Integral Calculus and Differential Equations Test

Result

Integral Calculus and Differential Equations Test - 3

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Question 1
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A function is given by f(t) = sin² t + cos 2t. Which of the following is true?

A
f has frequency components at 0 and 1/2π Hz.

B
f has frequency components at 0 and 1/π Hz.

C
f has frequency components at 1/2π and 1/π Hz.

D
f has frequency components at 0.1/2π and 1/π Hz.

Solution
Question 2
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Following are the values of a function y(x) : y(-1) = 5, y(0), y(1)   as per Newton ’s central difference scheme is:

A
0

B
1.5

C
2.0

D
3.0

Solution
Question 3
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A
∞

B
2

C
0

D
1

Solution

the squeeze theorem for this. Recall that sinx is only defined on −1 ≤sinx ≤1. Therefore
Question 4
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The function f(x) = |x+1| on the interval [-2, 0]

A
Continuous and differentiable

B
Continuous on the integral but not differentiable at all points

C
Neither continuous nor differentiable

D
Differentiable but not continuous

Solution

f(x ) = x+ 1
f is continuous in [−2, 0]
but not differentiable at
x =−1 because we can draw
infinite number of tangents at x = −1
Question 5
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If y=|x| for x <0 and y=x for x ≥0, then

A
y is discontinuous at x = 0

B

C
y is not defend at x = 0

D

Solution
Question 6
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What is the derivative of f(x) = |x| at x = 0?

A
1

B
-1

C
0

D
Does not exist

Solution

f( x ) = x .
At x = 0, we can draw infinitely many tangents at x=0.So limit does not exists.
Question 7
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A
1

B
∞

C
0

D
-1

Solution

= 1.0= 1
Question 8
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A
0

B
1/6

C
1/3

D
1

Solution
Question 9
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The function Y=| 2-3x |

A
is continuous ∀x ∈R and differentiable ∀x ∈R

B
is continuous ∀x ∈R and differentiable ∀x ∈R except at x = 3/2

C
is continuous ∀x ∈R and differentiable ∀x ∈R except at x = 2/3

D
is continuous ∀x ∈R except at x = 3 and differentiable ∀x ∈R
Question 10
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A
Indeter minate

B
0

C
1

D
2
Question 11
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The function f(x) = x3 - 6x2 + 9x + 25 has

A
a maxima at x= 1 and a minima at x = 3

B
a maxima at x = 3 and a minima at x = 1

C
no maxima, but a minima at x = 1

D
a maxima at x = 1, but no minima

Solution
Question 12
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A
1

B
–1

C
∞

D
–∞

Solution
Question 13
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The value of the integral   is

A
1

B
π

C
2

D
2 π

Solution
Question 14
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The following plot shows a function y which varies linearly with x. The value of the integral I =

A
1.0

B
2.5

C
4.0

D
5.0

Solution

Here the points (0,1) and (-1,0) are on the time
∴The equn of the line is
Question 15
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The value of the integral

A
−π

B
−π/2

C
π/2

D
π

Solution

=π
Question 16
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The length of the curve between x = 0 and x = 1 is

A
0.27

B
0.67

C
1

D
1.22

Solution

Length of the wire
Question 17
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A
-1/3

B
1/4

C
1/2

D
1

Solution
Question 18
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A continuous-time system is described by y (t) = e ^{−x (t)} where y (t) is the output and x (t) is the input. y(t) is bounded.

A
only when x(t) is bounded

B
only when x(t) is non-negative

C
only or t ≥0 if x (t) is bounded for t ≥0

D
even when x(t) is not bounded

Solution
Question 19
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A
0

B
1

C
In 2

D

Solution

Since
Question 20
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The value of the quantity P, where

A
0

B
1

C
e

D
1/e

Solution