# Engineering Mathematics (Test 4)

## Gate Exam : Ce Civil Engineering

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Q.1
The z-Transform of a sequence x[n] is given as X(z) = 2z+4-4/z+3z2. If y[n] is the first difference of x[n], then Y(z) is given by
A. B. C. D. Explaination / Solution:

y(n) is first difference of x(n) So
g(n)=x(n)-x(n-1) Workspace
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Q.2
The integral when evaluated by using Simpson’s 1/3 rule on two equal subintervals each of length 1, equals
A. 1.000
B. 1.098
C. 1.111
D. 1.120
Explaination / Solution: Workspace
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Q.3
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. P(A ∩ B) = P(A) P(B)
B. C. P( A U B) = P(A) + P(B)
D. Explaination / Solution:

A and B are two independent events with probabilities,  P(A) ≠ 0 and P(B) ≠ 0
Now, we check the given options. True True False True

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Q.4
Find the sum of the expression A. 7
B. 8
C. 9
D. 10
Explaination / Solution:

The expression can be written as Workspace
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Q.5
The solution of the partial differential equation is of the form
A. B. C. D. Explaination / Solution: Workspace
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Q.6
The binary operation ∨ is defined as follows ; Which one of the following is equivalent to P∨Q?
A. ~Q Λ ~P
B. P ν ~Q
C. ~P Λ Q
D. ~P ν ~Q
Explaination / Solution:
No Explaination.

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Q.7
Consider a hash table with 100 slots. Collisions are resolved using chaining. Assuming simple uniform hashing, what is the probability that the first 3 slots are unfilled after the first 3 insertions?
A. (97 × 97 × 97)/1003
B. (99 × 98 × 97)/1003
C. (97 × 96 × 95)/1003
D.  (97 × 96 × 95)/(3! × 1003)
Explaination / Solution: Workspace
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Q.8
An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x, y) +iv(x, y) where i = √-1. If u = xy, the expression of v should be
A. ((x + y)2/2) + k
B. ((x2 - y2)/2) + k
C. ((y2 - x2)/2) + k
D. ((x - y)2/2) + k
Explaination / Solution:
No Explaination.

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Q.9
All the four entries of the 2 × 2 matrix are nonzero, and one of its eigenvalue is zero. Which of the following statements is true?
A. P11P12 – P12P21 = 1
B. P11P22 – P12P21 = -1
C. P11P22 – P12P21 = 0
D. P11P22 + P12P21 = 0
Explaination / Solution:

The product of Eigen value is equal to the determinant of the matrix. Since one of the Eigen value is zero, the product of Eigen value is zero, thus determinant of the matrix is zero.

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Q.10
For matrices of same dimension M , N and scalar c, which one of these properties DOES NOT ALWAYS hold ?
A.

(MT)T = M

B.

(cM)T = c(M)T

C.

(M + N)T = MT + NT

D. MN = NM
Explaination / Solution:

Let the matrices i.e. the property holds always  i.e. property does not hold always.

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