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QUESTION: 1

Consider the system of equations given below:

x + y = 2

2x + 2y = 5

This system has

Solution:

(b) This can be written as AX = B Where A

Angemented matrix

rank(A) ≠ rank(). The system is inconsistant .So system has no solution.

QUESTION: 2

For what value of a, if any, will the following system of equations in x, y and z have a solution?

2x + 3y = 4

x+y+z = 4

x + 2y - z = a

Solution:

(b)

If a = 0 then rank (A) = rank() = 2. Therefore the system is consistant

∴ The system has sol^{n} .

QUESTION: 3

Solution for the system defined by the set of equations

4y + 3z = 8;

2x – z = 2

and 3x + 2y =5 is

Solution:

Ans.(d)

Consider the matrix A = ,Now det (A) = 0

So byCramer's Rule the system has no solution

QUESTION: 4

For what values of α and β the following simultaneous equations have an infinite numberof solutions?

x + y + z = 5; x + 3y + 3z = 9; x + 2y + αz = β

Solution:

(d)

=

For infinite solution of the system

α − 2 = 0 and β − 7 = 0

⇒ α = 2 and β = 7.

QUESTION: 5

Let A be a 3 × 3 matrix with rank 2. Then AX = 0 has

Solution:

(b)

We know , rank (A) + Solution space X(A) = no. of unknowns.

⇒2 + X(A) = 3 . [Solution space X(A)= No. of linearly independent vectors]

⇒ X(A) =1.

QUESTION: 6

A is a 3 x 4 real matrix and A x = b is an inconsistent system of equations. The highest possible rank of A is

Solution:

(b). Highest possible rank of A= 2 ,as Ax = b is an inconsistent system.

QUESTION: 7

Consider the matrices X _{(4 × 3)}, Y_{ (4 × 3)} and P_{ (2 × 3)}. The order or P (X^{T}Y)^{–1}P^{T}] ^{T} will be

Solution:

(a)

QUESTION: 8

Given matrix [A] = the rank of the matrix is

Solution:

(c)

∴Rank(A) = 2

QUESTION: 9

The Laplace transform of

Solution:

Ans. (b)False

Laplace transform of

QUESTION: 10

There are two containers, with one containing 4 Red and 3 Green balls and the other containing Blue and 4 Green balls. One bal is drawn at random form each container.The probability that one of the ball is Red and the other is Blue will be

Solution:

QUESTION: 11

If L defines the Laplace Transform of a function, L [sin (at)] will be equal to

Solution:

Ans. (b)

⇒

QUESTION: 12

The Inverse Laplace transform of is

Solution:

Ans. (c)

QUESTION: 13

Laplace transform for the function f(x) = cosh (ax) is

Solution:

Ans. (b)

It is a standard result that

L (cosh at) =

QUESTION: 14

If F(s) is the Laplace transform of function f (t), then Laplace transform of

Solution:

Ans. (a)

QUESTION: 15

Laplace transform of the function sin ωt

Solution:

Ans. (b)

QUESTION: 16

Laplace transform of (a + bt)^{2} where ‘a’ and ‘b’ are constants is given by:

Solution:

Ans.(c)

QUESTION: 17

A delayed unit step function is defined as Its Laplace transform is

Solution:

Ans. (d)

QUESTION: 18

The Laplace transform of the function sin^{2} 2t is

Solution:

Ans.(a)

QUESTION: 19

Find the rank of the matrix** **

Solution:

To find out the rank of the matrix first find the |A|

If the value of the |A| = 0 then the matrix is said to be reduced

But, as the determinant of A has some finite value, then the rank of the matrix is 3.

QUESTION: 20

The running integrator, given by

Solution:

Ans. (b)

QUESTION: 21

The state transition matrix for the system X^{-} = AX with initial state X(0) is

Solution:

Ans. (c)

QUESTION: 22

The Fourier transform of x(t) = e^{–at} u(–t), where u(t) is the unit step function

Solution:

Ans. (d)

QUESTION: 23

The fundamental period of the discrete-time signal is

Solution:

Ans. (b)

or

or

QUESTION: 24

u(t) represents the unit step function. The Laplace transform of u(t – ζ) is

Solution:

Ans. (c)

f(t) = u(t – ζ)

L{f(t)} = L{u(t – ζ)}

QUESTION: 25

The fundamental period of x(t) = 2 sin πt + 3 sin 3πt, with t expressed in seconds, is

Solution:

Ans. (d)

H.C.F. of 2π and 3π is 6π.

Then, fundamental frequency = 6π

∴ Period, T = = 3 sec

QUESTION: 26

If the Fourier transform of x[n] is X(e^{jω}), then the Fourier transform of (–1)n x[n] is

Solution:

Ans. (c)

QUESTION: 27

Given f(t) and g(t) as shown below:

g (t) can be expressed as

Solution:

QUESTION: 28

Given f(t) and g(t) as shown below:

The Laplace transform of g(t) is

Solution:

Ans. (c)

QUESTION: 29

The Laplace transform of g(t) is

Solution:

Ans. (c)

QUESTION: 30

Let Y(s) be the Laplace transformation of the function y (t), then final value of the function is

Solution:

Ans. (c)

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