## Kerala Plus Two Maths Model Question Paper 2

**Time : 2 1/2 Hours**

**Cool off time : 15 Minutes**

**Maximum : 80 Score**

**General Instructions to Candidates :**

- There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
- Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
- Read questions carefully before you answering.
- Read the instructions careully.
- When you select a question, all the sub-questions must be answered from the same question itself.
- Calculations, figures and graphs should be shown in the answer sheet itself.
- Malayalam version of the questions is also provided.
- Give equations wherever necessary.
- Electronic devices except non programmable calculators are not allowed in the Examination Hall.

**QUESTIONS**

**Question 1 to 7 carry 3 scores each. Answer any six questions only
**

Question 1.

Question 2.

Question 3.

Consider the differential equation \(\frac { { d }^{ 2 }{ y } }{ { dx }^{ 2 } } +y=0\)

a. write its order and degree

b. Verify that y = a cos x + b sin x, where a, b ∈ R is a solution of the given differential equation

Question 4.

Consider the differential equation x \(\frac { dy }{ dx } \) + 2y = x2 ; x ≠ 0

a. What is its integrating factor?

b. Obtain its general solution.

Question 5.

The foot of the perpendicular drawn from origin to a plane is (4,-2, 5).

a. How far is the plane from the origin?

b. Find a unit vector perpendicular to that plane.

c. Obtain the equation of the plane in general form.

Question 6.

Question 7.

**Question 8 to 17 carry 4 scores each. Answer any eight questions only
**

Question 8.

Question 9.

Question 10.

Question 11.

Consider the points A (2, 2, -1), B (3, 4, 2) and C (7, 0, 6). Find the vector and Cartesian equation of the plane passing through these points.

Question 12.

a. Find the cartesian equation of the plane through the point (1,2,-3) and perpen-dicular to the vector 2\(\widehat { i } \) – \(\widehat { j } \) + 2\(\widehat { k } \).

b. Find the angle between the above plane and the line \(\frac { x-1 }{ 2 } \) = \(\frac { y-3 }{ 3 } \) = \(\frac { z }{ 6 } \)

Question 13.

a. Let R be the relation on the set N of . natural numbers given by R = {(a,b): a – b > 2, b > 3}.

Choose the correct answer.

A. (4,1) ∈ R

B. (5,8) ∈ R

C. (8,7) ∈ R

D.(10,6) ∈ R

b. If f (x) = 8x^{3} and g (x) = x^{1/3}, find g (f(x)) and f (g (x)).

c. Let * be a binary operation on the set Q of rational numbers defined by a*b= \(\frac { ab }{ 3 } \).

Check whether * is commutative and associative?

Question 14.

X | 1 | 2 | 3 | 4 | 5 |

P(X) | 1/2 | 1/4 | 1/8 | 1/16 | P |

The probability distribution of a random variable X taking values 1, 2, 3, 4, 5 is given.

a. Find the value of P.

b. Find the mean of X.

c. Find the variance of X.

X | 1 | 2 | 3 | 4 | 5 |

P(X) | 1/2 | 1/4 | 1/8 | 1/16 | P |

Question 15.

a. Consider the family of all circles having their centre at the point (1,2). Write the equation of the family.

Write the corresponding differential equation.

b. Write the integrating factor of the differential equation,

Question 16.

Consider the functions: f (x) = |x|-1 and g(x) = 1- | x|

a. Sketch their graphs and shade the closed region between them.

b. Find the area of their shaded region.

Question 17.

a. What is the value of sin^{-1} (sin 160°)?

**Question 18 to 24 carry 6 scores each. Answer any 5 questions only**

Question 18.

Question 19.

integrate the following :

Question 20.

a. Choose the correct statement related to the matrices A = \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)

Question 21.

a. Which of the following functions is always increasing?

i. x + sin 2x

ii. x – sin 2x

iii. 2x + sin 3x

iv. 2x – sin x

b. The radius of a cylinder increases at a rate of 1 cm/s and its height decreases at a rate of 1 cm/s. Find the rate of change of its volume when the radius is 5 cm and the height is 15 cm.

If the volume should not change even when the radius and height are changed, what is the relation between the radius and height?

c. Write the equation of tangent at (1,1) on the curve 2x^{2} + 3y^{2} = 5.

Question 22.

Consider the linear programming problem :

Minimize Z = 3x + 9y

Subject to the constraints :

x+3y < 60 ; x + y > 10 ; x < y ; x > 0 , y > 0

a. Draw its feasible region.

b. Find the vertices of the feasible region.

c. Find the minimum value of Z subject to the given constraints.

Minimize Z = 3x + 9y

Subject to the constraints :

x + 3y **<** 60 ; x + y > 10 ; x < y ; x > 0, y > 0

Question 23.

Consider the following figure :

a. Find the point of intersection P, of the circle, x^{2}+ y^{2} = 32 and the line y = x.

b. Express the area of the shaded portion as a Sum of two definite integrals.

c. Find the area of the shaded portion.

Question 24.

**ANSWERS**

Answer 1.

Answer 2.

Answer 3.

Answer 4.

Answer 5.

Answer 6.

Answer 7.

Answer 8.

Answer 9.

Answer 10.

Answer 11.

Answer 12.

Answer 13.

Answer 14.

Answer 15.

Answer 16.

Answer 17.

Answer 18.

Answer 19.

Answer 20.

Answer 21.

Answer 22.

Answer 23.

Answer 24.

c. 1. f (x) is continuous at (-2, 2)

2. f (x) is differentiable at (-2, 2)

3. If f (b) = f (a) then there exist a point at (-2, 2) such that f^{1} (c) = 0

f (b) = f (-2) = (-2)^{2} + 2 = 6

f (a) = f (2) = 2^{2} + 2 = 6

∴ Then there exist a point C such that

f^{1} (c) = 0

f^{1} (x) = 2x

f^{1} (x) = 2c = 0

∴ c = 0 at (-2,2)

∴ Rolle’s theorem verified.