NDA  >  NDA I - Mathematics Question Paper 2016

# NDA I - Mathematics Question Paper 2016 - NDA

Test Description

## 120 Questions MCQ Test NDA (National Defence Academy) Past Year Papers - NDA I - Mathematics Question Paper 2016

NDA I - Mathematics Question Paper 2016 for NDA 2023 is part of NDA (National Defence Academy) Past Year Papers preparation. The NDA I - Mathematics Question Paper 2016 questions and answers have been prepared according to the NDA exam syllabus.The NDA I - Mathematics Question Paper 2016 MCQs are made for NDA 2023 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for NDA I - Mathematics Question Paper 2016 below.
Solutions of NDA I - Mathematics Question Paper 2016 questions in English are available as part of our NDA (National Defence Academy) Past Year Papers for NDA & NDA I - Mathematics Question Paper 2016 solutions in Hindi for NDA (National Defence Academy) Past Year Papers course. Download more important topics, notes, lectures and mock test series for NDA Exam by signing up for free. Attempt NDA I - Mathematics Question Paper 2016 | 120 questions in 150 minutes | Mock test for NDA preparation | Free important questions MCQ to study NDA (National Defence Academy) Past Year Papers for NDA Exam | Download free PDF with solutions
 1 Crore+ students have signed up on EduRev. Have you?
NDA I - Mathematics Question Paper 2016 - Question 1

### Suppose ω is a cube root of unity with ω≠1. Suppose P and Q are the points on the complex plane defined by ω and ω2. If O is the origin, then what is the angle between OP and OQ?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 1

P and Q are points on complex plane. Angle between OP and OQ is

NDA I - Mathematics Question Paper 2016 - Question 2

### Suppose there is a relation * between the positive numbers x and y given by x * y if and only if x ≤ y2. Then which one of the following is correct?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 2

x and y are positive numbers.
x ≤ y2
x < x  positive numbers.
Hence relation is reflexive.
Transitive -​

Thus relation is not transitive.
Symmetric
1 ≤ (2)2 while 2  (I)2
Hence relation is not symmetric.
Thus x ≤ y2  positive numbers is reflexive, but not transitive and symmetric.

NDA I - Mathematics Question Paper 2016 - Question 3

### If x2+ px + 4  for all real values of x, then which one of thefollowing is correct?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 3

(a) x2-px + 4> 0  real values of x.
If b2 - 4ac < 0
⇒p2 - 4(1)(4)<0
⇒p2 < 16 ⇒|p| < 4

NDA I - Mathematics Question Paper 2016 - Question 4

If z=x + iy= , where i = √-1, then what is the fundamental amplitude of

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 4

z = x + iy

NDA I - Mathematics Question Paper 2016 - Question 5

If f(x1)- f(x2) is for x1, x2 ∈ (-1,1), then what is f(x) equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 5

NDA I - Mathematics Question Paper 2016 - Question 6

What is the range of the function  2 ,where X ∈R?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 6

NDA I - Mathematics Question Paper 2016 - Question 7

A straight line intersects x and y axes at P and Q respectively If (3,5) is the middle point of PQ, then what is the area ofthe triangle OPQ?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 7

As we know that line PQ intersects x-axis andy-axis at Rand Q.
∵ M is the mid point of PQ

⇒ x = 6 and y = 10
Hence area of triangle OPQ

NDA I - Mathematics Question Paper 2016 - Question 8

If a circle of radius b units with centre at (0, b) touches the line y = x — a√2 , then what is the value of b?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 8

Distance from the centre to the point of line which touches circle is OM = radius

NDA I - Mathematics Question Paper 2016 - Question 9

Consider the function f(θ)= 4(sin2 θ+ cos4 θ)

Q. What is the maximum value of the function f(θ)?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 9

f(θ) = 4 (sin2θ + cos4 θ)
= 4 (sinθ + cos2 θ(1- sin2 θ))
= 4 (sin2 θ + cos2 θ - sin2 θ cos2 θ)

For maximum value of f(θ), sin22θ should be minimum.
i.e. sin22θ = 0
f(θ)lmax=4(l1-0) = 4

NDA I - Mathematics Question Paper 2016 - Question 10

Consider the function f(θ)= 4(sin2 θ+ cos4 θ)

Q. What is the minimum value of the function f(θ)?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 10

f(θ) = 4 (sin2θ + cos4 θ)
= 4 (sinθ + cos2 θ(1- sin2 θ))
= 4 (sin2 θ + cos2 θ - sin2 θ cos2 θ)

For minimum value ot f(θ), sin22θ should be maximum i.e. sin22θ= 1.

NDA I - Mathematics Question Paper 2016 - Question 11

Consider the function f(θ)= 4(sin2 θ+ cos4 θ)

Consider the following statements:
f(θ) = 2 has no solution.
f(θ) =  has a solution.

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

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 11

f(θ) = 4 (sin2θ + cos4 θ)
= 4 (sinθ + cos2 θ(1- sin2 θ))
= 4 (sin2 θ + cos2 θ - sin2 θ cos2 θ)

Since sin θ cannot have vlaue greater than 1 & less than -1.
Hence f(θ) = 2 has no solution.

NDA I - Mathematics Question Paper 2016 - Question 12

Consider the curves

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 12

Hence f(x ) and g(x) intersects at ( -1 , -2 ) and (2 ,3 ).

NDA I - Mathematics Question Paper 2016 - Question 13

What is the area bounded by the curves

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 13

NDA I - Mathematics Question Paper 2016 - Question 14

Consider the function

How many solutions does the function f(x) = 1 have?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 14

This is a cubic equation.
If we put y = then ( 3y + 1) = 0 is a factor o f cubic equation.

NDA I - Mathematics Question Paper 2016 - Question 15

Consider the function

How many solutions does the function f(x) = -1 have?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 15

Similarly for (f) = -1 we will get 27y3 - 27y2 - 4 = 0 and after solving it we will find that it has two solutions.
y0=1.1184,-0.05922.

NDA I - Mathematics Question Paper 2016 - Question 16

Consider the functions
f(x) = xg(x) and g(x =
Where [•] is the greatest integer function.

What is  equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 16

As g(x) is a gretest integer function so value of g(x) in integral limit will be

NDA I - Mathematics Question Paper 2016 - Question 17

What is  equal to ?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 17

The value of g(x) in value  will be 2 and in range

form(l)

NDA I - Mathematics Question Paper 2016 - Question 18

Consider the function f ( x ) = | x - 1 |+ x2 , w here x ∈R .

Which one of the following statements is correct?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 18

f(x) = | x - l |  + x2  x  ∈R
f1 (x ) = |x - l |, f2(x) = x2
f1 (x) and f2{x) both are continuous.
Hence f(x) is continuous.
f(x) in differentiable at x = 0
f1(x) is not differentiable at x = 1.
Hence(fx) is continuous but not differentiable at x= 1

NDA I - Mathematics Question Paper 2016 - Question 19

Consider the function f ( x ) = | x - 1 |+ x2 , w here x ∈R .

Which one of the following statements is correct?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 19

As we know,

f(x) is in quadratic form (parabola). Hence f(x) is decreasing in  and increasing

NDA I - Mathematics Question Paper 2016 - Question 20

Which one of the following statements is correct?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 20

f(x) has local minimum at one point only in (-∞ ,∞ ).

NDA I - Mathematics Question Paper 2016 - Question 21

What is the area of the region bounded by x-axis, the curve y = f(x) and the two ordinates and x = 1 ?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 21

Hence area required for given region is

NDA I - Mathematics Question Paper 2016 - Question 22

What is the area of the region bounded by x-axis, the curve y = f(x) and the two ordinates x = 1 and

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 22

Area required for given region is

NDA I - Mathematics Question Paper 2016 - Question 23

Given that an
Consider the following statements:
1. The sequence {a2n} is in AP with common difference zero.
2. The sequence {a2n+1} is in AP with common difference zero.

Which of the above statements is/are correct?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 23

Since it is a definite integral will have a definite value. The sequence {a2n} is in AP with common difference. Statement (1) is correct.
The sequence {a2n + 1} is also in AP with common difference.
Statement (2) is correct.

NDA I - Mathematics Question Paper 2016 - Question 24

Given that an =

What is an-1 - an-4​ equal to ?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 24

∵ given sequence an also AP with no difference.
Thus an-1 - an-4 = 0

NDA I - Mathematics Question Paper 2016 - Question 25

Consider the equation x + |y| = 2y.

Which of the following statements are not correct?
1. y as a function of x is not defined for all real x.
2. y as a function of x is not continuous at x = 0.
3. y as a function of x is differentiable for all x.

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

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 25

x+ | y |= 2y
x = 2y - |y |
2y-| y | = x

∵ by checking
y as a function of x is continuous at x = 0, but not differentiable at x = 0.
So all of the statements are not correct.

NDA I - Mathematics Question Paper 2016 - Question 26

Consider the equation x + |y| = 2y.

What is the derivative of y as a function of x with respect to x for x < 0?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 26

Option (d) is correct.

NDA I - Mathematics Question Paper 2016 - Question 27

Consider the lines y = 3x, y = 6x and y = 9

What is the area of the triangle formed by these lines?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 27

OAB is triangle

NDA I - Mathematics Question Paper 2016 - Question 28

Consider the lines y = 3x, y = 6x and y = 9

The centroid of the triangle is at which one of the following points?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 28

Coordinates o f O, A, B are (0, 0) respectively.

NDA I - Mathematics Question Paper 2016 - Question 29

Consider the function f(x) = (x - l )2 ( x + 1) (x - 2)3

Q. What is the number of points of local minima of the function f(x)?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 29

f(x) = (x-l)2(x + l ) (x-2)3
f'(x) = 2(x - l)(x + l)(x - 2)3+ ( x - l)2(x - 2)3+(x - 1)2 (x + l)3(x - 2)2
= (x - l)(x -1)2 [2(x+1)(x - 2 ) +( x - l) ( x - 2) + 3 ( x - l ) ( x + l)]
f '(x) = ( x - l)(x - 2 )2[2x2 - 2x - 4 + x2 - 3x + 2 + 3x2 - 3]
= (x - l)(x - 2)2 [6x2 - 5x - 5]
For maxima and minima
f'(x )= 0
(x - l)(x - 2)2 [6x2 - 5x - 5] = 0

The change in signs of f(x) for dififrent values of x is shown:

∵ Local Minima are

NDA I - Mathematics Question Paper 2016 - Question 30

Consider the function f(x) = (x - l )2 ( x + 1) (x - 2)3

What is the number of points of local maxima of the function f(x) ?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 30

Local Maxima is [x = 1 ]

NDA I - Mathematics Question Paper 2016 - Question 31

Let f(x) and g(x) be twice differentiable functions on [0,2] satisfying f"(x) = g"(x), f'(1) = 4, g'(1) = 6,f(2) =3 and g(2) = 9. Then what is f(x) - g(x) at x = 4 equal to ?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 31

again integrating equation (1)

Rearranging equation (3) again, we get

NDA I - Mathematics Question Paper 2016 - Question 32

Consider the curves y = | x — 1 | and |x| = 2

Q. What is/are the point(s) of intersection of the curves ?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 32

and x=2
Hence curves intersect at (-2,3) and (2,1).

NDA I - Mathematics Question Paper 2016 - Question 33

Consider the curves y = | x — 1 | and |x| = 2.

Q. What is the area of the region bounded by the curves and x-axis?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 33

Bounded region is shaded.
So area of bounded region has two triangles ACB and BDE.

Area of region bounded by curves and x-axis is

NDA I - Mathematics Question Paper 2016 - Question 34

Consider the function

Q. What is the value of f (0)?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 34

NDA I - Mathematics Question Paper 2016 - Question 35

Consider the function

Q. What is the value of p for which f"(0)=0?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 35

NDA I - Mathematics Question Paper 2016 - Question 36

Consider a tiangle ABC in which

Q. What is the value of sin

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 36

NDA I - Mathematics Question Paper 2016 - Question 37

Consider a tiangle ABC in which

Q. What is the value of

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 37

As we know that

NDA I - Mathematics Question Paper 2016 - Question 38

Given that tan α and tanβ are the roots of the equation x2 + bx + c = 0 with b ≠ 0.

Q. What is tan(α + β) equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 38

NDA I - Mathematics Question Paper 2016 - Question 39

Given that tan α and tanβ are the roots of the equation x2 + bx + c = 0 with b ≠ 0.

What is sin(α+ β)sec α see β equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 39

sin( α + β) sec α see β

= tan α + tan β
= -b

NDA I - Mathematics Question Paper 2016 - Question 40

Consider the two circles (x-l)2 + ( y-3)2 = r2 and x2 + y2 - 8x + 2y + 8 = 0

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

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 40

Given equation of circles
{h1, k1) = coordinates of centre = (1,3)
x2 + y2 - 8x + 2y + 8 = 0
(x - 4 )2 + (y+1)2 =(3)2
(h2 , k2) = coordinates o f centre = (4 ,-1)
Distance between centres of two circles

NDA I - Mathematics Question Paper 2016 - Question 41

Consider the two circles (x-l)2 + ( y-3)2 = r2 and x2 + y2 - 8x + 2y + 8 = 0

Q. If the circles intersect at two distinct points, then which one of the following is correct?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 41

Given equation of circles
{h1, k1) = coordinates of centre = (1,3)
x2 + y2 - 8x + 2y + 8 = 0
(x - 4 )2 + (y+1)2 =(3)2
(h2 , k2) = coordinates o f centre = (4 ,-1)
Radius of circle one = r1 = r
Radius of circle two = r2 = 3
∵ Circle intersects at two points so distance between circle is d < r1 + r2
5 < r + 3
r> 2

NDA I - Mathematics Question Paper 2016 - Question 42

Consider the two lines x + y + 1 = 0 and 3x + 2y + 1 = 0

Q. What is the equation of the line passing through the point of intersection of the given lines and parallel to x-axis?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 42

Equations of lines
x+ y + 1 = 0
3x+2y+ 1=0

Points of intersection (1, -2).
Equation ofx-axis
y=0
Line parallel to x axis is
y = k
If this line passes through (1, -2) then
⇒ y = - 2
⇒ y + 2 = 0
Equation of line passing through (1, -2) and parallel to x-axis is
y + 2 = 0

NDA I - Mathematics Question Paper 2016 - Question 43

Consider the two lines x + y + 1 = 0 and 3x + 2y + 1 = 0

Q. What is the equation of the line passing through the point of intersection of the given lines and parallel to y-axis?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 43

Equations of lines
x+ y + 1 = 0
3x+2y+ 1=0

Points of intersection (1, -2).
Equation of y-axis
x = 0
Equation of line parallel to x -axis is
x = k
If this line passes through (1,-2 then)
x = 1
Hence equation of line which passes through point of intersection of given line (1, -2) and parallel to y-axis
x = 1
⇒ x-1= 0

NDA I - Mathematics Question Paper 2016 - Question 44

Consider the equation
k sinx + cos 2x = 2k - 7

If the equation possesses solution, then what is the minimum value of k?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 44

K sin x + cos 2x = 2 K - 7
K sin x + (1 -2 sin2 x ) = (2K - 7)
2 sin2 x - K sin x + (2k - 8) = 0
This is a quadratic equation in sin x.

For minimum value of k
sin x = -l

Squaring both sides, we get
K2 - 16 K+ 64 = K2 + 16 + 8 K
24K=48
K=2

NDA I - Mathematics Question Paper 2016 - Question 45

Consider the equation
k sinx + cos 2x = 2k - 7

If the equation possesses solution, then what is the maximum value of k?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 45

For maximum value of K
sin x = 1

NDA I - Mathematics Question Paper 2016 - Question 46

Consider the function f (x)  where [•] denotes the greatest integer function.

Q. What is  equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 46

NDA I - Mathematics Question Paper 2016 - Question 47

Consider the function f (x)  where [•] denotes the greatest integer function.

Q. What is equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 47

NDA I - Mathematics Question Paper 2016 - Question 48

Let z1, z2 and z3 be non-zero com plex num bers satisfying  , where i =√- 1 .

Q. What is Z1 + z2 + z3 equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 48

Given
Let us suppose that z - x + iy

x2 - y2 +2xyi = ix+y
Comparing real and imaginary part of both sides
x2 - y2 = y and 2 xy = x.
Taking 2xy=x
(2y - l)x = 0

ifx = 0

Since given numbers are non zero complex numbers.
So, z1 - 0 + ( - 1 )i = - i

NDA I - Mathematics Question Paper 2016 - Question 49

Let z1, z2 and z3 be non-zero com plex num bers satisfying  , where i =√- 1 .
Consider the following statements:
1. z1z2z3 is purely imaginary.
2. z1z2 + z2z3 + z3z1 is purely real.

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

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 49

Hence z1z2 + z2z3 + z3Z1 = 0 is purely real.
Hence both statements are correct.

NDA I - Mathematics Question Paper 2016 - Question 50

Given that logx y, logz x, logy z are in GP, xyz = 64 and x3, y3, zare inA.P.

Q. Which one of the following is correct ?
x,y and z are

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 50

Thus x, y z are in A.P. and G.P. both.

NDA I - Mathematics Question Paper 2016 - Question 51

Given that logx y, logz x, logy z are in GP, xyz = 64 and x3, y3, zare inA.P.

Q. Which one of the following is correct?
xy, yz and zx are

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 51

Similarly xy,yz, zx are also in A.P. and G.P. both.

NDA I - Mathematics Question Paper 2016 - Question 52

Let z be a complex number satisfying

Q. What is |z| equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 52

⇒|z-4| = |z-8|
Let z = x + iy
| x + iy - 4 | = | x + iy -8 |
Squaring both sides, we get
[ ( x - 4 )2 + y2] = [(x-8)2 + y2]
(x-4)2 =(x-8)2
x2 + 16 - 8x = x2 + 64 - 1 6x
8x = 48 ⇒ x= 6

Squaring both sides, we get
4(x2 + y2) = 9 [ (x - 2)2 + y2]
⇒ 4 X2 + 4 y2 = 9x2 + 36 - 36 x + 9 y2
⇒ 5x2 +5y2 -36x+36 = 0
as we know x = 6
5(6)2 + 5y2 - 36 x 6 + 36 = 0
⇒ 5y2 = 0 ⇒ y = 0.
Hence x = 6 and y = 0.
⇒ z = 6
|z| = 6

NDA I - Mathematics Question Paper 2016 - Question 53

Let z be a complex number satisfying

Q. What is  equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 53

NDA I - Mathematics Question Paper 2016 - Question 54

A function f(x) is defined as follows:

Consider the following statements:
1. The function f(x) is continuous at x = 0.
2. The function f(x) is continuous at x =

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

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 54

Given

For continuity,

f(0)=L.HL. = RHL
Hence function is continuous at x = 0.
Statement (1) is correct.

Hence fimction is continuous at
Statement (2) is correct.

NDA I - Mathematics Question Paper 2016 - Question 55

A function f(x) is defined as follows:

Consider the following statements:
1. The function f(x) is differentiable at x = 0. 71
2. The function f(x) is differentiable at x =

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

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 55

For differentiability,
L.HD. = RH.D.
Thus at x= 0

L.HD.≠ RHD.
So at x = 0 function is not differentiable. Statement (1) is not correct.

Hence function is not differentiable at x =
Statement (2) is not correct.

NDA I - Mathematics Question Paper 2016 - Question 56

Let α and β  (α < β ) be th e roots of the equatio n x2 + bx + c = 0, where b > 0 and c < 0.
Consider the following:
1. β < - α
2. P < | a |

Q. Which of the above is/are correct?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 56

x2 + bx + c = 0 and roots are α and β.
where a < p. Hence roots of given quadratic equation are

NDA I - Mathematics Question Paper 2016 - Question 57

Consider the following:
1. α + β + αβ > 0
2. α2 β + β2α > 0

Which of the above is/are correct?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 57

Sum of roots = α + β = -b
Multiplication of roots = αβ = c
Hence

NDA I - Mathematics Question Paper 2016 - Question 58

Consider a parallelogram whose vertices are A (1,2), B (4, y), C (x, 6) and D (3,5) taken in order.

Q. What is the value o f AC2 - BD2 ?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 58

Suppose Mid point of AC and BD is M (a, b).

NDA I - Mathematics Question Paper 2016 - Question 59

Consider a parallelogram whose vertices are A (1,2), B (4, y), C (x, 6) and D (3,5) taken in order.

Q. What is the point of intersection of the diagonals?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 59

Point of intersection (a, b) is

NDA I - Mathematics Question Paper 2016 - Question 60

Consider a parallelogram whose vertices are A (1,2), B (4, y), C (x, 6) and D (3,5) taken in order.

Q. What is the area of the parallelogram?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 60

Area of parallelogram=2 area of Δ ADB

NDA I - Mathematics Question Paper 2016 - Question 61

Let f : RR be a function such that
f(x ) = x3 + x2 f '(1) + xf "(2)+ f "'(3)
for x ∈ R

What is f(1) equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 61

NDA I - Mathematics Question Paper 2016 - Question 62

Let f : R → R be a function such that
f(x ) = x3 + x2 f '(1) + xf "(2)+ f "'(3)
for x ∈ R

Q. What is f '(1) equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 62

f'(l)=-5

NDA I - Mathematics Question Paper 2016 - Question 63

Let f : R → R be a function such that
f(x ) = x3 + x2 f '(1) + xf "(2)+ f "'(3)
for x ∈ R

Q. What is f'""(10) equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 63

f "'(10)=6

NDA I - Mathematics Question Paper 2016 - Question 64

Consider the following:
1. f(2 ) = f(1) - f(0)
2. f "(2) - 2f '(1) = 12

Q. Which of the above is/are correct?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 64

NDA I - Mathematics Question Paper 2016 - Question 65

A plane P passes through the line of intersection of the planes 2x - y + 3z = 2, x + y - z = 1 and the point (1 ,0 ,1 ).

Q. What are the direction ratios of the line of intersection of the given planes?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 65

Hence direction ratios ofthe line of intersection of given plane <2,-5,-3 >

NDA I - Mathematics Question Paper 2016 - Question 66

A plane P passes through the line of intersection of the planes 2x - y + 3z = 2, x + y - z = 1 and the point (1 ,0 ,1 ).

Q. What is the equation of the plane P?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 66

NDA I - Mathematics Question Paper 2016 - Question 67

A plane P passes through the line of intersection of the planes 2x - y + 3z = 2, x + y - z = 1 and the point (1 ,0 ,1 ).

Q. If the plane P touches the sphere x2 + y2 + z2 = r2, then what is r equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 67

Plane P touches the sphere x2 + y2 + z2 = r2 then r=Distane between centre of sphere (0,0,0) to plance P.

NDA I - Mathematics Question Paper 2016 - Question 68

Consider th e function f (x ) = | x2 - 5x + 6 |

Q. What is f '(4) equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 68

NDA I - Mathematics Question Paper 2016 - Question 69

Consider th e function f (x ) = | x2 - 5x + 6 |

Q. What is f"(2.5) equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 69

f"(2.5) =-2

NDA I - Mathematics Question Paper 2016 - Question 70

Let f(x) be the greatest integer function and g(x) be the modulus function.

Q. What is  equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 70

f(x) → greatest integer function
f(x)=[x]
g(x) → modulus fuction
g(x)=| x |

NDA I - Mathematics Question Paper 2016 - Question 71

Let f(x) be the greatest integer function and g(x) be the modulus function.

Q. What is equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 71

f(x) → greatest integer function
f(x)=[x]
g(x) → modulus fuction
g(x)=| x |

NDA I - Mathematics Question Paper 2016 - Question 72

Consider a circle passing through the origin and the points (a, b) and (-b, -a).

Q. On which line does the centre of the circle lie?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 72

Suppose; x2 + y2 + 2gx + 2fy + c = 0 is the eq. of the circle.
Since; it passes through

x + y =0 is the line which passes through (f, -f)

NDA I - Mathematics Question Paper 2016 - Question 73

Consider a circle passing through the origin and the points (a, b) and (-b, -a).

Q. What is the sum of the squares of the intercepts cut off by the circle on the axes?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 73

The two intercepts are : -2g & -2f
∵ from eq (l) &( 2) we get;
is sum of squares of intercepts

NDA I - Mathematics Question Paper 2016 - Question 74

Let  be two unit vectors and 0 be the angle between them.

Q. What is cos  equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 74

NDA I - Mathematics Question Paper 2016 - Question 75

Let  be two unit vectors and 0 be the angle between them.

What is sin equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 75

NDA I - Mathematics Question Paper 2016 - Question 76

Consider the following statements:
1. There exists
2. sin-1
Q. Which of the above statements is/are correct ?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 76

Hence, statement (2) is correct.

NDA I - Mathematics Question Paper 2016 - Question 77

Consider the following statements:​
1.
2. There exist x ,y ∈[-l, 1], where x ≠ y such that

Which of the above statements is/are correct ?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 77

Statement-1

Statement (1) is wrong.
Statement 2,

Only when x = y
Here x ≠ y .
Statement (2) is also wrong.

NDA I - Mathematics Question Paper 2016 - Question 78

What are the order and degree respectively of the differential equation whose solution is y = cx + c2 - 3c3/2 + 2, where c is a parameter?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 78

Given:
Solution of differential equation is
y - cx + c- 3c3/2 + 2 ...........(1)
To find order and degree of differential equation, we will find differential equation first.
Now differentiating equation (1) w.r.t. x and putting value of c to remove it, we get

Hence order of differential equation is 1 and degree is 4.

NDA I - Mathematics Question Paper 2016 - Question 79

What is

equal to, where [•] is the greatest integer function?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 79

NDA I - Mathematics Question Paper 2016 - Question 80

If  then what is  equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 80

NDA I - Mathematics Question Paper 2016 - Question 81

If , where a ≠ 0 then what is  equal to ?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 81

NDA I - Mathematics Question Paper 2016 - Question 82

What is  equal to ?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 82

NDA I - Mathematics Question Paper 2016 - Question 83

If A is a square matrix, then what is adj(A-1) - (adj A)-1 equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 83

NDA I - Mathematics Question Paper 2016 - Question 84

What is the binary equivalent ofthe decimal number 0.3125?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 84

NDA I - Mathematics Question Paper 2016 - Question 85

Let R be a relation on the set N of natural numbers defined by 'nRM n is a factor of m'. Then which one of the following is correct?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 85

NDA I - Mathematics Question Paper 2016 - Question 86

What is  equal to ?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 86

NDA I - Mathematics Question Paper 2016 - Question 87

What is the number of natural numbers less than or equal to 1000 which are neither divisible by 10 nor 15 nor 25?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 87

Let A, B & C be the sets of numbers divisible by 10,15 & 25 respectively
No. divisible by 10 = 100 = n(A)
No. divisible by 15 = 66 = n (B)
No. divisible by 25 = 40 = n (C)
No. divisible by (10 & 15) = 33 = n(A B)
No. divisible by (15 & 25) = 13 = n (B C)
No. divisible by (25 & 10) = 20 = n (A C)
No. divisible by (10,15 & 2 5 ) = 6 = n ( A B C )
No. divisible by 10,15 and 25 = n ( A B C )
= 100 + 66 + 4 0 -3 3 -1 3 -2 0 + 6=146
Thus, no. which are neither divisible by 10 nor 15 nor
25=1000-146 = 854

NDA I - Mathematics Question Paper 2016 - Question 88

(a, 2b) is the mid-point of the line segment joining the points
(10, -6) and (k, 4). If a - 2b = 7, then what is the value of k?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 88

M = mid point of line segment PQ

Put the values of a & ab in eq (1), we get

NDA I - Mathematics Question Paper 2016 - Question 89

Consider the following statements:
1. If ABC is an equilateral triangle, then 3tan( A+B) tan C =1.
2. If ABC is a triangle in which A= 78°, B =66°, then

3. If ABC is any triangle, then

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

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 89

∵ ABC is an equilateral triangle.
∴A = B = C=60°
L.H.S. = 3 tan (A + B) tan C
= 3 tan 120° tan 60°
= 3(-√3)(√3)
=-9 ≠ 1
Hence statement (1) is incorrect.
Statement-2
ABC is a triangle such that A=78° and B = 66°
C = 180 - (78 + 66) = 180 -144 = 36°

Hence statement (2) is correct.
Statement (3)

We can see that statement (3) is not correct. Hence only 2nd statement is correct.

NDA I - Mathematics Question Paper 2016 - Question 90

if  and  then what is  equal to ?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 90

NDA I - Mathematics Question Paper 2016 - Question 91

What is the mean deviation from the mean of the numbers 10,9,21,16,24?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 91

Given numbers-10,9,21,16,24

NDA I - Mathematics Question Paper 2016 - Question 92

Three dice are thrown simultaneously. What is the probability that the sum on the three faces is at least 5?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 92

As we know that 3 dice are thrown. We want prob. of sum on three faces at least 5 i.e. some may be 5 or more. We will find prob. of sum on three faces not 5 or less, i.e. sum on faces is 3 and 4 (1,2 is not possible because of 3 dice).
No. of ways for sum on faces not 5 or more = 4
[(1, 1, 1) , ( 1, 2, 1) , ( 1, 1, 2),(2, 1, 1)]
Total out comes = 216
Prob. of not 5 or more
Prob. of sum on three faces at least 5

NDA I - Mathematics Question Paper 2016 - Question 93

Two independent events A and B have P(A)  and P(B)  What is the probability that exactly one of the two events A or B occurs?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 93

A and B are independent.

We want to find probability that exactly one of the two events^ or B occurs i.e. when^4 occurs B does not and vice-versa.
Lets take desired prob. is P.

NDA I - Mathematics Question Paper 2016 - Question 94

A coin is tossed three times. What is the probability of getting head and tail alternately?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 94

Coin is tossed three tim es i.e. total outcomes = 23 = 8 [(H, H, H), (H, H, T), (H, T, H), (H, T, T), (T, H, H),
(T, H, T), (T, T, H), (T, T, T)]
Alternate head and tail are coming two times only.
Thus prob. of getting head and tail alternately

NDA I - Mathematics Question Paper 2016 - Question 95

If the total number of observations is 20  and  then what is the variance of the distribution?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 95

Total no. of observation (n) = 20

NDA I - Mathematics Question Paper 2016 - Question 96

A card is drawn from a wel-shuffled deck of 52 cards. What is the probability that it is queen of spade?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 96

Prob. of getting queen of spade

NDA I - Mathematics Question Paper 2016 - Question 97

If two dice are thrown, then what is the probability that the sum on the two faces is greater than or equal to 4?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 97

Since two dice are thrown so number of outcomes are 36.
No. of ways when sum on two faces less than 4 = 3.
[(1, 1), ( 1, 2), (2, 1)]
Hence prob of getting sum on two faces less than 4

Thus required prob. that sum on the two faces is greater
than or equal to 4

NDA I - Mathematics Question Paper 2016 - Question 98

A certain type of missile hits the target with probability p = 0.3. What is the least number of missiles should be fired so that there is at least an 80% probability that the target is hit?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 98

Probability of hittiy the forget = 0.3
If'n' is the no. of times that the Missile is fired.
Probability of hitting at least once = 1-[1-0.3]n = 0.8
0.7n=0.2
n log 0.7 = log 0.2
⇒n=4.512
for n = 4 ; p < 0 . 8
taken = 5

NDA I - Mathematics Question Paper 2016 - Question 99

For two mutually exclusive events A and B, P(A) = 0.2 and  = 0.3. What is  equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 99

Events A and B are mutually exclusive.

NDA I - Mathematics Question Paper 2016 - Question 100

What is the probability of 5 Sundays in the month of December?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 100

In month of December 31 days i.e. (28 + 3) days.
In 28 days will get 4 Sundays.
If we get any Sunday in first 3 days of December than only we can get 5 Sundays in month.
n (5th Sunday) = 3     [4 weeks + 3 days]
n(5) = 7
Hence prob. of 5 Sundays in month of December =

NDA I - Mathematics Question Paper 2016 - Question 101

If m is the geometric mean of

then what is the value of m?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 101

Three terms are

NDA I - Mathematics Question Paper 2016 - Question 102

A point is chosen at random inside a rectangle measuring 6 inches by 5 inches. What is the probability that the randomly selected point is at least one inch from the edge of the rectangle?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 102

Probability that the randomly selected point is at least one inch from the edge of the rectangle

NDA I - Mathematics Question Paper 2016 - Question 103

The mean of the series x1,x2, . . . , xn is  If x2 is replaced by λ , th en what is the new mean?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 103

Mean of series (x1, x2, x3.....xn)

Now we will replace x2 by λ so no. of elements in series will not change.
New series will include λ and exclude x2 Hence new series sum :

NDA I - Mathematics Question Paper 2016 - Question 104

For the data
3, 5, 1, 6, 5, 9, 5, 2, 8, 6
the mean, median and mode are x, y and z respectively. Which one of the following is correct?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 104

Given data 3,5,1,6,5,9,5,2,8,6 and mean, median and mode are x, y, z respectively.
Rearranging data
1,2,3,5,5,5,6,6,8,9

Mode (z) = most frequently occuring value = 5 Hence x=y=z.

NDA I - Mathematics Question Paper 2016 - Question 105

Consider the following statements in respect of a histogram:
1. The total area of the rectangles in a histogram is equal to the total area bounded by the corresponding frequnecy polygon and the x-axis.
2. When class intervals are unequal in a frequency distribution, the area of the rectangle is proportional to the frequency.

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

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 105

Statement (1) is correct because total area of the rectangles in a histogram is equal to the total area bounded by the corresponding frequency polygon and x-axis. Statement (2) is also correct.

NDA I - Mathematics Question Paper 2016 - Question 106

A fair coin is tossed 100 times. What is the probability of getting tails an odd number of times?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 106

Let x denote number of tails. Then, X is a binomial variate with parameters:

NDA I - Mathematics Question Paper 2016 - Question 107

What is the number of ways in which 3 holiday travel tickets are to be given to 10 employees of an organization, if each employee is eligible for any one or more of the tickets?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 107

No. of ways in which 3 holiday travel tickets are to be given to 10 employees = 103 = 1000

NDA I - Mathematics Question Paper 2016 - Question 108

If one root of the equation (1 - m) x2 +1 x +1 = 0 is double the other and 1 is real, then what is the greatest value of m?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 108

Given equation is

Roots are α, β.
∵ One root is double the other.
β = 2α
Sum of roots = α + β

NDA I - Mathematics Question Paper 2016 - Question 109

What is the number of four-digit decimal numbers (<1) in which no digit is repeated?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 109

Let the given 4 digit decimal number is
Places after decimal can be filled in the following ways:

Total number of ways = 7 x 8 x 9 x 9 = 4536

NDA I - Mathematics Question Paper 2016 - Question 110

What is a vector of unit length orthogonal to both the vectors

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 110

Vector of unit length orthogonal to both the vectors

NDA I - Mathematics Question Paper 2016 - Question 111

If  are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then which one of the following is correct?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 111

Position vectors o f vertices A, B and C are

∵ triangle is equilateral.
∴ Centroid and orthocenter will coincide. Centroid = orthocenter position vector

∵ given in question orthocenter is at origin.
Hence

NDA I - Mathematics Question Paper 2016 - Question 112

What is the area of the parallelogram having diagonals

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 112

NDA I - Mathematics Question Paper 2016 - Question 113

Consider the following in respect of the matrix

Q. Which of the above is/are correct?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 113

NDA I - Mathematics Question Paper 2016 - Question 114

Which of the following determinants have value ‘zero’?

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

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 114

two columns are same so value of determinant is zero.

∵ diagonal is zero so value of determinant is zero.

NDA I - Mathematics Question Paper 2016 - Question 115

What is the acute angle between the lines represented by the equations

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 115

NDA I - Mathematics Question Paper 2016 - Question 116

The system of linear equations kx + y + z =1, x + ky + z = 1 and x + y + kz = 1 has a unique solution under which one of the following conditions?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 116

Linear equations

Linear equantion will have unique solution when A-1x exist:

NDA I - Mathematics Question Paper 2016 - Question 117

What is the number of different messages that can be represented by three 0’s and two 1 ’s?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 117

Number of different messages that can be represented by three 0's and two l's is 10.
Option (a) is correct.

NDA I - Mathematics Question Paper 2016 - Question 118

If loga(ab) = x, then what is logb(ab) equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 118

NDA I - Mathematics Question Paper 2016 - Question 119

then what is   equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 119

NDA I - Mathematics Question Paper 2016 - Question 120

Suppose ω1, and ω2 are two distinct cube roots of unity different from 1. Then what equal to?

Detailed Solution for NDA I - Mathematics Question Paper 2016 - Question 120

Cube root of unity are
w1 and w2 are two distinct cube roots of unity different from 1.

## NDA (National Defence Academy) Past Year Papers

2 docs|24 tests
Information about NDA I - Mathematics Question Paper 2016 Page
In this test you can find the Exam questions for NDA I - Mathematics Question Paper 2016 solved & explained in the simplest way possible. Besides giving Questions and answers for NDA I - Mathematics Question Paper 2016, EduRev gives you an ample number of Online tests for practice

2 docs|24 tests