NDA I - Mathematics Question Paper 2015


120 Questions MCQ Test NDA (National Defence Academy) Past Year Papers | NDA I - Mathematics Question Paper 2015


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This mock test of NDA I - Mathematics Question Paper 2015 for Defence helps you for every Defence entrance exam. This contains 120 Multiple Choice Questions for Defence NDA I - Mathematics Question Paper 2015 (mcq) to study with solutions a complete question bank. The solved questions answers in this NDA I - Mathematics Question Paper 2015 quiz give you a good mix of easy questions and tough questions. Defence students definitely take this NDA I - Mathematics Question Paper 2015 exercise for a better result in the exam. You can find other NDA I - Mathematics Question Paper 2015 extra questions, long questions & short questions for Defence on EduRev as well by searching above.
QUESTION: 1

Let θ be a positive angle. If the number of degrees in θ is divided by the number of radians in θ, then an irrational number 180/π results. If the number of degrees in θ is71multiplied by the number of radians in θ, then an irrational number 125π/9 results. The angle θ must be equal to

Solution:

QUESTION: 2

In a triangle ABC, a = (1 + a/3) cm, b = 2 cm and angle C = 60°.Then the other two angles are

Solution:

QUESTION: 3

Let α be the root of the equation 25cos2θ+ 5cosθ - 12 = 0 where 

Q. What is tan α equal to?

Solution:

QUESTION: 4

Let a be the root of the equation 25cos2θ + 5cosθ - 12 = 0,where 

Q. What is sin 2 α equal to?

Solution:

 

QUESTION: 5

The angle of elevation of the top of a tower from a point 20 m away from its base is 45°. What is the height of the tower?

Solution:

QUESTION: 6

The equation tan-1 (1 +x) + tan-1 (1-x) = π/2 is satisfied by

Solution:

QUESTION: 7

The angles of elevation of the top of a tower standing on a horizontal plane from two points on a line passing through the foot of the tower at distances 49 m and 36 m are 43° and 47° respectively. What is the height of the tower?

Solution:

QUESTION: 8

(1 - sinA + cos A)2 is equal to

Solution:

QUESTION: 9

What is  equal to?

Solution:

QUESTION: 10

Consider 

Q. What is x equal to?

Solution:

 

QUESTION: 11

Consider 

Q. What is x - y equal to?

Solution:

QUESTION: 12

Consider 

Q. What is x - y + z equal to?

Solution:

QUESTION: 13

Consider the triangle ABC with vertices A (-2,3), B (2,1) and C (0,2)

Q. What is the circumcentre of the triangle ABC?

Solution:

A circumcentre is a point at which perpendicular bisectors meet each other.
Here, ‘E’ represents circumcentre

QUESTION: 14

Consider the triangle ABC with vertices A (-2,3), B (2,1) and C (0,2).

Q. What is the centroid of the tirnalge ABC?

Solution:

QUESTION: 15

Consider the triangle ABC with vertices A (-2,3), B (2,1) and C (0,2).

Q. What is the foot of the altitude from the vertex A of the triangle ABC?

Solution:

QUESTION: 16

Let X be the set of all persons living in a city. Persons x,y in X are said to be related as x < y ify is at least 5 years older than x. Which one of the following is correct?

Solution:

QUESTION: 17

Which one of the following matrices is an elementary matrix?

Solution:

An elementary matrix has each diagonal element 1. So, option (b) is correct answer.

QUESTION: 18

Consider the following statements in respect of the given equation:
(x2 + 2)2 + 8x2 = 6x(x2 + 2)
1. All the roots of the equation are complex.
2. The sum of all the roots of the equation is 6.

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

Solution:

QUESTION: 19

In solving a problem that reduces to a quadratic equation, one student makes a mistake in the constant term and obtains 8 and 2 for roots. Another student makes a mistake only in the coefficient of first-degree term and finds -9 and-1 for roots. The correct equation is

Solution:

QUESTION: 20

If  then that is A + 3A-1 equal to?

Solution:

QUESTION: 21

 In a class of 60 students, 45 students like music, 50 students like dancing, 5 students like neither. Then the number of students in the class who like both music and dancing is

Solution:

QUESTION: 22

If log10 2, log10 (2x- 1) and log10(2x+ 3 ) are three consecutive terms of an A.P, then the value of x is

Solution:

QUESTION: 23

The matrix  is

Solution:

The given matrix is not symmetric option (a) is wrong.
From option (b): For Skew-symmetric matrix

Given matrix is not skew-symmetric option (b) is wrong.
From option (c): For Hermitian matrix

option (c) is wrong.
From option (d): For Skew-Hermitian matrix The diagonal element of a skew-hermitian matrix are pure imaginary or zero

Here, diagonal element indicates that the given matrix is skew-hermitian matrix, option (d) is correct.

QUESTION: 24

Let Z be the set of integers and aRb, where a, b ∈ Z if and only if (a - b) is divisible by 5.
Consider the following statements:
1. The relation R partitions Z into five equivalent classes.
2. Any two equivalent classes are either equal or disjoint.

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

Solution:
QUESTION: 25

If where  then argument 

Solution:

On comparing real and imaginary part on both sides, we get
r cos θ = - l ...(i)
r sin θ = - l ...(ii)
On dividing eq. (ii) by (i), we get

QUESTION: 26

If m and n are the roots of the equation (x+p)(x + q)=0, then the roots of the equation (x - m) (x - n) + k = 0 are

Solution:

QUESTION: 27

If 1, ω, ω2 are the cube roots of unity, then the value of(l + ω)(l + ω2)(l + ω4)(l + ω8) is 

Solution:

 

QUESTION: 28

What is the sum of the series 0.5 + 0.55 + 0.555 + ... to n terms?

Solution:

QUESTION: 29

Let A = { 1,2,3,4,5,6,7,8,9,10}. Then the number of subsets of A containing exactly two elements is

Solution:

A={ 1,2,3,4,5,6,7,8,9,10} Number of subsets of A containing two elements = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + l
= 45

Option (c) is correct
Alternate Method
The number of subsets of A containing exactly two elements is

QUESTION: 30

What is the square root of i, where i =√-1?

Solution:

QUESTION: 31

The point on the parabola y2 = 4ax nearest to the focus has its abscissa

Solution:

Here, 'S’ represents focus 0(0,0) is a point which is on parabola y2 = 4ax and nearest to focus (a, 0)

abscissa of O (0,0) is x = 0
Option (a) is correct.

QUESTION: 32

A line passes through (2,2) and is perpendicular to the line 3x + y = 3. Its y-intercept is

Solution:

A line passes through (2,2) and is perpendicular to the line3x+y=3
Slope of line 3x + y=3 is-3

Slope of line which passes through (2,2) is 1/3
Equation of line passes through (2,2) and having slope (1/3) is

QUESTION: 33

The hyperbola  Passes through the point  and the length of its latus rectum is 4/3 units. The length of the conjugate axis is

Solution:

 

QUESTION: 34

The Perpendicular distance between the straight lines 6x + 8y+ 15 = 0 and 3x + 4y+9 = 0 is

Solution:

6 x + 8 y + 1 5 = 0                          (1.)
and 3x+4y+9=0                              (2.)

Multiply equation (ii) by 2, we get
6x+8y+ 18=0
Distance between the straight lines

Option (b) is correct.

QUESTION: 35

The area of a triangle, whose vertices are (3,4), (5,2) and the point of intersection of the lines x = a and y= 5, is 3 square units. What is the value of a?

Solution:

Area of ΔABC = 3 sq. unit

QUESTION: 36

The length of perpendicular from the origin to a line is 5 units and the line makes an angle 120° with the positive direction of x-axis.The equation of the line is

Solution:

QUESTION: 37

The equation of the line joining the origin to the point of interesection of the lines 

Solution:

From solving equations (i) and (ii), we get the intersection point. 

Now, equation o f line jo in in g (0, 0) and

Here, slope of line = 1
y=x .
 x—y = 0
Option (a) is correct.

QUESTION: 38

The projections of a directed line segment on the coordinate axes are 12,4,3 respectively.​

Q. What is the length of the line segment?

Solution:

The projection of a directed line segment on the co-ordinate axes are 12,4,3, respectively

QUESTION: 39

The projections of a directed line segment on the coordinate axes are 12,4,3 respectively.

Q. What are the direction cosines of the line segment?

Solution:

Direction cosine of line segment= 

Option (a) is correct.

QUESTION: 40

From the point P(3,- 1,11 ) , a perpendicular is drawn on the line L given by the equation   Let Q be the foot of the perpendicular.

Q. What are the direction ratios of the line segment PQ2

Solution:

Equation of line passing through P(3, -1, 11) and perpendicular to  is:

The direction ratio are (-1,6, - 4)
Option (b) is correct.

QUESTION: 41

From the point P(3, -1 ,11 ) , a perpendicular is drawn on the line L given by the equation  Let Q be the foot of the perpendicular.

Q. What is the length of the line segment PQ?

Solution:

QUESTION: 42

A triangular plane ABC with centroid (1,2,3) cuts the coordinate axes at A, B, C respectively.

Q. What are the intercepts made by the plane ABC on the axes?

Solution:

x1 = 3 ,y1 = 6 and z1 = 9
Intercept made by plane on the axes are 3, 6 and 9, respectively.
Option (a) is correct.

QUESTION: 43

A triangular plane ABC with centroid (1,2,3) cuts the coordinate axes at A, B, C respectively.

Q. What is the equation of plane ABC?

Solution:

The plane passes through the point A (3,0,0), B(0, 6, 0) and C(0,0,9). So, it should satisfy the equation given in option for all the three points.
From option (a)
For point A (3,0,0)
x+2y+3z= 1
3 + 0 + 0 ≠ l .
option (a) is wrong.
From option (b)
For point A(3,0,0)
3x+2y+z = 3
3(3) + 0 + 0 ≠ 3
option (b) is wrong.
From option (c)
For point A(3,0,0)
2x+3y+6z= 18
2(3) +0 + 0 ≠ 18
option (c) is wrong.
From option (d)
For point A (3,0,0)
6x+3y+2z= 18
6(3) + 0 + 0 = 18 
For point B(0,6,0)
6x+3y+2z= 18
0 + 3(6)+0=18
For point C(0,0,9)
6x+3y+2z= 18
0 + 0 + 2x9=18
Option (d) is correct.

QUESTION: 44

A point P (1,2,3) is one vertex of a cuboid formed by the coordinate planes and the planes passing through P and parallel to the coordinate planes.

Q. What is the length of one of the diagonals of the cuboid?

Solution:

Length of one of the diagonal of cube

Option (b) is correct.

QUESTION: 45

A point P (1,2,3) is one vertex of a cuboid formed by the coordinate planes and the planes passing through P and parallel to the coordinate planes.

Q. What is the equation of the plane passing through P(1,2,3) and parallel to xy-plane?

Solution:

Equation of plane passing through (1,2,3) and parallel to xy-plane is z = 3.
Option (c) is correct.

QUESTION: 46

The decimal number (127. 25)10, when converted to binary nunber, takes the form

Solution:

QUESTION: 47

Consider the following in respect of two non-singular matrices A and B of same order:
1. det (A + B) = det A + det B
2. (A + B)-1=A-1+B-1

Q. Which of the above is/are correct?

Solution:

Non-singular matrix is a matrix whose determinate Value is non-zero.

QUESTION: 48

If  satisfy the equation AX = B , then the matrix A is equal to

Solution:

QUESTION: 49

What is  equal to?

Solution:

QUESTION: 50

How many words can be formed using all the letters of the word ‘NATION’ so that all the three vowels should never come together? 

Solution:

The given word is ‘NATION’.
Total number of words that can be formed from given word ‘NATION’

Now numbers of words that can be formed from given word ‘NATION’, so that all vowels never come together.

QUESTION: 51

(x3 - 1) can be factorised as 

Solution:

As we know that cube root of unity is 1, ω and ω2
x3- 1 = ( x - 1) (x - ω ) (x - ω2)
Option (b) is correct.

QUESTION: 52

What is 

where i =√-1, equal to?

Solution:

QUESTION: 53

Let  If AB = C, then what is A2 equal to?

Solution:

QUESTION: 54

The value of 

Solution:

QUESTION: 55

If A = {x: x is a multiple of 3} and
B = {x: x is a multiple of 4} and
C= {x: x is a multiple of 12}, then which one of the following is a null set?

Solution:

A = {x: x is a multiple of 3}
A = { 3,6 ,9 ,1 2 ,1 5 ,1 8 ,2 4 ,....}

B = {x: x is a multiple of 4}
B= {4,8,12,16,20,24,28,32,.... }
C= {x: x is a multiple of 12}
C = {12,24,36,48,60,72,84,96,.....}

QUESTION: 56

If (11101011)2 is converted decimal system, then the resulting number is

Solution:

QUESTION: 57

What is the real part of (sin x + i cos x)3 where i = √-1 ?

Solution:

QUESTION: 58

If  (θ) = , then E(α)E(β) is equal to

Solution:

QUESTION: 59

Let A = {x, y, z}and B= {p,q, r, s}. What is the number of distinct relations from B to A?

Solution:

QUESTION: 60

If 2p + 3q = 18 and 4p2 + 4pq - 3q2 - 36 = 0, then what is (2p + q) equal to?

Solution:

QUESTION: 61

Given that 

Q. What is the value of A?

Solution:

QUESTION: 62

Given that 

Q. What is the value of B?

Solution:

QUESTION: 63

Given that 

Q. What is the value of A?

Solution:

QUESTION: 64

Given that 

Q. What is the value of B?

Solution:

QUESTION: 65

What is the solution of the differential equation 

Solution:

QUESTION: 66

What is the solution of the differential equation 

Solution:

QUESTION: 67

What is the solution of the differential equation 

Solution:

This is a linear differential equation of the form 

QUESTION: 68

What is  equal to?

Solution:

QUESTION: 69

The adjacent sides AB and AC of a triangle ABC are represented by the vectors respectively. The area of the triangle ABC is

Solution:

QUESTION: 70

A force  is applied at the point P, whose position vector is What is the magnitude of the moment of the force about the origin?

Solution:

QUESTION: 71

Given that the vectors  re non-collinear. The values of x and y for which  holds true if 

Solution:

QUESTION: 72

If  is equal to

Solution:

QUESTION: 73

Let α , β, γ be distinct real numbers. The points with position vectors 

Solution:

 

QUESTION: 74

If  then which of the following is/are correct?

Q. Select the correct answer using the code given below 

Solution:

According to statements (1) and (2),

QUESTION: 75

If   then which one o f the following is correct?

Solution:

QUESTION: 76

If  equal to?

Solution:

QUESTION: 77

Consider the following statements:
 is an increasing function on [ 0 ,∞).
 is an increasing function on (-∞ ∞).

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

Solution:

From statement -1

The given function is increasing on interval [0,∞]
From Statement -2

Hence, the given function is increacing from [-∞, ∞]
Both statement are correct
Option (c) is correct.

QUESTION: 78

For each non-zero real number x, let (x) =   The range of is

Solution:

For a non-zero real number x

QUESTION: 79

Consider the following statements:

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

Solution:

From statement-1
From the definition of greatest integer function
f (x) = [x] is discontinuous at x = n for any value o f n ∈Z 
Statement 1 is correct
From statement-2

QUESTION: 80

What is the derivative of  with respect to tan-1x?

Solution:

QUESTION: 81

If then what is  equal to?

Solution:

QUESTION: 82

Given a function 

where a, b are constants. The function is continuous everywhere. 

Q. What is the value of a?

Solution:

QUESTION: 83

Given a function 

where a, b are constants. The function is continuous everywhere. 

Q. What is the value of b?

Solution:

QUESTION: 84

Q. Which of the above functions have inverse defined on their ranges?

Solution:
QUESTION: 85

The integral 

Q. What is r equal to?

Solution:

QUESTION: 86

The integral 

Q. What is α equal to?

Solution:

QUESTION: 87

Consider the function 

Q. At what value of x does f(x) attain minimum value?

Solution:

In order to find the value of ‘ x ’ , where f(x) is maximum or minimum; equation f '( x ) equal to zero.

Now equating f '( x ) to zero, we get
4x=0
x=0
Hence, f (x) attain minimum value at x = 0
Option (b) is correct.

QUESTION: 88

Consider the function ,Where 

Q. What is the minimum value of f(x)

Solution:

f(x) is minimum at x = 0

minimum value of f(x) is -1
Option (c) is correct.

QUESTION: 89

Consider the function 

Which is continuous at  where α is a constant.

Q. What is the value of α ?

Solution:

QUESTION: 90

Consider the function Which is continuous at  where α is a constant.

What is  equal to?

Solution:

QUESTION: 91

The mean and the variance 10 observations are given to be 4 and 2 respectively. If every observation is multiplied by 2, the mean and the variance of the new series will be respectively

Solution:

QUESTION: 92

Which one of the following measures of central tendency is used in construction of index numbers?

Solution:

Geometric mean is used in construction of index numbers.
option (b) is correct.

QUESTION: 93

The correlation coefficient between two variables Xand Yis found to be 0.6. All the observations on X and Y are transformed using the transformations U= 2 - 3X and V= 4 Y + 1. The correlation coefficient between the transformed variables U and Kwill be

Solution:
QUESTION: 94

Which of the following statements is/are correct in respect of regression coefficients?
1. It measures the degree of linear relationship between two variables.
2. It gives the value by which one variable changes for a unit change in the other variable.

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

Solution:

Regression coefficients measures the degree of linear relationship between two variables and does not give the value by which one variable changes for a unit change in the other variable.
option (a) is correct.

QUESTION: 95

A set of annual numerical data, comparable over the years, is given for the last 12 years.
1. The data is best represented by a broken line graph, each corner (turning point) representing the data of one year.
2. Such a graph depicts the chronological change and also enables one to make a short-term forecast.

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

Solution:

The annual numerical data for comparable for last 12 years is represented by broken line graph, where each turning point represent the data of a particular year, while such graph do not depict the chronological change.
Option (a) is correct.

QUESTION: 96

Two men hit at a target with probabilities 1/2 and 1/3 respectively. What is the probability that exactly one of them hits the target?

Solution:

QUESTION: 97

Two similar boxes Bi (i = 1,2) contain (i + 1) red and (5 — i — 1) black balls. One box is chosen at random and two balls are drawn randomly. What is the probability that both the balls are of different colours?

Solution:
QUESTION: 98

In an examination, the probability of a candidate solving a question is 1/2 Out of given 5 questions in the examination, what is the probability that the candidate was able to solve at least 2 questions?

Solution:

QUESTION: 99

I f , th e n w hich one o f the follow ing is not correct?

Solution:
QUESTION: 100

The mean and the variance in a binomial distribution are found to be 2 and 1 respectively. The probability P(X= 0) is

Solution:

QUESTION: 101

The mean of five numbers is 30. If one number is excluded, their mean becomes 28. The excluded number is

Solution:

Mean of 5 numbers = 30
Total sum of 5 numbers = 30 x 5 = 150
After excluded one number
Mean of 4 numbers will be = 28
Total sum of 4 numbers = 4x28=112
Thus, excluded number
= (sum of 5 numbers - sum of 4 numbers)
= 150-112 = 38 
Option (d) is correct.

QUESTION: 102

If A and B are two events such that  P (A ∪ B) = 3/4 , P (A ∩ B) = 1/4 and  = 2/3, then what is P(B) equal to?

Solution:

Option (b) is correct.

QUESTION: 103

The ‘less than’ ogive curve and the ‘more than’ ogive curve intersect at

Solution:

The ‘less than’ ogive curve and the ‘more than’ ogive curve intersect at median.
Option (a) is correct. 

QUESTION: 104

In throwing of two dice, the number of exhaustive events that ‘5’ will never appear on any one of the dice is

Solution:

Required number of exhausistance events = ( 6 - 1) x ( 6 - 1)=5 x 5=25
Option (c) is correct.

QUESTION: 105

Two cards are drawn successively without replacement from a wellshuffled pack of 52 cards. The probability of drawing two aces is

Solution:

Option (b) is correct.

QUESTION: 106

Consider the line  and the circle x2 + y2 = 4

Q. What is the area of the region in the first quadrant enclosed by the x-axis, the line x = √3 and the circle?

Solution:

QUESTION: 107

Consider the line  and the circle x2 + y2 = 4.

Q. What is the area of the region in the first quadrant enclosed by the x-axis, the line x = √3y and the circle?

Solution:

QUESTION: 108

Consider the curves y = sin x and y = cos x.

Q. What is the area of the region bounded by the above two curves and the lines x = 0 and x = π/4?

Solution:

QUESTION: 109

Consider the curves y = sin x and y = cos x.

Q. What is the area of the region bounded by the above two curves and the lines x = π/4 and π/2?

Solution:
QUESTION: 110

Consider the function 

Q. What is the maximum value of the function?

Solution:

QUESTION: 111

Consider the following statements:
1. The function attains local minima at x = - 2 and x = 3.
2. The function increases in the interval (-2,0).

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

Solution:

From Statement 1:
Function attain local m inim a at x = - 2 and x = 3
As we have,

QUESTION: 112

Consider the parametric equation

Q. What does the equation represent?

Solution:

Hence, the given equation represent a circle of radius(a) .
Option (b) is correct.

QUESTION: 113

Consider the parametric equation 

Q. What is dy/dx equal to?

Solution:

QUESTION: 114

Consider the parametric equation 

Q. What is d2y/dxequal to? 

Solution:

QUESTION: 115

Consider the following statements: 

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

Solution:

Hence, the differential equation is of order -1 and degree-2.
both Statements 1 and 2 are correct.
Option (c) is correct.

QUESTION: 116

What is  equal to?

Solution:

QUESTION: 117

Consider the integral where m is a positive integer

Q. What is I1 equal to?

Solution:

QUESTION: 118

Consider the integral here m is a positive integer.

Q. What is I2 + I3 equal to?

Solution:

QUESTION: 119

Consider the integral  where m is a positive integer

Q. What is Im equal to?

Solution:

QUESTION: 120

Consider the integral  where m is a positive integer

Consider the following:

Q. Which of the above is /are correct?

Solution:

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