Engineering Mathematics | GATE Subject Wise
Question 1 |
The number of distinct eigenvalues of the matrix 
is equal to __________.
is equal to __________.
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 __________.
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 _____________.
evaluated over the unit circle |z| = 1 is _____________.
Fill in the Blank Type Question |
Question 3 Explanation:
Question 4 |
Which one of the following functions is analytic over the entire complex plane?
E1/z | |
In(z) | |
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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
(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
Hyperbolas and Circles | |
Circles and Hyperbolas | |
Parabolas and Circles | |
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 _______________.
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 _______________.
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]?
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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
It is given that A has only one real Eigen value. Then the real Eigen value of A is
-2.5 | |
0 | |
15 | |
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
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
is 0 | |
1 | |
2 | |
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
= 5(0-12) – 10(6-6) +10(6-0)= -60-0+60=0
But a 2×2 minor,
=0-10=-10≠0 Rank = 2Question 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?
Which one among the following is correct? | |
Both I and III are true | |
Both II and IV are true | |
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
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) _____.
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
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.
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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
Where x, y, z are real. The volume of R (up to two decimal places) isFill 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
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
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
where and are constants. If at and then the relation between and is
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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?
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?
S1 implies S2 | |
S1 implies S3 | |
S2 implies S1 | |
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"
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:
  (with initial rest conditions) | |
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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.
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
around has the form
The coefficient 
(correct to two decimal places) is equal to ____________.
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 ___________.
and vector The number of distinct real values of for which the equation has infinitely many solution is ___________.
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 ___________.
and Assume that and are independent variables. At the value (correct to two decimal places) of is ___________.
Fill in the Blank Type Question |
Question 20 Explanation:
Question 21 |
The position of a particle 
is described by the differential equation:
is described by the differential equation:
The initial conditions are 
and 
The position (accurate to two decimal places) of the particle at 
is _____________.
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 
where 
The value of the integral 
is __________.
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
and satisfies the differential equation The equation that describes the curve is  | |
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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,
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__________
Fill in the Blank Type Question |
Question 24 Explanation:
Question 25 |
The residues of a function f(z) 
are
are
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Question 25 Explanation:
Question 26 |
The values of the integrals 
and 
are
and are
same and equal to 0.5 | |
same and equal to -0.5 | |
0.5 and – 0.5, respectively | |
- 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
If C is defined as |z|=3, then the value of I is
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Question 27 Explanation:
Question 28 |
If the vector function
is irrotational, then the values of the constants k1, k2 and k3 respectively, are
is irrotational, then the values of the constants k1, k2 and k3 respectively, are
0.3, –2.5, 0.5 | |
0.0, 3.0, 2.0 | |
0.3, 0.33, 0.5 | |
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 _____.
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 = ________.
n the interval occurs at x = ________.
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:
M4k+1 | |
M4k + 2 | |
M4k+3 | |
M4k |
Question 31 Explanation:
Question 32 |
The second moment of a Poisson-distributed random variable is 2. The mean of the random variable is _________.
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
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
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
P is true, Q is false; R is false | |
P is false, Q is true, R is true | |
P is false, Q is true; R is false | |
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
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: 
The solutions have neither maxima nor minima anywhere except at the boundaries. | |
The solutions are not separable in the coordinates. | |
The solutions are not continuous. | |
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)?
Suppose . Which one of the following is a graph of F(x)?
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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.
-> 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 _______.
where D denotes the disc: evaluates to _______.
Fill in the Blank Type Question |
Question 36 Explanation:
Method-I : -
Method – II:-
(Changing into polar coordinates by
) 
Another Approach
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 ______
The value of the integral is ______
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
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
–1/2 | |
–1/3 | |
1/3 | |
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 ?
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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?
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?
The residue of  at z = 1 is 1/2 | |
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 (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 ________.
is an eigenvector of the matrix is ________. Fill in the Blank Type Question |
Question 41 Explanation:
Question 42 |
The solution of the differential equation 
with y(0 = y’(0) = 1 is
with y(0 = y’(0) = 1 is (2-t)et | |
(1+2t)e-t | |
(2+t)e-t | |
(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.
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 
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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 _______.
is _______.
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_____.
has zero as an eigenvalue is_____.
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
where z is a complex variable. The value of b for which the function f(z) is analytic is 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
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?
f(x) increase monotonically | |
f(x) increases, then decreases and increases again | |
f(x) decreases, then increases and decreases again | |
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.
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 ?
1 | |
2 | |
3 | |
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
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
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
with C oriented anti-clockwise.
The value of over the curve C equals 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
has det A. = 100 and trace A=14 The value of
is
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
square matrix Where x is unknown. If the eigenvalues of the matrix A are and the x is equal to
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Question 52 Explanation:
Given
Product of eigen values = Det of A
Product of eigen values = Det of A
Question 53 |
For 
the residue of the pole at z = 0 is _____.
the residue of the pole at z = 0 is _____. 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. _______
Fill in the Blank Type Question |
Question 54 Explanation:
P(Head) = 0.3
P(Tail) = 0.7
Required probability = TTTTH
= (0.7)(0.7)(0.7)(0.7)(0.3)
=0.07203
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 ______.
is equal to ______.
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 ____
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 ____
Fill in the Blank Type Question |
Question 56 Explanation:
Question 57 |
The solution of the initial value problem given below is 
With
and 
With
and
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Question 57 Explanation:
Question 58 |
For A = 
, the determinant of AT A–1 is
, the determinant of AT A–1 is
sec2 x | |
cos 4x | |
1 | |
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
y = 2 | |
x = 2 | |
x + y = 4 | |
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,
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 
2pnj | |
0 | |
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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______
Fill in the Blank Type Question |
Question 61 Explanation:
Question 62 |
The value of 
is_________.
is_________. Fill in the Blank Type Question |
Question 62 Explanation:
Let 
Now multiply equ(i) with 1/2
Now subtrac eq(ii) from (i)
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________.
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 _____.
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
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 ________
. Given x(0) = 20 and x(1) = 10/e, where e = 2.718,
the value of x(2) is ________
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
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 _____.
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 onS
is _____.
Fill in the Blank Type Question |
Question 66 Explanation:
In the given vector field,
By using divergence theorem
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?
47/48 | |
1/4 | |
13/48 | |
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
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
is
tan y – cot x = c | |
tan x – cot y = c | |
tan y + cot x = c | |
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 _______________
. If f(z 1) = f(z2) for all z1≠ z2, a = 2, b = 4 and c = 5, then d should be equal to _______________
150 | |
10 | |
50 | |
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 ___
1.5 | |
4 | |
0.25 | |
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,
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
with initial condition x (0) = 1. The response x(t) for t > 0 is  | |
 | |
 | |
 |
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?
 | |
(cM)T = c(M)T | |
 | |
 |
Question 72 Explanation:
Matrix multiplication is not commutative in general.
All other are properties of matrix.
Refer: www.khanacademy.org
All other are properties of matrix.
Refer: www.khanacademy.org
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 _____
0.42 | |
0.52 | |
0.62 | |
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
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 _____
0.42 | |
0.52 | |
0.62 | |
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
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
dz is
 | |
 | |
 | |
 |
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 __________.
1 | |
2 | |
4 | |
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 ________
0.23 | |
0.27 | |
0.31 | |
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
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________
0 | |
0.2 | |
0.4 | |
0.5 |
Question 78 Explanation:
Question 79 |
The Taylor series expansion of 3 sin x + 2 cos x is
 | |
 | |
 | |
 |
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 _________.
644 | |
720 | |
748 | |
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 ___________.
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 ___________.
0 | |
1 | |
2 | |
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
denoting the mean values of X and X2, respectively. The relation which always holds true is  | |
 | |
 | |
 |
Question 82 Explanation:
For a random variable X, we define the variance as
i.e., variance cannot be negative .So ,we have
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)]
where the random phase is uniformly distributed in the interval
[O,2π].
The auto-correlation
E[X(t1) X(t2)]
 | |
 | |
 | |
 |
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 _________.
1 | |
2 | |
3 | |
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 ________.
100 | |
8 | |
1000 | |
200 |
Question 85 Explanation:
Determinant of matrix AB will be
Determinant of (A) * Determinant of (B)
5*40 = 200
so answer = 200
Determinant of (A) * Determinant of (B)
5*40 = 200
so answer = 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 __________.
40 | |
50 | |
60 | |
80 |
Question 86 Explanation:
Question 87 |
The value of
is
is
ln2 | |
1.0 | |
e | |
 |
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
has two equal roots, then the values of a are
± 1 | |
0,0 | |
± j | |
±1/ 2 |
Question 88 Explanation:
For equal roots, Discriminant B2 − 4AC = 0
Question 89 |
The system of linear equations
a unique solution | |
infinitely many solutions | |
no solution | |
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
 | |
 | |
 | |
 |
Question 90 Explanation:
Question 91 |
The maximum value of the determinant among all 2×2 real symmetric matrices with trace 14 is ________.
40 | |
45 | |
49 | |
52 |
Question 91 Explanation:
Question 92 |
If 
1 | |
2 | |
3 | |
4 |
Question 92 Explanation:
Question 93 |
The value of the integral
0.2 | |
0.4 | |
0.5 | |
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 _____.
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 _____.
0.25 | |
0.35 | |
0.45 | |
0.50 |
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=______.
0 | |
1 | |
2 | |
-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?
 | |
 | |
 | |
 |
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.
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.
P1—M3, P2—M2, P3—M4, P4—M1 | |
P1—M3, P2—M1, P3—M4, P4—M2 | |
P1—M4, P2—M1, P3—M3, P4—M2 | |
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
0.067 | |
0.073 | |
0.082 | |
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,
So, The required probability,
Question 99 |
The maximum value of
in the interval 
is _________.
in the interval is _________.
2 | |
4 | |
6 | |
8 |
Question 99 Explanation:
Question 100 |
Which one of the following statements is NOT true for a square matrix?
If A is upper triangular, the eigenvalues of A are the diagonal elements of it | |
If A is real symmetric, the eigenvalues of A are always real and positive | |
If A is real, the eigenvalues of A and AT are always the same | |
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 __________________.
1 | |
3 | |
4 | |
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
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 ________.
be independent and identically distributed random variables with the uniform distribution on [0, 1]. The probability is ________.
0.12 | |
0.16 | |
0.20 | |
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
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 _______.
The value of H(Y1) + H(Y2) in bits is _______.
2 | |
5 | |
8 | |
6 |
Question 103 Explanation:
Question 104 |
The series 
converges to
converges to
 | |
 | |
2 | |
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
7 | |
9 | |
12 | |
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 _____.
in the direction of the unit vector at an angle of with y-axis at point (1,1), is given by _____.
3 | |
5 | |
8 | |
4 |
Question 106 Explanation:
Question 107 |
With initial values y(0) = y’(0)=1 the solution of the differential equation
at x = 1 ________
at x = 1 ________
0.44 | |
0.48 | |
0.54 | |
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_________.
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_________.
0.22 | |
0.33 | |
0.44 | |
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)
12 | |
36 | |
30 | |
45 |
Question 109 Explanation:
Let
x (opposite side)
y (adjacent 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,
x (opposite side)
y (adjacent 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 ______________
If the output is 0, the probability that the input is also 0 equals ______________
0.4 | |
0.6 | |
0.7 | |
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
10 ° | |
18 ° | |
50 ° | |
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
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
is
0 | |
1/3 | |
1 | |
3 |
Question 112 Explanation:
Question 113 |
A polynomial 
with all coefficients positive has
with all coefficients positive has no real roots | |
no negative real root | |
odd number of real roots | |
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
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
and respectively. The probability is 4/9 | |
½ | |
2/3 | |
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)
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
determinant where is the k × k identity matrix. Using the above property, the determinant of the matrix given below is
2 | |
5 | |
8 | |
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
change the initial condition to –y(0) and the forcing function 2x(t) | |
change the initial condition to 2y(0) and the forcing function to –x(t) | |
change the initial condition to  and the forcing function to  | |
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)
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
. If C is a counter clockwise path in the z-plane such that | z + 1| = 1, the value of is -2 | |
-1 | |
1 | |
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
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
, then the value of xx is  | |
 | |
x | |
1 |
Question 118 Explanation:
Question 119 |
Consider the differential equation 
with
and 
. The numerical value of 
is
with and . The numerical value of is 2 | |
- 1 | |
0 | |
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
1/3 | |
1/2 | |
2/3 | |
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
21 | |
25 | |
41 | |
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.
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
and , the value of A3 is 15 A + 12 I | |
19A + 30 I | |
17A + 15 I | |
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
(-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.
 | |
 | |
 | |
 |
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
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)?
(0, 0, 0) | |
(0, -1, 0) | |
(0, 1, 0) | |
(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.