# Engineering-Mathematics | Subject-Wise Solved Questions

## Engineering Mathematics | GATE Subject Wise

 Question 1
The number of distinct eigenvalues of the matrix is equal to __________.
 A Fill in the Blank Type Question
Question 1 Explanation:
 Question 2
Let Z be an exponential random variable with mean That is, the cumulative distribution function of Z is given by

Then Pr(Z > 2 |Z > 1), rounded off to two decimal places, is equal to __________.

 A Fill in the Blank Type Question
Question 2 Explanation:
 Question 3
The value of the contour integral evaluated over the unit circle |z| = 1 is _____________.
 A Fill in the Blank Type Question
Question 3 Explanation:
 Question 4
Which one of the following functions is analytic over the entire complex plane?
 A E1/z B In(z) C D Cos (z)
Question 4 Explanation:
(a) e1/z is NOT analytic at z = 0
(b) ℓnz is NOT analytic in Domain D = {z / x ≤ 0 , y = 0}
(c) 1/(1-z) is not analytic at z=1
(d) cos (z) =1 - z2 + z4/4! - z6/6! + z8/8!
∴ cosz is analytic every where in the complex plane z
 Question 5
The families of curves represented by the solution of the equation

for n = - 1 and n = + 1, respectively, are

 A Hyperbolas and Circles B Circles and Hyperbolas C Parabolas and Circles D Hyperbolas and Parabolas
Question 5 Explanation:
 Question 6
Consider the homogeneous ordinary differential equation with y(x) as a general solution. Given that y(1) = 1 and y(2) = 14 the value of y(1.5), rounded off to two decimal places, is _______________.
 A Fill in the Blank Type Question
Question 6 Explanation:
 Question 7
Consider a differentiable function f(x) on the set of real numbers such that f(-1) = 0 and |f’(x)| ≤ 2. Given these conditions, which one of the following inequalities is necessarily true for all x [-2, 2]?
 A B C D
Question 7 Explanation:
 Question 8
Consider the 5 × 5 matrix It is given that A has only one real Eigen value. Then the real Eigen value of A is
 A -2.5 B 0 C 15 D 25
Question 8 Explanation:
If sum of all the rows or columns are same, then that
sum = one eigen value of that matrix

Sum of all rows
(15-x) * |Matrix| = 0
15 is a factor .

Another Approach
 Question 9
The rank of the matrix M = is
 A 0 B 1 C 2 D 3
Question 9 Explanation:
|M |=
= 5(0-12) – 10(6-6) +10(6-0)= -60-0+60=0
But a 2×2 minor, =0-10=-10≠0
Rank = 2
 Question 10

Consider the following statement about the linear dependence of the real valued functions y1= 1, y2= x and y 3 =x2, over the field of real numbers.
I. y1, y2 and y3 are linearly independent on -1 x 0
II. y1 , y2 and y3 are linearly dependent on 0 x 1
III. y1 , y2 and y3 are linearly independent on 0 x 1
IV. y1, y2 and y3 are linearly dependent on -1 x 0
Which one among the following is correct?

 A Which one among the following is correct? B Both I and III are true C Both II and IV are true D Both III and IV are true
Question 10 Explanation:
Given
y1= 1, y 2= x, y 3= x2
Consider

= =2 ≠0

y1, y2, y3 are linearly independent x
 Question 11
Three fair cubical dice are thrown simultaneously. The probability that all three dice have the same number of dots on the faces showing up is (up to third decimal place) _____.
 A Fill in the Blank Type Question
Question 11 Explanation:
From given data, n(s) = 6x6x6 = 216
the number of outcomes =
{ (1,1,1), (2,2,2), (3,3,3), (4,4,4), (5,5,5), (6,6,6) }
n(p) =6
therefore, probability = n(p) /n(s) = 6/216 =0.0277 =~0.028
 Question 12
Let f(x) = ex+x2 for real x. From among the following, choose the Taylor series approximation of f(x) around x = 0, which included all powers of x less than or equal to 3.
 A B C D
Question 12 Explanation:
 Question 13
A three dimensional region R of finite volume is described by Where x, y, z are real. The volume of R (up to two decimal places) is
 A Fill in the Blank Type Question
Question 13 Explanation:
 Question 14
Let I where x, y, z are real, and let C be the straight line segment from point A : (0, 2,1) to point B : (4,1, -1). The value of I is
 A Fill in the Blank Type Question
Question 14 Explanation:
 Question 15
Let where and are constants. If at and then the relation between and is
 A B C D
Question 15 Explanation:
 Question 16
Let M be a real matrix. Consider the following statements:
S1: has 4 linearly independent eigenvectors.
S2: has 4 distinct eigenvalues.
S3: is non-singular (invertible).
Which one among the following is TRUE?
 A S1 implies S2 B S1 implies S3 C S2 implies S1 D S3 implies S2
Question 16 Explanation:
The Correct Answer Among All the Options is C
Eigen vectors corresponding to distinct eigen values are linearly independent.
So, “S2 implies S1”.
->We know that if a matrix A has'n' distinct eigen values then A has 'n' linearly independent eigenvectors
. Therefore"S2 implies S1"
 Question 17
Let the input be U and the output be Y of a system, and the other parameters are real constants. Identify which among the following systems is not a linear system:
 A (with initial rest conditions) B C D
Question 17 Explanation:
Presence of a constant term introduces non-linearity. Thus,  is a non-linear system.
Linear system must result zero output for zero input.
It must satisfy super position and homogenity principle
Which option c does not satisfy so Option (C) is Correct Answer.
 Question 18
Taylor series expansion of around has the form

The coefficient (correct to two decimal places) is equal to ____________.

 A Fill in the Blank Type Question
Question 18 Explanation:
 Question 19
Consider matrix and vector The number of distinct real values of for which the equation has infinitely many solution is ___________.
 A Fill in the Blank Type Question
Question 19 Explanation:
 Question 20
Let and Assume that and are independent variables. At the value (correct to two decimal places) of is ___________.
 A Fill in the Blank Type Question
Question 20 Explanation:
 Question 21
The position of a particle is described by the differential equation:

The initial conditions are and The position (accurate to two decimal places) of the particle at is _____________.

 A Fill in the Blank Type Question
Question 21 Explanation:

Given condition,

This can be solved easily in laplace domain,

By taking inverse Laplace transform we get y(t),

Now its value at

 Question 22
The contour C given below is on the complex plane where

The value of the integral is __________.

 A Fill in the Blank Type Question
Question 22 Explanation:
 Question 23
A curve passes through the point and satisfies the differential equation The equation that describes the curve is
 A B C D
Question 23 Explanation:
Given Differential equation,

We need to use suitable substitution here,
Put,

After simplification we obtain the following relation,

Given that the curve passes through points,
, we can obtain the value of constant C.

So,
 Question 24
The smaller angle (in degrees) between the planes x + y + z =1 and 2x – y + 2z = 0 is__________
 A Fill in the Blank Type Question
Question 24 Explanation:
 Question 25
The residues of a function f(z) are
 A B C D
Question 25 Explanation:
 Question 26
The values of the integrals and are
 A same and equal to 0.5 B same and equal to -0.5 C 0.5 and – 0.5, respectively D - 0.5 and 0.5, respectively
Question 26 Explanation:

 Question 27
An integral I over a counter clock wise circle C is given by If C is defined as |z|=3, then the value of I is
 A B C D
Question 27 Explanation:
 Question 28
If the vector function is irrotational, then the values of the constants k1, k2 and k3 respectively, are
 A 0.3, –2.5, 0.5 B 0.0, 3.0, 2.0 C 0.3, 0.33, 0.5 D 4.0, 3.0, 2.0
Question 28 Explanation:
 Question 29
Passengers try repeatedly to get a seat reservation in any train running between two stations until they are successful. If there is 40% chance of getting reservation in any attempt by a passenger, then the average number of attempts that passengers need to make to get a seat reserved is _____.
 A Fill in the Blank Type Question
Question 29 Explanation:
 Question 30
The minimum value of the function occurs at x = n the interval occurs at x = ________.
 A Fill in the Blank Type Question
Question 30 Explanation:

at x =1, f( x) has local minimum.

has local maximum
For x = 1, local minimum value
Finding f(- 100) = -333433.33
f (100) = 333233.33
( x =100, -100 are end points of interval)
Minimum occurs at x= - 100
 Question 31
Let M4 = I, (where I denotes the identity matrix) and M ≠ I, M2 ≠I and M3 ≠I. Then, for any natural number k, M-1  equals:
 A M4k+1 B M4k + 2 C M4k+3 D M4k
Question 31 Explanation:
 Question 32
The second moment of a Poisson-distributed random variable is 2. The mean of the random variable is _________.
 A Fill in the Blank Type Question
Question 32 Explanation:
We know that if X is parameter of poison’s distribution

Therefore Mean of random variables = First moment = 1
 Question 33
Given the following statements about a function f: R —>R, select the right option.
P: If f(x) is continuous at x = xo, then it is also differentiable at x = xo.
Q: If f(x) is continuous at x = xo, then it may not be differentiable at x = xo.
R: If f(x) is differentiable at x = xo, then it is also continuous at x = xo
 A P is true, Q is false; R is false B P is false, Q is true, R is true C P is false, Q is true; R is false D P is true, Q is false, R is true
Question 33 Explanation:
Since continuous function may not be differentiable. But differentiable function is always continuous. i.e.,
Every differentiable function is continuous but converse need not be true
 Question 34
Which one of the following is a property of the solutions to the Laplace equation:
 A The solutions have neither maxima nor minima anywhere except at the boundaries. B The solutions are not separable in the coordinates. C The solutions are not continuous. D The solutions are not dependent on the boundary conditions.
Question 34 Explanation:
 Question 35
Consider the plot of f(x) versus x as shown below. Suppose . Which one of the following is a graph of F(x)?
 A B C D
Question 35 Explanation:
->Integration of an odd function is even in this logic A
-> B cannot be the answer as they are odd functions.

->C and D are even functions but the integration of a linear curve has to be parabolic in nature and it cannot be a constant function. Based on this Option C is correct.
 Question 36
The integral where D denotes the disc: evaluates to _______.
 A Fill in the Blank Type Question
Question 36 Explanation:
Method-I : -

Method – II:-

(Changing into polar coordinates by)
Another Approach
 Question 37
In the following integral, the contour C enclose the points 2πj and -2πj

The value of the integral is ______
 A Fill in the Blank Type Question
Question 37 Explanation:
 Question 38
A function is defined in the closed interval [–1, 1]. The value of x, in the open interval (–1, 1) for which the mean value theorem is satisfied, is
 A –1/2 B –1/3 C 1/3 D 1/2
Question 38 Explanation:
 Question 39

Suppose A and B are two independent events with probabilities P(A) ≠ 0 and P(B) ≠ 0.

Let and be their complements.

Which one of the following statements is FALSE ?

 A B C D
Question 39 Explanation:
 Question 40
Let be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is NOT TRUE?
 A The residue of at z = 1 is 1/2 B C D (complex conjugate of z) is an analytical function
Question 40 Explanation:

 Question 41
The value of p such that the vector is an eigenvector of the matrix is ________.
 A Fill in the Blank Type Question
Question 41 Explanation:

 Question 42
The solution of the differential equation with y(0 = y’(0) = 1 is
 A (2-t)et B (1+2t)e-t C (2+t)e-t D (1-2t)et
Question 42 Explanation:
Differential equation is s2 + 2s + 1 = 0
roots are equal s1 = s2 = - 1
So, y = c1e-t + c2t e-t
y(0) = C1 = 1
yï = - C1e-t – C2t e-t + C2e-t
yï(0) = - C1 + C2 = 1
C2 = 2
So solution is y = e-t + 2te-t
y = (1 + 2t) e-t.
 Question 43
Which one of the following graphs describes the function
 A B C D
Question 43 Explanation:

Putting f’(x) = 0, we get x = 0, or x = 1

At x = 0, f"(x) = 1 (so we have a minimum)
At x = 1,
(so we have a maximum), curve B. shows a single local minimum = 0 and a single local maximum at x = 1.

OR

f(1) = e-x (x2+x+1)
f(0) = 1
f(0.5) = 1.067
For positive values of x, function never goes negative
 Question 44
The maximum area (in square units) of a rectangle whose vertices lie on the ellipse is _______.
 A Fill in the Blank Type Question
Question 44 Explanation:
 Question 45
The value of x for which the matrix A =has zero as an eigenvalue is_____.
 A Fill in the Blank Type Question
Question 45 Explanation:
 Question 46
Consider the complex valued function where z is a complex variable. The value of b for which the function f(z) is analytic is
 A Fill in the Blank Type Question
Question 46 Explanation:
f(z) = 2z3 + b1|z|3
Given that f(z) is analytic.
which is possible only when b = 0
since |z|3 is differentiable at the origin but not analytic.
2z3 is analytic everywhere
∴ f(z) = 2z3 + b|z|3 is analytic
only when b = 0

Lets take an Example

 Question 47
As x varies from —1 to +3, which one of the following describes the behavior of the function f (x) = x3-3x2 + 1?
 A f(x) increase monotonically B f(x) increases, then decreases and increases again C f(x) decreases, then increases and decreases again D f(x) increases and then decreases
Question 47 Explanation:
Substitute X vales from -1 to 3
f (- 1) = − 3,
f (0) = 1
f (1) =− 1
f (2) =− 3
f (3) = 1
Correct Option is B

We can plot for various valve of x

f(x) increases, decreases and again increases.
 Question 48
How many distinct values of x satisfy the equation sin(x) = x/2, where x is in radians ?
 A 1 B 2 C 3 D 4 or more
Question 48 Explanation:

 Question 49
Consider the time-varying vector in Cartesian coordinates,
Where is a constant. When the vector magnitude |I| is at its minimum value, the angle 0 that I make with the x axis (in degrees, such that) is
 A Fill in the Blank Type Question
Question 49 Explanation:
,
The minimum magnitude will be 5
At '5' magnitude angle is 90 °.
 Question 50
Suppose C is the closed curve defined as the circle with C oriented anti-clockwise. The value of over the curve C equals
 A Fill in the Blank Type Question
Question 50 Explanation:
Using Green's Theorem
 Question 51
The matrix has det A. = 100 and trace A=14
The value of is
 A Fill in the Blank Type Question
Question 51 Explanation:
 Question 52
Consider a square matrix Where x is unknown. If the eigenvalues of the matrix A are and the x is equal to
 A B C D
Question 52 Explanation:
Given
Product of eigen values = Det of A
 Question 53
For the residue of the pole at z = 0 is _____.
 A Fill in the Blank Type Question
Question 53 Explanation:

Residue = coefficient of =1
 Question 54
The probability of getting a “head” in a single toss of a biased coin is 0.3. The coin is tossed repeatedly till a “head” is obtained. If the tosses are independent, then the probability of getting “head” for the first time in the fifth toss is. _______
 A Fill in the Blank Type Question
Question 54 Explanation:
P(Tail) = 0.7
Required probability = TTTTH
= (0.7)(0.7)(0.7)(0.7)(0.3)
=0.07203
 Question 55
The integral is equal to ______.
 A Fill in the Blank Type Question
Question 55 Explanation:
 Question 56
Consider the first order initial value problem

with exact solution y(x) = x2 + ex For x = 0.1, the percentage difference between the exact solution and the solution obtained using a single iteration of the second-order Runge-Kutta method with step-size h = 0.1 is ____
 A Fill in the Blank Type Question
Question 56 Explanation:
 Question 57
The solution of the initial value problem given below is
With and
 A B C D
Question 57 Explanation:
 Question 58
For A = , the determinant of AT A–1 is
 A sec2 x B cos 4x C 1 D 0
Question 58 Explanation:
 Question 59
The contour on the x-y plane, where the partial derivative of x2 + y2 with respect to y is equal to the partial derivative of 6y + 4x with respect to x, is
 A y = 2 B x = 2 C x + y = 4 D x – y = 0
Question 59 Explanation:
Given
Partial derivative of with respect to y =0+2Y
= partial derivative of with respect to x=0+4
So,
 Question 60
If C is a circle of radius r with centre z0, in the complex z-plane and if n is a non-zero integer, then
 A 2pnj B 0 C D 2pn
Question 60 Explanation:
 Question 61
Consider the function g(g) = e–t sin (2πt) u (t) where u (t) is the unit step function. The area under g(t) is______
 A Fill in the Blank Type Question
Question 61 Explanation:
 Question 62
The value of is_________.
 A Fill in the Blank Type Question
Question 62 Explanation:
Let

Now multiply equ(i) with 1/2

Now subtrac eq(ii) from (i)
 Question 63
The Newton-Raphson method used to solve the equation f(x) = x3 – 5x2 + 6x – 8 = 0. Taking the initial guess as x = 5, is________.
 A Fill in the Blank Type Question
Question 63 Explanation:

By Newton-Raphson method,
 Question 64
A fair die with faces {1, 2, 3, 4, 5, 6} is thrown repeatedly till ‘3’ is observed for the first time. Let X denote the number of times the die is thrown. The expected value of X is _____.
 A Fill in the Blank Type Question
Question 64 Explanation:
Given,
Probability of getting
Probability of not getting
Now, the random variable X  does represent number of throws required for getting 3. So,

Let

Equation (1) – (2),

Hence, the expected value of X is
 Question 65
Consider the differential equation
.
Given x(0) = 20 and x(1) = 10/e, where e = 2.718,
the value of x(2) is ________
 A Fill in the Blank Type Question
Question 65 Explanation:
Given,

This is a homogeneous equation. So, particular solution is zero. We get auxiliary equation as

Solving equations (1) and (2),

Taking t = 2
 Question 66
A vector field D = exists inside a cylindrical region enclosed by the surfaces p = 1, z = 0 and z = 5. Let S be the surface bounding this cylindrical region. The surface integral of this field on
S is _____.
 A Fill in the Blank Type Question
Question 66 Explanation:
In the given vector field,

By using divergence theorem
 Question 67
Ram and Ramesh appeared in an interview for two vacancies in the same department. The probability of Ram’s selection is 1/6 and that of Ramesh is 1/8. What is the probability that only one of them will be selected?
 A 47/48 B 1/4 C 13/48 D 35/48
Question 67 Explanation:
P(Ram) = 1/6, P(Ramesh = 1/8)

Probability that only one of them will be selected ,
is given by
P(only one) = P(Ram) × P(not Ramesh) + P(not Ram) × P(Ramesh)
= 1/6 × 7/8 + 5/6 × 1/8 = 12/48 = 1/4
 Question 68
The general solution of the differential equation is
 A tan y – cot x = c B tan x – cot y = c C tan y + cot x = c D tan x + cot y = c
Question 68 Explanation:
 Question 69
Let f(z) = . If f(z ­1) = f(z2) for all z1 z2, a = 2, b = 4 and c = 5,
then d should be equal to _______________
 A 150 B 10 C 50 D 25
Question 69 Explanation:

Given
 Question 70
Let the random variable X represent the number of times a fair coin needs to be tossed till two consecutive heads appears for the first time. The expectation of X is ___
 A 1.5 B 4 C 0.25 D 1.6
Question 70 Explanation:
Let X be random variable which denote number of tosses to get two heads.

Subtracting equation (2) from (1),

Hence,
 Question 71
Consider the differential equation with initial condition x (0) = 1. The response x(t) for t > 0 is
 A B C D
Question 71 Explanation:

For the differential equation, we have
 Question 72
For matrices of same dimension M, N and scalar c, which one of these properties DOES NOT ALWAYS hold?
 A B (cM)T = c(M)T C D
Question 72 Explanation:
Matrix multiplication is not commutative in general.
All other are properties of matrix.

 Question 73
In a housing society, half of the families have a single child per family, while the remaining half have two children per family. The probability that a child picked at random, has a sibling is _____
 A 0.42 B 0.52 C 0.62 D 0.67
Question 73 Explanation:
Let X = number of families
E1 = 1 children family
E2 = 2 children family
A = picking a child
Then by Baye’s theorem, Required probability is
 Question 74
In a housing society, half of the families have a single child per family, while the remaining half have two children per family. The probability that a child picked at random, has a sibling is _____
 A 0.42 B 0.52 C 0.62 D 0.67
Question 74 Explanation:
Let X = number of families
E1 = 1 children family
E2 = 2 children family
A = picking a child
Then by Baye’s theorem, Required probability is
 Question 75
C is a closed path in the z-plane given by |z| = 3. The value of the integral
dz is
 A B C D
Question 75 Explanation:
 Question 76
A real (4 × 4) matrix A satisfies the equation A2 = I, where I is the (4 × 4) identity matrix. The positive eigen value of A is __________.
 A 1 B 2 C 4 D 8
Question 76 Explanation:
 Question 77
Let X1, X2, and X3 be independent and identically distributed random variables with the uniform distribution on [0, 1]. The probability P{X1 is the largest} is ________
 A 0.23 B 0.27 C 0.31 D 0.33
Question 77 Explanation:
There are 6 cases possible
P(x1>x2>x3),
P(x1>x3>x2),
P(x2>x1>x3),
P(x2>x3>x1),
P(x3>x2>x1),
P(x3>x1>x2).

All cases are equally likely and hence P(x1) being largest is P(x1>x2>x3) or P(x1>x3>x2).

Hence, P(x1 being largest)=2/6=1/3=0.33
 Question 78
The capacity of a Binary Symmetric Channel (BSC) with cross-over probability 0.5 is________
 A 0 B 0.2 C 0.4 D 0.5
Question 78 Explanation:
 Question 79
The Taylor series expansion of 3 sin x + 2 cos x is
 A B C D
Question 79 Explanation:
 Question 80
The volume under the surface z(x, y) = x + y and above the triangle in the x-y plane defined by { 0 ≤ y ≤ x and 0 ≤ x ≤ 12 } is _________.
 A 644 B 720 C 748 D 864
Question 80 Explanation:
We have the triangular surface in x -y plane as shown in figure below.
 Question 81
Consider the matrix:

Which is obtained by reversing the order of the columns of the identity matrix I6.
Let P= I6 + αJ6 where α is a non-negative real number.
The value of α for which det(P) = 0 is ___________.
 A 0 B 1 C 2 D 4
Question 81 Explanation:

Since, α is non-negative real number. Hence,
α = 1
 Question 82
Let x be a real-valued random variable with denoting the mean values of X and X2, respectively. The relation which always holds true is
 A B C D
Question 82 Explanation:
For a random variable X, we define the variance as

i.e., variance cannot be negative .So ,we have
 Question 83
Consider a random process where the random phase is uniformly distributed in the interval [O,2π]. The auto-correlation E[X(t1) X(t2)]
 A B C D
Question 83 Explanation:
 Question 84
A fair coin is tossed repeatedly until a ‘Head’ appears for the first time. Let L be the number of tosses to get this first ‘Head’. The entropy H(L) in bits is _________.
 A 1 B 2 C 3 D 4
Question 84 Explanation:
 Question 85
The determinant of matrix A is 5 and the determinant of matrix B is 40. The determinant of matrix AB is ________.
 A 100 B 8 C 1000 D 200
Question 85 Explanation:
Determinant of matrix AB will be
Determinant of (A) * Determinant of (B)
5*40 = 200
 Question 86
Let X be a random variable which is uniformly chosen from the set of positive odd numbers less than 100. The expectation E[X]is __________.
 A 40 B 50 C 60 D 80
Question 86 Explanation:
 Question 87
The value of is
 A ln2 B 1.0 C e D
Question 87 Explanation:
Make the limit of (1+(1/x))x as x approaches infinity equal to any variable e.g. y, k. and take the natural logarithm of both sides.
 Question 88
If the characteristic equation of the differential equation

has two equal roots, then the values of a are
 A ± 1 B 0,0 C ± j D ±1/ 2
Question 88 Explanation:
For equal roots, Discriminant B2 − 4AC = 0
 Question 89
The system of linear equations
 A a unique solution B infinitely many solutions C no solution D exactly two solutions
Question 89 Explanation:

Clearly rank(A)=2 , rank(A/B) = 2 , number of unknowns = 3
so rank (A) = rank (A / B) =2
Since, rank (A) = rank (A / B) < number of unknowns
Equations have infinitely many solutions.
 Question 90
The real part of an analytic function f(z) where z = x + jy is given by e-y cos(x). The imaginary part of f(z) is
 A B C D
Question 90 Explanation:
 Question 91
The maximum value of the determinant among all 2×2 real symmetric matrices with trace 14 is ________.
 A 40 B 45 C 49 D 52
Question 91 Explanation:
 Question 92
If
 A 1 B 2 C 3 D 4
Question 92 Explanation:
 Question 93
The value of the integral
 A 0.2 B 0.4 C 0.5 D 0.8
Question 93 Explanation:

We can use pasrevalis theorem
 Question 94
The input to a 1-bit quantizer is a random variable X with pdf
fx(x) =2e-2x for x ≥ 0
and
fx(x) =0 for x < 0.
For outputs to be of equal probability, the quantizer threshold should be _____.
 A 0.25 B 0.35 C 0.45 D 0.5
Question 94 Explanation:

One bit quantizer will give two levels. Both levels have probability of Pd of input X is

Let xT be the threshold
Where x1 and x2 are two levels
 Question 95
The maximum value of the function f(x) = ln(1 + x) - x (where .x > - 1) occurs at x=______.
 A 0 B 1 C 2 D -1
Question 95 Explanation:
 Question 96
Which ONE of the following is a linear non-homogeneous differential equation, where x and y are the independent and dependent variables respectively?
 A B C D
Question 96 Explanation:
A) is a first order linear equation (non-homogeneous)
B) is a first order linear equation and homogeneous
C)The equation in option (C) is non-linear and non-homogenous.
D)The equation in option (D) is non-linear and homogenous.
 Question 97
Match the application to appropriate numerical method.
 A P1—M3, P2—M2, P3—M4, P4—M1 B P1—M3, P2—M1, P3—M4, P4—M2 C P1—M4, P2—M1, P3—M3, P4—M2 D P1—M2, P2—M1, P3—M3, P4—M4
Question 97 Explanation:
 Question 98
An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is
 A 0.067 B 0.073 C 0.082 D 0.091
Question 98 Explanation:
P[fourth head appears at the tenth toss] = P [getting 3 heads in the first 9 tosses and one head at tenth toss]
So, The required probability,
 Question 99
The maximum value of in the interval is _________.
 A 2 B 4 C 6 D 8
Question 99 Explanation:
 Question 100
Which one of the following statements is NOT true for a square matrix?
 A If A is upper triangular, the eigenvalues of A are the diagonal elements of it B If A is real symmetric, the eigenvalues of A are always real and positive C If A is real, the eigenvalues of A and AT are always the same D If all the principal minors of A are positive, all the eigenvalues of A are also positive
Question 100 Explanation:
 Question 101
A fair coin is tossed repeatedly till both head and tail appear at least once. The average number of tosses required is __________________.
 A 1 B 3 C 4 D 2
Question 101 Explanation:
let the random variable X be number of tosses. Here we need at least two tosses to get a success

with 2 tosses HT and TH are valid cases out of HH,HT,TH,TT
with 3 tosses HHT and TTH are valid cases out of HHH,HHT,HTH,HTT,THH,THT,TTH,TTT
And so on

X                    2             3            4               5                    ..........

Prob(X=i)        2/4        2/8          2/16           2/32                 ........

E(x)=             2*2/4 + 3 *2/8 + 4*2/16 +......  (i)

E(x) /2=                     2*2/8 + 3*2/16 +.........(ii)

(i)-(ii) is 2*2/4 + (3-2)2/8 + (4-3)2/16 +.....

E(x)/2 => 1+2/8+2/16+2/32+...

E(x)/2=>1+ 2(1/8 + 1/16 + 1/32 +...)

E(x)/2=>1 +2* (1/8 / (1 -  1/2))

E(x)/2=>1+1/2

E(x)/2=>3/2

so E(x)=>3

 Question 102
Let be independent and identically distributed random variables with the uniform distribution on [0, 1]. The probability is ________.
 A 0.12 B 0.16 C 0.2 D 0.24
Question 102 Explanation:
Given be independent and identically distributed with uniform distribution on [0,1]
Let z = x1 + x2 - x3
P{x1 + x2 ≤ x3} = P{x1 + x2 - x3 ≤ o}
=P{z ≤ o}

Let us find probability density function of random variable z.
Since Z is summation of three random variable
Overall pdf of z is convolution of the pdf of

pdf of –x3 is

 Question 103
A binary random variable X takes the value of 1 with probability 1/3. X is input to a cascade of 2 independent identical binary symmetric channels (BSCs) each with crossover probability 1/2. The outputs of BSCs are the random variables Y1 and Y2 as shown in the figure.

The value of H(Y1) + H(Y2) in bits is _______.
 A 2 B 5 C 8 D 6
Question 103 Explanation:
 Question 104
The series converges to
 A B C 2 D e
Question 104 Explanation:
 Question 105
The magnitude of the gradient for the function f(x,y,z) = x2 + 3y2 + z3 at the point (1,1,1) is
 A 7 B 9 C 12 D 5
Question 105 Explanation:
 Question 106
The directional derivative of in the direction of the unit vector at an angle of with y-axis at point (1,1), is given by _____.
 A 3 B 5 C 8 D 4
Question 106 Explanation:
 Question 107
With initial values y(0) = y’(0)=1 the solution of the differential equation
at x = 1 ________
 A 0.44 B 0.48 C 0.54 D 0.62
Question 107 Explanation:
 Question 108
Parcels from sender S to receiver R pass sequentially through two post-offices. Each post-office has a probability of losing an incoming parcel, independently of all other parcels. Given that a parcel is lost, the probability that it was lost by the second post-office is_________.
 A 0.22 B 0.33 C 0.44 D 0.55
Question 108 Explanation:
 Question 109
For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle, the angle between the hypotenuse and the side is________(o)
 A 12 B 36 C 30 D 45
Question 109 Explanation:
Let
x (opposite side)
z (hypotenuse) of a right angled triangle.

Given Z + y = K(constant) ......(1) and angle between them say then Area,

In order to have maximum area,
 Question 110
Consider the Z-channel given in the figure. The input is 0 or 1 with equal probability.

If the output is 0, the probability that the input is also 0 equals ______________
 A 0.4 B 0.6 C 0.7 D 0.8
Question 110 Explanation:
 Question 111
The maximum value of θ until which the approximation sin θ ≈ θ  holds to within 10% error is
 A 10 ° B 18 ° C 50 ° D 90 °
Question 111 Explanation:
For 10% error, we can check for θ= 18 ° first,

Now, we check it for θ= 50 °

From the above we can conclude that the error is more than 10%. Hence, for error less than 10%,  till θ= 18 ° we have the approximation
 Question 112
The divergence of the vector field is
 A 0 B 1/3 C 1 D 3
Question 112 Explanation:
 Question 113
A polynomial with all coefficients positive has
 A no real roots B no negative real root C odd number of real roots D at least one positive and one negative real root
Question 113 Explanation:
Given polynomial,

Since, all the coefficients are positive so, the roots of equation is given by

It will have at least one pole in right hand plane as there will be least one sign change from (a1) to (a0) in the Routh matrix 1st column. Also, there will be a corresponding pole in left hand plane
i.e., at least one positive root (in R.H.P.)
i.e., at least one negative root (in L.H.P.)
Rest of the roots will be either on imaginary axis or in L.H.P
 Question 114
Let U and V be two independent zero mean Gaussian random variables of variances and respectively. The probability is
 A 4/9 B ½ C 2/3 D 5/9
Question 114 Explanation:
Since U and V are given to be normal random variables, therefore their difference will also be a normal random variable.
Here, let Y=3V-2U, where Y is also a normal random variable with mean = 0 and variance = 32 (1/9) + 22 (1/4) = 2
So it will be symmetric about mean that is 0.
P(Y>=0)= 1/2 (by symmetry property)
 Question 115
Let A be an m × n matric and B an n × m matric. It is given that determinant determinant where is the k × k identity matrix. Using the above property, the determinant of the matrix given below is
 A 2 B 5 C 8 D 16
Question 115 Explanation:
 Question 116
A system described by a linear, constant coefficient, ordinary, first order differential equation lies an exact solution given by y(t) for t > 0, when the forcing function is x(t) and the initial condition is y(0). If one wishes to modify the system so that the solution becomes –2X0 for t > 0, we need to
 A change the initial condition to –y(0) and the forcing function 2x(t) B change the initial condition to 2y(0) and the forcing function to –x(t) C change the initial condition to and the forcing function to D change the initial condition to and the forcing function to
Question 116 Explanation:
The solution of a system described by a linear, constant coefficient, ordinary, first order differential equation with forcing function x(t) is y(t) so, we can define a function relating x(t) and y(t) as below

where P, Q, K are constant. Taking the Laplace transform both the sides, we get
…(1)
Now, the solutions becomes

or,
So, Eq. (1) changes to

or, …(2)
Comparing Eq. (1) and (2), we conclude that

This makes the two equations to be same. Hence, we require to change the initial condition to -2y(0) and the forcing equation to – 2x(t)
 Question 117
Given . If C is a counter clockwise path in the z-plane such that | z + 1| = 1, the value of is
 A -2 B -1 C 1 D 2
Question 117 Explanation:
Given

Poles are at -1 and -3 i.e. (-1,0) and (-3,0).
From figure below of |z + 1| = 1, we see that (-1,0) is inside the circle and (- 3, 0) is outside the circle.

Residue theorem says,
Residue of those poles
which are inside (C)
So the required integral is given
by the residue of function at pole (-1, 0) (which is inside the circle).
This residue is
 Question 118
If , then the value of xx is
 A B C x D 1
Question 118 Explanation:
 Question 119
Consider the differential equation with and . The numerical value of is
 A 2 B - 1 C 0 D 1
Question 119 Explanation:

taking Laplace transform on both the sides we have

 Question 120
A fair coin is tossed till a head appears for the first time. The probability that the number of required tosses is odd, is
 A 1/3 B 1/2 C 2/3 D 3/4
Question 120 Explanation:
P (odd tosses) = P (H) + P(TTH) + P(TTTTH) + .....

 Question 121
The maximum value of f(x) = x3 - 9x2 + 24x + 5 in the interval [1, 6] is
 A 21 B 25 C 41 D 46
Question 121 Explanation:
We need absolute maximum of
f(x) = x3 – 9x2 + 24x + 5 in the interval [1,6]
First find local maximum if any by putting
f'(x) = 0
i.e. f’(x) = 3x2 – 18x + 24 = 0
i.e. x2 – 6x + 8 = 0
x = 2,4
Now f”(x) = 6x – 18
f”(2) = 12 – 18 - -6 < 0
(So x = 2 is a point of local maximum) and
f”(4) = 24 – 18 = +6 > 0
(So x = 4 is a point of local minimum) Now tabulate the values of f at end point of interval and at local maximum point, to find absolute maximum in given range, as shown below:

Clearly the absolute maxima is at x = 6 and absolute maximum value is 41.
 Question 122
Given that and , the value of A3 is
 A 15 A + 12 I B 19A + 30 I C 17A + 15 I D 17 A + 21I
Question 122 Explanation:
Characteristic equation of A is

(-5-λ)(-λ) + 6 = 0

So,
(by Cayley Hamilton theorem)
A2 = –5A – 6I
Multiplying by A on both sides, we have,
A3 =-5A2-6A
A3 = -5(-5A-6I)-6A
= 19A + 30I
 Question 123
Find the variance of the distribution shown in the figure.
 A B C D
Question 123 Explanation:
The Correct Answer Among All the Options is D
As it has constant pdf that means it has uniform distribution. And for uniform distribution of pdf , we have formula
Variance= mean square value
Mean =
Mean square value =
here, b=
therefore, variance =
Refer the Topic Wise Question for Statistics Engineering Mathematics
 Question 124
A cube of side 1 unit is placed in such a way that the origin coincides with one of its top vertices and the three axes along three of its edges. What are the coordinates of the vertex which is diagonally opposite to the vertex whose coordinates are (1,0,1)?
 A (0, 0, 0) B (0, -1, 0) C (0, 1, 0) D (1, 1, 1)
Discuss      ISRO CS 2014     Engineering-Mathematics     Geometry
Question 124 Explanation:

As per the above diagram the coordinates of the vertex which is diagonally opposite to the vertex whose coordinates are (1, 0, 1) = (0, -1, 0).
There are 124 questions to complete.