Time: 3 hours                                                                             Maximum marks: 80

General Instructions:

1. This Question paper contains - five sections A, B, C, D and E. Each section is compulsory. However, there are internal choices in some questions.

2. Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark each.

3. Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.

4. Section C has 6 Short Answer (SA)-type questions of 3 marks each.

5. Section D has 4 Long Answer (LA)-type questions of 5 marks each.

6. Section E has 3 source based/case based/passage based/integrated units of assessment of 4 marks each with sub-parts.

 

Section –A

(Multiple Choice Questions)

Question 1:

If A² – 2A – I = 0, then A^-1 is equal to  

(a) I

(b) A + 2

(c) A - 2

(d) A

 

Question 2:

The determinant of 3 rows (R₁, R₂, R₃) and 3 columns (C₁, C₂, C₃) has a value 15. If two columns C₂ and C₃ are interchanged then the new determinant value will be

(a) 15

(b) -15

(c) 45

(d) -45

 

Question 3:

The value of cross product of two vectors (a - b) and (a + b) is  

(a) a² – b²

(b) 2(a * b)

(c) a + b

(d) b * a

 

Question 4:

If |a| = 4 and -3 ≤ k ≤ 2, then the range of |ka| is

(a) [0, 8]

(b) [-12, 8]

(c) [0, 12]

(d) [8, 12]

 

Question 5:

If a line has direction ratios 2, –1, –2, then its direction cosines are

(a)

(b)

(c)

(d)

 

Question 6:

Given that the events A and B are such that P(A) = , P(A ∩ B) =  and P(B) = p. Find p if they are independent.

(a)

(b)

(c)

(d)

 

Question 7:

The degree of differential equation

(a) 1.5

(b) 1

(c) 3

(d) 0.5

 

Question 8:

If a and b are unit vectors, then what is the angle between a and b for √3a – b to be a unit vector?

(a)  

(b)

(c)

(d)

 

Question 9:

If ∫ sec²(7 – 4x)dx = a tan (7 – 4x) + C, then value of a is 

(a)

(b)

(c)

(d)

 

Question 10:

The relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is.

(a) reflexive, symmetric and transitive

(b) reflexive, not symmetric, transitive

(c) not reflexive, symmetric, not transitive

(d) not reflexive, not symmetric, not transitive

 

Question 11:

A = [a_ij]_m*n is a square matrix, if

(a) m < n                               

(b) m > n                                 

(c) m = n                    

(d) None of these

 

Question 12:

If f(x) =  ax² + b, b ≠ 0,  x ≤ 1

               bx² + ax + c,    x > 1, then f(x) is continuous and differentiable at x = 1 if

(a) c = 0, a = 2b

(b) c ∈ R, a = b

(c) a = b, c = 0

(d) a = b, c ≠ 0

Question 13:

The equation x² + y² = 0 in three dimensional space represents

(a) a point

(b) the empty set

(c) z-axis

(d) none of these

 

Question 14:

If θ be the angle between two vectors a and b, then a.b ≥ 0 only when

(a) 0 < θ <

(b) 0 ≤ θ ≤

(c) 0 < θ < π

(d) 0 ≤ θ ≤ π

 

Question 15:

If [x      1]   2       0    = 0 then x equals

                  -2       0  

(a) 0

(b) 2

(c) -2

(d) -1

 

Question 16:

The integrating factor of differential equation cos x *  + y * sin x = 1 is

(a) cos x

(b) tan x

(c) sec x

(d) sin x

 

Question 17:

If A and B are two events such that P(A) ≠ 0 and P(B|A) = 1, then

(a) A ⊂ B

(b) B ⊂ A

(c) B = φ

(d) A = φ

 

Question 18:

The corner points of the feasible region determined by the following system of linear inequalities: 2x + y ≤ 10, x + 3y ≤ 15, xy ≥ 0 are (0, 0), (5, 0), (3, 4) and (0, 5). Let Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both (3, 4) and (0, 5) is                                                                                                     

(a) p = q                              

(b) p = 2q                        

(c) p = 3q                          

(d) q = 3p

 

ASSERTION-REASON BASED QUESTIONS

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 choices.

(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.

 

Question 19:

If A = {1, 2, 3}, B = {4, 5, 6, 7} and f = {(1, 4), (2, 5), (3, 6)} is a function from A to B.

ASSERTION (A): f(x) is a one-one function.

REASON (R): f(x) is an onto function.

 

Question 20:

The total revenue received from the sale of x units of a product is given by

R(x) = 3x² + 36x + 5 in rupees.

ASSERTION (A): The marginal revenue when x = 5 is 66

REASON (R): Marginal revenue is the rate of change of total revenue with respect to the number of items sold at an instance.

 

Section – B

Question 21:

What is the value of sec² (tan^-1 2)?

 

Question 22:

Show that f(x) = tan x – x is always increasing.

 

Question 23:

Find the slope of the tangent to the curve y = x³ – 3x + 2 at the point whose x-coordinate is 3.

 

Question 24:

Evaluate:

 

Question 25:

If the curve ay + x² = 7 and x³ = y cut orthogonally at (1, 1) then find the value of a.

 

Section – C

Question 26:

Evaluate:  

 

Question 27:

The random variable X has probability distribution P(X) of the following form, where k is some number:

X = x	0	1	2	3	4	5	6	7
p(x)	0	k	2k	2k	3k	k2	2k2	k2 + k

 

(a) Determine the value of k                            

(b) Find P(0 < X < 5)

 

Question 28:

Find the value of

 

Question 29:

Find the general solution of  + 2x1 +  =

 

Question 30:

Maximize Z = − x + 2y

subject to the constraints:

x ≥ 3, x + y ≥ 5, x + 2y ≥ 6, y ≥ 0

 

Question 31:

Find the approximate value of  

 

Section – D

Question 32:

Find the area bounded by curves {(x, y): y ≥ x² and y = |x|}.

 

Question 33:

Show that the function f : R → R defined by f(x) = (x - 1)(x - 2)(x - 3) is onto but not one-one

 

Question 34:

Solve the following system of equations

x − y + z = 4                                  

2x + y − 3z = 0                                           

x + y + z = 2

 

Question 35:

Let a = i + 4j + 2k, b = 3i – 2j + 7k and c = 2i – j + 4k. Find a vector d which is perpendicular to both a and b, and c.d = 15.

 

Section – E

Question 36:

Teams A, B and C went for playing a tug of war game. Teams, A, B, C have attached a rope to a metal ring and are trying to pull the ring into their own area as shown in the given figure.

Team A pulls the force F₁ = 4i + 0j KN

Team B pulls the force F₂ = -2i + 4j KN

Team C pulls the force F₃ = -3i - 3j KN

Based on the above information, answer the following questions.

(i) Which team will win the game?

(a) B                      (b) A                      (c) C                          (d) A and C

(ii) What is the magnitude of the teams combined force?

(a) 7 KN                      (b) 1.4 KN                      (c) 5 KN                          (d) 10 KN

(iii) In which direction is the ring getting pulled?

(a)  radian            (b)  radian                (c)  radian                (d)  radian

 

Question 37:

A real estate company is going to build a new residential complex. The land they have purchased can hold at most 4500 apartments. Also, if they make x apartments, then the monthly maintenance cost for the whole complex would be as follows:

Fixed cost = Rs 50, 00, 000

Variable cost = Rs (160x – 0.04x²)

Based on the above information given, answer the following questions.

(i) The maintenance cost as a function of x will be

(a) 160x – 0.04x²                                  (b) 5000000                     

(c) 5000000 + 160x – 0.04x²            (d) 5000000 – (160x – 0.04x²)

(ii) If C(x) denotes the maintenance cost function, then maximum value of C(x) occurs at x = _____

(a) 100                          (b) 2000                      (c) 3500                      (d) 4000

(iii) If the minimum maintenance cost is attain then the maintenance cost for each apartment would be

(a) 1000                      (b) 1091.11                   (c) 200                        (d) 2000.11

 

Question 38:

If there is a statement involving the natural number n such that

(i) The statement is true for n = 1

(ii) When the statement is true for n = k (where k is some positive integer), then the statement is also true for n = k + 1

Then, the statement is true for all natural numbers n.

Also, if A is a square matrix of order n then A² is defined as AA

In general, A^m = AAAA……..A (m times), where m is any positive integer.

Based on the above information given, answer the following questions.

(i) If A =

     , then for any positive integer n, Aⁿ will be

 

(ii) Let A = 

     and Aⁿ = [a_ij]_3*3  then cofactor of a¹³ is  

                   

(a) aⁿ                      (b) -aⁿ                      (c) 2aⁿ                          (d) 0

(iii) If A is a square matrix such that |A| = 2, then for any positive integer n, |Aⁿ|is equal to

(a) 0                      (b) 2n                      (c) 2ⁿ                          (d) n²

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