Engineering Mathematics Test - 1 - PDF Flipbook
Engineering Mathematics Test - 1
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GATE
EEE
Engineering
Mathematics
Test-01Solutions
ENGINEERING MATHEMATICS
1. If A and B are real symmetric matrices of order n then which of
the following is true.
a) AAT = I
b) A = A-1
c) AB = BA
d) (AB)T = BTAT
Answer: (d)
Solution:
By the properties of transpose of matrices option (d) is correct.
i.e. By the reversal law of the transpose of the product matrices,
we have (AB)T = BTAT 11� are
2. The Eigen values of the matrix �
a) (a + 1), 0
b) a, 0
c) (a – 1), 0
d) 0, 0
Answer: (a)
Solution:
= � 11�
⇒ | − | = 0
⇒ � − 1 � = 0
−
1
⇒ 2 − ( + 1) + 0 = 0
∴ = 0, + 1 ℎ
1
3. If for a matrix, rank equals both the number of rows and number
of columns, then the matrix is called
a) Non-singular
b) Singular
c) Transpose
d) Minor
Answer: (a)
Solution:
Given that both rows and columns are equal.
⇒ The given matrix is square matrix and also given that ( )=
number of rows of A = number of columns of A
∴ The matrix order is same as rank of A.
Hence, the matrix is non – singular matrix.
13 2
4. If the determinant of the matrix �0 5 −6� is 26, then the
8
27
27 8
determinant of the matrix �0 5 −6� is
13 2
a) -26
b) 26
c) 0
d) 52
Answer: (a)
Solution:
By the properties of determinant of the matrices, if two rows are
interchanged in a determinant then the value of the determinant
2
does not change but sign will change (In this problem R1 and
R3 are interchanged.)
5. In matrix algebra AS = AT (A, S, T, are matrices of appropriate
order) implies S = T only if
a) A is symmetric
b) A is singular
c) A is non-singular
d) A is skew symmetric
Answer: (c)
Solution:
By left cancellation law of matrix multiplication,
we have,
AS = AT ⇒ S = T only if A is non- singular
6. The eigen values and the corresponding eigen vectors of a 2 × 2
matrix is given by
Eigen value Eigen vector
1 = 8 1 = �11�
2 = 4 2 = �−11�
The matrix is
a) �62 62�
b) �64 46�
3
c) �24 42�
d) �48 84�
Answer: (a)
Solution:
Given 1 = 8, 2 = 4 and 1 = �11� , 2 = �−11�
A = PDP-1
Where = [ 1 2] and = �80 40� −11�−1
= �11 −11� �80 04� �11
∴ = �88 −44� �11 −11� 1
2
= 1 �142 142� = �62 26�
2
7. The function = 2 + 250 at x = 5 attains
a) Maximum
b) Minimum
c) Neither
d) 1
Answer: (b)
Solution:
′ = 2 − 250 = 0 ⇒ = 5 ℎ
2
′′ = 2 + 500 ⇒ ′′(5) = 2 + 4 = 6 > 0
3
⇒ = 5
4
8. The function f(x) = x3 – 6x2 + 9x + 25 has
a) maxima at x=l and minima at x = 3
b) maxima at x=3 and minima at x = 1
c) no maxima, but minima at x = 3
d) maxima at x = 1, but no minima
Answer: (a)
Solution:
′( ) = 3 2 − 12 + 9 = 0 ⇒ = 1, 3
′′( ) = 6 − 12
′′(1) = −6 < 0 ⇒ f(x) has a maximum at x = 1
′′(3) = −6 > 0 ⇒ f(x) has a minimum at x = 3
9. If a function is continuous at a point its first derivative
a) may or may not exist
b) exists always
c) will not exist
d) has a unique value
Answer: (a)
Solution:
Every differentiable function is continuous but a continuous
function may or may not be differentiable.
10. A discontinuous real function can be expressed as
a) Taylor's series and Fourier's series
b) Taylor's series and not by Fourier's series
c) neither Taylor's series nor Fourier's series
d) not by Taylor's series, but by Fourier's series
5
Answer: (d)
Solution:
Taylor's series exist only for continuous and differentiable
functions and Fourier's series exist even though the function
have finite no. of discontinuous points
11. The expression curl (grad f) where f is a scalar function is
a) Equal to ∇2
b) Equal to div (grad f)
c) A scalar of zero magnitude
d) A vector of zero magnitude
Answer: (d)
Solution:
curl (grad f) = ̅
12. For the function ϕ = ax2 y – y3 to represent the velocity
potential of an ideal fluid, ∇2ϕ should be equal to zero. In that
case, the value of ‘a’ has to be
a) -1
b) 1
c) -3
d) 3
Answer: (d)
Solution:
∅ = 2 − 3 = ∇2∅
= 2∅ + 2∅ + 2∅ = 0
2 2 2
⇒ 2 − 6 = 0 ⇒ = 3
6
13. The directional derivative of the following function at (1, 2) in
the direction of (4i + 3j) is: f(x, y) = x2 + y2
a) 4/5
b) 4
c) 2/5
d) 1
Answer: (b)
Solution:
f(x, y) = x2 + y2, � = 4 + 3 , = (1, 2)
(∇ ) = (2 ̅ + 2 � )(1,2) = 2 + 4
Directional derivative = (∇ ) . � = (2 + 4 ). (4 +3 ) = 4
| � | 5
14. For the scalar field = 2 + 2 , the magnitude of the gradient
2 3
at the point (1, 3) is
a) �193
b) �29
c) √5
d) 9
2
Answer: (c)
Solution:
∇ = + 2
3
∇ (1,3) = + 2
|∇ | = √1 + 4 = √5
7
15. ∇ × (∇ × ) where P is a vector is equal to
a) × ∇ × − ∇2
b) ∇2 + ∇(∇. )
c) ∇2 + (∇ × )
d) ∇(∇. ) − ∇2
Answer: (d)
Solution:
We have the vector identity
Curl (Curl P) = ∇ × (∇ × ) = grad (div P) – ∇2
= ∇(∇. ) − ∇2
16. The probability that two friends share the same birth-month is
a) 1/6
b) 1/12
c) 1/144
d) 1/24
Answer: (b)
Solution:
One of two persons may born in any month.
Then the probability that the second person also may born in the
same month is 1/12. ∴Required probability P(E) = 1/12
17. Suppose that the expectation of a random variable X is 5.
Which of the following statements is true?
a) There is a sample point at which X has the value= 5
b) There is a sample point at which X has the value >5
c) There is a sample point at which X has a value 5
8
d) None of the above
Answer: (c)
Solution:
Option ‘c’ is the correct statement because if all sample points
are less than 5 then expectation cannot be 5.
18. Four arbitrary points (x1, y1), (x2, y2), (x3, y3), (x4, y4), are
given in the xy - plane using the method of least squares. If
regression of y upon x gives the fitted line y = ax + b; and
regression of x upon y gives the fitted line x = cy + d, then
a) the two fitted lines must coincide
b) the two fitted lines need not coincide
c) it is possible that ac=0
d) A must be 1/c
Answer: (b)
Solution:
Other cases are not possible.
19. Four fair coins are tossed simultaneously. The probability that
at least one heads and at least one tails turn up is
a) 1
16
b) 1
8
c) 7
8
d) 15
16
Answer: (c)
9
Solution:
( ) = 16
Probability of all heads or all tails appearing = 1 + 1 = 1
16 16 8
∴ Required probability = 1 – 1 = 7
8 8
20. A hydraulic structure has four gates which operate
independently. The probability of failure of each gate is 0.2.
Given that gate l has failed, the probability that both gates 2 and
3 will fail is
a) 0.240
b) 0.200
c) 0.040
d) 0.008
Answer: (c)
Solution:
( ) = 0.2: ( = 1, 2, 3, 4)
( 2 ∩ 3/ 1) = ( 2 ∩ 3 ∩ 1)/ ( 1)
= 0.2×0.2×0.2 = 0.04
0.2
21. The necessary and sufficient condition for the differential
equation of the form M (x, y)dx + N(x, y)dy = 0 to be exact is
a) M =N
b) =
c) =
d) 2 = 2
2 2
10
Answer: (c)
Solution:
By a theorem, the necessary and sufficient for the differential
equation of the form M dx + N dy = 0 to be exact is = .
22. The solution of a differential equation
y" + 3y' + 2y = 0
a) c1ex + c2e2x
b) c1e-x + c2e3x
c) c1e-x + c2e-2x
d) c1e-2x + c22-x
Answer: (c)
Solution:
Given y" + 3y' + 2y = 0 …... (1)
⇒ ( 2 + 3 + 2) = 0
Consider 2 + 3 + 2 = 0
= −1, −2
∴ = 1 − + 2 −2 is the general solution of (1)
23. For the differential equation ( , ) + ( , ) = 0 to be
exact is
a) =
b) =
c) f = g
d) 2 = 2
2 2
11
Answer: (b)
Solution:
Given ( , ) + ( , ) = 0
⇒ ( , ) + ( , ) = 0
⇒ ( , ) + ( , ) = 0
Now, the condition for exactness is =
24. The equation 2 − ( 2 + 4 ) + = 8 −8
2
a) partial differential equation
b) non-linear differential equation
c) non-homogeneous differential equation
d) ordinary differential equation
Answer: (c)
Solution:
According to classification of differential equations, the given
differential equation is linear-non-homogeneous ordinary
differential equation.
25. The solution of the differential equation 2 + 2 = 0
2
a) = 1
+
b) = − 3 +
3
c) c ex
d) unsolvable as equation is non-linear
Answer: (a)
12
Solution:
Given dy + 2 = 0
dx
⇒ ∫ = − ∫ + ⇒ − 1 = − +
2
⇒ 1 = − ⇒ = 1 ( = − )
+
26. (s+1)-2 is the Laplace transform of
a) t2
b) t3
c) e-2t
d) te-t
Answer: (d)
Solution:
{( + 1)−2} = −
27. Let F(s) = L [f(t)] denote the Laplace transform of the function
f(t). Which of the following statements is correct?
a) L[df /dt] = 1/s F(s);
�∫01 ( ( ))� = ( ) − (0)
b) L[df /dt] = s F(s) – F(0);
�∫01 ( ( ))� = − /
c) L[df /dt] = s F(s) – F(0);
�∫01 ( ( ))� = ( − )
d) L[df /dt] = s F(s) – F(0);
�∫01 ( ( ))� = 1 ( )
13
Answer: (d)
Solution:
� � = ( ) − (0)
�∫0 ( ) � = ( )
28. The inverse Laplace transform of the function +5 is
( +1)( +3)
________.
a) 2e-t – e-3t
b) 2e-t + e-3t
c) e-t – 2e-3t
d) e-t +2e-3t
Answer: (a)
Solution:
+5 = 2 − 1
( +1)( +3) +1 +3
−1 �( +1 +)(5 +3)� = 2 −1 � +11� − −1 � +13�
= 2 − − −3
29. The value of the contour integral ∫| − |=2 1 in the
2+4
positive sense is
a)
2
b) −
2
c) −
2
d)
2
14
Answer: (d)
Solution:
Given = ∫ 1 where C is | − | = 2
2+4
The integrand is not analytic at = ±2 and z = 2i lies insides
C. By Cauchy integral formula
= ∫ � +12 � = 2 � +12 � =2
−2
= 2 �2 +1 2 � =
2
30. Starting from x0 = 1, one step of Newton-Raphson method in
solving the equation x3 + 3x – 7 = 0 gives the next value x1 as
a) x1 = 0.5
b) x1 = 1.406
c) x1 = 1.5
d) x1 = 2
Answer: (c)
Solution:
( ) = 3 + 3 − 7 0 = 1
⇒ 1( ) = 3 2 + 3
Newton Raphson’s formula is
+1 = − ( )
1( )
1 = 0 − ( 0) = 1 − (1) = 1 − (−3) = 3 = 1.5
1( 0) 1(1) 6 2
∴ 1 = 1.5
15
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