Engineering Mathematics Test - 4 - PDF Flipbook
Engineering Mathematics Test - 4
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Test-04Solutions
ENGINEERING MATHEMATICS
1. The number of substrings (of all lengths inclusive) that can be
formed from a character string of length n is
a) n
b) n2
c) ( −1)
2
d) ( +1)
2
Answer: (d)
2. How many 4 digit even numbers have all 4 digits distinct?
a) 2240
b) 2296
c) 2620
d) 4536
Answer: (b)
Solution:
Number of 4- digits even numbers ending with zero = 9.8.7 =
504
Number of 4-digits numbers not ending with zero = 8.8.7 =
1792 required number of 4-digits even numbers = 504 + 1792 =
2296
1
3. In a connected graph, a bridge is an edge whose removal
disconnects a graph. Which one of the following statements is
true?
a) A tree has no bridge
b) A bridge cannot be part of a simple cycle
c) Every edge of a clique with size > 3 is a bridge (A clique is
any complete sub graph of a graph)
d) A graph with bridge cannot have a cycle
Answer: (b)
Solution:
In a connected graph, an edge is a bridge iff the edge is not a
part of any cycle in G.
∴ Option (b) is correct
4. Let R be a non-empty relation on a collection of sets defined by
ARB if and only if A ∩ B ≠ ∅. Then, (pick the true statement)
a) R is reflexive and transitive
b) R is symmetric and not transitive
c) R is an equivalence relation
d) R is not reflexive and not symmetric
Answer: (b)
Solution:
A ∩ A = A ≠
⟹ A is not related to A
R is not reflexive
Let AR B ⟹ (A ∩ B)
2
⟹ (B ∩ A)
⟹ BR A
R is not reflexive
Let A = {1, 2}, B = {3, 4}, C = {1, 5}
Here ARB and BRC but A is not related to C
R is not transitive
5. Let A be a set with n elements. Let C be a collection of distinct
subsets of A such that for any two subsets S1 and S2 in C, either
S1⊂S2 or S2 ⊂ S1. What is the maximum cardinality of C?
a) n
b) n + 1
c) 2n-1 + 1
d) n!
Answer: (b)
Solution:
Let A = {a1, a2, … an)
To meet the given conditions, the subsets are { , {a1}, {a1, a2},
{a1, a2, a3}…., A}
6. How many different non-isomorphic Abelian groups of order 4
are there?
a) 2
b) 3
c) 4
d) 5
Answer: (a)
3
Solution:
Let G = {e, a, b, c} in a group of order 4 w.r.t.* where e is
identity element.
Any group with 4 elements is abelian the 4 possible Abelian
group are
(i) e-1 = e, a-1 = a, b-1 = b, c-1 = c
(ii) e-1 = e, a-1 = b, b-1 = c, c-1 = c
(iii) e-1 = e, a-1 = a, b-1 = c, c-1 = b
(iv) e-1 = e, a-1 = c, b-1 = b, c-1 = a
The groups defined in ii, iii&iv are isomorphic
Only two non-isomorphic Abelian of order 4 are possible
7. For the composition table of a cyclic group shown below
Which one of the following choices is correct?
a) a, b are generators
b) b, c are generators
c) c, d are generators
d) d, a are generators
Answer: (c)
Solution:
A is identify element and therefore cannot be a generator
B2 = b *b = a
4
i.e order of b = 2, therefore cannot be a generator
i.e order of b = 2, therefore b is not a generator
Number of generator = (4) = 2
Where = Euler function the generators are c and d
8. Consider the following relation on subsets of the set S integers
between 1 and 2014. For two distinct subsets U and V of S, we
say U < V if the minimum element in the symmetric difference
of the two sets is in U.
Consider the following two statements:
S1: There is a subset of S that is larger than every other subset.
S2: There is a subset of S that is smaller than every other subset.
Which one of the following is CORRECT?
a) Both S1 and S2 are true
b) S1 is true and S2 is false
c) S2 is true and S1 is false
d) Neither S1 nor S2 is true
Answer: (a)
Solution:
If we choose U = {1} and V = S – U, then U is less than every
other subset
Both S1 and S2 are true.
9. Let R be a relation on the set of ordered pairs of positive
integers such that ((p, q), (r, s)) ∈ R if and only if p-s = q-r.
Which one of the following is true about R?
a) Both reflexive and symmetric
5
b) Reflexive but not symmetric
c) Not reflexive but symmetric
d) Neither reflexive nor symmetric
Answer: (c)
Solution:
Given that
(p, q)R (r, s) ⇔ (p ‒ s) = (q ‒ r)
Now (p ‒ q) ≠ (q ‒ p)
⟹ (p, q) is not related to (p, q)
⟹ R is not reflexive
Let (p, q)R (r, s)
⟹p‒s=q‒s
⟹r‒q=s‒p
⟹ (r, s)R (p, q)
⟹ R is symmetric
R is symmetric but not reflexive
10. A binary relation R on N*N is defined as follows: (a, b) R (c,
d) if a ≤ c or b ≤ d. Consider the following propositions:
P: R is reflexive
Q: R is transitive
Which one of the following statements is TRUE?
a) Both P and Q are true.
b) P is true and Q is false.
c) P is false and Q is true.
d) Both P and Q are false.
6
Answer: (b)
Solution:
P: we have a ≤ a and b ≤ b
⟹ (a, b) R(a, b)
⟹ R is reflexive
P is true
Q: we have (2, 2) R(1, 2) and (1, 1)
But (2, 2) is not related to (1, 1)
⟹ R is non-transitive
11. For A = �− 1 tan �, the determinant of ATA-1 is
tan 1
a) sec2x
b) cos4x
c) 1
d) 0
Answer: (c)
Solution:
| −1| = | || −1|
= | |. 1 = 1
| |
∴ | | = (1 + 2 ) = 2 = 1
12. We have a set of 3 linear equations in 3 unknowns. 'X ≡ Y'
means X and Y are equivalent statements and 'X ≢ Y' means X
and Y are not equivalent statements.
P: There is a unique solution.
Q: The equations are linearly independent.
7
R: All eigen values of the coefficient matrix are non-zero.
S: The determinant of the coefficient matrix is nonzero.
Which one of the following is TRUE?
a) P ≡ Q ≡ R ≡ S
b) P ≡ R ≢Q ≡ S
c) P ≡ Q ≢R ≡ S
d) P ≢Q ≢ R ≢S
Answer: (a)
Solution:
If there is a unique solution, then | | ≠ 0
⇒ All eigenvalues of ‘A’ are non-zero and the given equations
are linearly independent.
13. The smallest and largest Eigen values of the following matrix
3 −2 2
are:�4 −4 6�
2 −3 5
a) 1.5 and 2.5
b) 0.5 and 2.5
c) 1.0 and 3.0
d) 1.0 and 2.0
Answer: (d)
Solution:
The characteristic equation of the given matrix is
− 3 + 4 2 − 5 + 2 = 0
⇒ ( − 2)(− 2 + 2 − 1) = 0
∴ = 2,1
8
14. The two Eigen Values of the matrix �21 1 �have a ratio of 3:1
for p=2. What is another value of ' p' for which the Eigen values
have the same ratio of 3:1?
a) ‒2
b) 1
c) 7 /3
d) 14/3
Answer: (d)
Solution:
Let the eigen values be &
Given that : = 3 : 1
i.e., =3 .....(1)
Moreover ( + ) = (2 + p)
4 = (2 + p) ....... (2) (using (1))
= (2p – 1)
3β2 = (2p - 1) .... (3) (using (1))
By solving (2) & (3) for 'p'
we get p = 2, 14
3
15. At least one eigenvalue of a singular matrix is
a) positive
b) zero
c) negative
d) imaginary
Answer: (b)
9
Solution:
Product of eigen values
= Determinant of the matrix
= 0 (∴ singular matrix)
∴At least one eigenvalue is zero.
16. For a given matrix P = �4 + 3 4 −− 3 � where = √−1, the
inverse of matrix P is
a) 1 �4 − 3 4 3 �
24 − +
b) 1 �4 3 4 −− 3 �
25 +
c) 1 �4 + 3 4 −− 3 �
24
d) 1 �4 + 3 4 −− 3 �
25
Answer: (a)
Solution:
| | = (4 + 3 )(4 − 3 ) + 2
= 16 + 9 − 1 = 24
∴ P-1 = = 1 �4 − 3 4 3 �
| | 24 − +
17. The value of the integral ∫02 ( −1)2 ( −1) is
( −1)2+ ( −1)
a) 3
b) 0
c) -1
d) -2
Answer: (b)
10
Solution:
Let (x – 1) = t
⇒ x = (t + 1)
∶ 0 → 2 ⇒ : −1 → 1
∴ ∫02 ( −1)2 ( −1) = ∫−11 � 2 2+ � = 0
( −1)2+ ( −1)
∵ Integrand is an odd function
(or)
Use the property
∫02 ( ) = 0{ (2 − ) = − ( )}
18. The value of the integral ∫02 ∫0 + is
a) 1 ( − 1)
2
b) 1 ( 2 − 1)2
2
c) 1 ( 2 − )
2
d) 1 � − 1 �2
2
Answer: (b)
Solution:
∫02 ∫0 + = ∫02 ( )0
= ∫02 ( − 1)
= ∫02( 2 − )
= �� 22 � − 2
�
0
= �� 2 4 − 2� − �21 − 1��
11
= 4 − 2 + 1
2 2
= � 2−1�2
2
19. = 0 + 1 −1 + − − − − − + −1 −1 +
where ai (i = 0 to n) are constants then + is
a)
b)
c) nf
d) �
Answer: (c)
Solution:
Given
= 0 + 1 −1 + − − − − − + −1 −1 +
⇒ f is a homogenous polynomial in x and y of degree 'n'
∴ By Euler's theorem for homogenous function, we have
∂f + y ∂f = nf
∂x ∂y
20. A function f(x) is linear and has a value of 29 at x = ‒ 2 and 39
at x = 3. Find its value at X = 5.
a) 59
b) 45
c) 43
d) 35
Answer: (c)
12
Solution:
Let f(x) = ax + b.......... (i)
f (‒2) = 29
⇒ ‒2a + b = 29......... (ii)
f(3) = 39
⇒3a + b = 39........ (iii)
By solving equation (ii) and (iii)
a = 2 and b = 33
∴ f(x) = 2x + 33
⇒ f(5) = 43
21. The maximum area (in square units) of a rectangle whose
vertices lie on the ellipse x2 + 4y2 = 1 is ____
Answer: 1
Solution:
Let (2x) & (2y) be the length & breadth of the rectangle
Let A = 2x × 2y = 4xy be the area of the rectangle.
Then A2 = 4x2y2 = x2(1 ‒ x2) = x2 ‒ x4
Let f(x) = x2 ‒ x4
Then f '(x) = 2x ‒ 4x3 & f "(x) = 2 ‒ 12x2
For maximum, we have
f'(x) = 0
13
⇒ 2x(1 ‒ 2x2) = 0 ⇒ = 0, 1 , −1
√2 √2
Here ′′(0) > 0, ′′ �√12� < 0
∴ = 4 × = 4 × √1− 2
2
= 2 √1 − 2 = 2 × 1 × �1 − 1 = 1
√2 2
22. The value of ∑∞ =0 �21� is______.
Answer: 2
Solution:
∑ ∞ =0 �21� = �21� + 2 �12�2 + 3 �12�3 + 4 �21�4 + ⋯ ….
= 1 �1 + 2 �21� + 3 �21�2 + 4 �12�3 + ⋯ �
2
= 1 �1 − 21�−2 = 2
2
23. For a vector E, which one of the following statements is NOT
TRUE?
a) If ∇. = 0, E is called solenoidal
b) If ∇ × = 0, E is called conservative
c) If ∇ × = 0, E is called irrotational
d) If ∇. = 0, E is called irrotational
Answer: (d)
Solution:
If div E = or ∇.E = 0 then E is called solenoidal vector
If Curl E = 0 or ∇×E = 0 then E is called irrotational & E is
conservative
14
From the above, ∇.E = 0 ⟹ E is an irrational vector is a wrong
statement
24. Given a vector field � = y2x � – yz � – x2 � , the line integral
∫ . evaluated along a segment on the x axis from x = 1 t o x
= 2 is
a) 2.33
b) 0
c) 2.33
d) 7
Answer: (b)
Solution:
Given F� = y2 x � x – yz � y – x2 � z
Along x-axis, we have y = 0 & z = 0
⟹dy = 0 & dz = 0
Hence x : 1 to 2
∫ � .d ̅ =∫ 1 =∫ 2 =1 2 = 22 2 = 0
25. Two cards are drawn at random in succession with
replacement from a deck of 52 well shuffled cards Probability of
getting both 'Aces' is
a) 1
169
b) 2
169
c) 1
13
d) 2
13
15
Answer: (a)
Solution:
Probability of drawing an ace card from a pack of cards = 4 1
52 1
Required probability = P(E) = 4 1 × 4 1 = 1
52 1 52 1 169
26. An examination consists of two papers, paper 1 and paper 2.
The probability of failing in paper 1 is 0.3 and that in paper 2 is
0.2. Given that a student has failed in paper 2, the probability of
failing in paper 1 is 0.6. The probability of a student failing
in both the papers is
a) 0.5
b) 0.18
c) 0.12
d) 0.06
Answer: (c)
Solution:
Given that P(1) = 0.3, P(2) = 0.2, P(1/2) = 0.6
Using conditional probability the probability that a student is
failed in paper 2 is given by P(1/2) = (1∩2)
(2)
⟹ 0.6 = (1∩2)
0.2
Required probability = (1 ∩ 2) = 0.12
16
27. Px(X) = Me(−2| |) + (−3| |) is the probability density
function for the real random variable X, over the entire x-axis,
M and N are both positive real numbers. The equation relating
M and N is
a) + 2 = 1
3
b) 2 + 1 = 1
3
c) + = 1
d) + = 3
Answer: (a)
Solution:
∫−∞∞ ( ) = 1
∫−∞∞[ −2| | + −3| |]dx = 1
⟹ 2M ∫0∞ −2 + 2 ] ∫0∞ −3 = 1
⟹ M + 2 3 = 1
28. Three values of x and y are to be fitted in a straight line in the
form = + by the method of least squares. Given Σ =
6, Σy = 21, Σ 2 = 14 Σxy = 46, the values of a and b are
respectively
a) 2, 3
b) l, 2
c) 2, 1
d) 3, 2
Answer: (d)
17
Solution:
Σ = 6, Σy = 21, Σ 2 = 14
Σxy = 46 and three values of x and y to fit the stright line
Let y = a + bx
Then the normal equation are Σ =na + b Σ
Σ = a Σ + b Σ 2
Where n = number of points
21 = 3a + 6b
46 = 6a + 14b
a=3
29. A screening test is carried out to detect a certain disease. It is
found that 12% of the positive reports and 15% of the negative
reports are incorrect. Assuming that the probability of a person
getting positive report is 0.01, the probability that a person
tested gets an incorrect report is
a) 0.0027
b) 0.0173
c) 0.1497
d) 0.2100
Answer: (c)
Solution:
Given probability of getting positive report = 0.01
And probability of getting positive report = 0.99
Required probability = probability of getting incorrect report
when it is test positive or negative = (0.01) (0.12) + (0.99) (0.15)
18
= 12 +9190×00105
10000
= 0.1497
30. An observer counts 240vah/h at a specific highway location.
Assume that the vehicle arrival at the location is Poisson
distributed, the probability of having one vehicle arriving over a
30-second time interval is_____
Answer: 0.27
Solution:
Given that = 240 veh/h = 240 veh/min = 4 veh/min = 2veh/sec
60
The required probability = P(X = 1) = − = 2 −2 = 0.27
19
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