# EC GATE 2009 (Test 5)

Tag: ec gate 2009
Q.1
Consider the system

where p and q are arbitrary real numbers. Which of the following statements about the controllability of the system is true ?
A.
The system is completely state controllable for any nonzero values of p and q
B. Only p = 0 and q = 0 result in controllability
C. The system is uncontrollable for all values of p and q
D. We cannot conclude about controllability from the given data
Explaination / Solution:

Since S is singular, system is completely uncontrollable for all values of p and q .

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Q.2
Consider two independent random variables X and Y with identical distributions. The variables X and Y take values 0, 1 and 2 with probabilities 1/2,1/4 and 1/4 respectively. What is the conditional probability P(X + Y = 2 X - Y = 0) ?
A. 0
B. 1/16
C. 1/6
D. 1
Explaination / Solution:

We have

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Q.3
The Taylor series expansion of is given by
A.
B.
C.
D.
Explaination / Solution:

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Q.4
Given that F(s) is the one-side Laplace transform of f (t), the Laplace transform
A.
B.
C.
D.
Explaination / Solution:

By property of unilateral laplace transform

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Q.5
Match each differential equation in Group I to its family of solution curves from Group II

A. A - 2, B - 3,C - 3,D - 1
B. A - 1, B - 3,C - 2,D - 1
C. A - 2, B - 1,C - 3,D - 3
D. A - 3, B - 2,C - 1,D - 2
Explaination / Solution:

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Q.6
The Eigen values of following matrix are

A. 3, 3 + 5j, 6 - j
B. -6 + 5j, 3 + j, 3 - j
C. 3 + j, 3 - j, 5 + j
D. 3, - 1 + 3j, - 1 - 3j
Explaination / Solution:

Sum of the principal diagonal element of matrix is equal to the sum of Eigen values. Sum of the diagonal element is -1 - 1 + 3 = 1.In only option (D), the sum of Eigen values is 1.

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Q.7
The 4-point Discrete Fourier Transform (DFT) of a discrete time sequence {1,0,2,3} is
A. [0, -2 + 2j , 2, -2 - 2j ]
B. [2, 2 + 2j , 6, 2 - 2j ]
C. [6, 1 - 3j , 2, 1 + 3j ]
D. [6, -1 + 3j , 0, -1 - 3j ]
Explaination / Solution:

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Q.8
A discrete random variable X takes values from 1 to 5 with probabilities as shown in the table. A student calculates the mean X as 3.5 and her teacher calculates the variance of X as 1.5. Which of the following statements is true ?

A. Both the student and the teacher are right
B. Both the student and the teacher are wrong
C. The student is wrong but the teacher is right
D. The student is right but the teacher is wrong