Integral Calculus and Differential Equations Test

Result

Integral Calculus and Differential Equations Test - 1

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Question 1
2 / -0.83

to

A

B

C

D

Solution
Question 2
2 / -0.83

A

B

C

D

Solution
Question 3
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A
0

B
1

C
u

D
eu

Solution

f is a homogeneous function of degree one
Question 4
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A
nu

B
n(n - 1)u

C

D

Solution

It is a homogeneous function of degree n
Question 5
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Match the List –I with List –II.

A
A

B
B

C
C

D
D

Solution

It is a homogeneous function of
degree 2.
Question 6
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If an error of 1% is made in measuring the major and minor axes of an ellipse, then the percentage error in the area is approximately equal to

A
1%

B
2%

C
π%

D
4%

Solution

Let 2 a and 2 b be the major and minor axes of the ellipse
Question 7
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Consider the Assertion (A) and Reason (R) given below:
Reason (R): Given function u is homogeneous of degree 2 in x and y.
Of these statements

A
Both A and R are true and R is the correctexplanation of A

B
Both A and R are true and R is not a correctexplanation of A

C
A is true but R is false

D
A is false but R is true

Solution

Given that u = xyf(y/x) Since it is a homogeneous function of degree 2.
Question 8
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If u = x log xy, where x³ + y³ + 3xy = 1, then du/dx is equal to

A

B

C

D

Solution

Given that u = x log xy ... (i)
Question 9
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f(x) = x² e^-x is increasing in the interval

A
] -∞, ∞[

B
] -2 , 0 [

C
] 2 , ∞[

D
] 0, 2 [

Solution
Question 10
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The minimum distance from the point (4, 2) to the parabola y² =8x is

A
√2

B
2 √2

C
2

D
3 √2

Solution

Let the point closest to (4, 2) be (2t² ,4)
Question 11
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The co-ordinates of the point on the parabola y = x² + 7x + 2 which is closest to the straight line y = 3x - 3, are

A
(-2, -8)

B
(2, -8)

C
(-2, 0)

D
None of these

Solution

Let the required point be P(x, y). Then, perpendicular distance of P(x, y) from y - 3x - 3 = 0 is
Question 12
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The shortest distance of the point (0, c), where 0 ≤c ≤5, from the parabola y = x² is

A

B

C

D
none of these

Solution

Let A (0,c) be the given point and P (x, y) be any point on y = x²
Question 13
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The set of equations
has infinite solutions, if a=

A
-3

B
3

C
4

D
-4

Solution

The given system of equations can be expressed in the matrix form:
The augmented matrix is
For infinite solutions
Question 14
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The maximum value of ( 1/x)^x is

A
e

B
e^1/e

C
(1/e)^e

D
none of these

Solution

f (x) = (1 / x)^x
f ’(x) = (1 / x)^x (log (1 / x) –1))
f ’(x) = 0
log (1 / x) –1 = log e
1 / x = e
x = 1 / e
The maximum value of function is e^1/e .
Question 15
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The minimum value of 3x+5y such that:
is ___________.

A
0

B
1

C
2

D
3

Solution

Comparing (ii) and (iii)
Hence, Checking at corner points:
At (0, 0) Z=0 so minimum value will be 0
Question 16
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The condition for which the eigenvalues of the matrix

are positive, is

A
k >1/2

B
k >-2

C
k >0

D
k <-1/2

Solution

All Eigen values of are positive
2 >0
∴2 ×2 leading minor must be greater than zero
Question 17
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The maximum value of f ( x) = (1 + cos x) sin x is

A
3

B
3 √3

C
4

D
3 √3/4

Solution
Question 18
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The solution to the system of equations

A
6,2

B
-6,2

C
-6,-2

D
6,-2

Solution
Question 19
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The greatest value of
on the interval [0, π/2] is

A
1/√2

B
√2

C
1

D
-√2

Solution
Question 20
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For what value of a, if any, will the following system of equations in x, y and z have a solution
2x + 3y = 4
x + y + z = 4
x + 2y - z = a

A
Any real number

B
0

C
1

D
There is no such value

Solution

Augmented matrix is
will have solution if a=0 as Rank A= Rank (aug. A)