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## Plus Two Maths Previous Year Question Papers and Answers 2018

Board | SCERT |

Class | Plus Two |

Subject | Maths |

Category | Plus Two Previous Year Question Papers |

**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**

**Questions 1 to 7 carry 3 scores each. Answer any Six questions. (6 × 3 = 18) **

Question 1.

If f(x) = , x ≠ 1

a. Find fof (x)

b. Find the inverse of f.

Question 2.

Using elementary row operations, find the inverse of the matrix [ ]

Question 3.

a. f (x) is a strictly increasing function, if f’ (x) is _____

i. positive

ii. negative

iii. 0

iv.None of these

b. Show that the function f given by f (x) = x^{3} – 3x^{2} + 4x, x ∈ R is strictly increasing.

Question 4.

Question 5.

Find the area of the region bounded by the Curve y^{2} = x, x-axis and the lines x=1 and x=4.

Question 6.

Find the general solution of the differential equation x + 2y = x^{2} log x

Question 7.

A manufacturer produces nuts and bolts. It takes 1 hour of work on Machine A and 3 hours on Machine B to produce a package of nuts. It take 3 hours on Machine A and 1 hour on Machine B to produce a package of bolts. He earns a profit of ₹ 17.50 per package on nuts and ₹ 7.00 per package on bolts. Formulate the above L.P.P., if the machines operates for at most 12 hours a day.

**Questions 8 to 17 carry 4 scores each. Answer any eight. **

Question 8.

Let A=N x N and ‘*’ be a binary operation on A defined by (a,b)*(c,d)=(a+c, b+d)

a. Find (1,2)*(2,3)

b. Prove that ‘*’ is commutative.

c. Prove that ‘*’ is associative.

Question 9.

a. Identify the function from the above graph.

i. tan^{-1 }x

ii. sin^{-1 }x

iii. cos^{-1 }x

iv.cosec^{-1} x

b. Find the domain and range of the fun-ction represented in above graph.

c. Prove that tan^{-1} +tan^{-1} =tan^{-1}

Question 10.

Question 11.

a. Find the slope of the tangent to the curve y = (x-2)^{2} at x=1

b. Find a point at which the tangent to the curve y = (x-2)^{2} is parallel to the chord joining the points A(2,0) and B(4, 4).

c. Find the equation of the tangent to the above curve and parallel to the line AB.

Question 12.

Question 13.

Consider the following figure :

Question 14.

a. The degree of the differential equation

b. Find the general solution of the differential equation sec^{2} x tan y dx + sec^{2} y tan x dy = 0

Question 15.

Question 16.

a. Find the equation of a plane which makes x, y, z intercepts respectively as 1, 2, 3.

b. Find the equation of a plane passing through the point (1,2,3) which is parallel to above plane

Question 17.

Solve the L.P.P. given below graphically :

**Questions from 18 to 24 carry 6 scores each. Answer any five. (5 × 6 = 30) **

Question 18.

Question 19.

Question 20.

a. Prove that the function defined by f (x) = cos (x^{2}) is a continuous function

Question 21.

Evaluate the following :

Question 22.

Question 23.

a. Find the angle between the lines

b. Find the shortest distance between the pair of lines

Question 24.

a. The probability distribution of a random variable is given by P(x). What is ∑ P(x)?

b. The following is a probability distribution function of a random variable.

i. Find k

ii. Find P(x>3)

iii. Find p(-3<x<4)

iv. Find p(x<-3)

**ANSWERS**

Answer 1.

Answer 2.

Answer 3.

a. positive

b. f(x) = x^{3}-3x^{2}+4x

f(x) is strictly increasing for all values of x at which

f(x)=3x^{2}-6x+4

= 3(x-1) 2 + 1 > 0 for all x ∈ R

∴ f(x) is strictly increasing on R.

Answer 4.

Answer 5.

Answer 6.

Answer 7.

i. Let x = no. of packets of nuts

y = no. of packets of bolts

ii. LPP is maximise, z = 17.5x + 7y

subject to x + 3y < 12, 3x + y < 12 x > 0, y > 0

(4, 0),(3,3), (0, 4), (0,0) are the comers among these profit is maximum at (3, 3). Maximum profit = 17.5 x 3+7 x 3 = 73.5

Answer 8.

a. (a,b) * (c,d) = (a+c, b+d)

(1,2) * (2,3) = (1+2, 2+3) = (3,5)

b. (a,b) * (c,d) = (a+c, b+d)

(c,d) * (a,b) = (c+a, d+b) = (a+c, b+d)

(a,b) * (c,d) = (c,d) * (a,b)

* is commutative

c. Let (a,b), (c,d), (e,f) ∈ A

(a,b) * [ (c,d) * (e,f)]

= (a,b)*[(c+ d), (d + f)]

= (a+c+e, b+d+f)

[(a,b)*(c,d)*(e,f) = (a+c, b+d)*(e,f) = (a+c+e. b+d+f)

i.e (a,b) * [(c,d)*(e,f)] = [(a,b) * (c,d)]*(e,f)

.’. * is associative

Answer 9.

Answer 10.

Answer 11.

Answer 12.

By definition

Answer 13.

The given eqn are

x^{2} + y^{2} = 50 _______

y = x _______

(1) ⇒ x^{2}+x^{2} = 50

2x^{2} = 50

x^{2} = 25, x=5

x lies on first quadrient so x= 5, y = 5 The point p (5,5).

Answer 14.

Answer 15.

Answer 16.

a. + + = 1

+ + = 1

6x + 3y + 2z = 6

b. The normal vector for the plane is

(6.3.2) . The general equation of a plane is

n. (x-x_{0}, y-y_{0}, z-z_{0}) = 0

(6.3.2) . (x-1, y-2, z-3)=0

6 ( x – 1 ) + 3 (y – 2) + 2 (z – 3) = 0

6x+3y+2z-6-6-6=0

6x+3y+2z-18=0

6x+3y+2z =18

Answer 17.

Answer 18.

Answer 19.

Answer 20.

a. f(x) = cos(x^{2}). The domain of F is R Let g(x) = cos x and h(x) = x^{2}. Then g(x) and h(x) are continuous functions

(goh) (x) = g(h(x))

= g(x^{2}) =cos(x^{2}) = f(x)

Since g and h are continuous, goh is also continuous. Hence f is continous.

Answer 21.

a. We know that derivative of mx is m. Thus, we make the substitution mx=t so that mdx=dt

Answer 22.

Answer 23.

Answer 24.

a The probability distribution of a random variable is given by p(x).

then P(x) = 1 then ∑ P(x) = 1

ie, sum of all the probabilities in a prob¬ability distribution must be one.

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