Sample paper maths

Sample Paper
Class 12 Maths
Sample Paper 5 | Class 12 Maths

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)

(Q1 - Q10) are multiple choice type questions. Select the correct option.

Question 1:

The value of is

(a)

(b)

(c)

(d)

Question 2:

If a matrix X of order 3 * 3 has determinant value 5, then the value of |3X| is

(a) 15

(b) 105

(d) 225

Question 3:

The points with position vectors 60i + 3j, 40i – 8j, ai – 52j are collinear if

(a) a = -20

(b) a = -40

(d) a = -80

Question 4:

A and B are two events such that P (A) ≠ 0. If A is a subset of B then the value of P( ) is

(a) 0

(b) 1

(c)

(d)

Question 5:

The x coordinate of a point on the line joining the points p(2, 2, 1) and q(5, 1, -2) is 4. Then its z coordinate is

(a) 1

(b) -1

(c)

(d)

Question 6:

The order and degree of differential equation + + + y⁵ = 0 is

(a) 3, 2

(b) 4, 1

(d) Not defined

Question 7:

The value of

(a)

(b)

(c)

(d) 1

Question 8:

Given that the events A and B are such that P(A) = , P(A ∩ B) = and P (B) = p. The value of p if events A and B mutually exclusive, is

(a)

(b)

(c)

(d)

Question 9:

The Principal value of cos^-1(cos 680⁰) is

(a) 20⁰

(b) 40⁰

(d) 80⁰

Question 10:

If , x ≠ , then the value of fof(x) is

(a) x

(b) x²

(c)

(d)

Question 11:

If A and B are two invertible matrices such that AB = C, then det(B) =

(a)

(b)

(c)

(d) None of these

Question 12:

The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 13x² + 26x + 15. Then the marginal revenue when x = 7, is

(a) Rs. 108

(b) Rs. 208

(d) Rs. 408

Question 13:

The value of is

(a) 0

(b) 1

(c)

(d) x + y + z

Question 14:

The points of discontinuity of f, where f is defined by

f(x) = ,if x < 0

-1 ,if x ≥ 0 is

(a) at x = 0

(b) at x > 0

(d) No point of discontinuity

Question 15:

A balloon, which always remains spherical, has a variable diameter . The rate of change of its volume with respect to x is

(a)

(b)

(c)

(d)

Question 16:

The area of the region bounded by the curves y = x² + 2, y = x, x = 0 and x = 3 is

(a) units

(b) units

(d) units

Question 17:

and A + A’ = I, then the value of α is

(a)

(b)

(c)

(d)

Question 18:

The value of

(a) 0

(b) 1

(c)

(d)

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): The function f(x) = x² + bx + c, where b and c are real constant, describes onto mapping.

REASON (R): Let A = {1, 2, 3, …., n} and B = {a, b}. Then the number of surjective from A into B is 2ⁿ – 2.

Question 20:

ASSERTION (A): We can write

REASON (R): Any value in the range of principla value branch is called principal value of that inverse trigonometric function.

SECTION – B

Question 21:

Test whether the relation is reflexive ,symmetric or transitive on the set specified R = {(m, n) : m - n ≥ 7} on z.

Question 22:

Write the function , |x| < a in the simplest form.

Question 23:

Find the values of k so that the function f is continuous at the indicated point.

Question 24:

If y = 500e^7x + 600e^-7x, show that .

Question 25:

Let f be a function defined on [a, b] such that f ′(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).

Question 26:

Prove that (a * b) * c = a * (b * c) if (c * a) * b = 0.

SECTION - C

Question 27:

Find the coordinates of the point where the line through (3, -4, -5) and (2, -3, 1) crosses the plane determined by points A(1, 2, 3), B(2, 2, 1) and C(-1, 3, 6).

Question 28:

Three distinguishable balls are distributed in three cells. Find the probability that all three occupy the same cell, given that at least two of them are in the same cell.

Question 29:

Find the equation of the tangent line to the curve y = x² − 2x + 7 which is perpendicular to the line 5y − 15x = 13.

Question 30:

Evaluate:

Question 31:

Find the area of the circle 4x² + 4y² = 9 which is interior to the parabola x² = 4y.

Question 32:

A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below:

(a) If unit sale prices of x, y and z are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra.

(b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit.

SECTION - D

Question 33:

, find A^–1. Using A^–1 solve the system of equations

2x – 3y + 5z = 11

3x + 2y – 4z = – 5

x + y – 2z = – 3

Question 34:

Minimize and Maximize Z = x + 2y subject to

x + 2y ≥ 100, 2x − y ≤ 0, 2x + y ≤ 200, x ≥ 0, y ≥ 0.

Question 35:

(a) The two adjacent sides of a parallelogram are 2i – 4j + 5k and i – 2j – 3k. Find the unit vector parallel to its diagonal. Also, find its area.

(b) If

Question 36:

A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by cab, metro, bike or by bus means of transport are respectively 0.3, 0.2, 0.1 and 0.4. The probabilities that he will be late are 0.25, 0.30, 0.35 and 0.10 if he comes by cab, metro, bike and other means of transport respectively.

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

(i) When the doctor arrives late, what is the probability that he will come by metro?

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

(ii) When the doctor arrives late, what is the probability that he will come by cab?

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

(iii) When the doctor arrives late, what is the probability that he will come by other means of transport?

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

Question 37:

The owner of an electric bike rental company have determined that if they charge customers Rs x per day to rent a bike, where 50 ≤ x ≤ 200, then the number of bikes (n), they rent per day can be shown by linear function n(x) = 2000 – 10x. If they charge Rs 50 per day or less, they will rent all their bikes. If they charge Rs 200 or more per day, they will not rent any bike.

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

(i) Total revenue R as a function of x can be represented as

(a) (b)

(ii) If R(x) demotes the revenue, then the maximum value of R(x) occur when x equals

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

(iii) At x = 260, the revenue collected by the company is

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

Question 38:

A child cut a pizza with a knife. Pizza is circular in shape which is represented by and sharp edge of knife represents a straight line given by .

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

(i) The point(s) of intersection of the edge of knife (line) and pizza in the figure is/are

(a) (b)

(ii) Which of the following shaded portion represent the smaller area bounded by pizza and edge of knife in first quadrant?

(iii) Area of each slice of pizza when child cut the pizza into 4 equal pieces is

(a) sq. units (b) sq. units

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