Differential Calculus
Multiple Choice Question
1.
The first three terms in the power series for log(1 + sin 𝑥) are
1
1
1
1
a)𝑥 − 2 𝑥 3 + 4 𝑥 5
b) 𝑥 + 2 𝑥 3 + 4 𝑥 5
1
1
1
c)−𝑥 − 2 𝑥 3 + 4 𝑥 5
2.
3.
4.
5.
1
d) 𝑥 − 2 𝑥 2 + 6 𝑥 3
In the Taylor series expansion of 𝑒 𝑥 + sin 𝑥 about the point 𝑥 = 𝜋 the co-efficient of (𝑥 − 𝜋)2 is
a)𝑒 𝜋
b)0.5𝑒 𝜋
c)𝑒 𝜋 + 1
d)𝑒 𝜋 − 1
Which of the following functions would have only odd powers of 𝑥 in its Taylor series expansion about the
point 𝑥 = 0.
a)sin(𝑥 3 ) b)sin(𝑥 2 )
c)cos(𝑥 2 )
d)cos(𝑥 3 )
The first three terms in expansion of 𝑥 4 − 3𝑥 3 + 2𝑥 2 − 𝑥 + 1 in powers of (𝑥 − 3) is
a) 16 − 38(𝑥 − 3) + 29(𝑥 − 3)2 b) 16 + 38(𝑥 − 3) − 29(𝑥 − 3)2
c) 16 − 38(𝑥 − 3) − 29(𝑥 − 3)2 d) 16 + 38(𝑥 − 3) + 29(𝑥 − 3)2
tan 𝑥
Expansion of log ( 𝑥 )
𝑥2
7
𝑥2
𝑥3
7
𝑥5
𝑥2
7
6.
a) 3 − 90 𝑥 4 + ⋯b)− 3 − 90 𝑥 4 + ⋯c)𝑥 + 3 − 30 + ⋯d) 3 + 90 𝑥 4 + ⋯
Expansion of sin 𝑥 cosh 𝑥
7.
a) 𝑥 + 3 + 30 + ⋯
Expansion of sin(𝑒 𝑥 − 1) is
𝑥3
𝑥2
8.
𝑥5
𝑥3
𝑥5
𝑥2
5
b) 𝑥 − 3 − 30 + ⋯
5
𝑥3
𝑥5
𝑥2
5
c)𝑥 + 3 − 30 + ⋯
a)𝑥 + 2 − 24 𝑥 4 + ⋯
b) 𝑥 + 2 + 24 𝑥 4 + ⋯ c)𝑥 − 2 − 24 𝑥 4 + ⋯
𝑥3
𝑥3
𝑥4
𝑥2
𝑥3
d) none
d) none
Expansion of log(1 + 𝑥 + 𝑥 2 + 𝑥 3 + 𝑥 4 ) is
𝑥2
𝑥4
𝑥2
a)− 𝑥 − 2 − 3 − 4 + ⋯b)𝑥 + 2 + 3 + 4 + ⋯
𝑥4
𝑥2
𝑥4
𝑥5
c) 𝑥 − 2 + 3 − 4 + ⋯d) 𝑥 + 2 + 3 + 4 + ⋯
9.
𝑥3
𝑥5
𝑥7
𝜋
The limit of the series 𝑓(𝑥) = 𝑥 − 3! + 5! − 7! + ⋯ as 𝑥 approaches 2 is
2𝜋
10.
a) 11.
12.
13.
𝜋
a) 3
b)𝜋/2
c)3
The Taylor series expansion of 𝑠𝑖𝑛(𝑥) at 𝑥 = 𝜋 is given by
(𝑥−𝜋)
1!
+
(𝑥−𝜋)3
3!
+ ⋯…
b)−1 −
(𝑥−𝜋)2
3!
+⋯
d)1
c) 1 −
(𝑥−𝜋)2
3!
+⋯
The limit of the series 𝑓(𝑥) = 1 − 𝑥 + 𝑥 2 − 𝑥 3 + 𝑥 4 − ⋯ as 𝑥 approaches ½ is
a) 2/3
b) 1/3
c) 1
d) 4/3
4
3
2
Representation of (𝑥 − 2) − 3(𝑥 − 2) + 4(𝑥 − 2) + 5 in powers of 𝑥 is
a) 61 − 84𝑥 + 4𝑥 2 − 11𝑥 3 + 𝑥 4
b) 61 + 84𝑥 − 4𝑥 2 − 11𝑥 3 + 𝑥 4
2
3
4
c) 61 + 84𝑥 + 4𝑥 − 11𝑥 + 𝑥
d) 61 + 84𝑥 + 4𝑥 2 − 11𝑥 3 − 𝑥 4
The limit of the series 𝑓(𝑥) = 1 + 𝑥 + 𝑥 2 + 𝑥 3 + 𝑥 4 + ⋯ as 𝑥 approaches ½ is
a) 4
b) ∞
c) 3
d) 2
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d)−1 +
(𝑥−𝜋)2
3!
+⋯
𝑝𝑥+sin 𝑥
is finite
𝑥2
𝑥 →0
14.
The integer p for which the lim
d)2
15.
a)0
b)-1
c)1
1+2+3+⋯+𝑛
The value of lim
is
𝑛2
a)0
c) -1
𝑛 →0
16.
b)1
The value of lim 𝑥 𝑥 is
𝑥 →0
a)0
17.
b)-1
d) -52
𝑥 →0
b)- log 2
c) log 3
sin 𝑥
The value of lim (
) 1𝑥 is
𝑥
𝑥 →0
1
If lim
tan(𝑥 2 )
𝑥𝑛
d) 0
1
b)1
c) 2
d) 2
= 1 then value of n is
a)0
b)1
The value of lim 𝑥 log 𝑥 is
c)2
d) 3
𝑥 →0
a) 1
23.
c) 2
The value of lim ( 𝑥 − log (2 − 𝑥)) is
𝑥 →0
22.
d) -2
−1
b) 5
a) 𝑒
21.
1
c) 2
1−2 cos 𝑥+cos 2𝑥
Find the value of lim ( 1−cos 2𝑥 )
𝑥 →0
a) Log 1
20.
d)none
b) 1
a) -1
19.
c)1
x− sin 𝑥
The value of lim ( 𝑥𝑠𝑖𝑛 𝑥 ) is
𝑥 →0
a) 0
18.
1
d) 2 .
1
b) -1
sin 𝑥
tan 𝑥
The value of lim 𝑥 − 𝑥 – cosh 𝑥 is
𝑥 →0
a) -1
b) 1
𝑥
c) 2
d) 0
c) -2
d)-3
𝑥
⁷24. The value of lim (2 − 3 ) is
𝑥
2
a) Log(3)
25.
𝑥 →0
d)
𝑒
b) 0
c) 2
log 𝑥
The value of lim ( 𝑐𝑜𝑡𝑥 ) is
𝑥 →0
log 3
b) -1
d) 1
c) 0
d) -2
𝑥 𝑎 − 𝑎𝑥
The value of lim 𝑥 𝑥− 𝑎𝑎 is
1+log 𝑎
a) 1−log 𝑎
28.
log 2
𝑥 →0
a) 1
27.
c)
The value of lim (cot 𝑥 sin 𝑥 ) is
a) e
26.
3
b) log (2)
𝑥 →𝑎
1
b) 2(1 + log a)
1−log 𝑎
c) 1+log 𝑎
d) 0
1
The value of lim (cos(𝑥) 𝑥 2 is
𝑥 →0
1
a)- 2
−𝑒
b) 2
−1
c) 1
d) 𝑒 2
29.
Find the value of c (a point where slope of a tangent to curve is zero) if f(x) = Sin(x) is continuous over
interval [ 0, π] and differentiable over interval (0, π) and c ∈ (0, π)
𝜋
𝜋
𝜋
𝑎) 2
b) π
c) 4
d) 6
30.
Find the value of c if f(x) = x(x-3) e3x, is continuous over interval [0,3] and differentiable over interval (0,
3) and c ∈ (0,3)
a) 0.369
b) 2.703
d)3
d)0
Find the value of c using Lagrange Mean value theorem for 𝑓(𝑥) = 𝑥 2 + 3𝑥 + 2 in [1, 2]
a)0.5
b)1.5
c)0.75
d)0
31.
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32.
Find the value of c using Lagrange Mean value theorem for 𝑓(𝑥) = 𝑒 𝑥 in [0, 1]
a) (𝑒 − 1)
b)log(𝑒 − 1)
c) 𝑒
d) 0
33.
Find the value of c using Lagrange Mean value theorem for 𝑓(𝑥) = log(𝑥) in [0, e]
a) (𝑒 − 1)
b)log(𝑒 − 1)
c) 𝑒
d) 0
34.
If f(x) is continuous on [a, b] and differentiable on (a, b) then there exists c in (a, b) such that
𝑓(𝑏)−𝑓(𝑎)
a)𝑓 ′ (𝑐) =
𝑏−𝑎
𝑓(𝑏)−𝑓(𝑎)
′
b) 𝑓 (𝑥) = 𝑏−𝑎
𝑓(𝑏)−𝑓(𝑎)
c) 𝑓 ′ (𝑐) = 𝑓(𝑏)
𝑓(𝑏)−𝑓(𝑎)
d) 𝑓 ′ (𝑐) = 𝑏+𝑎
35.
If f(x) is continuous on [a, b], differentiable on (a, b) and f(a)=f(b) then there exists c in (a, b) such that
𝑓(𝑏)−𝑓(𝑎)
a)𝑓 ′ (𝑐) =
𝑏−𝑎
1
b) 𝑓 ′ (𝑐) = 𝑏−𝑎
c) 𝑓 ′ (𝑐) = 𝑎
d) 𝑓 ′ (𝑐) = 0
36.
If f(x) and g(x) are continuous on [a, b], differentiable on (a, b) then there exists c in (a, b) such that
𝑓(𝑏)−𝑓(𝑎)
𝑏−𝑎
𝑓′ (𝑐)
𝑓(𝑏)−𝑓(𝑎)
b) 𝑔′ (𝑐) = 𝑔(𝑏)−𝑔(𝑎)
c) 𝑓 ′ (𝑐) = 𝑎
a)𝑓 ′ (𝑐) =
d) 𝑓 ′ (𝑐) = 0
If f(x)=ex and g(x)=e-x are continuous on [a, b], differentiable on (a, b) then the value of c satisfying
37.
𝑓′ (𝑐)
𝑓(𝑏)−𝑓(𝑎)
= 𝑔(𝑏)−𝑔(𝑎)
𝑔′ (𝑐)
a)0
b)𝑎 + 𝑏
c) 𝑎 − 𝑏
𝑎+𝑏
d) 2
Answers
1.
2.
3.
4.
5.
6.
7.
8.
d
a
a
d
d
c
a
b
9.
10.
11.
12.
13.
14.
15.
16.
d
a
a
a
d
b
d
c
17.
18.
19.
20.
21.
22.
23.
24.
a
c
b
b
c
d
a
a
25.
26.
27.
28.
29.
30.
31.
32.
studymedia.in/fe/m1-mcqs/
d
b
c
a
a
b
b
b
33.
34.
35.
36.
37.
a
a
d
b
d
STES’s
SKN Sinhgad Institute of Technology and Science, Lonavala.
F.E., Semester - I
Engineering Mathematics I
Multiple Choice Questions
Unit I (Mean Value Theorems)
Question
According to Rolle’s mean value theorem, f  x  is continuous in  a, b , differentiable in
 a, b  such that f  a   f b  , then there exits c   a, b  such that
A
f c  0
B
f 'c  0
C
f ''  c   0
D
f 'c  0
Answer
B
Question
According to Lagrange’s mean value theorem, f  x  is continuous in  a, b , differentiable
in  a, b  , then there exits c   a, b  such that
A
f 'c 
f b  f  a 
ba
B
f c 
f b   f  a 
ba
C
f ''  c  
f ' b   f '  a 
ba
D
f 'c  0
Answer
A
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Question
According to Cauchy’s mean value theorem, f  x  & g  x  are continuous in  a, b ,
differentiable in  a, b  such that g  a   g  b  , then there exits c   a, b  such that
A
f '  c  f ' b   f '  a 

g '  c  g ' b   g ' b 
B
f ''  c  f '  b   f '  a 

g ''  c  g '  b   g '  b 
C
f '  c  f b   f  a 

g '  c  g b   g b 
D
f '  c  f b   f  a 

g '  c  g b   g b 
Answer
C
Question
If f  x   x 2  10 x  16 such that f  3  f  7  , then according to Rolle’s theorem c 
A
2
B
3
C
4
D
5
Answer
D
Question
If f  x   x3  12 x defined in 0, 2 3  such that f  0   f 2 3 , then according to Rolle’s

theorem c 
A
2
B
-2
C
1
D
0
studymedia.in/fe/m1-mcqs/

Answer
A
Question
If f  x   sin x in the interval 0, 2  , then according to Rolle’s theorem c 
A
B
C
only

2
only
3
2
both

2
,
3
2
D
none of the above
Answer
C
Question
If f  x   x3  4 x defined in 0, 2 such that f  0   f  2  , then according to Rolle’s
theorem c 
A
B
2
3

2
3
C
0
D
1
Answer
A
Question
If f  x   x  x  2  defined in 1, 3 such that f 1  1, f  3  3 , then according to
Lagrange’s mean value theorem, c 
A
0
B
1
studymedia.in/fe/m1-mcqs/
C
2
D
3
Answer
C
Question
If f  x   x 2 defined in 1, 5 such that f 1  1, f  5  25 , then according to Lagrange’s
mean value theorem, c 
A
0
B
1
C
2
D
3
Answer
D
Question
If f  x   x  x  1 defined in 1, 2 such that f 1  0, f  2   2 , then according to
Lagrange’s mean value theorem, c 
A
1
2
B
3
2
C

D
3
4
Answer
B
Question
If f  x   x 2  3x  2 defined in  1, 2 such that f  1  6, f  2   0 , then according to
3
2
Lagrange’s mean value theorem, c 
A
1
2
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B
1
C
3
2
D
2
Answer
A
Question
If f  x   x 2 , g  x   x 3 defined in 0, 1 such that
f  0  0, f 1  1, g  0   0, g 1  1 , then according to Cauchy’s mean value theorem,
f 'c

g 'c
A
0
B
1
C
-1
D
2
Answer
B
Question
If f  x   x 2  1, g  x   x 3  1 defined in 0, 1 such that
f  0  1, f 1  2, g  0   1, g 1  2 , then according to Cauchy’s mean value theorem,
f 'c

g 'c
A
0
B
1
C
2
D
0.5
Answer
B
Question
If f  x   2 x 2 , g  x   3 x 3 defined in 0, 1 such that
f  0   0, f 1  2, g  0   0, g 1  3 , then according to Cauchy’s mean value theorem,
studymedia.in/fe/m1-mcqs/
f 'c

g 'c
A
2
3
B
3
2
C

2
3
D

3
2
Answer
A
Question
If f  x   3 x 2 , g  x   4 x 3 defined in 0, 1 such that
f  0   0, f 1  3, g  0   0, g 1  4 , then according to Cauchy’s mean value theorem,
f 'c

g 'c
A
0
B
0.5
C
0.75
D
1
Answer
C
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Sequence & series
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( ) where n is positive
1) The sequence
integer is
a) Bounded above
b) bounded below
c) a)& b) both are correct
d) a )& b) both are wrong
2)
The series ∑
∑
a) convergent
c) oscillatory
is
11) The general term of the series
is
a)
(
)
(
c)
)
d) (
)
is
b) divergent
d) None of the above
c)
√
√
is
√
a) (
a) convergent
c) oscillatory
)
12) The general term of the series
b) divergent
d) None of the above
3) The sequence
b) (
b)(
)√
(
d)
)√
)√
(
)√
13) The general term of the series
4) The series ∑
is convergent if
a)
c)
is
b)
d)
5) The series ∑
⁄
a) convergent
c) oscillatory
is
⁄
a) convergent
c) oscillatory
b)
c)
d)
is
b) divergent
d) None of the above
6) The series ∑
a)
b) divergent
d) None of the above
14) The general term of the series
is
a) (
) (
c) (
)
)
(
)
b) (
)
d) (
) (
(
)
)
15) The general term of the series
is
7) The series ∑
(
a) convergent
c) oscillatory
)
a)
is
b) divergent
d) None of the above
( ) is
b) divergent
d) None of the above
8) The sequence
a) convergent
c) oscillatory
9) The general term of the series
is
a) (
)(
)
b) (
)(
)
c) (
) (
)
d) (
) (
)
is
c)
(
)
(
)
b)
d)
) (
b) (
)
)
(
)
c) (
)
(
)
d) (
)
(
)
16) Let ∑
be the series of positive terms. if
then the series is
a) Always divergent
b) Always convergent
c) May or may not be convergent
d) None of the above
17) Let ∑
10) The general term of the series
a)
(
(
)
(
)
be the series of positive terms. if
then the series is
a) Always divergent
b) Always convergent
c) May or may not be convergent
d) None of the above
18) If the alternating series ∑ (
convergent then the sequence
studymedia.in/fe/m1-mcqs/
)
is
must be
a) Monotonically decreasing
b) Monotonically increasing
c) Oscillatory
d) constant
25) The series ∑
is
b)
c)
d)
(
(
( )
)( )(
(
(
)
)
)(
)
(
)(
)
is
(
(
)
)
b)
(
(
)
)
c)
(
(
)
)
d)
(
(
)
)
(
)
23) If
c)
d)
b)
c)
d)
28) If
then
b)
c)
d)
is
b) 1
c)
d)
*
(
)
a)
c)
(
)
b)
d)
then
+
then
) ⁄ is
a)
b)
c)
d) e
then
is
a)
b)
c)
d)
*
(
(
)
+
)
a) 1
c)
32) If
24) If
is
a) 0
31) If
a)
then
a)
30) If
then
is
√
(
22) The general term of the series
a)
b) 1
)
)(
then
)
a) 0
29) If
)
)(
(
)(
)
is
21) The general term of the series
a)
(
(
27) If
b) convergent
d) None of the above
( )
)(
b)
d)
26) If
is
a) Divergent
c) Oscillatory
is divergent if
)√
a)
c)
19) If the alternating series ∑ ( )
is
convergent then the sequence
is
a) divergent
b) Convergent to 0
c) Convergent to non-zero finite number
d) Oscillatory
20) The series ∑
(
then
is
b) 0
d) 2/3
then
a)
b)
c)
d) 1
is
is
33) If
then the series ∑
divergent if
a)
b)
c)
d)
studymedia.in/fe/m1-mcqs/
is
34) If
then the series is
43) The series ∑ (
convergent if
a)
of
b)
c)
d)
35) If
(
c)
d)
[√
36) If
∑
b)
then the series∑
divergent if
a)
b)
c)
d)
is
b)∑
√
c) ∑
b) Diverges
d) None of these
(
(
)
d) ∑ (
)(
)
is
is
b)
d)
47) The series
a)
c)
Convergent
Oscillating
b) Divergent
d) None of these
then the series ∑
48) If
38) Which of the following series converges?
a) ∑
is
a)
c)
d)
37) If
d) None of these
a) Convergent
b) Divergent
c) Oscillating
d) None of these
46) The range of convergence of the series
then
is
c)
c)
45) The series ∑
]
a)
b)
a) Converges
c) Oscillating
is
b)
a)
44) The series ∑
then
)
a)
) converges for the value
)
divergent if
a)
c)
b)
d)
49) If
(
then
)
is
39) Which of the following series diverges?
a)
b)
d) ∑
c)
d)
40) The ratio test fails for the series
50) If
a) ∑ (
b) ∑
)
c) ∑
a) ∑
b) ∑
c) ∑
d) ∑
41) The series ∑ (
a)
c)
)
(
(
(
)
converges if
is convergent
b)
d)
)
+
then
) ⁄ is
a)
b)
c)
d) e
51) The series ∑
b)
d)
42) The series
if
a)
c)
*
(
is divergent if
)√
a)
b)
c)
d)
52) If
a)
studymedia.in/fe/m1-mcqs/
then
is
b)
is
c)
1
d)
c)
53) The series
is
a) Convergent
c) Oscillating
54) If ∑
∑
√
b) Divergent
d) None of these
then
d)
63) The general term of the series
is
a)
(
c) (
is
a) ∑
b) ∑
c) ∑
d) None of these
)(
)
) (
)
b)
(
d) (
)(
)
) (
)
64) The general term of the series
is
the series ∑
55) By ratio test,
is convergent if
a)
c)
c)
b)
d) None of these
65) Let ∑
the series ∑
56) By ratio test,
is divergent if
a)
c)
a)
)
(
)
b)
d)
(
)
(
)
be the series of positive terms. if
then the series is
Always divergent
Always convergent
May or may not be convergent
None of the above
a)
b)
c)
d)
b)
d) None of these
(
66) Which of the following series converges?
a) ∑
b) ∑
√
d) ∑ (
c)∑
57) For a series ∑
( ), the auxiliary series ∑
is
b) ∑
a)∑
b) ∑
c) ∑
d) None of these
c) ∑
d) ∑
59) The series ∑
( )
a)
c)
60) The series
is
a) Convergent
c) Oscillatory
is
b) Divergent
d) None of these
converges if
b)
d)
b)
d)
a)
b)
)
b)
d)
is
b) Diverges
d) None of these
70) The range of convergence of the series
∑
(
)
is
a)
c)
71) If
diverges if
(
is
convergent if
a)
c)
a) Converges
c) Oscillating
converges if
62) The series
68) The series
69) The series ∑
b) Divergent
d) None of these
61) The series
a)
c)
)
67) The ratio test fails for the series
a) ∑ √
58) The series
a) Convergent
c) Oscillatory
)(
divergent if
a)
c)
studymedia.in/fe/m1-mcqs/
b)
d)
then the series ∑
b)
d)
is
a)
b)
c)
d)
80) The general term of the series
is
72) The general term of the series
is
a)
(
)(
b)
)
(
)(
a) (
) (
(c)(
)
)
(
)
b)(
)
d)(
) (
(
)
)
)
81) The general term of the series
c)(
) (
d)(
)
) (
)
is
73) Which of the following series diverges?
a)
∑
c)
∑
(
b)∑
)
a) (
) (
c) (
)
(
74) The series ∑ (
) converges for the value
of
a)
b)
c)
d)None of these
75) If
(
then
)
a)
b)
c)
d)
76) The series ∑
∑
a) convergent
c) oscillatory
c)
(
)
b)divergent
d)None of the above
b)(
d)(
78) The general term of the series
79)
a) (
c) (
d)(
)
(
)
b) 1
c)
d)
)
b)
c) (
)
d) (
)
(
is
)
c)
(
)
d) (
c)
(
)
d)
(
86) The
)√
)√
d) (
)√
√
)
derivative of
(
)
)
√
is
a) (
(
)√
The general term of the series
derivative of
b)
b) (
is
is
a) 0
85) The
a)
is
√
)
84) The
derivative of
)
a) (
b)
is
is
)
(
then
83) The
The general term of the series
(
)
Successive Differentiation
is
a)
)
b)(
d)∑
82) If
77)
)
) is
(
) is
)
)
)
derivative of
a)
(
b)
c)
(
(
d)
(
)
)
)
87) The
derivative of (
is any positive integer and
a)
(
)
c)
studymedia.in/fe/m1-mcqs/
(
)
) , where
is
b)
d) 0
88) The
(
)
a) (
c)
(
(
derivative of
)
(
(
(
)
)
(
(
)
b. (√ )
(
)
c.
(
)
d.
(
)
)
√
98) If
(
a)
c) 1
(
92) The
derivative of (
is any positive integer is
)(
(
b)
(
)(
)
then
)
)
) (
)
) (
)
)(
) (
)
then
)
)
(
)
(
101) If
a) (
c) 1
then
)
is
102) If
then
is
c)
) (
is
)
b) 0
d)
(
)
(
) (
(
)
(
(
)
b)
(
)
d)
)
)
)
)
(
(
d)(√ )
)
)
and ( )
)
, where
is
b) ( )
d) ( )
94) If
is
) then
a)
(
)(
(
(
(
b)
)
(
(
d)
then ( )
b) 1
d) 3
a) (√ )
)
(
b)
d) 0
and ( )
100) If
)
)
(
is
),
)
)
(
(
(
)
( ), is
(
(
a)
then ( )
)
then
)
c)
95) If
a) 0
c)
)
) is
and
)
b)
c)
d)
93) If
a)
c) (
(
)
99) If
is
a) 0
c) 2
derivative of
c)
)
)
(
a. (√ )
where
(
and ( )
)
d) 0
derivative of
91) The
is
b)
d)
97) If
is
) is
b) (
)
90) The
)
then
)
d) 0
)
(
c)
b) (
)
89) The
a)
96) If
a) (
c)
derivative of
103) If
a)
then ( )
)
)is
b)1
d)
then
(
c)
(
104) If
a)(
b) (
c) (
d)(
) (
)
is
b)
)
d)
(
)
)
)
)
studymedia.in/fe/m1-mcqs/
(
(
(
(
)
) then
)
)
)
)
105) If
24
25
26
27
then
√
)
)
)
)
a)(
b)(
c)(
d)(
106) If (
a)(
(
)
c
b
b
a
51
52
53
54
then
)
(
)
(
)
)
c
d
c
b
105
106
107
c
b
c
[02]
[02]
)
)
(
)
(
)
)
)
(
)
107) If
a)
b)
c)
d)
(
) then
(
)
(
)
(
(
)
[02]
)
Infinite series & successive differentiation
Answer Key
An
s
C
B
A
B
A
B
C
C
A
C
B
C
A
C
C
C
B
A
B
B
c
a
c
78
79
80
81
)
b)(
(
c) (
(
d) (
Q.No
.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
b
d
b
c
Q.No
.
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
An
s
c
b
d
a
c
d
a
b
d
a
b
a
d
b
a
b
a
b
a
a
c
b
b
Q.No
.
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
An
s
b
c
b
b
d
c
c
d
a
c
c
d
d
a
a
d
d
a
d
b
b
b
a
Q.No
.
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
An
s
c
c
d
b
a
d
a
a
b
b
c
b
a
a
b
d
a
a
a
a
c
b
a
studymedia.in/fe/m1-mcqs/
Taylor’s and Maclaurin’s series
a) 1
c) 2
b) 4/3!
d) 3!
1) The expansion of ( ) in ascending powers of
about
is
7) The coefficient of
(
) is
a)
c)
in the expansion of
8) The expansion of
three terms is
(
)
a)
(
)
b)
about
a)
( )
( )
( )
( )
b)
( )
( )
( )
( )
( )
c)
d)
( )
( )
( )
( )
2) Expansion of
( )
( )
( )
in powers of
is
a)
c)
(
)
d)
(
)
b)
d)
(
(
upto first
)
)
(
)
(
)
b)
9) In the expansion of
in
powers of (
) the constant term is
a)
b)
c)
d)
c)
d)
3) Expansion of
(
ascending powers of
)
(
) in
is
b)
10) The expansion of
a)
b)
c)
c)
d)
d)
a)
4) The coefficient of (
of
) in the expansion
about is
) is
(
11) The expansion of
) is
a)
a) √
b) √
c)
d)
b)
c)
d)
5) The expansion of (
powers of is
) in ascending
12) In the expansion of
a) ( )
( )
( )
( )
b) ( )
( )
( )
( )
) is
a) 0
c) 1
b)
d)
(
c) ( )
( )
( )
( )
d) ( )
( )
( )
( )
c)
d)
6) The coefficient of
in the expansion of
) in ascending powers of
about
coefficient of (
13) The expansion of
powers of is
a)
b)
(
(
is
studymedia.in/fe/m1-mcqs/
, the
) in ascending
14) The coefficient of (
) in the expansion of
in powers of (
)
is
a) 1
c) 9
c)
d)
b)
d)
15) Expansion of
is
b)
in ascending powers of
22) In the Taylor series expansion of
about the point
the co-efficient of
(
) is
a)
b)
c)
d)
a)
b)
c)
d)
16) Expansion of
is
23) Which of the following functions would have
only odd powers of in its Taylor series
expansion about the point
.
(
)
a)
b) ( )
( )
c)
d)
( )
in ascending powers of
a)
b)
c)
d)
17) The
24) The limit of the series ( )
term in the expansion of
a)
(
)
) (
b) (( )
)
c)
)
d) (( )
as
is
approaches is
a)
b)
c)
d) 1
25) The Taylor series expansion of
18) The coefficient of
is
a) 0
c) 1
in the expansion of
is given by
a)
b)
d)1/2
c)
) is
d)
a)
(
)
b) (
c)
(
)
d) (
20) First two terms in expansion of
is
a)
)
)
(
)
)
(
b)
19) The constant term in the expansion of
(
(
(
)
)
(
26) Expansion of
(
a)
b)
b)
c)
c)
d)
d)
21) The first three terms in the power series for
(
) are
27) Expansion of
a)
studymedia.in/fe/m1-mcqs/
)
) is
at
(
(
c)
d)
a)
)
)
(
(
)
)
b)
34) The Maclaurin series of
( )
(
) is
a)
b)
c)
d)
None
c)
d) None
28) Expansion of
(
)
a)
b)
Indeterminate form
c)
d)
35)
29) Expansion of
(
is
) is
a) 0
c)
a)
b)1
d)2
b)
36) If
c)
of
d) None
30) The limit of the series
( )
as approaches ½ is
a)
b)
c)
d)
is
a) 0
c)
2/3
1/3
1
4/3
b)1
d)2
(
37)
)
is
a) 0
c)
as
b) 1
d) 2
(
(
38)
31) The limit of the series
( )
)
is
)
a) 0
c) 1
Approaches is
a)
b)
c)
d)
is finite then the value
b)
d) 2
4
39)
is
a) 0
c) 1
3
2
(
40)
32) Representation of
(
)
(
)
powers of is
(
)
a)
b)
c)
d)
in
b)
d) 2
) is
a) 0
c) 1
b)
d) 2
41) The value of
is
a)
c)
b)1
d)
42) The value of
( )
33) The first three terms in expansion of
in powers of
a)
b)
(
(
is
)
)
(
(
)
)
a)
c) 1
studymedia.in/fe/m1-mcqs/
b) 0
d) 2
is
43) The value of
a)
c) 1
is
a) 1
c)
b) 0
d) 2
44) The value of
a)
c) 1
b)
d) 2
55) The value of
is
a) 1
c) -3
is
b) 0
d) 2
b) -1
d) 2
56) The value of
45) The value of
a)
c) 1
is
57) The value of
is
b)
c)
d)
(
)
a) 0
c)
58) The value of
is
(
) is
b)
d) 1
(
49) The value of
a) 2
c) 4
50) The value of
)
59) The value of
is
60) The value of
+
61) If
)
is
b)
d) 2
(
)
is
b)
d)
(
)
is finite
52) The value of
is finite then value of
b) -2
d) -1
62) If
then
a)
b)
c)
d)
b)
d)
is finite then
and
and
and
and
is
53) The value of
a)
c)
54) The value of
b) 2
d)
(
is equal
b)
d)
a) 2
c) 1
then
value of
a) 1
c)
(
is
51) The value of
a)
c)
is equal to
b) 0
d) ⁄
to
a) -1
c)
b) √
d) √
a)
c)
)
a) 5
c) -5
b)
d)
*
(
a) 1
c) -3
b)
d) 1
48) The value of
a)
c) 0
b) -1
d) 2
a) 1
c) 1/2
a)
47) The value of
a) 1
c) -3
b) 0
d) 2
46) The value of
is
)
63) The value of
is
a)-1
b)0
c)1
d)
is
b) 0
d) √
64) The value of
a) -1
c) 1
is
studymedia.in/fe/m1-mcqs/
is
b) 0
d)
is
65) The value of
a) 1
c)
is
b) 0
d)
78) If
66) It is given that ( )
67)
following is true
a)
c)
( )
( )
and
a) 1
c)
, then value of (
b) 0
d)
then which of the
) is
is equal to
79) If
then which of the
following is true
a)
b)
c)
d) none
a)
b) 0
c)
d) 1
80) The value of
is
)
68) The value of
)
)
)
b)
d)
is
)
)
)
)
81) The value of
69) The value of
b)
c)
d)
)
(
(
(
)
)
is
)
)
)
)
(
)
)
)
)
77) If
the following is true
a)
b)
c)
d)
is
)
)
)
)
85) If ( ) and ( ) are 2 functions such that
is
75) The value of
84) The value of
)
is
)
)
74) The value of
)
)
then
)
(
(
)
76) The value of
)
)
b)
d)
a)
b)
c)
d)
)
)
73) The value of
)
is
)
72) The value of
)
then
83) If
)
)
)
)
)
71) The value of
)
)
)
a)
c)
is
)
)
(
82) If
70) The value of
)
)
is
a)
is
( )
and ( )
to
( )
( )
( )
( )
)
is
)
)
is
86) The value of
)
exists then which of
( )
is equal
( )
then
a)
b)
)
is
c)
87) The value of
a)
b)
studymedia.in/fe/m1-mcqs/
( )
( )
( )
( )
)
d)
is
c) 1
d) -1
(
88) If
)
a)
c)
then
( )
( )
89)
)
( ) is equal
, then
to
)
)
)
)
)
)
104) The value of
)
91)
)
(
is
)
)
)
)
)
)
)
92)
)
is equal to
)
)
93)
)
)
)
is equal to
)
)
(
95)
) (
is equal to
)
)
96)
a) 1
(
)
)
)
)
97)
)
is equal to
)
)
)
98)
)
)
99)
)
)
100)
)
)
101)
)
)
(
b) 8
)
c) 9
is
d) none
Q.No.
Ans
Q.No.
Ans
Q.No.
Ans
Q.No.
Ans
1
b
28
d
55
b
82
b
2
b
29
a
56
a
83
a
3
b
30
a
57
c
84
a
4
d
31
d
58
b
85
d
5
b
32
c
59
d
86
b
6
c
33
d
60
c
87
c
7
a
34
a
61
b
88
a
8
d
35
d
62
a
89
d
9
c
36
c
63
c
90
c
10
c
37
b
64
b
91
a
11
c
38
c
65
a
92
b
12
a
39
a
66
b
93
d
13
d
40
a
67
d
94
d
a
41
c
68
a
95
a
15
d
42
b
69
a
96
a
)
16
c
43
c
70
a
97
a
17
b
44
b
71
c
98
a
18
c
45
b
72
a
99
c
19
a
46
a
73
b
100
a
20
a
47
c
74
c
101
a
21
d
48
a
75
d
102
a
22
b
49
b
76
b
103
b
23
a
50
b
77
c
104
a
24
d
51
c
78
b
105
c
25
d
52
d
79
a
106
c
26
b
53
c
80
c
107
c
27
c
54
b
81
b
)
)
)
is equal to
)
)
14
is equal to
)
)
is
is equal to
is equal to
)
)
)
Answer Key
is equal to
)
(
)
Taylor's & Maclaurin's Theorem , Indeterminant Form
)
)
)
is
)
107) The value of
)
is equal to
)
)
94)
)
)
)
)
106) The value of
)
)
(
105) The value of
is equal to
)
)
is
is equal to
)
) is
)
103) The value of
)
)
)
)
90)
(
102) The value of
b)
d)
)
studymedia.in/fe/m1-mcqs/
Partial Differentiation
1) If
(
) then
D)
Ans :C
5) If
is
A)
A)
B)
B)
C)
C)
D)
D)
Ans : B
Ans:A
2) If
(
) then
is
6) If
B)
B)
C)
C)
D)
D)
Ans:B
Ans:A
7) If
(
) then
is
and
and
is
C)
C)
D)
)
D)
Ans:B
Ans:D
8) If
( ) and
A)
( )
B)
( )
C)
( )
then
is
A)
C)
then
B)
B)
B)
then
A)
A)
4) If
is
A)
A)
3) If
then
studymedia.in/fe/m1-mcqs/
(
) then
is
12) ) If
( )
D)
then
( ) is
Ans: A
( ) and
9) If
then
and
A)
√
is
A)
( )
B)
( )
B)
C)
D)
( )
C)
Ans : C
D) ( )
13)If is a homogeneous function of
of degree then
Ans: D
( ) and
10) If
then
A)
A)
√
is
B)
( )
C)
( )
B)
D)
C)
( )
Ans:B
D)
( )
14) If is a homogeneous function of
of degree then
Ans: A
11) If
and
then
A)
( ) is
(
A)
B)
B)
)
(
C)
C)
D)
Ans :B
D)
(
)
Ans:D
studymedia.in/fe/m1-mcqs/
)
15) If is a homogeneous function of
( ) then
of degree and
+ then
A) u is a homogeneous function of degree
one.
( )
A)
*
18) If
( )
B)
is a homogeneous function of
degree one.
B)
( )
C)
C) u is a homogeneous function of degree
zero.
( )
( )
D)
( )
D)
is a homogeneous function of
degree zero.
Ans:A
16) If is a homogeneous function of
( ) then
of degree and
Ans: B
A)
√
( )
( )
19)If
( )
where ( )
( )
where ( )
( )
B)
C)
C)
( )
( )
√
√ and
√
are constants then
where
( ) is
A)
( )
√ and
√
are constants then
( )
B)
( )
√
( )
where ( )
( )
√
√
√
D)
√
Ans: C
D)
( )
( )
where ( )
( )
Ans:C
17)
20) If
√
where
( ) is
(
) is a
A)
√
A) Non-homogeneous function.
B)
B) Homogeneous function of degree zero.
C) Homogeneous function of degree one.
C)
D) Homogeneous function of degree two.
D)
√
√
√
Ans: B
studymedia.in/fe/m1-mcqs/
Ans: A
then ( )
21)If
24) If and are independent variables
and is a function of
then
is
is
A)
A)
B)
B)
C)
C)
D)
D)
Ans: B
Ans: D
then ( )
22) If
(
25) If
is
A)(
)
A)
B) (
)
C) (
)
D) (
)
B)
C)
D)
Ans: A
Ans: A
23) If and are independent variables
and is a function of
then
is
A)
B)
C)
D)
Ans: C
studymedia.in/fe/m1-mcqs/
) then
is
studymedia.in/fe/m1-mcqs/
Jacobian
1
If u = f(x,y)and v= g(x,y) , the Jacobian of u, v w. r. t. x, y is given by
(
(
a)
2
)
)
If
b)
If (
)
(
)
)
(
(
d)
)
)
(
(
(
(
)
)
(
(
(
)
)
(
)
)
)
) (
(
(
)
a
) d)2(
)
(
)
(
)
)
) (
(
(
(
)
)
(
)
)
) (
(
(
) then
(
(
)
)
c
)
)
b
be the implicit functions of u,v w.r.t x,y then
(
(
(
(
)
)
)
)
)
b)(
(
(
(
(
)
)
)
)
)
c)(
(
(
(
(
)
)
)
)
d)(
)
(
(
(
(
)
)
)
)
(
,
(
(
)(
)(
) b)(
)
)(
)(
) c)(
)(
(
(
a) (
d) (
If
a
)
c
)
a
(
(
)(
)(
)
=
)
)(
)(
)(
) (
(
) b)(
)
,
)
)(
)(
)
)(
)(
)(
)(
)
)
)
)
b)
(
(
(
(
(
)(
)(
)(
)(
(
)(
)(
d)( ) (
)(
b)
(
)
)
)
)
)
)
c)
)(
(
(
(
(
) c)(
)(
)(
)
(
)
(
Let (
functions of u,v,w into x,y,z then
(
(
(
(
)
)
)
)(
)(
(
(
)(
(
,
then
)
)
)
)(
)(
then
)
)
)
)
)
)
If
c)(
(
(
d)
b
)
)
(
If
a)
)
)
c
d)
)
a) (
d) (
10
)
)
)
=
)
then
9
(
(
) c)2(
c)
Let (
a) (
8
(
(
c)
c)
If u = x(1-y), v = xy then
(
(
7
)
)
) b)4(
a) (
6
)
)
, v = 2xy , then
a)
5
(
(
b)
a)4(
4
(
(
b)
If u = f(x,y)and v= g(x,y) , the Jacobian of inverse function is given by
(
(
3
)
)
a
)
)
)
)
)
)
)
)
)
(
)
=
d)
(
(
(
(
)
)
)
)
studymedia.in/fe/m1-mcqs/
be the implicit
a
13
functions u,v….of x,y ….are said to be functionally dependent,
a) if the corresponding inverse Jacobian is zero b) if the corresponding Jacobian is zero
c) if the corresponding Jacobian is nonzero d) none of these.
+2xy+2x+2y ,
are functionally dependent then the relation
between them is
a) (
)
= u b) (
)
=u
c) (
)
= u d) (
)
=v
If
,
functionally dependent, then the relation
14
between them is
a)u = sin v b) v = sinu c) u = tan v d) v = tan u
(
) be the function of x,y then we write,
If
11
12
)
15
)
(
If
16
17
18
19
20
21
22
23
24
25
) be the function of x,y then we write,
c) s =
b)
c)
b
a
b
d) none of these
If ( )
then the minimum value of the function at (-3,0) is
a)10 b)12 c)-10 d) -12
(
) is having maximum values at (a,b) if at (a,b)
The function
a)
b)
c)
, r < 0 d)
,r > 0
(
) is having minimum values at (a,b) if at (a,b)
The function
a)
b)
c)
, r < 0 d)
,r > 0
If ( )
then p =
a) (
) b) (
) c)(
) d)- (
)
(
) the minimum value of f(a/3, a/3) is
Let
a)
b
) =
c)p
)s=
a) s
b
b
c
d
b
b
d)
The stationary values of the function
)
,
b)
c)
,
d)
, where x + y + z =1is given by
b
,
Let u = f(x,y,z) be a function of x, y and z.
dx, dy, dz, du are known as……… errors in x, y, z and u respectively
a) Relative b) Absolute c) Percentage d) none of these
Let u = f(x,y,z) be a function of x, y and z.
,
,
are known as ……. errors w. r. t. x, y, z and u respectively.
a)Relative b) Absolute c) Percentage d) none of these
In calculating the volume of a right circular cylinder, errors of 2% and 1% are found in
measuring height and base radius respectively. Then the percentage error in calculated
volume of the cylinder is
a)3 b) 1 c) 2 d) 4
When the errors of 2% and 3% are made in measuring its major and minor axes of ellipse
respectively then the percentage error in the area of an ellipse is
a) 4 % b) 5 % c) 3 % d) 1 %
studymedia.in/fe/m1-mcqs/
b
a
d
b
Matrices (RANK AND NORMAL FORM)
01.
[
The rank of matrix
c)
] is
d) None of these
[
a) 3
b) 1
c) 2 d) None of these
06.
02.
[
The rank of matrix
a) 2
b) 1
c) 3
03.
The matrix [
if
is
a) 4
]is
] will be singular
c) 6
+
07.
The rank of
*
a) 1
b) 3
c) 2
+ is
d) 0
The rank of the matrix
[
,
] is
a) 2
b) 1
c) 3
09.
If the rank of
[
then
a)
c)
d) 4
] is 1
b)
d)
] is
a)
b)
[
*
b)
d)
d) 12
The cofactor of the matrix
[
+
a)
c)
08.
b) 8
*
For
d) 4
04. For a rectangular matrix of order
rank of
a) Min (m,n)
b) Lowest order of the minor of matrix,
c) Max (m,n)
d) All of these
05.
]
]
[
10.
The rank of
a) 3
b) 1
]
studymedia.in/fe/m1-mcqs/
[
c) 2
] is
d) None of these
11.
The rank of the matrix [
a) 0
b) 2
12.
The rank of a matrix
c) 1
a) 1
b)2
c) 3
13.
Adjoint of
[
a) [
c) Other than
d) None of these
16. If
rank of
a) Rank of
c) Rank of
are non- singular then the
is that of
b) Rank of
d) None of these
] is
d) 3
[
] is
d) None of these
17. If the rank of matrix
are same
then the matrices
are
a) Equal
b) Equivalent
c) Not equal
d) None of these
18.
] is
] b) [
The rank of
1
1
A
2

3
]
2
1
3
2
4
7
3
4
2
2
4

6
is
c) [
]
d) [
]
19.
14.
[
If
]is
orthogonal
then
a)
[
]
a) 1
b)2
c)
3
If
*
+then
d)4
(
)
a) *
+
b) *
c) *
+
d) None of these
20.
The matrix [
+
] will be singular if
is equal to
b)
[
]
a) -2
21.
c)
[
]
d) None of these
15. If
is non-singular then rank of
that of
a) Rank of
b) Rank of
is
b)3
c) 4
d) -3
If a matrix A has at least one minor of
order r is non zero and every minors of
order (r+1) are zero then
a)  (A)  r
c)  (A)  r
studymedia.in/fe/m1-mcqs/
b)  (A)  r
d) none of these
22.
For matrix A order m  n , the rank r of
matrix A is
a) r minimum of m and n
b)
maximum of m and n
c)
minimum of m and n
d)
maximum of m and n
23.
24.
If a matrix A has all its minors of order
(r + 1) are zero then
a) ( )
b) ( )
c) ( )
d) none of these
For non singular matrix A of order n×n ,
rank r of A is
a) r  n
b) r = n
c) r > n
d) none of these
25. The rank of matrix of order m  n is
a) highest order of its non-vanishing minor
b) smallest order of its non- vanishing minor
c) highest order of its vanishing minor
d) smallest order of its vanishing minor
26.
a)
b)
c)
d)
The rank of matrix does not alter by
Elementary row transformation
elementary column transformation
Taking transpose
all the above
27. Which of the following
elementary transformation?
a)
b)
c)
28.
is
29. By
performing
elementary
transformation if any non- zero matrix A of
order 4 X 5 is reduced to the normal form
I2
0

a) 4
a) 4
c)  I 2
0
d) *
b) 2
c) 5
d) 1
31. By
performing
elementary
transformation if any non- zero matrix A of
order 3 X 4 is reduced to the normal form
 I3
0 then the rank of A is equal to
a) 4
b) 2
c) 3
d) 1
32. By
performing
elementary
transformation if any non- zero matrix A of
order 4 X 3 is reduced to the normal form
not
b) 2
c) 3
d) 1
33. For non-singular matrix A, there exist
two non- singular matrices P and Q such that
PAQ is in Normal form, then A1 is equal
to
a) PQ b)
c) QP d)
34.
1 0 
 is
0 2 
Normal form of matrix A = 
a)  I 2
c)
 I2 
d) 1
 I4  then the rank of A is equal to
is
b)
c) 5
30. By
performing
elementary
transformation if any non- zero matrix A of
order 4 X 4 is reduced to the normal form
a) 4
1 0 0 
Normal form of matrix A =  0 1 0 


 0 0 1 
+
b) 2
 I3 
 0  then the rank of A is equal to
 
d)
a) *
0
then the rank of A is equal to
0
0
 I3 
+
studymedia.in/fe/m1-mcqs/
b)
 I2 
 I1 
0
d)  
35.
1 0 0 
Normal form of matrix A =  0 0 1 


1 0 0 
is
a)
b)
I2
d) 
0
c)
36.
4 0 0
The rank of matrix A =  0 3 0 


 0 0 5 
is equal to
a) 4
b) 3
37.
d) 1
b) 3
c) 2
equal to
b) 3
rank
c) 2
40.
The
of
matrix
1
0

0

0
2 3 4 5
1 2 3 4 
is equal to
0 1 2 3

0 0 0 0
41.
The
rank
1
2

 3
3
6
9
4
b) 3
42.
The
rank
1
3

 1
0
1
1
0
1
2
44.
45.
of
matrix
d)
1
A
=
6
1 4  is equal to
7 10 
a)
43.
c) 5
8
c) 2
of
d) 1
matrix
A
=
1
2  is equal to
0 
b) 1
c) 3
d) 4
1 3 6 
The rank of matrix A = 1 4 5  is


1 5 4 
b) 1
c) 0
d) 2
1 1 1 
The rank of matrix A =  2 3 4  is


 2 2 3 
c) 2
d) 4
1
2
The rank of matrix A = 
0

0
is equal to
a) 1
b) 4
d) 1
A
3
equal to
a) 1 b) 3
d) 1
1 2 3
The rank of matrix A =  2 2 2  is


 3 3 3 
a) 4
b)
equal to
a) 3
c) 2 d) 1
b) 3
4
a) 2
1 1 1 
The rank of matrix A =  2 2 2  is


 3 3 3 
equal to
a) 4
39.
c) 5
1 2 3
The rank of matrix A = 
 is
3 1 2
equal to
a) 4
38.
0
0
a)
c) 3
1 1 1
3 4 5
1 2 3

1 2 3
d) 2
=
SYSTEM OF LINEAR ALGEBRAIC EQUATIONS
studymedia.in/fe/m1-mcqs/
46. Homogeneous
system
equations
a) Is always inconsistent
b) Is always consistent
c) has always infinite solution
d) none of these
of
linear
47. Non-homogeneous system of linear
equations A X = B is consistent if
( | )
a) ( )
(
)
( | )
b)
c)  (A) >number of unknown
d) none of these
48. Non-homogeneous system of linear
equations A X = B is inconsistent if
( | )
( | )
a) ( )
b) ( )
c) ( ) number of unknown d) none of
these
49. For consistent m  n non-homogeneous
system of linear equations A X =B ( )
( )
then the system possesses
a) Unique solution
b) No solutions
c) Infinitely many solutions
d) n - r solutions
50. An n  n homogeneous system of linear
equations
A X = 0 is given.
If the rank of A is
, then the system has
a)
independent solutions
b)
independent solutions
c) no solutions
d)
independent solutions
51. For consistent m  n non-homogeneous
system of linear equations A X = B, if rank
of A = r < number of unknowns, then the
system possesses
a) unique solution
b) no solutions
c) infinitely many solutions
d) n- r solutions
52. The condition for unique solution of
m  n non- homogenous system of linear
equations AX=B is
a)
rank of A= r = number of unknowns
b)
rank of A = r < number of unknowns
c) rank of A= r > number of unknowns
d) none of the above
53. The condition for infinitely many
solutions of m  n non-homogenous system
of linear equations AX=B is
a)
rank of A= r = number of unknown
b) rank of A = r < number of unknowns
c) rank of A= r > number of unknowns
d) not defined
54. An n  n homogeneous system of linear
equations A X = B with A is non-singular
matrix has
a)
unique solution
b)
no solutions
c)
infinitely many solutions
d)
n- r solutions
55. An n  n homogeneous system of linear
equations
A X = 0 with A is non-singular matrix has
a)
n- r solutions
b)
non-trivial solution
c)
infinitely many solutions
d)
trivial solution
56. An n  n homogeneous system of linear
equations
A X = 0 with A is singular matrix has
a)
trivial solution
b)
non-trivial solution
c)
no solution
d)
n - r solutions
studymedia.in/fe/m1-mcqs/
57.
given
system
of
linear
equations
3x + 2y + z = 0, x + 4y + z = 0, 2x + y+ 4z = 0
x + 2y - z = 0, 3x + 8y - 3z = 0, 2x + 4y + (k-3)z = 0
has
a)
No solution
b)
only trivial solution
c)
Infinite solutions
d)
none of these
58.
given
system
of
Has infinitely many solutions?
a) k = 0 b) k = 1 c) k = 2
3
linear
equations
x + 2y + 3z = 0, 2x + 3y + z = 0, 4x + 5y+ 4z = 0
has
a) no solution
solution
c) Infinite solutions
64. For what values of k, the homogeneous
system
b)
only
trivial
d) none of these
65. For what values of λ, the system of
linear
equations
x + y + z = 6, x + 2y + 3z = 10, x + 2y +  z = 10
Has infinitely many solutions?
a) λ = 1 b) λ = 3
c) λ = -3 d) λ = 10
66.
For what values of µ  8 the system of
linear
59. Given
system
of
linear
equations
x - 4y + 5z = 0, 2x - y + 3z = 0, 3x + 2y+ z = 0
has
a) No solution
solution
c) Infinite solutions
b)
only
trivial
61.
Given system of linear equations
x + y + z = 1, x + 2y + 4z = 2, x + 4y + 10 z = 4 has
a) No solution
c) Infinite solutions
b) unique solution
d) n-r solutions
62. Given system of linear equations
x - 4 y + 5z = - 1, 2x - y + 3z = 1, 3x + 2y + z = 3
has
a) No solution
b) unique solution
c) Infinite solutions
d) n-r solutions
2x - y + 3z = 2, x + y + 2z = 2, 5x - y + µz = 2
has
a) No solution
b) unique solution
c) Infinite many solutions d) x = 0, y = 0,
z=0
67. The system
consistent if it has
a) Many solution
b) No solution
c) At least one solution
d) Unique solution
system
of
linear
equations
x + y + z = 3, x + 2y + 3z = 4, x + 4y + 9z = 6
has
a) No solution
c) Infinite solutions
b) unique solution
d) n-r solutions
is said to be
68. The composite transformation for the
given transformation
and
Which expresses
a)
b)
c)
d)
69.
Given
equations
d) none of these
60. Given system of linear equations
x + 3y + z = 0, 2x - 2y - 6z = 0, 3x + y - 5 z = 0
has
a) No solution
b) only trivial solution
c) Infinite solutions
d) none of these
63.
d) k =
in terms of
is
The system of linear equations
has
a) Unique solution b) An infinite solution
c) No solution
d) Exactly two
distinct solution
studymedia.in/fe/m1-mcqs/
70.
If
are linearly dependent vectors
then
a)
c)
b)
d)
[
72. If
)
)are linearly
[
77. If
71. The system of linear equation
has
a) Unique solution
c) Infinitely solution
(
76. If the vectors
(
)
(
dependent then the relation is
a)
b)
c)
d)
]
is
orthogonal then
b) No solution
d) None of these
] is orthogonal
a)
[
]
b)
[
c)
[
d)
None of these
]
then
a)
b)
[
]
a)
c)
orthogonal then
for which the
a) [
]
has many solutions are
b)
d)
b) [
]
c)
74. The values of
system
[
78. If the matrix
c)
d) None of this
73. The value of
system
]
[
]
] is
is
d) None of these
for which the
has unique solution
a)
b)
c)
d)
79. If the matrix
[
orthogonal then the value of
75. For the system
a)
b)
c)
d)
the
values of
a) 2, 3 , 1
c) 0 , 0 , 0
are
b) 1 , 1 , 1
d) 1 , 2 , 3
studymedia.in/fe/m1-mcqs/
is
]
is
constants
80. If
is orthogonal then
[
]
a)
b)
c)
d) None of these
(
)
81. If the vectors
(
)
(
) are
linearly independent then
a)
b)
)
c) (
d) None of these
82.
If
the
√
√
√
√
is
√
√
83. Given
the
*
]
b)
√
d) None of these
transformation
+ * + the values of
independent if
a)
b)
c)
d)
86.
b)
c)
d) none of these
*
+
are
84. Among the following , the pair of vectors
orthogonal to each other is
a)
b)
c)
d)
For
c1 x1  c2 x2  c3 x3  0 where
For
x1 , x2 , x3 non- zero vectors and c1 , c2,c3 are
constants
x1 , x2 , x3
then
are
linearly
independent if
a) c1  0, c2  0, c3  0
b) not all c1 , c2, c3 are zero
(
),
(
)
87. The vectors
are
a) linearly dependant (b) linearly independent
c) mutually orthogonal d) none of these
(
)
(
)
The vectors
(
) are
a) linearly dependant ( b) linearly independent
c) mutually orthogonal d) none of these
88.
89.
a)
85.
linearly
orthogonal
then
a)
c)
are
c) c1  , c2  , c3   d) none of these
√
√
[
matrix
x1 , x2 , x3
then
c1 x1  c2 x2  c3 x3  0 where
x1 , x2 , x3 non- zero vectors and c1 , c2,c3 are
For
an
orthogonal
matrix
1 0 1 
1 
A=
0 1 0  , A 1 is

2
1 0 1 
a)
√
[
]
c)
[
]
90.
For
an
A=
1  12 5  1
, A is
13  5 12
a)
*
√
studymedia.in/fe/m1-mcqs/
]
b)[
d) does not exit
orthogonal
+
b)
*
matrix
+
c) *
91.
+
96.
d) does not exit
The matrix of linear transformation
y1  2 x1  x2  x3 , y2  x1  x2  2 x3 , y3  x1  2 x3
For what values of k, the matrix
1
2
A= 
 -k


k
 is an orthogonal matrix
1
2 
a) [
]
b) [
]
a) 
3
3
b) 
4
2
c) [
]
d) [
]
97.
For what values of b, the matrix
A=
1  b 5
is an orthogonal matrix
13  5 b 
92. The
linear
transformation
 4 5 1   x1 
y   3 1 2   x2  is
1 4 1   x3 
a) Nonsingular
c) singular
93.
98.
b) composite
d) none of these
The
linear
Whether
)
 cos 
A=
  sin 
a) yes
b) no
For what values of λ, the matrix
2
1 2  is an orthogonal matrix
2 1 
a)
b)
c)
linear
transformation
2
d)
The
a) Orthogonal b) composite
d) none of these
b) orthogonal
d) none of these
 y1 , y2 , y3  in
corresponding to  1,3,0  in X are
95.
d) 16
is
 0 1   x1 
Y= 
   is
 1 0   x2 
coordinates
)
c) 12
d) 1

1
A=  2
3
 2
99.
 y1   2 1 1   x1 
94. For the transformation  y   1 1 2   x 
 2 
 2
 y3  1 0 2  x3 
a) (
c) (
b) 13
1
2
transformation
 2 1 3  x1 
y   3 2 1   x2  is
1 4 5  x3 
a) Nonsingular
c) singular
a) 5
c) 
b) (
d) (
the
Y
)
)
matrix
sin  
is orthogonal.
cos  
c) can’t say d) none of these
c) singular
100. f Y = AX and Z = BY be two linear
transformation then the composite
transformation which takes X to Z is
given by
a)Z=(AB)X
b)Z=(BA)X
c)X=(BA)Z
d)X=(AB)Z
101. If Y =
transformations
transformation is
a) X = A-1 Y
c) X = Y A-1
AX
then
is
non-singular
its
inverse
b) Y = A-1 X
d) does not exit
102. For square matrix A to be an orthogonal
matrix
a)
b)
studymedia.in/fe/m1-mcqs/
c)
d)
103. If A is an orthogonal matrix then A-1
equals to
a) A b) A T c) A 2 d) I
104. If A is an orthogonal matrix then
determinant of A is
a) 0
b) 3
c) 2
d) 1
105. If A is an orthogonal matrix then
a) A T is orthogonal
b) A-1 is orthogonal
c) A  1
d) all are correct
c) *
111.
+
If
d) *
the
 1

Y=  2
 3

 2
linear
+
transformation
3

2   x1 
  is an orthogonal linear
1   x2 

2 
transformation
then its inverse transformation is
√
a)
[
]* +
√
√
106. If Y = AX is orthogonal linear
transformation then the matrix A is
a) Nonsingular
b) orthogonal
c) singular
d) none of these
107. For
an
 cos 
A=
  sin 
sin   1
, A is
cos  
a) *
orthogonal
[
]* +
√
√
c)
[
]* +
√
matrix
√
d)
+ b) *
[
]* +
√
+
[
108. If the matrix
b) y
] then the
Eigen values of are
a) 2,1,2 b) 4,5,2 c) 3,3,3
d) 3,2,5
c)
*
d)
*
*
+
112. The Eigen vectors for the matrix
*
109. If
+
+ is
+ then the Eigen
a) * + * +
values are
a)
c)
c) * + *
b)
d)
b) * + * +
+
d) None of these
113. The Eigen values for the matrix
110. Find
a)
b)
of
*
*
+
*
+ is
+
[
a) 2 , 1 , 1
c) 2 , 4 , 5
studymedia.in/fe/m1-mcqs/
] are
b) -2 , 3 , 6
d) None of these
114. The Eigen values for the matrix
*
+ are
*
a) 2 , -1
c) 2 , 3
b) 2 , 2
d) 3 , 4
*
115. If
of
+ then the Eigen values
are
a) 4 , 6
c) 3 , 2
119. The Eigen values of a matrix
+ is
a)
c)
120.
b)
d)
The spectrum of a matrix
[
] is
a)
c)
2,1,1
2,2,2
b) 2 , 2
d) 1 , 2
116. The product of the Eigen values of
[
] is
a) 24
b) 36
c) 21
d) 32
117. The Eigen vectors for the matrix
[
121. The Eigen values and the corresponding
Eigen vectors of a
matrix are given by
Eigen values
and Eigen vectors
* +
*
] are
a) [ ] [ ]
b) [ ]
c) [
d) [ ]
] [ ]
b) 1 , 1 , 3
d) None of this
+ then the matrix is
a)
*
+
b) *
+
c)
*
+
d) *
+
122. For the matrix
[
] , one of
the Eigen value is equal to
corresponding Eigen vector is
then the
118. The Eigen vectors for the matrix
[
] are
a.
[
b.
[ ] [
c)
] [ ],[
[ ] [
],[
],[ ]
a) [
]
b) [
c) [
]
d) [ ]
]
]
123. Eigen values of a matrix
]
d) None of these
are
matrix
a)
c)
*
+
then the Eigen values of the
is
1 and 25
6 and 4
studymedia.in/fe/m1-mcqs/
b) 5 and 1
d) 2 and 10
a.
[ ] [
b.
[ ] [ ],[ ]
a)
b)
c)
d)
125. The minimum and maximum Eigen
c)
[ ] [ ],[ ]
values of the matrix [
130. The characteristics equation of a matrix
124. The Eigen value of [
]
are
respectively then the other
Eigenvalue is
[
a) 5
c) 3
a.
b.
c.
d.
b) 1
d) -1
126. The
Eigen
[
] are
a)
c)
values
of
the
matrix
]
d) None of these
] is
None of these
131. The Eigen values of a matrix
[
b)
d)
127. The characteristic equation of the
[
matrix
a.
b.
c)
],[
] are
] is
] are
a) 1 , 5 , 1
c) 3 , 2 , 1
b) 2 , 2 , 1
d) None of these
132. The
spectrum
[
] are
of
a
matrix
b)
d) None of these
a)
c)
b)
d)
128. Using Cayley Hamilton theorem to
*
133. The minimum and maximum Eigen
+ we get
values of the matrix
[
a)
c)
b)
d) None of these
are
then the other Eigen value is
129. For the matrix
[
a) 6
c) 5
b)4
d) None of these
Eigen vectors are
]
the
]
134. The characteristic equation for the square
matrix A is
studymedia.in/fe/m1-mcqs/
a) A + λ I = 0
b) A 2 - λ I = 0
c) A - λ I = 0
d) none of these
135.
If 1 , 2 , 3 are Eigen values of matrix
A then Eigen values of A-1 are
a)
b)
c)
d)
1 1 1
, ,
1 2 3
136. If 1 , 2 , 3 are Eigen values of matrix
A then trace of A equal to
a)
b) 1  2  3
c) 1  2  3
d)
2
2
2
142. Eigen vectors of a real symmetric matrix
A are orthogonal if the Eigen values are
a) repeated
b) non- repeated
c) complex
d) none of these
143. Eigen vectors corresponding to distinct
Eigen values of real symmetric matrix are A
are
a) linearly dependent
b) equal
c) orthogonal
d) none of these
144. If 1 , 2 , 3 are Eigen values of matrix
A of order three then the Eigen values of
matrix Am are
b) 1 2  2 3  1 3
m
a)
c) 1 , 2 , 3
m
m
m
m
m
m
d) 1 , 2 , 3
m
m
m
m
m
137. The sum of Eigen values of a matrix is
equal to
a) rank of matrix
b) determinant of matrix
c) trace of matrix
d) none of these
145. If 1 , 2 , 3 are Eigen values of matrix
138. The product of the Eigen values of a
matrix is equal to
a) rank of matrix
b) determinant of matrix
c) trace of matrix
d) none of these
a)
b)
c)
d)
139. The Eigen values of a upper triangular
matrix are
a) its principal diagonal elements
b) 0,0,0
c) 1,1,1
d) none
of these
A of order three and k is any non zero
constant
then the Eigen values of matrix kA are
146. The
Eigen
values
of
matrix
8 4 
A= 
 are
2 2 
a)
c)
147. The
b)
d) none of these
Eigen
values
of
matrix
 1 2
A= 
 are
 5 4 
140. The Eigen values of a lower triangular
matrix are
a) its principal diagonal elements
b) 0,0,0 c) 1,1,1
d) none of these
a)
c)
141. If an Eigen value of a square A is
  0 then
a) A is nonsingular
b) A is orthogonal
c) A is singular
d) none of these
 1 0 0
148. For matrix, A=  1 2 0  , the Eigen


 4 0 3 
values of A are
studymedia.in/fe/m1-mcqs/
b)
d) none of these
a)-1 ,-2 ,-3
c) 1 ,-1 , 4
b) 1 ,-1 , 2
d)1 ,2 ,3
155. The characteristic equation of the
3 5
 is
 2 4 
matrix 
149. For
matrix
1 2 1
A= 0 3 2  ,
0 0 5 
Eigen values of A are
a)1 ,-2 ,-1
b)0 , 3, 2
c)-1 ,-3 ,-5
d)1 ,3 ,5
150. For
matrix
a)
b)
c)
d)
the
156. The characteristic equation of the
 1 0 0 
A=  2 3 0  , the
 1 4 2 
Eigen values of A2 are
a) -1 , -9 , -4
b) 1 , 9 , 4
c) -1 , -3 , 2
d) 1 , 3 ,-2
3 1 1
matrix  1 5 1 is


 1 1 3 
a)
b)
c)
d)
157. If the characteristic equation for the
151. If A is any non-zero matrix of order
2  2 with trace of A = -1 and A  2
matrix A is   6  11  6  0
3
2
then the Eigen values Of A are
a) 2 , -1 b) -3 , 2 c) -2 , -1 d) -2 , 1
then the Eigen values of the matrix are
a) 1,2,3
b) -1,-2,-3
c) 1,2,-3
d) none of these
152. The sum and product of Eigen values of
158. If the characteristic equation for the
2 2 1
the matrix A=  1 3 1  are respectively


 1 2 2 
matrix A is   2    0
a) 7 and 7 b) 7 and 5 c) 5 and 6 d)5 and 8
3
2
then the Eigen values of the matrix are
a) 0,1,1
b) 0,-1,-1
c) 0,2,2
d)none of these
159. If the characteristic equation for the
153. If
are Eigen values of matrix
 2 9 5 
A=  5 10 7  then
 9 21 14 
a)-16
b)2
c)-6
is
matrix A is   3  3  1  0
3
2
then the Eigen values of the matrix are
a. 1,-1,-1
b) 1,2,3
d)-14
c)1,1,1
154. Two Eigen values 0f 3  3 whose
determinant is equal to 4 are -1 and 2.
The third Eigen value of the matrix is equal
to
a)1 b)-1
c)-2
d)2
d)none of these
160. The Eigen vector
Eigen value
λ1 = 1
studymedia.in/fe/m1-mcqs/
corresponding to
4 6 6
for the matrix A=  1
3 2  is obtained by solving

 1 4 3
studymedia.in/fe/m1-mcqs/
a)
a. [
][ ]
[ ]
[
][ ]
[ ]
c.
2
b) A - 3 A - 3I
c)
d)
166. Using Cayley Hamilton theorem, A-1 for the
b. [
][ ]
[ ]
d.
None of these
1 4
 is calculated from
 2 3
matrix A= 
a) (
CAYLEY HAMILTON THEOREM
161. Cayley Hamilton theorem states that
a) Sum of Eigen values of matrix is equal to trace of
matrix
b) Product of Eigen values of matrix is equal to
determinant of the matrix
c) Every square matrix satisfies its own
characteristic equation
d) Eigen values of a matrix and its transpose is
same
162. If the characteristic equation of matrix A of
order
c) (
c)
d)
163. If the characteristic equation of matrix A of
d) (
)
)
1 2 
 is calculated from
1 1 
matrix A= 
a) (
c) (
)
b) (
d) (–
)
)
)
168. Using Cayley Hamilton theorem, A3 for the
1 4
 is calculated from
 2 3
matrix A= 
a)
c)
b)
d)
169. If
b)
)
b) (
167. Using Cayley Hamilton theorem, A-1 for the
2  2 is  2  2  1  0 then by C-H theorem
a)
)
 3  5 2  9  1  0 is
characteristic
equation of matrix A
then by Cayley Hamilton theorem, A 4 is calculated
from
a)
b)
order 3  3 is   5  9  I  0
3
2
then by Cayley Hamilton theorem
a)
b)
c)
d)
164. If the characteristic equation of matrix A of
order 2  2 is   9  1  0
2
then by Cayley Hamilton theorem A-1 is equal to
a) A - 9
b) A + 9 I
c) – A – 9 I
d) A2 – 9 A – I
165. If the characteristic equation of matrix A of
order 3  3 is   3  3  1  0
3
c)
d)
170. If the characteristic equation of matrix A of
order 2  2 is   3  2  0 .Using Cayley
2
Hamilton Theorem, simplified expression of
is
a) 3 A – 5 I
b) - 5 A + 3 I
c) 5 A - 3 I
d) none of these
171. If the characteristic equation of matrix A of
order 3  3 is  -6  9  4  0 Using Cayley
3
2
Hamilton theorem, simplified expression of
is
2
then by Cayley Hamilton theorem A-1 is equal to
a) 5 A + 3 I
c) 5 A - 3 I
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b) - 5 A + 3 I
d) none of these
Matrices
Answer Key
Q. No.
Ans.
Q. No.
Ans.
Q. No.
Ans.
Q. No.
Ans.
Q. No.
Ans.
1
a
36
b
71
b
106
d
141
c
2
b
37
c
72
c
107
d
142
b
3
a
38
d
73
a
108
d
143
c
4
a
39
c
74
d
109
d
144
d
5
c
40
b
75
c
110
d
145
d
6
a
41
c
76
c
111
a
146
c
7
c
42
a
77
d
112
d
147
c
8
c
43
d
78
d
113
d
148
d
9
d
44
b
79
c
114
a
149
d
10
a
45
d
80
d
115
a
150
b
11
b
46
a
81
a
116
c
151
d
12
b
47
a
82
a
117
b
152
b
13
a
48
b
83
c
118
a
153
b
14
c
49
a
84
b
119
a
154
c
15
d
50
d
85
d
120
c
155
d
16
a
51
c
86
a
121
a
156
a
17
b
52
a
87
a
122
d
157
a
18
c
53
b
88
a
123
a
158
a
19
b
54
a
89
c
124
a
159
c
20
a
55
d
90
b
125
c
160
a
21
b
56
b
91
a
126
d
161
c
22
c
57
b
92
a
127
a
162
b
23
c
58
a
93
c
128
c
163
d
24
b
59
c
94
b
129
a
164
a
25
a
60
c
95
a
130
a
165
d
26
d
61
c
96
a
131
a
166
b
27
c
62
c
97
c
132
a
167
b
28
b
63
b
98
d
133
a
168
a
29
b
64
b
99
a
134
c
169
a
30
a
65
b
100
b
135
d
170
a
31
c
66
b
101
a
136
d
171
c
32
c
67
c
102
a
137
c
33
d
68
a
103
b
138
b
34
b
69
c
104
d
139
a
35
d
70
b
105
d
140
a
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Engineering Mathematics-I
Multiple Choice Questions
UNIT I
Taylor’s and Maclaurin’s Series
1) The expansion of
powers of about
in ascending
is
5) The expansion of
powers of is
a)
a)
b)
b)
c)
c)
d)
2) Expansion of
d)
in powers of
is
6) The coefficient of
(
a)
in the expansion of
) in ascending powers of
is
b)
c)
d)
3) Expansion of
in ascending powers of
a) 1
c) 2
b) 4/3!
d) 3!
7) The coefficient of
of
is
a)
c)
in the expansion
8) The expansion of
first three terms is
a)
b)
about
b)
d)
is
a)
b)
c)
upto
c)
d)
4) The coefficient of (
expansion of
in ascending
) in the
about is
a) √
b) √
c)
d)
d)
9) In the expansion of
in
powers of
the constant term is
a)
b)
c)
d)
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Page 1
Engineering Mathematics-I
10) The expansion of
a)
b)
Multiple Choice Questions
is
16) Expansion of
powers of is
in ascending
a)
c)
b)
d)
c)
11) The expansion of
d)
is
a)
17) The
b)
d)
12) In the expansion of
about
, the coefficient of (
13) The expansion of
ascending powers of
a)
b)
) is
b)
d)
in
in the expansion of
b)
d)1/2
19) The constant term in the expansion of
(
) is
a)
b)
c)
d)
20) First two terms in expansion of
is
d)
15) Expansion of
powers of is
18) The coefficient of
is
a) 0
c) 1
is
c)
14) The coefficient of
expansion of
in powers of
a) 1
c) 9
is
a)
b)
c)
d)
c)
a) 0
c) 1
term in the expansion of
a)
in the
b)
c)
is
d)
b)
d)
in ascending
21) The first three terms in the power series
for
are
a)
a)
b)
b)
c)
c)
d)
d)
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Page 2
Engineering Mathematics-I
Multiple Choice Questions
22) In the Taylor series expansion of
about the point
the coefficient of
is
a)
b)
c)
d)
27) Expansion of
23) Which of the following functions would
have only odd powers of in its Taylor
series expansion about the point
.
a)
b)
c)
d)
28) Expansion of
24) The limit of the series
29) Expansion of
as
a)
b)
c)
d) None
(
)
a)
b)
c)
d)
is
a)
approaches is
b)
a)
b)
c)
d) 1
c)
d) None
25) The Taylor series expansion of
is given by
at
30) The limit of the series
as
a)
approaches ½ is
a)
b)
c)
d)
b)
c)
2/3
1/3
1
4/3
d)
31) The limit of the series
as
26) Expansion of
is
Approaches is
a)
b)
c)
d)
a)
b)
c)
4
3
2
d)
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Page 3
Engineering Mathematics-I
Multiple Choice Questions
32) Representation of
in powers of
38)
is
a) 0
c) 1
is
a)
b)
c)
d)
39)
b)
d) 2
is
a) 0
c) 1
33) The first three terms in expansion of
in powers of
(
40)
b)
d) 2
) is
a) 0
c) 1
is
a)
b)
c)
d)
b)
d) 2
41) The value of
34) The Maclaurin series of
is
a)
b)
c)
d)
None
35)
is
a) 0
c)
b)1
d)2
36) If
is finite then the
value of
a) 0
c)
37)
a)
c)
b)1
d)
42) The value of
( )
a)
c) 1
Indeterminate form
is
is
b) 0
d) 2
43) The value of
a)
c) 1
b) 0
d) 2
44) The value of
a)
c) 1
b) 0
d) 2
45) The value of
a)
c) 1
is
is
is
b) 0
d) 2
is
b)1
d)2
is
a) 0
c)
b) 1
d) 2
46) The value of
is
a)
b)
c)
d)
47) The value of
a) 0
c)
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is
b)
d) 1
Page 4
Engineering Mathematics-I
Multiple Choice Questions
48) The value of
a)
c) 0
is
49) The value of
b)
d)
*
58) The value of
+
is
59) The value of
51) The value of
is
finite then
a)
c)
52) The value of
is
is
(
)
is
b)
d) 2
(
)
a) 5
c) -5
is
b)
d)
is
equal to
a) -1
c)
61) If
b) 2
d)
b)
d)
is finite then value of
is
a) 2
c) 1
53) The value of
a)
c)
b) 0
d) √
54) The value of
is
is
b) 0
d) ⁄
60) The value of
b)
d)
finite then value of
a) 1
c)
)
a) 1
c) -3
b) √
d) √
a)
c)
(
equal to
a) 1
c) 1/2
is
a) 2
c) 4
50) The value of
57) The value of
b)
d) 1
b) -2
d) -1
is
a) 1
c)
62) If
a)
b)
c)
d)
b)
d) 2
55) The value of
is
a) 1
c) -3
b) -1
d) 2
56) The value of
is
a) 1
c) -3
b) -1
d) 2
then
63) The value of
and
and
and
and
is
a)-1
b)0
c)1
d)
64) The value of
a) -1
c) 1
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is
b) 0
d)
Page 5
Engineering Mathematics-I
65) The value of
a) 1
c)
Multiple Choice Questions
76) The value of
is
b) 0
d)
is
77) If
66) It is given that
and
exists then
which of the following is true
a)
b)
c)
d)
, then value of
is
a) 1
c)
67)
b) 0
d)
78) If
of the following is true
a)
b)
c)
d)
is equal to
a)
b) 0
c)
d) 1
68) The value of
is
69) The value of
is
79) If
b)
c)
d)
then which
of the following is true
a)
b)
c)
a)
then which
d) none
80) The value of
70) The value of
is
is
81) The value of
71) The value of
is
is
82) If
72) The value of
a)
c)
is
then
b)
d)
83) If
73) The value of
is
74) The value of
is
a)
b)
c)
d)
84) The value of
75) The value of
then
is
is
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Page 6
Engineering Mathematics-I
85) If
that
and
Multiple Choice Questions
are 2 functions such
and
then
(
95)
) (
96)
86) The value of
b)
c)
89)
is equal to
97)
is equal to
d)
98)
is equal to
is
b)
c) 1
88) If
a)
c)
)
is
87) The value of
a)
(
is equal to
is equal to
a)
)
d) -1
99)
is equal to
then
b)
d)
, then
100)
is
is equal to
101)
is equal to
equal to
90)
91)
is equal to
is equal to
102) The value of
(
103) The value of
is
104) The value of
92)
93)
94)
) is
is
is equal to
105) The value of
is
106) The value of
is
107) The value of
is
is equal to
is equal to
a) 1
Sinhgad College of Engineering
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b) 8
c) 9
d) none
Page 7
Engineering Mathematics-I
Multiple Choice Questions
Unit - IV
Taylor's & Maclaurin's Theorem , Indeterminant Form
Answer Key
Q.No.
Ans
Q.No.
Ans
Q.No.
Ans
Q.No.
Ans
1
b
28
d
55
b
82
b
2
b
29
a
56
a
83
a
3
b
30
a
57
c
84
A
4
d
31
d
58
b
85
d
5
b
32
c
59
d
86
b
6
c
33
d
60
c
87
c
7
a
34
a
61
b
88
a
8
d
35
d
62
a
89
d
9
c
36
c
63
c
90
c
10
c
37
b
64
b
91
a
11
c
38
c
65
a
92
b
12
a
39
a
66
b
93
d
13
d
40
a
67
d
94
d
14
a
41
c
68
a
95
a
15
d
42
b
69
a
96
a
16
c
43
c
70
a
97
a
17
b
44
b
71
c
98
a
18
c
45
b
72
a
99
c
19
a
46
a
73
b
100
a
20
a
47
c
74
c
101
a
21
d
48
a
75
d
102
a
22
b
49
b
76
b
103
b
23
a
50
b
77
c
104
a
24
d
51
c
78
b
105
c
25
d
52
d
79
a
106
c
26
b
53
c
80
c
107
c
27
c
54
b
81
b
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Page 8
Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
((MARKS))
1
QUESTION IS OF
HOW MANY
MARKS? (1 OR 2
OR 3 UPTO 10)
((QUESTION))
According to Rolle’s mean value theorem, f  x  is continuous in  a, b ,
ENTER
CONTENT. QTN
CAN HAVE
IMAGES ALSO
differentiable in  a, b  such that f  a   f  b  , then there exits
((OPTION_A))
f c  0
c   a, b  such that
THIS IS
MANDATORY
OPTION
((OPTION_B))
f 'c  0
THIS IS ALSO
MANDATORY
OPTION
((OPTION_C))
f ''  c   0
This is optional
((OPTION_D))
f 'c  0
This is optional
((OPTION_E))
This is optional.
If optional keep
empty so that
system will skip
this option
((CORRECT_CH B
OICE)) Either A
or B or C or D or
E
((EXPLANATION
)) This is also
optional
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Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
studymedia.in/fe/m1-mcqs/
Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
((MARKS))
1
QUESTION IS OF
HOW MANY
MARKS? (1 OR 2
OR 3 UPTO 10)
((QUESTION))
According to Lagrange’s mean value theorem, f  x  is continuous in
ENTER
CONTENT. QTN
CAN HAVE
IMAGES ALSO
 a, b , differentiable in  a, b  , then there exits c   a, b  such that
((OPTION_A))
f 'c 
THIS IS
MANDATORY
OPTION
((OPTION_B))
THIS IS ALSO
MANDATORY
OPTION
((OPTION_C))
This is optional
((OPTION_D))
f c 
f ''  c  
f b  f  a 
ba
f b  f  a 
ba
f ' b  f '  a 
ba
f 'c  0
This is optional
((OPTION_E))
This is optional.
If optional keep
empty so that
system will skip
this option
((CORRECT_CH A
OICE)) Either A
or B or C or D or
E
((EXPLANATION
)) This is also
optional
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Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
((MARKS))
1
QUESTION IS OF
HOW MANY
MARKS? (1 OR 2
OR 3 UPTO 10)
((QUESTION))
According to Cauchy’s mean value theorem, f  x  & g  x  are
ENTER
CONTENT. QTN
CAN HAVE
IMAGES ALSO
continuous in  a, b , differentiable in  a, b  such that g  a   g  b  , then
((OPTION_A))
f 'c
THIS IS
MANDATORY
OPTION
((OPTION_B))
THIS IS ALSO
MANDATORY
OPTION
((OPTION_C))
This is optional
((OPTION_D))
This is optional
there exits c   a, b  such that
g 'c

f ' b  f '  a 
g ' b  g ' b 
f ''  c  f '  b   f '  a 

g ''  c  g '  b   g '  b 
f '  c  f b   f  a 

g '  c  g b  g b
f '  c  f b   f  a 

g '  c  g b  g b
((OPTION_E))
This is optional.
If optional keep
empty so that
system will skip
this option
((CORRECT_CH C
OICE)) Either A
or B or C or D or
E
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Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
((EXPLANATION
)) This is also
optional
((MARKS))
1
QUESTION IS OF
HOW MANY
MARKS? (1 OR 2
OR 3 UPTO 10)
((QUESTION))
If f  x   x2  10 x  16 such that f  3  f  7  , then according to Rolle’s
ENTER
CONTENT. QTN
CAN HAVE
IMAGES ALSO
theorem c 
((OPTION_A))
2
THIS IS
MANDATORY
OPTION
((OPTION_B))
3
THIS IS ALSO
MANDATORY
OPTION
((OPTION_C))
4
This is optional
((OPTION_D))
5
This is optional
((OPTION_E))
This is optional.
If optional keep
empty so that
system will skip
this option
((CORRECT_CH
D
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Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
OICE)) Either A
or B or C or D or
E
((EXPLANATION
)) This is also
optional
((MARKS))
1
QUESTION IS OF
HOW MANY
MARKS? (1 OR 2
OR 3 UPTO 10)
((QUESTION))
ENTER
CONTENT. QTN
CAN HAVE
IMAGES ALSO
((OPTION_A))

according to Rolle’s theorem c 
2
THIS IS
MANDATORY
OPTION
((OPTION_B))
-2
THIS IS ALSO
MANDATORY
OPTION
((OPTION_C))
1
This is optional
((OPTION_D))

If f  x   x3  12 x defined in 0, 2 3  such that f  0   f 2 3 , then
0
This is optional
((OPTION_E))
This is optional.
If optional keep
empty so that
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Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
system will skip
this option
((CORRECT_CH A
OICE)) Either A
or B or C or D or
E
((EXPLANATION
)) This is also
optional
((MARKS))
1
QUESTION IS OF
HOW MANY
MARKS? (1 OR 2
OR 3 UPTO 10)
((QUESTION))
ENTER
CONTENT. QTN
CAN HAVE
IMAGES ALSO
((OPTION_A))
If f  x   sin x in the interval 0, 2  , then according to Rolle’s theorem
c
only

2
only
3
2
THIS IS
MANDATORY
OPTION
((OPTION_B))
THIS IS ALSO
MANDATORY
OPTION
((OPTION_C))
This is optional
((OPTION_D))
both

2
,
3
2
none of the above
This is optional
((OPTION_E))
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Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
This is optional.
If optional keep
empty so that
system will skip
this option
((CORRECT_CH C
OICE)) Either A
or B or C or D or
E
((EXPLANATION
)) This is also
optional
((MARKS))
1
QUESTION IS OF
HOW MANY
MARKS? (1 OR 2
OR 3 UPTO 10)
((QUESTION))
If f  x   x3  4 x defined in  0, 2 such that f  0   f  2  , then
ENTER
CONTENT. QTN
CAN HAVE
IMAGES ALSO
according to Rolle’s theorem c 
((OPTION_A))
2
3
THIS IS
MANDATORY
OPTION
((OPTION_B))
THIS IS ALSO
MANDATORY
OPTION
((OPTION_C))

2
3
0
This is optional
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Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
((OPTION_D))
1
This is optional
((OPTION_E))
This is optional.
If optional keep
empty so that
system will skip
this option
((CORRECT_CH A
OICE)) Either A
or B or C or D or
E
((EXPLANATION
)) This is also
optional
((MARKS))
1
QUESTION IS OF
HOW MANY
MARKS? (1 OR 2
OR 3 UPTO 10)
((QUESTION))
If f  x   x  x  2  defined in 1, 3 such that f 1  1, f  3  3 , then
ENTER
CONTENT. QTN
CAN HAVE
IMAGES ALSO
according to Lagrange’s mean value theorem, c 
((OPTION_A))
0
THIS IS
MANDATORY
OPTION
((OPTION_B))
1
THIS IS ALSO
MANDATORY
OPTION
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Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
((OPTION_C))
2
This is optional
((OPTION_D))
3
This is optional
((OPTION_E))
This is optional.
If optional keep
empty so that
system will skip
this option
((CORRECT_CH C
OICE)) Either A
or B or C or D or
E
((EXPLANATION
)) This is also
optional
((MARKS))
1
QUESTION IS OF
HOW MANY
MARKS? (1 OR 2
OR 3 UPTO 10)
((QUESTION))
If f  x   x 2 defined in 1, 5 such that f 1  1, f  5  25 , then
ENTER
CONTENT. QTN
CAN HAVE
IMAGES ALSO
according to Lagrange’s mean value theorem, c 
((OPTION_A))
0
THIS IS
MANDATORY
OPTION
((OPTION_B))
1
studymedia.in/fe/m1-mcqs/
Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
THIS IS ALSO
MANDATORY
OPTION
((OPTION_C))
2
This is optional
((OPTION_D))
3
This is optional
((OPTION_E))
This is optional.
If optional keep
empty so that
system will skip
this option
((CORRECT_CH D
OICE)) Either A
or B or C or D or
E
((EXPLANATION
)) This is also
optional
((MARKS))
1
QUESTION IS OF
HOW MANY
MARKS? (1 OR 2
OR 3 UPTO 10)
((QUESTION))
If f  x   x  x  1 defined in 1, 2 such that f 1  0, f  2   2 , then
ENTER
CONTENT. QTN
CAN HAVE
IMAGES ALSO
according to Lagrange’s mean value theorem, c 
((OPTION_A))
1
2
THIS IS
MANDATORY
studymedia.in/fe/m1-mcqs/
Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
OPTION
((OPTION_B))
THIS IS ALSO
MANDATORY
OPTION
((OPTION_C))
3
2

This is optional
((OPTION_D))
This is optional
3
2
3
4
((OPTION_E))
This is optional.
If optional keep
empty so that
system will skip
this option
((CORRECT_CH B
OICE)) Either A
or B or C or D or
E
((EXPLANATION
)) This is also
optional
((MARKS))
1
QUESTION IS OF
HOW MANY
MARKS? (1 OR 2
OR 3 UPTO 10)
((QUESTION))
If f  x   x 2  3x  2 defined in  1, 2 such that f  1  6, f  2   0 ,
ENTER
CONTENT. QTN
CAN HAVE
IMAGES ALSO
then according to Lagrange’s mean value theorem, c 
studymedia.in/fe/m1-mcqs/
Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
((OPTION_A))
THIS IS
MANDATORY
OPTION
((OPTION_B))
1
2
1
THIS IS ALSO
MANDATORY
OPTION
((OPTION_C))
This is optional
3
2
((OPTION_D))
2
This is optional
((OPTION_E))
This is optional.
If optional keep
empty so that
system will skip
this option
((CORRECT_CH A
OICE)) Either A
or B or C or D or
E
((EXPLANATION
)) This is also
optional
((MARKS))
1
QUESTION IS OF
HOW MANY
MARKS? (1 OR 2
OR 3 UPTO 10)
((QUESTION))
If f  x   x 2 , g  x   x3 defined in  0, 1 such that
ENTER
f  0  0, f 1  1, g  0   0, g 1  1 , then according to Cauchy’s
studymedia.in/fe/m1-mcqs/
Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
CONTENT. QTN
CAN HAVE
IMAGES ALSO
mean value theorem,
((OPTION_A))
0
f 'c
g 'c

THIS IS
MANDATORY
OPTION
((OPTION_B))
1
THIS IS ALSO
MANDATORY
OPTION
((OPTION_C))
-1
This is optional
((OPTION_D))
2
This is optional
((OPTION_E))
This is optional.
If optional keep
empty so that
system will skip
this option
((CORRECT_CH B
OICE)) Either A
or B or C or D or
E
((EXPLANATION
)) This is also
optional
((MARKS))
1
QUESTION IS OF
HOW MANY
MARKS? (1 OR 2
studymedia.in/fe/m1-mcqs/
Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
OR 3 UPTO 10)
((QUESTION))
If f  x   x2  1, g  x   x3  1 defined in  0, 1 such that
ENTER
CONTENT. QTN
CAN HAVE
IMAGES ALSO
f  0  1, f 1  2, g  0   1, g 1  2 , then according to Cauchy’s
((OPTION_A))
mean value theorem,
f 'c
g 'c

0
THIS IS
MANDATORY
OPTION
((OPTION_B))
1
THIS IS ALSO
MANDATORY
OPTION
((OPTION_C))
2
This is optional
((OPTION_D))
0.5
This is optional
((OPTION_E))
This is optional.
If optional keep
empty so that
system will skip
this option
((CORRECT_CH B
OICE)) Either A
or B or C or D or
E
((EXPLANATION
)) This is also
optional
((MARKS))
1
studymedia.in/fe/m1-mcqs/
Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
QUESTION IS OF
HOW MANY
MARKS? (1 OR 2
OR 3 UPTO 10)
((QUESTION))
If f  x   2 x2 , g  x   3x3 defined in  0, 1 such that
ENTER
CONTENT. QTN
CAN HAVE
IMAGES ALSO
f  0  0, f 1  2, g  0   0, g 1  3 , then according to Cauchy’s
((OPTION_A))
THIS IS
MANDATORY
OPTION
((OPTION_B))
THIS IS ALSO
MANDATORY
OPTION
((OPTION_C))
mean value theorem,
g 'c

2
3
3
2

2
3

3
2
This is optional
((OPTION_D))
f 'c
This is optional
((OPTION_E))
This is optional.
If optional keep
empty so that
system will skip
this option
((CORRECT_CH A
OICE)) Either A
or B or C or D or
E
((EXPLANATION
)) This is also
optional
studymedia.in/fe/m1-mcqs/
Engineering Mathematics-I 2019-course Unit-I Mean value Theorem
((MARKS))
QUESTION IS OF
HOW MANY
MARKS? (1 OR 2
OR 3 UPTO 10)
1
((QUESTION))
If f  x   3x2 , g  x   4 x3 defined in  0, 1 such that
ENTER CONTENT.
QTN CAN HAVE
IMAGES ALSO
f  0  0, f 1  3, g  0   0, g 1  4 , then according to Cauchy’s
mean value theorem,
((OPTION_A))
0
f 'c
g 'c

THIS IS
MANDATORY
OPTION
((OPTION_B))
0.5
THIS IS ALSO
MANDATORY
OPTION
((OPTION_C))
0.75
This is optional
((OPTION_D))
1
This is optional
((OPTION_E)) This
is optional. If
optional keep
empty so that
system will skip
this option
((CORRECT_CHOIC C
E)) Either A or B or
C or D or E
((EXPLANATION))
This is also
optional
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