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Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2


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30 Questions MCQ Test GATE Mechanical (ME) 2023 Mock Test Series | Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2

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Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 1

Consider the system of equations given below: 
x + y =  2
2x + 2y = 5
This system has

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 1

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

Angemented matrix 

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

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - 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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 2

(b)


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

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 3

Solution for the system defined by the set of equations
4y + 3z = 8;
2x – z = 2
and 3x + 2y =5 is 

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 3

Ans.(d)
Consider the matrix A =   ,Now det (A) = 0
So byCramer's Rule the system has no solution

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - 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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 4

(d)

 =


For infinite solution of the system
α − 2 = 0 and β − 7 = 0
⇒ α = 2 and β = 7.

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 5

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 5

(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.

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - 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  

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 6

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

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 7

Consider the matrices X (4 × 3), Y (4 × 3) and P (2 × 3). The order or P (XTY)–1PT] T will be 

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 7

(a)


Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 8

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 8

(c)


∴Rank(A) = 2

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 9

The Laplace transform of  

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 9

Ans. (b)False
Laplace transform of  

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 10

There are two containers, with one containing 4 Red and 3 Green balls and the other containing 3 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       

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 11

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 11

Ans. (b)


⇒ 

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 12

The Inverse Laplace transform of   is         

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 12

Ans. (c)



Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 13

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 13

Ans. (b)
It is a standard result that
L (cosh at) =  

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 14

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 14

Ans. (a)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 15

Laplace transform of the function sin ωt            

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 15

Ans. (b)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 16

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 16

Ans.(c)   


Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 17

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 17

Ans. (d)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 18

The Laplace transform of the function sin2 2t is  

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 18

Ans.(a)


 

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 19

 Find the rank of the matrix 

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 19

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.

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 20

The running integrator, given by             

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 20

Ans. (b)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 21

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 21

Ans. (c)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 22

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 22

Ans. (d)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 23

The fundamental period of the discrete-time signal is 

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 23

Ans. (b)

or  

or   

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 24

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 24

Ans. (c)
f(t) = u(t – ζ)
L{f(t)} = L{u(t – ζ)}

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 25

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 25

Ans. (d)
H.C.F. of 2π and 3π is 6π.
Then, fundamental frequency = 6π
∴ Period, T = = 3 sec

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 26

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 26

Ans. (c)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 27

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

g (t) can be expressed as                      

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 28

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

The Laplace transform of g(t) is                  

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 28

Ans. (c)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 29

The Laplace transform of g(t) is     

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 29

Ans. (c)

Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 30

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

Detailed Solution for Test: Systems of Linear Equations, Matrix Algebra & Transform Theory- 2 - Question 30

Ans. (c)

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