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QUESTION: 1

The maximum value of logx/x in (2, ∞) is

Solution:

1/e

let f(x)=logx/x

to get max of f(x)... fi(x)=0 u get some value of x

substitute it in f(x) u get max value of f(x)

QUESTION: 2

The area enclosed by the parabola y^{2} = 8x and the line y = 2x is

Solution:

QUESTION: 3

Locus of a point P equidistant from two fixed points A and B is ____________

Solution:

The line which makes equal distance from the two fixed points will definitely pass through the midpoint of line joining the two points and will definitely perpendicular to the line formed by joining the two points.

QUESTION: 4

The real part of 1/(1-cosθ + i sin θ) is

Solution:

QUESTION: 5

Solution:

QUESTION: 6

If the matrix is singular, then θ =

Solution:

QUESTION: 7

Solution:

QUESTION: 8

If f x = x 2 | x | for x ≠ 0, f 0 = 0, then f x at x = 0 is

Solution:

QUESTION: 9

If f( x) = sin π [ x ] then f ′ (1 − 0) is equal to

Solution:

**ANSWER :- b**

**Solution :- **f′(1-0)= lim h→0 f(1-0-h)−f(1-0)

= lim h→0 (sinπ[1-0-h]−sinπ[1-0])/h

= lim h→0 (sinπ−sinπ)/h=0

QUESTION: 10

The family of curves, in which the subtangent at any point to any curve is double the abscissa, is given by

Solution:

QUESTION: 11

Observe the following statements

Which of the following is a correct statement?

Solution:

QUESTION: 12

The solution of

is

Solution:

QUESTION: 13

The solution of the equation (dy/dx)=cotx coty is

Solution:

QUESTION: 14

The differential of sin^{-1}[(1-x)/(1+x)] w.r.t. √x is equal to

Solution:

QUESTION: 15

Suppose that g(x) = 1 + √x and f(g(x)) = 3 + 2√x + x, then f(x) is

Solution:

QUESTION: 16

Solution:

QUESTION: 17

∫[(1)/(1-x^{2})]dx=

Solution:

QUESTION: 18

If sin^{-1}x + sin^{-1}y = 2π/3, then cos^{-1} x + cos^{-1} y is equal to

Solution:

QUESTION: 19

if is

Solution:

QUESTION: 20

If p : A man is happy

q : A man is rich

Then , the statement , ' if a man is not happy , then he is not rich ' is written as

q : A man is rich

Then , the statement , ' if a man is not happy , then he is not rich ' is written as

Solution:

QUESTION: 21

If f : ℝ → ℝ is defined by f x =

Solution:

QUESTION: 22

If B is a non-singular matrix and A is a square matrix, then det (B^{-1}AB) =

Solution:

QUESTION: 23

m [ − 3 4 ] + n [ 4 − 3 ] = [ 10 − 11 ]

so, 3 m + 7 n =

Solution:

m[ -3 4 ] + n[ 4 -3 ] = [ -3m 4m ] + [ 4n -3n ]

= [ (-3m)+4n 4m+(-3n) ] = [ 10 -11 ]

• we can compare both sides,

-3m + 4n = 10 .... (1)

And 4m - 3n = -11 .... (2)

• Solving both equations

multiply equation (1) by 4 and equation (2) by 3

-12m + 14n = 40

and 12m - 9n = -33

• Add equation (1) and (2)

-12m + 14n + 12m - 9n = 40 - 33

5n = 7

n = 7/5

• Substitute value of n in equation (1)

-3m + 4×(7/5) = 10

-3m + 28/5 = 10

-15m + 28 = 50

-15m = 50 - 28

m = -22/15

• Substitute value of m and n in 3m + 7n

3(-22/15) + 7(7/5) = -22/5 + 49/5

= 27/5

**• Hence, value of 3m + 7n is 27/5**

QUESTION: 24

If the probability that A and B will die within a year are p and q respectively, then the probability that only one of them will be alive at the end of the year, is

Solution:

QUESTION: 25

We have (n+1) white balls and (n+1) black balls. In each set the balls are numbered from 1 to (n+1). If these balls are to be arranged in a row so that two consecutive balls are of different colours, then the number of these arrangements is

Solution:

Between (n + 1) white balls there are (n + 2) gaps in which (n + 1) black ball are to arranged.

No. of reqd arrangements = (n + 1)! (n + 1)! = [(n + 1)!]^{2}

Now between (n + 1) black balls (n + 1) white balls are to be filled no. of ways

= (n + 1)! (n + 1)!

∴ Reqd ways = 2[(n + 1)!]^{2}

QUESTION: 26

A bag contains 6 white and 4 black balls. Two balls are drawn at random. The probability that they are of the same colour is

Solution:

QUESTION: 27

A fair dice is tossed repeatedly until six shows up 3 times. The probability that exactly 5 tosses are needed is

Solution:

QUESTION: 28

If in ∆ABC ∠A=45^{o}, ∠C=60^{o}, then a+c√2=

Solution:

QUESTION: 29

Let the harmonic means and the geometric mean of two positive numbers be in the ratio 4 : 5. The two numbers are in the ratio

Solution:

QUESTION: 30

Which one of the following is a source of data for primary investigations?

Solution:

QUESTION: 31

If each observation of a raw data, whose variance is σ^{2}, is multiplied by λ, then the variance of new set is

Solution:

QUESTION: 32

The degree of the equation is ____

Solution:

QUESTION: 33

If the roots of the equation (a^{2} + b^{2})x^{2} - 2(ac + bd)x + (c^{2} + d^{2}) = 0 are equal is

Solution:

(a^{2} + b^{2})x^{2} - 2 (ac + bd)x + (c^{2} + d^{2}) = 0

For equal root, Discriminant (D) = 0

or [- 2 (ac + bd)]^{2} - 4.(a^{2} + b^{2}) (c^{2} + d^{2}) = 0

or 4 (ac + bd)^{2} - 4 (a^{2} + b^{2}) (c^{2} + d^{2}) = 0

or a^{2}c^{2} + b^{2}d^{2} + 2acbd - (a^{2} + b^{2}) (c^{2} + d^{2}) = 0

or a^{2}c^{2} + b^{2}d^{2} + 2acbd - a^{2}c^{2} - a^{2}d^{2} - b^{2}d^{2}

or 2acbd - a^{2}d^{2} - b^{2}c^{2} = 0

or (ad - bc)^{2} = 0

or ad = bc

QUESTION: 34

If A (1,2,3),B(-1,-1,-1) be the pts., then the distance AB is

Solution:

QUESTION: 35

The plane of intersection of x^{2}+y^{2}+z^{2}+2x+2y+2z+2=0 and 4x^{2}+4y^{2}+4z^{2}+4x+4y+4z-1=0 is

Solution:

QUESTION: 36

The equation 3sin^{2}x + 10cosx - 6 = 0 is satisfied if

Solution:

QUESTION: 37

The angle between the two straight lines represented by 6y^{2} - xy - x^{2} + 30y + 36 = 0 is

Solution:

QUESTION: 38

The length of the intercept made by the circle x^{2} + y^{2} + 10x - 6y + 9 = 0 on the X-axis is

Solution:

QUESTION: 39

If makes an acute angle with , then is equal to

Solution:

QUESTION: 40

If |a+b|=|a-b|, then the angle between a and b is

Solution:

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