NDA I - Mathematics Question Paper 2016


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


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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?

Solution:



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


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?

Solution:

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.

QUESTION: 3

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

Solution:

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

QUESTION: 4

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

Solution:

z = x + iy




 

QUESTION: 5

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

Solution:



QUESTION: 6

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

Solution:


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?

Solution:

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
 

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?

Solution:

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

QUESTION: 9

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

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

Solution:

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

QUESTION: 10

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

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

Solution:

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.

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?

Solution:

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.

QUESTION: 12

Consider the curves

Solution:


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

QUESTION: 13

What is the area bounded by the curves

Solution:




QUESTION: 14

Consider the function

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

Solution:




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

QUESTION: 15

Consider the function 

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

Solution:


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.

QUESTION: 16

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

What is  equal to?

Solution:


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

QUESTION: 17

What is  equal to ?

Solution:


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

form(l)


 

QUESTION: 18

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

Which one of the following statements is correct?

Solution:

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

QUESTION: 19

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

Which one of the following statements is correct?

Solution:

As we know,

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

QUESTION: 20

Which one of the following statements is correct?

Solution:

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

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 ?

Solution:


Hence area required for given region is

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 

Solution:

Area required for given region is

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?

Solution:


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.

QUESTION: 24

Given that an = 

What is an-1 - an-4​ equal to ?

Solution:

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

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.

Solution:

 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.

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?

Solution:


Option (d) is correct.

QUESTION: 27

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

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

Solution:

OAB is triangle

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?

Solution:

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


 

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)?

Solution:

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

 

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) ?

Solution:

Local Maxima is [x = 1 ]

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 ?

Solution:


again integrating equation (1) 

Rearranging equation (3) again, we get 

QUESTION: 32

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

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

Solution:


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

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?

Solution:

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

QUESTION: 34

Consider the function

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

Solution:




QUESTION: 35

Consider the function

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

Solution:


QUESTION: 36

Consider a tiangle ABC in which

Q. What is the value of sin

Solution:




QUESTION: 37

Consider a tiangle ABC in which

Q. What is the value of 

Solution:

As we know that


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?

Solution:

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?

Solution:

sin( α + β) sec α see β

= tan α + tan β 
= -b

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?

Solution:

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

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?

Solution:

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

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?

Solution:

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

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?

Solution:

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

QUESTION: 44

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

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

Solution:

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

QUESTION: 45

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

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

Solution:

For maximum value of K
sin x = 1

QUESTION: 46

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

Q. What is  equal to?

Solution:

QUESTION: 47

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

Q. What is equal to?

Solution:

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? 

Solution:

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

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?

Solution:



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

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

Solution:



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

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

Solution:

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

QUESTION: 52

Let z be a complex number satisfying

Q. What is |z| equal to?

Solution:


⇒|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

QUESTION: 53

Let z be a complex number satisfying

Q. What is  equal to?

Solution:

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?

Solution:

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.

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 ?

Solution:

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.
 

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?

Solution:

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

QUESTION: 57

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

Which of the above is/are correct?

Solution:

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

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 ?

Solution:

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


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?

Solution:

Point of intersection (a, b) is 

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?

Solution:

Area of parallelogram=2 area of Δ ADB

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?

Solution:


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?

Solution:

f'(l)=-5

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?

Solution:

f "'(10)=6

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?

Solution:

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?

Solution:



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

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?

Solution:

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?

Solution:

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

QUESTION: 68

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

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

Solution:


QUESTION: 69

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

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

Solution:

f"(2.5) =-2

QUESTION: 70

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

Q. What is  equal to?

Solution:

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

QUESTION: 71

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

Q. What is equal to?

Solution:

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

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?

Solution:

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)

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?

Solution:

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

QUESTION: 74

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

Q. What is cos  equal to?

Solution:



QUESTION: 75

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

What is sin equal to?

Solution:

QUESTION: 76

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

Solution:



Hence, statement (2) is correct. 

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 ?

Solution:

Statement-1





Statement (1) is wrong.
Statement 2,

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

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?

Solution:

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.

QUESTION: 79

What is

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

Solution:

QUESTION: 80

If  then what is  equal to?  

Solution:


QUESTION: 81

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

Solution:

QUESTION: 82

What is  equal to ?

Solution:

QUESTION: 83

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

Solution:


QUESTION: 84

What is the binary equivalent ofthe decimal number 0.3125?

Solution:

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?

Solution:


QUESTION: 86

What is  equal to ?

Solution:


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?

Solution:

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

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?

Solution:

M = mid point of line segment PQ

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

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?

Solution:

∵ 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.

QUESTION: 90

if  and  then what is  equal to ?

Solution:

QUESTION: 91

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

Solution:

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


QUESTION: 92

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

Solution:

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

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?

Solution:

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.

QUESTION: 94

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

Solution:

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 

QUESTION: 95

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

Solution:

Total no. of observation (n) = 20 

QUESTION: 96

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

Solution:

Prob. of getting queen of spade 

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?

Solution:

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 

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?

Solution:

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

QUESTION: 99

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

Solution:

Events A and B are mutually exclusive.

QUESTION: 100

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

Solution:

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 = 

QUESTION: 101

If m is the geometric mean of

then what is the value of m?

Solution:

Three terms are


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?

Solution:


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

QUESTION: 103

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

Solution:

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 :

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?

Solution:

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.

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?

Solution:

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.

QUESTION: 106

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

Solution:

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

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?

Solution:

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

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?

Solution:

Given equation is

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

QUESTION: 109

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

Solution:

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 

QUESTION: 110

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

Solution:


Vector of unit length orthogonal to both the vectors 

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? 

Solution:

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 

QUESTION: 112

What is the area of the parallelogram having diagonals 

Solution:


QUESTION: 113

Consider the following in respect of the matrix

Q. Which of the above is/are correct?

Solution:


QUESTION: 114

Which of the following determinants have value ‘zero’?

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

Solution:


two columns are same so value of determinant is zero. 

∵ diagonal is zero so value of determinant is zero. 

QUESTION: 115

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

Solution:

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?

Solution:

Linear equations

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

QUESTION: 117

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

Solution:

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

QUESTION: 118

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

Solution:

QUESTION: 119

  then what is   equal to?

Solution:


QUESTION: 120

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

Solution:

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