# Engineering Mathematics (Test 3)

## Gate Exam : Ce Civil Engineering

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Q.1
If f(x) is an even function and a is a positive real number, then equals
A. 0
B. a
C. 2a
D. Explaination / Solution: Workspace
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Q.2
With K as a constant, the possible solution for the first order differential equation is
A. B. C. D. Explaination / Solution: Workspace
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Q.3
Which one of the following statements is TRUE about every n × n matrix with only real eigenvalues?
A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.
B. If the trace of the matrix is positive, all its eigenvalues are positive.
C. If the determinant of the matrix is positive, all its eigenvalues are positive.
D. If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.
Explaination / Solution:

If the trace of the matrix is positive and the determinant of the matrix is negative then atleast one of its eigen values is negative. Since determinant = product of eigen values.

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Q.4
The value of the integral ( where C is a closed curve given by |z| = 1) is
A. πi
B. πi/5
C. 2πi/5
D. -πi
Explaination / Solution:
No Explaination.

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Q.5
An unbalanced dice (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is 90% of the probability that the face value is even. The probability of getting any even numbered face is the same. If the probability that the face is even given that it is greater than 3 is 0.75, which one of the following options is closest to the probability that the face value exceeds 3?
A. 0.453
B. 0.468
C. 0.485
D. 0.492
Explaination / Solution:
No Explaination.

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Q.6
The value of where δ(t) is the Dirac delta function, is
A. 1/2e
B. 2/e
C. 1/e2
D. 1/2e2
Explaination / Solution: Workspace
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Q.7
Let A be a 4 × 3 real matrix with rank 2. Which one of the following statement is TRUE?
A.  Rank of AT A is less than 2.
B.  Rank of AT A is equal to 2.
C.  Rank of AT A is greater than 2
D.  Rank of AT A can be any number between 1 and 3.
Explaination / Solution: Workspace
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Q.8
The respective expressions for complimentary function and particular integral part of the solution of the differential equation are
A. [c1 + c2x + c3 sin√3x + c4 cos√3x] and [3x4 - 12x2 + c]
B. [c2x+ csin√3x + c4 cos√3x] and [5x4 - 12x2 + c]
C. [c1 + c3 sin√3x + c4 cos√3x] and [3x4 - 12x2 + c]
D. [c1 + c2x + c3 sin√3x + c4 cos√3x] and [5x4 - 12x2 + c]
Explaination / Solution: Workspace
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Q.9
The distance between the origin and the point nearest to it on the surface z2 = 1 + xy is
A. 1
B. √3/2
C. √3
D. 2
Explaination / Solution:
No Explaination.

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Q.10
Newton-Raphson method is used to compute a root of the equation x2 - 13 = 0 with 3.5 as the initial value. The approximation after one iteration is
A. 3.575
B. 3.676
C. 3.667
D. 3.607