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:

Let X, Y, Z, W and P are matrices of order 2 * n, 3 * k, 2 * p, n * 3 and p * k respectively. If n = p then the order of the matrices 7X – 5Z is

(a) p * 2

(b) 2 * n

(c) n * 3

(d) p * n

 

Question 2:

If A is a skew symmetric matrix of order 3 then the value of |A| is

(a) 0

(b) 3

(c) 9

(d) 27

 

Question 3:

The value of i.(j * k) + j.(i * k) + k.(i * j)  

(a) 0

(b) -1

(c) 1

(d) 3

 

Question 4:

Let A = {1, 2, 3, …., n} and B = {a, b}, then the number of surjections from A into B is

(a) ⁿP₂

(b) 2ⁿ - 2

(c) 2ⁿ - 1

(d) None of these

 

Question 5:

If the direction cosines of a line are k, k, k, then

(a) k > 0

(b) 0 < k < 1

(c) k = 1

(d) k =

 

Question 6:

Two cards are drawn from a well shuffled deck of 52 playing cards with replacement. The probability that both cards are queens, is

(a)

(b)  

(c)  

(d)

 

Question 7:

The numbers of arbitrary constants in the particular solution of a differential equation of a third order are

(a) 3

(b) 2

(c) 1

(d) 0

 

Question 8:

The value of λ for which the vectors 3i – 6j + k and 2i – 4j + λk are parallel, is

(a)

(b)

(c)

(d)

 

Question 9:

If f(x) = , then f’(x) is 

(a) cos x + x * sin x

(b) x * sin x

(c) x * cos x

(d) sin x + x * cos x

 

Question 10:

Let S be the set of all real numbers. Then the relation R = {(a, b): a² + b² = 1} is

(a) symmetric and transitive

(b) symmetric and reflexive but not transitive

(c) symmetric but neither reflexive nor transitive

(d) symmetric, reflexive and transitive

 

Question 11:

If A is a skew symmetric matrix and n is even positive integer, then Aⁿ is

(a) a symmetric matrix

(b) a skew symmetric matrix

(c) a diagonal matrix

(d) None of these

 

Question 12:

Suppose f(x) and g(x) are two real functions continuous at a real number c.

Then which one is/are correct?

(a) f(x) ± g(x) is continuous at x = c.

(b) f(x) * g(x) is continuous at x = c.

(c)  is continuous at x = c. (provided g (c) ≠ 0)

(d) All of above

 

Question 13:

The reflection of the point (α, β, γ) in the xy-plane is  

(a) (α, β, 0)

(b) (0, 0, γ)

(c) (-α, -β, γ)

(d) (α, β, -γ)

 

Question 14:

Which of the following statement is true?

(a) Vectors a and –a are collinear.

(b) Two collinear vectors are always equal in magnitude.

(c) Two vectors having same magnitude are collinear.

(d) Two collinear vectors having the same magnitude are equal.

 

Question 15:

A 2 * 2 matrix, X = [a_ij], whose elements are given by a_ij = , is

 

     

 

Question 16:

 is equal to  

(a) e^x * cos x + C

(b) e^x * sec x + C

(c) e^x * sin x + C

(d) e^x * tan x + C

 

Question 17:

If A and B are two independent events with P(A) =  and P(B) = , then P(A’ ꓵ B’) is

(a)

(b)

(c)

(d)

 

Question 18:

The equation of the line which passes through the point (1, 2, 3) and is parallel to the vector 3i + 2j – 2k is                                                                                                     

(a) r = (i - 2j + 3k) + λ (3i + 2j – 2k)                              

(b) r = (i + 2j - 3k) + λ (3i + 2j – 2k)                        

(c) r = (i + 2j + 3k) + λ (3i + 2j – 2k)                          

(d) r = (i + 2j + 3k) + λ (3i + 2j + 2k)

 

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:

ASSERTION (A): P is a point on the line segment joining the points (3, 2, -1) and (6, -4, -2). If x coordinate of P is 5, then y coordinate is -2.

REASON (R): The two lines x = ay + b, z = cy + d and x = a’y + b’, z = c’y + d’ will be perpendicular iff aa’ + bb’ + cc’ = 0

 

Question 20:

ASSERTION (A): For the constraints of a LPP problem is given by

x₁ + 2x₂ ≤ 2000

x₁ + x₂ ≤ 1500

x₁ ≤ 600

x₁, x₂ ≥ 0

The points (1000, 0), (0, 500), (2, 0) lie in the positive bounded region but (2000, 0) does not lie in the positive bounded region.

REASON (R): From the given graph, it is clear that the point (2000, 0) is outside.

 

Section – B

Question 21:

What is the value of tan^-1 √3 - sec^-1 (-2)?

 

Question 22:

Show that the function f: R -> R defined as f(x) = 2x + cos x is an increasing function.

 

Question 23:

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector 3i + 5j – 6k.

 

Question 24:

Evaluate:

 

Question 25:

A particle moves along the curve x² = 2y. Find the point at which ordinate increases at the same rate as the abscissa.

 

Section – C

Question 26:

Evaluate:

 

Question 27:

The random variable X can take only the values 0, 1, 2, 3. Given that P(X = 0) =

P(X = 1) = p and P(X = 2) = P(X = 3) such that  = 2 * , find the value of p.

 

Question 28:

Find the value of

 

Question 29:

Find the particular solution of the differential equation satisfying the given

condition:  = y * tan x, given that y = 1, when x = 0.

 

Question 30:

The feasible solution of a LPP is shown in the given figure. Let Z = 3x – 4y be the objective function. Find the value of Max Z – Min Z.

 

Question 31:

Find the approximate value of   

 

Section – D

Question 32:

Find the area of the region enclosed by the parabola x² = y, the line y = x + 2 and x-axis.

 

Question 33:

Let T be the set of all triangles in the Euclidean plane and let a be the relation R on T be defined as aRb if a is congruent to b for all a. Show that R is an equivalence relation.

 

Question 34:

Using the property of determinants, prove that:

 

Question 35:

Find the vector equation of the line passing through the point (1, 2, −4) and

perpendicular to the two lines:  and

 

Section – E

Question 36:

An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats for the executive class. However, at least 4 times as many passengers prefer to travel by economy class, than by executive class. It is given that the number of executive class tickets is x and that of economy class tickets is y.

Based on the above information, answer the following questions.

(i) The relation between x and y is

(a) x < y                      (b) y > 80                 (c) x ≥ y                      (d) y ≥ 4x

(ii) Which among is not a constraint for this LPP?

(a) x ≥ 0                     (b) x + y ≤ 200         (c) x ≥ 80                    (d) 4x - y ≤ 0

(iii) The profit when x = 20 and y = 80 is

(a) Rs 60000            (b) Rs 68000             (c) Rs 64000              (d) Rs 136000

 

Question 37:

In a school project, Suneeta was asked to construct a triangle and name as ABC. Two angles A and B were given to be equal to  and  respectively.

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

(i) The value of sin A is

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

(ii) cos(A + B + C) = ______

(a) 1                                (b)                                   (c) -1                             (d)

(iii) The value of angle C is

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

 

Question 38:

A group of people start playing cards and as we know a well shuffled pack of cards contains a total of 52 cards. Then two cards are drawn simultaneously (or successively without replacement). 

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

 

(i) If x = number of kings = 0, 1, 2. Then P(x = 0) = ?

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

(ii) Find the mean number of kings.

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

(i) If x = number of kings = 0, 1, 2. Then P(x = 2) = ?

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

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