NDA I - Mathematics Question Paper 2016
120 Questions MCQ Test NDA (National Defence Academy) Past Year Papers | NDA I - Mathematics Question Paper 2016
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?
P and Q are points on complex plane. Angle between OP and OQ is
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?
x and y are positive numbers.
x ≤ y2
x < x2 
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.
If x2+ px + 4 for all real values of x, then which one of thefollowing is correct?
(a) x2-px + 4> 0 
real values of x.
If b2 - 4ac < 0
⇒p2 - 4(1)(4)<0
⇒p2 < 16 ⇒|p| < 4
If z=x + iy= 
, where i = √-1, then what is the fundamental amplitude of 
z = x + iy
If f(x1)- f(x2) is 
for x1, x2 ∈ (-1,1), then what is f(x) equal to?
What is the range of the function 
2 ,where X ∈R?
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?
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
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?
Distance from the centre to the point of line which touches circle is OM = radius
Consider the function f(θ)= 4(sin2 θ+ cos4 θ)
Q. What is the maximum value of the function f(θ)?
f(θ) = 4 (sin2θ + cos4 θ)
= 4 (sin2 θ + 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
Consider the function f(θ)= 4(sin2 θ+ cos4 θ)
Q. What is the minimum value of the function f(θ)?
f(θ) = 4 (sin2θ + cos4 θ)
= 4 (sin2 θ + 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.
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?
f(θ) = 4 (sin2θ + cos4 θ)
= 4 (sin2 θ + 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.
Consider the curves
Hence f(x ) and g(x) intersects at ( -1 , -2 ) and (2 ,3 ).
What is the area bounded by the curves
Consider the function 
How many solutions does the function f(x) = 1 have?
This is a cubic equation.
If we put y = 
then ( 3y + 1) = 0 is a factor o f cubic equation.
Consider the function
How many solutions does the function f(x) = -1 have?
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.
Consider the functions
f(x) = xg(x) and g(x = 
Where [•] is the greatest integer function.
What is 
equal to?
As g(x) is a gretest integer function so value of g(x) in integral limit will be
What is 
equal to ?
The value of g(x) in value 
will be 2 and in range
form(l)
Consider the function f ( x ) = | x - 1 |+ x2 , w here x ∈R .
Which one of the following statements is correct?
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
Consider the function f ( x ) = | x - 1 |+ x2 , w here x ∈R .
Which one of the following statements is correct?
As we know,
f(x) is in quadratic form (parabola). Hence f(x) is decreasing in 
and increasing 
Which one of the following statements is correct?
f(x) has local minimum at one point only in (-∞ ,∞ ).
What is the area of the region bounded by x-axis, the curve y = f(x) and the two ordinates 
and x = 1 ?
Hence area required for given region is
What is the area of the region bounded by x-axis, the curve y = f(x) and the two ordinates x = 1 and 
Area required for given region is
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?
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.
Given that an =
What is an-1 - an-4 equal to ?
∵ given sequence an also AP with no difference.
Thus an-1 - an-4 = 0
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.
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.
Consider the equation x + |y| = 2y.
What is the derivative of y as a function of x with respect to x for x < 0?
Option (d) is correct.
Consider the lines y = 3x, y = 6x and y = 9
What is the area of the triangle formed by these lines?
OAB is triangle
Consider the lines y = 3x, y = 6x and y = 9
The centroid of the triangle is at which one of the following points?
Coordinates o f O, A, B are (0, 0) 
respectively.
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)?
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
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) ?
Local Maxima is [x = 1 ]
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 ?
again integrating equation (1)
Rearranging equation (3) again, we get
Consider the curves y = | x — 1 | and |x| = 2
Q. What is/are the point(s) of intersection of the curves ?
and x=2
Hence curves intersect at (-2,3) and (2,1).
Consider the curves y = | x — 1 | and |x| = 2.
Q. What is the area of the region bounded by the curves and x-axis?
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
Consider the function
Q. What is the value of f (0)?
Consider the function
Q. What is the value of p for which f"(0)=0?
Consider a tiangle ABC in which
Q. What is the value of sin
Consider a tiangle ABC in which
Q. What is the value of 
As we know that
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?
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?
sin( α + β) sec α see β
= tan α + tan β
= -b
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?
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
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?
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
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?
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
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?
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
Consider the equation
k sinx + cos 2x = 2k - 7
If the equation possesses solution, then what is the minimum value of k?
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
Consider the equation
k sinx + cos 2x = 2k - 7
If the equation possesses solution, then what is the maximum value of k?
For maximum value of K
sin x = 1
Consider the function f (x) 
where [•] denotes the greatest integer function.
Q. What is 
equal to?
Consider the function f (x) where [•] denotes the greatest integer function.
Q. What is 
equal to?
Let z1, z2 and z3 be non-zero com plex num bers satisfying 
, where i =√- 1 .
Q. What is Z1 + z2 + z3 equal to?
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
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?
Hence z1z2 + z2z3 + z3Z1 = 0 is purely real.
Hence both statements are correct.
Given that logx y, logz x, logy z are in GP, xyz = 64 and x3, y3, z3 are inA.P.
Q. Which one of the following is correct ?
x,y and z are
Thus x, y z are in A.P. and G.P. both.
Given that logx y, logz x, logy z are in GP, xyz = 64 and x3, y3, z3 are inA.P.
Q. Which one of the following is correct?
xy, yz and zx are
Similarly xy,yz, zx are also in A.P. and G.P. both.
Let z be a complex number satisfying
Q. What is |z| equal to?
⇒|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
Let z be a complex number satisfying
Q. What is 
equal to?
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?
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.
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 ?
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.
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?
Given quadratic equation,
x2 + bx + c = 0 and roots are α and β.
where a < p. Hence roots of given quadratic equation are
Consider the following:
1. α + β + αβ > 0
2. α2 β + β2α > 0
Which of the above is/are correct?
Sum of roots = α + β = -b
Multiplication of roots = αβ = c
Hence
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 ?
Suppose Mid point of AC and BD is M (a, b).
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?
Point of intersection (a, b) is 
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?
Area of parallelogram=2 area of Δ ADB
Let f : R → R be a function such that
f(x ) = x3 + x2 f '(1) + xf "(2)+ f "'(3)
for x ∈ R
What is f(1) equal to?
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?
f'(l)=-5
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?
f "'(10)=6
Consider the following:
1. f(2 ) = f(1) - f(0)
2. f "(2) - 2f '(1) = 12
Q. Which of the above is/are correct?
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?
Hence direction ratios ofthe line of intersection of given plane <2,-5,-3 >
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?
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?
Plane P touches the sphere x2 + y2 + z2 = r2 then r=Distane between centre of sphere (0,0,0) to plance P.
Consider th e function f (x ) = | x2 - 5x + 6 |
Q. What is f '(4) equal to?
Consider th e function f (x ) = | x2 - 5x + 6 |
Q. What is f"(2.5) equal to?
f"(2.5) =-2
Let f(x) be the greatest integer function and g(x) be the modulus function.
Q. What is 
equal to?
f(x) → greatest integer function
f(x)=[x]
g(x) → modulus fuction
g(x)=| x |
Let f(x) be the greatest integer function and g(x) be the modulus function.
Q. What is 
equal to?
f(x) → greatest integer function
f(x)=[x]
g(x) → modulus fuction
g(x)=| x |
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?
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)
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?
The two intercepts are : -2g & -2f
∵ from eq (l) &( 2) we get;
is sum of squares of intercepts
Let 
be two unit vectors and 0 be the angle between them.
Q. What is cos 
equal to?
Let be two unit vectors and 0 be the angle between them.
What is sin 
equal to?
Consider the following statements:
1. There exists 
2. sin-1 
Q. Which of the above statements is/are correct ?
Hence, statement (2) is correct.
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 ?
Statement-1
Statement (1) is wrong.
Statement 2,
Only when x = y
Here x ≠ y .
Statement (2) is also wrong.
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?
Given:
Solution of differential equation is
y - cx + c2 - 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.
What is
equal to, where [•] is the greatest integer function?
If 
then what is 
equal to?
If 
, where a ≠ 0 then what is 
equal to ?
What is 
equal to ?
If A is a square matrix, then what is adj(A-1) - (adj A)-1 equal to?
What is the binary equivalent ofthe decimal number 0.3125?
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?
What is 
equal to ?
What is the number of natural numbers less than or equal to 1000 which are neither divisible by 10 nor 15 nor 25?
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
(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?
M = mid point of line segment PQ
Put the values of a & ab in eq (1), we get
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?
∵ 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.
if 
and 
then what is 
equal to ?
What is the mean deviation from the mean of the numbers 10,9,21,16,24?
Given numbers-10,9,21,16,24
Three dice are thrown simultaneously. What is the probability that the sum on the three faces is at least 5?
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
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?
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.
A coin is tossed three times. What is the probability of getting head and tail alternately?
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 
If the total number of observations is 20 
and 
then what is the variance of the distribution?
Total no. of observation (n) = 20
A card is drawn from a wel-shuffled deck of 52 cards. What is the probability that it is queen of spade?
Prob. of getting queen of spade 
If two dice are thrown, then what is the probability that the sum on the two faces is greater than or equal to 4?
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 
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?
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
For two mutually exclusive events A and B, P(A) = 0.2 and 
= 0.3. What is 
equal to?
Events A and B are mutually exclusive.
What is the probability of 5 Sundays in the month of December?
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 = 
If m is the geometric mean of
then what is the value of m?
Three terms are
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?
Probability that the randomly selected point is at least one inch from the edge of the rectangle
The mean of the series x1,x2, . . . , xn is 
If x2 is replaced by λ , th en what is the new mean?
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 :
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?
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.
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?
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.
A fair coin is tossed 100 times. What is the probability of getting tails an odd number of times?
Let x denote number of tails. Then, X is a binomial variate with parameters:
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?
No. of ways in which 3 holiday travel tickets are to be given to 10 employees = 103 = 1000
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?
Given equation is
Roots are α, β.
∵ One root is double the other.
β = 2α
Sum of roots = α + β
What is the number of four-digit decimal numbers (<1) in which no digit is repeated?
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
What is a vector of unit length orthogonal to both the vectors
Vector of unit length orthogonal to both the vectors
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?
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 
What is the area of the parallelogram having diagonals 
Consider the following in respect of the matrix
Q. Which of the above is/are correct?
Which of the following determinants have value ‘zero’?
Q. Select the correct answer using the code given below,
two columns are same so value of determinant is zero.
∵ diagonal is zero so value of determinant is zero.
What is the acute angle between the lines represented by the equations 
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?
Linear equations
Linear equantion will have unique solution when A-1x exist:
What is the number of different messages that can be represented by three 0’s and two 1 ’s?
Number of different messages that can be represented by three 0's and two l's is 10.
Option (a) is correct.
If loga(ab) = x, then what is logb(ab) equal to?
then what is 
equal to?
Suppose ω1, and ω2 are two distinct cube roots of unity different from 1. Then what 
equal to?
Cube root of unity are 
w1 and w2 are two distinct cube roots of unity different from 1.
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