Maths Sure Shot 2024
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nikhilyadav94566Sure Shot Questions
Chapter – 01
Relations and Functions
MCQ (d) Neither symmetric nor transitive
7. The identity element for the binary operation
1. Let T be the set of all triangles in the Euclidean
ab
plane, and let a relation R on T be defined as aRb if defined on Q – {0} as a * b a, b Q {0} is
2
a is congruent to ba, b T . Then R is
(a) 1 (b) 0
(a) Reflexive but not transitive (c) 2 (d) None of these
(b) Transitive but not symmetric
(c) Equivalence 8. If the set A contains 5 elements and the set B
(d) None of these contains 6 elements, then the number of one – one
and onto mappings from A to B is
2. Consider the non-empty set consisting of children (a) 720 (b) 120
in a family and a relation R defined as aRb if a is (c) 0 (d) None of these
brother of b. Then R is
(a) symmetric hut not transitive 9. Let A = {1, 2, 3, ….., n} and B = {a, b}. Then the
(b) transitive hut not symmetric number of surjections from A into B is
(c) neither symmetric nor transitive
(a) n P2 (b) 2n 2
(d) both symmetric and transitive
(c) 2n 1 (d) None of these
3. The maximum number of equivalence relations on
the set A = {1, 2, 3} are 1
10. Let f : R R be defined by f (x) xR .
(a) 1 (b) 2 x
(c) 3 (d) 5 Then f is
(a) one – one (b) onto
4. If a relation R on the set {1, 2, 3} be defined by R = (c) bijective (d) f is not defined
{(1, 2)}, then R is
(a) reflexive (b) transitive 11. Let f : R R be defined by f (x) 3x 5 and
2
(c) symmetric (d) None of these x
g : R R by g (x) . Then g o f (x) is
x 1
2
5. Let us define a relation R in R as aRb if a b . Then
R is
3x 2 5 3x 2 5
(a) (b)
(a) An equivalence relation 9 x 4 30 x 2 26 9 x 4 6 x 2 26
(b) Reflexive, transitive but not symmetric 3x 2 3x 2
(c) 4 (d)
(c) Symmetric, transitive but not reflexive x 2 x2 4 9 x 4 30 x 2 2
(d) Neither transitive nor reflexive but symmetric
12. Which of the following functions from Z into Z are
6. Let A = {1, 2, 3} and consider the relation R = {(1, 1), bijective?
(2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}. Then R is (a) f (x) x (b) f (x) x 2
3
(a) Reflexive but not symmetric
(c) f (x) 2 x 1 (d) f (x) x 1
2
(b) Reflexive but not transitive
(c) Symmetric and transitive
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13. Let f : R R be the function defined by 19. The number of equivalence relations in the set {1,
f (x) x 3 5 . Then f 1 (x) is 2, 3} containing the elements (1, 2) and (2, 1) is
1 1
[Term I, 2021 – 22]
(a) (x 5) 3
(b) (x 5) 3
1
(a) 0 (b) 1
(c) (5 x) 3
(d) 5 x (c) 2 (d) 3
14. Let f : A B and g : B C be the bijective 20. A relation R is defined on Z as aRb if and only if a2 –
functions. Then (g o f)-1 is
7ab + 6b2 = 0. Then, R is
1
(a) f o g 1 (b) f o g
[Term I, 2021 – 22]
1 1
(c) g o f (d) g o f
(a) Reflexive and symmetric
3
15. Let f : R R be defined by (b) Symmetric but not reflexive
5
(c) Transitive but not reflexive
3x 2
f (x) . Then
5x 3 (d) Reflexive but not symmetric
1
(a) f (x) f(x)
1
(b) f (x) f(x) 21. Let A = {1, 3, 5}. Then the number of equivalence
(c) (f o f ) x x relations in A containing (1, 3) is [2020]
1 (a) 1 (b) 2
(d) f 1 (x) f (x)
19 (c) 3 (d) 4
16. Let A = {3, 5}. Then number of reflexive relations
22. The relation R in the set {1, 2, 3} given by R = {(1,
on A is [2023]
2), (2, 1), (1, 1)} is [2020]
(a) 2 (b) 4
(a) Symmetric and transitive, but not reflexive
(c) 0 (d) 8
(b) Reflexive and symmetric, but not transitive
(c) Symmetric, but neither reflexive nor transitive
17. Let R be a relation in the set N given by
(d) An equivalence relation
R = {(a, b): a = b – 2, b > 6}. Then [2023]
(a) (8, 7) R (b) (6,8) R
(c) (3,8) R (d) (2, 4) R
18. A relation R is defined on N. Which of the following
is the reflexive relation?
[Term I, 2021 – 22]
(a) R {(x, y) : x y, x, y N}
(b) R {(x, y) : x y 10, x, y N}
(c) R = {(x, y): xy is the square number, x, y N }
(d) R {(x, y) : x 4 y 10, x, y N}
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Assertion-Reasoning (1 mark) d) A is false but R is true.
27. Let R be the relation in the set of integers Z
23. Consider the function f : R→ R defined as f(x) = x3 given by R = {(a, a) : 2 divides (a - a)}
Assertion (A): R is a reflexive relation.
Assertion (A): f(x) is a one - one function.
Reason (R): A relation is said to be reflexive. if x R x, ∀
Reason (R): f(x) is a one - one function if co – domain x∈z
= range.
a) Both A and R are true and R is the correct a) Both A and R are true and R is the correct
explanation of A. explanation of A.
b) Both A and R are true but R is not the correct b) Both A and R are true but R is not the correct
explanation of A.
explanation of A.
c) A is true but R is false.
c) A is true but R is false.
d) A is false but R is true.
d) A is false but R is true.
28. Assertion (A): 𝑓(𝑥) = 1 + 𝑥 2 is a one to one
function from 𝑅 + → 𝑅.
24. Assertion (A): A relation 𝑅 = {(𝑎, 𝑏): |𝑎 − 𝑏𝑙 < Reason (R): Every strictly monotonic function is a one
3} defined to
on the set 𝐴 = {1, 2, 3, 4} is reflexive. one function.
Reason (R): A relation R on the set A is said to be
reflexive if for (a, b) ∈ R and (b, c) ∈ R, we have (a, a) Both A and R are true and R is the correct
c) ∈ R. explanation of A.
b) Both A and R are true but R is not the correct
a) Both A and R are true and R is the correct explanation of A.
explanation of A. c) A is true but R is false.
b) Both A and R are true but R is not the correct d) A is false but R is true.
explanation of A.
c) A is true but R is false.
d) A is false but R is true. 29. Assertion (A): A function f: N → N be defined
𝑛
𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛
25. Assertion (A): The relation R in the set A = {1, 2, 3, 2
by 𝑓(𝑛) = {(𝑛+1)
4, 5, 6} defined as R = {(x, y) : y is divisible by x} is
2
𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑
not an equivalence relation. for all n ∈ N; is one - one.
Reason (R): The relation R will be an equivalence Reason (R): A function f: A → B is said to be injective
relation, if it is reflexive, symmetric and transitive. if a ̸= b then f(a) ̸= f(b).
a) Both A and R are true and R is the correct a) Both A and R are true and R is the correct
explanation of A. explanation of A.
b) Both A and R are true but R is not the correct b) Both A and R are true but R is not the correct
explanation of A. explanation of A.
c) A is true but R is false. c) A is true but R is false.
d) A is false but R is true. d) A is false but R is true.
26. Assertion (A): The relation R in the set A = (1, 2, 30. Assertion (A): The Relation R given by R = {(1, 3), (4,
3, 4) defined as R = {(x, y): y is divisible by x) is an 2), (2, 4), (2, 3), (3, 1) on set A = {1, 2, 3, 2} is
equivalence relation. symmetric.
Reason (R): A relation R on the set A is equivalence if Reason (R): For symmetric Relation
it is reflexive, symmetric and transitive. 𝑅 = 𝑅 −1
a) Both A and R are true and R is the correct
a) Both A and R are true and R is the correct explanation of A.
explanation of A. b) Both A and R are true but R is not the correct
b) Both A and R are true but R is not the correct explanation of A.
explanation of A. c) A is true but R is false.
c) A is true but R is false. d) A is false but R is true.
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31. Assertion (A):Every function is invertible.
Reason (R): Only bijective functions are invertible.
a) Both A and R are true and R is the correct
explanation of A.
b) Both A and R are true but R is not the correct
explanation of A.
c) A is true but R is false.
d) A is false but R is true.
32. Assertion (A): The Greatest Integer Function f:
R→ R, given by f(x) = [x]is one - one.
Reason (R): A function f: A → B is said to be injective
if f(a) = f(b) ⇒ a = b.
a) Both A and R are true and R is the correct
explanation of A.
b) Both A and R are true but R is not the correct
explanation of A.
c) A is true but R is false.
d) A is false but R is true.
Case Study Question
34. Read the text carefully and answer the questions:
Sherlin and Danju are playing Ludo at home during
33. Read the text carefully and answer the questions: Covid - 19. While rolling the dice, Sherlin’s sister
Students Raji observed and noted the possible outcomes of
of Grade 9, planned to plant saplings along straight the throw every time belongs to set {1, 2, 3, 4, 5,
lines, 6}. Let A be the set of players while B be the
parallel to each other to one side of the playground set of all possible outcomes.
ensuring
that they had enough play area. Let us assume that
they
planted one of the rows of the saplings along the line y
=
x - 4. Let L be the set of all lines which are parallel on
A = {S, D}, B = {1, 2, 3, 4, 5, 6}
the ground and R be a relation on L.
(a) Let R B→ B be defined by R = {(x, y): y is divisible by
x} is
a) Not reflexive but symmetric and transitive
b) Equivalence
c) Reflexive and transitive but not symmetric
d) Reflexive and symmetric and not transitive
(b) Raji wants to know the number of functions from A
to B. How many number of functions are possible?
a) 6! b) 62
c) 212 d) 26
(c) Let R be a relation on B defined by R = {(1,2), (2,2),
(1,3), (3,4), (3,1), (4,3), (5,5)}. Then R is:
a) Transitive b) Reflexive
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c) None of these d) Symmetric 41. Show that the function 𝑓: 𝑅 → 𝑅 given by 𝑓(𝑥) =
𝑥 3 + 1 for all 𝑥 ∈ 𝑅 is bijective.
(d) Raji wants to know the number of relations possible
from A to B. How many numbers of relations are 42. Check whether the relation R defined on the set
possible? 𝐴 = {1,2,3,4,5,6} 𝑎𝑠 𝑅 = {(𝑎, 𝑏): 𝑏 = 𝑎 + 1} is
reflexive, symmetric or transitive.
a) 212 b) 6!
c) 62 d) 26
43. Let 𝐴 = {𝑥 ∈ 𝑍: 0 ≤ 𝑥 ≤ 12}. Show that that 𝑅 =
(e) Let R : B→ B be defined by R = {(1,1), (1,2), (2,2), {(𝑎, 𝑏): 𝑎, 𝑏 ∈ 𝐴, |𝑎 − 𝑏| is divisible by 4} is an
(3,3), (4,4), (5,5), (6,6)}, then R is: equivalence relation. Find the set of all elements
related to 1. Also write the equivalence class .
a) Symmetric
b) Transitive and symmetric 44. Show that 𝑓: 𝑁 → 𝑁, given by 𝑓(𝑥) =
c) Reflexive and Transitive 𝑥 + 1, 𝑖𝑓 𝑥 𝑖𝑠 𝑜𝑑𝑑
{ is one –one and onto.
d) Equivalence 𝑥 − 1, 𝑖𝑓 𝑥 𝑖𝑠 𝑒𝑣𝑒𝑛
45. State the reason why the relation 𝑅 = {(𝑎, 𝑏) ∶
Questions 𝑎 ≤ 𝑏 2 } on the set R of real numbers is not
reflexive.
35. Check whether the relation R defined on the set A
46. Show that the relation R in the set N x N defined by
= {1, 2, 3, 4, 5, 6} as
(a, b)R(c, d) if a2 + d2 = b2 + c2∀ a, b, c, d ∈ N, is an
R = {(a, b) : b = a + 1} is reflexive, symmetric or equivalence relation.
transitive. [2019]
47. Show that the relation S defined on set
𝑁 𝑥 𝑁 𝑏𝑦 (𝑎, 𝑏)𝑆(𝑐, 𝑑) 𝑖𝑓 𝑎 + 𝑑 = 𝑏 + 𝑐 is
36. Show that the relation S in the set an equivalence relation.
A {x Z : 0 x 12} given by
48. Let N denote the set of all natural numbers and R
S {(a, b) : a, b Z | a b | is divisible by 3} is an be the relation on 𝑁 𝑥 𝑁 defined by
(𝑎, 𝑏)𝑅(𝑐, 𝑑) 𝑖𝑓 𝑎𝑑(𝑏 + 𝑐) = 𝑏𝑐(𝑎 + 𝑑). Show
equivalence relation. [Al 2019]
that R is an equivalence relation.
𝑥2
37. If 𝑦 = 𝑓(𝑥) = , is the function one-one and 49. Prove that the relation R in the set
1+𝑥 2
onto provided 𝑓: 𝑅 → 𝑅?
𝐴 = {1, 2, 3, 4, 5} given by
𝑅 = {(𝑎, 𝑏) ∶ |𝑎 − 𝑏| is even} is an equivalence
38. Give an example of relation R on 𝐴 = {1, 2, 3} relation.
which is reflexive but neither symmetric nor
transitive.
50. Let 𝐴 = 𝑅 − {3}, 𝐵 = 𝑅 − {1}. Let 𝑓: 𝐴 → 𝐵
39. Is the function 𝐹: 𝑍 → 𝑍 such that 𝑓(𝑥) = 𝑥 2 + 𝑥 defined by
injective, surjective or bijective? 𝑥−2
𝑓(𝑥) = 𝑥−3 ∀𝑥 ∈ 𝐴. Then show that 𝑓 is bijective.
40. If 𝑓: 𝑅 → 𝑅 be the function defined by 𝑓(𝑥) =
2
4𝑥 3 + 7, show that 𝑓(𝑥) is bijection. 51. Show that the function f in 𝐴 = 𝑅 − {3}defined as
4𝑥+3
𝑓(𝑥) = 6𝑥−4 is one-one and onto.
52. Let 𝑓: 𝑅 → R be defined by (i) 𝑓(𝑥) = 𝑥 + |𝑥| (ii)
𝑓(𝑥) = 𝑥 +1. Determine whether or not 𝑓 is onto.
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58. Let N be the set of all natural numbers and R be a
53. Write the domain of the relation R defined on the relation in N defined by 𝑅 =
set Z of integers as follows: (𝑎, 𝑏) ∈ 𝑅 ⇔ 𝑎2 + {(𝑎, 𝑏): 𝑎 𝑖𝑠 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑏}, then show that R is
𝑏 2 = 25 reflexive and transitive but not symmetric.
54. Let 𝑓 ∶ 𝑅 → R be defined by 𝑓(𝑥) = 𝑥 2 + 1. Find 59. Classify the following functions as injective,
the pre-image of 17 and (-3). surjective or bijective. (i) 𝑓: 𝑅 → 𝑅 Rdefined by
𝑓(𝑥) = 𝑠𝑖𝑛𝑥 (ii) 𝑓: 𝑅 → 𝑅 defined by
1 𝑓(𝑥) = 𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 2 𝑥
2
55. Show that 𝑓: 𝑅 + → 𝑅 + defined by 𝑓(𝑥) = 2𝑥 is
bijective, where 𝑅 + is the set of all non zero
60. If A = {1, 2, 3, 4], define relations on A which have
positive real number.
properties of being: (i) reflexive, transitive but not
symmetric. (ii) symmetric but neither reflexive nor
𝜋 𝜋
56. Let 𝐴 = {𝑥: 𝑥 ∈ 𝑅, − 2 ≤ 𝑥 ≤ 2 } and 𝐵 = {𝑦: 𝑦 ∈ transitive. (iii) reflexive, symmetric and transitive.
𝑅, −1 ≤ 𝑦 ≤ 1}. Show that the function 𝑓: 𝐴 → 𝐵
such that 𝑓(𝑥) = 𝑠𝑖𝑛𝑥 is bijective. 61. Show that the function 𝑓: 𝑅 → 𝑅 such that 𝑓(𝑥) =
1,𝑖𝑓 𝑥 𝑖𝑠 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙
{ −1,𝑖𝑓 𝑥 𝑖𝑠 𝑖𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 is many one and not onto. Find
57. Show that the function 𝑓: 𝑅 → 𝑅 defined by 𝑥 3 + 1
(i) 𝑓 (2) (ii) 𝑓(√2) (iii) 𝑓 (iv) 𝑓 (2 + √3)
𝑥 is a bijection.
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Sure Shot Questions
Chapter – 02
Inverse Trigonometric Functions
MCQ (1 mark) 2
7. If cos sin 1 cos 1 x 0 , then x is equal to
1. Which of the following is the principal value branch 5
of cos-1x? 1 2
(a) (b)
5 5
(a) , (b) 0,
2 2 (c) 0 (d) 1
(c) 0, (d) 0,
2 8. The value of sin(2 tan 1 (0.75)) is equal to
2. Which of the following is the principal value branch (a) 0.75 (b) 1.5
(c) 0.96 (d) sin 1.5
of cossec1 x ?
(a) , (b) 0, 3
2 2 2 9. The value of cos 1 cos is equal to
2
(c) , (d) , {0} 3
2 2 2 2 (a) (b)
2 2
5 7
3. If 3tan 1 x cot 1 x , then x equals (c) (d)
2 2
(a) 0 (b) 1
1
1 10. The value of expression 2sec1 2 sin 1 is
(c) -1 (d) 2
2
5
(a) (b)
6 6
33
4. The value of sin 1 cos is 7
5 (c) (d) 1
6
3 7 4
(a) (b) 11. If tan 1 x tan 1 y 1 1
, then cot x cot y
5 5 5
equals
(c) (d)
10 10 2
(a) (b)
5 5
5. The domain of the function cos1 2 x 1 is 3
(c) (d)
(a) [0, 1] (b) [-1, 1] 5
(c) (-1, 1) (d) [0, ] 2a 1 1 a 2x
2
12. If sin 1 2
cos 2
tan 1 2
1 a 1 a 1 x
6. The domain of the function defined by
, where a, x ]0,1[ , then the value of x is
f (x) sin 1 x 1 is
a
(a) [1, 2] (b) [-1, 1] (a) 0 (b)
2
(c) [0, 1] (d) none of these
2a
(c) a (d)
1 a2
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7 19. In the given question, a statement of assertion (A)
13. The value of cot cos 1 is is followed by a statement of reason (R). Choose
25
the correct answer out of the following choices.
25 25
(a) (b) Assertion (A): The domain of the function sec-12x
24 7
1 1
24 7 is , , .
(c)
25
(d)
24 2 2
1 2 Reason (R): sec1 2
14. The value of expression tan cos 1 is 4
2 5 (a) Both A and R are true and R is the correct
(a) 2 5 (b) 5 2 explanation of A.
(b) Both A and R are true but R is not the correct
52
(c) (d) 5 2 explanation of A.
2
(c) A is true but R is false.
(d) A is false but R is true. [2022 – 23]
2x
15. If | x | 1, t8hen 2 tan 1 x sin 1 2
is equal 1
1 x 20. sin sin 1 is equal to [Term I, 2021 –
3 2
to
22]
(a) 4 tan 1 x (b) 0
1 1
(c) / 2 (d) (a) (b)
2 3
(c) -1 (d) 1
16. If cos1 cos1 cos1 3 , then
( ) + ( ) ( ) equals 21. sin(tan 1x) , where |x| < 1, is equal to
(a) 0 (b) 1
x 1
(c) 6 (d) 12 (a) (b)
1 x2 1 x2
17. The number of real solutions of the equation 1 x
(c) (d)
1 x2 1 x2
1 cos 2 x 2 cos (cosx)in , is
1
[Term I, 2021 – 22]
2
(a) 0 (b) 1
(c) 2 (d) infinite 22. Simplest form of
1 cos x 1 cos x 3
tan 1 , x is
1 cos x 1 cos x 2
18. If cos1 x sin 1 x , then [Term I, 2021 – 22]
1
x 1 (b) 0 x
1 x 3 x
(a) (a) (b)
2 2 4 2 4 2
1 x x
(c) 1 x (d) x 0 (c) (d)
2 2 2
23. If tan 1 x y , then [Term I, 2021 – 22]
(a) 1 y 1 (b) y
2 2
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(c)
y
(d) y
, .
28. The principal value of cot 1 3 is
2 2 2 2 [2020]
(a) (b)
1
24. sin sin 1 is equal to [2023] 6 6
3 2 2 5
(c) (d)
1 3 6
(a) 1 (b)
2
1 1 3
(c) (d) 29. tan 1 3 tan 1 tan 1 is valid for what
3 4 1 3
values of ? [2020]
3 1 1 1
25. If f (x) | cosx |, then f is (a) , (b)
4 3 3 3
[2023] 1
(a) 1 (b) -1 (c) (d) All real values of
3
1 1
(c) (d)
2 2 3
30. The principal value of tan 1 tan is
5
26. Two statements are given, one labelled Assertion [2020]
(A) and the other labelled Reason (R). Select the 2 2
correct answer from the codes (a), (b), (c) and (d) (a) (b)
5 5
as given below.
3 3
Assertion (A): All trigonometric functions have (c) (d)
5 5
their inverse over their respective domains.
Reason (R): The inverse of tan-1x exists for some
x . [2023] Assertion-Reasoning (1 mark)
(a) Both Assertion (A) and Reason (R) are true
and Reason (R) is the correct explanation of
Assertion (A).
31. Assertion (A): Function f : R → R given by f(x) = sin
x is not a bijection.
(b) Both Assertion (A) and Reason (R) are true
Reason (R): A function f : A → B is said to be bijection if
and Reason (R) is not the correct explanation it is one - one and onto.
of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false. a) Both A and R are true and R is the correct
(d) Assertion (A) is false but Reason (R) is true. explanation of A.
b) Both A and R are true but R is not the correct
explanation of A.
13
27. The value of sin 1 cos is c) A is true but R is false.
5 d) A is false but R is true.
[Term I, 2021 – 22]
3
(a) (b) 32. Assertion (A): Principal value of 𝑡𝑎𝑛−1 (−√3) 𝑖𝑠 −
5 10 𝜋
.
3 3
𝜋 𝜋
(c) (d) Reason (R): 𝑡𝑎𝑛−1 1: 𝑅 → (− 2 , 2 )
5 10
so for any x ∈ R,
𝜋 𝜋
tan - 1(x) represent an angle in (− 2 , 2 ).
a) Both A and R are true and R is the correct
explanation of A.
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b) Both A and R are true but R is not the correct
explanation of A.
c) A is true but R is false.
d) A is false but R is true.
33. Assertion (A): Domain of 𝑓(𝑥) = 𝑠𝑖𝑛−1 𝑥 +
𝑐𝑜𝑠 𝑥 𝑖𝑠 [− 1,1].
Reason (R): Domain of a function is the set of all
possible values for which function will be defined.
a) Both A and R are true and R is the correct
explanation of A.
b) Both A and R are true but R is not the correct
explanation of A.
c) A is true but R is false.
d) A is false but R is true.
34. Assertion (A): We can write
𝑠𝑖𝑛−1 𝑥 = (𝑠𝑖𝑛 𝑥 −1 ).
Reason (R): Any value in the range of principal value
branch
is called principal value of that inverse
trigonometric function.
a) Both A and R are true and R is the correct
explanation of A.
b) Both A and R are true but R is not the correct
explanation of A.
c) A is true but R is false.
d) A is false but R is true. 36. Read the text carefully and answer the questions:
The Government of India is planning to fix a hoarding
Case Study Question board at the face of a building on the road of a busy
market for awareness on COVID - 19 protocol. Ram,
Robert, and Rahim are the three engineers who are
35. Read the text carefully and answer the working on this project. A is considered to be a
questions: of Suresh is standing at the top of a person viewing the hoarding board 20 metres away
building of 100 from the building, standing at the edge of a pathway
m height. Suresh is new for car driving so his father nearby. Ram, Robert and Rahim suggested to the firm
wants to monitor speed of car from the top of the to place the hoarding board at three different
building. locations namely C, D and E. C is at the height of 10
metres from
the ground level. For the viewer, A, the angle of
elevation of “D” is double the angle of elevation of C
The angle of elevation of E is triple the angle of
elevation of C for the same viewer. Look at the figure
given:
Suresh started his car from point Q as shown in the
picture when he completes the distance of 100 m, the
angle of elevation of the car from the top of the
building is 𝛼° . After 15 sec from the original point Q
and reaching at B the angle of elevation changes to β◦ .
(we take BQ = 100 √ 3m)
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1 1
43. Find the value of 𝑐𝑜𝑠 −1 (2) + 2 𝑠𝑖𝑛−1 (2).
44. Find the domain and range of 𝑠𝑖𝑛−1 𝑥.
45. Find the number of solutions of the equation
11𝜋
2𝑐𝑜𝑠 −1 𝑥 + 𝑠𝑖𝑛−1 𝑥 = , 𝑖𝑓 𝑠𝑖𝑛−1 𝑥 + 𝑐𝑜𝑠 −1 𝑥
6
𝜋
=
2
−√3 𝜋
46. Evaluate: 𝑐𝑜𝑠 {𝑐𝑜𝑠 −1 ( 2
) + }.
6
3 4
47. Simplify :𝑐𝑜𝑠 −1 (5 𝑐𝑜𝑠𝑥 + 5 𝑠𝑖𝑛𝑥).
𝑎−𝑥
48. write in the simplest form : 𝑡𝑎𝑛−1 √𝑎+𝑥 .
49. find the number of triplets (𝑥, 𝑦, 𝑧) satisfying the
equation
Questions 3𝜋
𝑠𝑖𝑛−1 𝑥 + 𝑠𝑖𝑛−1 𝑦 + 𝑠𝑖𝑛−1 𝑧 = 2
.
7
cos 1 cos
37. Evaluate: 3 50. Solve for 𝑥: 𝑐𝑜𝑠(𝑡𝑎𝑛−1 𝑥) =
3
𝑠𝑖𝑛 (𝑐𝑜𝑡 −1 ).
[2023] 4
9 9 1 1 9 1 2 2 51. Find the value of sin-1(𝑐𝑜𝑠
43𝜋
).
sin sin 5
8 4 3 4 3
38. Prove that: .
1 1
[2020] 52. Write the principle value of cos-1(2) + 2𝑠𝑖𝑛−1 (2)
1 1 31
2 tan 1 tan 1 tan 1 √1+𝑥 2 −1
39. Prove that: 2 7 17 . 53. Write the simplest form of 𝑡𝑎𝑛−1 [ 𝑥
].
[2020C]
40. Prove that: 54. Express in the simplest form:
12 3 56 𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥 𝜋 𝜋
cos1 sin 1 sin 1 . 𝑡𝑎𝑛−1 [ ],− < 𝑥 <
13 5 65 𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥 4 4
[Al 2019]
41. Prove that: 3
55. Write the value of sin(2𝑠𝑖𝑛−1 5).
1 x 1 x 1
tan 1 1
cos x −√3 𝜋
1 x 1 x 4 2 56. Evaluate: 𝑐𝑜𝑠 [𝑐𝑜𝑠 −1 ( 2
) + 6]
1
; x 1. [2019C] 57. Find the value of 𝑠𝑒𝑐 2 (𝑡𝑎𝑛−1 2) +
2
𝑐𝑜𝑠𝑒𝑐 2 (𝑐𝑜𝑡 −1 3).
42. Find the principal value branch of 𝑐𝑜𝑠 −1 𝑥.
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1 7𝜋
62. Evaluate: 𝜋 {216 𝑠𝑖𝑛−1 (𝑠𝑖𝑛 6
) +
58. Find the range of 𝑓(𝑥) = 𝑠𝑖𝑛−1 𝑥 + 𝑡𝑎𝑛−1 𝑥 + 2𝜋 5𝜋
27 𝑐𝑜𝑠 −1 (𝑐𝑜𝑠 ) + 28 𝑡𝑎𝑛 −1
(𝑡𝑎𝑛 )+
𝑠𝑒𝑐 −1 𝑥. 3 4
−𝜋
200𝑐𝑜𝑡 −1 (𝑐𝑜𝑡 4 )}
3𝜋
59. If 𝑐𝑜𝑠𝑒𝑐 −1 𝑥 + 𝑐𝑜𝑠𝑒𝑐 −1 𝑦 + 𝑐𝑜𝑠𝑒𝑐 −1 𝑧 = − 2
,
𝑥 𝑦 𝑧 5 3 63
find the 2 value of + + . 63. Show that 𝑠𝑖𝑛−1 13 + 𝑐𝑜𝑠 −1 5 = 𝑡𝑎𝑛−1 16.
𝑦 𝑧 𝑥
60. Find the value of tan (𝑐𝑜𝑠 −1 𝑥) and hence evaluate 64. What is the domain of the function defined by
8 𝑓(𝑥) = 𝑠𝑖𝑛−1 √𝑥 − 1 ?
tan (𝑐𝑜𝑠 −1 (17)).
65. Find the domain of the function 𝑐𝑜𝑠 −1 (2𝑥 − 1).
5𝜋
61. Find the value of tan−1 (𝑡𝑎𝑛 ) +
6
13𝜋 14𝜋
−1
𝑐𝑜𝑠 (𝑐𝑜𝑠 6 ). 66. Find the value of 𝑐𝑜𝑠 −1 (𝑐𝑜𝑠 3
)
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Sure Shot Questions
Chapter – 03
Matrices
MCQ (1 mark) (c) O (d) None of these
0 0 4
1. The matrix P 0 4 0 is
1 0 0
4 0 0
7. The matrix 0 2 0 is a
(a) square matrix (b) diagonal matrix 0 0 0
(c) unit matrix (d) None of these
(a) Identity matrix
(b) Symmetric matrix
2. Total number of possible matrices of order 3 x 3
(c) Skew symmetric matrix
with each entry 2 or 0 is
(d) None of these
(a) 9 (b) 27
(c) 81 (d) 512
0 5 8
2 x y 4 x 7 7 y 13
3. If , then the values 8. The matrix 5
0 12 is a
5x 7 4 x y x 6
8 12 0
of x, y respectively are
(a) 3, 1 (b) 2, 3 (a) Diagonal matrix
(c) 2, 4 (d) 3, 3 (b) Symmetric matrix
(c) Skew symmetric matrix
4. If A and B are two matrices of the order 3 x m and (d) Scalar matrix
3 x n, respectively, and m = n, then the order of
matrix (5A – 2B) is
(a) m x 3 (b) 3 x 3 9. If A is a matrix of order m x n and B is a matrix such
(c) m x n (d) 3 x n that AB’ and B’A are both defined, then order of
matrix B is
0 1 (a) m m (b) n n
5. If A , then A2 is equal to (c) n m (d) m n
1 0
0 1 1 0 10. If A and B are matrices of same order, then (AB’ –
(a) (b)
1 0 1 0 BA’) is a
0 1 1 0 (a) Skew symmetric matrix
(c) (d) (b) Null matrix
0 1 0 1
(c) Symmetric matrix
1 if i j
6. If matrix A [a ij ]22 , where aij (d) Unit matrix
0 if i j
then A2 is equal to
(a) I (b) A
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11. If A is a square matrix such that A2 = I, then 17. If A is a square matrix and A2 = A, then
(A I)3 (A I)3 7A is equal to (l A)2 3A is equal to [2023]
(a) A (b) I – A (a) l (b) A
(c) I + A (d) 3A (c) 2A (d) 3l
4 2
18. If A , then (A 2l)(A 3l ) is equal to
12. For any two matrices A and B, we have
1 1
(a) AB = BA (b) AB BA [Term I, 2021 – 22]
(c) AB = O (d) None of the above (a) A (b) l
(c) 5l (d) O
13. On using elementary column operation
C2 C2 2C1 in the following matrix equation 19. If order of matrix A is 2 x 3, of matrix B is 3 x 2, and
1 3 1 1 3 1 of matrix C is 3 x 3, then which one of the following
2 4 0 1 2 4 , we have; is not defined? [Term I, 2021 – 22]
(a) C(A + B’) (b) C(A + B’)’
1 5 1 1 3 5
(a) (c) BAC (d) CB + A’
0 4 2 2 2 0
1 5 1 1 3 5
(b) 2 1 1 1
0 4 0 1 0
20. If A 1 1 1 , then A5 A4 A3 A2 is
1 5 1 3 3 1
(c) 4 1 1 1
2 0 0 1 2
1 5 1 1 3 5 equal to [Term I, 2021 – 22]
(d) 0 (a) 2A (b) 3A
2 0 0 1 2
(c) 4A (d) O
1 0 x 0 21. If A is a square matrix such that A2 = A, then
14. If A , B 2
1 1 and A = B , then x
2 1 (l A)3 A is equal to [2020]
equals [2023]
(a) l (b) O
(a) 1 (b) -1 (c) l A (d) l A
(c) 1 (d) 2
3
1 1 1 x 6
22. If A [2 3 4], B 2 , X = [1 2 3] and
15. If 0 1 1 y 3 , then the value of (2x + y
2
0 0 1 z 2
2
– z ) is [2023]
Y 3 , then AB + XY equals
1 2 4 4
16. If x y , then [2023]
[2020]
2 5 9
(a) [28] (b) [24]
(a) x = 1, y = 2 (b) x = 2, y = 1
(c) 28 (d) 24
(c) x = 1, y = -1 (d) x = 3, y = 2
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23. If for a square matrix A, A2 A l O , then A-1 [Term I, 2021 – 22]
equals [2023] (a) A (b) l + A
(a) A (b) A l (c) l – A (d) l
(c) l A (d) A l
30. Given that A and A2 3l , the
24. If A [a ij ] is a skew – symmetric matrix of order n,
then [2022 – 23] [Term I, 2021 – 22]
(a) 1 0
2
1
(a) aij i, j (b) aij 0 i, j
a ji (b) 1 0
2
(c) aij 0 , where i j (c) 3 0
2
(d) aij 0 where i j (d) 3 0
2
2a b a 2b 4 3 31. If A, B are non – singular square matrices of the
25. If , then value of a
5c d 4c 3d 11 24
1 1
same order, then (AB ) =
+ b – c + 2d is [Term I, 2021 – 22] [2022 – 23]
(a) 8 (b) 10 (a) A1B (b) A1B 1
(c) 4 (d) -8
(c) BA1 (d) AB
26. Given that matrices A and B are of order 3 x n and
m x 5 respectively, then the order of matrix C = 5A 1 1 0 2 2 4
32. If A 2 3 4 and B 4 2 4 , then
+ 3B is [Term I, 2021 – 22]
(a) 3 x 5 and m = n (b) 3 x 5 0 1 2 2 1 5
(c) 3 x 3 (d) 5 x 5 [Term I, 2021 – 22]
1
(a) A B (b) A1 6B
1
27. If A [a ij ] is a square matrix of order 2 such that (c) B 1 B (d) B 1 A
6
1, when i j
aij , then A2 is
0, when i j 5 x
33. If A and A AT , where AT is the
[Term I, 2021 – 22]
y 0
1 0 1 1 transpose of the matrix A, then [2023]
(a) (b)
1 0 0 0 (a) x = 0, y = 5 (b) x = y
1 1 1 0 (c) x + y = 5 (d) x = 5, y = 0
(c) (d)
1 0 0 1
34. If a matrix A = [1 2 3], then the matrix AA’ (where
0 2 0 3a
28. If A and kA , then the
A’ is the transpose of A) is [2023]
3 4 2b 24 1 0 0
values of k, a and b respectively are
(a) 14
(b) 0 2 0
[Term I, 2021 – 22]
0 0 3
(a) -6, -12, -18 (b) -6, -4, -9
(c) -6, 4, 9 (d) -6, 12, 18 1 2 3
(c) 2 3 1
(d) [14]
2
29. If A is square matrix such that A = A, then 3 1 2
(l A)3 7A is equal to
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40.
35. If P is a 3 x 3 matrix such that P’ = 2P + l, where P’ is
the transpose of P, then
[Term I, 2021 – 22]
(a) P l (b) P l
(c) P 2l (d) P 2l
cos sin
36. If A and A A ' l , then the
sin cos
value of is [Term I, 2021 – 22]
(a) (b)
6 3
3 Case Study Question
(c) (d)
2
Assertion-Reasoning (1 mark) 41. Read the text carefully and answer the questions:
Two farmers Ramakishan and Gurucharan Singh
37. cultivate only three
varieties of rice namely Basmati, Permal and Naura.
The sale (in rupees) of these varieties of rice by both
the farmers in the month of September and October
are given by the following matrices A and B.
38.
39.
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42. Read the text carefully and answer the questions:
A trust fund has |35000 that must be invested in
two different types of bonds, say X and Y. The first
bond pays 10% interest 2 p.a. which will be given
to an old age home and second one pays 8%
interest p.a. which will be given to WWA (Women
Welfare Association). Let A be a 1× 2 matrix and B
be a 2 × 1 matrix, representing the investment and
interest rate on each bond respectively.
Questions
3 2 1 0
1. If A and l , find scalar k
1 1 0 1
so that A l kA .
2
[2020]
2 0 1
2. If A 2 1 3 , find A 5 A 4l and
2
1 1 0
hence find a matrix such that
A2 5 A 4l X O . [Delhi 2015]
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1 3 2 12. If A is a square matrix such that 𝐴2 = 𝐼, then find
the simplified value of (𝐴 − 𝐼)3 + (𝐴 + 𝐼)3 − 7𝐴.
3. If A 2 0 1 , then show that
1 2 3
A3 4 A2 3 A 11l O . Hence find A-1. 13. If A is a square matrix such that A2 = A, then write
the value of
[2020]
0 6 5x 7A − (𝐼 +
4. If the matrix 3
x 3
2
is symmetric, find 𝐴) , 𝑤ℎ𝑒𝑟𝑒 𝐼 𝑖𝑠 𝑎𝑛 𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦 𝑚𝑎𝑡𝑟𝑖𝑥
x
the value of x. [2021]
𝑎 + 4 3𝑏 2𝑎 + 2 𝑏 + 2
14. If[ ]=[ ], write the
1 5 8 −6 8 𝑎 − 8𝑏
5. For the matrix A , verify that value of 𝑎 − 2𝑏.
6 7
[2020C]
(i) (A A') is a symmetric matrix. 15. If 𝑚𝑎𝑡𝑟𝑖𝑥 𝐴 = [
3 −3
] and 𝐴2 = 𝜆𝐴, then write
−3 3
(ii) (A A') is a skew – symmetric matrix. the value of 𝜆.
4 1 16. Find the value of X and Y if
6. If 𝑋 = [ ], show that 6𝑋 − 𝑋 2 = 9𝐼,
−1 2 2 3 6 5
where 𝐼 is the unit matrix. 𝑋+𝑌 =[ ],𝑋 − 𝑌 = [ ].
5 1 7 3
7. Find matrices 𝑋 𝑎𝑛𝑑 𝑌 𝑖𝑓 2𝑋 + 𝑌 = 2 0 1
2 −3 1 0 3 5 17. If A= [2 1 3], find A2−5𝐴 + 4𝐼 and hence find
[ ] 𝑎𝑛𝑑 𝑋 − 𝑌 = [ ]
1 2 3 −2 −4 1 1 −1 0
a matrix X such that
A2−5𝐴 + 4𝐼 + 𝑋 = 0.
8. Find the value of 𝜆, a non-zero scalar, if
1 3 5 2 5 6 7 19 27 0 𝑎 𝑏
𝜆[ ] + 2[ ]=[ ] 1 1
2 4 6 1 3 5 8 18 28 18. Find 2 (𝐴 + 𝐴′)and 2 (𝐴 − 𝐴′). If A= [−𝑎 0 𝑐]
−𝑏 −𝑐 0
2 0 1
9. If 𝐴 = [2 1 3],then find (𝐴2 − 5𝐴). 2 4 −6
1 −1 0 19. Express the matrix A = [7 3 5 ] as the sum of
1 −2 4
a symmetric and skew symmetric matrix.
2 −1
10. Find matrix A such that [ 1 0 ]𝐴 = 3 −2 −4
20. Express the matrix A = [ 3 −2 −5] as the sum
−3 4
−1 −8 −1 1 2
[ 1 −2]. of a symmetric and skew symmetric matrices and
9 22 verify your result.
11. find a matrix A such that 2𝐴 − 3𝐵 + 5𝐶 = 0,
−2 2 0 21. Show that the matrix BAB is symmetric or skew-
where𝐵 = [ ] 𝑎𝑛𝑑 𝐶 =
3 1 4 symmetric accordingly when A is symmetric or
2 0 −2
[ ]. skew-symmetric.
7 1 6
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0 2𝑦 𝑧
27. Find x, y and z, if 𝐴 = [𝑥 𝑦 −𝑧] satisfies
1 1 1 𝑥 −𝑦 𝑧
22. If A = [1 1 1] then prove that An = 𝐴′ = 𝐴−1 .
1 1 1
3𝑛−1 3𝑛−1 3𝑛−1
[3𝑛−1 3𝑛−1 3𝑛−1 ] , then n∈ 𝑁.
3𝑛−1 3𝑛−1 3𝑛−1 1 0 2
28. If 𝐴 = [0 2 1] and 𝐴3 − 6𝐴2 + 7𝐴 +
2 0 3
2 𝑘𝑙3 = 𝑂, then find k.
23. If [𝑥 −2 4𝑥 𝑥 2 ] = [ −3 1
], then find x.
𝑥 𝑥3 −𝑥 + 2 1
2 −1 2
29. If 𝐴 = [3 √3 2]and 𝐵 = [ ],
4 2 0 1 2 4
𝑝 𝑞 then verify that (i) (A’)’=A(ii) (A+B)’=A’+B’ (iii)
24. Express [ ] as the sum of a symmetric and (kB)’=kB’, where k is any constant.
𝑟 𝑠
a skew-symmetric matrix.
30. For What value of 𝑥, is the matrix
25. If A = diag [2 -1 3] and 6 = diag [3 0 -1], then 0 1 −2
find 4A + 2B. [−1 0 3 ], a skew-symmetric matrix?
𝑥 −3 0
26. If 𝐴 = [𝑎𝑖𝑗 ] is a matrix given
4 −2 1 3
by[ 5 7 9 6 ], then write the order
21 15 15 −25
of A. Also, show that 𝑎32 = 𝑎23 + 𝑎24 ..
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Sure Shot Questions
Chapter – 04
Determinants
MCQ (1 mark) 6. If A, B and C are angles of a triangle, then the
1 cosC cos B
2 x 5 6 2 determinant cos C 1 cos A
1. If , then value of x is
8 x 7 3 cos B cos A 1
(a) 3 (b) 3 (a) 0 (b) -1
(c) 6 (d) 6 (c) 1 (d) None of these
a b bc a cos t t 1
2. The value of determinant b c c a b is f (t)
7. Let f (t) 2sin t t 2t , then lim 2 is equal
t 0 t
ca ab c sin t t t
(a) a3 b3 c3 (b) 3abc to
(c) a3 b3 c3 3abc (d) None of these (a) 0 (b) -1
(c) 2 (d) 3
3. The area of a triangle with vertices (-3, 0), (3, 0)
and (0, k) is 9 sq. units. The value of k will be 8. The maximum value of
(a) 9 (b) 3 1 1 1
(c) -9 (d) 6
1 sin 1 is ( is real number)
1
1 cos 1 1
b 2 ab b c bc ac
4. The determinant ab a a b b 2 ab equals 1 3
2
(a) (b)
bc ac c a ab a 2 2 2
(a) abc(b c)(c a)(a b) 2 3
(c) 2 (d)
(b) (b c)(c a)(a b) 4
(c) (a b c)(b c)(c a)(a b)
0 x a x b
(d) None of these
9. If f (x) x a 0 x c , then
5. The number of distinct real roots of xb xc 0
sin x cos x cos x (a) f(a) = 0 (b) f(b) = 0
cos x sin x cos x 0 in the interval (c) f(0) = 0 (d) f(1) = 0
cos x cos x sin x
2 3
x is 10. If A 0 2 5 , then A1 exists if
4 4
(a) 0 (b) 2 1 1 3
(c) 1 (d) 3 (a) 2 (b) 2
(c) 2 (d) None of these
11. If A and B are invertible matrices, then which of
the following is not correct?
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(a) adj A | A | .A
1
2
17. If A and | A | 27 , then the value of
3
1
(b) det(A) [det(A)]
1
2
(c) (AB) B A
1 1 1 is [Term I, 2021 – 22]
1 1 1
(a) 1 (b) 2
(d) (A B) B A
(c) 5 (d) 7
12. If x, y, z are all different from zero and
5 3 1
1 x 1 1
18. If 7 x 3 0 , then the value of x is
1 1 y 1 0 , then value of
9 6 2
1 1 1 z
[Term I, 2021 – 22]
x1 y 1 z 1 is (a) 3 (b) 5
1 1 1 (c) 7 (d) 9
(a) xyz (b) x y z
(c) x y z (d) -1
yk y y
19. The determinant y yk y is equal to
13. The value of the determinant y y yk
x x y x 2y
[Term I, 2021 – 22]
x 2y x x y is (a) k(3 y k )
2
(b) 3y k
3
x y x 2y x (c) 3y k
2
(d) k (3 y k)
2
(a) 9 x (x y) (b) 9 y (x y)
2 2
(c) 3 y (x y) (d) 7 x (x y)
2 2
1 2 3
14. The are two values of a which makes determinant, 20. The value of 2 2 3 3 4 4 is
1 2 5 3 4 5
2 a 1 86 , then sum of these number [Term I, 2021 – 22]
0 4 2a (a) 12 (b) -12
(c) 24 (d) -24
is
(a) 4 (b) 5
(c) -4 (d) 9
21. If A is a non – singular square matrix of order 3
such that A2 = 3A, then value of |A| is
2 7 1 [2020]
15. The value of the determinant 1 1 1 is (a) -3 (b) 3
(c) 9 (d) 24
10 8 1
[2023]
x 0 8
(a) 47 (b) -79
(c) 49 (d) -51 22. The roots of the equation 4 1 3 0 are
2 0 x
3 4 [2020C]
16. If 1 2 1 = 0, then the value of is (a) -4, 4 (b) 2, -4
(c) 2, 4 (d) 2, 8
1 4 1
[2023]
(a) 1 (b) 2 23. If A is a square matrix of order 3 and |A| = 5, then
(c) 3 (d) 4 the value of |2A’| is [2020]
(a) -10 (b) 10
(c) -40 (d) 40
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24. If A is a skew – symmetric matrix of order 3, then (a) 64 (b) 16
the value of |A| is [2020] (c) 0 (d) -8
(a) 3 (b) 0
(c) 9 (d) 27 31. If A is a square matrix of order 3, such that A (adj
A) = 10 l, then |adj A| is equal to [2020]
(a) 1 (b) 10
25. If A is a 3 x 3 matrix such that |A|= 8, then |3A| (c) 100 (d) 101
equals [2020]
(a) 8 (b) 24 2 4 2x 4
(c) 72 (d) 216 32. If , then the possible value(s) of ‘x’
5 1 6 x
is/are [2022 – 23]
4 3 (a) 3 (b) 3
26. The inverse of is
7 5 (c) - 3 (d) 3, 3
[Term I, 2021 – 22]
5 3 5 3
(a) (b) 33. If A is a square matrix of order 3, [A’] = -3, then
7 4 7 4 |AA’| = [2022 – 23]
5 7 5 3
(c) (d)
3 4 7 4
(a) 9 (b) -9
(c) 3 (d) –3
k 8
1 0 0 34. Value of k, for which A is a singular
27. If A 0 1 0 , then A-1 4 2k
matrix is [Term I, 2021 – 22]
59 69 1
[Term I, 2021 – 22] (a) 4 (b) -4
(a) is A (b) is (-A) (c) 4 (d) 0
(c) is A2 (d) does not exist
1 2 4 35. Given that A is a non – singular matrix of order 3
28. If A 2 1 3 is the adjoint of a square
such that A2 = 2A, then value of |2A| is
[Term I, 2021 – 22]
4 2 0 (a) 4 (b) 8
matrix B, then B-1 is equal to (c) 64 (d) 16
[Term I, 2021 – 22]
(a) A (b) 2A 36. Given that A [a ij ] is a square matrix of order 3 x
3
1 1
(c)
2
B (d)
2
A 3 and |A| = -7, then the value of a
i 1
i2 Ai 2 , where
29. If Aij denotes the cofactor of element aij is
1
1 tan 1 tan a b [Term I, 2021 – 22]
tan ,
1 tan 1
b a (a) 7 (b) -7
then [Term I, 2021 – 22] (c) 0 (d) 49
(a) a = 1 = b (b) a cos 2 , b sin 2
37. If A is a square matrix of order 3 and |A| = 5, then
(c) a sin 2 , b cos (d) a cos , b sin |adj A| = [2022 – 23]
(a) 5 (b) 25
2 0 0 1
30. If A 0 2 0 , then the value of |adj A|
(c) 125 (d)
5
0 0 2
is [2020]
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38. Given that A is a square matrix of order 3 and |A|
= -4, then |adj A| is equal to 43.
[Term I, 2021 – 22]
(a) -4 (b) 4
(c) -16 (d) 16
2 5
39. For matrix A , (adjA) ' is equal to
11 7
[Term I, 2021 – 22]
2 5 7 5
(a) (b)
11 7 11 2
7 11 7 5 44.
(c) (d)
5 2 11 2
3 1
40. For A , then 14A-1 is given by
1 2
[Term I, 2021 – 22]
2 1 4 2
(a) 14 (b)
1 3 2 6
2 1 3 1
(c) 2 (d) 2 45.
1 3 1 2
Assertion-Reasoning (1 mark)
41.
Case Study Question
46. Read the text carefully and answer the questions:
Three shopkeepers Sunil, Vinod and Neeraj are
using polythene bags, handmade bags (prepared
42. by prisoners) and newspapers envelope as carry
bags. It is found that the shopkeepers Sunil, Vinod
and Neeraj are using (20, 30, 40), (30, 40, 20) and
(40, 20, 30) polythene bags, handmade bags and
newspapers envelopes respectively. The
shopkeepers Sunil, Vinod and Neeraj spent |250,
|270 and |200 on these carry bags respectively.
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to award its 4, 1 and 3 students on the respective
values (by giving the same award money to the
three values as school A). The total amount of
award for one prize on each value is Rs 1200.
(a) What is the award money for Honesty?
a) Rs 350 b) Rs 300
c) Rs 400 d) Rs 500
(b) What is the award money for Punctuality?
a) Rs 300 b) Rs 500
c) Rs 280 d) Rs 450
(a) What is the cost of one polythene bag? (c) What is the award money for Hard work?
a) Rs 550 b) Rs 500
a) Rs 3 b) Rs 2 c) Rs 400 d) Rs 300
c) Rs 5 d) Rs 1
(d) If a matrix P is both symmetric and skew -
(b) What is the cost of one handmade bag? symmetric, then |P| is equal to
a) Rs 2 b) Rs 5 a) 0 b) None of these
c) Rs 1 d) Rs 3 c) 1 d) – 1
(c) What is the cost of one newspaper envelope? (e) If P and Q are two matrices such that PQ = Q and
QP = P, then |Q2| is equal to
a) Rs 2 b) Rs 5
c) Rs 1 d) Rs 3 a) 1 b) |P|
c) |Q| d) 0
(d) Keeping in mind the social conditions, which
shopkeeper is better?
a) Neeraj b) Sunil
Questions
c) Vinod d) None of these 2 3
48. For the matrix A , verify the
4 6
(e) Keeping in mind the environmental conditions,
following: A(adj A) (adjA) A | A | l .
which shopkeeper is better?
[2020C]
a) Sunil b) Neeraj
c) None of these d) Vinod
2 3 1
49. Given A , compute l and show that
4 7
47. Read the text carefully and answer the questions:
Two schools A and B want to award their selected 2 A1 9l A . [2018]
students on the values of Honesty, Hard work and
Punctuality. School A wants to award each, each 50. If A is a skew – symmetric matrix of order 3, then
and each for the three respective values to its 3, 2 prove that 𝑑𝑒𝑡𝐴 = 0.
and 1 students respectively with a total award
money of |2200. School B wants to spend Rs 3100
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51. Using matrix method, solve 𝑥 − 2𝑦 = 4 𝑎𝑛𝑑 − 1 2 1
3𝑥 + 5𝑦 = −7. 63. If 𝐴 = [−1 1 1], find 𝐴−1 . Hence solve the
1 −3 1
system of equations:
5 2 − 13 𝑥 + 2𝑦 + 𝑧 = 4, −𝑥 + 𝑦 + 𝑧 = 0, 𝑥 − 3𝑦 + 𝑧 = 4.
52. Let 𝐴 = [ ] 𝑎𝑛𝑑 𝐵 = [ ], find a matrix
3 −1 1
𝑋 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐴𝑋 = 𝐵.
1 −1 0 2 2 −4
64. If 𝐴 = [2 3 4]and 𝐵 = [−4 2 −4]are
3 −4 0 1 2 2 −1 5
53. If 𝐴 = [ ], find matrix B such that 𝐴𝐵 = 𝐼. square matrices, find 𝐴 ∙ 𝐵 and hence solve the
−1 2
system of equations:
𝑥 − 𝑦 = 3, 2𝑥 + 3𝑦 + 4𝑧 = 17 𝑎𝑛𝑑 𝑦 + 2𝑧 = 7.
1 5
54. If 𝐴 = [ ], 𝑓𝑖𝑛𝑑 𝑎𝑑𝑗 𝐴.
6 7 65. If a, b, c are real numbers, then prove that
𝑎 𝑏 𝑐
|𝑏 𝑐 𝑎| = −(𝑎 + 𝑏 + 𝑐)(𝑎 + 𝑏𝜔 + 𝑐𝜔2 )(𝑎 +
55. If A is a square matrix satisfying 𝐴′ 𝐴 = 𝐼, write the 𝑐 𝑎 𝑏
value of |A|. 𝑏𝜔2 + 𝑐𝜔)
56. If A is a 3x3 matrix, |𝐴| ≠0 |3𝐴|= 𝑘|𝐴|, then write 66. In a triangle ABC, if
the value of k. 1 1 1
| 1 + 𝑠𝑖𝑛 𝐴 1 + 𝑠𝑖𝑛 𝐵 1 + 𝑠𝑖𝑛 𝐶 | =
𝑠𝑖𝑛 𝐴 + 𝑠𝑖𝑛2 𝐴 𝑠𝑖𝑛 𝐵 + 𝑠𝑖𝑛2 𝐵 𝑠𝑖𝑛 𝐶 + 𝑠𝑖𝑛2 𝐶
0 then prove that 𝛥𝐴𝐵𝐶 is an isosceles triangle.
4 6
57. If A = ( ), then what is the value of A. (adj A) ?
7 5
67. A shopkeeper has 3 varieties of pen ‘A’, ‘B’ and ‘C’.
58. For what value of 𝑥, the given matrix A = Meenu purchased 1 pen of each variety for a total
3 − 2𝑥 𝑥 + 1 of Rs. 21. Jeevan purchased 4 pens of ‘A’ variety, 3
( )is singular matrix?
2 4 pens of ‘B’ variety and 2 pens of ‘C’ variety for Rs.
60. While Shikha purchased 6 pens of ‘A’ variety, 2
pens of ‘B’ variety and 3 pens of ‘C’ variety for Rs.
1+𝑥 7 70. Using matrix method, find cost of each variety
59. For what value of 𝑥, the matrix ( )is a
3−𝑥 8 of pen.
singular matrix?
68. A school wants to award its students for the values
of honesty, regularity and hard work with a total
60. If A,B are square matrices of the same order, then
cash award of Rs. 6,000. Three times the award
prove that
money for hard work added to that given for
𝑎𝑑𝑗(𝐴𝐵) = (𝑎𝑑𝑗𝐵)(𝑎𝑑𝑗𝐴).
honesty amounts to Rs. 11,000. The award money
given for honesty and hard work together is double
𝑐𝑜𝑠 𝛼 −𝑠𝑖𝑛 𝛼 0 the one given for regularity. Represent the above
61. If A = ( 𝑠𝑖𝑛 𝛼 𝑐𝑜𝑠 𝛼 0) find 𝑎𝑑𝑗 𝐴 and verify situation algebraically and find the award money
0 0 1 for each value, using matrix method. Apart from
that these values, suggest one more value which the
A(𝑎𝑑𝑗 𝐴) = (𝑎𝑑𝑗 𝐴) 𝐴 = |𝐴|𝐼3 . school must include for awards.
69. Find the equation of the line joining A(1, 3) and B(0,
3 2 1 0) using determinants and find the value of k if D(k,
62. If 𝐴 = [4 −1 2 ], then find 𝐴−1 and hence 0) is a point such that area of ∆ABD is 3 square
7 3 −3 units.
solve the following system of equation: 3𝑥 + 4𝑦 +
7𝑧 = 14. 2𝑥 − 𝑦 + 3𝑧 = 4. 𝑥 + 2𝑦 − 3𝑧 = 0
1 −2 3
70. If 𝐴 = [ 0 −1 4],find (𝐴′)−1
−2 2 1
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71. Find the adjoint of the matrix 𝐴 = 𝑐𝑜𝑠𝛼 −𝑠𝑖𝑛𝛼 0
−1 −2 −2 73. If 𝐴 = [ 𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼 0] find adj A and verify
[2 1 −2]and hence show that A. (adj 0 0 1
2 −2 1 that 𝐴(𝑎𝑑𝑗𝐴) = (𝑎𝑑𝑗𝐴)𝐴 = |𝐴|𝐼3 .
A)=|𝐴|𝑙3 .
Here
2 −3 5
2 −1 74. If A = -4 then find 𝐴 = [3 2 −4], then find 𝐴−1
72. If 𝐴 = [ ] and 𝐼 is the identity matrix of
−1 2 1 1 −2
order 2, then show that 𝐴2 = 4𝐴 − 3𝐼. Hence find Using 𝐴−1 , solve the following system of equations:
𝐴−1 . 2𝑥 − 3𝑦 + 5𝑧 = 11,3𝑥 + 2𝑦 − 4𝑧 = −5 𝑥 + 𝑦 −
2𝑧 = −3
For Solutions
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Sure Shot Questions
Chapter – 05
Continuity and Differentiability
MCQ (1 mark) (d) None of these
1
x2 6. If f (x) x 2 sin , where x 0 , then the value
1. If f(x) = 2x and g (x) 1 , then which of the x
2
of the function f at x = 0, so that the function is
following can be a discontinuous function?
continuous at x = 0, is
(a) f (x) g(x) (b) f (x) g(x)
(a) 0 (b) -1
g (x) (c) 1 (d) None of these
(c) f (x).g(x) (d)
f (x)
4 x 2 mx 1, if x 2
2. The function f (x) is 7. If f (x) , is continuous at
4 x x3 sin x n, if x
(a) Discontinuous at only one point 2
(b) Discontinuous at exactly two points x / 2 , then
(c) Discontinuous at exactly three points n
(a) m 1, n 0 (b) m 1
(d) None of these 2
m
(c) n (d) m n
2 2
3. The set of points where the function f given by f(x)
= |2x – 1| sinx is differentiable is 8. Let f(x) = |sinx|. Then
1 (a) f is everywhere differentiable
(a) R (b) R
2 (b) f is everywhere continuous but not
(c) (0, ) (d) None of these differentiable at x n , n Z
(c) f is everywhere continuous but not
4. The function f(x) = cotx is discontinuous on the set
differentiable at x (2 n 1) ,nZ
(a) {x n ; n Z} 2
(b) {x 2 n ; n Z} (d) None of these
(c) x (2 n 1) ;nZ 1 x2
dy
2 9. If y log 2
, then is equal to
n 1 x dx
(d) x ; nZ
2 4 x3 4 x
(a) (b)
1 x4 1 x4
5. The function f (x) e is 4 x3
|x|
1
(c) (d)
(a) Continuous everywhere but not differentiable 4 x4 1 x4
at x = 0
(b) Continuous and differentiable everywhere dy
10. If y sin x y , then is equal to
(c) Not continuous at x = 0 dx
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cos x cos x 17. The function f(x) = [x], where [x] is the greatest
(a) (b)
2 y 1 1 2 y integer function that is less than or equal to x, is
continuous at [Term I, 2021 – 22]
sin x sin x
(c) (d) (a) 4 (b) -2
1 2 y 2 y 1
(c) 1.5 (d) 1
11. The derivative of cos 1 2 x 2 1 w.r.t. cos-1 x is 18. The function f(x) = |x| is [2023]
1 (a) Continuous and differentiable everywhere.
(a) 2 (b) (b) Continuous and differentiable nowhere.
2 1 x2
(c) Continuous everywhere, but differentiable
2
(c) (d) 1 x 2 everywhere except at x = 0.
x (d) Continuous everywhere, but differentiable
nowhere.
d2y
12. If x t , y t , then
2 3
is
dx 2 19. The derivative of x 2 x w.r.t. x is [2023]
3 3
(a) x 2 x 1
2x
(a) (b) (b) 2 x log x
2 4t
(c) 2 x (1 logx) (d) 2 x (1 logx)
2x 2x
3 3
(c) (d)
2t 4
dy
20. If y (2 x) x , then
2 3
is equal to
13. The value of c in Rolle’s theorem for the function dx (1,1)
f(x) = x3 – 3x in the interval [0, 3] is [Term I, 2021 – 22]
(a) 1 (b) -1 (a) 2 (b) -2
3 1 (c) 3 (d) -3/2
(c) (d)
2 3
1 x2 for x 1
14. For the function f (x) x , x [1,3] , the value 21. The function f (x) is
x 2 x for x 1
of c for mean value theorem is [Term I, 2021 – 22]
(a) 1 (b) 3 (a) Not differentiable at x = 1
(c) 2 (d) None of these (b) Differentiable at x = 1
(c) Not continuous at x = 1
15. The function f(x) = [x], where [x] denotes the (d) Neither continuous nor differentiable at x = 1
greatest integer less than or equal to x, is
continuous at [2023] 1 x dy
22. If sec1 a , then is equal to
(a) x = 1 (b) x = 1.5 1 y dx
(c) x = -2 (d) x = 4 [2020C]
x 1 x 1
3x 8, if x 5 (a)
y 1
(b)
y 1
16. If the function f (x) is
2k , if x 5
y 1 y 1
Continuous, then the value of k is (c) (d)
x 1 x 1
[Term I, 2021 – 22]
(a) 2/7 (b) 7/2 dy
23. If y log(sine ) , then
x
(c) 3/7 (d) 4/7 is [2023]
dx
x
(a) cot e x (b) cos ec e
x x x x
(c) e cot e (d) e cos ec e
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1
24. If y tan (e ) , then
2x dy 1 cos 4 x
is equal to , if x 0
dx f (x) 8 x 2 is continuous at x
[Term I, 2021 – 22]
k , if x 0
2e 2 x 1 = 0 is [2022 – 23]
(a) (b)
1 e4 x 1 e4 x (a) 0 (b) -1
2 1 (c) 1 (d) 2
(c) 2 x (d) 2 x
e e2 x e e2 x
31. The value of k(k < 0) for which the function
d 2x 1 cos kx
25. If x A cos 4t B sin 4t , then is equal to x sin x , x 0
dt 2 f defined as f (x) is
[2023] 1, x0
(a) x (b) -x 2
(c) 16x (d) -16x continuous at x = 0 is [Term I, 2021 – 22]
(a) 1 (b) -1
2
d y 1 1
x
26. If y e , then is equal to (c) (d)
dx 2 2 2
[Term I, 2021 – 22]
(a) -y (b) y 32. The point(s), at which the function f given by
(c) x (d) -x x
,x 0
d y 2 f (x) | x | is continuous, is/are
27. If x t 1, y 2at , then
2
at t = a is 1, x 0
dx 2
[Term I, 2021 – 22] [Term I, 2021 – 22]
1 1 (a) x R (b) x = 0
(a) (b)
a 2a 2 (c) x R {0} (d) x = -1 and 1
1
(c) (d) 0
2a 2 dy
33. If e x e y e x y , then is
dx
1
28. If y sin(2sin x) , then (1 x 2 ) y2 is equal to [Term I, 2021 – 22]
[Term I, 2021 – 22] (a) e y x (b) e y x
(a) xy1 4 y (b) xy1 4 y (c) e y x (d) 2e x y
(c) xy1 4 y (d) xy1 4 y
dy
34. If y log(cose ) , then
x
is
dx
x2 d2y [Term I, 2021 – 22]
29. If y log e 2
, then equals [2020]
e dx 2 (a) cos e x 1 (b) e x cos e x
(d) e tan e
x x x x
1 1 (c) e sin e
(a) (b)
x x2
(c)
2
x2
2
(d) 2
x
35. The derivative of sin 1 2 1 x 2 w.r.t.
1
sin 1 x, x 1 , is [Term I, 2021 – 22]
30. The value of ‘k’ for which the function 2
(a) 2 (b) 2
2
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b) Both A and R are true but R is not the correct
(c) (d) -2 explanation of A.
2
c) A is true but R is false.
d) A is false but R is true.
d2y
36. If y 5cos x 3sin x , then is equal to
dx 2 41. Assertion (A): |sin x| is a continuous function.
[Term I, 2021 – 22] Reason (R): If f(x) and g(x) both are continuous
(a) -y (b) y functions, then g of (x) is also a continuous function.
(c) 25y (d) 9y a) Both A and R are true and R is the correct
explanation of A.
b) Both A and R are true but R is not the correct
d2y explanation of A.
37. If x a sec , y b tan , then 2
at is
dx 6 c) A is true but R is false.
[Term I, 2021 – 22] d) A is false but R is true.
3 3b 2 3b
(a) (b) 42. Assertion (A): Every differentiable function is
a2 a continuous but converse is not true.
3 3b b Reason (R): Function f(x) = |x| is continuous.
(c) (d) a) Both A and R are true and R is the correct
a 3 3a 2
explanation of A.
b) Both A and R are true but R is not the correct
Assertion-Reasoning (1 mark) explanation of A.
c) A is true but R is false.
d) A is false but R is true.
38. Assertion (A): f(x) = |x - 3| is continuous at x = 0.
Reason (R): f(x) = |x - 3| is differentiable at x = 0.
Case study [4 Marks]
a) Both A and R are true and R is the correct
explanation of A. Here, question 28(i) to (iii) is a case study based
b) Both A and R are true but R is not the correct question of 4 marks.
explanation of A.
c) A is true but R is false. 43. Let f(x) be a real valued function. Then its
d) A is false but R is true. Left Hand Derivative (L.H.D.):
f (a h) f(a)
Lf '(a) lim
39. Assertion (A): The function f(x) = |cos x| is h 0 h
continuous function. Right Hand Derivative (R.H.D.):
Reason (R): The function f(x) = cos |x| is a continuous
f (a h) f(a)
function. Rf '(a) lim
h 0 h
a) Both A and R are true and R is the correct Also, a function f(x) is said to be differentiable at
explanation of A. x = a if its L.H.D. and R.H.D. at x = a exist and
b) Both A and R are true but R is not the correct both are equal.
explanation of A.
| x 3 |, x 1
c) A is true but R is false.
For the function f (x) x 2 3x 13
d) A is false but R is true.
, x 1
4 2 4
40. Assertion (A): The function defined by f(x) = cos(x2) Answer the following questions:
is a continuous function. (i) What is R.H.D. of f(x) at x = 1?
Reason (R): The sine function is continuous in its (ii) What is L.H.D. of f(x) at x = 1?
domain i.e. x ∈ R.
(iii) Check if the function f(x) is differentiable at x =
a) Both A and R are true and R is the correct 1.
explanation of A.
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OR 1 x2 1 x2
51. Differentiate tan 1 with respect
(iii) Find f’(2) and f’(-1). [2023] 1 x 2 1 x 2
to cot 1 x 2 . [Al 2019]
Questions
52. Differentiate
44. Find the value(s) of ' ' , if the function 1 x2 1 2x
tan 1 w.r.t .sin 1 , if
sin 2 x x 1 x 2
, if x 0
f (x) x 2 is continuous at x = 0.
1, x (1,1) .
if x 0
[Foreign 2016, Delhi 2014]
[2023]
53. Find the relationship between a and b so that the
dy function ‘f’ defined by:
45. If y a a x , then find .
dx 𝑎𝑥 + 1, 𝑖𝑓 𝑥 ≤ 3
𝑓(𝑥) = { is continuous at 𝑥 = 3.
[2020C] 𝑏𝑥 + 3, 𝑖𝑓𝑥 > 3
54. Find the value of a, if the function 𝑓(𝑥) is defined
46. Find the values of p and q, for which by:
1 sin 3 x 2𝑥 − 1, 𝑥 < 2
2
, if x / 2 𝑓(𝑥) = { 𝑎, 𝑥 = 2 is continuous at 𝑥 = 2.
3cos x 𝑥 + 1, 𝑥 > 2
f (x) p, if x / 2 is continuous
q (1 sinx) 55. What is the value of k for which the function
, if x / 2 𝑠𝑖𝑛 2𝑥
( 2 x)
2 ,𝑥 ≠0
𝑓(𝑥) = { 5𝑥 is continuous at 𝑥 = 0?
at x / 2 . [Delhi 2016] 𝑘, 𝑥 = 0
56. Show that the function 𝑓(𝑥) = |𝑥 − 3|, 𝑥 ∈ 𝑅, is
47. Find the values of a and b, if the function f defined continuous but not differentiable at 𝑥 = 3.
x 2 3 x a, x 1
by f (x) 57. Find the value of a and b such that the following
bx 2, x 1
function 𝑓(𝑥) is a continuous function: (𝑥) =
Is differentiable at x = 1. [Foreign 2016] 5, 𝑥≤2
{𝑎𝑥 + 𝑏 2 < 𝑥 < 10 .
21, 𝑥 ≥ 10
dy
48. If y e (cosx) x , then find
2
x cos x
.
dx 58. Determine the value of the constant ‘k’ so that the
[2020] function
𝑘𝑥
y , 𝑖𝑓 𝑥<0
49. If log(x 2 y 2 ) 2 tan 1 , show that 𝑓(𝑥) = {|𝑥| is continuous at 𝑥 = 0.
x 3, 𝑖𝑓 𝑥 ≥ 0
dy x y
. [Delhi 2019] 59. Find the values of p and q, for which 𝑓(𝑥) =
dx x y
1−𝑠𝑖𝑛3 𝑥
3 𝑐𝑜𝑠2 𝑥
, 𝑖𝑓 𝑥 < 𝜋/2
dy 𝑝 , 𝑖𝑓 𝑥 = 𝜋/2 is continuous at x= /2.
50. If y (x) (cosx)sin x , then find
cos x
. 𝑞(1−𝑠𝑖𝑛 𝑥)
dx , 𝑖𝑓 𝑥 > 𝜋/2
{ (𝜋−2𝑥)2 }
[Al 2019]
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60. Show that the function 𝑓(𝑥) = 2𝑥 − |𝑥| is 𝑑𝑦
68. If (cos x)y = (cosy)x , then find 𝑑𝑥 .
continuous but not differentiable at x=0.
√1+𝑠𝑖𝑛𝑥 + √1−𝑠𝑖𝑛𝑥
61. Find the value of k, for which 69. If y = 𝑐𝑜𝑡 −1 [ ], then find the value
√1+𝑠𝑖𝑛 𝑥 − √1−𝑠𝑖𝑛𝑥
𝑑𝑦
of 𝑑𝑥
√1+𝑘𝑥−√1−𝑘𝑥
𝑥
, 𝑖𝑓 − 1 ≤ 𝑥 <0
𝑓(𝑥) = { 2𝑥+1
} is 70. If 𝑥 = 𝑎 𝑠𝑖𝑛 2𝑡 (1 + 𝑐𝑜𝑠 2𝑡) and 𝑦 =
𝑥−1
, 𝑖𝑓 0 ≤ 𝑥 < 1 𝑏 𝑐𝑜𝑠 2𝑡 (1 − 𝑐𝑜𝑠2𝑡), thyen show that
continuous at x = 0. 𝑑𝑦 𝑑𝑦
(𝑑𝑥 )𝑎𝑡 𝑡=𝜋/4 = 𝑏/𝑎. Also find the value of (𝑑𝑥 ) at t
𝜋
= 3.
62. Find the value of ‘a’ for which the function f
defined as
𝜋 𝑑𝑦
𝑎 𝑠𝑖𝑛 2 (𝑥 + 1), 𝑥 ≤ 0 71. If 𝑦 = 𝑥 𝑠𝑖𝑛𝑥 + (𝑠𝑖𝑛𝑥)𝑐𝑜𝑠 𝑥 , then find .
𝑑𝑥
𝑓(𝑥) = { 𝑡𝑎𝑛 𝑥−𝑠𝑖𝑛 𝑥 } is continuous
𝑥3
, 𝑥 > 0
72. Differentiate the following function with respect
at x =0. to
x: 𝑦 = (𝑠𝑖𝑛 𝑥)𝑥 + 𝑠𝑖𝑛−1 √𝑥
63. Examine the continuity of the following function:
𝑥
, 𝑥≠0 73. If 𝑥 = 𝑎𝑐𝑜𝑠 𝜃 + 𝑏 𝑠𝑖𝑛 𝜃 and 𝑦 = 𝑎 𝑠𝑖𝑛𝜃 − 𝑏𝑐𝑜𝑠𝜃,
2|𝑥|
𝑓(𝑥) = {1 } at x=0. 𝑑2𝑦 𝑑𝑦
, 𝑥 = 0 then show that 𝑦 2 −𝑥 + 𝑦 = 0.
2 𝑑𝑥 2 𝑑𝑥
𝑡
64. Find the values of k so that the function f is 74. If 𝑥 = 𝑎 (𝑐𝑜𝑠𝑡 + 𝑙𝑜𝑔 𝑡𝑎𝑛 2) , 𝑦 = 𝑎𝑠𝑖𝑛𝑡, then
continuous at the indicated point for: 𝑓(𝑥) = 𝑑2 𝑦 𝑑2 𝑦
𝑘 𝑐𝑜𝑠 𝑥 𝜋 find and
, 𝑖𝑓 𝑥 ≠ 2 𝑑𝑡 2 𝑑𝑥 2
𝜋−2𝑥 𝜋
{ 𝜋 } at x = 2 .
3, 𝑖𝑓 𝑥 = 2 𝑥+𝑒𝑥+..𝑡𝑜∞ 𝑑𝑦 𝑦
75. If 𝑦 = 𝑒 𝑥+𝑒 ,prove that 𝑑𝑥 = 1−𝑦.
65. Show that the function ‘f’ defined by f(x) =
3𝑥 − 2, 0 < 𝑥 ≤ 1
√𝑥−𝑥
{2𝑥 2 − 𝑥, 1 < 𝑥 ≤ 2} is continuous at x =2, but 76. Differentiate 𝑡𝑎𝑛−1 (1+𝑥3/2 )w.r.t.x.
5𝑥 − 4, 𝑥 > 2
not differentiable. 𝑑2 𝑦
77. If 𝑦 = 𝐴𝑒 −𝑘𝑡 𝑐𝑜𝑠 (𝑝𝑡 + 𝑐), prove that 𝑑𝑡 2 +
𝑑𝑦
66. Show that the function 𝑓(𝑥) = |𝑥 − 1| + 2𝑘 𝑑𝑡 + 𝑛2 𝑦 = 0, where 𝑛2 = 𝑝2 + 𝑘 2
|𝑥 + 1|, for all x 𝜖 𝑅, is not differentiable at the
points x= -1 and x = 1.
𝑑𝑦 𝑙𝑜𝑔 𝑥
67. If 𝑥 𝑦 = 𝑒 𝑥−𝑦 , then show that 𝑑𝑥 = {𝑙𝑜𝑔 (𝑥𝑒)}2
.
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Sure Shot Questions
Chapter – 06
Application of Derivatives
MCQ (1 mark) 6. If y x 10 and if x changes from 2 to 1.99, what
4
is the change in y?
1. The sides of an equilateral triangle are increasing (a) 0.32 (b) 0.032
at the rate of 2 cm/sec. The rate at which the area (c) 5.68 (d) 5.968
increases, when side is 10 cm is
7. The equation of tangent to the curve
(a) 10 cm2/s (b) 3 cm2/s
10 2
y(1 x 2 ) 2 x , where it crosses x-axis is
2
(c) 10 3 cm / s (d) cm /s (a) x + 5y = 2 (b) x – 5y = 2
3
2. A ladder, 5 meter long, standing on a horizontal (c) 5x – y = 2 (d) 5x + y = 2
floor, leans against a vertical wall. If the top of the
ladder slides downwards at the rate of 10 cm/sec, 8. The points at which the tangents to the curve
then the rate at which the angle between the floor y x3 12 x 18 are parallel to x-axis are
and the ladder is decreasing when lower end of (a) (2, -2), (-2, -34) (b) (2, 24), (-2, 0)
ladder is 2 metres from the wall is (c) (0, 34), (-2, 0) (d) (2, 2), (-2, 34)
1 1
(a) radian/sec (b) radian/sec
9. The tangent to the curve y e at the point (0, 1)
10 20 2x
(c) 20 radian/sec (d) 10 radian/sec meets x-axis at
1
3. The curve y x
1/5
at (0, 0) has (a) (0, 1) (b) , 0
2
(a) A vertical tangent (parallel to y – axis) (c) (2, 0) (d) (0, 2)
(b) A horizontal tangent (parallel to x – axis)
(c) An oblique tangent
10. The slope of tangent to the curve x t 2 3t 8 ,
(d) No tangent
y 2t 2 2t 5 at the point (2, -1) is
4. The equation of normal to the curve 3x y 8 ,
2 2 22 6
(a) (b)
7 7
which is parallel to the line x + 3y = 8 is
6
(a) 3x y 8 (b) 3x y 8 0 (c) (d) -6
7
(c) x 3 y 8 0 (d) x 3 y 0
11. The two curves x 3xy 2 0 and
3 2
5. If the curve ay x 7 and x y , cut
2 3
3x2 y y3 2 0 intersect at an angle of
orthogonally at (1, 1), then the value of a is
(a) 1 (b) 0
(a) (b)
(c) -6 (d) 6 4 3
(c) (d)
2 6
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12. The interval on which the function 20. Maximum slope of the curve
f (x) 2 x3 9 x2 12 x 1 is decreasing is y x3 3x2 9 x 27 is
(a) [1, ) (b) [2, 1] (a) 0 (b) 12
(c) (, 2] (d) [1,1] (c) 16 (d) 32
13. Let t8he f : R R be defined by 21. The interval in which the function
f (x) 2 x cosx, then f f (x) 2 x3 9 x2 12 x 1 is decreasing, is
(a) Has a minimum at x [2023]
(b) Has a maximum, at x = 0 (a) (1, ) (b) (2, 1)
(c) Is a decreasing function (c) (, 2) (d) [-1, 1]
(d) Is an increasing function
22. The function f(x) = x3 + 3x is increasing in interval
14. y x(x 3) decreases for the values of x given by
2
[2023]
(a) 1 < x < 3 (b) x < 0 (a) (,0) (b) (0, )
3 (c) R (d) (0, 1)
(c) x > 0 (d) 0 x
2
23. The interval, in which function y x 6 x 6 is
3 2
15. Which of the following functions is decreasing on
increasing, is
(a) (, 4) (0, ) (b) (, 4)
0, ?
2 (c) (-4, 0) (d) (, 0) (4, )
(a) sin2x (b) tanx [Term I, 2021 – 22]
(c) cosx (d) cos3x
16. The function f(x) = tanx – x 24. The function (x - sinx) decreases for
(a) Always increases [Term I, 2021 – 22]
(b) Always decreases
(c) Never increases (a) all x (b) x
2
(d) Sometimes increases and sometimes
decreases. (c) 0 x (d) no value of x
4
17. If x is real, the minimum value of x 2 8x 17 is
25. The value of x for which (x x ) is maximum, is
2
(a) -1 (b) 0
(c) 1 (d) 2 [Term I, 2021 – 22]
(a) ¾ (b) 1/2
(c) 1/3 (d) ¼
18. The smallest value of f (x) x 18x 96 x in
3 2
[0, 9] is
26. A wire of length 20 cm is bent in the form of a
(a) 126 (b) 0
sector of a circle. The maximum area that can be
(c) 135 (d) 160
enclosed by the wire is [Term I, 2021 – 22]
(a) 20 sq. cm (b) 25 sq. cm
19. The maximum value of sinx . cosx is
(c) 10 sq. cm (d) 30 sq. cm
1 1
(a) (b)
4 2
(c) 2 (d) 2 2
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Assertion-Reasoning (1 mark)
27. Assertion (A): The function f(x) =𝑥 2 - 4x+6 is strictly
increasing in the interval (2,∞).
32. Assertion (A): the absolute maximum value of the
function 2𝑥 3 − 24𝑥 in the interval [1, 3] is 89.
28. Assertion (A): f(x) =2𝑥 3 − 2𝑥 2 + 12𝑥 − 3 is
increasing outside the interval (1,2).
CASE STUDY
33. Case study – Some young entrepreneur started a
29. Assertion (A): 𝑓(𝑥) = 𝑡𝑎𝑛 𝑥 − 𝑥 always increases. industry “young achievers” for casting metal into
Reason (R): Any function y = f(x) is increasing if various shapes. They put up an advertisement
𝑑𝑦
𝑑𝑥
>0 online stating the same and expecting order to cast
metal for toys, sculptures, decorative pieces and
more. A group of friends wanted to make
innovative toys and hence contracted the “young
achievers” to order them to cast metal into solid
half cylinders with a rectangular base and semi –
circular ends.
30. Assertion (A): if x is real, then the minimum value
of 𝑥 2 − 8𝑥 + 17 is 1.
Based on the above information, answer the following
questions (29 to 33):
1. The volume (V) of the casted half cylinder will be
[Term I, 2021 – 22]
1 2
(a) r 2 h (b) r h
3
31. The function f be given by f(x) = 2𝑥 2 + 6𝑥 + 5. 1 2
(c) r h (d) r 2 (r h)
2
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2. The total surface area (S) of the casted half
cylinder will be [Term I, 2021 – 22]
(a) rh 2 r 2 rh
(b) rh r 2 2rh
(c) 2 rh r 2 2rh
(d) rh r 2 rh
Assume the speed of the train as v km/h.
3. The total surface area S can be expressed in
terms of V and r as [Term I, 2021 – Based on the given information, answer the following
22] questions.
2V ( 2)
(a) 2 r 6. Given that the fuel cost per hour is k times the
r square of the speed the train generates in
2V km/h, the value of k is
(b) r
r 16 1
(a) (b)
2V ( ) 3 3
(c) r 2
r 3
(c) 3 (d)
2V ( 2) 16
(d) 2 r 2
r
7. If the train has travelled a distance of 500 km,
4. For the given half – cylinder of volume V, the
then the total cost of running the train is given
total surface area S is minimum, when
by function
[Term I, 2021 – 22]
15 600000
(a) ( 2) V r
2 3 (a) v
16 v
(b) ( 2) V r
2 2
375 600000
(b) v
(c) 2( 2) V r 4 v
2 3
5 2 1500000
(d) ( 2) V r
2 (c) v
16 v
3 6000
5. The ratio h : 2r for which S to be minimum will (d) v
16 v
be equal to [Term I, 2021 –
22]
(a) 2 : 2 (b) 2 : 1 8. The most economical speed to run the train is
(c) : 1 (d) : 2 (a) 18 km/h (b) 5 km/h
(c) 80 km/h (d) 40 km/h
34. Case Study: The fuel cost per hour for running a 9. The fuel cost for the train to travel 500 km at
train is proportional to the square of the speed it the most economical speed is
generates in km per hour. If the fuel costs Rs.48 (a) Rs. 3750 (b) Rs.750
per hour at speed 16 km per hour and the fixed (c) Rs.7500 (d) Rs.75000
charges to run the train amount is Rs.1200 per
hour. 10. The total cost of the train to travel 500 km at
the most economical speed is [Term I, 2021 –
22]
(a) Rs.3750 (b) Rs.75000
(c) Rs.7500 (d) Rs.15000
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Case study based questions are compulsory. Attempt (a) 0 m (b) 30 m
any 4 sub parts from question. Each sub – part carrie (c) 50 m (d) 80 m
1 marks. (v) The extra area generated if the area of
35. An architect designs a building for a multi-national the whole floor is maximized is
company. The floor consists of a rectangular region 3000
(a) m2
with semicircular ends having a perimeter of 200 m
as shown below: 5000
(b) m2
7000
(c) m2
(d) No change, Both areas are equal
[2020 – 21]
Based on the above information answer the
36. Case-study : Sooraj’s father wants to construct a
following: rectangular garden using a brick wall on one side
(i) If x and y represents the length and of the garden and wire fencing for the other three
breadth of the rectangular region, then sides as shown in the figure. He has 200 metres of
the relation between the variables is
fencing wire.
(a) x y 100
(b) 2 x y 200
(c) x y 50
(d) x y 100
(ii) The area of the rectangular region A Based on the above information, answer the
expressed as a function of x is following questions:
(a)
2
100x x 2 (i) Let ‘x’ metres denote the length of the side of
the garden perpendicular to the brick wall and
(b)
1
100x x 2 'y' metres denote the length of the side parallel
to the brick wall. Determine the relation
x representing the total length of fencing wire
(c) 100 x
and also write A(x), the area of the garden,
100 x x
2 (ii) Determine the maximum value of A(x).
(d) y 2 2
[2023]
(iii) The maximum value of area A is
37. Case- Study: Read the following passage and
(a) m2 answer the questions given below.
3200
3200 In a elliptical sport field the authority wants to
(b) m2 design a rectangular soccer field with the
maximum possible area. The sport field is given
5000
(c) m2 x2 y 2
by the graph of 2 2 1 .
1000 a b
(d) m2
(iv) The CEO of the multi – national company
is interested in maximizing the area of
the whole floor including the semi –
circular ends. For this to happen the
value of x should be
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41. Show that the height of the cylinder of maximum
volume that can be inscribed in a sphere of radius R
2R
is . Also, find the maximum volume.
3
[2019]
42. If the sum of lengths of the hypotenuse and a side
of a right angled triangle is given, show that the
area of the triangle is maximum, when the angle
(i) If the length and the breadth of the
between them is .
rectangular field be 2x and 2y 3
respectively, then find the area function [NCERT Exemplar, Delhi 2017, Al 2016, 2014]
in terms of x.
(ii) Find the critical point of the function. 43. Find the maximum and minimum values of
(iii) Use First Derivative Test to find the 𝑓(𝑥) = −|𝑥 − 1| + 5 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑅.
length 2x and width 2y of the soccer field
(in terms of a and b) that maximize its
44. Find the intervals in which the function
area. 𝑥4
𝑓(𝑥) = − 𝑥 3 − 5𝑥 2 + 24𝑥 + 12 is (i) strictly
OR 4
increasing, (ii) strictly decreasing.
Use Second Derivative Test to find the
length 2x and width 2y of the soccer field
45. An expensive square piece of golden colour board
(in terms of a and b) that maximize its of side 24 centimetres, is to be made into a box by
area. [2022 – 23] cutting a square from each corner and folding the
flaps to form a box. What should be the side of the
square piece to be cut from each corner of the
Questions board to hold maximum volume and minimize the
wastage?
38. A ladder 13 m long is leaning against a vertical wall.
The bottom of the ladder is dragged away from the 46. An open tank with a square base and vertical sides
wall along the ground at the rate of 2 cm/sec. How is to be constructed from a metal sheet so as to
fast is the height on the wall decreasing when the hold a given quantity of water. Show that the cost
foot of the ladder is 5 m away from the wall? [Al of material will be least when depth of the tank is
2019] half of its width.
47. AB is the diameter of a circle and C is any point on
39. Find the intervals in which the function
the circle. Show that the area of triangle ABC is
x4 maximum, when it is an isosceles triangle.
f (x) x3 5 x 2 24 x 12 is [2018]
4
(a) Strictly increasing
(b) Strictly decreasing 48. Find the intervals in which the function
𝑓(𝑥) = 3𝑥 4 − 4𝑥 3 − 12𝑥 2 + 5 is (a) strictly
increasing (b) strictly decreasing.
40. Find the intervals in which the function
f (x) 3x 4 4 x3 12 x2 5 is 49. Show that the function 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 3𝑥,
[Delhi 2014] 𝑥 ∈ 𝑅 is increasing on R.
(a) Strictly increasing
(b) Strictly decreasing 50. A tank with rectangular base and rectangular sides
open at the top is to be constructed so that its
depth is 3 m and volume is 75 𝑐𝑚3 .If building of
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tank costs Rs. 100 per square metre for the base
and Rs. 50 per square metre for the sides, find the 60. The sides of an equilateral triangle are increasing at
cost of least expensive tank. the rate of 2 cm/sec. Find the rate at which the area
increases, when the side is 10 cm.
51. A figure consists of a semi-circle with a rectangle on
its diameter. Given the perimeter of the figure, find 61. The volume of a sphere is increasing at the rate of
its dimensions in order that the area may be 3 cubic centimetre per second. Find the rate of
maximum. increase of its surface area, when the radius is 2 cm.
52. Show that semi-vertical angle of a cone of 62. The total cost C(x) associated with the production
maximum volume and given slant height is of x units of an item is given by C(x) =
𝑐𝑜𝑠 −1 (1/√3) 0.005𝑥 2 _ 0.02𝑥 2 + 30𝑥 + 5000. Find the
marginal cost when 3 units are produced, where by
marginal cost we mean the instantaneous rate of
53. Prove that the least perimeter of an isosceles change of total cost at any level of output.
triangle in which a circle of radius r can be inscribed
is 6√3 r. 63. Find the intervals in which the function 𝑓(𝑥) =
𝑥4
− 𝑥 3 + 5𝑥 2 + 24𝑥 + 12 is
4
(a) strictly increasing (b) strictly
54. Show that the altitude of the right circular cone of
decreasing
maximum volume that can be inscribed in a sphere
of radius r is 4r/3. Also show that the maximum
64. Find the intervals in which 𝑓(𝑥) = 𝑠𝑖𝑛 3𝑥 −
volume of the cone is 8/27 of the volume of the
𝑐𝑜𝑠 3𝑥, 0 < 𝑥 < 𝜋, is strictly increasing or strictly
sphere.
decreasing.
55. Prove that the height of the cylinder of maximum
65. Prove that the function f defined by 𝑓(𝑥) = 𝑥 2 −
volume, that can be inscribed in a sphere of radius
𝑥 + 1 is neither increasing nor decreasing in (-
R is 2R/√3. Also find the maximum volume. 1,1). Hence, find the intervals in which 𝑓(𝑥) is (i)
strictly increasing (ii) strictly decreasing.
56. Show that the right-circular cone of least curved
surface and given volume has an altitude equal to 66. A kite is 120 m high and 130 m of string is out. If
√2 times the radius of the base. the kite is moving away horizontally at the rate of
52 m/sec, find the rate at which the string is being
57. If lengths of three sides of a trapezium other than pulled out.
base are equal to 10cm, then find the area of the
trapezium when it is maximum. 67. 𝑥 and 𝑦 are the sides of two squares such that 𝑦 =
𝑥 − 𝑥 2 . Find the rate of change of the area of
58. A ladder 13 m long is leaning against a vertical wall. second square with respect to the area of first
The bottom of the ladder is dragged away from the square.
wall along the ground at the rate of 2 cm/sec. How
fast is the height on the wall decreasing when the
foot of the ladder is 5 m away from the wall?
59. The side of an equilateral triangle is increasing at
the rate of 2 cm/s. At what rate is its area
increasing when the side of the triangle is 20 cm?
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Sure Shot Questions
Chapter – 07
Integrals
MCQ (1 mark) x9
5. dx is equal to
cos 2 x cos 2 (4 x 2 1)6
1. cos x cos
dx is equal to
5
1 1
(a) 2(sinx x cos x ) C (a) 4 2 C
5x x
(b) 2(sinx xcos ) C 5
(c) 2(sinx 2 xcos ) C
1 1
(b) 4 2 C
5 x
(d) 2(sinx 2 x cos ) C
1
1 4 C
5
(c)
10 x
dx
2. sin(x a)sin(x b) is equal to 1 1
5
(d) 2 4 C
10 x
sin(x b)
(a) sin(b a) log C
sin(x a)
dx
(b) cos ec(b a) log
sin(x a)
C
6. If (x 2)(x 2
1)
a log |1 x 2 |
sin(x b) 1
b tan x 1/ 5log | x 2 | C , then
sin(x b) 1 2
(c) cos ec(b a) log C (a) a ,b
sin(x a) 10 5
sin(x a) 1 2
(d) sin(b a) log C (b) a , b
sin(x b) 10 5
1 2
(c) a ,b
10 5
tan
1
3. xdx is equal to
1 2
1 (d) a , b
(a) (x 1) tan x x C 10 5
1
(b) x tan x x C
x3
(c) x x tan 1 x C 7. dx is equal to
x 1
(d) x (x 1) tan 1 x C
x 2 x3
(a) x log |1 x | C
2 3
1 x
2
4. e x
2
dx is equal to (b) x
x 2 x3
log |1 x | C
1 x 2 3
ex e x x 2 x3
(a) C (b) C (c) x log |1 x | C
1 x2 1 x2 2 3
ex e x x 2 x3
(c) C (d) C (d) x log |1 x | C
(1 x 2 )2 (1 x 2 )2 2 3
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x sin x e x (1 x)
8. 1 cos x dx is equal to 10. cos2 (xe x ) dx is equal to [2020]
(a) log |1 cosx | C
(b) log | x sinx | C
(a) tan(xe x ) c (b) cot(xe x ) c
x (c) cot(e x ) c (d) tan[e x (1 x)] c
(c) x tan C
2
x x log x 1
(d) x.tan C
e
x
11. dx is equal to [2020]
2 x
ex
3
x dx (a) log(e logx) c
x
(b) c
9. If 1 x2
a(1 x 2 )3/2 b 1 x 2 C , then x
1 1 (c) x log x e x c (d) e x log x c
(a) a , b 1 (b) a ,b 1
3 3 ex
12. [1 (x 1) log(x 1)]dx equals [2020C]
1 1 x 1
(c) a , b 1 (d) a , b 1
3 3 ex
(a) c
/4 x 1
dx
10. 1 cos 2 x
is equal to (b) e x
x
c
/4 x 1
(a) 1 (b) 2
(c) e x log(x 1) e x c
(c) 3 (d) 4
(d) e x log(x 1) c
/2
11. 1 sin 2 xdx is equal to 1
| x 2 |
dx, x 2 is equal to
0
13. [2023]
(a) 2 2 (b) 2 2 1
x2
(a) 1 (b) -1
(c) 2 (d) 2 2 1 (c) 2 (d) -2
sec x
12. sec x tan xdx equals [2023]
(a) sec x tan x c (b) sec x tan x c 4
(e x)dx is equal to
2x
14. [2023]
(c) tan x sec x c (d) (secx tanx) c
0
15 e8 16 e8
e
5logx
13. dx is equal to [2023] (a) (b)
2 2
x5 x6 e 15
8
e8 15
(a) C (b) C (c) (d)
5 6 2 2
(c) 5x 4 C (d) 6x5 C
/8
tan
2
15. (2 x) dx is equal to [2020]
x e dx equals
3
2 x
14. [2020]
0
(a)
1 x3
e C (b)
1 x4
e C 4 4
(a) (b)
3 3 8 8
1 3
(c) e x C
1 2
(d) e x C 4 4
(c) (d)
2 2 4 2
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20.
/4
16.
/4
sec2 xdx is equal to [2020]
(a) -1 (b) 0
(c) 1 (d) 2
Assertion-Reasoning (1 mark)
17. In the following questions, a statement of Assertion
(A) is followed by a statement of Reason (R).
Choose the correct answer out of the following Case Study Question
choice [2023]
21. Read the text carefully and answer the questions:
10 x
8
Rajni and Priyanka practice the problems based on
Assertion (A):
2 x 10 x
dx 3
integrals. They
b b
Reason (R) :
a
f (x)dx f (a b x)dx
a
(a) Both (A) and (R) are true and (R) is the
correct explanation of (A).
(b) Both (A) and (R) is true, but (R) is not the
correct explanation of the (A).
(c) (A) is true and (R) is false.
(d) (A) is false, but (R) is true.
18.
22. Read the text carefully and answer the questions:
The given integral∫ f(x) dx can be transformed into
another form by changing the independent variable
x to t by substituting x = g(t)
19.
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Questions
3𝑥+5
15. Evaluate: ∫ 𝑑𝑥
sin x √𝑥 2 −8𝑥+7
1. Find: dx [Term II, 2021 – 22C]
sin(x 2a)
(5𝑥−2)
16. Find: ∫ (3𝑥 2 +2𝑥+1) 𝑑𝑥
(3sin 2) cos
2. Find 5 cos 2
4sin
d . [Delhi 2016]
𝑥 2 +1
17. Evaluate: ∫ 𝑒 𝑥 𝑑𝑥
(𝑥+1)2
/3
sin x cos x
3. Evaluate:
/6 sin 2 x
dx [2020C, Al
sin(𝑥−𝑎)
18. Evaluate: ∫ 𝑑𝑥
2014C] sin(𝑥+𝑎)
4. Evaluate e2 x .sin x dx . [Delhi 2016]
0 4 19. Evaluate: ∫ 𝑥 𝑠𝑖𝑛−1 𝑥 𝑑𝑥
/2
dx
5. Find:
0
3
cos x 2sin 2 x
. [Al 2015] 20. Evaluate: ∫ 𝑒 𝑥 (
sin 4𝑥−4
1−𝑐𝑜𝑠4𝑥
) 𝑑𝑥
5𝑥+3
𝑑𝑥 21. Evaluate: ∫ 𝑑𝑥
6. Evaluate ∫ 𝑑𝑥 √𝑥 2 +4𝑥+10
√5−4𝑥−2𝑥 2
1−𝑥 2 𝑥2
7. Find ∫ 𝑑𝑥 22. Evaluate: ∫ (𝑥 2 +4)(𝑥 2 +9) 𝑑𝑥
𝑥(1−2𝑥)
23. Evaluate: ∫(√𝑐𝑜𝑡𝑥 + √𝑡𝑎𝑛𝑥)𝑑𝑥
𝑠𝑖𝑛2𝑥
8. Evaluate: ∫ (1+𝑠𝑖𝑛 𝑥)(2+𝑠𝑖𝑛 𝑥) 𝑑𝑥 𝑥+2
24. Find: ∫ (𝑥 2 +3𝑥+3) 𝑑𝑥
√𝑥+1
log 𝑥
9. Find (∫ (𝑥+1)2 𝑑𝑥.)
5𝑥 𝑥
25. Find: ∫ 55 . 55 . 5𝑥 . 𝑑𝑥
cos 2x − cos 2𝛼
10. Evaluate :∫ 𝑑𝑥
cos 𝑥−cos 𝛼
1−√𝑥
𝑑𝑥
26. Find: ∫ √ 𝑑𝑥
1+√𝑥
11. Evaluate ∫ 𝑥(𝑥 5 +3)
(𝑥−4)𝑒 𝑥
12. Evaluate ∫ 𝑠𝑖𝑛−1 √𝑥−𝑐𝑜𝑠−1 √𝑥
(𝑥−2)3 27. Find : ∫ 𝑠𝑖𝑛−1 𝑑𝑥, 𝑥 ∈ [0′1]
√𝑥+𝑐𝑜𝑠−1 √𝑥
𝑥𝑠𝑖𝑛−1 𝑥
13. Given ∫ 𝑒 𝑥 (tan 𝑥 + 1) 𝑠𝑒𝑐𝑥𝑑𝑥 = 𝑒 𝑥 𝑓(𝑥) + 28. Find:∫ 𝑑𝑥
√1−𝑥 2
𝐶 Write f(x) satisfying the above.
𝑥 2 −3𝑥+1
29. Integrate the following w.r.t.x :
tan 𝜃+𝑡𝑎𝑛3 𝜃 √1−𝑥 2
14. Evaluate: ∫ 1+𝑡𝑎𝑛3 𝜃
𝑑𝜃
30. Evaluate: ∫ 𝑒 2𝑥 . 𝑠𝑖𝑛(3𝑥 + 1)𝑑𝑥
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Sure Shot Questions
Chapter – 08
Application of Integrals
MCQ [1 Marks] 6. The area of the region bounded by parabola y2 = x
and the straight line 2y = x is
4
1. The area of the region bounded by the y-axis, (a) sq. unit (b) 1 sq. unit
3
y cos x and y sin x, 0 x is 2
(c) sq. unit
1
(d) sq. unit
2 3 3
(a) 2 sq. units (b) 2 1 sq. units
(c) 2 1 sq. unit (d) 2 2 1 sq. units
7. The area of the region bounded by the curve
y sin x between x 0, x and the x-axis is
2. The area of the region bounded by the curve 2
x 2 4 y and the straight line x = 4y – 2 is (a) 2 sq. units (b) 4 sq. units
3 5 (c) 3 sq. units (d) 1 sq. unit
(a) sq. unit (b) sq. unit
8 8
7 9
(c) sq. unit (d) sq. unit
8 8 8. The area of the region bounded by the ellipse
x2 y 2
1 is
25 16
3. The area of the region bounded by the curve (a) 20 sq. units (b) 20 2 sq. units
y 16 x 2 and x – axis is (c) 3 sq. units (d) 4 sq. units
(a) 8 sq. units (b) 20 sq. units
(c) 16 sq. units (d) 256 sq. units
9. The are of the region bounded by the circle
x 2 y 2 1is
4. Area of the region in the first quadrant enclosed by (a) 2 sq. unit (b) sq. units
the x-axis, the line y = x and the circle x y 32
2 2 (c) 3 sq. units (d) 4 sq. units
is
(a) 16 sq. units (b) 4 sq. units
(c) 32 sq. units (d) 24 sq. units 10. The area of the region bounded by the curve
y x 1 and the lines x = 2 and x = 3 is
7 9
(a) sq. units (b) sq. units
5. Area of the region bounded by the curve 2 2
y cos x between x 0 and x is (c)
11
sq. units (d)
13
sq. units
(a) 2 sq. units (b) 4 sq. units 2 2
(c) 3 sq. units (d) 1 sq. unit
Required area = area of shaded region
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11. The area of the region bounded by the curve 14. Assertion (A): The area of the region bounded by
x 2 y 3 and the lines y = 1 and y = -1 is the curve
3
3 y = x2 and the line y = 4 is 32 .
(a) 4 sq. units (b) sq. units Reason (R): Since the given curve represented by the
2
equation y = 𝑥 2 is a parabola symmetrical about y - axis
(c) 6 sq. units (d) 8 sq. units
only, is given by
therefore, from figure, the required area of the region
AOBA
Assertion-Reasoning (1 mark)
12. Assertion (A):
a)Both A and R are true and R is the correct
explanation of A.
b) Both A and R are true but R is not the correct
explanation of A.
c) A is true but R is false.
d) A is false but R is true.
15. Assertion (A):
13. Assertion (A):
16. Assertion (A): he area enclosed by the circle 𝑥 2 +
𝑦 2 = 𝑎2 𝑖𝑠 𝜋 𝑎2 .
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18. Read the text carefully and answer the questions:
The location of the three houses of a society is
represented by points A ( - 1, 0), B( 1, 3), and C(3,
2) as shown in the figure.
Case Study Question
17. Read the text carefully and answer the questions:
The bridge connects two hills 100 feet apart. The
arch on the bridge is in a parabolic form. The
highest point on the bridge is 10 feet above the
road at the middle of the bridge as seen in the
figure.
Questions
19. Using integration, find the area of the region
bounded by lines x – y + 1 = 0, x = -2, x = 3 and x –
axis. [Term II, 2021 – 22]
20. Using integration, find the area of the region
bounded by the curves: [Delhi 2014C]
y | x 1| 1, x 3, x 3, y 0
1 3
Required area = ( x)dx (x 2) dx
3 1
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21. Using method of integration, find the area of the 𝑥2 𝑦2 𝑥 𝑦
+ = 1 and the line 3 + 2 = 1.
region enclosed between two circles 𝑥 2 + 9 4
𝑦 2 𝑎𝑛𝑑 (𝑥 − 2)2 + 𝑦 2 = 4.
22. Using integration, find the area of the region in the 31. Find the area of the region {(𝑥, 𝑦): 𝑦 2 ≤ 4𝑥, 4𝑥 2 +
first quadrant enclosed by the X-axis, the line 𝑦 = 𝑥 4𝑦 2 ≤ 9}using method of integration.
and the circle 𝑥 2 + 𝑦 2 = 32.
32. Using integration, find the area of the region
2
23. Find the area bounded by the circle𝑥 + 𝑦 = 16 2 bounded by the two parabolas 𝑦 2 = 4𝑥 and 𝑥 2 =
and the line √3𝑦 = 𝑥 in the first quadrant, using 4𝑦.
integration.
33. Find the area of the region {(𝑥, 𝑦): 𝑥 2 + 𝑦 2 ≤ 4,
24. Using integration find the area of the triangular 𝑥 + 𝑦 ≥ 2. }
region whose sides have equations 𝑦 = 2𝑥 +
1, 𝑦 = 3𝑥 + 1 𝑎𝑛𝑑 𝑥 = 4.
25. A farmer has a field of shape bounded by 𝑥 = 34. Using integration, find the area of the following
𝑦 2 and 𝑥 = 3, he wants to divide this into his two region:
sons equally by a straight line 𝑥 = 𝑐 Can you find c? {(𝑥, 𝑦): 𝑦 2 ≥ 𝑎𝑥, 𝑥 2 + 𝑦 2 ≤ 2𝑎𝑥, 𝑎 > 0}
26. Using integration, find the area of the region
{(𝑥, 𝑦): 𝑥 2 + 𝑦 2 ≤ 2𝑎𝑥, 𝑦 2 > 𝑎𝑥, 𝑥, 𝑦 ≥ 0}. 35. Find the area of the region {(𝑥, 𝑦): 𝑥 2 ≤ 𝑦 ≤ |𝑥|}.
27. Prove that the curves 𝑦 2 = 4𝑥 and 𝑥 2 = 4𝑦 divide
the area of square bounded by x = 0, x = 4, y = 4 and 36. Find the area bounded by the curve 𝑦 =
y = 0 into three equal parts. 𝑥|𝑥|, 𝑥 −axis and the lines 𝑥 = −3 and 𝑥 = 3.
28. Using integration, find the area of the region
bounded by the curve 37. Find the area of the region bounded by 𝑦 2 =
𝑥 2 = 4𝑦 and the line 𝑥 = 4𝑦 − 2. 2𝑥 + 1 and 𝑋 − 𝑦 = 1.
29. Using integration, find the area of the region 𝑥2 𝑦2
bounded by the triangle whose vertices are 38. The ellipse 2 + 2 = 1 is divided into two parts by
𝑎 𝑏
(−1,2), (1,5)𝑎𝑛𝑑 (3,4). a b the line 2𝑥 = 𝑎. Find the area of the smaller
part.
30. Find the area of the smaller region bounded by the
ellipse
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Sure Shot Questions
Chapter – 09
Differential Equations
MCQ (1 mark) d2y
(d) y 0
dx 2
1. The degree of the differential equation
2 5. Solution of differential equation xdy – ydx = 0
d 2 y dy
2
dy
2 x sin is represents
dx dx dx (a) A rectangular hyperbola
(a) 1 (b) 2 (b) Parabola whose vertex is at origin
(c) 3 (d) not defined (c) Straight line passing through origin
(d) A circle whose centre is at origin
2. The degree of the differential equation
3 6. Integrating factor of the differential equation
dy 2 2 d 2 y
1 2 is cos x
dy
y sin x 1 is
dx dx dx
3 (a) cosx (b) tanx
(a) 4 (b)
2 (c) secx (d) sinx
(c) not defined (d) 2
7. Solution of the differential equation
tan y sec2 xdx tan x sec2 ydy 0 is
3. The order and degree of the differential equation (a) tan x tan y k
1
d 2 y dy 4 1 (b) tan x tan y k
x 5
0 , respectively, are
dx 2 dx (c)
tan x
k
(a) 2 and not defined tan y
(b) 2 and 2 (d) tan x.tan y k
(c) 2 and 3
(d) 3 and 3
8. Family y Ax A of curves is represented by the
3
differential equation of degree
4. The differential equation for
(a) 1 (b) 2
y A cos x B sin x , where A and B are
(c) 3 (d) 4
arbitrary constants is
d2y xdy
(a) 2
2 y 0 9. Integrating factor of y x 4 3x is
dx dx
d2y (a) x (b) log x
(b) 2
2 y 0
dx 1
(c) (d) -x
d y 2 x
(c) y 0
dx 2
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dy (a) y e x (x 1) (b) y xe x
10. Solution of y 1 , y(0) = 1 is given by
dx (c) y xe x 1 (d) y (x 1) e x
(a) xy e x (b) xy e x
(c) xy 1 (d) y 2e x 1
dy y 1 17. Integrating factor of the differential equation
11. The number of solutions of when y(1) =
dx x 1 dy
y tan x sec x 0 is
2 is dx
(a) none (b) one (a) cosx (b) sec x
(c) two (d) infinite (c) ecos x (d) esec x
18. The solution of the differential equation
12. Which of the following is a second order
dy 1 y 2
differential equation? is
dx 1 x 2
(a) (y')2 x y 2 1
(a) y tan x (b) y x k (1 xy)
(b) y ' y " y sin x 1
(c) x tan y (d) tan(xy) k
(c) y "' (y") y 0
2
(d) y ' y
2
19. The integrating factor of the differential equation
dy 1 y
y is
dx x
13. Integrating factor of the differential equation x ex
(a) (b)
dy ex x
(1 x 2 ) xy 1 is
dx (c) xe
x
(d) e
x
x
(a) -x (b)
1 x2
1 mx
20. y ae be satisfies which of the following
1 x2
mx
(c) (d) log(1 x 2 )
2
differential equations?
dy
(a) my 0
dx
14. The general solution of
dy
e x cos ydx e x sin y dy 0 is (b) my 0
dx
(a) e cos y k (b) e sin y k
x x
d2y
(c) m2 y 0
(c) e k cos y (d) e k sin y
x x 2
dx
dy
(d) my 0
dx
15. The degree of the differential equation
d 2 y dy
3 21. The solution of the differential equation cos x sin y
6 y 5 0 is dx + sin x cos y dy = 0 is
dx dx
2
sin x
(a) 1 (b) 2 (a) c
(c) 3 (d) 5 sin y
(b) sin x sin y c
dy (c) sin x sin y c
16. The solution of y e x , y (0) 0 is
dx (d) cos x cos y c
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dy 28. The order and degree of the differential equation
22. The solution of x y e x is 2
d3y
4
dx d2y dy
ex k 3 3 2 2 y are
(a) y dx dx dx
x x (a) 1, 4 (b) 3, 4
(b) y xe x cx (c) 2, 4 (d) 3, 2
(c) y xe x k
29. The order and degree of the differential equation
ey k
(d) x dy 2 d 2 y
y y 1 2 are
23. The differential equation of the family of curves dx dx
x2 y 2 2ay 0 , where a is arbitrary constant, (a) 2,
3
(b) 2, 3
is 2
dy (c) 2, 1 (d) 3, 4
(a) (x 2 y 2 ) 2 xy
dx
30. Which of the following is the general solution of
(b) 2 x 2 y 2
dy
xy d2y dy
dx
2
2 y 0?
dx dx
(c) 2 x 2 y 2
dy
xy
dx (a) y (Ax B) e x
(b) y (Ax B) e x
(d) x 2 y 2
dy
2 xy
x
dx (c) y Ae Be
x
(d) y A cos x B sin x
24. Family y Ax A of curves will respond to a
3
differential equation of order
dy
(a) 3 (b) 2 31. General solution of y tan x sec x is
dx
(c) 1 (d) not defined
(a) y sec x tan x c
(b) y tan x sec x c
dy
2 x e x y is
2
25. The general solution of (c) tan x y tan x c
dx
y y
(d) x sec x tan y c
c (b) e e c
2 2
x x
(a) e
y
(c) e e c c
2 2
y x x
(d) e 32. Solution of the differential equation
dy y
sin x is
dx x
26. The curve for which the slope of the tangent at any (a) x(y cosx) sinx c
point is equal to the ratio of the abscissa to the
(b) x(y cosx) sinx c
ordinate of the point is
(a) An ellipse (c) xy cos x sin x c
(b) Parabola (d) x(y cosx) cosx c
(c) Circle
(d) Rectangular hyperbola 33. The general solution of the differential equation
(e x 1) ydy (y 1) e x dx is
dy
y e x , y (0) 0 is (a) (y 1) k(e 1)
x
27. The solution of
dx
(b) y 1 e 1 k
x
x
(a) y e (x 1) (b) y xe
x
(c) y log{k(y 1)(e 1)}
x
x x
(c) y xe 1 (d) y xe
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e x 1 39. The number of arbitrary constants in the particular
(d) y log k solution of a differential equation of second order
y 1
is (are) [2020]
(a) 0 (b) 1
34. The solution of the differential equation
(c) 2 (d) 3
dy
e x y x 2e y is
dx
(a) y e x y x 2e y c 40. The integrating factor for solving the differential
3
x dy
(b) e y e x c equation x y 2 x 2 is [2023]
3 dx
x3 (a) e y (b) e e
(c) e e
x
cy
3 1
(c) x (d)
x3 x
(d) e x e y c
3
41. The integrating factor of the differential equation
dy
35. The solution of the differential equation (x 3 y 2 ) y is [2020]
dy 2 xy 1 dx
is (a) y (b) -y
dx 1 x 2
(1 x 2 ) 2
1 1 1
(a) y(1 x ) c tan x
2
(c) (d) -
y y
y
(b) c tan 1 x
1 x2
1
(c) y log(1 x ) c tan x
2
42. If m and n, respectively, are the order and the
1
(d) y(1 x ) c sin x
2
degree of the differential equation
4
d dy
0 , then m + n = 0 [2022
36. The sum of the order and the degree of the dx dx
d dy
3
– 23]
differential equation is [2023]
dx dx (a) 1 (b) 2
(a) 2 (b) 3 (c) 3 (d) 4
(c) 5 (d) 0
Assertion-Reasoning (1 mark)
37. The order and the degree of the differential
2 43. Assertion (A): Integrating factor of
dy d3y
equation 1 3 4 3 respectively are
dx dx
[2023]
2
(a) 1, (b) 3, 1
3
(c) 3, 3 (d) 1, 2
38. The number of solutions of the differential
dy y 1
equation , when y(1) = 2, is
dx x 1
[2023] 44. Assertion (A): Homogeneous system of linear
(a) zero (b) one equations is always consistent.
(c) two (d) infinite
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Reason (R): x = 0, y = 0 is always a solution of the Case Study [4 Marks]
homogeneous system of equation.
a) Both A and R are true and R is the correct 48. Case study: An equation involving derivatives of
explanation of A. the dependent variable with respect to the
b) Both A and R are true but R is not the correct independent variables is called a differential
explanation of A. equation A differential equation of the form
c) A is true but R is false. dy
d) A is false but R is true. F (x, y) is said to be homogeneous if F(x, y)
dx
is a homogeneous function of degree zero,
45. Assertion (A): The order and degree of the
whereas a function F(x, y) is a homogeneous
differential equation
function of degree n if F ( x, y) = n F (x, y) . To
solve a homogeneous differential equation of the
dy y
type F (x, y) g , we make substitution
dx x
y = vx and then separate the variables.
Based on the above, answer the following
questions.
Show that (x y )dx 2 xydy 0 is a
2 2
(i)
differential equation of the type
46. Assertion (A): The order of the differential dy y
equation given g .
dx x
(ii) Solve the above equation to find its
general solution. [2023]
Questions
1. Find the product of the order and the degree of the
d dy
differential equation (xy 2 ) . y 0 .
dx dx
[2022]
47.
2. Find the integrating factor of
dy
x (1 x cotx) y x .
dx
3. Find the particular solution of the differential
dy
equation (1 x 2 ) 2 xy tan x , given
dx
y(0) = 1. [Term II, 2021 – 22C]
4. Solve the following differential equation:
y dy y
x cos y cos x ; x 0 .
x dx x
[Al 2019C, 2014C]
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𝑑𝑦
5. Prove that x 2 y 2 C (x 2 y2 )2 is the general 16. Solve the differential equation: (𝑥 2 − 1) 𝑑𝑥 +
2
solution of the differential equation 2𝑥𝑦 = 𝑥 2 −1
, where
(x3 3xy2 ) dx (y3 3x 2 y) dy . Where C is a
𝑥 ∈ (−∞, −1)⋃(1, ∞).
parameter. [NCERT, Delhi 2017]
𝑑𝑦
17. Solve the differential equation: (1 + 𝑥 2 ) 𝑑𝑥 +
6. Solve the following differential equation : (1 + 𝑦 = 𝑡𝑎𝑛−1 𝑥
𝑑𝑦
𝑥)2 𝑑𝑥 + 𝑦 = 𝑡𝑎𝑛−1 𝑥.
𝑑𝑦 𝑦
18. Solve the differential equation: 𝑥 sin ( ) +
𝑑𝑥 𝑥
7. Solve the following differential equation : 𝑦
(1 + 𝑥 2 )𝑑𝑦 + 2𝑥𝑦 𝑑𝑥 = 𝑐𝑜𝑡 𝑥 𝑑𝑥; 𝑥 ≠ 0. 𝑥 − 𝑦 sin (𝑥 ) = 0 is homogeneous. Find the
particular solution of this differential equation,
𝑑𝑦 given that 𝑥 = 1, when 𝑦 = 𝜋/2.
8. Solve the differential equation :(𝑥 + 1) 𝑑𝑥 =
2𝑒 −1 − 1; 𝑦(0) = 0. 𝑑𝑦
19. Solve the differential equation: 𝑐𝑜𝑠 2 𝑥 𝑑𝑥
+𝑦 =
9. Write the order & degree of the following tan 𝑥
differential equations
𝑑3 𝑦 𝑑2𝑦 𝑑𝑦 20. Solve the differential equation:
(i) 𝑑𝑥 3
+ 2 (𝑑𝑥 2 ) − 𝑑𝑥 + 𝑦 = 0 𝑑𝑦
√1 + 𝑥 2 + 𝑦 2 + 𝑥 2 𝑦 2 + 𝑥𝑦 𝑑𝑥 = 0
2
𝑑2𝑦 𝑑𝑦 3
(ii) (𝑑𝑥 2 ) + 𝑥 2 (𝑑𝑥 ) = 0
21. Show that the following differential equation is
homogeneous and then solve it. 𝑦𝑑𝑥 +
𝑦
𝑥𝑙𝑜𝑔 (𝑥 ) 𝑑𝑦 − 2𝑥𝑑𝑦 = 0
10. Find order & degree of differential equation:
𝑑4 𝑦 𝑑3 𝑦
𝑑𝑥 4
+ 𝑆𝑖𝑛 (𝑑𝑥 3 ) = 0 𝑑𝑦
22. Show that the differential equation (𝑥 − 𝑦) =
𝑑𝑥
𝑥 + 2𝑦, is homogeneous and solve it.
11. Solve the differential equation.𝑦𝑙𝑜𝑔𝑦𝑑𝑥 − 𝑥𝑑𝑦 =
0.
23. Solve the differential equation: (𝑐𝑜𝑡 −1 𝑦 +
𝑥)𝑑𝑦 = (1 + 𝑦 2 )𝑑𝑥
12. Find the general solution of the following
differential equation: 𝑒 𝑥 tan 𝑦 𝑑𝑥 + (1 −
𝑒 𝑥 )𝑠𝑒𝑐 2 𝑦 𝑑𝑦 = 0 24. Find the particular solution of the differential
equation
𝑑𝑦
(1 + 𝑥 3 ) + 6𝑥 2 𝑦 = (1 + 𝑥 2 ), given that 𝑦 =
𝑑𝑥
13. Find the particular solution of the following
1 𝑤ℎ𝑒𝑛 𝑥 = 1.
differential equation :
𝑑𝑦
𝑑𝑥
= 1 + 𝑥 2 + 𝑦 2 + 𝑥 2 𝑦 2 , given that y=1 when 25. Show that the differential equation 2𝑦𝑒 𝑥/𝑦 𝑑𝑥 +
𝑥
x=0
(𝑦 − 2𝑥𝑒 𝑦 ) 𝑑𝑦 0 is homogeneous. Find the
𝑥 𝑥
𝑥 particular solution of this differential equation,
14. Solve : (1 + 𝑒 𝑦 )𝑑𝑥 + 𝑒 𝑦 (1 − 𝑦)𝑑𝑦
given that 𝑥 = 0 𝑤ℎ𝑒𝑛 𝑦 = 1.
26. Find the particular solution of the differential
15. Solve the differential equation: (𝑡𝑎𝑛−1 𝑦 − equation
𝑑𝑦
=
𝑥(2𝑙𝑜𝑔𝑥+1)
given that 𝑦 =
𝑥)𝑑𝑦 = (1 + 𝑦 2 )𝑑𝑥 𝑑𝑥 sin 𝑦+𝑦 cos 𝑦
𝜋/2 𝑤ℎ𝑒𝑛 𝑥 = 1.
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27. Solve the following differential equation: 𝑑𝑦
𝑥 + 𝑦 − 𝑥 + 𝑥𝑦 cot 𝑥 = 0; 𝑥 ≠ 0, given that
𝑦 𝑑𝑦 𝑦 𝑑𝑥
𝑥 𝑐𝑜𝑠 (𝑥 ) 𝑑𝑥 =𝑦 𝑐𝑜𝑠 (𝑥 ) + 𝑥; 𝑥 ≠ 0. 𝜋
when 𝑥 = 2 , 𝑦 = 0.
30. Find the general solution of the following
28. Solve the differential equation: (𝑡𝑎𝑛−1 𝑦 − differential equation:
𝑥)𝑑𝑦 = (1 + 𝑦 2 )𝑑𝑥. −1 𝑑𝑦
(1 + 𝑦 2 ) + (𝑥 − 𝑒 𝑡𝑎𝑛 𝑦) =0
𝑑𝑥
29. Find the particular solution of the differential
equation
For Solutions
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Sure Shot Questions
Chapter – 10
Vector Algebra
MCQ (1 mark) 5
(c) (d)
2 2
1. The vector in the direction of the vector
^ ^ ^ 5. Find the value of such that the vectors
i 2 j 2 k that has magnitude 9 is ^ ^ ^ ^ ^ ^
^ ^ ^ a 2 i j k and b i 2 j 3 k are
^ ^ ^ i 2 j 2 k orthogonal.
(a) i 2 j 2 k (b)
3 (a) 0 (b) 1
^ ^ ^
^ ^ ^
3 5
(c) 3 i 2 j 2 k (d) 9 i 2 j 2 k (c) (d)
2 2
^ ^ ^
6. The value of for which the vectors 3 i 6 j k
2. The position vector of the point which divides the ^ ^ ^
and 2 i 4 j k are parallel is
join of points 2 a 3 b and a b in the ratio 3 : 1
2 3
is (a) (b)
3 2
3 a 2 b 7 a 8 b 5 2
(a) (b) (c) (d)
2 4 2 5
3a 5a
(c) (d)
4 4
7. The vectors from origin to the points A and B are
^ ^ ^ ^ ^ ^
a 2 i 3 j 2 k and b 2 i 3 j k ,
3. The vector having initial and terminal points as (2, respectively, then the area of triangle OAB is
5, 0) and (-3, 7, 4) respectively is
(a) 340 (b) 25
^ ^ ^
(a) i 12 j 4 k 1
(c) 229 (d) 229
^ ^ ^ 2
(b) 5 i 2 j 4 k
^ ^ ^ 8. For any vector a , the value of
(c) 5 i 2 j 4 k ^ ^ ^
^ ^ ^ ( a i )2 ( a j )2 ( a k )2 is equal to
(d) i j k 2 2
(a) a (b) 3 a
2 2
4. The angle between two vectors a and b with (c) 4 a (d) 2 a
magnitudes 3 and 4, respectively and
9. If | a | 10, | b | 2and a . b 12 , then the value of
a . b 2 3 is
| a b | is
(a) (b)
6 3 (a) 5 (b) 10
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(c) 14 (d) 16 ^ ^ ^
16. Two vectors a a1 i a2 j a3 k and
^ ^ ^ ^ ^ ^ ^ ^ ^
10. The vectors i j 2 k , i j k and b b1 i b2 j b3 k are collinear if [2023]
^ ^ ^ (a) a1b1 a2b2 a3b3 0
2i j k are coplanar if
(a) 2 (b) 0
a1 a2 a3
(b)
(c) 1 (d) 1 b1 b2 b3
(c) a1 b1 , a2 b2 , a3 b3
(d) a1 a2 a3 b1 b2 b3
11. If a , b , c are unit vectors such that a b c 0
^ ^ ^
, then the value of a . b b . c c . a a is 17. The value of p for which p(i j k ) is a unit
(a) 1 (b) 3 vector is [2020]
3 1
(c) (d) None of these (a) 0 (b)
2 3
12. Projection vector of a and b is (c) 1 (d) 3
a b
(a) b
a .b
(b)
| b |2
18. ABCD is a rhombus, whose diagonals intersect at E.
|b|
^ Then EA EB EC ED equals [2020]
a.b
(d) b
a .b
(c)
| a |2 (a) 0 (b) AD
|a|
(c) 2 BC (d) 2 AD
13. If a , b , c are three vectors such that a b c 0
^ ^
19. A unit vector along the vector 4 i 3 k is [2023]
and | a | 2, | b | 3, | c | 5 , then value of
1 ^ ^ 1 ^ ^
(a) (4 i 3 k ) (b) (4 i 3 k )
a . b b . c c . a is 7 5
(a) 0 (b) 1 1 ^ ^ 1 ^ ^
(c) (4 i 3 k ) (d) (4 i 3 k )
(c) -19 (d) 38 7 5
20. If is the angle between two vectors a and b ,
14. If | a | 4 and 3 2 , then the range of | a |
then a . b 0 only when [2023]
is
(a) [0, 8] (b) [-12, 8] (a) 0 (b) 0
(c) [0, 12] (d) [8, 12] 2 2
(c) 0 (d) 0
^ ^ ^
15. The number of vectors of unit length perpendicular 21. The magnitude of the vector 6 i 2 j 3 k is
^ ^ ^ ^ ^
to the vectors a 2 i j 2 k and b j k is [2023]
(a) one (b) two (a) 1 (b) 5
(c) three (d) infinite (c) 7 (d) 12
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^ ^ ^
26.
22. If the projection of a i 2 j 3 k on
^ ^
b 2 i k is zero, then the value of is [2020]
(a) 0 (b) 1
2 3
(c) (d)
3 2
^ ^ ^
23. If i , j , k are unit vectors along three mutually
perpendicular directions, then [2020]
^ ^ ^ ^
27.
(a) i . j 1 (b) i j 1
^ ^ ^ ^
(c) i . k 0 (d) i k 0
Assertion-Reasoning (1 mark)
24. Assertion (A): The position of a particle in a
rectangular
28.
Reason (R) is correct. The displacement vector of the
particle that moves from point P(2, 3, 5) to point Q(3,
4, 5) Case Study Questions
29. Read the text carefully and answer the questions:
A building is to be constructed in the form of a
25. Assertion (A): Multiplying any vector by an scalar is triangular pyramid, ABCD as shown in the figure.
a meaningful operations.
Reason (R): In uniform motion speed remains
constant.
a) Both A and R are true and R is the correct
explanation of A.
b) Both A and R are true but R is not the correct
explanation of A.
c) A is true but R is false.
d) A is false but R is true.
Let its angular points are A(0, 1, 2), B(3, 0, 1), C(4, 3,
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6) and D(2, 3, 2) and G be the point of intersection of
the medians of △ BCD.
Questions
^ ^ ^ ^ ^ ^
31. If a 4 i j k and b 2 i 2 j k , then find a
unit vector along the vector a b . [2023]
32. Find the area of a parallelogram whose adjacent
^ ^ ^
sides are determined by the vectors a i j 3 k
^ ^ ^
and b 2 i 7 j k . [2023]
30. Read the text carefully and answer the questions:
A plane started from airport O with a velocity of ^ ^ ^ ^ ^
120 m/s towards east. Air is blowing at a velocity 33. The two vectors j k and 3 i j 4 k represent the
of 50 m/s towards the north As shown in the two sides AB and AC, respectively of a ABC . Find
figure. the length of the median through A.
The plane travelled 1 hr in OA direction with the [Delhi 2016, Foreign 2015]
resultant velocity. From A and B travelled 1 hr with
keeping velocity of 120 m/s and finally landed at B.
34. If 𝑎⃗ = 2𝑖̂ + 4𝑗̂ − 5𝑘̂ 𝑎𝑛𝑑 𝑏⃗⃗ = 𝑖̂ + 2𝑗̂ + 3𝑘̂, Find the
unit vector in the direction of 𝑎⃗ + 𝑏⃗⃗.
35. If 𝐴𝐵𝐶𝐷 is a quadrilateral and E and F are the mid-
points of AC and BD, prove that
36. Using vectors find the area of triangle ABC with
vertices 𝐴(1, 2, 3), 𝐵(2, −1, 4) 𝑎𝑛𝑑 𝐶(4, 5, −1).
37. Write the position vector of the point which divides
the join of points with position vector 3𝑎⃗ −
2𝑏⃗⃗ 𝑎𝑛𝑑 2𝑎⃗ + 3𝑏⃗⃗ in the ratio 2:1.
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38. The two adjacent sides of a parallelogram are
2𝑖̂ − 4𝑗̂ − 5𝑘̂ 𝑎𝑛𝑑 2𝑖̂ + 2𝑗̂ + 3𝑘̂. 49. Find the direction cosines of the vector joining the
Find the two unit vectors parallel to its diagonals. points A(1,2, −3) and B(−1, −2,1) directed from B
Using the diagonal vectors, find the area of the to A.
parallelogram.
50. Find the area of parallelogram whose adjacent sides
are determined by the vector 𝑎⃗ = 𝑖̂ − 𝑗̂ +
39. Find a unit vector in the direction of 𝑎⃗ = 3𝑖̂ − 2𝑗̂ + 2𝑘̂ 𝑎𝑛𝑑 𝑏⃗⃗ = 2𝑖̂ − 𝑗̂ − 𝑘̂
6𝑘̂.
51. If 𝑎⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂ and 𝑏⃗⃗ = 𝑗̂ − 𝑘̂ , find a vector 𝑐⃗ such
that
40. Find the projection of the vector 𝑖̂ − 𝑗̂ on the 𝑎⃗ × 𝑐⃗ = 𝑏⃗⃗and 𝑎⃗ ∙ 𝑐⃗ = 3.
vector𝑖̂ + 𝑗̂.
52. Find a unit vector perpendicular to both of the
41. Find the projection of 𝑎⃗ on 𝑏⃗⃗ , if 𝑎⃗ ∙ 𝑏⃗⃗ = 8 and 𝑏⃗⃗ = vector
2𝑖̂ + 6𝑗̂ + 3𝑘̂ 3𝑎⃗ + 2𝑏⃗⃗ 𝑎𝑛𝑑 3𝑎⃗ − 2𝑏⃗⃗, where 𝑎⃗ = 𝑖̂ + 𝑗̂ +
𝑘̂ 𝑎𝑛𝑑 𝑏⃗⃗ = 𝑖̂ + 2𝑗̂ + 3𝑘̂ .
42. Find ‘λ’ when the projection of 𝑎⃗ = 𝜆𝑖̂ + 𝑗̂ + 4𝑘̂ on 53. If the vector −𝑖̂ + 𝑗̂ − 𝑘̂ bisects the angle between
𝑏⃗⃗ = 2𝑖̂ + 6𝑗̂ + 3𝑘̂ is 4 units. the vector 𝑐⃗and the vector 3𝑖̂ + 4𝑗̂, then find the
unit vector in the direction of 𝑐⃗.
43. If 𝑎⃗, 𝑏⃗⃗are two vectors such that |𝑎⃗ + 𝑏⃗⃗| = |𝑎⃗|, then 54. If 𝑎⃗, 𝑏⃗⃗are unit vectors such that the vector 𝑎⃗ + 3𝑏⃗⃗ is
prove that perpendicular to 7𝑎⃗ − 5𝑏⃗⃗ 𝑎𝑛𝑑 𝑎⃗ − 4𝑏⃗⃗ is
2𝑎⃗ + 𝑏⃗⃗ is perpendicular to 𝑏⃗⃗. perpendicular to7𝑎⃗ − 2𝑏⃗⃗, then find the angle
between 𝑎⃗ 𝑎𝑛𝑑 𝑏⃗⃗.
44. Write the direction ratio’s of the vector 𝑎⃗ = 𝑖̂ + 𝑗̂ −
2𝑘̂ and hence calculate its direction cosines. 55. Using vectors, find the area of the triangle ABC
with vertices A(1,2,3), B(2, -1,4) and C(4,5,-1)
45. Find the vector 𝑝⃗which is perpendicular to both
𝛼⃗ = 4𝑖̂ + 5𝑗̂ − 𝑘̂ and 𝛽⃗ = 𝑖̂ − 4𝑗̂ + 5𝑘̂ and 𝑝⃗ ∙ 𝑞⃗ = 56. If 𝑖̂ − 𝑗̂ + 𝑘̂ , 2 𝑖̂ − 5 𝑗̂, 3𝑖̂ − 2 𝑗̂ − 3𝑘̂ and 𝑖̂ − 6𝑗̂ − 𝑘̂
21, where 𝑞⃗ = 3𝑖̂ + 𝑗̂ − 𝑘̂. respectively are the position vectors of points A, B,
C and D, then find the angle between the straight
lines AB and CD. Find whether 𝐴𝐵 ̅̅̅̅ and 𝐶𝐷
̅̅̅̅ are
collinear or not.
46. If 𝑎⃗ = 𝑖̂ − 𝑗̂ + 7𝑘̂and 𝑏⃗⃗ = 5𝑖̂ − 𝑗̂ + 𝜆𝑘̂ , then find the
57. Let 𝑎⃗ = 4𝑖̂ + 5𝑗̂ − 𝑘̂ , 𝑏⃗⃗ = 𝑖̂ − 4𝑗̂ + 5𝑘̂ and 𝑐⃗ =
value of λ so that 𝑎⃗ + 𝑏⃗⃗and 𝑎⃗ − 𝑏⃗⃗are perpendicular
3𝑖̂ − 𝑗̂ − 𝑘̂.Find a vector 𝑑⃗ which is perpendicular
vectors.
to both 𝑐⃗ and 𝑏⃗⃗ and 𝑑⃗ .𝑎⃗ = 21.
47. If 𝑎⃗, 𝑏⃗⃗, 𝑐⃗are three vectors such that |𝑎⃗| = 3, |𝑏⃗⃗| =
4 𝑎𝑛𝑑 |𝑐⃗| = 5 and each one of them is
perpendicular to the sum of the other two, then 58. Vectors 𝑎⃗, ⃗⃗⃗
𝑏 and 𝑐⃗ are such that 𝑎⃗ + 𝑏⃗⃗ − 𝑐⃗ = 0 and
find |𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗|. |𝑎⃗| = 3, |𝑏⃗⃗| = 5 and |𝑐⃗| = 7 . Find the angle
between 𝑎⃗ and 𝑏⃗⃗.
48. If vectors 𝑎⃗ = 2𝑖̂ + 2𝑗̂ + 3𝑘̂ , 𝑏⃗⃗ = −𝑖̂ + 2𝑗̂ +
𝑘̂ 𝑎𝑛𝑑 𝑐⃗ = 3𝑖̂ + 𝑗̂ are such that 𝑎⃗ + 𝜆𝑏⃗⃗ is
perpendicular to 𝑐⃗, then find the value of 𝜆
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Sure Shot Questions
Chapter – 11
Three-Dimensional Geometry
MCQ (1 mark) (a) 9 sq. units (b) 18 sq. units
(c) 27 sq. units (d) 81 sq. units
1. Distance of the point ( , , ) from y-axis is
7. The locus represented by xy + yz = 0 is
(a) (b) | | (a) A pair of perpendicular lines
(c) | | | | (d) 2 2 (b) A pair of parallel lines
(c) A pair of parallel planes
(d) A pair of perpendicular planes
2. If the directions cosines of a line are k, k, k, then
8. The plane 2x – 3y + 6z – 11 = 0 makes an angle
(a) k > 0 (b) 0 < k < 1
1 1 sin 1 ( ) with x-axis. The value of is equal to
(c) k = 1 (d) k or
3 3 3 2
(a) (b)
2 3
2^ 3 ^ 6 ^ 2 3
3. The distance of the plane r . i j k 1 (c) (d)
7 7 7 7 7
from the origin is
9. Distance of the point (p,q,r) from y-axis is [2023]
(a) 1 (b) 7
(a) q (b) |q|
1
(c) (d) None of these (c) |q| + |r| (d) p2 r 2
7
4. The sine of the angle between the straight line 10. The length of the perpendicular drawn from the
x 2 y 3 z 4 point (4, -7, 3) on the y – axis is [2020]
and the plane (a) 3 units (b) 4 units
3 4 5
(c) 5 units (d) 7 units
2 x 2 y z 5 is
10 4 11. The vector equation of XY – plane is [2020]
(a) (b)
6 5 5 2 ^ ^
(a) r . k 0 (b) r . j 0
2 3 2 ^
(c) (d)
5 10 (c) r . i 0 (d) r . n 1
5. The reflection of the point ( , , ) in the xy-
plane is 1 1 1
12. If the direction cosines of a line are , , ,
(a) ( , ,0) (b) (0, 0, ) a a a
(c) ( , , ) (d) ( , , ) then [2023]
(a) 0 < a < 1 (b) a > 2
6. The area of the quadrilateral ABCD, where A(0, 4, (c) a > 0 (d) a 3
1), B(2, 3, -1), C(4, 5, 0) and D(2, 6, 2), is equal to
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13. If a line makes angles of 900, 1350 and 450 with the Case Study Questions
x, y and z axes respectively, then its direction
cosines are [2023] 17. Read the text carefully and answer the questions:
1 1 1 1 Consider
(a) 0, , (b) , 0,
2 2 2 2 the following diagram, where the forces in the cable
are
1 1 1 1
(c) , 0, (d) 0, , given.
2 2 2 2
Assertion – Reasons
Is Assertion and Reason based question carrying 1
mark. Two statements are given, one labelled Assertion
(A) and the other labelled Reason (R). Select the correct
answer from the codes (a), (b), (c) and (d) as given
below.
(a) What is the equation of the line along cable AD?
14. Assertion (A): The lines (b) What is length of cable DC?
(c) Find vector DB
r a1 b1 and r a2 b2 are perpendicular, (d) What is sum of vectors along the cable?
when b1 . b2 0 .
18. Read the text carefully and answer the questions: If
Reason (R): The angle between the lines
𝑎1 , 𝑏1 ,
r a1 b1 and r a2 b2 is given by
(a) Both A and R are true and R is the correct
explanation of A.
(b) Both A and R are true and R is not the correct
explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
15. The angle between the lines 2 x 3 y z and 6x
= -y = -4z is [2023]
(a) 00 (b) 30 0
(c) 450 (d) 900
y z
16. If the two lines L1 : x 5,
3 2
y z
L2 : x 2, are perpendicular, then the
1 2
value of is [2020C]
2
(a) (b) 3
3
7
(c) 4 (d)
3
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Questions 9. Write the Cartesian equation of the following line
given in vector form.
𝑟⃗ = (2𝑖̂ + 𝑗̂ − 4𝑘̂ ) + 𝜆(𝑖̂ − 𝑗̂ − 𝑘̂ )
1. The equations of a line are 5x - 3 = 15y -r 7 = 3 - 10z.
Write the direction cosines of the line and find the 10. Find the equation of a plane that cuts the
coordinates of a point through which it passes. coordinates axes at
[2023] (𝑎, 0, 0) , (0, 𝑏, 0) 𝑎𝑛𝑑 (0, 0, 𝑐).
11. Write the vector equation of the line passing
2. The cartesian equation o a line AB is: through the point (1, −1,2) and parallel to the line
𝑥−3 𝑦−1 𝑧+1
2x 1 y 2 z 3 whose equations are 1 = 2 = −2
12 2 3
Find the direction cosines of a line parallel to line 12. If the equation of the line AB is
3−𝑥 𝑦+2 𝑧+2
AB. [Term II, 2021 – 22] = =
3 −2 6
13. Find the shortest distance between the following
x 1 y 3 z 5 pair of skew lines:
3. Show that the lines and
3 5 7 𝑥−1 2−𝑦 𝑧+1 𝑥+2 𝑦−3 𝑧
= = , = =
x2 y 4 z 6 2 3 4 −1 2 3
intersect. Also find their ^ ^ ^
1 3 5 17 i 10 j k
point of intersection. [Delhi 2014]
14. Show that the lines
𝑟⃗ = (𝑖̂ + 𝑗̂ − 𝑘̂ ) + 𝜆(3𝑖̂ − 𝑗̂) and
4. Find the equation of a line passing through the 𝑟⃗ = (4𝑖̂ − 𝑘̂ ) + 𝜇(2𝑖̂ + 3𝑘̂ ) intersect. Also find their
point (1, 2, -4) and perpendicular to two lines point of intesection.
𝑟⃗ = (8𝑖̂ − 19𝑗̂ + 10𝑘̂) + 𝜆(3𝑖̂ − 16𝑗̂ + 7𝑘̂ )
𝑎𝑛𝑑 𝑟⃗ = (15𝑖̂ − 29𝑗̂ + 5𝑘̂ ) + 𝜇 (3𝑖̂ +
̂
8𝑗̂ − 5𝑘 ). 15. Find the coordinates of the point where the line
through the points (3, −4, −5) and (2, −3, 1)
crosses the plane 2𝑥 + 𝑦 + 𝑧 = 7.
5. Find the equation of the plane passing through the
points (−1, 2, 0), (2, 2, −1) and parallel to the line
𝑥−1 2𝑦+1 𝑧+1
1
= 2 = −1 .
16. Find the shortest distance between the following
lines :
𝑥−3 𝑦−5 𝑧−7 𝑥+1 𝑦+1
6. Find the value of p, so that the lines 𝑙1 : = = 𝑎𝑛𝑑 =
1−𝑥 7𝑦−14 𝑧−3 7−7𝑥 𝑦−5 6−𝑧
1 −2 1 7 −6
= 𝑝 = 2 𝑎𝑛𝑑 𝑙2 : 3𝑝 = 1 = 5 and 𝑧+1
3 =
perpendicular to each other. Also find the 1
equations of a line passing through a point (3,2,-4)
17. Find the vector and cartesian equations of the line
and parallel to line 𝑙1 .
passing through the point (1, 2, −4) and
perpendicular to the two lines
𝑥 − 8 𝑦 + 19 𝑧 − 10 𝑥 − 15 𝑦 − 29
7. Write the direction ratios of the vector 3𝑎⃗ + = = 𝑎𝑛𝑑 =
3 −16 7 3 8
2𝑏⃗⃗where 𝑧−5
=
𝑎⃗ = 𝑖̂ + 𝑗̂ − 2𝑘̂ and 𝑏⃗⃗ = 2𝑖̂ − 4𝑗̂ + 5𝑘.
̂ −5
8. A line passes through the point with position vector
̂ and makes angles 60°, 120°, and 45°
2𝑖̂ − 3𝑗̂ + 4𝑘.
with 𝑥, 𝑦 𝑎𝑛𝑑 𝑧 − 𝑎𝑥𝑖𝑠 respectively. Find the
equation of the line in the Cartesian form.
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18. Find the coordinates of the foot of perpendicular are at right angles. Also, find whether the lines are
and the length of the perpendicular drawn from the intersecting or not.
point P(5, 4, 2) to the line
𝑟⃗ = (−𝑖̂ + 3𝑗̂ + 𝑘̂ ) + 𝜆(2𝑖̂ + 3𝑗̂ − 𝑘̂ ). Also find the 24. Find the vector and cartesian equations of the line
image of P in this line. through the point (1, 2, -4) and perpendicular to the
two lines
𝑟⃗ = (8𝑖̂ − 19𝑗̂ + 10𝑘̂) + 𝜆(3𝑖̂ − 16𝑗̂ + 7𝑘̂
19. Prove that the line through A(0, -1, -1) and B(4, 5, and
1) intersects the line through C(3, 9, 4) and D(- 𝑟⃗ = (15𝑖̂ + 29𝑗̂ + 5𝑘̂ + 𝜇(3𝑖̂ + 8𝑗̂ − 5𝑘̂ ).
4,4,4).
1−𝑥
25. Find the value of p, so that the lines 𝑙1 : 3
=
𝑥+1 𝑦+3 𝑧+5 𝑥−2
20. Show that the lines = = and = 7𝑦−14 𝑧−3 7−7𝑥 𝑦−5 6−𝑧
3 5 7 1
𝑝
− 2 and 𝑙2 : 3𝑝 = 1
= 5
are
𝑦−4 𝑧−6
3
= 5
in intersection. perpendicular to each 3p 1 5 other. Also find the
equation of a line passing through a point (3,2, -4)
21. Show that the lines 𝑟⃗⃗⃗ = 3𝑖̂ + 2𝑗̂ − 4𝑘̂ + 𝜆(𝑖̂ + and parallel to line 𝑙1 .
2𝑗̂ + 2𝑘̂ ): 𝑟⃗ = 5𝑖̂ − 2𝑗̂ + 𝜇(3𝑖̂ + 2𝑗̂ + 6𝑘̂ ) are
𝑥+2 2𝑦−7
intersecting. Hence find their point of intersection. 26. Find the direction cosines of the line 2
= 6
=
5−𝑧
6
.
Also, find the vector equation of 2 6 6 the line
22. If a line makes 60° and 45° angles with the positive
through the point A(-1, 2,3) and parallel to the
directions of 𝑥 −axis and 𝑧 −axis respectively, then
given line.
find the angle that it makes with the positive
direction of 𝑦 −axis. Hence, write the direction
cosines of the line. 27. Find the shortest distance between the lines 𝑟⃗ =
(𝑖̂ + 2𝑗̂ + 𝑘̂ ) + 𝜆(𝑖̂ − 𝑗̂ + 𝑘̂ )and 𝑟⃗ = (2𝑖̂ − 𝑗̂ −
1−𝑥 7𝑦−14 𝑘̂ ) + 𝜇(2𝑖̂ + 𝑗̂ + 2𝑘̂
23. Find the value of 𝜆, So that the lines = =
3 𝜆
𝑧−3 7−7𝑥 𝑦−5 6−𝑧 7−7𝑥 𝑦−5 6−𝑧
𝑎𝑛𝑑 = = and = =
2 3𝜆 1 5 3𝜆 1 5
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Sure Shot Questions
Chapter – 12
Linear Programming
MCQ (1 mark)
1. The corner points of the feasible region
determined by the system of linear constraints are
(0, 0), (0, 40), (20, 40), (60, 20), (60, 0). The
objective function is Z = 4x + 3y.
Compare the quantity in Column A and Column B
Column A Column B
(a) 0 (b) 8
Maximum of Z 325 (c) 12 (d) -18
(a) The quantity in column A is greater
(b) The quantity in column B is greater
(c) The two quantities are equal 4. Corner points of the feasible region for an LPP are
(d) The relationship cannot be determined on the (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5).
basis of the information suppled. Let F = 4x + 6y be the objective function.
The minimum value of F occurs at
2. The feasible region for a LPP is shown shaded in (a) (0, 2) only
the figure. Let Z = 3x – 4y be the objective function. (b) (3, 0) only
Minimum of Z occurs at (c) The mid – point of the line segment joining the
points (0, 2) and (3, 0) only
(d) Any point on the line segment joining the
points (0, 2) and (3, 0)
5. Corner points of the feasible region determined by
the system of linear constraints are (0, 3), (1, 1)
and (3, 0). Let Z = px + qy, where p, q > 0. Condition
on p and q so that the minimum of Z occurs at (3,
(a) (0, 0) (b) (0, 8) 0) and (1, 1) is
(c) (5, 0) (d) (4, 10) q
(a) p 2q (b) p
2
3. The feasible region for an LPP is shown shaded in (c) p 3q (d) p q
the figure. Let F = 3x – 4y be the objective function.
Maximum value of F is
6. Which of the following points satisfies both the
inequations 2x + y 10 and x 2 y 8 ?
[2023]
(a) (-2, 4) (b) (3, 2)
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(c) (-5, 6) (d) (4, 2) (c) x 2 y 104 and 2 x y 76
(d) x 2 y 104 and 2 x y 38
7. The solution set of the inequation 3x + 5y < 7 is
[2023]
(a) Whole xy-plane except the points lying on the
12. If the minimum value of an objective function Z =
line 3x + 5y = 7.
ax + by occurs at two points (3, 4) and (4, 3) then
(b) Whole xy-plane along with the points lying on
[Term I, 2021 – 22]
the line 3x + 5y = 7.
(a) a + b = 0 (b) a = b
(c) Open half plane containing the origin except
(c) 3a = b (d) a = 3b
the points of line 3x + 5y = 7.
(d) Open half plane not containing the origin.
13. For the following LPP, maximise Z = 3x + 4y subject
[2023]
to constraints x y 1, x 3, x 0, y 0 the
maximum value is [Term I, 2021 – 22]
8. If the corner points of the feasible region of an LPP
(a) 0 (b) 4
are (0, 3), (3, 2) and (0, 5), then the minimum value
(c) 25 (d) 30
of Z = 11x + 7y is [Term I, 2021 – 22]
(a) 21 (b) 33
14. The corner points of the feasible region
(c) 14 (d) 35
determined by the system of linear inequalities are
(0, 0), (4, 0), (2, 4) and (0, 5). If the maximum value
of z = ax + by, where a, b > 0 occur at both (2, 4)
9. The number of solutions of the system of
and (4, 0), then [2020]
inequations x 2 y 3,3x 4 y 12, x 0, y 1
(a) a = 2b (b) 2a = b
is [Term I, 2021 – 22] (c) a = b (d) 3a = b
(a) 0 (b) 2
(c) finite (d) infinite
15. In an LPP, if the objective function z = ax + by has
10. The maximum value of Z = 3x + 4y subject to the
the same maximum value on two corner points of
constraints x 0, y 0 and x y 1 is
the feasible region, then the number of points at
[Term I, 2021 – 22] which zmax occurs is [2020]
(a) 7 (b) 4 (a) 0 (b) 2
(c) 3 (d) 10 (c) finite (d) infinite
11. The feasible region of an LPP is given in the 16. The feasible region for an LPP is shown below:
following figure [Term I, 2021 – 22] Let z = 3x – 4y be the objective function.
Minimum of z occurs at [NCERT Exemplar,
2020]
Then, the constraints of the LPP are x 0, y 0
and
(a) 2 x y 52 and x 2 y 76
(b) 2 x y 104 and x 2 y 76
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(a) (0, 0) (b) (0, 8) (b) (3, 0) only
(c) (5, 0) (d) (4, 10) (c) (0.6, 1.6) and (3.0) only
(d) At every point of the line – segment joining
the points (0.6, 1.6) and (3.0)
17. The graph of the inequality 2x + 3y > 6 is
[2020]
(a) Half plane that contains the origin
(b) Half plane that neither contains the origin nor 21. Based on the givens shaded region as the feasible
the points of the line 2x + 3y = 6. region in the graph, at which point(s) is the
(c) Whose XOY – plane excluding the points on objective function Z = 3x + 9y maximum?
the line 2x + 3y = 6. [Term I, 2021 – 22]
(d) Entire XOY – plane.
18. The objective function of an LPP is [2020]
(a) A constant
(b) A linear function to be optimized
(c) An inequality
(d) A quadratic expression
19. The solution set of the inequality 3x + 5y < 4 is (a) Point B
[2022 – 23] (b) Point C
(a) An open half – plane not containing the (c) Point D
origin. (d) Every point on the line segment CD
(b) An open half – plane containing the origin.
(c) The whole XY – plane not containing the line 22. In the given graph, the feasible region for a LPP is
3x + 5y = 4. shaded. The objective function Z = 2x – 3y, will be
(d) A closed half plane containing the origin. minimum at [Term I, 2021 – 22]
20. The corner points of the shaded unbounded
feasible region of an LPP are (0, 4), (0.6, 1.6) and
(3, 0) as shown in the figure. The minimum value of
the objective function Z = 4x + 6y occurs at
(a) (4, 10) (b) (6, 8)
(c) (0, 8) (d) (6, 5)
23. In the feasible region, the minimum value of Z
occurs at [Term I, 2021 – 22]
(a) A unique point
[2022 – 23] (b) No point
(a) (0.6, 1.6) only (c) Infinitely many points
(d) Two points only
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28. Assertion (A): The maximum value of Z = 11x + 7y
24. For an objective function Z = ax + by, where a, b > subject to the constraints 2x + y ≤ 6; x ≤ 2; x ≥ 0, y ≥
0; the corner points of the feasible region 0
occurs at the comer point (0, 6).
determined by a set of constraints (linear
Reason (R): If the feasible region of the given LPP is
inequalities) are (0, 20), (10, 10), (30, 30) and (0, bounded, then the maximum and minimum value of
40). The condition on a and b such that the the objective function occurs at corner points.
maximum Z occurs at both the points (30, 30) and a) Both A and R are true and R is the correct
(0, 40) is [Term I, 2021 – 22] explanation of A.
(a) b – 3a = 0 (b) a = 3b b) Both A and R are true but R is not the correct
(c) a + 2b = 0 (d) 2a – b = 0 explanation of A.
c) A is true but R is false.
d) A is false but R is true.
25. In a linear programming problem, the constraints
on the decision variables x and y are 29. Assertion (A): Consider the linear programming
problem. Maximise Z = 4x + y Subject to
x 3 y 0, y 0, 0 x 3 . The feasible region constraints x + y ≤ 50, x + y ≥ 100, and x, y ≥ 0.
[Term I, 2021 – 22] Then, maximum value of Z is 50.
(a) Is not in the first quadrant Reason (R): If the shaded region is not bounded
(b) Is bounded in the first quadrant then maximum value cannot be determined.
(c) Is unbounded in the first quadrant a) Both A and R are true and R is the correct
explanation of A.
(d) Does not exist
b) Both A and R are true but R is not the correct
explanation of A.
Assertion-Reasoning (1 mark) c) A is true but R is false.
d) A is false but R is true.
26. Assertion (A): Maximum value of Z = 3x + 2y,
subject to the constraints x + 2y ≤ 2; x ≥ 0; y ≥ 0 will
30. Assertion (A): The maximum value of Z = x + 3y.
be obtained at point (2, 0).
Such
Reason (R): In a bounded feasible region, it always
that 2x + y ≤ 20, x + 2y ≤ 20, x, y ≥ 0 is 30.
exist a maximum and minimum value.
Reason (R): The variables that enter into the
a) Both A and R are true and R is the correct
problem are called decision variables.
explanation of A.
a) Both A and R are true and R is the correct
b) Both A and R are true but R is not the correct
explanation of A.
explanation of A.
b) Both A and R are true but R is not the correct
c) A is true but R is false.
explanation of A.
d) A is false but R is true.
c) A is true but R is false.
d) A is false but R is true.
27. Assertion (A): The maximum value of Z = 5x + 3y,
satisfying the conditions x ≥ 0, y ≥ 0 and 5x + 2y ≤
10, is 15. Case Study Questions
Reason (R): A feasible region may be bounded or
unbounded. 31. Read the text carefully and answer the questions:
a) Both A and R are true and R is the correct Corner points of the feasible region for an LPP are
explanation of A. (0, 3), (5, 0), (6, 8), (0, 8). Let Z = 4x - 6y be the
b) Both A and R are true but R is not the correct objective function.
explanation of A.
c) A is true but R is false.
d) A is false but R is true.
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Subject to the constraints
[2023]
2 x y 4
x y 3
x 2y 2
x 0, y 0
34. Solve the following linear programming problem
(a) At which corner point the minimum value of Z graphically.
occurs? Maximize: P = 70x + 40y
(b) At which corner point the maximum value of Z Subject to: 3x 2 y 9, 3x y 9, x 0, y 0 .
occurs?
[2023]
(c) What is the value of (maximum of Z - minimum of
Z)?
(d) The corner points of the feasible region determined 35. Find graphically, the maximum value of 𝑧 = 2𝑥 +
by the system of linear inequalities are 5𝑦, subject to constraints given below:
2𝑥 + 4𝑦 ≤ 8
3𝑥 + 𝑦 ≤ 6
32. Read the text carefully and answer the questions: 𝑥+𝑦 ≤4
Dinesh is having a jewelry shop at Green Park, 𝑥 ≥ 0, 𝑦 ≤ 0
normally he does not sit on the shop as he remains
busy in political meetings. The manager Lisa takes
care of jewelry shop where she sells earrings and 36. Solve the following linear programming problem
necklaces. She gains profit of Rs 30 on pair of graphically :
earrings & Rs 40 on neckless. It takes 30 minutes to Maximize Z = 7𝑥 + 10𝑦
make a pair of earrings and 1 hour to make a Subject to the constraints
necklace, and there are 10 hours a week to make 4𝑥 + 6𝑦 ≤ 240
jewelry. In addition, there are only enough 6𝑥 + 3𝑦 ≤ 240
𝑥 ≥ 10
materials to make 15 total of jewelry items per
𝑥 ≥ 0, 𝑦 ≥ 0
week.
37. Solve the following LPP graphically: Minimize 𝑧 =
5𝑥 + 7𝑦 Subject to the constraints 2𝑥 + 𝑦 ≥
8 𝑥 + 2𝑦 ≥ 10 𝑥, 𝑦 ≥ 0
38. Solve the following LPP graphically: Minimise 𝑍 =
(a) Formulate the above information 5𝑥 + 10𝑦 Subject to constraints 𝑥 + 2𝑦 ≤
mathematically. 120, 𝑥 + 𝑦 ≥ 60, 𝑋 − 2𝑦 ≥ 0 and 𝑥, 𝑦 ≥ 0
(b) Graphically represent the given data.
(c) To obtain maximum profit how many pair of
earing and neckleses should be sold? 39. Maximise 𝑍 = 𝑥 + 2𝑦 Subject to the constraints:
(d) What would be the profit if 5 pairs of earrings 𝑥 + 2𝑦 ≥ 100,2𝑥 − 𝑦 < 0 2𝑥 + 𝑦 ≤ 200 𝑥, 𝑦 ≥
and 5 0 Solve the above LPP graphically.
necklaces are made?
Questions 40. Find graphically, the maximum value of 𝑧 = 2𝑥 +
5𝑦, subject to constraints given below: 2𝑥 + 4𝑦 ≤
8, 3𝑥 + 𝑦 ≤ 6, 𝑥 + 𝑦 ≤ 4; 𝑥 ≥ 0, 𝑦 ≥ 0
33. Solve the following linear programming problem
graphically:
Maximise z = -3x – 5y
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41. Maximise 𝑧 = 8𝑥 + 9𝑦 subject to the constraints
given below: 2𝑥 + 3𝑦 ≤ 6,3𝑥 − 2𝑦 ≤ 6, 𝑦 ≤
1; 𝑥, 𝑦 ≥ 0 43. Let 𝑧 = 𝑋 + 𝑦, then the maximum of z subject to
constraints 𝑦 ≥ |𝑥| − 1, 𝑦 ≤ 1 − |𝑥|, 𝑥 ≥
0, 𝑦 ≥ 0 𝑖𝑠
42. Minimise 𝑍 = 30𝑥 + 20𝑦 subject to 𝑥 + 𝑦 ≤
8, 𝑥 + 4𝑦 ≥ 12,5𝑥 + 8𝑦 ≥ 20, 𝑥, 𝑦 ≥ 0
For Solutions
Download Arvind Academy App
Click Link: http://bit.ly/2kTWbkj
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Sure Shot Questions
Chapter – 13
Probability
MCQ (1 mark) 5. If A and B are two events such that
1 1 1
1. If P(A)
4
and P(A B)
7
, then P(B | A) is P(A) , P(B) , P(A | B) , then
5 10 2 3 4
equal to P(A' B') equals to
1 1 1 3
(a) (b) (a) (b)
10 8 12 4
7 17 1 3
(c) (d) (c) (d)
8 20 4 16
6. If P(A) = 0.4, P(B) = 0.8 and P(B|A) = 0.6, then
7 17 P(A B) is equal to
2. If P(A B) and P(B) , then P(A|B)
10 20 (a) 0.24 (b) 0.3
equals (c) 0.48 (d) 0.96
14 17
(a) (b)
17 20
7 1 7. If A and B are two events and A , B , then
(c) (d)
8 8 (a) P(A | B) P(A).P(B)
P(A B)
(b) P(A | B)
P(B)
3 2 3
3. If P(A) , P(B) and P(A B) , then (c) P(A | B).P(B | A) 1
10 5 5
P(B | A) P(A | B) equals (d) P(A | B) P(A) | P(B)
1 1
(a) (b)
4 3
5 7 8. A and B are events such that P(A) = 0.4, P(B) = 0.3
(c) (d)
12 12 and P(A B) 0.5 . Then P( B ' A) equals
2 1
(a) (b)
3 2
2 3 1 3 1
4. If P(A) , P( B) and P(A B) , then (c) (d)
5 10 5 10 5
P(A' | B').P(B' | A') is equal to
5 5
(a) (b)
6 7 9. You are given that A and B are two events such
25 3 1 4
(c) (d) 1 that P(B) , P(A | B) and P(A B) ,
42 5 2 5
then P(A) equals
3 1
(a) (b)
10 5
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1 3 (b) The sum of their probabilities must be equal to
(c) (d)
2 5 1
(c) Both (a) and (b) are correct
(d) None of these
10. Refer to question 9 above P(B|A’) equals
1 3
(a) (b)
5 10 3
16. Let A and B be two events such that P(A) ,
1 3 8
(c) (d)
2 5 5 3
P(B) and P(A B) . Then
8 4
P(A | B).P(A' | B') is equal to
3 1 4 2 3
11. If P(B) , P(A | B) and P(A B) , (a) (b)
5 2 5 5 8
then P(A B) ' P(A' B) 3 6
(c) (d)
1 4 20 25
(a) (b)
5 5
1 17. If the events A and B are independent, then
(c) (d) 1
2 P(A B) is equal to
(a) P(A) + P(B) (b) P(A) – P(B)
(c) P(A) . P(B) (d) P(A) | P(B)
7 9 4
12. Let P(A) , P(B) and P(A B) .
13 13 13
Then P(A' | B) is equal to 18. Two events E and F are independent. If P(E) = 0.3,
6 4 P(E F) 0.5 , then P(E | F) P(F | E) equals
(a) (b)
13 13 2 3
(a) (b)
4 5 7 35
(c) (d)
9 9 1 1
(c) (d)
70 7
13. If A and B are events such that P(A) > 0 and
P(B) 1 , then P(A' | B') equals 19. A bag contains 5 red and 3 blue balls. If 3 balls are
(a) 1 P(A | B) (b) 1 P(A' | B) drawn at random without replacement the
1 P(A B) probability of getting exactly one red ball is
(c) (d) P(A') | P(B') 45 135
P(B') (a) (b)
196 392
15 15
14. If A and B are two independent events with (c) (d)
56 29
3 4
P(A) and P(B) , then P(A' B')
5 9 20. Refer to question 19 above. The probability that
equals exactly two of the three balls were red, the first
4 8 ball being red, is
(a) (b)
15 45 1 4
1 2 (a) (b)
(c) (d) 3 7
3 9 15 5
(c) (d)
28 28
15. If two events are independent, then
(a) They must be mutually exclusive
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21. Three persons, A, B and C, fire at a target in turn, 27. Two dice are thrown. If it is known that the sum of
starting with A. Their probability of hitting the numbers on the dice was less than 6, the
target are 0.4, 0.3 and 0.2 respectively. The probability of getting a sum 3, is
probability of two hits is 1 5
(a) (b)
(a) 0.024 (b) 0.188 18 18
(c) 0.336 (d) 0.452 1 2
(c) (d)
5 5
22. Assume that in a family, each child is equally likely
to be a boy or a girl. A family with three children is
chosen at random. The probability that the eldest 28. Which one is not a requirement of a binomial
child is a girl given that the family has at least one distribution?
girl is (a) There are 2 outcomes for each trial
1 1 (b) There is a fixed umber of trials
(a) (b)
2 3 (c) The outcomes must be dependent on each
2 4 other
(c) (d)
3 7 (d) The probability of success must be the same
for all the trials
23. A die is thrown and card is selected at random
from a deck of 52 playing cards. The probability of
getting an even number on the die and a spade 29. Two cards are draw from a well shuffled deck of 52
card is playing cards with respectively. The probability,
1 1 that both cads are queen, is
(a) (b)
2 4 1 1 1 1
1 3
(a) (b)
(c) (d) 13 13 13 13
8 4 1 1 1 4
(c) (d)
13 17 13 51
24. A box contains 3 orange balls, 3 green balls and 2
blue balls. Three balls are drawn at random from
the box without replacement. The probability of 30. The probability of guessing correctly at least 8 out
drawing 2 green balls and one blue ball is of 10 answers on a true – false type examination is
3 2 7 7
(a) (b) (a) (b)
28 21 64 128
1 167 45 7
(c) (d) (c) (d)
28 168 1024 41
25. A flashlight has 8 batteries out of which 3 are dead. 31. The probability that a person is not a swimmer is
If two batteries are selected without replacement 0.3. The probability that out of 5 persons 4 are
and tested, the probability that both are dead is swimmers is
33 9 5
(a) (b) (a) C4 (0.7)4 (0.3) (b) 5 C1 (0.7)(0.3)4
56 64
(c)
1
(d)
3 (c)
5
C4 (0.7)(0.3)4 (d) (0.7)4 (0.3)
14 28
26. Eight coins are tossed together. The probability of
32. The probability distribution of a discrete random
getting exactly 3 heads is
variable X is given below:
1 7
(a) (b) X 2 3 4 5
256 32
5 3 P(X) 5/k 7/k 9/k 11/k
(c) (d)
32 32
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The value of k is 38. For two events A and B, If P(A) = 0.4, P(B) = 0.8 and
(a) 8 (b) 16 P(B/A) = 0.6, then P(A B) is [2023]
(c) 32 (d) 48 (a) 0.24 (b) 0.3
(c) 0.48 (d) 0.96
33. For the following probability distribution:
X -4 -3 -2 -1 0 39. If for any two events A and B,
4 7
P(X) 0.1 0.2 0.3 0.2 0.2 P(A) and P(A B) , then P(B/A) is
5 10
E(X) is equal to [2023]
(a) 0 (b) -1 1 1
(c) -2 (d) -1.8 (a) (b)
10 8
7 17
(c) (d)
8 20
34. For the following probability distribution
X 1 2 3 4 40. In the following questions, a statements are
Assertion (A) is followed by a statement of Reason
P(X) 1/10 1/5 3/10 2/5
(R). Choose the correct answer out of the following
E(X2) is equal to choices:
(a) 3 (b) 5 Assertion (A): Two coins are tossed
(c) 7 (d) 10 simultaneously. The probability of getting two
heads, if it is known that at least one head comes
1
up, is .
35. Suppose a random variable X follows the binomial 3
distribution with parameters n and p, where 0 < p Reason (R): Let E and F be two events with a
< 1. If P(x = r)/P(x = n – r) is independent of n and r, P(E F)
random experiment, then P(F/ E) .
then p equals P(E)
1 1 [2023]
(a) (b)
2 3 (a) Both (A) and (R) are true and (R) is the correct
1 1 explanation of (A)
(c) (d)
5 7 (b) Both (A) and (R) are true, but (R) is not the
correct explanation of the (A)
36. Five fair coins are tossed simultaneously. The (c) (A) is true, and (R) is False.
probability of the events that atleast one head (d) (A) is false, but (R) is true.
comes up is [2023]
27 5 41. A card is picked at random from a pack of 52
(a) (b)
32 32 playing cards. Given that the picked card is a
31 1 queen, the probability of this card to be a card of
(c) (d)
32 32 spade is [2020]
1 4
(a) (b)
37. A die is thrown once. Let A be the event that the 3 13
number obtained is greater than 3. Let B be the 1 1
(c) (d)
event that the number obtained is less than 5. 4 2
Then P(A B) is [2020]
2 3 42. If A and B are two independent events with
(a) (b) 1 1
5 5 P(A) and P(B) , then P(B’|A) is equal to
(c) 0 (d) 1 3 4
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[2020] 46. A man P speaks truth with probability p and an
1 1 other man Q speaks truth with probability 2p.
(a) (b) Assertion (A): If P and Q contradict each other with
4 3 1
probability then there are two values of p.
3 2
(c) (d) 1 Reason (R): A quadratic equation with real coefficients
4
has two real roots.
a) Both A and R are true and R is the correct
explanation of A.
Assertion-Reasoning (1 mark) b) Both A and R are true but R is not the correct
explanation of A.
c) A is true but R is false.
43. Let A and B be two independent events.
d) A is false but R is true.
Assertion (A): If P(A) = 0.3 and P(A ∪ 𝐵̅ )= 0.8, then
2
P(𝐵̅) is 7
Reason (R): P(𝐸̅ ) = 1 - P(𝐸̅ ), where E is any event.
a) Both A and R are true and R is the correct
Case Study Questions
explanation of A.
Case study – based questions are compulsory.
b) Both A and R are true but R is not the correct
explanation of A. Attempt any 4 sub parts from each question. Each sub
c) A is true but R is false. – part carries 1 mark.
d) A is false but R is true. 47. In an office three employees Vinay, Sonia and Iqbal
process incoming copies of a certain form. Vinay I
process 50% of the forms, Sonia processes 20%
44. Assertion (A): Consider the experiment of drawing and Iqbal the remaining 30% of the forms. Vinay
a card from a deck of 52 playing cards, in which the has an error rate of 0.06, Sonia has an error rate of
elementary events are assumed to be equally 0.04 and Iqbal has an error rate of 0.03.
likely. If E and F denote the events the card drawn
is a spade and the card drawn is an ace
1 1
respectively, then P(E|F) = 4 and P(F|E) =3
Reason (R): E and F are two events such that the
probability of occurrence of one of them is not
affected by occurrence of the other. Such events
are called independent events.
a) Both A and R are true and R is the correct
explanation of A.
b) Both A and R are true but R is not the correct
explanation of A. Based on the above information answer the
c) A is true but R is false. following:
d) A is false but R is true. (i) The conditional probability that an error is
committed in processing given that Sonia
processed the form is
45.
(a) 0.0210 (b) 0.04 (c) 0.47 (d) 0.06
(ii) The probability that Sonia processed the form
and committed an error is
(a) 0.005 (b) 0.006 (c) 0.008 (d) 0.68
(iii) The total probability of committing an error
in processing the form is
(a) 0 (b) 0.047
(c) 0.234 (d) 1
(iv) The manager of the company wants to do a
quality check. During inspection he selects a form
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at random from the days output of processed improve profits of the company are 0.8, 0.5 and 0.3
forms. If the form selected at random has an respectively. If the change does not take place, find
error, the probability that the form is not the probability that it is due to the appointment of
C.
processed by Vinay is
30 53. A and B throw a pair of dice alternately. A wins the
(a) 1 (b)
47 game if he gets a total of 7 and B wins the game if
20 17 he gets a total of 10. If A starts the game, then find
(c) (d) the probability that B wins.
47 47
(v) Let A be the event of committing an error in
processing the form and let E1, E2 and E3 be the 7 9 4
54. If 𝑃(𝐸) = , 𝑃(𝐹) = and 𝑃(𝐸̅ /𝐹̅ ) = , then
events that Vinay, Sonia and Iqbal processed the 13 13 13
3
evaluate :
form. The value of P(E
i 1
i | A) is (i)
(ii)
𝑃(𝐸̅ /𝐹)
𝑃(𝐸/𝐹)
[2020 – 21]
(a) 0 (b) 0.03
(c) 0.06 (d) 1 55. A DIE IS ROLLED. If E = (1, 3, 5), F = (2, 3) and G = {2,
3, 4, 5}, find (a) P(𝐸 ∪ 𝐹)/𝐺)
(b) P(𝐸 ∩ 𝐹)/𝐺)
Questions
1 56. If P(F) = 1/5 and P(𝐸) =1/2 , E and F are
48 .The probability that A hits the target is and the independent events. Find P(𝐸𝑈𝐹).
3
2
probability that B hits it, is . If both try to hit the
5 57. A problem in mathematics is given to 4 students A,
target independently, find the probability that the B, C, D. Their chances of solving the problem,
1 1 1 2
target is hit. [Term II, 2021 – 22] respectively are, 3 , 4 , 5 𝑎𝑛𝑑 3. What is the
probability that (i) the problem will be solved? (ii) at
49. There are two bags. Bag contains 1 red and 3 white most one of them solve the problem?
balls, and Bag II contains 3 red and 5 white balls. A
bag is selected at random and a ball is drawn from 58. A speaks truth in 75% of the cases, while B in 90%
it. Find the probability that the ball so drawn is red of the cases. In what percent of cases are they likely
to contradict each other in stating the same fact?
in colour. [Term II, 2021 – 22]
Do you think that statement of B is always true?
50. A purse contains 3 silver and 6 copper coins and a
second purse contains 4 silver and 3 copper coins. If 59. A bag contains (2𝑛 + 1) coins. It is known that (𝑛 −
a coin is drawn at random from one of the two 1) of these coins have a head on both sides,
purses, find the probability that it is a silver coin. whereas the rest of the coins are fair. A coin is
[2020] picked up at random from the bag and is tossed. If
the probability that the toss results in a head is
31
51. A die, whose faces marked 1, 2, 3 in red and 4, 5, 6
42
, determine the value of n.
in green, is tossed. Let A be the event “number
obtained is even” and B be the event “number 60. A and B throw a pair of dice alternatively, till one of
obtained is red”. Find if A and B are independent them gets a total of 10 and wins the game. Find
events. their respective probabilities of winning, if A starts
first.
52. Three persons A, B and C apply for a job of Manager
61. A bag A contains 4 black and 6 red balls and bag B
in a private Company. Chances of their selection
contains 7 black and 3 red balls. A die is thrown. If 1
(A,B and C) are in the ratio 1:2:4, the probabilities
or 2 appears on it, then bag is chosen, otherwise
that A, B and C can introduce changes to improve to
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bag. If two balls are drawn at random (without 67. A card from a pack of 52 playing cards is lost. From
replacement) from the selected bag, find the the remaining cards of the pack, three cards are
probability of one of them being red and other drawn at random (without replacement) and are
black. found to be all spades. Find the probability of the lost
card being spade.
62. Of the students in a school, it is known that 30%
have 100% attendance and 70% students are 68. Of the students in a college, it is known that 60%
irregular. Previous year results report that 70% of reside in hostel and 40% are day scholars (not
all students who have 100% attendance attain A residing in hostel). Previous year results report that
grade and 10% irregular students attain A grade in 30% of all students who reside in hostel attain A
their annual examination. At the end of the year, grade and 20% od day scholars attain A grade in their
one student is chosen at random from the school annual examination. At the end of the year, one
and he was found to have an A grade. What is the student is chosen at random from the college and he
probability that the student has 100% attendance? has an ‘A’ grade. What is the probability that the
student is a hosteler.
63. Bag A contains 5 black and 3 red balls while bag B
contains 4 black and 4 red balls. Two balls are 69. Out of a group of 30 honest people, 20 always speak
transferred at random from bag A to bag B and then the truth. Two persons are selected at random from
a ball is drawn from bag B at random. If the ball the group. Find the probability distribution of the
drawn from bag B is found to be red, find the number of selected persons who speak the truth.
probability that two red balls were transferred from Also find the mean of the distribution. What values
A to B. are described in this question?
64. In a factory which manufactures bolts, machines A,
B and C manufacture respectively 30%, 50% and 70. Two numbers are selected at random (without
20% of the bolts. Of their outputs, 3%, 4% and 1% replacement) from the first six positive integers. Let
respectively are defective bolts. A bolt is drawn at X denote the larger of the two numbers obtained.
random from the product and is found to be Find the probability distribution of the random
defective. Find the probability that this is not variable X, and hence find the mean of the
manufactured by machine B. distribution.
Soln. Let X be the random variable.
65. Three machines E1, E2, E3 in a certain factory X can take values 2, 3, 4, 5 or 6.
produces 50%, 25% and 25% respectively, of the
Total number of ways = 6 C2 15
total daily output of electric tubes. It is known that
4% of the tube produced on each of machines E1 The probability distribution of a random variable X
and E2 are defective and that 5% of those produced is given by
on E3 are defective. If one tube is picked up at X 2 3 4 5 6
random from a day’s production, calculate the
probability that it is defective. P( 1/1 2/1 3/1 4/1 5/1
X) 5 5 5 5 5
66. Let X denote the number of hours you study during
a randomly selected school day. The probability
Mean = XP(X)
that X can take the values 𝑥, has the following form, 1 2 3 4 5
2 3 4 5 6
where 𝑘 is some unknown constant. 15 15 15 15 15
0.1 𝑖𝑓 𝑥 = 0 2 6 12 20 30 70 14
𝑘𝑥, 𝑖𝑓 𝑥 = 1 𝑜𝑟 2 .
𝑃(𝑋 = 𝑥) = { 15 15 15 15 15 15 3
𝑘(5 − 𝑥), 𝑖𝑓 𝑥 = 3 𝑜𝑟 4
X P(X) XP(X)
2 2
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 And variance =
a. Find the value of 𝑘. 2
b. What is the probability that you study (i) At least 70 14 70 196 14
two hours? (ii) Exactly two hours? (iii) At most two 3 3 3 9 9
hours?
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71. Assume that each born child is equally likely to be a 72. A bag contains 3 red and 7 black balls. Two balls
boy or a girl. If a family has two children, what is the are selected at random one-by-one without
conditional probability that both are girls? Given replacement. If the second selected ball happens to
that be red, what is the probability that the first selected
(i) the youngest is a girl, ball is also red? [Delhi 2014C]
(ii) atleast one is a girl.
[Delhi 2014]
For Solutions
Download Arvind Academy App
Click Link: http://bit.ly/2kTWbkj
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