Integral Calculus and Differential Equations Test

Result

Integral Calculus and Differential Equations Test - 2

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Question 1
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As x is increased from –∞to ∞, the function

A
Monotonically increases

B
Monotonically decreases

C
Increases to a maximum value and then decreases

D
Decreases to a minimum value and then increases

Solution
Question 2
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A
0.5

B
1

C
2

D
not defined

Solution
Question 3
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A
1 or 4

B
4 only

C
1 only

D
Undefined

Solution

Given
Square both side, we get
Question 4
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A
0

B
2

C
1

D
∞

Solution
Question 5
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A
∞

B
0

C
2

D
Does not exist

Solution

cos x and sin x are finite whatever x may be
Question 6
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A
∞

B
0

C
2

D
1

Solution
Question 7
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Which of the following functions is not differentiable in the domain [-1,1]?

A
f(x) = Maximum (x,-x)

B
f(x) = x-1

C
f(x) = 2

D
f(x) = x²

Solution
Question 8
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If f(x) =   then lim_x-►3 f(x) will be

A
–1/3

B
5/18

C
0

D
2/5

Solution
Question 9
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A
1/16

B
1/4

C
1/8

D
1/12

Solution

= 1/12
Question 10
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What should be the value of λsuch that the function defined below is continuous at x = π/2?

A
0

B
2/ π

C
1

D
π/2

Solution

By the given condition
From (1), λ= 1
Question 11
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The value of the function

A
0

B
-1/7

C
1/7

D
∞

Solution
Question 12
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Consider the function f(x) = |x|³ , where x is real. Then the function f(x) at x = 0 is

A
Twice differentiable but not thrice

B
Once differentiable but not twice

C
Continuous but not differentiable

D
Thrice differentiable

Solution
Question 13
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The expression e^{–ln x} for x >0 is equal to

A
–x

B
x

C
x^–1

D
–x^–1

Solution
Question 14
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Consider the following two statements about the function f(x) = |x|
P: f(x) is continuous for all real values of x
Q: f(x) is differentiable for all real values of x
Which of the f oll owi ng is TRU E?

A
P is true and Q is false

B
P is false and Q is true

C
Both P and Q are true

D
Both P and Q are false

Solution

The graph of f(x) is
f(x) is continuous for all real values of x Lim |x| = Lim |x| = 0
as can be seen from graph of |x|.
and Lim f(x) = +1 as can be seen from graph of |x|
x →0+
Left deriva tive ≠Rig ht derivative
So |x| is continuous but not differentiable at x = 0.
Question 15
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What is the value of

A
0

B
e^–2

C
e^–1/2

D
1

Solution
Question 16
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The integral is given by

A
1/2

B
2/3

C
4/3

D
8/3

Solution
Question 17
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Which one of the following function is strictly bounded?

A

B

C

D

Solution

For a strictly bounded function f(x), limit should be finite
Question 18
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A

B

C

D
Zero

Solution
Question 19
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Which of the following integrals is unbounded?

A

B

C

D

Solution
Question 20
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What is the value of the definite integral,

A
0

B
a/2

C
a

D
2a

Solution