## Kerala Plus Two Maths Model Question Paper 4

**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.

a. The slope of the tangent to the curve given by

x = 1- cos θ , y = θ – sin θ at θ = \(\frac { \pi }{ 2 }\) is

(i) 0

(ii) -1

(iii) 1

(iv) Not defined

b. Find the intervals in which the function f (x) = x^{2} – 4x + 6 is strictly decreasing.

Question 2.

Question 3.

Question 4.

a. Write the Cartesian equation of the straight line through the point (1, 2, 3) and along the vector 3\(\widehat { i }\) + \(\widehat { j }\) + 2\(\widehat { k }\).

b. Write a general point on this straight line.

c. Find the distance from (1, 2, 3) to the plane 2x + 3y – z + 2 = 0.

Question 5.

Question 6.

Solve the system of linear equations :

x + 2y + z = 8

2x + y – z = 1

x – y + z = 2

x + 2y + z = 8

2x + y – z = 1

Question 7.

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

Question 8.

Question 9.

Question 10.

i. Find the equation of the plane through the points (3,-1,2), (5,2,4) and (-1,-1,6).

ii. Find the perpendicular distance from the point (6,5,9) to this plane.

Question 11.

a. Express the euqations of the lines into vector form.

b. Find the shortest distance between the lines.

Question 12.

Question 13.

a. Find the equation of a plane with intercepts 2, 3 and 4 on X, Y and Z axes respectively.

b. Find the distance of the point (-1,-2, 3) from the plane r.(2\(\widehat { i }\) – \(\widehat { j }\)) + 4 \(\widehat { k }\) = 4.

Question 14.

Question 15.

Question 16.

Question 17.

a. For two independent events A and B, which of the following pair of events need not be independent?

i. A’, B’

ii. A, B’

iii. A’, B

iv. A-B, B-A

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

Question 18.

Consider the following L.P.P.

Maximize Z = 3x+2y

Subject to the constraints

x+2y < 10

3x+y < 15

x, y > 0

a. Draw its feasible region.

b. Find the comer points of the feasible region.

c. Find the maximum value of Z.

Question 19.

a. y=a cos x+b sin x is the solution of the differential equation

Question 20.

Question 21.

Question 22.

Question 23.

Question 24.

a. If \(\overline { a } \),\(\overline { b } \),\(\overline { c } \),\(\overline { d } \) respectively are the position vectors representing the vertices A, B, C, D of a parallelogram, then write \(\overline { d } \) in terms of \(\overline { a } \), \(\overline { b } \) and \(\overline { c } \) .

b. Find the projection vector of \(\overline { b } \) = \(\widehat { i }\) + 2 \(\widehat { j }\) + \(\widehat { k }\) along the vector \(\overline { a } \) = 2i +j + 2k. Also write \(\overline { b } \) as the sum of a vector along \(\overline { a } \) and a vector perpendicular to \(\overline { a } \) .

c. Find the area of a parallelogram for which the vectors 2 \(\widehat { i }\) + \(\widehat { j }\) and 3 \(\widehat { i }\) + \(\widehat { j }\) + 4 \(\widehat { k }\) are adjacent sides.

**ANSWERS**

Answer 1.

The point x = 2 divides the real line into two disjoint intervals namly (-∞, 2) and (2,∞). In the interval(-∞,2), f'(x)=2x-4 < 0

∴ f is strictly decreasing in this interval.

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.