## Kerala Plus Two Maths Model Question Paper 5

**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 carefully.
- 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.

a. Let R be a relation on the set {1,2,3} given by R= {(1,1), (2,2), (1,2), (2,1), (2,3)}. Which among the following element to be inclu¬ded to R so that R becomes Symmetric?

i. (3,3)

ii. (3,2)

iii. (1,3)

iv. (3,1)

b. If * is defined by a * b=a-b^{2} and ⊕ is de-fined by a ⊕ is b=a^{2}+b, where a and b are integers. Then find the value of (3⊕4)*5.

Question 2.

Question 3.

Find rate of change of area of a circle.

a. With respect to the radius, when r = 10cm

b. With respect to the time when the radius is increasing at the rate of 0.7cm/s. Given that r=5cm.

Question 4.

Question 5.

Find area of a circle with centre (0, 0) and radius “a” using integration.

Question 6.

Consider the differential equation \(\frac { dy }{ dx } \) = \(\frac { x+y }{ x } \)

a. Write the order of the differential equation.

b. Solve above given differential equation.

Question 7.

Following Table shows a brief description about manufacturing process of a company. Time required in hours per unit of the product and maximum availability of machines is also given in the table

a. Write the objective function.

b. Whether it is a maximisation case or a minimisation case Justify,

c. Write the contraints.

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

Question 8.

a. Afunctionf: A → B whereA= {1,2,3} and B= {4,5,6} defined by f (1) = 5, f (2) = 6, f (3) = 4, Check whether f is a bijection. If it is bijection, write f^{1} as set of ordered pairs.2

b. The operation table for an operation * is given below. Given that I is the identify element, then which among the following is true regarding the elements in first column?

Question 9.

Question 10.

a. Find th relation between ‘a’ and ‘b’ so that the function defined by

b. “All continuous function are not differ-entiable”. Justify this statement with an example.

Question 11.

a. Find the equation of the tangent to the curve y = x^{2} – 2x + 7 at (2,7)

b. Find the maximum value of the function.

Question 12.

Question 13.

Consider the differential equation \(\frac { xdy }{ dx } \) + y = \(\frac { 1 }{ { x }^{ 2 } } \)

a. Find the integrating factor

b. Solve the above differential equation.

Question 14.

Question 15.

Question 16.

a. Find the cartesian equation of a line passing through the origin and (5,-2,3)

b. The point P(x,y,z) lies in the First Octant and its distance from origin is 12 units. If the position vector of P makes angles 45°, 60° with x and y axes respectively, find co-ordinates of P.

Question 17.

Solve graphically :

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

Question 18.

Question 19.

a. Without expanding prove that

b. Consider the following system of equations 2x – 3y + 5z = 11, 3x + 2y – 4z = -5, x + y-2z = -3

i. Express the system in the form Ax = B.

ii. Solve the system by matrix method a.

Question 20.

Question 21.

Evaluate the following.

Question 22.

Consider the parabolas y^{2} = 4x and x^{2} = 4y

a. Draw rough figure for the above parabolas.

b. Find the point of intersection of the two parabolas.

c. Find the area bounded by these two pa-rabolas.

Question 23.

a. Find the shortest distance between the lines whos vector equations are

b. If a plane meets positive x axis at a distance of 2 units from origin, positive y axis at a distance of 3 units from orign and positive z axiz at a distance of 4 units from origin. Find the equation of the plane.

c. Find the prependicular distance of (0,0,0) from the plane obtained in part (b).

Question 24.

a. A die is thrown twice let the event A be ‘odd number on First throw’ and B be ‘odd number on the second throw’ check whether A and B are independent.

b. Coloured balls are distributed in three boxes as shown in the following table.

A box is selected at random and a ball is taken out. If the ball taken is of red colour, What is the probabability that the other ball in the box is also of red colour?

**ANSWERS**

Answer 1.

Answer 2.

Answer 3.

Answer 4.

Answer 5.

Answer 6.

Answer 7.

Lex x = Machine G and y = Machine H

a. Objective function: Z= 20x + 30y

b. It is a maximization problem.

c. Constraints are:

3x + 4y < 10 ; 5x + 6y < 1 5 ; x *>* 0 ; y > 0

Answer 8.

a. A = {1,2,3}

B = {4,5,6}

f(1) = 5 ; f (2) = 6 ; f (3) = 4

Since it is one-one as well as onto, f is bijective.

∴ f = {(1,5), (2,6), (3,4)}

∴ f^{-1} ={(5,1), (6,2), (4,3)}

b. (ii) 1,2,3

1 * 1 = 2,

2 * 1=2

Further 3 * 2 = 3

2 * 3 = 3

∴ * is commutative.

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.