Practice Questions Class XII Session: 2021-22 Mathematics ...

Practice Questions Class – XII

Session: 2021-22 Mathematics (Code – 041)

TERM 1

1) Given below is a relation R from the set X = {x, y, z} to itself.

R = {(x, x), (x, y), (y, x), (y, z), (x, z)}

Which of the following is true about the relation R?

A) R is reflexive and transitive but not symmetric.

B) R is symmetric and transitive but not reflexive.

C) R is transitive but neither reflexive nor symmetric.

D) R is not reflexive, not symmetric and not transitive.

2) P and Q are two matrices such that, PQ and QP, both exist.

Which of the following can be DEFINITELY concluded from the above statement?

A) PQ is equal to QP.

B) P and Q are square matrices.

C) P and Q are of the same order.

D) P and Q' are of the same order.

3) Shown below is the graph of a trigonometric function.

Which of the following represents the inverse function of the above trigonometric function in

its principal value branch?

4) Kiara is twice as old as Juhi and thrice as old as Udita. The sum of all their ages is 33 years.

If k, j and u represent the ages of Kiara, Juhi and Udita respectively, which matrix equation

represents the given scenario?

5) The function 𝑓(𝑥) = 𝑠𝑖𝑛−1 (sin 𝑥+cos 𝑥

√2) is increasing in the interval (

−𝜋

2,

𝜋

4).

Which of the following describes the nature of the function in the interval (−𝜋

2, 𝜋)?

A) The function is constant.

B) The function is increasing.

C) The function is decreasing.

D) The function is both increasing and decreasing.

6) X and Y are symmetric matrices of order 3.

Which of these is DEFINITELY a symmetric matrix?

A) XY – YX

B) XY + YX

C) XX' + YY'

D) X(Y + Y')

7) The objective function of a linear programming problem, along with the graph of its

constraints, is given below.

Z = 3x + 4y

What is the maximum value of Z and where is it obtained?

A) Z max = 30 obtained at (2, 6)

B) Z max = 34 obtained at (6, 4)

C) Z max = 38 obtained at (10, 2)

D) Z max = 42 obtained at (6, 6)

8) If Fc is the set of all continuous functions and Fd is the set of all differentiable functions, then

which among the following correctly represents the relationship between the two sets?

9) Three students Abha, Bhakti and Chirag were asked to define a ONE-ONE function from set

X = {1, 3, 5, 7, 9} to set Y = {0, 2, 4, 6, 8}. Their responses are shown below:

Aabha: f = {(1, 0), (1, 2), (1, 4), (1, 6), (1, 8)}

Bhakti: f = {(1, 0), (3, 2), (5, 4), (7, 6), (9, 8)}

Chirag: f = {(1, 4), (3, 4), (5, 4), (7, 4), (9, 4)}

Who defined a ONE-ONE function correctly?

A) only Bhakti

B) only Aabha and Bhakti

C) only Bhakti and Chirag

D) only Chirag and Aabha

10) What is the value of 𝑡𝑎𝑛 (𝑐𝑜𝑠𝑒𝑐−1 (−2√3

3))?

A) -√3

B) −1

√3

C) 1

√3

D) √3

11) A solid sphere of gold is being melted such that the radius is decreasing uniformly. When its

radius is 1 cm, the rate of change of its volume is 2 cm3/s.

Which of the following is the rate of change of its surface area at the same time?

A) 2 cm2/s

B) 4 cm2/s

C) 4π cm2/s

D) 8π cm2/s

12) Observe the function f(x) graphed below.

Which of the following represent section(s) where f ''(x) is zero?

A) only BC

B) only CD

C) only BC and EF

D) only AB, CD and DE

13) Read the linear programming problem given below.

“A pharmaceutical company manufactures two drugs - drug A and drug B. The process

involves two steps - synthesis and testing. Each lot of drug A requires 15-man hours for

synthesis and 3-man hours for testing. Each lot of drug B requires 5-man hours for synthesis

and 2-man hours for testing. For synthesizing and testing, the maximum man hours available

per week are 390 and 24 respectively. The company makes a profit of Rs 3500 on each lot of

drug A and Rs 8000 on each lot of drug B. How many lots of drug A and drug B should be

manufactured each week to make a maximum profit?”

What is the objective function of the above linear programming problem?

(Note: Consider the number of lots of drug A to be x and drug B to be y.)

A) 3x + 2y ≤ 24

B) 15x + 5y ≤ 390

C) 390x + 24y

D) 3500x + 8000y

14) Which of the following is an equivalence relation on the set P = {1, 4, 9}?

A) R1 = {(1, 1), (4, 4), (9, 9)}

B) R2 = {(1, 1), (4, 4), (1, 4), (4, 1)}

C) R3 = {(4, 4), (9, 9), (1, 1), (9, 1), (1, 9), (1, 4), (4, 1)}

D) R4 = {(1, 4), (4, 4), (9, 4), (4, 1), (1, 1), (9, 9), (9, 1)}

15) The normal to the curve y = f(x) at the point (5, 7) makes an angle of 𝜋

4 with the x-axis in the

positive direction.

Which of the following is the value of f '(5)?

A) -1

B) 1

C) 7

D) (cannot say without knowing f(x).)

16) What is the value of 𝑐𝑜𝑠 (𝑐𝑜𝑡−1 (7

24))?

A) 7

24

B) 7

25

C) 24

7

D) 24

25

17) Assuming x ≥ 0 and y ≥ 0, which of the following sets of constraints will result in an

unbounded feasible region for a linear programming problem?

A) x + 2y ≤ 2

x + 2y ≤ 6

B) x + 2y ≥ 2

x + 2y ≤ 6

C) x + 2y ≤ 2

x + 2y ≥ 6

D) x + 2y ≥ 2

x + 2y ≥ 6

18) Which of the following is true about the function defined as f(x) = (x3 - 9x)?

A) √3 is a point of local maximum.

B) √3 is a point of local minimum.

C) √3 is the point of absolute maximum.

D) the function has neither a local minimum nor a local maximum.

19) Given below are two matrices, P and Q:

For what value(s) of a, is P = Q2?

A) 1

B) 2, -2

C) 1, 2 and -2

D) (P = Q2 is not possible for any value of a.)

20) Kabir observed the price of a particular share in the stock market for 16 months. Based on

his observation, he formulated a function, p = 16t – t2 + 8, t ∈ [0, 16], where p is the price of

the share (in Rs) and t is the time (in months).

Which of the following is the maximum price of the share during the period Kabir observed

it?

A) Rs 8

B) Rs 56

C) Rs 72

D) Rs 200

21) A and B are two sets with m elements and n elements respectively (m < n).

How many onto functions can be defined from set A to set B?

A) 0

B) m!

C) n!

D) nm

22) Which quadrant will have the point of the intersection of ALL of the following functions?

I) sin-1 x

II) cos-1 x

III) tan-1 x

A) only quadrant I

B) only quadrant III

C) both quadrants I and II

D) (no common point of intersection.)

23) Kushal, Harsh, Mridul and Asha were asked to simplify 5AB + 2(BA + AB) - 3BA, where A and

B are matrices of orders 2 × 3 and 3 × 2 respectively. Shown below are their responses.

Kushal: 6AB

Harsh: 7AB – BA

Mridul: 8AB Asha: 6AB – 2BA Whose gave the correct answer?

A) Kushal

B) Harsh

C) Mridul

D) Asha

24) Shiv is planning to build a triangular garden. The sketch of the garden is shown below.

What will be the area, in square units, of the triangular garden?

A) 15

B) 20

C) 37

D) 74

25) Three function 𝑢(𝑥), 𝑣(𝑥) and 𝑤(𝑥) are defined below for 𝑥 > 0.

𝑢(𝑥) = 𝑥2

𝑣(𝑥) = sin 𝑥 𝑤(𝑥) = 𝑥 + 1

Find the expression for 𝑑(𝑢(𝑣(𝑤(𝑥))))

𝑑𝑥 for 𝑥 > 0.

A) sin 2𝑥 B) 𝑠𝑖𝑛 (2𝑥 + 2)

C) 2 sin(𝑥 + 1)

D) 2 sin(𝑥 + 1) + cos(𝑥 + 1) + 1

26) Which of the following is/are the critical points of the function 𝑦 = 𝑡𝑎𝑛−1(sec 𝑥) in the

interval (−𝜋2

,𝜋2

)?

A) only c = 0

B) only c = 𝜋

2 and c =

−𝜋

2

C) only c = 0, c = 𝜋

2 and c =

−𝜋

2

D) (The function has no critical points in the given interval.)

27) The objective function of a linear programming problem is given by Z = ax + by; where a and

b are constants.

If the maximum of Z occurs at two points (40, 60) and (60, 50), which of the following

represents the relationship between the coefficients of the objective function?

A) a = 2b

B) b = 2a

C) b = -2a

D) (cannot be found using the given information.)

28) A relation R is defined on the set of integers (Z) by,

R = {(a, b) : 4 divides a – b}

How many equivalence classes does R divide Z into?

A) 3

B) 4

C) 5

D) Infinitely many

29) What is the principal value of 𝑐𝑜𝑡−1(√3) + 𝑠𝑒𝑐−1(−2)?

A) −𝜋

2

B) −𝜋

6

C) 0

D) 5𝜋

6

30) Given matrix M = [0 33 2

]

Which of the following represents matrix N, such that MN = NM?

A) [−2 11 −4

]

B) [2 −3

−2 0]

C) [3 −1

−1 8]

D) [−1 00 −1

]

31) A curve in the xy - plane is given by the parametric form x = at2 and y = 2at with ‘t’ as a

parameter, for all t > 0.

What is the value of 𝑑𝑥

𝑑𝑦?

A) 1

B) t

C) 1

𝑡

D) 𝑡

2

32) Chinmay's badminton court had a width of 6.1 m. He measured the length of the court as

13.2 m with an error of 0.02 m.

Which of the following is the approximate error in calculating the area of the court?

A) 0.02 m2

B) 0.122 m2

C) 0.264 m2

D) 1.61 m2

33) Aditi is making a circular dosa. She is spreading the dosa batter such that its radius is

increasing at a rate of 2cm/s.

Which of the following is the rate of change of the area of the dosa, in terms of π, when its

radius is 6 cm?

(Note: Assume that the dosa has negligible thickness.)

A) 4π sq cm

B) 12π sq cm

C) 24π sq cm

D) 72π sq cm

34) What is the range of 3𝑐𝑜𝑠−1(𝑥)?

A) [-3π, 3π]

B) [-3, 3]

C) [0, 3π]

D) [0,𝜋

3]

35) Which of these matrices is equal to its inverse?

36) A linear programming problem (LPP) along with the graph of its constraints is shown below.

Minimize: Z = 3x + 2y

Which of the following is true about the above LPP?

A) The optimal solution does not exist as the feasible region is unbounded.

B) (0, 3) is the optimal solution as the inequality 3x + 2y < 6 has points in common with the

feasible region.

C) (2, 1) is the optimal solution as the inequality 3x + 2y > 6 has points in common with the

feasible region.

D) (0, 3) is the optimal solution as the inequality 3x + 2y < 6 does NOT have any point in

common with the feasible region.

37) QS = I, where Q is a diagonal matrix of order greater than 1 and I is an identity matrix.

What can be DEFINITELY concluded about the type of matrix S?

A) Row matrix

B) Scalar matrix

C) Identity matrix

D) Diagonal matrix

38) Which of the following statements is correct with reference to a linear programming

problem?

A) At optimal solution, all resources are completely utilised.

B) Every linear programming problem has at least one optimal solution.

C) A linear programming problem can have infinitely many optimal solutions.

D) If a linear programming problem has an unbounded feasible region, then it has NO solution

39) A curve is represented by x = 3sec θ, y = 3tan θ where 0 < θ < 𝜋

2.

Which of the following is the y intercept of the tangent to the curve at (2√3, √3)?

A) −3√3

B) √3

C) 2

D) 3√3

40) Shown below is the top view of a rescue boat moving in a river, traveling in a straight-line

path represented by the line y = 3x - 4. It spots a person standing on the riverside at (6, 3).

(Note: Assume that the person is stationary.)

Which of the following is the x-coordinate of the point on the path where the person is

closest to the rescue boat?

A) 2.7

B) 3

C) 3.5

D) 2.4

Section C

Section C consists of 10 questions based on two Case Studies. Attempt any 4 questions under each

case study. A total of 8 questions to be attempted. Each question carries 1 mark.

Answer the questions based on the information given.

Given below are two matrices, E = [eij] and F = [fij]. The minor of element e22 in E is equal to

the negative of the minor of element f11 in F.

41) What is the value of k?

A) -12

B) -3

C) 3

D) 12

42) What is the cofactor of element e12 in matrix E?

A) -9

B) -6

C) 6

D) 9

43) Which of the following is the adjoint of matrix E?

A) P

B) Q

C) R

D) S

44) Which of the following represents the inverse of matrix E?

45) Which of the following relationship is true for matrices E and F?

A) (Adj E) - F = 0

B) (Adj F) - E = 0

C) (Adj E) + F = 0

D) (Adj E) + (Adj F) = E + F

Study the given information and answer the questions that follow.

When a rocket is propelled for its journey into outer space, it happens over multiple stages.

During these stages, the rocket experiences different magnitudes of acceleration. Shown

below is an example of the graph of effective acceleration as a function of time during its

propulsion.

Effective acceleration (in m/s2) is given by, a(t) = c(t) - g

where, c(t) = upward acceleration due to thrust

g = downward acceleration due to gravity

The different stages of the propulsion can be algebraically represented by the effective

acceleration function given below:

where k is a constant.

The fuel consumed (in kg) is dependent on the acceleration function, a(t), and the

relationship is given below:

46) The acceleration function a(t) is definitely continuous at all the points in which of the

following sets?

A) [0, 40]

B) [0, ∞) - {40}

C) [0, 25]

D) (25, 40) ∪ (40, ∞)

47) During which stage(s) of the propulsion, is the rate of effective acceleration decreasing over

time?

A) only [0, 10)

B) only [15, 25)

C) [0, 10) ∪ [15, 25)

D) [7, 10) ∪ [15, 25)

48) For what value of k, is the acceleration function continuous at t = 25?

A) −1110

B) 4

30

C) 11

10

D) (no such value of k exists.)

49) Find the fuel consumed at t = 9 s.

A) 2500e4.5 kg

B) 2500e kg

C) 2500 kg

D) 2500e-1 kg

50) What is the rate of fuel usage at t = 7 s?

A) 2500e(-22.05) kg/s

B) 2500 kg/s

C) 1250 kg/s

D) (no fuel is used because the acceleration is at its maxima.)

x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x

Answer key

Sl.no Key answer Sl.no Key answer

1 D 26 A

2 D 27 B

3 B 28 D

4 D 29 D

5 C 30 D

6 C 31 B

7 B 32 B

8 C 33 C

9 A 34 C

10 A 35 C

11 B 36 D

12 C 37 D

13 D 38 C

14 A 39 A

15 A 40 A

16 B 41 C

17 D 42 A

18 B 43 B

19 D 44 D

20 C 45 C

21 A 46 D

22 D 47 A

23 B 48 C

24 C 49 B

25 B 50 C