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QUESTION: 1

If u = x^{m} y^{n } then

Solution:

Given that u = x^{m }y^{n}

Taking logarithm of both sides, we get log u = m log x + n log y Differentiating with respect to x,we get

QUESTION: 2

to

Solution:

QUESTION: 3

Solution:

QUESTION: 4

Solution:

f is a homogeneous function of degree one

QUESTION: 5

Solution:

It is a homogeneous function of degree n

QUESTION: 6

Match the List–I with List–II.

Solution:

It is a homogeneous function of

degree 2.

QUESTION: 7

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

Solution:

Let 2 a and 2 b be the major and minor axes of the ellipse

QUESTION: 8

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

Solution:

Given that u = xyf(y/x) Since it is a homogeneous function of degree 2.

QUESTION: 9

If u = x log xy, where x^{3} + y^{3} + 3xy = 1, then du/dx is equal to

Solution:

Given that u = x log xy ... (i)

QUESTION: 10

Solution:

The given function is homogeneous of degree 2.

QUESTION: 11

If a < 0, then f(x) = e^{ax} + e^{-ax} is decreasing for

Solution:

QUESTION: 12

f(x) = x^{2}e^{-x} is increasing in the interval

Solution:

QUESTION: 13

The least value of a for which f(x) = x^{2} + ax + 1 is increasing on ] 1, 2, [ is

Solution:

f'(x) = (2x + a)

QUESTION: 14

The minimum distance from the point (4, 2) to the parabola y^{2} = 8x is

Solution:

Let the point closest to (4, 2) be (2t^{2},4)

QUESTION: 15

The co-ordinates of the point on the parabola y = x^{2} + 7x + 2 which is closest to the straight line y = 3x - 3, are

Solution:

Let the required point be P(x, y). Then, perpendicular distance of P(x, y) from y - 3x - 3 = 0 is

QUESTION: 16

The shortest distance of the point (0, c), where 0 ≤ c ≤ 5, from the parabola y = x^{2} is

Solution:

Let A (0,c) be the given point and P (x, y) be any point on y = x^{2}

QUESTION: 17

The maximum value of ( 1/x)^{x} is

Solution:

QUESTION: 18

Solution:

QUESTION: 19

The maximum value of f ( x) = (1 + cos x) sin x is

Solution:

QUESTION: 20

The greatest value of

on the interval [0, π/2] is

Solution:

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