Real Number

1. If 𝑝 and 𝑞 are two distinct prime numbers, then their HCF is

(a) 2 (b) 0 (c) 𝑒𝑖𝑡ℎ𝑒𝑟 1 𝑜𝑟 2 (d) 1
2. If 𝑝 and 𝑞 are two distinct prime numbers, then LCM (𝑝, 𝑞) is
(a) 1 (b) 𝑝 (c) 𝑞 (d) 𝑝𝑞
3. Let 𝑝 be a prime number. The sum of its factors is
(a) 𝑝 (b) 1 (c) 𝑝 + 1 (d) 𝑝 − 1
4. Let 𝑝 be a prime number. The quadratic equation having its factors as zeros is
(a) 𝑥 2 − 𝑝𝑥 + 𝑝 = 0 (b) 𝑥 2 − (𝑝 + 1)𝑥 + 𝑝 = 0
(c) 𝑥 2 + (𝑝 + 1)𝑥 + 𝑝 = 0 (d) 𝑥 2 − 𝑝𝑥 + (𝑝 + 1) = 0
5. The LCM of the smallest two-digit composite number and the smallest composite
number is
(a) 12 (b) 20 (c) 4 (d) 44
6. The HCF of smallest prime number and the smallest composite number is
(a) 2 (b) 4 (c) 6 (d) 8
7. If the HCF of 85 and 153 is expressible in the form 85𝑛 − 153, then the value of n
is
(a) 3 (b) 2 (c) 4 (d) 1
8. If two positive integers 𝑎 and 𝑏 are written as 𝑎 = 𝑝3 𝑞4 and 𝑏 = 𝑝2 𝑞3 , where 𝑝
and 𝑞 are prime numbers, such that HCF (𝑎, 𝑏) = 𝑝𝑚 𝑞𝑛 and LCM (𝑎, 𝑏) ≠ 𝑝𝑟 𝑞 𝑠
then (𝑚 + 𝑛)(𝑟 + 𝑠) equal to
(a) 15 (b) 30 (c) 35 (d) 72
9. The smallest number which when divided by 17, 23 and 29 leaves a remainder 11
in each
(a) 493 (b) 11350 (c) 11339 (d) 667

10. The largest number which divides 70 and 125, leaving remainders 5 and 8,
respectively, is
(a) 13 (b) 65 (c) 875 (d) 1750
11. The LCM of two numbers is 1200. Which of the following cannot be their HCF?
(a) 4 (b) 5 (c) 6 (d) 3
12. The sum of the exponents of the prime factors in the prime factorization of 196,
is
(a) 1 (b) 2 (c) 4 (d) 6
13. The HCF of two numbers is 18 and their product is 12960. Their LCM will be
(a) 420 (b) 600 (c) 720 (d) 800
14. The HCF and the LCM of 12, 21, 15 respectively are
(a) 3,140 (b) 12,420 (c) 3,420 (d) 420,3
15. The least number that is divisible by all the numbers from 1 to 10 (both inclusive)
is
(a) 10 (b) 100 (c) 504 (d) 2520
16. The ratio of LCM and HCF of the least composite number and the least prime
number is
(a) 1: 2 (b) 2: 1 (c) 1: 1 (d) 1: 3
17. Which of these numbers always end with the digits 6.
(a) 4 𝑛 (b) 2𝑛 (c) 6𝑛 (d) 6𝑛
18. The numbers 525 and 3000 are divisible by 3, 5, 15, 25 and 75. What is the HCF
of 525 and 3000?
19. Write the prime factor of 2 × 7 × 11 × 13 × 17 + 21
20. If 7560 = 23 × 3𝑝 × 𝑞 × 7, find 𝑝 and 𝑞.
21. Explain why:
(i) 11 × 13 × 17 + 17 is a composite number.
(ii) 1 × 2 × 3 × 5 × 7 + 3 × 7 is a composite number.
22. The LCM of two numbers is 14 times their HCF. The sum of LCM and HCF is 600. If
one of the numbers is 280. Find the other number.
23. If the sum of LCM and HCF of two numbers is 1260 and their LCM is 900 more
than their HCF, then the product of two numbers is
(a) 203400 (b) 194400 (c) 198400 (d) 205400
24. Assertion: If 11 divides 627264, then 11 divides 792.
Reason: Let 𝑝 be a prime number and a be a positive integer, if 𝑝 divides 𝑎², then 𝑝
divides
(a) Assertion and reason are correct and reason is correct explanation for assertion
(b) Assertion and reason are correct but reason is not correct explanation for
Assertion
(c) Assertion is correct but reason is false
(d) Both Assertion and reason are false.
25. On a morning walk, three person’s steps off together and their steps measure 40
cm, 42 cm, and 45 cm respectively. What is the minimum distance each should walk,
so that each can cover the same distance in complete steps?
26. During a sale, colour pencils were being sold in the pack of 24 each and crayons in
the pack of 32 each. If you want full packs of both and the same number of pencils
and crayons, how many packets of each would you need to buy?
27. A street shopkeeper prepares 396 Gulab jamuns and 342 ras-gullas. He packs
them, in combination. Each containter consists of either gulab jamuns or ras- gullab
but have equal number of pieces. Find the number of pieces he should put in each
box so that number of boxes are least. How many boxes will be packed in all?
28. In seminar, the no. of participants in Hindi, English and Mathematics are 60, 84
and 108 respectively. Find the minimum number of rooms required if in each room
the same number of participants are to be seated and all of them being of the same
subject.
29. Prove that √𝑝 + √𝑞 is irrational, where 𝑝, 𝑞 are primes.

30. Prove that √3 + √5 is irrational

ANSWER’S
Q1. D Q16. B
Q2. D Q17. C
Q3. C Q18. 75
Q4. B Q19. 5, 7, 139
Q5. B Q20. p=3, q=5
Q6. A Q22. 80
Q7. B Q23. B
Q8. C Q24. A
Q9. B Q25. 2520
Q10. A Q26. 3 packs of crayons, 4 Packs of
Q12. C pencil
Q13. C Q27. 41
Q14. C Q28. 21
Q15. D
 Polynomials
1. If one zero of the quadratic polynomial 𝑥 2 + 3𝑥 + 𝑘 is 2, then the value of 𝑘 is
(a) 10 (b) −10 (c) 5 (d) −5
2. If one zero of the polynomial 𝑃(𝑥) = 5𝑥 2 + 13𝑥 + 𝐾 is reciprocal of the other,
then value of 𝑘 is
1
(a) 0 (b) 5 (c) (d) 6
6

3. If the zeroes of the quadratic polynomial 𝑥 2 + (𝑎 + 1)𝑥 + 𝑏 are 2 and – 3, then

(a) 𝑎 = −7, 𝑏 = −1 (b) 𝑎 = 5, 𝑏 = −1
(c) 𝑎 = 2, 𝑏 = −6 (d) 𝑎 = 0, 𝑏 = −6
4. The quadratic polynomial 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, 𝑎 ≠ 0 is represented by this graph then 𝑎
is

(a) 𝑁𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑜. (b) 𝑊ℎ𝑜𝑙𝑒 𝑛𝑜.

(c) 𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝐼𝑛𝑡𝑒𝑔𝑒𝑟 (d) 𝐼𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑜.
5. The zeroes of the quadratic polynomial 𝑥 2 + 99𝑥 + 127 are
(a)𝐵𝑜𝑡ℎ 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 (b) 𝑏𝑜𝑡ℎ 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
(c)𝑂𝑛𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑎𝑛𝑑 𝑜𝑛𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 (d) 𝑏𝑜𝑡ℎ 𝑒𝑞𝑢𝑎𝑙
6. The zeroes of the quadratic polynomial 𝑥 2 + 𝑘𝑥 + 𝑘, 𝑘 ≠ 0,
(a)Cannot both be positive (b) cannot both be negative
(c) Are always unequal (d) are always equal
7. If the zeroes of the quadratic polynomial 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, 𝑐 ≠ 0 are equal, then
(A) 𝑐 and 𝑎 have opposite signs (B) 𝑐 and 𝑏 have opposite signs
(C) c and a have the same sign (D) c and b have the same sign
8. If one of the zeroes of a quadratic polynomial of the form x 2 + ax + b is the
negative of the other, then it
(A) has no linear term and the constant term is negative.
(B) has no linear term and the constant term is positive.
(C) Can have a linear term but the constant term is negative.
(D) Can have a linear term but the constant term is positive.
9. Which of the following is not the graph of a quadratic polynomial?

A B

c D
10. In Given figure show the graph of the polynomial f(x) = ax 2 + bx + c. Which of
the following option is correct
(a) 𝑎 < 0, 𝑏 < 0 𝑎𝑛𝑑 𝑐 > 0 (b) 𝑎 < 0, 𝑏 < 0 𝑎𝑛𝑑 𝑐 < 0
(c) 𝑎 < 0, 𝑏 > 0 𝑎𝑛𝑑 𝑐 > 0 (d) 𝑎 < 0, 𝑏 > 0 𝑎𝑛𝑑 𝑐 < 0

11. In given figure shows the graph of the polynomial f(x) = ax 2 + bx + c. which of
the following options is correct,
(a) 𝑎 < 0, 𝑏 > 0 𝑎𝑛𝑑 𝑐 > 0 (b) 𝑎 > 0, 𝑏 < 0 𝑎𝑛𝑑 𝑐 > 0
(c) 𝑎 < 0, 𝑏 < 0 𝑎𝑛𝑑 𝑐 < 0 (d) 𝑎 > 0, 𝑏 > 0 𝑎𝑛𝑑 𝑐 > 0

12. If 𝛼, 𝛽 are the zeros of the polynomial 𝑓(𝑥 ) = 𝑥 2 − 𝑝(𝑥 + 1) − 𝑐 such that
(𝛼 + 1)(𝛽 + 1) = 0, then 𝑐 =
(A) 1 (B) 0 (C) −1 (D) 2
13. Assertion: If zeroes of the polynomial 𝑓(𝑥 ) = 5𝑥 2 − 11𝑥 − (𝑘 − 3)
are reciprocal of each other then 𝑘 = −2
c
Reason: The product of the zeroes of the polynomial ax 2 + bx + c is −
a

(a) If both assertion and reason are true and reason is the correct explanation of
assertion.
(b) If both assertion and reason are true but reason is not the correct explanation of
assertion.
(c) If assertion is true but reason is false.
(d) If assertion is false but reason is true.
14. If 𝛼 and 𝛽 are the zeroes of the polynomial 𝑝(𝑥) = 𝑥 2 – 𝑝(𝑥 + 1) – 𝑐 such that
(𝛼 + 1) (𝛽 + 1) = 0, the 𝑐 =
1 1
15. If 𝛼 and 𝛽 are the zeros of the polynomial 𝑓(𝑥 ) = 𝑥 2 + 𝑥 + 1, then + =
𝛼 𝛽

16. Find the quadratic polynomial whose zeros are (5 + 2√3) and (5 − 2√3)
17. If one zero of 𝑝(𝑥) = 4𝑥 2 – (8𝑘 2 – 40𝑘)𝑥– 9 is negative of the other, find values of
𝑘.
18. What will be the number of real zeros of the polynomial 𝑥 2 + 1?
19. If 𝛼 and 𝛽 are zeros of polynomial 6𝑥 2 – 7𝑥– 3, then form a quadratic polynomial
where zeros are 2𝛼 and 2𝛽.
20. What will be the number of zeros of the polynomials whose graphs are parallel to
(i) 𝑦 − 𝑎𝑥𝑖𝑠 (ii) 𝑥 − 𝑎𝑥𝑖𝑠?
21. What will be the number of zeros of the polynomials whose graphs are either
touching or intersecting the axis only at the points: (i) (–3, 0), (0, 2) & (3, 0) (ii) (0, 4),
(0, 0) and (0, –4)
22. If zeros of 𝑥 2 – 𝑘𝑥 + 6 are in the ratio 3: 2, find 𝑘.
23. If 𝛼 and 𝛽 are the zeros of the polynomial 𝑥 2 – 5𝑥 + 𝑚 such that 𝛼– 𝛽 = 1, find
𝑚.
𝑚 𝑛
24. If 𝑚 and 𝑛 are the zeros of the polynomial 3𝑥 2 + 11𝑥– 4, find the value of + .
𝑛 𝑚

25. Obtain zeros of 4√3𝑥 2 + 5𝑥 − 2√3 and verify relation between its zeroes and
coefficients.
26. Form a quadratic polynomial one of whose zero is 2 + √5 and sum of the zeros is
4.
27. If 𝛼 and 𝛽 are the zeros of the quadratic polynomial 𝑓 (𝑥 ) = 𝑥 2 − 4𝑥 + 3, find
the value of (𝛼 4 𝛽 2 + 𝛼 2 𝛽 4 ).
28. If 𝛼 and 𝛽 are the zeros of the polynomial 𝑝(𝑥 ) = 2𝑥 2 + 5𝑥 + 𝑘, satisfying the
21
relation 𝛼 2 + 𝛽 2 + 𝛼𝛽 = the find the value of 𝑘.
4

29. If the sum of the squares of zeroes of the polynomial 𝑓(𝑥 ) = 𝑥 2 − 8𝑥 + 𝑘 is 40,
find the value of 𝑘.
30. Find the zeroes of the polynomial 𝑥 2 − 3𝑥 − 𝑚(𝑚 + 3).

ANSWER’S
Q1. B Q13. C Q25. 25𝑥 2 − 30𝑥 + 4
Q2. B Q14. 1 Q26. −
2 √3
,
Q3. D Q15. −1 √3, 4
7
Q4. C Q16. 𝑥 2 − 10𝑥 + 13 Q27.
4
Q5. B Q17. 0,5 Q28. 1
1
Q6. A Q18. 0 Q29. [3𝑥 2 − 16𝑥 +
3
Q7. A Q19. [3𝑥 2 − 7𝑥 − 6]𝑘 16]
Q8. A Q20. 1,0 Q30. 2 − √5
Q9. D Q21. 2,1 Q31. 90
Q10. A Q22. −5,5 Q32. K=2
Q11. B Q23. 6 Q33. K=12
Q12. A 145
Q24. − Q34. (m+3) and -m
12
 Linear Equations

1. If a pair of linear equations is consistent, then the lines will be

(A) Parallel (B) always coincident
(C) Intersecting or coincident (D) always intersecting
2. The pair of equations 𝑦 = 0 and 𝑦 = – 7 has
(A) One solution (B) two solutions
(C) Infinitely many solutions (D) no solution
3. The pair of equations 𝑥 = 𝑎 and 𝑦 = 𝑏 graphically represents lines which are
(A) Parallel (B) Intersecting at (𝑏, 𝑎)
(C) Coincident (D) intersecting at (a, b)
4. If the lines given by 3𝑥 + 2𝑘𝑦 = 2 and 2𝑥 + 5𝑦 + 1 = 0 are parallel, then the
value of 𝑘 is
−5 2 15 3
(A) (B) (C) (D)
4 5 4 2

5. The value of c for which the pair of equations 𝑐𝑥 – 𝑦 = 2 and 6𝑥 – 2𝑦 = 3 will

have infinitely many solutions is
(A) 3 (B) – 3 (C) – 12 (D) 𝑛𝑜 𝑣𝑎𝑙𝑢𝑒
6. One equation of a pair of dependent linear equations is – 5x + 7y = 2. The
second equation can be
(A) 10𝑥 + 14𝑦 + 4 = 0 (B) – 10𝑥 – 14𝑦 + 4 = 0
(C) – 10x + 14y + 4 = 0 (D) 10x – 14y = – 4
7. The father’s age is six times his son’s age. Four years hence, the age of the father
will be four times his son’s age. The present ages, in years, of the son and the father
are, respectively
(A) 4 𝑎𝑛𝑑 24 (B) 5 𝑎𝑛𝑑 30 (C) 6 𝑎𝑛𝑑 36 (D) 3 𝑎𝑛𝑑 24
8. If 𝑎𝑥 + 𝑏𝑦 = 𝑐 and 𝑙𝑥 + 𝑚𝑦 = 𝑛 has unique solution then the relation between
the coefficient will be
(A) 𝑎𝑚 ≠ 𝑙𝑏 (B) 𝑎𝑚 = 𝑙𝑏 (C) 𝑎𝑏 = 𝑙𝑚 (D) 𝑎𝑏 ≠ 𝑙𝑚
9. In ∆𝐴𝐵𝐶, ∠𝐶 = 3∠𝐵, ∠𝐶 = 2(∠𝐴 + ∠𝐵) then ∠𝐴, ∠𝐵, ∠𝐶 are respectively.
(A) 30°, 60°, 90° (B) 20°, 40°, 120°
(C) 45°, 45°, 90° (D) 110°, 40°, 50°
10. For which values of 𝑝 and 𝑞, will the following pair of linear equations have
infinitely many solutions?
4𝑥 + 5𝑦 = 2
(2𝑝 + 7𝑞)𝑥 + (𝑝 + 8𝑞)𝑦 = 2𝑞 − 𝑝 + 1
11. Solve the following pair of linear equations:
21𝑥 + 47𝑦 = 110
47𝑥 + 21𝑦 = 162
12. For which value(s) of 𝑘 will the pair of equations
𝑘𝑥 + 3𝑦 = 𝑘 − 3
12𝑥 + 𝑘𝑦 = 𝑘
Have no solution?
13. For which values of 𝑎 and 𝑏, will the following pair of linear equations have
infinitely many solutions?
𝑥 + 2𝑦 = 1
(𝑎 − 𝑏 )𝑥 + (𝑎 + 𝑏 )𝑦 = 𝑎 + 𝑏 − 2
14. Find the values of 𝑥 and 𝑦 in the following rectangle [see Fig. 3.2]

𝒙 + 𝟑𝒚

𝟑𝒙 + 𝒚
𝟕

𝟏𝟑

𝑥 𝑦 𝑥 𝑦
15. Find the solution of the pair of equations + − 1 = 0 and + = 15. Hence,
10 5 8 6
find λ, if 𝑦 = 𝜆𝑥 + 5.
16. Draw the graph of the pair of equations 2𝑥 + 𝑦 = 4 and 2𝑥 – 𝑦 = 4. Write the
vertices of the triangle formed by these lines and the y-axis. Also find the area of this
triangle.
17. Write an equation of a line passing through the point representing solution of the
pair of linear equations 𝑥 + 𝑦 = 2 and 2𝑥– 𝑦 = 1. how many such lines can we find?
18. The age of the father is twice the sum of the ages of his two children. After 20
years, his age will be equal to the sum of the ages of his children. Find the age of the
father.
19. Two numbers are in the ratio 5: 6. If 8 is subtracted from each of the numbers,
the ratio becomes 4:5. Find the numbers.
20. There are some students in the two examination halls A and B. To make the
number of students equal in each hall, 10 students are sent from A to B. But if 20
students are sent from B to A, the number of students in A becomes double the
number of students in B. Find the number of students in the two halls.
21. A shopkeeper gives books on rent for reading. She takes a fixed charge for the
first two days, and an additional charge for each day thereafter. Latika paid Rs 22 for
a book kept for six days, while Anand paid Rs 16 for the book kept for four days. Find
the fixed charges and the charge for each extra day.
22. The angles of a cyclic quadrilateral ABCD are
∠A = (6x + 10)°, ∠B = (5x)°
∠C = (x + y)°, ∠D = (3𝑦 – 10) °
Find x andy, and hence the values of the four angles
23. Jamila sold a table and a chair for Rs 1050, thereby making a profit of 10% on the
table and 25% on the chair. If she had taken a profit of 25% on the table and 10% on
the chair she would have got Rs 1065. Find the cost price of each.
24. Determine, graphically, the vertices of the triangle formed by the lines
𝑦 = 𝑥, 3𝑦 = 𝑥, 𝑥 + 𝑦 = 8
25. A two-digit number is obtained by either multiplying the sum of the digits by 8 or
then subtracting 5 or by multiplying the difference of the digits by 16 and then
adding 3. Find the number.
26. A railway half ticket costs half the full fare, but the reservation charges are the
same on a half ticket as on a full ticket. One reserved first class ticket from the station
A to B costs Rs 2530. Also, one reserved first class ticket and one reserved first class
half ticket from A to B costs Rs 3810. Find the full first class fare from station A to B,
and also the reservation charges for a ticket.
27. Vijay had some bananas, and he divided them into two lots A and B. He sold the
first lot at the rate of Rs 2 for 3 bananas and the second lot at the rate of Re 1 per
banana, and got a total of Rs 400. If he had sold the first lot at the rate of Re 1 per
banana, and the second lot at the rate of Rs 4 for 5 bananas, his total collection
would have been Rs 460. Find the total number of bananas he had.
28. 𝐴𝐵𝐶𝐷𝐸 is a pentagon with 𝐵𝐸 ∥ 𝐶𝐷 and 𝐵𝐶 ∥ 𝐷𝐸, 𝐵𝐶 is perpendicular to 𝐶𝐷 if
the perimeter of 𝐴𝐵𝐶𝐷𝐸 is 21 𝑐𝑚, Find 𝑥 and 𝑦.

29. A and B are two points 150 km apart on a highway. Two cars start with different
speeds from A and B at same time. If they move in same direction, they meet in 15
hours. If they move in opposite direction, they meet in one hour. Find their speeds
30. The area of a rectangle gets reduced by 67 square metres, when its length is
increased by 3 m and breadth is decreased by 4 m. If the length is reduced by 1 m
and breadth is increased by 4 m, the area is increased by 89 square metres. Find the
dimensions of the rectangle
ANSWER’S
Q1. C Q16. (2,0), (0,4), (0, −4), 8 𝑠𝑞. 𝑢𝑛𝑖𝑡
Q2. D Q17. 𝑥 = 𝑦, 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦 𝑚𝑎𝑛𝑦 𝑙𝑖𝑛𝑒
Q3. D Q18. 40 Years
Q4. C Q19. 40,48
Q5. D Q20. 100 𝑖𝑛 ℎ𝑎𝑙𝑙 𝐴, 80 𝑖𝑛 ℎ𝑎𝑙𝑙 𝐵
Q6. D Q21. 𝑅𝑠. 10, 3
Q7. C Q22. 𝑥 = 20, 𝑦 = 30, 𝐴 = 130°,
Q8. A 𝐵 = 100°, ∠𝐶 = 50°, ∠𝐷 = 80°
Q9. B Q23. Table=500 rs, Chair=400 rs
Q10. 𝑝 = −1, 𝑞 = 2 Q24. (0,0), (4,4)(6,2)
Q11. 𝑥 = 3, 𝑦 = 1 Q25. 83
Q12. 𝑘 = −6 Q26. 𝑅𝑠. 2500, 30
Q13. 𝑎 = 3, 𝑏 = 1 Q27. 500
Q14. 𝑥 = 1, 𝑦 = 4 Q28. 𝑥 = 5, 𝑦 = 0
Q15. 𝑥 = 340, 𝑦 = Q29. 80km/hr, 70km/hr
1
165, 𝜆 = − Q30. 𝑙 = 28𝑚, 𝑏 = 19𝑚
2
 Quadratic Equations
1. Which of the following is a quadratic equation?
2
(A) 𝑥 2 + 2x + 1 = (4 − 𝑥)2 + 3 (B) –2𝑥 2 = (5 – x) (2𝑥 − )
5
3
(C) (k + 1)𝑥 2 + x = 7, where k = –1 (D) 𝑥 3 – 𝑥 2 = (𝑥 − 1)3
2

2. Which of the following is not a quadratic equation?

(A) 2(x − 1)2 = 4x 2 – 2x + 1 (B) 2𝑥 – x 2 = x 2 + 5

(C) (√2x + √3)2 + x 2 = 3x 2 - 5𝑥 (D) (x 2 + 2x)2 = x 4 + 3 + 4x 3

3. Which of the following equations has 2 as a root?
(A) 𝑥 2 – 4x + 5 = 0 (B) 𝑥 2 + 3x – 12 = 0
(C) 2𝑥 2 – 7x + 6 = 0 (D) 3𝑥 2 – 6x – 2 = 0
1 5
4. If is a root of the equation 𝑥 2 + 𝑘𝑥 – = 0, then the value of 𝑘 is
2 4
1 1
(A) 2 (B) −2 (C) (D)
4 2

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

equation 𝑝(𝑥 2 + 𝑥) + 𝑘 = 0 has equal roots find the value of 𝑘.
2𝑥 1 3𝑥+9 −3
6. Solve for 𝑥: + + = 0, 𝑥 ≠ 3,
𝑥−3 2𝑥+3 (𝑥−3)(2𝑥+3) 2

7. Solve for 𝑥: 4𝑥 2 + 4𝑏𝑥 − (𝑎2 − 𝑏2 ) = 0

8. Solve for 𝑥: 4𝑥 2 − 2(𝑎2 + 𝑏2 )𝑥 + 𝑎2 𝑏2 = 0
9. Solve by using quadratic formula
𝑎𝑏𝑥 2 + (𝑏2 – 𝑎𝑐) 𝑥 – 𝑏𝑐 = 0.
10. Solve 9𝑥 2 – 6𝑎2 𝑥 + 𝑎4 – 𝑏4 = 0 using quadratic formula.
11. A train travels at a certain average speed for a distance of 63 km and then travels
a distance of 72 km at an average speed of 6 km/h more than its original speed. If it
takes 3 hours to complete the total journey, what is its original average speed?
12. Find a natural number whose square diminished by 84 is equal to thrice of 8 more
than the given number.
13. A natural number, when increased by 12, equals 160 times its reciprocal. Find the
number.
14. A train, travelling at a uniform speed for 360 km, would have taken 48 minutes
less to travel the same distance if its speed were 5 km/h more. Find the original
speed of the train.
15. If Zeba were younger by 5 years than what she really is, then the square of her
age (in years) would have been 11 more than five times her actual age. What is her
age now?
16. Find two consecutive positive integers, sum of whose squares is 365.
17. At 𝑡 minutes past 2 pm, the time needed by the minute’s hand of a clock to show
𝑡2
3 pm was found to be 3 minutes less than minutes. Find 𝑡.
4

18. 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.
19. Two water taps together can fill a tank in 6 hours. The tap of larger diameter
takes 9 hours less than the smaller one to fill the tank separately. Find the time in
which each tap can separately fill the tank.
20. 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.
21. If the price of a book is reduced by rs.5, a person can buy 5 more books for rs.
300. Find the original list price of the book.
22. In a flight of 600 km, an aircraft was slowed down due to bad weather. Its
average speed was reduced by 200 km/hr and the time of flight increased by 30
minutes. Find the duration of flight.
23. The speed of a boat in still water is 15 km/hr. It can go 30 km upstream and
return downstream to the original point in 4 hrs 30 minutes. Find the speed of the
stream.
24. Sum of areas of two squares is 400 cm2. If the difference of their perimeter is 16
cm. Find the side of each square.
25. If the roots of the quadratic equation
(𝑏 – 𝑐)𝑥 2 + (𝑐 – 𝑎)𝑥 + (𝑎 – 𝑏) = 0 are equal, prove 2𝑏 = 𝑎 + 𝑐.
26. If the equation (1 + 𝑚2 )𝑛2 𝑥 2 + 2𝑚𝑛𝑐𝑥 + (𝑐 2 – 𝑎2 ) = 0 has equal roots, prove
that 𝑐 2 = 𝑎2 (1 + 𝑚2 ).
27. A rectangular park is to be designed whose breadth is 3 m less than its length. Its
area is to be 4 square metres more than the area of a park that has already been
made in the shape of an isosceles triangle with its base as the breadth of the
rectangular park and of altitude 12 m. find the length and breadth of the park.
28. An aeroplane left 30 minutes later than its scheduled time and in order to reach
its destination 1500 km away in time, it had to increase its speed by 250 km/hr from
its usual speed. Determine its usual speed.
29. The hypotenuse of a right-angled triangle is 6 cm more than twice the shortest
side. If the third side is 2 cm less than the hypotenuse, find the sides of the triangle.
30. The difference of two natural numbers is 3 and the difference of their reciprocals
3
is Find the numbers.
28
 Q14. 45 km/h
ANSWER’S
Q15. 14 Years
Q1. D
Q16. 13 & 14
Q2. C
Q17. 14 min.
Q3. C
Q18. 6 articles, Rs. 15
Q4. A
7
Q19. 9hrs.
Q5.
4 Q20. 7 Years
−3
Q6. 𝑥 = −1, 𝑥 ≠ Q21. Rs. 20
2
(𝑎+𝑏) 𝑎−𝑏
Q7.𝑥 = − ,𝑥 = Q22. 1 hour
2 2
𝑏2 𝑎2 Q23. 5 km/hr
Q8. 𝑥 = ,
𝑎 𝑏
𝑏 𝑐 Q24. 16 & 12 cm
Q9. 𝑥 = − ,
𝑎 𝑏
Q27. L=7m, B=4 m
𝑎2 +𝑏2
Q10. Q28. 750 km/h
3

Q11. 42 km/h Q29. 10, 26, 24

Q12. 12 Q30. 7& 4
Q13. 8
 Arithmetic Progression

1. The first three terms of an A.P. respectively are 3y – 1, 3y + 5 and 5y + 1 then y

equals:
(A) −3 (B) 4 (C) 5 (D) 2
2. What is the common difference of an A.P. in which 𝑎18 – 𝑎14 = 32 ?
(A) 8 (B) −8 (C) −4 (D) 4
3. The nth term of the A.P. (1 + √3), (1 + 2√3), (1 + 3√3), … is

(A) 1 + 𝑛√3 (B) 𝑛 + √3 (C) 𝑛(1 + √3) (D) 𝑛√3

4. Show that (a – b)2, (a2 + b2) and (a + b)2 are in A.P.
5. Which term of the A.P. 5, 15, 25, ....... will be 130 more than its 31st term?
6. How many terms of the A.P. 22, 20, 18, ....... should be taken so that their sum is
zero.
7. If 10 times of 10th term is equal to 20 times of 20th term of an A.P. Find its 30th
term.
8. Solve 1 + 4 + 7 + 10 + … + 𝑥 = 287
1 1 1
10. If , and are in A.P. find x.
𝑥+2 𝑥+3 𝑥+5

11. In an A.P. find Sn, where an = 5n – 1. Hence find the sum of the first 20 terms.
12. Which term of the A.P. : 121, 117, 113 ... is the first negative terms ?
13. Find the 20th term from the last term of the A.P. 3, 8, 13, ... 253.
14. Find the middle terms of the A.P. 7, 13, 19, ......., 241.
1 1
15. If the mth term of an A.P. be and nth term be , show that its (𝑚𝑛)𝑡ℎ is 1.
𝑛 𝑚

16. If the pth term A.P. is q and the qth term is p, prove that its nth term is (p + q – n).
17. Find the number of natural numbers between 101 and 999 which are divisible by
both 2 and 5.
18. In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in
the third and so on. There are 5 rose plants in the last row. How many rows are there
in the flower bed?
19. For what value of n, are the nth term of two A.P’s 63, 65, 67 ....... and 3, 10, 17 .....
are equal ?
21. Find the sum of integers between 10 and 500 which are divisible by 7.
22. Split 207 into three parts such that these are in AP and the product of the two
smaller parts is 4623.
23. The sum of the first n terms of an AP whose first term is 8 and the common
difference is 20 is equal to the sum of first 2n terms of another AP whose first term is
– 30 and the common difference is 8. Find n.
24. The sum of first 7 terms of an A.P. is 63 and the sum of its next 7 term is 161. Find
the 28th term of this A.P.
25. If the sum of the first seven terms of an A.P. is 49 and the sum of its first 17 terms
is 289. Find the sum of first n terms of an A.P.
27. In an AP prove 𝑆12 = 3 (𝑆8 – 𝑆4 ) where 𝑆𝑛 represent the sum of first 𝑛 terms of
an A.P.
28. The sum of four consecutive numbers in A.P. is 32 and the ratio of the product of
the first and last term to the product of two middle terms is 7 : 15. Find the numbers.
29. The ratio of the 11th term to the 18th term of an AP is 2 : 3. Find the ratio of the 5th
term to the 21st term, and also the ratio of the sum of the first five terms to the sum
of the first 21 terms.
30. Show that the sum of an AP whose first term is a, the second term b and the last
(𝑎+𝑐)(𝑏+𝑐−2𝑎)
term c, is equal to
2(𝑏−𝑎)

32. The ratio of the sums of first m and first n terms of an AP is 𝑚²: 𝑛². Show that the
ratio of its mth and nth terms is (2𝑚 − 1): (2𝑛 − 1)
33. If the ratio of the sum of the first 𝑛 terms of two APs is (7𝑛 + 1): (4𝑛 + 27) then
find the ratio of their 9th terms.
34. A thief runs with a uniform speed of 100 m/min. After 1 minute, a policeman runs
after the thief to catch him. He runs with a speed of 100 m/min in the first minute
and increases his speed by 10 m/m in every succeeding minute. After how many
minutes will the policeman catch the thief?
ANSWER’S
Q1. C Q19. 𝑛 = 13
Q2. A Q20. AP 3, 1, -1, …
Q3. A
Q21. 17885
Q5. 44th terms
Q6. 𝑛 = 23 Q22. 67, 69, 71
Q7. 𝑎30 = 0 Q23. 11
Q8. 𝑎14 = 40
164 Q24. a28=57
Q9. 𝑛 is not natural number 𝑛 =
3
Q10. 𝑥 = 1 Q25. 𝑛2
Q28. 2, 6, 10, 14
Q11. S20=1030
Q29. 1:3, 5:49
Q12. 𝑛 = 32
Q13. 158 Q31. 12750
Q14. 121, 127 Q33. 24:19
Q17. 𝑛 = 89
Q34. 5 min
Q18. 𝑛 = 10
 Triangles

1. If ∆ABC ~ ∆EDF and ∆ABC is not similar to ∆DEF, then which of the following is not
true?
(A) BC. EF = AC. FD (B) AB. EF = AC. DE
(C) BC. DE = AB. EF (D) BC. DE = AB. FD
𝐴𝐵 𝐵𝐶 𝐶𝐴
2. If in two ∆ABC and ∆PQR, = = then
𝑄𝑅 𝑃𝑅 𝑃𝑄

(A) ∆𝑃𝑄𝑅 ~ ∆𝐶𝐴𝐵 (B) ∆𝑃𝑄𝑅 ~ ∆𝐴𝐵𝐶

(C) ∆CBA ~ ∆PQR (D) ∆BCA ~ ∆PQR
3. If in two triangles DEF and PQR, ∠D = ∠Q and ∠R = ∠E, then which of the following
is not true?
EF DF DE DF DE DF EF DF
(A) = (B) = (C) = (D) =
PR PQ PQ PQ QR PQ RP QR

4. In ∆ABC and ∆DEF, ∠B =∠E, ∠F =∠C and AB = 3DE. Then, the two triangles are
(A) Congruent but not similar
(B) Similar but not congruent
(C) Neither congruent nor similar
(D) Congruent as well as similar
5. It is given that ∆ABC ~ ∆DFE, ∠A =30°, ∠C = 50°, AB = 5 cm, AC = 8 cm and DF= 7.5
cm. Then, the following is true:
(A) DE = 12 cm, ∠F = 50°
(B) DE = 12 cm, ∠F = 100°
(C) EF = 12 cm, ∠D = 100°
(D) EF = 12 cm, D = 30°
𝐴𝐵 𝐵𝐶
6. If in triangles ABC and DEF, = , then they will be similar, when
𝐷𝐸 𝐹𝐷

(A) ∠B = ∠E (B) ∠A = ∠D (C) ∠B = ∠D (D) ∠A = ∠F

7. In the given fig., AB || DC and diagonals AC and BD intersects at O. If OA = 3x – 1
and OB = 2x + 1, OC = 5x – 3 and OD = 6x – 5, find the value of x.

D C

A B

8. In the given figure, ABCD is a parallelogram. AE divides the line segment BD in the
ratio 1 : 2. If BE = 1.5 cm find BC.

𝑎+𝑏
9. In the given figure, DE || AC. Which of the following is correct? 𝑥 = or
𝑎𝑦
𝑎𝑦
𝑥=
𝑎+𝑏

E D
𝑥
𝑏

C A
𝑦
10. In the given figure 𝐴𝐵 ∥ 𝑃𝑄 ∥ 𝐶𝐷, 𝐴𝐵 = 𝑥, 𝐶𝐷 = 𝑦 and 𝑃𝑄 = 𝑧. Prove that
1 1 1
+ =
𝑥 𝑦 𝑧
11. In the figure, a point O inside ∆ABC is joined to its vertices. From a point D on AO,
DE is drawn parallel to AB and from a point E on BO, EF is drawn parallel to BC. Prove
that 𝐷𝐹 || 𝐴𝐶.

𝐵𝐸 𝐵𝐶
12. In the given figure DE || AC and = . Prove that 𝐷𝐶|| 𝐴𝑃.
𝐸𝐶 𝐶𝑃

13. Legs (sides other than the hypotenuse) of a right triangle are of lengths 16cm and
8cm. Find the length of the side of the largest square that can be inscribed in the
triangle.
𝐴𝐷 𝐴𝐸
14. In the given figure ∠D = ∠E and = Prove that ∆BAC is an isosceles triangle.
𝐷𝐵 𝐸𝐶

15. Find the value of 𝑥 for which DE ∥ AB in Given Fig

A B

𝟑𝒙 + 𝟒

D E

𝒙+𝟑 𝒙

C
16. In Given Fig. if ∠1 = ∠2 and ∆𝑁𝑆𝑄 ≅ ∆𝑀𝑇𝑅, then prove that ∆𝑃𝑇𝑆 ~ ∆𝑃𝑅𝑄.

1 2
S T

Q R N
M

17. In Given Fig. if AB ∥ DC and AC and PQ intersect each other at the point O, prove
that OA . CQ = OC . AP.

18. If ∆ABC ~ ∆DEF, AB = 4 cm, DE = 6 cm, EF = 9 cm and FD = 12 cm, find the

perimeter of ∆ABC.
19. ABCD is a trapezium in which AB || DC and P and Q are points on AD and BC,
respectively such that PQ ∥ 𝐷𝐶. If 𝑃𝐷 = 18 𝑐𝑚, 𝐵𝑄 = 35 𝑐𝑚 and 𝑄𝐶 = 15 𝑐𝑚, find
𝐴𝐷.
20. It is given that ∆𝐴𝐵𝐶~∆𝐸𝐷𝐹 such that AB=5 cm, AC=7 cm, DF=15 cm and DE=12
cm. Find the lengths of the remaining sides of the triangles.
21. In Given Fig. If PQRS is a parallelogram and AB ∥ PS, then prove that OC ∥ SR.
22. A street light bulb is fixed on a pole 6 m above the level of the street. If a woman
of height 1.5 m casts a shadow of 3m, find how far she is away from the base of the
pole.
23. In Given fig. 𝑙 ∥ 𝑚 And line segments AB, CD and EF are concurrent at point P.
𝐴𝐸 𝐴𝐶 𝐶𝐸
Prove that = = .
𝐵𝐹 𝐵𝐷 𝐹𝐷

𝒍 𝒎

A D
P
E F
C
B

24. In Given Fig. 𝑃𝐴, 𝑄𝐵, 𝑅𝐶 and 𝑆𝐷 are all perpendiculars to a line 𝑙, 𝐴𝐵 = 6 𝑐𝑚,
𝐵𝐶 = 9 𝑐𝑚, 𝐶𝐷 = 12 𝑐𝑚 and 𝑆𝑃 = 36 𝑐𝑚. Find 𝑃𝑄, 𝑄𝑅 and 𝑅𝑆.

25. In Given Fig. line segment DF intersect the side AC of a triangle ABC at the point E
𝐵𝐷 𝐵𝐹
such that E is the mid-point of CA and ∠AEF = ∠AFE . Prove that =
𝐶𝐷 𝐶𝐸

[Hint: Take point G on AB such that CG ∥ DF.]

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

𝐴𝑂 𝑂𝐵
the point O. Using a similarity criterion for two triangles, show that = .
𝑂𝐶 𝑂𝐷
27. A girl of height 90 cm is walking away from the base of a lamp-post at a speed of
1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4
seconds.

28. In given fig., if ∆ABE ≅ ∆ACD, show that ∆ADE ~ ∆ABC.

29. Sides AB and BC and median AD of a triangle ABC are respectively proportional to
sides PQ and QR and median PM of ∆PQR (see fig.). Show that ∆ABC ~ ∆PQR.

30. In given fig., OB is the perpendicular bisector of the line segment DE, FA ⊥ OB
1 1 2
and FE intersects OB at the point C. Prove that + = .
𝑂𝐴 𝑂𝐵 𝑂𝐶
ANSWER’S 𝒂𝒚
Q9. 𝒙 =
Q1. C 𝒂+𝒃
16
Q2. A Q13. 𝑙 = 𝑐𝑚
3
Q3. B Q15. 𝑥 = 2
Q4. B Q18. 18 cm
Q5. B Q19. 60 𝑐𝑚
Q6. C Q20. 𝐵𝐶 = 6.25𝑐𝑚, 𝐸𝐹 = 16.8 𝑐𝑚
Q7. 𝑥 = 2 Q22. 9 𝑚
Q8. BC=3 cm Q24. 8 cm,12 cm, 16 cm
Q27. 1.6 m
 Coordinate Geometry
1. The distance of the point 𝑃 (2, 3) from the x-axis is
(A) 2 (B) 3 (C) 1 (D) 5
2. The distance of the point 𝑃 (– 6, 8) from the origin is
(A) 8 (B)2√7 (C) 10 (D) 6
3. The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is

(A) 5 (B)11 (C) 12 (D) 7 + √5

4. The points (– 4, 0), (4, 0), (0, 3) are the vertices of a
(A) Right triangle (B) isosceles triangle (C) Equilateral triangle (D) scalene triangle
5. The point which divides the line segment joining the points (7, –6) and (3, 4) in
ratio 1 : 2 internally lies in the
(A) I quadrant (B) II quadrant (C) III quadrant (D) IV quadrant
6. The fourth vertex D of a parallelogram ABCD whose three vertices are
𝐴 (– 2, 3), 𝐵 (6, 7) and 𝐶 (8, 3) is
(A) (0,1) (B) (0, −1) (C) (−1,0) (D) (1,0)
7. If the point 𝑃 (2, 1) lies on the line segment joining points 𝐴 (4, 2) and
𝐵 (8, 4), then
1 1 1
(A) AP = 𝐴𝐵 (B) 𝐴𝑃 = 𝑃𝐵 (C) PB = 𝐴𝐵 (D) AP = 𝐴𝐵
3 3 2

8. The coordinates of the point which is equidistant from the three vertices of the
∆𝐴𝑂𝐵 as shown in the fig. is
𝑥 𝑦 𝑦 𝑥
(A) (𝑥, 𝑦) (B) (𝑦, 𝑥) (C) ( , ) (D)( , )
2 2 2 2

9. If the distance between the points (4, 𝑝) and (1,0) is 5, then the value of 𝑝 is
(A) 4 𝑜𝑛𝑙𝑦 (B) ≠ 4 (C) −4 𝑜𝑛𝑙𝑦 (D)0
10. Assertion: Centroid of a triangle formed by the points
(a, b), (b, c) and (𝑐, 𝑎) is at origin, Then 𝑎 + 𝑏 + 𝑐 = 0.
Reason: Centroid of a ∆ABC with vertices A(x1,y1), B(x2, y2) and C(x3, y3) is given by
𝑥1 +𝑥2 +𝑥3 𝑦1 +𝑦2 +𝑦3
( , ).
3 3

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct
explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct
explanation of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assetion (A) is false but Reason (R) is true
11. If the distances of 𝑃(𝑥, 𝑦) from 𝐴(5, 1) and 𝐵(– 1, 5) are equal then prove that
3𝑥 = 2𝑦.
12. If the point 𝑃(3, 4) is equidistant from the points 𝐴(𝑎 + 𝑏, 𝑏 – 𝑎) and
𝐵(𝑎 – 𝑏, 𝑎 + 𝑏) then prove that 3𝑏 – 4𝑎 = 0.

13. Find the points on the x–axis which are at a distance of 2√5 from the point
(7, –4). How many such points are there?
14. What type of a quadrilateral do the points A (2, –2), B (7, 3), C (11, –1) and
D (6, –6) taken in that order, form?
15. Find the value of a, if the distance between the points A (–3, –14) and B (a, –5) is
9 units.
16. Find a point which is equidistant from the points A (–5, 4) and B (–1, 6)? How
many such points are there?
17. If P (9a – 2, –b) divides line segment joining A (3a + 1, –3) and B (8a, 5) in the ratio
3: 1, find the values of a and b.
18. If (a, b) is the mid-point of the line segment joining the points A (10, –6) and
B (𝑘, 4) and a – 2b = 18, find the value of k and the distance AB.
19. The line segment joining the points A (3, 2) and B (5,1) is divided at the point P in
the ratio 1:2 and it lies on the line 3𝑥 – 18𝑦 + 𝑘 = 0. Find the value of 𝑘.
20. 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 coordinates of the point of division.
21. Find the relation between 𝑥 and 𝑦 if 𝐴(𝑥, 𝑦), 𝐵(– 2, 3) and 𝐶(2, 1) form an
isosceles triangle with 𝐴𝐵 = 𝐴𝐶.
22. If (– 4, 3) and (4, 3) are two vertices of an equilateral triangle, find the
coordinates of the third vertex, given that the origin lies in the interior of the triangle.
23. The points A (x1,y1), B (x2,y2) and C (x3,y3) are the vertices of ∆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
24. If the points A (1, –2), B (2, 3) C (a, 2) and D (– 4, –3) form a parallelogram, find
the value of 𝑎 and height of the parallelogram taking AB as base.
25. Students of a school are standing in rows and columns in their playground for a
drill practice. A, B, C and D are the positions of four students as shown in given figure.
Is it possible to place Jaspal in the drill in such a way that he is equidistant from each
of the four students A, B, C and D? If so, what should be his position?

26. Ayush starts walking from his house to office. Instead of going to the office
directly, he goes to a bank first, from there to his daughter’s school and then reaches
the office. What is the extra distance travelled by Ayush in reaching his office?
(Assume that all distances covered are in straight lines). If the house is situated at
(2, 4), bank at (5, 8), school at (13, 14) and office at (13, 26) and coordinates are in
km.
27. The line segment joining the points A(2, 1) and B(5, -8) is trisected at the points P
and Q such that P is nearer to A. If P also lies on the line given by 2𝑥 − 𝑦 + 𝑘 = 0,
find the value of 𝑘.
28. If the point C(-1, 2) divides interplay the line-segment joining the points A(2, 5)
and B(x, y) in the ratio 3:4, find the value of x² + y².
29. If A(5, 2), B(2,-2) and C(-2, t) are the vertices of a right angled triangle with
∠𝐵 = 90°, then find the value of 𝑡.
30. The points A(4, 7), B(p, 3) and C(7, 3) are the vertices of a right triangle, right-
angled at B. Find the value of 𝑝.

ANSWER’S
Q1. B Q18. 𝐾 = 22, 𝐴𝐵 = 2√61
Q2. C Q19. 19
Q3. B 8
Q20. 8:1, ( , − )
1
3 9
Q4. B
Q22. 0,3 − 4√3
Q5. D
𝑥2 +𝑥3 𝑦2 +𝑦3
Q23. (i) ( , )
Q6. B 2 2
𝑥1 +𝑥2 +𝑥3 𝑦1 +𝑦2 +𝑦3
Q7. D (ii) ( , )
3 3

Q8. A 12√16
Q24. 𝑎 = −3, ℎ
13
Q9. B
Q25. Yes, (7, 5)
Q10. A
Q26. Extra Distance=2.4 km
Q13. (9, 0), (5, 0), 2 points
Q27. -8
Q14. Rectangle
Q28. 29
Q15. 𝑎 = −3
Q29. 1
Q16. (-3, 5), Infinity Number
Q30. 𝑃 = 4
Q17. 𝑎 = 1, 𝑏 = −3
 Trigonometry
5 sin 𝜃−4 cos 𝜃
1. If 5 tan 𝜃 − 4 = 0, then the value of is
5 sin 𝜃+4 cos 𝜃
5 5 1
(A) (B) (C) 0 (D)
3 6 6

2. In given fig., If 𝐴𝐷 = 4𝑐𝑚, 𝐵𝐷 = 3 𝑐𝑚 and 𝐶𝐵 = 12 𝑐𝑚, then cot𝜃 =

12 5 13 12
(A) (B) (C) (D)
5 12 12 13

𝑎
3. Given that sin𝜃 = , then cos𝜃 is equal to
𝑏

𝑏 𝑏 √𝑏2 −𝑎2 2
(A) (B) (C) (D)
√𝑏2 −𝑎2 𝑎 𝑏 √𝑏2 −𝑎2

1 1
4. Given that sin𝛼 = and cos 𝛽 = , then the value of ( 𝛼 + 𝛽 ) is
2 2

(A) 0° (B) 30° (C) 60° (D) 90°

1 𝑐𝑜𝑠𝑒𝑐 2 𝜃−𝑠𝑒𝑐 2 𝜃
5. Given tan 𝜃 = ,find the value of .
√3 𝑐𝑜𝑠𝑒𝑐 2 𝜃+𝑠𝑒𝑐 2 𝜃

6. Prove that:- 𝑠𝑒𝑐 4 𝜃 − 𝑠𝑒𝑐 2 𝜃 = 𝑡𝑎𝑛4 𝜃 + 𝑡𝑎𝑛2 𝜃

7. If 𝑥 = 𝑝 𝑠𝑒𝑐 𝜃 + 𝑞 𝑡𝑎𝑛 𝜃& 𝑦 = 𝑝 𝑡𝑎𝑛 𝜃 + 𝑞 𝑠𝑒𝑐 𝜃 then prove that
𝑥 2 – 𝑦 2 = 𝑝2 – 𝑞2 .
1
8. If 7 𝑠𝑖𝑛2 𝜃 + 3 𝑐𝑜𝑠 2 2𝜃 = 4 then show that tan 𝜃 = .
√3
1 1
9. If sin(𝐴 – 𝐵) = , cos(𝐴 + 𝐵) = then find the value of A and B.
2 2

10. If tan(3𝑥 − 15°) = 1 then find the value of 𝑥.

cos 𝜃 𝑐𝑜𝑠𝜃
11. Find the value of 𝜃, if + = 4, 𝜃 ≤ 90°.
1−sin 𝜃 1+sin 𝜃
tan 𝐴+sec 𝐴−1 1+sin 𝐴
12. Prove that: =
tan 𝐴−sec 𝐴+1 𝐶𝑜𝑠 𝐴
1 1 1 1
13. Prove that: − = −
sec 𝑥 −tan 𝑥 cos 𝑥 cos 𝑥 sec 𝑥+tan 𝑥

14. If 𝑠𝑖𝑛 𝜃 + 𝑠𝑖𝑛2 𝜃 = 1, prove that 𝑐𝑜𝑠 2 𝜃 + 𝑐𝑜𝑠 4 𝜃 = 1

15. If 𝑎 𝑐𝑜𝑠 𝜃 + 𝑏 𝑠𝑖𝑛 𝜃 = 𝑚 and 𝑎 𝑠𝑖𝑛 𝜃– 𝑏 𝑐𝑜𝑠 𝜃 = 𝑛. Prove that:
𝑎 2 + 𝑏 2 = 𝑚2 + 𝑛2 .
sin 𝜃 1+cos 𝜃
16. Prove: + = 2 𝑐𝑜𝑠𝑒𝑐 𝜃
1+cos 𝜃 sin 𝜃

17. Prove: (sin 𝛼 + cos 𝛼 )(tan 𝛼 + cot 𝛼 ) = sec 𝛼 + 𝑐𝑜𝑠𝑒𝑐 𝛼

𝑐𝑜𝑡 2 𝛼
18. Prove:- 1 + = 𝑐𝑜𝑠𝑒𝑐𝛼
1+𝑐𝑜𝑠𝑒𝑐 𝛼

19. Simplify (1 + 𝑡𝑎𝑛2 𝜃) (1– 𝑠𝑖𝑛 𝜃) (1 + 𝑠𝑖𝑛𝜃)

20. If 2𝑠𝑖𝑛2 𝜃 − 𝑐𝑜𝑠 2 𝜃 = 2,then find the value of 𝜃.
21. Prove that:
2 (𝑠𝑖𝑛6 𝜃 + 𝑐𝑜𝑠 6 𝜃) – 3 (𝑠𝑖𝑛4 𝜃 + 𝑐𝑜𝑠 4 𝜃) + 1 = 0
22. Prove that:
(1 + 𝑐𝑜𝑡 𝐴 + 𝑡𝑎𝑛 𝐴) (𝑠𝑖𝑛 𝐴 – 𝑐𝑜𝑠 𝐴) = 𝑠𝑖𝑛 𝐴 𝑡𝑎𝑛 𝐴 – 𝑐𝑜𝑡 𝐴 𝑐𝑜𝑠 𝐴
23. If 𝑠𝑖𝑛 𝜃 + 𝑐𝑜𝑠 𝜃 = 𝑚 and sec 𝜃 + 𝑐𝑜𝑠𝑒𝑐 𝜃 = 𝑛 then show that
𝑛(𝑚2 – 1) = 2𝑚.
𝑠𝑖𝑛2 𝜃−2𝑠𝑖𝑛4 𝜃
24. Prove that: 𝑠𝑒𝑐 2 𝜃 − =1
2𝑐𝑜𝑠4 𝜃−𝑐𝑜𝑠2 𝜃

25. If 𝑠𝑖𝑛 𝜃 + 𝑐𝑜𝑠 𝜃 = √3 , then prove that 𝑡𝑎𝑛 𝜃 + 𝑐𝑜𝑡 𝜃 = 1

26. If 𝑡𝑎𝑛 𝜃 + 𝑠𝑖𝑛 𝜃 = 𝑚, 𝑡𝑎𝑛 𝜃– 𝑠𝑖𝑛 𝜃 = 𝑛, then prove that 𝑚2 – 𝑛2 = 4√𝑚𝑛.
𝑝2 −1
27. If 𝑐𝑜𝑠𝑒𝑐 𝜃 + cot 𝜃 = 𝑝, then prove that 𝑐𝑜𝑠𝜃 = .
𝑝2 +1

28. Prove that: √𝑠𝑒𝑐 2 𝜃 + 𝑐𝑜𝑠𝑒𝑐 2 𝜃 = tan 𝜃 + cot 𝜃

1
29. If 1 + 𝑠𝑖𝑛2 𝜃 = 3 sin 𝜃 cos 𝜃, prove that tan 𝜃 = 1 𝑜𝑟 .
2
𝑙 2 +1
30. If tan 𝜃 + sec 𝜃 = 𝑙, then prove that sec 𝜃 = .
2𝑙
ANSWER’S
Q1. C
Q2. A
Q3. C
Q4. D
1
Q5.
2

Q9. 𝐴 = 45°, 𝐵 = 15°

Q10. 20°
Q11. 60°
Q19. 1
Q20. 90°
 Some Application of Trigonometry
1. The length of the shadow of a tower on the plane ground is √3 times the height of
the tower. The angle of elevation of sun is:
(A) 45° (B) 30° (C) 60° (D) 90°
2. The tops of the poles of height 16 m and 10 m are connected by a wire of length l
metres. If the wire makes an angle of 30° with the horizontal, then 𝑙 =
(A) 26 𝑚 (B) 16 𝑚 (C) 12 𝑚 (D) 10 𝑚
3. A ladder leaning against a wall makes an angle of 60° with the horizontal. If the
foot of the ladder is 2.5 m away from the wall, then the length of the ladder is
(A) 3 𝑚 (B) 4 𝑚 (C) 5 𝑚 (D) 6 𝑚
4. If a tower is 30 m high, casts a shadow 10√3𝑚 long on the ground, then the angle
of elevation of the sun is
(A) 30° (B) 45° (C) 60° (D) 90°
5. A tower is 50 m high. When the sun’s altitude is 45° then what will be the length of
its shadow?
50
6. The length of shadow of a pole 50 m high is 𝑚 find the sun’s altitude.
√3

7. In the figure, find the value of BC.

8. In the figure, find the value of CF.

9. The shadow of a vertical tower on level ground increases by 10 m when the
altitude of the sun changes from 45° to 30°. Find the height of the tower.[𝑈𝑠𝑒 √3 =
1.73]
10. An aeroplane at an altitude of 200 m observes angles of depression of opposite
points on the two banks of the river to be 45° and 60°, find the width of the river.
[𝑈𝑠𝑒 √3 = 1.732]
11. The angle of elevation of a tower at a point is 45°. After going 40 m towards the
foot of the tower, the angle of elevation of the tower becomes 60°. Find the height of
the tower. [𝑈𝑠𝑒 √3 = 1.732]
12. The upper part of a tree broken over by the wind makes an angle of 30° with the
ground and the distance of the foot of the tree from the point where the top touches
the ground is 25 m. What was the total height of the tree?
13. A vertical flagstaff stands on a horizontal plane. From a point 100 m from its foot,
the angle of elevation of its top is found to be 45°. Find the height of the flagstaff.
14. The length of a string between kite and a point on the ground is 90 m. If the string
3
makes an angle 𝛼 with the level ground and sin 𝛼 = . Find the height of the kite.
5
There is no slack in the string.
15. An aeroplane, when 3000 m high, passes vertically above another plane at an
instant when the angle of elevation of two aeroplanes from the same point on the
ground are 60° and 45° respectively. Find the vertical distance between the two
planes. [𝑈𝑠𝑒 √3 = 1.732]
16. From the top of a 7 m high building, the angle of elevation of the top of the tower
is 60° and the angle of depression of the foot of the tower is 45°. Find the height of
the tower.
17. A man standing on the deck of a ship, 10 m above the water level observes the
angle of elevation of the top of a hill as 60° and angle of depression of the bottom of
the hill as 30°. Find the distance of the hill from the ship and height of the hill.
18. From a window 60 m high above the ground of a house in a street, the angle of
elevation and depression of the top and the foot of another house on the opposite
side of the street are 60° and 45° respectively. Show that the height of opposite
house is 60(1 + √3) metres.
19. The angle of elevation of an aeroplane from a point A on the ground is 60°. After
a flight of 30 seconds, the angle of elevation changes to 30°. If the plane is flying at a
constant height of 3600√3 m, find the speed in km/hour of the plane.
20. A bird is sitting on the top of a tree, which is 80 m high. The angle of elevation of
the bird, from a point on the ground is 45°. The bird flies away from the point of
observation horizontally and remains at a constant height. After 2 seconds, the angle
of elevation of the bird from the point of observation becomes 30°. Find the speed of
flying of the bird. [𝑈𝑠𝑒 √3 = 1.732]
21. The angles of elevation of the top of a tower from two points on the ground at
distances 9 m and 4 m from the base of the tower are in the same straight line with it
are complementary. Find the height of the tower.
22. An observer from the top of a light house, 100 m high above sea level, observes
the angle of depression of a ship, sailing directly towards him, changes from 30° to
60°. Determine the distance travelled by the ship during the period of observation.
23. A 1.2m tall girl spots a balloon on the eve of Independence Day, 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 girl at an instant is 60°. After some time, the angle of
elevation reduces to 30°. Find the distance travelled by the balloon.
24. Two pillars of equal heights stand on either side of a roadway 150 m wide. From
a point on the roadway between the pillars, the angles of elevation of the top of the
pillars are 60° and 30°. Find the height of pillars and the position of the point.
25. A moving boat is observed from the top of a 150 m high cliff moving away form
the cliff. The angle of depression of the boat changes form 60° to 45° in 2 minutes.
Find the speed of the boat in m/h.
26. The rod AC of a TV disc antenna is fixed at right angles to the wall AB and a rod
CD is supporting the disc as shown in the figure. If AC = 1.5 m long and CD = 3 m, find
(i) tan 𝜃 (ii) sec 𝜃 + cosec 𝜃.
27. The angle of elevation of a cloud from a point h metres above the surface of a
lake is θ and the angle of depression of its reflection in the lake is 𝜙. Prove that the
𝑡𝑎𝑛𝜙+tan 𝜃
height of the cloud above the lake is ℎ ( ).
tan 𝜙−tan 𝜃

28. A vertical tower stands on a horizontal plane and is surmounted by a vertical flag
staff of height ℎ. At a point on the plane, the angles of elevation of the bottom and
the top of the flag staff are α and β, respectively. Prove that the height of the tower
ℎ tan 𝛼
is ( ).
tan 𝛽−tan 𝛼

29. A window of a house is h metres above the ground. From the window, the angles
of elevation and depression of the top and the bottom of another house situated on
the opposite side of the lane are found to be α and β, respectively. Prove that the
height of the other house is h ( 1 + tan α cot β ) metres.
30. The lower window of a house is at a height of 2 m above the ground and its upper
window is 4 m vertically above the lower window. At certain instant the angles of
elevation of a balloon from these windows are observed to be 60o and 30o,
respectively. Find the height of the balloon above the ground.

ANSWER’S
Q1. B Q15. 1268 m
Q2. C Q16. 7(√3) + 1 m
Q3. C Q17. 10√3𝑚, 40 𝑚
Q4. C Q19. 864 km/hr
Q5. 50 m Q20. 29.28 m
Q6. 60° Q21. 6 𝑚
Q7. 130 𝑚 Q22. 115.46 m
Q8. 25 m Q23. 58√3
Q9. 13.65 𝑚 Q24. ℎ𝑒𝑖𝑔ℎ𝑡 = 64.95 𝑚,
Q10. 315.46 m Q25. 1902 𝑚/ℎ (Approx)
Q11. 94.64 m 1
Q26. (i) tan 𝜃 =
Q12. 25√3 m √3
2
Q13. 100 m (ii) sec 𝜃 + 𝑐𝑜𝑠𝑒𝑐 𝜃 = +2
√3
Q14. 54 𝑚 Q30. 8 m
 Circles
1. In fig., ∆𝐴𝐵𝐶 is circumscribing a circle. Find the length of BC.

2. The length of the tangent to a circle from a point P, which is 25 cm away from the
centre, is 24 cm. What is the radius of the circle?
3. In fig., ABCD is a cyclic quadrilateral. If ∠𝐵𝐴𝐶 = 50° and ∠𝐷𝐵𝐶 = 60°, then
find∠𝐵𝐶𝐷.

4. In figure, O is the centre of a circle, PQ is a chord and the tangent PR at P makes an

angles of 50° with PQ. Find ∠𝑃𝑂𝑄.

5. If two tangents inclined at an angle 60° are drawn to a circle of radius 3 cm, then
find the length of each tangent.
6. If radii of two concentric circles are 4 cm and 5 cm, then find the length of the
chord of that circle which is tangent to the other circle.

7. In the given figure, PQ is tangent to outer circle and PR is tangent to inner circle. If
PQ = 4cm, OQ = 3 cm and OR = 2 cm then find the length of PR.
8. In the given figure, O is the centre of the circle, PA and PB are tangents to the circle
then find ∠𝐴𝑄𝐵.

9. In the given figure, If ∠𝐴𝑂𝐵 = 125° then find ∠𝐶𝑂𝐷.

10. If two tangent TP and TQ are drawn from an external point T such that
∠𝑇𝑄𝑃 = 60° then find∠𝑂𝑃𝑄.

11. In the given fig. AP = 4 cm, BQ = 6 cm and AC = 9 cm. Find the semi perimeter of
∆𝐴𝐵𝐶.

12. A circle is drawn inside a right angled triangle whose sides are a, b, c where c is
𝑎+𝑏−𝑐
the hypotenuse, which touches all the sides of the triangle. Prove 𝑟 = where 𝑟
2
is the radius of the circle.
13. Prove that the tangent at any point of a circle is perpendicular to the radius
through the point of contact.
14. In the given Fig., AC is diameter of the circle with centre O and A is the point of
contact, then find 𝑥.

15. In the given fig. KN, PA and PB are tangents to the circle. Prove that: KN = AK +
BN.

16. In the given fig. PQ is a chord of length 6 cm and the radius of the circle is 6 cm.
TP and TQ are two tangents drawn from an external point T. Find ∠𝑃𝑇𝑄.

17. In the given figure find AD, BE, CF where AB = 12 cm, BC = 8 cm and AC = 10 cm.

18. Two tangents PA and PB are drawn to a circle with centre O from an external
point P. Prove that ∠𝐴𝑃𝐵 = 2∠𝑂𝐴𝐵.
19. In the given fig. OP is equal to the diameter of the circle with centre O. Prove that
∠𝐴𝐵𝑃 is an equilateral triangle.

20. In the given fig., find PC. If AB = 13 cm, BC = 7 cm and 𝐴𝐷 = 15 𝑐𝑚.

21. In the given figure, find the radius of the circle.

22. In the given figure, a circle touches all the four sides of a quadrilateral ABCD. If AB
= 6 cm, BC = 9 cm and CD = 8 cm, then find the length of AD.
23. Prove that the angle between the two tangents drawn from an external point to a
circle is supplementary to the angle subtended by the line segment joining the points
of contact at the centre.
24. In figure, XP and XQ are tangents from X to the circle with centre O, R is a point
on the circle and AB is tangent at R. Prove that: 𝑋𝐴 + 𝐴𝑅 = 𝑋𝐵 + 𝐵𝑅

25. In the given figure, find the perimeter of ∆𝐴𝐵𝐶, if 𝐴𝑃 = 12 𝑐𝑚.

26. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
27. In a right triangle ABC in which ∠B = 90°, a circle is drawn with AB as diameter
intersecting the hypotenuse AC and P. Prove that the tangent to the circle at P
bisects BC.
28. In given Fig. tangents PQ and PR are drawn to a circle such that ∠RPQ = 30°. A
chord RS is drawn parallel to the tangent PQ. Find the ∠RQS.
[Hint: Draw a line through Q and perpendicular to QP.]
29. In given Fig. O is the centre of a circle of radius 5 cm, T is a point such that OT = 13
cm and OT intersects the circle at E. If AB is the tangent to the circle at E, find the
length of AB.

30. The tangent at a point C of a circle and a diameter AB when extended intersect at
P. If ∠𝑃𝐶𝐴 = 110°, find ∠𝐶𝐵𝐴 [see given Fig.].

ANSWER’S
Q1. 10 cm Q14. 40°
Q2. 7 cm Q16. 120°
Q3. 70° Q17. 7𝑐𝑚, 5𝑐𝑚, 3𝑐𝑚
Q4. 100° Q20. 5 cm
Q5. 3√3 cm Q21.11 𝑐𝑚
Q6. 6 cm Q22. 5 cm
Q7. √21 𝑐𝑚 Q25. 24 𝑐𝑚
Q8. 70° Q28. 30°
20
Q9. 55° Q29. 𝑐𝑚
3
Q10. 30° Q30. 70°
Q11. 15 cm
 Area Related to Circles
1. If the circumference of a circle and the perimeter of a square are equal, then
(A) Area of the circle = Area of the square
(B) Area of the circle > Area of the square
(C) Area of the circle < Area of the square
(D) Nothing definite can be said about the relation between the areas of the circle
and square.
2. If the perimeter of a circle is equal to that of a square, then the ratio of their areas
is
(A) 22 : 7 (B) 14 : 11 (C) 7 : 22 (D) 11: 14
3. A wire can be bent in the form of a circle of radius 35 cm. If it is bent in the form of
a square, then what will be its area?
4. If the difference between the circumference and radius of a circle is 37 cm, then
22
find the circumference of the circle. [use 𝜋 = ]
7

5. If diameter of a circle is increased by 40%, find by how much percentage its area
increases?
6. The perimeter of a sector of a circle of radius 14 cm is 68 cm. Find the area of the
sector.
7. Find the perimeter of the given fig, where AED is a semicircle and ABCD is a
rectangle.
8. In fig. OAPBO is a sector of a circle of radius 10.5 cm. Find the perimeter of the
sector.
9. Two circles touch externally. The sum of their areas is 130𝜋sq. cm and the distance
between their centres is 14 cm. Find the radii of the circles.
10. Find the radius of a circle whose circumference is equal to the sum of the
circumferences of two circles of radii 15 cm and 18 cm.
11. In Fig, AB is a diameter of the circle, AC = 6 cm and BC = 8 cm. Find the area of the
shaded region (Use 𝜋 = 3.14).

12. Find the area of the shaded region in Fig. where arcs drawn with centres A, B, C
and D intersect in pairs at mid-points P, Q, R and S of the sides AB, BC, CD and DA,
respectively of a square ABCD (Use 𝜋 = 3.14).

13. In Fig. arcs are drawn by taking vertices A, B and C of an equilateral triangle of
side 10 cm. to intersect the sides BC, CA and AB at their respective mid-points D, E
and F. Find the area of the shaded region (Use 𝜋 = 3.14).

14. In Fig. 11.12, arcs have been drawn with radii 14 cm each and with centres P, Q
and R. Find the area of the shaded region.
15. In Fig. 11.13, arcs have been drawn of radius 21 cm each with vertices A, B, C and
D of quadrilateral ABCD as centres. Find the area of the shaded region.

16. A piece of wire 20 cm long is bent into the form of an arc of a circle subtending an
angle of 60° at its centre. Find the radius of the circle.
17. A chord of a circle of radius 20 cm subtends an angle of 90° at the centre. Find the
area of the corresponding major segment of the circle. (Use 𝜋 = 3.14).
18. With the vertices A, B and C of a triangle ABC as centres, arcs are drawn with radii
5 cm each as shown in Fig. If AB = 14 cm, BC = 48 cm and CA = 50 cm, then find the
area of the shaded region. (Use 𝜋 = 3.14).

19. In Fig. 11.17, ABCD is a trapezium with AB || DC, AB = 18 cm, DC = 32 cm and

distance between AB and DC = 14 cm. If arcs of equal radii 7 cm with centres A, B, C
and D have been drawn, then find the area of the shaded region of the figure.

20. Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60°
21. Find the area of a quadrant of a circle whose circumference is 22 cm
22. The length of the minute hand of a clock is 14 cm. Find the area swept by the
minute hand in 5 minutes.
23. In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find:
(i) the length of the arc
(ii) area of the sector formed by the arc
(iii) area of the segment formed by the corresponding chord
24. A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the
areas of the corresponding minor and major segments of the circle. (Use 𝜋 = 3.14 and
√3 = 1.73)
25. A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find
the area of the corresponding segment of the circle. (Use 𝜋 = 3.14 and √3 = 1.73)
26. A horse is tied to a peg at one corner of a square shaped grass field of side 15 m
by means of a 5 m long rope (see Fig.) Find
(i) the area of that part of the field in which the horse can graze.
(ii) the increase in the grazing area if the rope were 10 m long instead of 5 m. (Use 𝜋
= 3.14)

27. An umbrella has 8 ribs which are equally spaced (see Fig.) Assuming umbrella to
be a flat circle of radius 45 cm, find the area between the two consecutive ribs of the
umbrella.
28. A round table cover has six equal designs as shown in Fig. 11.11. If the radius of
the cover is 28 cm, find the cost of making the designs at the rate of rs. 0.35 per cm2 .
(Use √3 = 1.7)

29. Area of a sector of a circle of radius 36 cm is 54 𝜋 cm2. Find the length of the
corresponding arc of the sector.
30. A piece of wire 11 cm long is bent into the form of an arc of a circle subtending an
angle of 45° at its centre. Find the radius of the circle.

ANSWER’S
Q1. B Q15. 1386 cm2 Q24. 20.4375 cm2,
60
Q2. B Q16. cm 686.0625 cm2
𝜋
Q3. 3025 cm2 Q17. 1142 cm2 Q25. 88.44 cm2
Q4. 44 cm Q18. 296.75 cm2
Q5. 96% Q26. i. 19.625 m2, ii.
Q6. 280 cm2 Q19. 196 cm2 58.875 cm2
132
Q7. 76 cm Q20. cm2 22275
7 Q27. cm2
Q8. 32 cm 77 28
Q21. cm2
8
Q9. 11 cm, 3 cm 154 2
Q28. Rs.162.68
Q10. 33 cm Q22. cm
3
Q29. 3 𝜋 cm
Q11. 54.5 cm2 Q23. i. 22cm ii. 231 cm 2

Q12. 30.96 𝑐𝑚2 Q30. 1

441√3
Q13. 39.25 cm2 iii. (231 − )cm2
4
Q14. 308 cm2
 Surface Area & Volume
1. The total surface area of a solid hemisphere of radius r is
(A) 𝜋𝑟 2 (B) 2𝜋𝑟 2 (C) 3𝜋𝑟 2 (D) 4𝜋𝑟 2
2. The volume and the surface area of a sphere are numerically equal, then the radius
of sphere is
(A) 0 𝑢𝑛𝑖𝑡𝑠 (B) 1 𝑢𝑛𝑖𝑡 (C) 2 𝑢𝑛𝑖𝑡𝑠 (D) 3 𝑢𝑛𝑖𝑡𝑠
3. A cylinder, a cone and a hemisphere are of the same base and of the same height.
The ratio of their volumes is
(A) 1: 2: 3 (B) 2: 1: 3 (C) 3: 1: 2 (D) 3: 2: 1
4. Volume of two spheres is in the ratio 64 : 125. Find the ratio of their surface areas.
5. A cylinder and a cone are of same base radius and of same height. Find the ratio of
the volumes of cylinder to that of the cone. I
N
A
6. If the volume of a cube is 1331 cm³, then find the length of its edge.
A

D
7. Two cones have their heights in the ratio 1 : 3 and radii in the ratio 3 : 1. What is
YA

the ratio of their volumes?

E
8. Find the height of largest right circular cone that can be cut out of a cube whose

E
G

volume is 729 cm³.

K
9. Two identical cubes each of volume 216 cm³ are joined together end to end. What
is the surface area of the resulting cuboid?
1
10. The volume of a right circular cylinder with its height equal to the radius is 25
7
3 22
cm . Find the height of the cylinder. Use 𝜋 = .
7

11. Find the depth of a cylindrical tank of radius 10.5 cm, if its capacity is equal to
that of a rectangular tank of size 15 cm × 11 cm × 10.5 cm.
12. A petrol tank is a cylinder of base diameter 28 cm and length 24 cm filted with
conical ends each of axis length 9 cm. Determine the capacity of the tank.
13. A solid is in the form of a cylinder with hemispherical ends. The total height of the
solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the
22
solid. Use 𝜋 =
7
14. A juice seller was serving his customers using glasses as shown in figure. The inner
diameter of the cylindrical glass was 5 cm but bottom of the glass had a
hemispherical raised portion which reduced the capacity of the glass. If the height of
a glass was 10 cm, find the apparent and actual capacity of the glass. [Use 𝜋 = 3.14]

15. A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm,
which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the
mass of the pole, given that 1 cm3 of iron has approximately 8 gm mass. (Use 𝜋=
3.14)
16. A right cylindrical container of radius 6 cm and height 15 cm is full of ice-cream,
I
which has to be distributed to 10 children in equal cones having hemispherical shape
N
on the top. If the height of the conical portion is four times its base radius, find the

A
A
radius of the ice-cream cone.

D
YA

17. A solid wooden toy is in the form of a hemi-sphere surmounted by a cone of

E
same radius. The radius of hemi-sphere is 3.5 cm and the total wood used in the

E
5
making of toy is 166 cm3 Find the height of the toy. Also, find the cost of painting
G

K
22
the hemi-spherical part of the toy at the rate of rs. 10 per cm². Use 𝜋 =
7

18. In the given figure, from a cuboidal solid metalic block of dimensions 15 cm × 10
cm × 5 cm a cylindrical hole of diameter 7 cm is drilled out. Find the surface area of
22
the remaining block. [Use 𝜋 = ].
7

19. A tent is in the shape of a right circular cylinder up to a height of 3 m and conical
above it. The total height of the tent is 13.5 m and radius of base is 14 m. Find the
cost of cloth required to make the tent at the rate of 80 per m2.
20. The rain water from a roof 22 m × 20 m drains into a cylindrical vessel having
diameter of base 2 m and height 3.5 m. If the vessel is just full, find the rainfall in cm.
21. The difference between outer and inner curved surface areas of a hollow right
circular cylinder, 14 cm long is 88 cm2. If the volume of the metal used in making the
cylinder is 176 cm3. Find the outer and inner diameters of the cylinder.
22. A hemispherical depression is cut out from one face of a cubical wooden block of
edge 21 cm, such that the diameter of the hemisphere is equal to edge of the cube.
Determine the volume of the remaining block.
23. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest
diameter the hemisphere can have? Find the surface area of the solid.
24. From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity
of the same height and same diameter is hollowed out. Find the total surface area of
the remaining solid to the nearest cm2.
I
25. A wooden article was made by scooping out a hemisphere from each end of a
N
A
solid cylinder, as shown in Fig. 12.11. If the height of the cylinder is 10 cm, and its
A
base is of radius 3.5 cm, find the total surface area of the article.

D
YA

E E
G

K
26. A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its
top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots,
each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of
the water flows out. Find the number of lead shots dropped in the vessel.
27. A cone of maximum size is carved out from a cube of edge 14 cm. Find the
surface area of the cone and of the remaining solid left out after the cone carved out.
28. A factory manufactures 120000 pencils daily. The pencils are cylindrical in shape
each of length 25 cm and circumference of base as 1.5 cm. Determine the cost of
coloring the curved surfaces of the pencils manufactured in one day at Rs 0.05 per
dm2.
29. A rocket is in the form of a right circular cylinder closed at the lower end and
surmounted by a cone with the same radius as that of the cylinder. The diameter and
height of the cylinder are 6 cm and 12 cm, respectively. If the slant height of the
conical portion is 5 cm, find the total surface area and volume of the rocket.
[use 𝜋 = 3.14]
30. A pen stand made of wood is in the shape of a cuboid with four conical
depressions and a cubical depression to hold the pens and pins, respectively. The
dimension of the cuboid are 10 cm, 5 cm and 4 cm. The radius of each of the conical
depressions is 0.5 cm and the depth is 2.1 cm. The edge of the cubical depression is 3
cm. Find the volume of the wood in the entire stand.

ANSWER’S
Q1. C Q18. 583 cm3
Q2. D
I Q19. 82720 rs
Q3. C
N
Q4. 16:25 Q20. 2.5 𝑐𝑚

A
A
Q5. 3:1 Q21.5 cm & 3 cm

D
Q6. 11 cm Q22. 6835.5 cm3
YA

E
Q7. 3:1 Q23. 7 cm, 332.5 cm2
Q8. 9 cm

E
Q24. 18 cm2
G

Q9. 360 cm2

Q25. 374 cm2

K
Q10. 2 cm
Q11. 5 cm Q26. 100
Q12. 18480 cm2
1 3
Q27. 154(√5 + 1) cm2,
Q13. 680 cm
6
Q14. 196.25 cm3, 163.54 cm3 (1022+ 154√5) cm2

Q15. 892.2624 kg Q28. 2250

Q16. 3 cm Q29. 301.44 cm2, 377.1 cm3

Q17. 6 cm, 770 rs Q30. 170.8 cm3

 Statistics
1. If the class intervals of a frequency distribution are 1 – 10, 11 – 20, 21 – 30, ....., 51
– 60, then the size of each class is:
(A) 9 (B) 10 (C) 11 (D) 5.5
2. Consider the frequency distribution.

The upper limit of median class is:

(A) 17 (B) 17.5 (C) 18 (D) 18.5
3. Daily wages of a factory workers are recorded as:

I
N
A
A

D
The lower limit of Modal class is:
YA

E
(A) ₹ 127 (B) ₹ 126 (C) ₹ 126.50 (D) ₹ 133

E
4. For the following distribution
G

K
The sum of Lower limits of the median class and modal class is
(A) 15 (B) 25 (C) 30 (D) 35
5. The median and mode respectively of a frequency distribution are 26 and 29.
Then, its mean is
(A) 27.5 (B) 24.5 (C) 28.4 (D) 25.8
∑ 𝑓𝑖 𝑑𝑖
6. In the formula 𝑥̅ = 𝑎 + ∑ 𝑓𝑖

for finding the mean of grouped data di ’s are deviations from a of

(A) Lower limits of the classes (B) Upper limits of the classes
(C) Mid points of the classes (D) Frequencies of the class marks
7. If xi ’s are the mid points of the class intervals of grouped data, fi ’s are the
corresponding frequencies and 𝑥̅ is the mean, then ∑(𝑓𝑖 𝑥𝑖 − 𝑥̅ ) is equal to
(A) 0 (B) −1 (C) 1 (D) 2
8. The times, in seconds, taken by 150 atheletes to run a 110 m hurdle race are
tabulated below:

The number of atheletes who completed the race in less than 14.6 seconds is:
(A) 11 (B) 71 (C) 82 (D) 130
9. Consider the following distribution:

Marks obtained Number of students

I
N
More than or equal to 0 63

A
A
More than or equal to 10 58

D
YA

More than or equal to 20 55

More than or equal to 30 51

E E
G

More than or equal to 40 48

More than or equal to 50
The frequency of the class 30-40 is
(A) 3 (B) 4 (C) 48
42
K (D) 51
10. What is the mean of first 12 prime numbers?
11. The mean of 20 numbers is 18. If 2 is added to each number, what is the new
mean?
12. The mean of 5 observations 3, 5, 7, x and 11 is 7, find the value of x.
13. What is the median of first 5 natural numbers?
14. What is the value of x, if the median of the following data is 27.5? 24, 25, 26, x +
2, x + 3, 30, 33, 37
15. What is the mode of the observations 5, 7, 8, 5, 7, 6, 9, 5, 10, 6?
16. The mean and mode of a data are 24 and 12 respectively. Find the median.
17. Write the class mark of the class 19.5 – 29.5.
18. Find the class-marks of the classes 10-25 and 35-55.
19. The mean of 11 observation is 50. If the mean of first Six observations is 49 and
that of last six observation is 52, then find sixth observation.
20. Find the mean of following distribution:

21. Find the median of the following distribution:

22. Mean of a frequency distribution (𝑥̅ ) is 45. If ∑ 𝑓𝑖 = 20 find ∑ 𝑓𝑖 𝑥𝑖

I
23. If the mean of the following distribution is 54, find the value of P.
N
A
A

D
YA

frequency 𝑥.
E E
24. The median of following frequency distribution is 24 years. Find the missing
G

K
25. Find the median of the following data:

26. Find mode of the following frequency distribution:

The mean of above distribution is 53. Use Empirical formula to find approximate
value of median.
27. The mean of the following data is 53, Find the values of f1 and f2.

28. If the median of the distribution given below is 28.5, find the values of x and y.

29. The mode of the frequency distribution is 36. Find the missing frequency (f).

30. Find the unknown entries a, b, c, d, e, f in the following distribution of heights of

students in a class :
I
N
A
A

D
YA

E E
G

K
ANSWER’S
Q1. B Q16. 20
Q2. B Q17. 24.5
Q3. C Q18. 17.5 and 45
Q4. B Q19. 56
Q5. B Q20. 20
Q6. C Q21.14
Q7. 𝐴 Q22. 900
Q8. C Q23. 11
Q9. 𝐴 Q24. 25
Q10. 16.4 approx. Q25. 30
Q11. 20 Q26. 58
Q12. 9 Q27. 𝑓1 = 18, 𝑓2 = 29
Q13. 3 Q28. 𝑥 = 20, 𝑦 = 7
Q14. 25 Q29. 𝑓 = 10
Q15. 5 Q30. a=12, b=13, c=35, d=8, e=5, f=50
 Probability
1. Which of the following cannot be the probability of an event?
2
(A) 0.7 (B) (C) −1.5 (D) 15%
3

2. Which of the following can be the probability of an event?

18 8
(A) −0.04 (B) 1.004 (C) (D)
23 7

3. An event is very unlikely to happen, its probability is closest to

(A) 0.0001 (B) 0.001 (C) 0.01 (D) 0.1
4. Out of one digit prime numbers, one number is selected at random. The
probability of selecting an even number is:
1 1 4 2
(A) (B) (C) (D)
2 4 9 5

5. When a die is thrown, the probability of getting an odd number less than 3 is:
1 1 1
(A) (B) (C) (D) 0
6 3 2

6. Rashmi has a die whose six faces show the letters as given below:

If she throws the die once, then the probability of getting C is:
1 1 1 1
(A) (B) (C) (D)
3 4 5 6

7. A card is drawn from a well shuffled pack of 52 playing cards. The event E is that
the card drawn is not a face card. The number of outcomes favourable to the event E
is:
(A) 51 (B) 40 (C) 36 (D) 12
8. In a family of 3 children, the probability of having atleast one boy is:
7 1 5 3
(A) (B) (C) (D)
8 8 8 4

9. The probability that a non-leap year selected at random will contains 53 Mondays
is:
 1 2 3 5
(A) (B) (C) (D)
7 7 7 7

10. One alphabet is chosen from the word MATHEMATICS. The probability of getting
a vowel is:
6 5 3 4
(A) (B) (C) (D)
11 11 11 11

11. Two coins are tossed simultaneously. The probability of getting at most one head
is
1 1 2 3
(A) (B) (C) (D)
4 2 3 4

12. A card is drawn at random from a pack of 52 playing cards. Find the probability
that the card drawn is neither an ace nor a king.
13. If 29 is removed from (1, 4, 9, 16, 25, 29), then find the probability of getting a
prime number.
14. Two dice are rolled simultaneously. Find the probability that the sum of the two
numbers appearing on the top is more than and equal to 10.
15. Find the probability of multiples of 7 in 1, 2, 3, .......,33, 34, 35.
16. An integer is chosen random between 1 and 100. Find the probability that (i) it is
divisible by 8, (ii) Not divisible by 8.
17. Three different coins are tossed together. Find the probability of getting (i)
exactly two heads, (ii) at least two heads.
18. Cards marked with number 3, 4, 5, .... 50 are placed in a box and mixed
thoroughly. A card is drawn at random from the box. Find the probability that the
selected cards bears a perfect square number.
19. If a number x is chosen at random from the numbers –3, –2, –1, 0, 1, 2, 3. What is
probability that 𝑥 2 ≤ 4 ?
20. A number x is selected at random from the numbers 1, 2, 3 and 4. Another
number y is selected at random from the numbers 1, 4, 9 and 16. Find the probability
that the product of x and y is less than 16.
21. In a lottery, there are 10 prizes and 25 are empty. Find the probability of getting a
prize. Also verify P(E) + P(𝐸̅ ) = 1 for this event.
22. The probability of a defective egg in a lot of 400 eggs is 0.035. Calculate the
number of defective eggs in the lot. Also calculate the probability of taking out a non-
defective egg from the lot.
23. Slips marked with numbers 3,3,5,7,7,7,9,9,9,11 are placed in a box at a game stall
in a fair. A person wins if the mean of numbers are written on the slip. What is the
probability of his losing the game?
24. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at
random from the box, find the probability that it bears
(i) a two digit number (ii) a perfect square number (iii) a number divisible by 5.
25. A die is thrown twice. Find the probability that:
(i) 5 will come up at least once (ii) 5 will not come up either time
26. Cards marked 1, 3, 5 .... 49 are placed in a box and mixed thoroughly. One card is
drawn from the box. Find the probability that the number on the card is:
(i) Divisible by 3 (ii) a composite number
(iii) Not a perfect square (iv) multiple of 3 and 5
27. Red queens and black jacks are removed from a pack of 52 playing cards. Find the
probability that the card drawn from the remaining cards is:
(i) a card of clubs or an ace (ii) a black king
(iii) neither a jack nor a king (iv) either a king or a queen
28. A box contain 100 red cards, 200 yellow cards and 50 blue cards. If a card is
drawn at random from the box, find the probability that it will be:
(a) a blue card (b) not a yellow card (c) neither yellow nor a blue card
29. A die has its six faces marked 0,1,1,1,6,6. Two such dice are thrown together and
the total score is recorded.
(i) How many different scores are possible?
(ii) What is the probability of getting a total of 7?
30. A bag contains 24 balls of which 𝑥 are red, 2𝑥 are white and 3𝑥 are blue. A ball is
selected at random. What is the probability that it is
(i) not red? (ii) white?
ANSWER’S
3 1
Q1. C Q17. (i) (ii)
8 2
Q2. C 1
Q18.
8
Q3. A
5
Q19.
Q4. B 7
1
Q5. A Q20.
2

Q6. A Q21.1
Q7. 𝐵 Q22. 14, 0.965
Q8. A 7
Q23.
10
Q9. 𝐴 9 1 1
Q24. (i) (ii) (iii)
10 10 5
Q10. D
16 25
Q25. (i) (ii)
Q11. D 25 36
8 2 21 2
Q12.
11 Q26. (i) (ii) (iii) (iv)
25 5 25 25
13
1 1 7 1
Q13. 0 Q27. (i) (ii) (iii) (iv)
3 24 8 8
1 1 3 2
Q14. Q28. (a) (b) (c)
6 7 7 7
1 1
Q15. Q29. (i) 8 scores (ii)
7 3
6 43 5 1
Q16. (i) (ii) Q30. (i) (ii)
49 49 6 3
5 1
0. (i) (ii)
6 3