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

Let X be the set of all citizens of India. Elements x, y in X aresaid to be related if the difference of their age is 5 years. Which one of the following is correct ?

Solution:

X = Set of all citizens of India

(R is not reflexive)

(R is symmetric)

(R is not transitive)

QUESTION: 2

2. Consider the following relations from A to B where A = {u, v, w, x, y, z} and B= {p,q, r, 5}.

1. {(u,p), (y,p), (w,p), (x, q), (y, q), (z, q)}

2. {(u,p),(y,q),(w,r),(z,s)}

3. {(«, s), (v, r), (w, q), (u,p), (v, q), (z, q),}

4. {(u, q), (v,p), (w, s), (x, r), (y, q), (z, $),}

Which of the above relations are not functions ?

Solution:

Given that, A = {u, v, x, y, z}; B = {p, q, r, s} As we know, a mapping f : x y is said to be a function, if each element in the set x has its image in set y. It is also possible that these are few elements in set y which are not the image of any element in set x. Every element in set x should have one and only one image.

(ii) and (iii) are not functions.

QUESTION: 3

If a and P are the roots of the equation ax2 + bx + c = 0, where a 0, then (aα + b){aβ+ b) is equal to:

Solution:

Given equation ax^{2} + bx + c = 0 (where a 0) α and β are roots of given equation.

(aα + b) (aβ + b) = a^{2} aβ + abα + abβ + b^{2}

= a^{2}αβ + ab (α + β) + b^{2}

From the given quadratic equation

QUESTION: 4

Let S denote set of all integers. Define a relation R on S as 'aRb if ab ≥ 0 where a, b ∈ S. Then R is :

Solution:

S = Set of all integers and

R = {(a, b), a , b ∈ S and a b≥ 0 }

For reflexive: aRa ⇒ a.a = a^{2} ⇒ 0

for all integers a. a ≥ 0

**For symmetric :** aRb ⇒ ab ⇒ 0 a, b ∈ S

If ab > 0, then ba ≥ 0 ⇒ bRa

**For transitive:**

If ab≥ 0 , be≥ 0, then also ac≥ 0

Rlation R is reflexive, symmetric and transitive.

Therefore relation is equivalence.

QUESTION: 5

The roots of the equation a^{2}x^{2} - 2 abx + b^{2} = 0 when a < 0 and b > 0 are:

Solution:

We have, 2a^{2} x^{2} - 2abx + b^{2} = 0

Discriminent, D=( - 2ab)^{2} - 4 (2a^{2}) (b^{2})

= 4a^{2}b^{2} - 8 a^{2}b^{2} = - 4 a^{2}b^{2} < 0

Roots are always complex.

QUESTION: 6

What is the sum of the two numbers (11110)_{2} and (1010)_{2} ?

Solution:

(11110)_{2} = 2^{4} x l + 2^{3} x l + 2^{2} x 1 + 2^{1} x l + 2^{0} x 0

= 16 + 8+4 + 2 + 0 = 30

(1010)_{2} = (2^{3} x 1 + 2^{2} x 0 + 2^{1} x 1 + 2^{0} x 0 = 8 + 0 + 2 + 0)= 10

Sum = 30 + 10=40

=(101000)_{2
}

QUESTION: 7

Let N denote the set of all non-negative integers and Z denote the set of all integers. The function Z → N given by is:

Solution:

f : Z → N and f ( x ) = |x|

When we drawa parallel line to x-axis.

It cuts the curve into more than one point. Therefore,f(x) = |x| is not one-one.

QUESTION: 8

If P and Q are two complex numbers, then the modulus of the quotient of P and Q is :

Solution:

The two complex numbers are

P =x+ iy and Q = α+iβ

Hence, the quotient of their modulus is equal to the quotient of their moduli.

QUESTION: 9

Let z = x + iy Where x, y are real variables i = √-i. If |2z -1| = |z - 2|, then the point z describes :

Solution:

Squaring both sides

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

⇒ 3x^{2} + 3y^{2} = 3

⇒ x^{2} + y^{2} = 1

It is the equation of a circle.

∴ The point z describes a circle.

QUESTION: 10

The sum of an infinite GP is x and the common ratio r is such that |r| < 1. If the first term of the GP is 2, then which one of the following is correct ?

Solution:

GP = x

x(where, a = 1st term and r = common ratio)

from equation (i) x > 1 Hence, 1 <x<

QUESTION: 11

A box contains 3 white and 2 black balls. Two balls aredrawn at random one after the other. If the balls are not replaced, what is the probability that both the balls are black?

Solution:

Total number of balls

= 5 Number of black balls

= 2 Required probability

QUESTION: 12

For two variables x and y, the two regression coefficients are b_{yx} = -3/2 and b_{xy} = - 1/6. The correlation coefficient between x and y is :

Solution:

b_{xy} and b_{yx} both have negative sign. Therefore we have to take negative sign

b_{xy} and b_{yx} both have negative sign. Therefore we have to take negative sign

Hence, correlation coefficient (r)

QUESTION: 13

The variance of numbers x_{1},x_{2},x_{3},.......x_{n}, is V Consider the following statements :

1. If every x_{i} is increased by 2, the variance of the new set of the new set of numbers is V.

2. If the numbers x_{i} is squared, the variance of the new set is V^{2}.

Which of the following statements is/are correct ?

Solution:

I : Variance is not dependent on change of origin. Therefore, if every x_{i} is increased by 2, the variance of the new set of numbers is not changed.

II: Variance is dependent on change of scale.

If the number x_{i} is squared the variance of the new set is V^{2}

V_{xi} x V_{xi} =V = V .V = v^{2}

QUESTION: 14

What is the mean of the squares of the first 20 natural numbers ?

Solution:

Mean of the squares of the first 20 natural number

QUESTION: 15

p, q, r, s, t, are five numbers such that the average ofp,q and r is 5 and that of s and t is 10. What is the average of all the five numbers ?

Solution:

According to question p + q + r = 5><3=15...(i)

s+t = 10 x 2 = 20 ...(ii)

From equations (i) and (ii), p + q + r + s+1 = 15 + 20 = 35

Average p, q, r, s and t =

QUESTION: 16

The cumulative frequency of the largest observed value must always be:

Solution:

The cumulative frequency of the largest observed value must always be less then the total number of observations.

QUESTION: 17

It has been found that if A and B play a game 12 times, A wins 6 times, B wins 4 times and they draw twice. A and B take part in a series of 3 games. The probability that they win alternately, is :

Solution:

QUESTION: 18

Out of 7 consonants and 4 vowels, words are to be formed by involving 3 consonants and 2 vowels. The number of such words formed is :

Solution:

Number of words =

QUESTION: 19

Let X denote the number of scores which exceed 4 in 18, tosses of a symmetrical die. Consider the following statements:

1. The arithmetic mean o f X is 6.

2. The standard deviation ofX is 2.

Which of the above statements is/are correct ?

Solution:

Statement 1 :

n (X) = 2

arithmetric mean of X = np

Statement 2: Standard deviation of

Hence, statements 1 and 2 both are correct.

QUESTION: 20

How many different words can be formed by taking four letters out of the letters of the word 'AGAIN' if each word has to start with A ?

Solution:

As 'A’ must be first letter of each word. Total number ofwords = 4! = 24

QUESTION: 21

The sum of the series formed by the sequence 3, √3, 1. upto infinity i s :

Solution:

This is a Geometric Progression with a = 3 ,

QUESTION: 22

then the locus o f z is:

Solution:

QUESTION: 23

The number 251 in decimal system is expressed in binary system by :

Solution:

Therefore, (251)_{10}=(11111011)_{2}

QUESTION: 24

What is the argum ent o f the complex number

Solution:

QUESTION: 25

Consider the following statements in respect of the matrix

1. The matrix A is skew-symmetric.

2. The matrix A is symmetric.

3. The matrix A is invertible.

Which of the above statements is/are correct ?

Solution:

QUESTION: 26

Consider two matrices

Which one of the following is correct ?

Solution:

QUESTION: 27

One of the roots of

Solution:

QUESTION: 28

If A is any matrix, then the product AA is defined only when A is a matrix o f order m x n where :

Solution:

A A is defined only when A is a matrix of order m x n where m = n.

A x A = (m x n) (m x n) = (m x n) (n x n) if m = n = mxn = nxnorm xm.

= A is a square matrix.

QUESTION: 29

The determinant of an odd order skew symmetric matrix is always:

Solution:

We know that, elements of principal diagonals of a skew-symmetric matrix are all zero.

QUESTION: 30

If any two adjacent rows or columns of a determinant are intercharged in position, the value of the determinant:

Solution:

If any two adjacent rows or columns of a determinant are interchanged in position, the value of the determinant changes its sign.

QUESTION: 31

In a survey of 25 students, it was found that 15 had taken Mathematics, 12 had taken Physics and 11 had taken Chemistry, 5 had taken Mathematics and Chemistry, 9 had taken Mathematics and Physics, 4 had taken Physics and Chemistry and 3 had taken all the three subjects.

The number of students who had taken only physics is :

Solution:

Only Physics = 12-(1+3 +6) = 2

QUESTION: 32

In a survey of 25 students, it was found that 15 had taken Mathematics, 12 had taken Physics and 11 had taken Chemistry, 5 had taken Mathematics and Chemistry, 9 had taken Mathematics and Physics, 4 had taken Physics and Chemistry and 3 had taken all the three subjects.

The number of students who had taken only two subjects is:

Solution:

Only two subjects = 6 + 2 + 1 = 9

QUESTION: 33

Consider the following statements:

1. The number of students who had taken only one subject is equal to the number of students who had taken only two subjects.

2. The number of students who had taken at least two subjects is four times the number of students who had taken all the three subjects.

Which of the above statements is/are correct ?

Solution:

**Statement 1:**

Students, who had taken only one subject = 2 + 5 + 4= 11

Students, who had taken only two subjects

=6+2+1=9

1 9

**Statement 2:**

Students who had taken atleast two subject = 1+2 + 6 + 3=12

Students who had taken all three subjects = 4x3 = 12

QUESTION: 34

In the expansion of where n is a positive integer, the sum of the coefficients o f x^{5} and x^{10} is 0.

What is n equal to ?

Solution:

For the coefficient x^{5}

QUESTION: 35

In the expansion of where n is a positive integer, the sum of the coefficients o f x^{5} and x^{10} is 0.

What is the value of the independent term ?

Solution:

For the coefficient x^{5}

QUESTION: 36

In the expansion of where n is a positive integer, the sum of the coefficients o f x^{5} and x^{10} is 0.

What is the sum of the coefficients of the two middle terms ?

Solution:

For the coefficient x^{5}

QUESTION: 37

Given that C( n,r ) : C(n,r+1) = 1:2 and C ( n ,r+1): C (n,r+2 ) = 2 :3.

What is n equal to ?

Solution:

Solving equations (i) and (ii),we get

n= 14, r=4

QUESTION: 38

Given that C( n,r ) : C(n,r+1) = 1:2 and C ( n ,r+1): C (n,r+2 ) = 2 :3.

What is r equal to ?

Solution:

Solution:

Solving equations (i) and (ii),we get

n= 14, r=4

QUESTION: 39

Given that C( n,r ) : C(n,r+1) = 1:2 and C ( n ,r+1): C (n,r+2 ) = 2 :3.

What is P(n, r) : C(n, r) equal to ?

Solution:

Solving equations (i) and (ii),we get

n= 14, r=4

P (n,r) : C (n,r) =

QUESTION: 40

The complete solution o f 3 tan^{2 }x=1 is given by:

where **n** ∈ Z

Solution:

QUESTION: 41

What is the value of cos 36° ?

Solution:

QUESTION: 42

Consider the following statements:

1. Value of sin 0 oscillates between -1 and 1.

2. Value of cos 0 oscillates between 0 and 1.

Which of the above statements is/are correct ?

Solution:

QUESTION: 43

If x and y are positive and xy > 1, then what is tan^{-x} + tan^{-1} equal to ?

Solution:

QUESTION: 44

Consider the following statements:

for all positive integers n≥ 2.

2. If x is any positive real number, then nx > 1 for all positive integers n ≥ 2.

Which of the above statements is/are correct ?

Solution:

**Statement 1:
**

**Statement 2
**

QUESTION: 45

Consider the following statements:

1. If 30 is an acute angle such that sin 30 = cos 20, then the m esurement o f 0 in radian equals to

2. One radian is the angle subtended at the centre of a circle by an arc of the same circle whose length is equal to the diameter of that circle.

Which of the above statements is/are correct ?

Solution:

**Statement: 1**

Statement: 2

One radian is the angle subtended at the centre of a circle by an arc of the same circle whose length is equal to radius of that circle.

Hence, statement 1 is correct.

QUESTION: 46

From an aeroplane above a straight road the angle of depression of two positions at a distance 20 m apart on the road are observed to be 30° and 45°. The height of the aeroplane above the ground is :

Solution:

Hence the height is 10(√3+l)m

QUESTION: 47

Consider the following statements:

1. There exists no triangle ABC for which sin A + sin B = sinC.

2. If the angle of a triangle are in the ratio 1 :2 : 3, then its sides will be in the ratio 1 : √3 :2.

Which of the above statements is/are correct ?

Solution:

Given, sin A + sin B = sin C

Here, the sum of two sides of AABC is equal to the third side, but it is not possible (Because by triangle inequality, the sum of the length of two sides of a triangle is always greater than the length of the third side)

Ratio of angles of a triangle

A : B : C = 1 :2 : 3

A + B + C = 180°

∴ A = 30°

B = 60°

C = 90°

the ratio in sides according to sine rule

a : b : c = sin A : sin B : sin C

= sin 30° : sin 60° : sin 90°

QUESTION: 48

Consider the following statements:

1. sin|x| + cos|x| is always positive.

2. sin(x2) + cos(x2) is always positive.Which of the above statements is/are correct ?

Solution:

**Statement 1 :** f_{1} (x) = sin |x| + cos |x|, the value o f |sin x| and (cos x) depends on its angles, sin |x| + cos |x| is not always positive.

**Statement 2** : f_{2} (x) = sin (x^{2}) + cos (x^{2}), the value o f x2 between any value which lies in the interval

then value of f_{2}(x) = sin (x^{2}) + cos (x^{2}) is always negative.

QUESTION: 49

What is equal to ?

Solution:

QUESTION: 50

What is equal to ?

Solution:

QUESTION: 51

Consider the following statements:

1. **tan ^{-1 }**+ tan

2. sin

Which of the above statements is/are correct ?

Solution:

QUESTION: 52

If A + B + C = π, then what is cos (A + B) + cos C equal to ?

Solution:

A + B + C=π

A + B = π - C

cos (A + B) = cos (π - C)

cos (A + B) = - cos C

or cos (A+ B) + cos C = 0

QUESTION: 53

What is cos 20° + cos 100° + cos 140° equal to ?

Solution:

cos 20° + cos 100° + cos 140°

= (cos 140° + cos 20°) + cos 100°

QUESTION: 54

What is sin^{-1} sin equal to ?

Solution:

QUESTION: 55

What is sin^{2} (3π) + cos^{2} (4π) + tan^{2} (5π) equal to ?

Solution:

sin^{2} (3π) + cos^{2} (4π) + tan^{2} (5π)

= sin^{2} (3π) + cos^{2} (π + 3π) + tan^{2}(5π)

= (sin^{2} (3π) + cos^{2} (3π)) + tan^{2} (2 x 2π + 7π)

= 1 + tan^{2} π = sec^{2}π = 1

QUESTION: 56

Which ofthe above lie on the line 3x+y= 5 and at a distance √10 from (1,2)?

Solution:

All three points (0 ,5 ), (2, - 1 ) and ( 3 , - 4 ) lie on 3x+y = 5

QUESTION: 57

What is the equation of the line through (1, 2) so that the segment ofthe line intercepted between the axes is bisected at this point ?

Solution:

Equation of line passing through (2,0) and (0,4)

QUESTION: 58

What is the equation of straight line passing through the point (4, 3) and making equal intercepts on the coordinate axes?

Solution:

Let equation of 1 ine be

line passing through (4,3), then a = 0 Required equation, x + y = 7

QUESTION: 59

What is the equation of the line mid-way between the lines 3x-4y + 12 = 0 and 3x-4y =6?

Solution:

Equation of line mid-way between these two lines

QUESTION: 60

What is the sum of the major and minor axes of the ellipse whose eccentricity is 4/5 and length of latus rectum is 14.4 unit?

Solution:

Let 2a and 2b be the length of major and minor axis respectively.

QUESTION: 61

A straight line passes through (1, - 2 , 3 ) and perpendicular to the plane 2x + 3y - z = 7.

What are the direction ratios of normal to plane ?

Solution:

Direction ratios o f normal to plane 2x + 3 y - z = 7 is < 2 ,3 ,- 1 >

QUESTION: 62

A straight line passes through (1, - 2 , 3 ) and perpendicular to the p lan e2x + 3y - z = 7.

Where does the line meet the plane ?

Solution:

Equation o f line,

Let P (2r + 1, 3r - 2, - r + 3) of the line meets the plane.

Then, 2(2r + l ) + 3 ( 3 r - 2 ) - ( - r + 3) = 0

4r + 2 + 9 r - 6 + r - 3 = 7

14r= 14

r= 1

P (3,1,2) meets the plane.

QUESTION: 63

A straight line passes through (1, - 2 , 3 ) and perpendicular to the p lan e2x + 3y - z = 7.

What is the image o f the point ( 1 , - 2 , 3 ) in the plane ?

Solution:

Let Q(x, y, z) is th e image o f (1, - 2 , 3 ) in the plane

QUESTION: 64

Consider the spheres x^{2} + y^{2} + z^{2} - 4y + 3 = 0 and x^{2} + y^{2} + z^{2} + 2x + 4 z - 4 = 0.

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

Solution:

x^{2} + y^{2} + z^{2}- 4y + 3 = 0

x^{2} + y^{2} - 4y+4 — 4 + z^{2}+ 3 = 0

x^{2} + ( y - 2 )^{2} + z^{2} = 1 ...

(i) Sphere with centre (0,2,0) and radius 1 unit.

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

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

(x + l)^{2} + y^{2} + (z + 2)^{2} = 32 ...(ii)

Sphere with centre (-1,0, -2) and radius 3 units.

QUESTION: 65

Consider the following statements:

1. The two spheres intersect each other.

2. The radius of first sphere is less than that of second sphere.

Which of the above statements is/are correct ?

Solution:

x^{2} + y^{2} + z^{2}- 4y + 3 = 0

x^{2} + y^{2} - 4y+4 — 4 + z^{2}+ 3 = 0

x^{2} + ( y - 2 )^{2} + z^{2} = 1 ...

(i) Sphere with centre (0,2,0) and radius 1 unit.

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

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

(x + l)^{2} + y^{2} + (z + 2)^{2} = 32 ...(ii)

Sphere with centre (-1,0, -2) and radius 3 units.

r_{1}+r_{2} = 3 + l= 4

C_{1}C_{2}<r_{1} + r2 ,

∴ Two spheres intersect each other.

QUESTION: 66

The vertices of a triangle ABC are A (2, 3, 1), B (-2, 2, 0), and C (0,1-1).

What is the cosine of angle ABC ?

Solution:

QUESTION: 67

What is the area of the triangle ?

Solution:

Area of triangle ABC =

QUESTION: 68

What is the magnitude of the line joining mid points of the sides AC and BC ?

Solution:

Mid-point of A and C,

Mid-point o f B and C

QUESTION: 69

Consider the vectors

What is the scalar projection of a on b ?

Solution:

QUESTION: 70

Consider the vectors

What is the vector perpendicular to both the vectors ?

Solution:

QUESTION: 71

What angle does make with z-axis ?

Solution:

cos^{2} α + cos^{2} β + cos^{2} y = 1

cos^{2} 60° + cos^{2} 30° + cos^{2} y = 1

cos^{2} y = 0 ⇒ y = 90°

QUESTION: 72

What are the direction cosines of

Solution:

QUESTION: 73

What is equal to ?

Solution:

QUESTION: 74

What is the angle between

Solution:

QUESTION: 75

A line passes through the points (6, -7, -1) and (2, -3, 1). What are the direction ratios of the line ?

Solution:

Direction ratios < (2 - 6), (-3 + 7), (1 + 1) > = < -4 ,4 ,2 >

QUESTION: 76

What is equal to ?

Solution:

QUESTION: 77

What is equal to ?

Solution:

Therefore limit does not exist.

QUESTION: 78

What is the derivative of

Solution:

QUESTION: 79

What is equal to ?

Solution:

QUESTION: 80

What is the slope o f the tangent to the curve y = sin^{1 }(sin^{2}x) at x = 0?

Solution:

QUESTION: 81

The solution of is :

Where c is an arbitary constant

Solution:

From (i) and (ii)

QUESTION: 82

What is the solution of satisfying y(0) = 0 ?

Solution:

QUESTION: 83

Consider the curve y = e^{2x}.

What is the slope of the tangent to the curve at (0, 1) ?

Solution:

QUESTION: 84

Consider the curve y = e^{2x}.

Where does the tangent to the curve at (0, 1) meet the x-axis ?

Solution:

Equation of line passing through (0, 1) and slope = 2

y - 1 = 2(x - 0)

y - 2 x + 1

Tangent to the curve at (0, 1) meets the x-axis at

QUESTION: 85

Consider an ellipse

What is the area of the greatest rectangle that can be inscribed in the ellipse ?

Solution:

Given equation o f ellipse,

Let A (a cos θ b sin θ) be any point on ellipse

(1st quadrant)

Coordinate of B =[(a cos (π - θ), b sin (π - θ)]

= (- a cos θ, b sin θ) (2nd quadrant)

Coordinate of C = [a cos (π + θ), b sin (π + θ)] (3rd quadrant)

Coordinate of D = [a cos (2π - θ), b sin (2π - θ)] = (a cos θ, - b sin θ) (4th quadrant)

Area of the rectangle ABCD

= (a cos θ + a cos θ) (b sin θ + b sin θ)

= 2a cos θ x 2b sin θ = 2ab sin 2θ

= 2ab x 1 =2ab

QUESTION: 86

Consider an ellipse

What is the area included between the ellipse and the greatest rectangle inscribed in the ellipse ?

Solution:

Area of shaded region= Area of ellipse - Area of rectangle

= π ab- 2ab = ab (π - 2)

QUESTION: 87

Consider the integrals

What is I_{1 }- I_{2} equal to ?

Solution:

QUESTION: 88

Consider the integrals

What is I_{1} equal to ?

Solution:

Adding I_{1 }and I_{2
}

QUESTION: 89

Consider the function

Where

What is equal to ?

Solution:

QUESTION: 90

Consider the function

Where

What is th e value of λ if the function is continuous at

Solution:

Function is continuous at x =

QUESTION: 91

If f( 9 ) = 9 and f'(9 )= 4 then what is equal to ?

Solution:

(ByL^{1} Hospital rule)

QUESTION: 92

What is x sin x dx equal to ?

Solution:

{x sin x is an even function}

QUESTION: 93

What is the general solution of the differential equation x d y - y dx = y^{2} ?

Where c is an arbitrary constant

Solution:

Differential equation x dy - y dx = y^{2}

= (y dx - x dy) = y^{2
}

QUESTION: 94

Consider the following statements:Which of the above statements is/are correct ?

1. The function is continuous at all x except at jc = 0.

2. The function is continuous at x =2.99 where [.]is the bracket function.

Solution:

QUESTION: 95

Consider the following statements:

1. The function is not differentiable at x = 1.

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

Which of the above statements is/are correct ?

Solution:

**Statement 1 :** f(x) = |x|

From the graph, the curve has sharp turn at x = 0. Therefore, the function f (x) = |x| is not differentiable only x = 0, it is differentiable at x = 1

**Statement 2:** f(x) = e^{x}

^{}

QUESTION: 96

then what is equal to ?

Solution:

QUESTION: 97

Consider the function

What is the maximum value of the function ?

Solution:

QUESTION: 98

Consider the function

What is the minimum value of the function ?

Solution:

QUESTION: 99

Let f(X) be a function defined in 1 ≤ x < by

Consider the following statements:

1. The function is continuous at every point in the interval ( 1 , ).

2. The function is differentiable at x = 1.5 .

Which of the above statements is/are correct ?

Solution:

Statement 1 : Given f (x) =

function defined in

the fiinction is polynomial, so it is continous and differentiable in its domain

**Statement 2:
**

QUESTION: 100

Let f(X) be a function defined in 1 ≤ x < by

What is the differentiable coefficient of f(X) at x = 3 ?

Solution:

QUESTION: 101

Consider the following statements:

1. f(2 + 0) does not exist.

2. f(2 - 0) does not exist.

Which of the above statements is/are correct ?

Solution:

QUESTION: 102

What is equal to ?

Solution:

QUESTION: 103

The general solution of the differential equation (x^{2} + x + 1) dy+(y^{2} + y+ l)dx = 0 is (x+y+ 1)=A(1 + Bx + Cy+Dxy) where B, C and D are constants and A is parameter.

What is B equal to ?

Solution:

(x^{2} + x + l ) dy + (y^{2} + y + l ) dx = 0

(x^{2} + x + l)dy = -(y^{2} + y + l)dx

B = - l

QUESTION: 104

The general solution of the differential equation (x^{2} + x + 1) dy+(y^{2} + y+ l)dx = 0 is (x+y+ 1)=A(1 + Bx + Cy+Dxy) where B, C and D are constants and A is parameter.

What is C equal to ?

Solution:

(x^{2} + x + l ) dy + (y^{2} + y + l ) dx = 0

(x^{2} + x + l)dy = -(y^{2} + y + l)dx

C = - l

QUESTION: 105

The general solution of the differential equation (x^{2} + x + 1) dy+(y^{2} + y+ l)dx = 0 is (x+y+ 1)=A(1 + Bx + Cy+Dxy) where B, C and D are constants and A is parameter.

What is D equal to ?

Solution:

(x^{2} + x + l ) dy + (y^{2} + y + l ) dx = 0

(x^{2} + x + l)dy = -(y^{2} + y + l)dx

= - 2

QUESTION: 106

Consider the following statements:

1. The function f(x) = sin x decreases on the interval (0, π/2).

2. The function f(x) = cos x increases on the interval (0, π/2).

Which of the above statements is/are correct ?

Solution:

sin x increases on the interval

QUESTION: 107

What is the number of arbitrary constants in the particular solution of differential equation of third order ?

Solution:

Particular solution do not have any constant

QUESTION: 108

What is the equation of a curve passing through (0, 1) and whose differential equation is given by dy = y tan x dx ?

Solution:

QUESTION: 109

Consider the following statements in respect of the differential equation

1. The degree of the differential equation is not defined.

2. The order of the differential equation is 2.

Which of the above statements is/are correct ?

Solution:

**Statement 1:** Differential equation is not a polynomial equation in its derivatives. So, its degree is not defined.

**Statement 2 :** The highest order derivative in the given polynomial is 2.

QUESTION: 110

What is the equation of parabola whose verted is at (0, 0) an d focus is at (0, - 2 ) ?

Solution:

Focus is (0 , - 2)

a = - 2 and parabola is along y-axis downward

x^{2} =4ay

x^{2} = - 8y

or x^{2} + 8y = 0

QUESTION: 111

Number of X is randomly selected from the set of odd numbers and Y is randomly selected from the set of even numbers of the set {1,2,3,4,5,6,7}.LetZ = (X+Y)..

What is P(Z= 5) equal to ?

Solution:

Set A = {1 ,2 ,3 ,4 ,5 ,6 ,7 } and z =x + y

x = set of odd numbers

y = set of even numbers

QUESTION: 112

Number of X is randomly selected from the set of odd numbers and Y is randomly selected from the set of even numbers of the set {1,2,3,4,5,6,7}.LetZ = (X+Y)..

What is P(Z= 10) equal to ?

Solution:

Set A = {1 ,2 ,3 ,4 ,5 ,6 ,7 } and z =x + y

x = set of odd numbers

y = set of even numbers

QUESTION: 113

Number of X is randomly selected from the set of odd numbers and Y is randomly selected from the set of even numbers of the set {1,2,3,4,5,6,7}.LetZ = (X+Y)..

What is P(Z= 11) equal to ?

Solution:

Set A = {1 ,2 ,3 ,4 ,5 ,6 ,7 } and z =x + y

x = set of odd numbers

y = set of even numbers

Z > 11 is only possible when x = 7 and y = 6

QUESTION: 114

What is P(Z is the product of two prime numbers) equal to ?

Solution:

Set A = {1 ,2 ,3 ,4 ,5 ,6 ,7 } and z =x + y

x = set of odd numbers

y = set of even numbers

Z = product of two prime numbers

Z = x + y = 7 + 6 = 13

n(E_{4})=3

QUESTION: 115

Number of telephone calls received in 245 succesive one minute intervals at an exchange is given below in the following frequency distribution.

What is the mean of the distribution ?

Solution:

QUESTION: 116

Number of telephone calls received in 245 succesive one minute intervals at an exchange is given below in the following frequency distribution.

What is the median of the distribution ?

Solution:

Required mean = 4

QUESTION: 117

Number of telephone calls received in 245 succesive one minute intervals at an exchange is given below in the following frequency distribution.

What is the mode of the distribution ?

Solution:

The higher frequency is 51

∴ mode = value of the variable corresponding to the higher frequency 154 = 4

QUESTION: 118

The mean and standard deviation of 100 items are 50,5 and that of 150 items are 40,6 respectively.

What is the combined mean of all 250 items ?

Solution:

Mean of 100 items

Mean of 150 items

Standard deviation of 100 items

Standard deviation of 150 items =

QUESTION: 119

The mean and standard deviation of 100 items are 50,5 and that of 150 items are 40,6 respectively.

What is the combined standard deviation of all 250 items ?

Solution:

Mean of 100 items

Mean of 150 items

Standard deviation of 100 items

Standard deviation of 150 items =

QUESTION: 120

The mean and standard deviation of 100 items are 50,5 and that of 150 items are 40,6 respectively.

What is the variance of all 250 items ?

Solution:

Mean of 100 items

Mean of 150 items

Standard deviation of 100 items

Standard deviation of 150 items =

Variance o fa ll 250 items

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