NDA I - Mathematics Question Paper 2019


120 Questions MCQ Test NDA (National Defence Academy) Past Year Papers | NDA I - Mathematics Question Paper 2019


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QUESTION: 1

What is the nth term of the sequence 25, -125, 625, -3125, ...?

Solution:

Substituting n = 1, 2, 3, 4, 5, on option (d), we obtain a5 = (−1)5−155+1 = 56 = 15625.

Therefore, the required terms are 25, – 125, 625, – 3125, and 15625.

QUESTION: 2

Suppose X = {1, 2, 3, 4} and R is a relation on X. If R = [(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}, then which one of the following is correct?

Solution:

Given X = {1, 2, 3, 4} and R = [(1, 1), (2, 2), (3, 3), (1, 2), (2,1), (2, 3), (3, 2)}
Reflexive Property states that for every real number x, x=x.
Here (1,1) ∈ R, (2,2) ∈ R, (3,3) ∈ R, but (4,4) ∉ R. So, R is not Reflexive.

Symmetric Property states that for all real numbers x and y, if x=y, then y=x.
(1, 2) ∈ R, (2, 1) ∈ R and, (2, 3) ∈ R, (3,2) ∈ R

So, R is symmetric.

Transitive Property states that for all real
numbers x, y and z, if x=y and y=z, then x=z.

Here (1, 2) ∈ R, (2, 1) ∈ R and (1, 1) ∈ R,

Similarly, (2, 3) ∈ R, (3, 2) ∈ R, (2, 2) ∈ R

But for (1, 2) ∈ R, (2, 3) ∈ R, (1, 3) ∉ R

So, R is not transitive.

Thus, R is neither reflexive nor transitive, but symmetric.

QUESTION: 3

A relation R is defined on the set N of natural numbers as xRy ⇒ x2 − 4xy + 3y2 = 0. Then which one of the following is correct?

Solution:


So, R is not transitive.
Hence, R is neither symmetric nor  transitive, but reflexive.

QUESTION: 4

If A= (x ∈ Z∶ x3 − 1 = 0) and B = (x ∈ Z ∶ x2 + x + 1 = 0), where Z is set of complex numbers, then what is A ∩ B equal to?

Solution:

QUESTION: 5

Consider the following statements for the two non-empty sets A and B:

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

Solution:

By drawing venn diagram

This statement is possible

The second statement is impossible.

QUESTION: 6

then what is adjoint of B equal to?

Solution:

QUESTION: 7

Let X be a non-empty set and let A, B, C be subsets of X. Consider the following statements:

Q. Which of the above statements are correct?

Solution:

QUESTION: 8

What are the roots of the equation | x2 - x - 6 | = x + 2?

Solution:

QUESTION: 9

 then the matrix A is a/an

Solution:

Hence matrix A is an Involutory matrix. 

QUESTION: 10

then what are the values of x and y respectively?

Solution:

By comparison both sides, we get

QUESTION: 11

The common roots of the equations z3 + 2z2 + 2z + 1 = 0 and z2017 + z2018 + 1 = 0 are

Solution:

Given


Now according to options there are four roots i.e. 1, –1, ω, ω2 But 1 and –1 are not satisfying the given equations. Hence ω, ω2 are the required roots.

QUESTION: 12

​If C(20, n + 2) = C(20, n − 2), then what is n equal to?

Solution:


QUESTION: 13

There are 10 points in a plane. No three of these points are in a straight line. What is the total number of straight lines which can be formed by joining the points?

Solution:

A straight line can be drawn by joining two points.

Hence the number of possible straight lines is

QUESTION: 14

​The equation px2 + qx + r = 0 (where p, q, r, all are positive) has distinct real roots a and b. Which one of the following is correct?

Solution:

Given

Here sum is negative & product is positive this is possible only when both the roots are negative. i.e. a < 0, b < 0.

QUESTION: 15

If A = {λ,{λ,μ}}, then the power set of A is

Solution:

Let A be a set, then the of all the possible subsets of is called the power set of A and is denoted by P(A).

QUESTION: 16

In a school, all the students play at least one of three indoor games – chess, carrom and table tennis. 60 play chess, 50 play table tennis, 48 play carrom, 12 play chess and carrom, 15 play carrom and table tennis, 20 play table tennis and chess.

Q. What can be the minimum number of students in the school?

Solution:

⇒ 50 + 28 + x + 12 − x + 21 + x
⇒ 111 + x (total) Minimum value when x=0 will be 111 + 0 = 111

QUESTION: 17

In a school, all the students play at least one of three indoor games – chess, carrom and table tennis. 60 play chess, 50 play table tennis, 48 play carrom, 12 play chess and carrom, 15 play carrom and table tennis, 20 play table tennis and chess.

Q. What can be the maximum number of students in the school?

Solution:

⇒ 50 + 28 + x + 12 − x + 21 + x
⇒ 111 + x (total)
The maximum value for x=12 111 + x = 0
111 + 12 = 0
The maximum value = 123

QUESTION: 18

If A is an identity matrix of order 3, then its inverse (A−1)

Solution:

A is an identity matrix of order 3, then its inverse (A−1)

Similarly, for A = A, A−1 = A

QUESTION: 19

A is a square matrix of order 3 such that its determinant is 4. What is the determinant of its transpose?

Solution:

A is a square matrix of order 

Determinant |A| = 4 Since, we know that  |A′ | = |A|

So, there is no value change when determinant to transpose, since both are equal.
Transpose |A′ | = 4

QUESTION: 20

From 6 programmers and 4 typists, an office wants to recruit 5 people. What is the number of ways this can be done so as to recruit at least one typist?

Solution:

From 6 programmers and 4 typists, an office wants to recruit 5 people.

The number of ways this can be done so as to recruit at least one typist= 252 – 6 = 246

QUESTION: 21

​What is the number of terms in the expansion of [(2x - 3y)2 (2x + 3y)2]2

Solution:

Given expression is 

Number of terms = 4 + 1 = 5

QUESTION: 22

In the expansion of (1 + ax)n , the first three terms are respectively 1, 12x and 64x2 . What is n equal to?

Solution:

Given:

Therefore, n = 9

QUESTION: 23

The numbers 1, 5 and 25 can be three terms (not necessarily consecutive) of

Solution:

The numbers 1,5 and 25 can be three terms (not necessarily consecutive) of one or many numbers of Arithmetic and Geometric Progressions.
To check the Arithmetic Progression: Let us assume that given number series 1, 5 and 25 are Ith, Jth and Qth terms in the AP series whose distance is equal to d.
Therefore,

Here x is a natural number.

So, the given three numbers can be terms in one or many numbers of

Arithmetic Progressions.

QUESTION: 24

The sum of (p+q)th and (p-q)th terms of an AP is equal to

Solution:

QUESTION: 25

If A is a square matrix of order n > 1, then which one of the following is correct?

Solution:

A is a square matrix of order n > 1. det (–A) = (−1)n det A is correct because, when 1 is substituted with I and if we multiplied the RHS then eventually equals to LHS

where −I is the matrix with −1 on the diagonal and 0 elsewhere. 

QUESTION: 26

What is the least value 25 cosec2 x + 36 sec2 x?

Solution:

QUESTION: 27

Let A and B be (3×3) matrices with det A = 4 and det B = 3

Q. What is det (2AB) equal to?

Solution:

Given det A = 4 and det B = 3 det(2AB)

= 23 det A det B 

= 8 × 4 × 3

= 96

QUESTION: 28

Let A and B be (3×3) matrices with det A = 4 and det B = 3

Q. What is det (3AB−1) equal to?

Solution:

QUESTION: 29

A complex number is given by 

Q. What is the modulus of z?

Solution:

Given complex number, 

QUESTION: 30

A complex number is given by 

What is the principal argument of z?

Solution:

We know that, Principal argument of z is

 

QUESTION: 31

What is the value of

Solution:

QUESTION: 32

tan 54° can be expressed as

Solution:

QUESTION: 33

Consider the following:

Q. What is the value of θ?

Solution:

QUESTION: 34

Consider the following

Q. What is the value of A?

Solution:

QUESTION: 35

Consider the following

Q. What is the value of B?

Solution:

QUESTION: 36

Consider the following
It is given that cos (θ - α) = a, cos (θ - β) = b 

Q. What is cos (α - β) equal to?

Solution:

QUESTION: 37

Consider the following

It is given that cos (θ - α) = a, cos (θ - β) = b 

Q. What is sin2 (α - β) + 2abcos (α - β) equal to?

Solution:

QUESTION: 38

If sin α + cos α​ = p, then what is cos 2 (2α)  equal to?

Solution:

Squaring on both sides, we get 

QUESTION: 39

What is the value of 

Solution:

=0

QUESTION: 40

 then what is x equal to?

Solution:

QUESTION: 41

 then what is the 
value of

Solution:

QUESTION: 42

then what is the value of 

Solution:

QUESTION: 43

What is the value of tan75° + cot75°?

Solution:

QUESTION: 44

What is the value of cos46°cos47°cos48°cos49°cos50° … cos135°?

Solution:

QUESTION: 45

 then what is sin θ equal to?

Solution:

QUESTION: 46

If the roots of the equation x2 + px + q = 0 are tan19° and tan26°, then which one of the following is correct?

Solution:

QUESTION: 47

What is the fourth term of an AP of n terms whose sum is n(n + 1)?

Solution:

QUESTION: 48

What is (1 + tanα tanβ)2 + (tanα − tanβ)2 − secαsec2β equal to?

Solution:

QUESTION: 49

If p = cosecθ − cotθ and q = (cosecθ + cotθ)−1, then which one of the following is correct?

Solution:

QUESTION: 50

If the angles of a triangle ABC are in the ratio 1 : 2 : 3, then the corresponding sides are in the ratio

Solution:

QUESTION: 51

Consider the following statements:

1. For an equation of a line, x cos θ + y sin θ = p, in normal form, the length of the perpendicular form the point (α, β) to the line is |α cos θ + β sin θ + p|.

2. The length of the perpendicular from the point (α, β) to the line 

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

Solution:

QUESTION: 52

A circle is drawn on the chord of a circle x2 + y2 = a2 as diameter. The chord lies on the line x + y = a. What is the equation of the circle?

Solution:

QUESTION: 53

The sum of the focal distances of a point on an ellipse is constant and equal to the

Solution:

The sum of the focal distance of any point on an ellipse is constant and equal to the length of the major axis of the ellipse.
Let P (x, y) be any point on the ellipse

Let MPM' be the perpendicular through P on directrices ZK and Z'K'. Now by definition we get,

Hence, the sum of the focal distance of a point P is constant and equal to the length of the major axis i.e., 2a of the ellipse.

QUESTION: 54

The equation 2x2 − 3y2 − 6 = 0 represents

Solution:

It is equation for hyperbola.

QUESTION: 55

The two parabolas y2 = 4ax and x2 = 4ay intersect

Solution:

The two parabolas y2 = 4ax and x2 = 4ay intersect at two points (0, 0) and (4a, 4a) on the line x = y.

QUESTION: 56

The points (1, 3) and (5, 1) are two opposite vertices of a rectangle. The other two vertices lie on the line y = 2x + c.

Q. What is the value of c?

Solution:

We know, the diagonals of a rectangle bisect each other. i.e. the mid point of (1, 3) and (5, 1) lie on the line y = 2x + c.

Since, (3, 2) lie on the line y=2x+c

C= -4

QUESTION: 57

If the lines 3y + 4x = 1, y = x + 5 and 5y + bx = 3 are concurrent, then what is the value of b? 

Solution:

If the lines 4x + 3y − 1 = 0, x − y + 5 = 0, bx + 5y − 3 = 0 are concurrent, then.

QUESTION: 58

What is the equation of the straight line which is perpendicular to y = x and passes through (3, 2)?

Solution:

QUESTION: 59

The straight lines x + y – 4 = 0, 3x + y – 4 = 0 and x + 3y – 4 = 0 form a triangle, which is

Solution:

QUESTION: 60

The circle x2 + y2 + 4x − 7y + 12 = 0, cuts an intercept on y-axis equal to

Solution:

QUESTION: 61

The centroid of the triangle with vertices A(2, –3, 3), B(5, –3, –4) and C(2, –3, –2) is the point

Solution:

QUESTION: 62

What is the radius of the sphere x2 + y2 + z2 − 6x + 8y − 10z + 1 = 0?

Solution:

We know that the general equation of the sphere is 

QUESTION: 63

The equation of the plane passing through the intersection of the planes 2x + y + 2z = 9, 4x –5y –4z = 1 and the point (3, 2, 1) is

Solution:

QUESTION: 64

The distance between the parallel planes 4x − 2y + 4z + 9 = 0 and 8x – 4y + 8z + 21 = 0 is 

Solution:

QUESTION: 65

What are the direction cosines of z-axis?

Solution:

Direction cosines of z-axis, assuming a three dimensional Cartesian coordinate system: (0, 0, 1). Meaning that the cosine of the angle between:

1. the z-axis and the x-axis is 0 (90 degrees);

2. the z-axis and the y-axis is 0 (90 degrees );

3. the z-axis and the z-axis is 1 ( 0 degrees).

QUESTION: 66

 then what is  equal to?

Solution:

= 5 − 15 – 96 = −106

QUESTION: 67

If the position vectors of points A and B are  respectively, then what is the length of 

Solution:

Given, position vector of point 

and position vector of point 

QUESTION: 68

If in a right-angled triangle ABC, hypotenuse AC = p, then what is  equal to? 

Solution:

QUESTION: 69

The sine of the angle between vectors 

Solution:

QUESTION: 70

What is the value of λ for which the vectors  are perpendicular?

Solution:

For perpendicular,

QUESTION: 71

What is the derivative of  with respect to x?

Solution:

QUESTION: 72

then what is   equal to? 

Solution:

QUESTION: 73

A function f defined by

Solution:

Hence it is an odd function.

QUESTION: 74

The domain of the function f defined by

Solution:

QUESTION: 75

  is equal to

Solution:

QUESTION: 76

For  is the ratio of perimeter to area of a circle of radius r. Then  is equal to

Solution:

QUESTION: 77

If  then is equal to

Solution:

QUESTION: 78

The number of real roots for the equation 

Solution:

By solving these equation we get x= 4, 5, –4, –5 but these values of x does not satisfy the given equation as |x| will always give positive value.
Hence the number of real roots of the given equation is zero.

QUESTION: 79

 is equal to

Solution:

QUESTION: 80

The domain of the function 

Solution:

For f(x) to have real values, the radicand (2 − x)(x − 3) must be positive. Hence (2 − x)(x − 3) ≥ 0
x = 2 and x = 3
Domain is [2, 3]

QUESTION: 81

The solution of the differential equation 

Solution:


Where c = e−a

QUESTION: 82

 is equal to

Solution:

QUESTION: 83

If y = a cos 2x + b sin 2x, then 

Solution:

QUESTION: 84

A given quantity of metal is to be cast into a half cylinder (i.e. with a rectangular base and semicircular ends). If the total surface area is to be minimum, then the ratio of the height of the half cylinder to the diameter of the semicircular ends is

Solution:

QUESTION: 85

 is equal to 

Solution:

QUESTION: 86

If  then what is  equal to?

Solution:

QUESTION: 87

What is ∫ ln(x2 )dx equal to?

Solution:

QUESTION: 88

The minimum distance from the point (4, 2) to y2 = 8x is equal to

Solution:

Let (x, y) be any point on y2 = 8x

Then the distance between (x, y) and (4, 2) is 

QUESTION: 89

The differential equation of the system of circles touching the y-axis at the origin is

Solution:

Equation of circle is

Differentiate with respect to x

Put this value in equation (1)

QUESTION: 90

Consider the following in respect of the differential equation:

1. The degree of the differential equation is 1.

2. The order of the differential equation is 2.

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

Solution:

The order is the highest numbered derivative in the equation, while the degree is the highest power to which a derivative is raised.

For example:  is a first degree second order differential equation, while is a second degree first order differential equation.

In the equation,

The degree of the differential equation is 1.

The order of the differential equation is 2.

QUESTION: 91

What is the general solution of the differential equation 

Solution:

This is a First Order Separable Differential Equation, we can "separate the variables" to give;

QUESTION: 92

The value of k which makes  continous at x = 0 is 

Solution:

QUESTION: 93

What is the minimum value of a2x + b2y where xy = c2 ?

Solution:

Differentiate with respect to x,

Now again differentiate eq (2)

QUESTION: 94

What is  equal to?

Solution:

QUESTION: 95

What is the area of one of the loops between the curve y = c sin x and x-axis?

Solution:

QUESTION: 96

If sinθ+ cosθ = √2cosθ, then what is (cosθ − sinθ) equal to?

Solution:

QUESTION: 97

In a circle of diameter 44 cm, the length of a chord is 22 cm. What is the length of minor arc of the chord? 

Solution:

QUESTION: 98

If  then in which quadrant does θ lie?

Solution:

As this diagram shows:

i) Every trigonometric function is positive in the first quadrant.

ii) In the second quadrant only sin is positive (and its inverse, cosec). Rest are negative.

iii) Third quadrant gives positive value only for tan (and again its inverse).

iv) Fourth quadrant means positive value only for cos.
Hence,   lie in third quadrant.

QUESTION: 99

How many three-digit even numbers can be formed using the digits 1, 2, 3, 4 and 5 when repetition of digits is not allowed?

Solution:

i.e. 4 × 3 × 2 = 24.

QUESTION: 100

The angle of elevation of a tower of height h from a point A due South of it is x and from a point B due East of A is y. If AB = z, then which one of the following is correct?

Solution:

Let the length of the tower OA = h Given AB= z

QUESTION: 101

From a deck of cards, cards are taken out with replacement. What is the probability that the fourteenth card taken out is an ace?

Solution:

QUESTION: 102

If A and B are two events such that  and  0.4, then what is ​equal to?

Solution:

QUESTION: 103

A problem is given to three students A, B and C whose probabilities of solving the problem are respectively. What is the probability that the problem will be solved if they all solve the problem independently?

Solution:

QUESTION: 104

A pair of fair dice is rolled. What is the probability that the second dice lands on a higher value than does the first? 

Solution:

QUESTION: 105

A fair coin is tossed and an unbiased dice is rolled together. What is the probability of getting a 2 or 4 or 6 along with head? 

Solution:

QUESTION: 106

If A, B, C are three events, then what is the probability that at least two of these events occur together? 

Solution:

QUESTION: 107

If two variables X and Y are independent, then what is the correlation coefficient between them?

Solution:

Variables X and Y are independent. Correlation and independence as it approaches zero there is less of a relationship (closer to uncorrelated).
If the variables are independent, Pearson's correlation coefficient is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two variables.
The correlation coefficient between them is 0.

QUESTION: 108

Two independent events A and B are such that  then what is P(B) equal to?

Solution:

QUESTION: 109

The mean of 100 observations is 50 and the standard deviation is 10. If 5 is subtracted from each observation and then it is divided by 4, then what will be the new mean and the new standard deviation respectively?

Solution:

Now the new standard deviation 

[As addition and subtraction does not affect σ].

QUESTION: 110

If two fair dice are rolled then what is the conditional probability that the first dice lands on 6 given that the sum of numbers on the dice is 8? 

Solution:

Sample space, S = {(2, 6), (6, 2), (3, 5), (5, 3), (4, 4)}
Probability (that the first dice lands on 6) 

QUESTION: 111

Two symmetric dice flipped with each dice having two sides painted red, two painted black, one painted yellow and the other painted white. What is the probability that both land on the same colour?

Solution:

Required probability 

QUESTION: 112

There are n socks in a drawer, of which 3 socks are red. If 2 of the socks are chosen randomly and the probability that both selected socks are red is , then what is the value of n?

Solution:

QUESTION: 113

Two cards are chosen at random from a deck of 52 playing cards. What is the probability that both of them have the same value?

Solution:

Two cards are chosen at random from a deck of 52 playing cards Probability that both of them have the same 

QUESTION: 114

In eight throws of a die, 5 or 6 is considered a success. The mean and standard deviation of total number of successes is respectively given by

Solution:

QUESTION: 115

A and B are two events such that A̅ and B̅ are mutually exclusive. If P(A) = 0.5 and P(B) = 0.6, then what is the value of P(A|B)? 

Solution:

QUESTION: 116

Consider the following statements:

1. The algebraic sum of deviations of a set of values from their arithmetic mean is always zero.

2. Arithmetic mean > Median > Mode for a symmetric distribution.

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

Solution:

1.  The sum of the deviations from the mean is zero.

2.  In a perfectly symmetrical, non-skewed distribution the mean, median and mode are equal.

Since 1 is alone correct, so, option (a) is correct.

QUESTION: 117

Let the correlation coefficient between X and Y be 0.6. Random variables Z and W are defined as Z = X + 5 and W are defined as  What is the correlation coefficient between Z and W?

Solution:

The correlation coefficient between X and Y be 0.6.

Correlation coefficient between Z and W = 0.6

QUESTION: 118

If all the natural numbers between 1 and 20 are multiplied by 3, then what is the variance of the resulting series?

Solution:

All the natural numbers between 1 and 20 are multiplied by 3

Using var  is the variance of the resulting series 399.25 

QUESTION: 119

What is the probability that an interior point in a circle is closer to the centre than to the circumference? 

Solution:

First of all, consider the radius of circle as r, then the points closer to center then boundary will lie within the radius of  
So, the favourable outcome would be the points inside the area of circle with radius Whereas the total possible outcomes could be all the points inside the area of circle with radius r.

Therefore, the probability is 

QUESTION: 120

If A and B are two events, then what is the probability of occurrence of either A or event B?

Solution:

If A and B are two events, the probability of occurrence of either A or event B is probability of having A or B.

Thus, the correct answer is union of A and B i.e. 

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