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# NDA I - Mathematics Question Paper 2017

## 120 Questions MCQ Test NDA (National Defence Academy) Past Year Papers | NDA I - Mathematics Question Paper 2017

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This mock test of NDA I - Mathematics Question Paper 2017 for Defence helps you for every Defence entrance exam. This contains 120 Multiple Choice Questions for Defence NDA I - Mathematics Question Paper 2017 (mcq) to study with solutions a complete question bank. The solved questions answers in this NDA I - Mathematics Question Paper 2017 quiz give you a good mix of easy questions and tough questions. Defence students definitely take this NDA I - Mathematics Question Paper 2017 exercise for a better result in the exam. You can find other NDA I - Mathematics Question Paper 2017 extra questions, long questions & short questions for Defence on EduRev as well by searching above.
QUESTION: 1

Solution:

QUESTION: 2

Solution:

QUESTION: 3

### The value of [C (7, 0) + C (7, 1)] + [C (7, 1) + C (7, 2)] +⋯ + [C (7, 6) + C (7, 7)] is

Solution:

QUESTION: 4

The number of different words (eight-letter words) ending and beginning with a consonant which can be made out of the letters of the word ‘EQUATION’ is

Solution:

QUESTION: 5

The fifth term of an AP of n terms, whose sum is n2 + n, is

Solution:

QUESTION: 6

The sum of all the two-digit odd numbers is

Solution:

QUESTION: 7

The sum of the first n terms of the series  is equal to

Solution:

QUESTION: 8

Consider the following in respect of sets A and B:

1. (A − B) ∪ B = A
2. (A − B) ∪ A = A
3. (A − B) ∩ B = ∅
4. A ⊆ B ⇒ A ∪ B = B
Which of the above are correct?

Solution:

By property,

QUESTION: 9

In the binary equation (1p101)2 + (10q1)2 = (100r00)2
Where p, q and r are binary digits, what are the possible values of p, q and r respectively?

Solution:

QUESTION: 10

If S = {x: x2 + 1 = 0, x is real}, then S is

Solution:

QUESTION: 11

The expansion (x − y)2, n ≥ 5 is done in the descending powers of x. If the sum of the fifth and sixth terms is zero, then x/y is equal to

Solution:

QUESTION: 12

If A =   and det(A3 ) = 125, then α is equal to

Solution:

QUESTION: 13

If B is a non-singular matrix and A is a square matrix, then the value of det(B−1AB) is equal to

Solution:

QUESTION: 14

If a ≠ b ≠ c , then one value of x which satisfies the equation is given by

Solution:

QUESTION: 15

If A =   then what is AAT equal to (where AT is the transpose of A)?

Solution:

QUESTION: 16

What is the value of tan 18°?

Solution:

QUESTION: 17

Let x, y, z be positive real numbers such that x, y, z are in GP and tan−1 , tan−1  and tan−1  are in AP. Then which one of the following is correct?

Solution:

QUESTION: 18

If tan(α + β) = 2 and tan(α − β) = 1, then tan(2α ) is equal to

Solution:

QUESTION: 19

Consider the following for triangle ABC:

Q. Which of the above are correct?

Solution:

QUESTION: 20

If sec θ − cosec θ = 4/3, then what is (sin θ − cos θ) equal to?

Solution:

QUESTION: 21

If a vertex of a triangle is (1, 1) and the midpoints of two sides of the triangle through this vertex are (-1, 2) and (3, 2), then the centroid of the triangle is

Solution:

QUESTION: 22

The incentre of the triangle with vertices A(1, √3), B(0, 0) and C(2, 0) is

Solution:

△ ABC is equilateral

⇒ Incentre = Centroid = (1, 1/√3)

QUESTION: 23

If the three consecutive vertices of a parallelogram are (-2, -1), (1,0) and (4, 3), then what are the coordinates of the fourth vertex?

Solution:

4th vertex = (4 − 2 − 1, 3 − 1 − 0) = (1,2)

QUESTION: 24

The two circles x2 + y2 = r2 and x2 + y2 − 10x + 16 = 0 intersect at two distinct points. Then which one of the following is correct?

Solution:

QUESTION: 25

What is the equation of the circle which passes through the points (3, -2) and (-2, 0) and having its centre on the line 2x − y − 3 = 0?

Solution:

(ℎ − 3)2 + (2ℎ − 1)2 = (ℎ + 2)2 + (2ℎ − 3)2

⟹ ℎ = − 3/2
⟹ 2ℎ − 3 = −6
∴ r 2 = 1/4 + 36
Equation of circle is
(x + 3/2)+ (y + 6)2 = 1/4+ 36
⟹ x2 + y2 + 3x + 12y + 2 = 0

QUESTION: 26

What is the ratio in which the point C = (− 2/7, − 20/7) divides the line joining the points A (−2, −2) and B (2, −4)?

Solution:

2k − 2/k + 1 = − 2/7

⟹ 8k = 6 ⟹ k = 3/4
ratio = 3: 4

QUESTION: 27

What is the equation of the ellipse having foci (±2, 0) and the eccentricity 1/4?

Solution:

2 = a/4 ⟹ a = 8

b2 = 64 − 4 = 60
Equation of ellipse is
x2/64 + y2/60 = 1

QUESTION: 28

What is the equation of the straight line parallel to 2x + 3y + 1 = 0 and passes through the point (-1, 2)?

Solution:

Let equation of line is 2x + 3y + λ = 0 putting (−1, 2), we get λ = −4
⟹ equation of line is 2x + 3y − 4 = 0

QUESTION: 29

What is the acute angle between the pair of straight lines √2x + √3y = 1 and √3x + 2y = 2?

Solution:

QUESTION: 30

If the centroid of a triangle formed by (7, x), (y, −6) and (9, 10) is (6, 3) , then the values of x and y are respectively

Solution:

QUESTION: 31

Let S be the set of all persons living in Delhi. We say that x, y in S are related if they were born in Delhi on the same day. Which one of the following is correct?

Solution:

R = {(x, y): x and y were born in Delhi on same day} R is an equivalence relation

QUESTION: 32

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

Solution:

Required no. of subsects = 10C2 + 10C3 = 45 + 120 = 165

QUESTION: 33

The value of   , where   is

Solution:

12n + 12n+1 + 12n+2 + 12n+3 = 0

QUESTION: 34

If the difference between the roots of the equation x2 + kx + 1 = 0 is strictly less than √5, where |k| ≥ 2, then k can be any element of the interval

Solution:

QUESTION: 35

If the roots of the equation x2 + px + q = 0 are in the same ratio as those of the equation x2 + lx + m = 0, then which one of the following is correct?

Solution:

Let α, kα be roots of x2 + px + q = 0 and β , kβ be roots of x2 + ℓx + m = 0 Clearly,

QUESTION: 36

The value of

Where n is not a multiple of 3 and  , is

Solution:

n is not multiple of 3

QUESTION: 37

Three-digit numbers are formed from the digits 1, 2, and 3 in such a way that the digits are not repeated. What is the sum of such three-digit numbers?

Solution:

Sum = 12(10° + 101 + 102 )

= 111 × 12 = 1332

QUESTION: 38

What is the sum of the series 0.3 + 0.33 + 0.333 + ⋯ n terms?

Solution:

QUESTION: 39

If ω, ω2 are the cube roots of unity, then (1 + ω)(1 + ω2 )(1 + ω3)(1 + ω + ω2 ) is equal to

Solution:

(1 + ω)(1 + ω2 )(1 + ω3)(1 + ω + ω2 ) = 0

QUESTION: 40

If the sum of m terms of an AP is n and the sum of n terms is m, then the sum of (m + n) terms is

Solution:

In an A.P

If Sm = n and Sn = m then Sm+n = -(m +n)

QUESTION: 41

The modulus and principal argument of the complex number are respectively

Solution:

Modulus = 1, Argument = 0

QUESTION: 42

If the graph of a quadratic polynomial lies entirely above the x-axis, then which one of the following is correct?

Solution:

If graph of quadratic lies entirely above x-axis then D  > 0. So, both roots are complex.

QUESTION: 43

If |z + 4| ≤ 3, then the maximum value of |z + 1| is

Solution:

|z + 1| = |z + 4 − 3|

≤ |z + 4| + |−3|
maximum value of |z + 1| = 6

QUESTION: 44

The number of roots of the equation z2 = 2z̅ is

Solution:

QUESTION: 45

If cot α and cot β are the roots of the equation x2 + bx + c = 0 with b ≠ 0, then the value of cot(α + β) is

Solution:

QUESTION: 46

The equations x + 2y + 3z = 12x + y + 3z = 25x + 5y + 9z = 4

Solution:

= 1(9 − 15) − 2(18 − 15) + 3(10 − 5)

= −6−6+15 = 3 ≠ 0

⟹ system has unique solution.

QUESTION: 47

If AB = C, then what is A2 equal to?

Solution:

QUESTION: 48

What is the value of the determinant

Solution:

QUESTION: 49

If  then which one of the following is correct?

Solution:

QUESTION: 50

Consider the set A of all matrices of order 3 × 3 with entries 0 or 1 only. Let B be the subset of A consisting of all matrices whose determinant is 1. Let C be the subset of A consisting of all matrices whose determinant is -1. Then which one of the following is correct?

Solution:

By symmetry B has as many elements as C.

QUESTION: 51

If A =  then what is A3 equal to?

Solution:

A =

⟹ A2 =

⟹ A3 =

QUESTION: 52

What is the order of

Solution:

(1 × 3)(3 × 3)(3 × 1)
= (1 × 1)

QUESTION: 53

If A = then the value of A2 is

Solution:

A =

⟹ A2 =

⟹ A4 = I =

QUESTION: 54

If A = 3/5 where 450° < 540°, then cos ��/2 is equal to

Solution:

sin A =

⟹ cos A = − 4/5 (∵ A lies in 2nd quadrant)

QUESTION: 55

What is 1/sin 10°− √3/cos10° equal to?

Solution:

QUESTION: 56

From the top of a lighthouse, 100 m high, the angle of depression of a boat is tan−1 ( 5/12). What is the distance between the boat and the lighthouse?

Solution:

tan �� = 5/12
⟹ 100/x = 5 / 12
⟹ x = 240 m

QUESTION: 57

The maximum value of sin (x + π/6) + cos (x + π/6) in the interval (0, π/2) is attained at

Solution:

f(x) is maximum when

QUESTION: 58

If K = sin ( π/18) sin (5π/18) sin (7π/18), then what is the value of K?

Solution:

K = sin 10° ∙ sin 50° ∙ sin 70°
= 1/4 sin 30° = 1/4×1/2 = 1/8

QUESTION: 59

The expression  is equal to

Solution:

QUESTION: 60

If sin θ = 3 sin(θ + 2α), then the value of tan(θ+ α) + 2 tan α is equal to

Solution:

QUESTION: 61

What is equal to?

Solution:

QUESTION: 62

Let f: [−6, 6] → R be defined by f(x) = x2 − 3. Consider the following:

Which of the above is/are correct?

Solution:

Both 1 and 2 are correct.

QUESTION: 63

Solution:

⟹ f(n) = g(q)

QUESTION: 64

If ��(x) = then what is equal to?

Solution:

QUESTION: 65

What is equal to?

Solution:

QUESTION: 66

Solution:

Clearly f − g is one-one and onto

QUESTION: 67

What is the length of the longest interval in which the function is increasing?

Solution:

QUESTION: 68

If   and  then what is equal to?

Solution:

QUESTION: 69

What is the maximum value of the function

Solution:

Maximum value = 4 + 1 = 5

QUESTION: 70

Let f(x) be an indefinite integral of sin2 x.
Consider the following statements:

Statement 1: The function f(x) satisfies f(x + π) = f (x) for all real x.

Statement 2: sin2(x + π) = sin2 x for all real x.

Q. Which one of the following is correct in respect of the above statements?

Solution:

Statement 2 true

QUESTION: 71

What are the degree and order respectively of the differential equation

Solution:

QUESTION: 72

What is the differential equation corresponding to  by eliminating a?

Solution:

QUESTION: 73

What is the general solution of the differential equation

Solution:

QUESTION: 74

Let . Then what is f′(5) equal to where f′(x) is the derivative of f(x)]?

Solution:

QUESTION: 75

If f (x) and g(x) are continuous functions satisfying f(x) = f(a − x) and g(x) + g(a − x) = 2, then what is  equal to?

Solution:

QUESTION: 76

For two department events A and B, it is given that P (A) = 0 ∙ 2 and P(B) = 0 ∙ 5. If A ⊆ B , then the values of conditional probabilities P(A|B) and P(B|A) are respectively.

Solution:

QUESTION: 77

A point is chosen at random inside a circle. What is the probability that the point is closer to the centre of the circle than to its boundary?

Solution:

Required probability

QUESTION: 78

If two regression lines between height (x) and weight (y) are 4y − 15x + 410 = 0 and 30x − 2y − 825 = 0, then what will be the correlation coefficient between height and weight?

Solution:

QUESTION: 79

In an examination, 40% of candidates got second class. When the data are represented by a pie chart, what is the angle corresponding to second class?

Solution:

QUESTION: 80

Consider the following statements:

Statement 1: Range is not a good measure of dispersion.

Statement 2: Range is highly affected by the existence of extreme values.

Q. Which of the following is correct in respect of the above statements?

Solution:

It is a fundamental concept. So, options ‘a’ is correct.

QUESTION: 81

A card is drawn from a well-shuffled ordinary deck of 52 cards. What is the probability that it is an ace?

Solution:

QUESTION: 82

If the data are moderately non-symmetrical, then which one of the following empirical relationships is correct?

Solution:

QUESTION: 83

Data can be represented in which of the following forms?

1. Textual form 2. Tabular form 3. Graphical form Select the correct answer using the code given below:

Solution:

Data can be represented in tabular and graphical form.

QUESTION: 84

For given statistical data, the graphs for less than ogive and more than ogive are drawn. If the point at which the two curves intersect is P, then abscissa of point P gives the value of which one of the following measures of central tendency?

Solution:

The abscissa of the point of intersection of less than and more than ogive is median.

QUESTION: 85

Consider the following statements:

1. Two events are mutually exclusive if the occurrence of one event prevents the occurrence of the other.

2. The probability of the union of two mutually exclusive events is the sum of their individual probabilities.
Which of the above statements is/are correct?

Solution:

Both statements are correct.

QUESTION: 86

If the regression coefficient of x and y and y on x are − 1/2and − 1/8respectively, then what is the correlation coefficient between x and y?

Solution:

Result

QUESTION: 87

A sample of 5 observations has mean 32 and median 33. Later it is found that an observation was recorded incorrectly as 40 instead of 35. If we correct the data, then which one of the following is correct?

Solution:

Median remains same but the mean will decrease.

QUESTION: 88

If two fair dice are thrown, then what is the probability that the sum is neither 8 nor 9?

Solution:

Required probability

QUESTION: 89

Let A and B are two mutually exclusive events with P (A) = 1/3 and P(B) = 1/4. What is the value of P(A̅ ∩ B̅ )?

Solution:

QUESTION: 90

The mean and standard deviation of a binomial distribution are 12 and 2 respectively. What is the number of trials?

Solution:

QUESTION: 91

A straight line with direction cosines 〈0, 1, 0〉 is

Solution:

Parallel to y-axis.

QUESTION: 92

(0, 0, 0), (a, 0, 0), (0, b, 0) and (0, 0, c) are four distinct points. What are the coordinates of the point which is equidistant from the four points?

Solution:

Let O(0, 0, 0) A(a, 0, 0), B(0, b, 0) and C(0, 0, c)  is equidistant from O, A, B and C...

QUESTION: 93

The points P(3, 2, 4), Q(4, 5, 2), R(5, 8, 0) and S(2, −1, 6) are

Solution:

P, Q, R, S are collinear.

QUESTION: 94

The line passing through the points (1, 2, −1) and (3, −1, 2) meets the yz plane at which one of the following points?

Solution:

QUESTION: 95

Under which one of the following conditions are the lines and  perpendicular?

Solution:

QUESTION: 96

and  are three coplanar vectors and then  which one of the following is correct?

Solution:

QUESTION: 97

Let ABCD be a parallelogram whose diagonals intersect at P and let O be the origin. What is ⃗ equal to?

Solution:

QUESTION: 98

ABCD is a quadrilateral whose diagonals are AC and BD. Which one of the following is correct?

Solution:

QUESTION: 99

If then which one of the following is correct?

Solution:

QUESTION: 100

If are perpendicular, then what is the value of λ?

Solution:

(2)(3) + (3)(2) − 4λ = 0 ⟹ λ = 3

QUESTION: 101

What is   equal to?

Solution:

QUESTION: 102

What is  equal to?

Solution:

QUESTION: 103

What is  equal to?

Solution:

QUESTION: 104

The function defined by f(x) = cos x, where x ∈ X, is one-one and onto if X and Y are respectively equal to

Solution:

f(x) = cos x is bijective for domain X = [0, π] and co-domain Y = [−1, 1]

QUESTION: 105

If  then what is  equal to?

Solution:

= f (a2)

QUESTION: 106

Question No. 106 ABCD

Solution:
QUESTION: 107

Question No. 107 ABCD

Solution:
QUESTION: 108

Questions 108 ABCD

Solution:
QUESTION: 109

Questions 109 ABCD

Solution:
QUESTION: 110

What is the derivative of log10 (5x2 + 3) with respect to x?

Solution:

QUESTION: 111

Let f(a) =

Q. Consider the following:

1.

2.

Which of the above is/are correct?

Solution:

QUESTION: 112

What is the maximum area of a triangle that can be inscribed in a circle of radius a?

Solution:

For area of △ to be maximum, triangle should be an equilateral triangle.

ℓ = length of side of equilateral triangle = √3a

QUESTION: 113

Let  where

Q. Then which one of the following is correct?

Solution:

QUESTION: 114

Suppose the function f(x) = xn , n ≠ 0 is differentiable for all x. Then n can be any element of the interval

Solution:

f(x) = xn , n ≠ 0.

⟹ f ′ (x) = nxn-1
f(x) to be differentiable,  n − 1 ≥ 0

⟹ n ≥ 1 ⟹ n ∈ [1, ∞]

QUESTION: 115

What is  dx equal to?

Solution:

QUESTION: 116

The variance of 20 observations is 5. If each observation is multiplied by 3, then what is the new variance of the resulting observations?

Solution:

New variance = 5 × (3)2 = 45

QUESTION: 117

The mean of a group of 100 observations was found to be 20. Later it was found that four observations were incorrect, which were recorded 21, 21, 18 and 20. What is the mean if the incorrect observations are omitted?

Solution:

Required mean

QUESTION: 118

A committee of two persons is constituted from two men and two women. What is the probability that the committee will have only women?

Solution:

Required probability

QUESTION: 119

A question is given to three students A, B and C whose chances of solving it are 1/2,1/3 and 1/4 respectively. What is the probability that the question will be solved?

Solution:

Required probability

QUESTION: 120

The mean weight of 150 students in a certain class is 60 kg. The mean weight of boys in the class is 70 kg and that of girls is 55 kg. What is the number of boys in the class?

Solution:

By aligation, ratio = 1: 2

∴ no. of boys = 1/3 × 150 = 50