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This mock test of NDA I - Mathematics Question Paper 2014 for Defence helps you for every Defence entrance exam.
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QUESTION: 1

Every quadratic equation ax^{2} +bx+c= 0 where a,b,c Every quadratic equation ax^{2} +bx+c= 0 where a,b,c

Solution:

QUESTION: 2

The relation S is defined on the set of integers Z as xSy if integer x devides integer y. Then

Solution:

QUESTION: 3

If are all positive, then the value of the

determinant

Solution:

QUESTION: 4

Which of the following statements are correct ?

1. A^{2} = A

2.B^{2} = B

3 .(AB)^{2}=AB

Q. Select the correct answer using the code given below:

Solution:

QUESTION: 5

What is (1001 )_{2 }equal to ?

Solution:

QUESTION: 6

What is equal to, where

Solution:

QUESTION: 7

Let z be a complex number such that |z| = 4 and arg z =

Q. WhereWhat is z equal to ?

Solution:

QUESTION: 8

If x + iy, where then what is x equal to ?

Solution:

QUESTION: 9

If are the roots of ax^{2} +bx + c = are the roots of px^{2} +qx + r = 0, then what is h equal to ?

Solution:

QUESTION: 10

If the matrix A is such that then what is A equal to ?

Solution:

QUESTION: 11

Consider the following statements :

1. Determinant is a square matrix.

2. Determinant is a number associated with a square

Q. Which of the above statements is/are correct ?

Solution:

1. We know that, determinant is not a square matrix, so it is not a true statement.

2. It is true that, determinant is a number associated with a square matrix.

Hence, Statement 2 is correct

QUESTION: 12

If A is an invertible matrix, then what is det (A~^{l}) equal to ?

Solution:

QUESTION: 13

From the matrix equation AB=AC, Where A, B, C are the square matrices of same order, we can conclude B = C provided

Solution:

From the matrix equation, A B = AC, where A, B and C are the square matrices of same order.

We can conclude B = C provided and A is non-singular.

QUESTION: 14

If A = is symmetric, th en w hat is x equal to ?

Solution:

QUESTION: 15

If then which one o f the following is correct ?

Solution:

QUESTION: 16

The function f : N → N , N being the set o f o f natural numbers, defined by f( x) = 2x + 3 is

Solution:

QUESTION: 17

What is equal to, where n is a natural number and

Solution:

QUESTION: 18

What is the number of ways in which one can post 5 letters in 7 letters boxes ?

Solution:

First letter can be put any 7 letters boxes = 7 ways

Similarly, 2nd, 3rd, 4th and 5th letters be put in 7 ways each, respectively

7 x 7 x 7 x 7 x 7= 7^{5}

QUESTION: 19

What is the number of ways that a cricket team of 11 players can be made out of 15 players ?

Solution:

Number of ways that a cricket team of 11 players can be made out o f 15 players

QUESTION: 20

A and B are two sets having 3 elements in common. Ifn(A) = 5, n(B) = 4, then what is n(A x B) equal to ?

Solution:

Here, n (A) = 5 and n(B) = 4

∴ n (A x B )= 5 x4 = 20

[ ∴ n(A) = m,n (B) = n ⇒ n(A x B) = mn]

QUESTION: 21

If f (x) = a x + b and g(x) = cx+d such that f [g(x)] = g [f(x)] then which one of the following is correct?

Solution:

f(x) = ax + bandg(x) = cx+d

f [(g(x))] = a ( e x + d) + b

= acx + ad + b

and g [f(x)] = c (ax+b) + d

= acx + dc + d

∴ f[(gx)] = gWd)]

⇒ad + b = bc + d

⇒f(d) = g(b)

QUESTION: 22

If A and B are square matrices of second order such that | A | = - l , |B| = 3, then w hat is |3AB| equal to ?

Consider the function

Solution:

We know that, |kA| = k^{n} | A |, where n is order of matrix A.

|3AB| = 3^{2} |A| |B|

(∴ |AB| = |A||B|)

= 9 (-l)(3 )

= -2 7

( ∴ |A |= - 1 , |B| = 3)

QUESTION: 23

What is f(2x) equal to ?

Solution:

QUESTION: 24

What is equal to ?

Solution:

QUESTION: 25

What is f(f(x)) equal to ?

Solution:

QUESTION: 26

Consider the expansion

Q. What is the independent term in the given expansion ?

Solution:

QUESTION: 27

What is the ratio of coefficient o f x^{15} to the term independent of x in the given expansion ?

Solution:

For coefficient o f x^{15},

30 - 3r = 15

⇒r = 5

the coefficient o f x 15 is ^{15}C_{5}.

and coefficient of independent of x is

30 - 3r = 0

⇒r = 10

So, coefficient o f independent o f x is ^{15}C_{10}.

QUESTION: 28

Consider the following statements:

1. There are 15 terms in the given expansion.

2. The coefficient o f x 12 is equal to the at of x^{3}.

Q. Which of the above statements is/are correct ?

Solution:

1. We know that, (a + b)^{n} have total number of terms is n+ 1

Hence, Statement 1 is false.

2. For coefficient o f x^{12}

30 - 3r = 12 ⇒ r = 6⇒ ^{15}C_{6}

and for coefficient of x^{3},

30 - 3r = 3 ⇒ r = 9⇒ ^{15}C_{9}

^{15}c_{6} = ^{15}c_{9}

Hence, statement 2 is correct.

QUESTION: 29

Consider the following statements:

1. The term containing x^{2} does not exist in the given expansion.

2. The sum of the coefficients of all the terms in the given expansion is 2^{15}.

Q. Which of the above statements is/are correct ?

Solution:

1. For coefficient of x^{2},

So, x^{2 }does not exist in the expansion

Hence, Statement 1 is correct.

Put x = 1 both sides, we get

( l + l )^{15} = ^{15}C_{0 }+ ^{15}C_{1} + .... + ^{15}C_{15}

⇒ 2^{15} = ^{15}C_{0} + ^{15}C_{1} +.... + ^{15}C_{15}

Hence, Statement 2 is correct

QUESTION: 30

What is the sum of the coefficients of the middle terms in the given expansion ?

Solution:

QUESTION: 31

What isequal to ?

Solution:

QUESTION: 32

A lamp post stands on a horizontal plane. From a point situated at a distance 150 m from its foot, the angle of elevation of the top is 30°. What is the height of the lamp post ?

Solution:

In ΔABC, we have BC = 150 m, AB = h ∠C=30°

QUESTION: 33

If cot A = 2 and cot B = 3, then what is the value of A + B ?

Solution:

cot A = 2 and cot B = 3

cot (A + B) =

⇒cot (A + B) = cot⇒ A + B=

QUESTION: 34

What is sin^{2}- sin^{2}equal to ?

Solution:

QUESTION: 35

What is equal to ?

Solution:

QUESTION: 36

What is equal to ?

Solution:

QUESTION: 37

In a triangle ABC, c = 2, A = 45°, a =, than what is C equal to ?

Solution:

According to sine rule,

QUESTION: 38

In a triangle ABC, sin A - cos B = cos C, then what is B equal to ?

Solution:

In a ΔABC, we have

sin A - cos B = cos C ⇒ sin A = cos B + cos C

[∵ cos (90° - θ) = sin θ]

⇒ A + C = B ...(i)

Also, A +C = 180°- B ...(ii)

So, 180°—B=180°

⇒2B=180°

∵ B=90°

QUESTION: 39

If then what isequal to ?

Solution:

Applying componendo and dividendo, we get

QUESTION: 40

If sin A sin (60° - A) sin (60°+A) = k sin 3A, then what is k equal to ?

Solution:

sin A • sin (60° - A) sin (60° + A) = k sin 3A

QUESTION: 41

The line meets the graph y = tan x, wherein k points. W hat is k equal to?

Solution:

Line and graph = y= tan x

Now, we have =tan x

⇒ tanx=tan 60°

⇒x=60°

Hence, one intersecting point is possible in the given domain i.e.,k= 1.

QUESTION: 42

Which one of the following is one of the solutions of the equation of the equation tan 2θ. tanθ = 1 ?

Solution:

tan 2θ. tanθ=1

⇒ 2 tan^{2}θ= 1 - tan^{2} 0 ⇒ 3 tan^{2}θ= 1

tan2θ = tan2 (30°)=tan^{2
}

QUESTION: 43

Given that 16sin^{5 }x =p sin 5x + q sin 3x + r sin x

Q. What is the value of p ?

Solution:

1 6 sin^{5} x = 16 (sin^{2} x)^{2}. sin x

= (6 + 2 cos 4 x - 8 cos 2x) sin x

= 6 sin x + 2 sin x cos 4x - 8 cos 2x. sin x

= 6 sin x + sin 5 x - sin 3 x - 4 (sin 3 x - sin x)

[∵ 2 sinA cos B = sin(A +B ) + sin(A -B )]

= 6 sin x + sin 5 x - sin 3 x - 4 sin 3x + 4 sin x

= sin 5x-5 sin 3x + 10 sin x

Clearly, p = 1, hence option (a) is correct.

QUESTION: 44

Given that 16sin^{5} x = psin 5 x + q sin 3 x + r sin x

What is the value of q ?

Solution:

1 6 sin^{5} x = 16 (sin^{2} x)^{2}. sin x

= (6 + 2 cos 4 x - 8 cos 2x) sin x

= 6 sin x + 2 sin x cos 4x - 8 cos 2x. sin x

= 6 sin x + sin 5 x - sin 3 x - 4 (sin 3 x - sin x)

[∵ 2 sinA cos B = sin(A +B ) + sin(A -B )]

= 6 sin x + sin 5 x - sin 3 x - 4 sin 3x + 4 sin x

= sin 5x-5 sin 3x + 10 sin x

Clearly, q = - 5 , hence option (d) is correct.

QUESTION: 45

Given that 16sin^{5 }x =p sin 5x + q sin 3x + r sin x

Wha t is the value of r ?

Solution:

1 6 sin^{5} x = 16 (sin^{2} x)^{2}. sin x

= (6 + 2 cos 4 x - 8 cos 2x) sin x

= 6 sin x + 2 sin x cos 4x - 8 cos 2x. sin x

= 6 sin x + sin 5 x - sin 3 x - 4 (sin 3 x - sin x)

[∵ 2 sinA cos B = sin(A +B ) + sin(A -B )]

= 6 sin x + sin 5 x - sin 3 x - 4 sin 3x + 4 sin x

= sin 5x-5 sin 3x + 10 sin x

Clearly, r = 10, hence option (c) is correct.

QUESTION: 46

LetS_{n }denote the sum of the n terms of an AP and 3Sn = S_{2n}.

What is S_{3n} = S_{n} equal to ?

Solution:

Given, S_{n} = Sum of first n terms of an AP.

According to direction, 3S_{n} = 2S_{2n}

Putting the value of S_{n} and S_{2n} in above equation.

6a+3 (n -l)d = 4 a + 2 (n - l)d

2 a = d ( n + 1)

QUESTION: 47

LetS_{n }denote the sum of the n terms of an AP and 3Sn = S_{2n}.

What is S_{3n} = S_{2n} equal to ?

Solution:

From explanation

QUESTION: 48

LetS_{n }denote the sum of the n terms of an AP and 3Sn = S_{2n}.

What is the length of the latus rectum of the ellipse 25x^{2 }+ 16y^{2}=400?

Solution:

Equation of ellipse is 25x^{2} + 16y^{2} = 400

Here, a^{2} = 16 and b^{2} = 25

∴Length of latus rectum

QUESTION: 49

Consider the circles x^{2} + y^{2 }+ 2ax + c = 0 and x^{2} + y^{2} + 2by + c = 0.

What is the distance between the centres of the two circles?

Solution:

Equations of circles are

x^{2} + y ^{2} + 2 a x + c = 0

and x^{2} + y^{2} + 2by + c = 0

Since, the centres of two circles are (-a, 0) and (0, -b)

∴ Distance between two centres =

QUESTION: 50

Consider the circles x^{2} + y^{2 }+ 2ax + c = 0 and x^{2} + y^{2} + 2by + c = 0.

The two circles touch each other if

Solution:

Two circles touch each other, iff distance between two centres = Sum of radius of two circles

QUESTION: 51

A (3,4) and B(5, -2) are two points and P is a point such that P A = PB. I f th e area o f triangle PAB is 10 square unit, what are the coordinates of P ?

Solution:

Given A (3,4) and B (5, -2)

Let, P (x, y)

Given that, PA= PB

⇒PA^{2} = PB^{2}

⇒ (x-3)2 + (y -4 )^{2} = (x-5)^{2} + (y + 2)^{2}

⇒x2-6 x + 9 + y^{2}-8y+ 16

= x^{2} - 1Ox + 25 + y^{2} + 4y + 4

⇒4x-12y=4

⇒x-3y = 1 ...(i)

⇒x(4 + 2 ) - y ( 3 - 5 ) + 1 (-6 -2 0 ) = ±20

⇒ 6x+2y- 26 = ± 20

⇒6x + 2y-26 = 20

or,6x+2y-26 = -20

⇒6x+2y=46 ...(ii) or6x+2y=6 ...(in)

From eqs. (i) and (ii), we get

x = 7,y=2

Similarly, from eqs. (i) and (iii),we get

x=l ,y =0

Hence, coordinates of P are (7,2) or (1,0)

QUESTION: 52

What is the product of the perpendiculars drawn from the points upon the line bx cos α+ ay sin α = ab?

Solution:

Given equation of line is bx cos α + ay sin α = ab

Perpendicular distance from point is

( ∴ distance from (x^{1}, y^{1}) to ax + by + c = 0 is

Similarly, perpendicular distance from point

By product of d_{1} and d_{2}, we get

Hence, product of the perpendiculars = - b_{2} = b_{2} (since, distance is positive)

QUESTION: 53

Which of the following is correc in respect of the equations

Solution:

Given equation of line is

and equation of second line is 2x + 3y = 5

∵ Slope of first line,

and slope of second line,

∵ m_{1}m_{2}= -1

Hence, two lines are perpendicular to each other.

QUESTION: 54

Consider a sphere passing through the origin and the points (2 ,1 , - 1 ) , (1 ,5 , - 4 ) , ( - 2 , 4 , - 6).

What is the radius of the sphere ?

Solution:

Equation of sphere passing through origin is

x^{2} + y^{2 }+ z^{2} + 2ux + 2vy + 2wz = 0

which passes through the points (2, 1, -1) (1,5, -4), and (-2,4,-6) .

∵ 4u + 2 v - 2 w = - 6 ...(i)

2u+10 v - 8w = - 42 ...(ii)

and - 4u + 8v - 12w = - 56 ...(iii)

From eqns (i), (ii) and (iii), we get

u= 1, v = -2 and w=3

QUESTION: 55

Consider a sphere passing through the origin and the points (2 ,1 , - 1 ) , (1 ,5 , - 4 ) , ( - 2 , 4 , - 6).

What is the centre of the sphere ?

Solution:

From explanation 54

Centre of sphere,

(-u, -v, -w) = (-1,2, -3)

QUESTION: 56

Consider a sphere passing through the origin and the points (2 ,1 , - 1 ) , (1 ,5 , - 4 ) , ( - 2 , 4 , - 6).

Consider the following statements:

1. The sphere passes through the point (0,4,0).

2. The point (1, 1, 1) is at a distance of 5 unit from the centre of the sphere.

Q. Which of the above statements is/are correct ?

Solution:

1) Equation of sphere is

x^{2} + y^{2} + z^{2} + 2x-4y + 6z = 0

Put the value (0,4,0),we get

0+16+0+0-16+0 = 0

So, the sphere passes through the point (0,4,0).

Hence, Statement 1 is correct.

2. Distance between the point (1, 1, 1) and centre of sphere (-1,2,-3)

Hence, statement 2 is not correct.

QUESTION: 57

The line joining the points (2, 1, 3) and (4, -2, 5) cuts the plane 2x + y-z = 3.

Where does the line cut the plane ?

Solution:

Equation of line passing through the points (2, 1, 3) and (4, - 2 , 5 ) is

⇒x = 2λ + 2 , y = - 3 λ + l andz = 2λ +3

So, (2λ + 2, - 3λ+ 1 , 2λ, + 3) satisfies the equation o f plane 2λ, + 2 - 3 λ , + l - 2v , - 3 = 3

⇒-3λ,=3

> λ = — 1

Hence, points are [2 (-1)+2, - 3 (-1) + 1,2 (-1) + 3] i.e., (0,4,1).

QUESTION: 58

The line joining the points (2, 1, 3) and (4, -2, 5) cuts the plane 2x + y-z = 3.

What is the ratio in which the plane divideds the line ?

Solution:

Let th e ratio is k : 1

QUESTION: 59

consider the plane passing through the points A (2,2,1), B (3,4,2) and C (7,0,6).

Which one of the following points lines on the plane ?

Solution:

We know that, equation of plane passing through three non-col linear points

(x_{1}, y_{1}, z_{1},), (x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, Z_{3}) is

Put the value of (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) we get

⇒ (x -2) (10+ 2)- (y-2 ) (5 -5 ) + (z -1 ) ( -2 -10) = 0

⇒12x- 12z= 12⇒x-z = 1

Hence the equation of plane parses through (1,0, 1)

QUESTION: 60

consider the plane passing through the points A (2,2,1), B (3,4,2) and C (7,0,6).

What are the direction ratios of the normal to the plane ?

Solution:

Direction ratios of the nomal to the plane x-z = 1 are (1,0,-1).

QUESTION: 61

Consider the function f ( x ) =

What is equal t o ?

Solution:

QUESTION: 62

Consider the function f ( x ) =

Consider the following statements:

1. The function is discontinuous at x = 3.

2. The function is not differentiable at x = 0.

What of the above statements is/are correct ?

Solution:

1.

2.Therefore f(x) is continuous at x = 4 2.

Given f(x ) = x^{2} - 5 x ≤ 3

f '( x ) = 2x

f '( 0) = 0

So, f (x) is differentiable at x = 0

Therefore, neither statement 1 nor 2 is correct.

QUESTION: 63

Consider the function f ( x ) =

What is the differential coefficient of f( x ) at x = 12 ?

Solution:

QUESTION: 64

The line 2y = 3x + 12 cuts the parabola 4y = 3x^{2}.

Where does the line cut the parabola ?

Solution:

Equation of line 2y = 3x + 12 „.(i)

Equation of parabola

4y= 3x^{2} „.(ii)

From eqs. (i) and (ii), we get

2(3x+ 12)=3x^{2}

3x^{2}- 6x- 2 4 = 0

x^{2}- 2x- 8 = 0

(x -4 )(x + 2 )= 0 x=4 and x =-2

Now putting the value of x in eqn (ii)

We get y = 12 and y = 3

Thus, the points (-2,3) and (4,12)

QUESTION: 65

The line 2y = 3x + 12 cuts the parabola 4y = 3x^{2}.

What is the area enclosed by the parabola and the line ?

Solution:

Equation of line 2y = 3x + 12, y =

Equation ofparabola4y = 3x^{2},

QUESTION: 66

The line 2y = 3x + 12 cuts the parabola 4y = 3x^{2}.

What is the area enclosed by the parabola, the line and the Y- axis in the first quadrant ?

Solution:

Equation of line 2y = 3x + 12 and equations of parabola 4y=3x^{2
}

= 3 x 4 + 24 - 16 = 36 - 16 = 20 sq. units.

∴Area enclosed by the parabola, the line and the y axis in first quadrant = 20 sq. units

QUESTION: 67

Consider the function

What is the non-zero value of k for which the function is continuous at x = 0?

Solution:

When function is continuous at x = 0, then

QUESTION: 68

Consider the following statements: 1. The fu n c tio n / (x) = [x]9 where [.] is the greatest integer

is continuous at all points except at x = 0.

2 . The function f( x) = sin|x| is continuous for all x ∈ R .

Which of the above statements is/are correct ?

Solution:

The greatest integer function is continuous at all statement points except integer. Hence, statement lis incorrect.

**Statement 2 :** Let h(a) = sin x and g(x) = |x| hog (x) = sin |x| ⇒ f(x) = hog (x) = sin |x| Therefore, g(x) is continuous, x ∈ Rand

h (x) is continous x ∈ R

When both are continuous then hog (x) is also continuous.

Thus, statement 2 is correct.

QUESTION: 69

Consider the curve x = a (cos θ + θ sin θ) and y = a (sin θ - θ cos θ).

What is equal to ?

Solution:

Givenx = a(eosθ + 0sinθ)

y = a (sinθ - θ cos θ) and we have = tanθ

According to question

= sec^{2}θ

QUESTION: 70

Consider the curve x = a (cos θ + θ sin θ) and y = a (sin θ - θ cos θ).

What is equal to ?

Solution:

Given,x = a(eosθ + θsinθ)and

y = a(sin θ -θ co s θ)

= a (-sin θ + θcosθ + sinθ) = aθcosθ

= a (cosθ + θ sin θ - cos θ) = a θ sin θ

QUESTION: 71

What is the area of the parabola y^{2} = 4bx bounded by its latus rectum ?

Solution:

∴area of parabola bounded by its latus rectum

QUESTION: 72

If y = x /n x + xe^{x}, then what is the value of at x = 1 ?

Solution:

y = x ln x + xe^{x}

After differentiating both sides with respect to x,

QUESTION: 73

Solution:

QUESTION: 74

Solution:

QUESTION: 75

Solution:

Note: 1+2 + 3 + ....+ n =

l^{2} + 2^{2} + 3^{2} + .... + n^{2} =

QUESTION: 76

Solution:

[∴ sin(π-x) = sinx]

Adding eqs. (i) and (ii), we get

When x = 0, then t = 1 and when x = then t = 2

According to the explanation, 1 = π

QUESTION: 77

Solution:

[∴ sin(π-x) = sinx]

Adding eqs. (i) and (ii), we get

When x = 0, then t = 1 and when x = then t = 2

QUESTION: 78

Solution:

[∴ sin(π-x) = sinx]

Adding eqs. (i) and (ii), we get

When x = 0, then t = 1 and when x = then t = 2

QUESTION: 79

Solution:

where, C is the constant of integration

(using integration by parts)

hence option (b) is correct.

QUESTION: 80

Solution:

where, C is the constant of integration

(using integration by parts)

QUESTION: 81

Solution:

QUESTION: 82

Solution:

QUESTION: 83

A rectangular box is to be made from a sheet of 24 inch length and 9 inch width cutting out identical squares of side length x from the four corners and turning up the sides

What is the value of x for width the vulume is maximum ?

Solution:

length o f the box = 24 - 2x

width of the box = 9 - 2x

Volume of the box=V

V = (24 - 2x ) (9 -2x ) .x (∵height o f box = x inch)

= ( 216 - 48 x - 18x +4x^{2}).x

V (x)= 4x^{3} - 66x^{2} + 2 16x

⇒ V'(x)= 12x^{2}-132x+ 216 For maximum, put V'(x)=0

⇒ 12x^{2}- 132x+216=0

⇒x2 - l l x + 18 = 0

⇒x2 - 9x - 2x + 18 = 0

⇒ x (x -9)-2(x-9)= 0

⇒ (x -9 )(x -2 ) = 0

⇒x=9 or x=2 Now V " ( x ) = 24 x - 132

V " (9 )= 2 1 6 - 132 = 84>0

and V" (2) = 4 8 - 1 3 2 = —84 < 0 Thus, volume is maximum when x = 2 inch.

QUESTION: 84

A rectangular box is to be made from a sheet of 24 inch length and 9 inch width cutting out identical squares of side length x from the four corners and turning up the sides

What is the maximum volume of the box?

Solution:

length o f the box = 24 - 2x

width of the box = 9 - 2x

Volume of box = (24 - 4) (9 - 4),2

= 20 x 5 x 2 = 200 cu inch

QUESTION: 85

What is the degree of the differential equation

Solution:

Consider the given differential equation,

In order to find degree, differential equation should be free from fractional indices.

Now, squaring eqn (i) both sides, we get

Since, the power of highest order derivative is 3, therefore degree = 3.

QUESTION: 86

What is the solution of the equation

Solution:

Consider the given differential equation

on separating the variables, we get

dy = e^{-x} dx,

on integrating both sides, we get

QUESTION: 87

Eliminating the arbitrary constants B and C in the expresion

Solution:

On differentiating both sides w.r.t. x, we get

On squaring both sides, we get

Now, on differentiating wr.t. x, we get

QUESTION: 88

Let f(x) = ax^{2} + bx + c such that f( 1 ) = f( - 1 ) and a , b, c are in Arithmetic Progression.

What is the value of b ?

Solution:

flx)=ax^{2} + bx+ c

f(1) = a + b + c

and f(-1) = a - b + c

⇒a + b + c = a - b + c⇒b = 0

QUESTION: 89

Let f(x) = ax^{2} + bx + c such that f( 1 ) = f( - 1 ) and a , b, c are in Arithmetic Progression.

f (a), f ' (b), f ' (c) are

Solution:

We have f '(x) = 2ax + b

f ' (a) = 2a^{2}, f '( b ) = 2ab = 0

and f'(c) = 2ac .........( ∵ b = 0 )

f'(a) = 2a^{2}

f'( b )=0

andf'(c)=-2a^{2 .............}(∵ 2b = a + c= > c= -a)

Hence f ' (a), f ' (b) an d f ' (c) are in AP.

QUESTION: 90

Let f(x) = ax^{2} + bx + c such that f( 1 ) = f( - 1 ) and a , b, c are in Arithmetic Progression.

f"(a), f" (b), f"(c)are

Solution:

f"(x)=2a

∵ f"(a) = f"(b) = f"(c)

QUESTION: 91

If then what is equal to ?

Solution:

QUESTION: 92

If then which one of the following is correct?

Solution:

QUESTION: 93

What is the area of the triangle OAB where O is the origin,

Solution:

QUESTION: 94

Which one of the following is the unit vector perpendicular to both

Solution:

QUESTION: 95

What is the interior acute angle of the parallelogram whose sides are represented by the vectorsand

Solution:

QUESTION: 96

For what value of λ are the vectors

perpendicular ?

Solution:

QUESTION: 97

What is the angle between ?

Solution:

QUESTION: 98

What is equal to ?

Solution:

QUESTION: 99

What is cosine of the angle between

Solution:

QUESTION: 100

What is equal to ?

Solution:

QUESTION: 101

What is equal to ?

Solution:

[divide numerator and denominator by cos^{2} x]

Put tanx = t

⇒sec^{2} xdx = dt

When x = 0, then t = 0 and when x =then t =

QUESTION: 102

A cylinder is inscribed in a sphere of radius r.

What is the height of the cylinder of maximum volume ?

Solution:

Let h be the height, R be the radius and V be the volume of cylinder.

In ΔOAB, we have

Clearly, V=πR^{2}h

For maximum put V' (h) = 0

Differentiating eq. (ii) w.r.t. h, we get

Thus, the volume is maximum when h =

QUESTION: 103

A cylinder is inscribed in a sphere of radius r.

Q. What is the radius of the cylinder of maximum volume ?

Solution:

Volume of cylinder is maximum when h =

QUESTION: 104

Consider the function f"(x) = sec^{4} x + 4 with f(0) = 0 and f'(0) = 0.

What is f'(x) equal to ?

Solution:

Put tan x = t in the integral I_{1},

then sec^{2} x dx = dt

QUESTION: 105

Consider the function f"(x) = sec^{4} x + 4 with f(0) = 0 and f'(0) = 0.

What is f(x) equal to?

Solution:

QUESTION: 106

Suppose A and B are two events. Event B has occurred and it is known that P(B) < 1. What is P(A|B^{C}) equal to ?

Solution:

QUESTION: 107

Consider events A, B, C, D, E o f the sample space S = { n : n is an integer such th a t 10 ≤ n ≤ 20} given by :

A is the set of all even numbers.

B is the set of all prime numbers.

C=(15).

D is the set of all integers ≤ 16

E is the set of all double digit numbers expressible as a power of 2.

A, Band D are

Solution:

A, B and D are exhaustive events but not mutually exclusive events.

QUESTION: 108

Consider events A, B, C, D, E o f the sample space S = { n : n is an integer such th a t 10 ≤ n ≤ 20} given by :

A is the set of all even numbers.

B is the set of all prime numbers.

C=(15).

D is the set of all integers ≤ 16

E is the set of all double digit numbers expressible as a power of 2.

A, B and C are

Solution:

A, B and C are exhaustive events but not mutually exclusive events.

QUESTION: 109

Consider events A, B, C, D, E o f the sample space S = { n : n is an integer such th a t 10 ≤ n ≤ 20} given by :

A is the set of all even numbers.

B is the set of all prime numbers.

C=(15).

D is the set of all integers ≤ 16

E is the set of all double digit numbers expressible as a power of 2.

B and C are

Solution:

B and C are mutually exclusive events but not exhaustive events.

QUESTION: 110

A is the set of all even numbers.

B is the set of all prime numbers.

C=(15).

D is the set of all integers ≤ 16

E is the set of all double digit numbers expressible as a power of 2.

C and E are

Solution:

C and E are mutually exclusive and elementary events.

QUESTION: 111

Consider the following statements in respect of histogram:

1. The histogram is a suitable representation of a frequency distribution of a continuous variable.

2. The area included under the whole histogram is the total frequency.

Which of the above statements is/are correct?

Solution:

1. It is true that, the histogram is a suitable representation of a frequency distribution of a continuous variable. Hence, Statement 1 is correct.

2. We know that, the area of histogram is proportional to the frequency, so it is not true statement.

QUESTION: 112

The regression lines will be perpendicular to each other if the coefficient of correlation r is equal to

Solution:

When regression lines perpendicular to each other then angle will be:

QUESTION: 113

For any two events A and B, which one of the following holds ?

Solution:

QUESTION: 114

The probability that in a random arrangement of the letters of the word 'UNIVERSITY', the two I's do **not** come together is

Solution:

QUESTION: 115

Threre are 4 white and 3 black balls in a box. In another box, there are 3 white and 4 black balls. An unbiased dice is rolled. If it shows a number less than or equal to 3, then a ball is drawn from the second box, otherwise from the first box. If the ball drawn is black then the possibility that the ball was drawn from the first box is

Solution:

Box I→4 W; 3 B

Box II→ 3 W ;4B

Probability for choosing first box =

Probability for choosing second box =

∴ Required probability =

QUESTION: 116

If are the means of two distrubutions such that and is the mean of the combined distrubution, then which one of the following statements is correct ?

Solution:

It is obvious that,

QUESTION: 117

What is the mean deviation about the mean for the data 4,7, 8,9,10,12,13,17?

Solution:

Mean =

∴ Mean deviation about mean =

QUESTION: 118

The variance of 20 observations is 5. If each observation is multiplied by 2, then what is the new varianve of the resulting observations ?

Solution:

Let x_{1}, x_{2}, , x_{20} be the given observations.

To find variance of2x_{1}, 2x_{2},2x_{3},...., 2x_{20},

Let denotes the mean of new observation,

Clearly,

Now, variance of new observation

QUESTION: 119

Two students X and Y appeared in an examination. The probability that X will qualify the examination is 0.05 and Y will qualify the examination is 0.10. The probability that both will qualify the examination is 0.02. What is the probability that only one of them will qualify the examination ?

Solution:

Let A and B be the events

= P (A) + P ( B ) - 2P (A B)

= 0.05 + 0.1-2(0.02)

= 0.15-0.04=0.11

Hence the pobability that only one of them will qualify the examination is 0.11.

QUESTION: 120

A fair coin is tossed four times. What is the probability that at most three tails occur ?

Solution:

n(S) = 2^{4}= 16

and n(E) = ^{4}C_{0} + ^{4}C_{1} , + ^{4}C_{2} + ^{4}C_{3}

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