Engineering Mathematics (Test 8)

Gate Exam : Ce Civil Engineering

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Q.1
The solution to the differential equation where k is a constant, subjected to the boundary conditions u(0)=0 and u(L)=U, is
A. u = U(x/L)
B. C. D. Explaination / Solution: Workspace
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Q.2
Consider the differential equation . The general solution with constant c is
A. y = tan x2/2 + tanc
B. y = tan2((x/2) + c)
C. y = tan2(x/2) + c
D. y = tan((x2/2) + c)
Explaination / Solution:
No Explaination.

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Q.3
The analytic function has singularity at
A. 1 and -1
B. 1 and i
C. 1 and -i
D. i and -i
Explaination / Solution:
No Explaination.

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Q.4
If the entries in each column of a square matrix M add up to 1, then an eigen value of M is
A. 4
B. 3
C. 2
D. 1
Explaination / Solution:

Consider the '2x2 ' square matrix Characteristic equation of M is
λ- (a + d)λ + (ad - bc) = 0......................(1)
Put λ =1, weget
1 - (a + d) + ad - bc = 0
1 - a - d + ad - (1 - d)(1 - a) = 0
1 - a - d + ad - 1 + a + d - ad = 0
0 = 0,which is true
λ = 1 Satisfies the equation (1) but λ = 2,3,4 does not satisfy the equation (1). For all possible values of a,d

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Q.5
The divergence of the vector field at a point (1,1,1) is equal to
A. 7
B. 4
C. 3
D. 0
Explaination / Solution:
No Explaination.

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Q.6
A source emits bit 0 with probability 1/3 and bit 1 with probability 2/3. The emitted bits are communicated to the receiver. The receiver decides for either 0 or 1 based on the received value R. It is given that the conditional density functions of R are as The minimum decision error probability is
A. 0
B. 1/12
C. 1/9
D. 1/6
Explaination / Solution:

Given the conditional density function of R as  Decision error probability that receiver decides 0 for a transmitted bit 1 is
fR/1 (r = 0) = 1/6
Again, the decision error probability that receiver decides 1 for a transmitted bit 0 is
fR/0 (r = 1) = 1/4
Hence, the minimum decision error probability is
fR/1 (r = 0) = 1/6

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Q.7
With respect to the numerical evaluation of the definite integral, where a and b are given, which of the following statements is/are TRUE?
(I) The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
(II) The value of K obtained using the Simpson’s rule is always equal to the exact value of the definite integral.
A. I only
B. II only
C. Both I and II
D. Neither I nor II
Explaination / Solution:  Workspace
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Q.8
The solution of x(dy/dx) + y = x4 with the condition y(1) = 6/5 is
A. y = (x4/5) + (1/x)
B. y = (4x4/5) + (4/5x)
C. y = (x4/5) + 1
D. y = (x5/5) + 1
Explaination / Solution:
No Explaination.

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Q.9
Consider a function f(x) = 1-|x| on -1≤ x ≤ 1. The value of x at which the function attains a maximum, and the maximum value of the function are.
A. 0,-1
B. -1,0
C. 0,1
D. -1,2
Explaination / Solution: Workspace
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Q.10
The Laplace transform of tet is
A. s/(s+1)2
B. 1/(s-1)2
C. 1/(s+1)2
D. s/(s-1) 