. ABCDEF is any hexagon with different vertices A, B, C, D, E and F as the

centre of circles with same radius 6 cm are drawn. Find the area of the

remaining part..

SURFACE AREAS AND VOLUMES

SECTION A: (1 MARK)

1. A cone and a cylinder are of same height.

If the radii of their bases are in the ratio 3 : 5, then find the ratio of their volumes .

2. Find the base area of the cylinder of radius 7 cm.

A

B

C D

3. If the radius of a sphere is 2 cm, then find the curved surface area of the sphere.

4. The surface area of the sphere is 2464 sq.cm. Find its radius.

5. How many litres of water will a hemispherical tank hold whose diameter is 4.2 m?

6. Radius and slant height of a cone are 20 cm and 29 cm respectively. Find its volume.

7. The surface area of the sphere is 2464 sq.cm. Find its radius.

8. The diameter of the base of a right circular cylinder is r and its height is equal to the radius

of its base. Find the volume of the cylinder.

9. If the circumference of the base of a solid right circular cone is 236 cm and its slant height

is 12 cm, find its curved surface area.

10. A sphere and a cube are of same height, find the ratio of their volumes if π = 22/7


11. A hemispherical bowl of radius 7 cm is full with water. The water from the bowl is to be

emptied into a cylindrical vessel of radius 3.5 cm. Calculate the height of cylindrical vessel.

12. A solid metallic sphere of radius 21 cm is melted and recasted to a no. of cones with radius

7 cm and height 28 cm. Find the no. of cones so formed.

13. A hemispherical iron of radius 14 cm is cast into a right circular cone of radius 98 cm. Find

the height of cone.

14. The C.S.A of hemisphere is 154 cm2. Find its volume.

15. The volume of the sphere is numerically equal to its surface area. Find its diameter.

16. The radius and the height of the cylinder are in the ratio 2:7. If the curved surface area of

the cylinder is 352 sq.cm, find it radius.

17. The radius of a wooden hemisphere is 10 cm. What is the its volume?

18. The perimeters of the ends of a frustum of a cone are 88 cm and 8.4 π cm. If the depth is 14

cm, then find its volume.

19. A solid cuboid of iron with dimensions 45 cm × 11 cm is melted and recast into a cylinder

pipe. The outer & inner diameters are 10 cm and 8 cm respectively. Find the length of pipe.

20. 120 spherical marbles each of diameter 2.1 cm are dropped into a cylindrical vessel of

diameter 7 cm containing some water which are completes immersed in water. What is the

rise in level of water in vessel?


21. A tent is in the form of a cylinder surmounted by a conical top. If the height and radius of

cylindrical part are 2.8 m and 2 m and slant height of the top is 3.2 m, find the area of

canvas used for the tent.

22. A sphere of radius 5 cm is dropped into a cylindrical vessel partly filled with water. The

diameter of the base of vessel is 20 cm. If the sphere is completely submerged, find the rise

of level of water.

23. 168 cones each of diameter 7 cm and height 6 cm are melted to form a metallic hemisphere. Find

the radius of hemisphere and the total surface area.

24. A well of diameter 10 m is 21 m deep is dug out and the earth from it is evenly spreaded to form a

platform 20 m by 16 m. Find the height of plat form.

25. A bucket is in the shape of a frustum of a cone & holds 18.304 litres of water. The radii of top &

bottom are 24 cm and 20 cm respectively, find the height of the bucket.

26. Water flows through a cylindrical pipe, of internal diameter 4 cm into a cylindrical tank of base

radius 30 cm at the rise of 0.5 m/s. Find the rise in level of water in the tank in 2 hours.

27. From a solid cylinder of height 8 cm and base diameter 10 cm, a conical cavity of same height and

same base diameter is hollowed out. What is the total surface area of the remaining solid.

28. A spherical tank with diameter 4 m full of water is to be emptied by a pipe at the rate of 5

7

2 litre

per second. How much time will it take to make the tank one third empty.

29. Volume of cylinder is 896 π with height 14 cm. Find the total surface area of cylinder.

30. A cylindrical container has diameter 12 cm and height 15 cm is full with ice cream. The ice cream

is to be filled into cones of height 12 cm & diameter 6 cm with hemispherical top. Find the no. of

cones which can be filled with ice cream.


31. The radii of the circular ends of a bucket of height 15 cm are 7 cm and R. If the volume of

bucket is 5390 cm3 find R.

32. A copper rod of diameter 4 cm and length 8 cm is drawn into a wire of length 40 m of

uniform thickness. Find the thickness of the wire.

33. A sector of circle of radius 15 cm has the angle 120°. It is rolled up so that two boundary

radii are joined together to form a cone. Find the volume of cone.

34. Using clay, a student made a right circular cone of height 48 cm and base radius 12 cm.

Another student reshapes it in the form of a sphere. Find the radius of the sphere.

35. The radii of the internal and external surfaces of a metallic spherical shell are 7 cm and 5

cm. It is melted & recast into a right circular cylinder of height 83

2cm. Find the diameter of

the base of the cylinder.

36. A conical vessel of ratio 12 cm and height

16 cm completely filled with water. A

sphere is lowered into the water & its size

is such that it touches the sides, as shown

in figure1. Find the fraction of water that

overflows.

fig 1

37. A 10 m wide cloth is used to make a conical tent of base diameter 21 m & height 30 m.

Find the cost of cloth used at the rate of Rs. 30 perimeter.

38. The circumference of the base and height of a right circular cylinder are 44 cm and 10 cm

respectively. Find its volume, curved surface area and total surface area.

39. The height of circular cone is trisected by two planes running parallel to the base. Find the

ratio of the volumes of the three cones thus formed.

40. Water flows through a cylindrical pipe of internal radius 3.5 cm at 5 m per sec. calculate the

volume of water in litres discharged by the pipe in one minute.

STATISTICS

1mark questions 1. Marks Number of students

Below 10

Below 20

Below 30

Below 40

Below 50

Below 60

3

12

27

57

75

80

For the above distribution , find the modal class.

2. Class Frequency

13.8-14.0

14.0-14.2

14.2-14.4

14.4-14.6

14.6-14.8

14.8-15.0

2

4

5

71

48

20

From the above distribution find the number of athletes who completed the race in less than 14.6 seconds.

3. Mean of n numbers x1, x2,… xn is m. If xn is replaced by x, what is the new mean?

4. Mean of 1,2,3,….n is 6n/11, then what is the value of n?

5. For the following distribution, find the sum of lower limits of the median class and modal class.

Class 0-5 5-10 10-15 15-20 20-25

Frequency 10 15 12 20 9

2 marks questions.

1. Convert the following data to a less than type distribution.

C.I. Frequency

50-55

55-60

60-65

65-70

70-75

75-80

2

8

12

24

38

16

2. If the mean value of the following distribution is 2.6, then the value of y will be?

Variable 1 2 3 4 5

Frequency 4 5 Y 1 2

3. Apply direct method to find arithmetic mean in the following:

Class Interval 100-120 120-140 140-160 160-180 180-200

Frequency 10 20 30 15 5

4. Convert the following data into more than type distribution:

Class

intervals

50-55 55-60 60-65 65-70 70-75 75-80

frequency 2 8 12 24 38 16

5. Find the median of the following:

Class

interval 0-10 10-20 20-30 30-40

40-50 total

Frequency 8 16 36 34 6 100

3marks questions

1. Calculate mode of the following data:

Marks

Obtained

No. of

Students

0-20

20-40

40-60

60-80

80-100

8

10

12

6

3

2.

The arithmetic mean of the following frequency distribution is 50. Find the value of p.

Class Frequency

0-20

20-40

40-60

60-80

80-100

17

P

32

24

19

3. Draw a less than type ogive of the following distribution:

Marks No. of students

0-10

10-20

20-30

30-40

40-50

50-60

5

4

8

10

15

18

4. The following table gives production yield per hectare of wheat of 100 farms of a village.

Production yield (in kg/ha) Number of farmers

50-55

55-60

60-65

65-70

70-75

75-80

2

8

12

24

38

16

Change the distribution to a more than type distribution, and draw its ogive.

5. Find the unknown entries a, b, c, d, e, f in the following distribution of heights of students in a class:

Height

(in cm)

Frequency Cumulative

Frequency

150-155

155-160

160-165

165-170

170-175

175-180

12

b

10

d

e

2

a

25

c

43

48

f

Total 50

6.

Apply direct method to find arithmetic mean in the following:

Class Interval 100-120 120-140 140-160 160-180 180-200

Frequency 10 20 30 15 5

4 marks questions

1.

Find the median of the following data:

Age at last birthday Number

15-19 4

20-24 20

25-29 38

30-34 24

35-39 10

40-44 9

2.

Compare the modal age of two groups of students A and B appearing for an entrance test:

Age (in years) Number of students in

16-18

18-20

20-22

22-24

24-26

Group A Group B

50

78

46

28

23

54

89

40

25

17

3.

The median of the following data is 52.5. Find the values of x and y, if the total frequency is 100.

Class Interval Frequency

0-10

10-20

20-30

30-40

40-50

50-60

60-70

2

5

x

12

17

20

y

70-80

80-90

90-100

9

7

4

4.

Find mean, median, mode of the following data:

Classes Frequency

0-20

20-40

40-60

60-80

80-100

100-120

120-140

6

8

10

12

6

5

3

5.

Find the missing frequencies and the median for the following distribution if the mean is 1.46.

No. of

accidents

Frequency

(No. of days)

0

1

2

3

4

5

46

x

y

25

10

5

Total 200

6.

To highlight child labor problem, some students organized a javelin throw competition. 50 students

participated in this competition. The distance (in metres) thrown are recorded below:

Distance (in m) Number of students

0-20

20-40

40-60

60-80

80-100

6

11

17

12

4

(a) Construct a cumulative frequency table.

(b) Draw cumulative frequency curve (less than type) and calculate the median distance thrown.

(c) Calculate the median distance by using the formula for median.

(d) Are the median distance calculated in (a) and (b) same?

(e) Which value is depicted by students?

7.

Apply deviation method to find arithmetic mean in the following:

Class Interval 0-10 10-20 20-30 30-40 40-50

Frequency 12 11 8 10 9

8. Apply assumed mean method to find arithmetic mean in the following:

Class

Interval 0-50 50-100 100-150 150-200 200-250 250-300

Frequency 8 15 32 26 12 7

9.

An incomplete frequency is given as follows:

Variable 10-20 20-30 30-40 40-50 50-60 60-70 70-80 Total

Frequency 12 30 ? 65 ? 25 18 229

Given that the median value is 46, determine the missing frequencies using the median formula.

10.

The daily expenditure of 100 families is given as under:

Expenditure 0-10 10-20 20-30 30-40 40-50

No. of families 14 ? 27 ? 15

The median and mode for the distribution are Rs. 26.67 and Rs. 29 respectively. Calculate the missing

frequencies.

INDIAN SCHOOL MUSCAT QUESTION BANK 2017-18...

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