Test: Functions Of One,Two Or Three Real Variables -1


20 Questions MCQ Test IIT JAM Mathematics | Test: Functions Of One,Two Or Three Real Variables -1


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QUESTION: 1

If f(x) is differentiable on interval l and such that |f'(x)| ≤ a on l, then f(x) is

Solution:

For xy ∈ l, by Lagrange’s mean value theorem
where x < c < y
implies f(x)- f(y) = (x - y) f '(c )
implies |f(x) - f(y) | = x - y || f'(c)|
for a given  such that |f(x) - f(y)| < ε, x, y ∈ l. Hence f(x) is uniformly continuous on l. We know that every uniformly continuous function is also continuous.

QUESTION: 2

The value of dydx by changing the order of integration is 

Solution:
QUESTION: 3

If f(x) is differentiable on (a, b) and f(a) = 0 and there exists a real number k such that |f'(x) ≤ k |f (x)| on [a, b], then f(x) is

Solution:

Choose x> a such that k(x1 - a) < 1,
let α = sup | f(t) | on a < t < x1 then by Lagrange’s mean value theorem

where a < t< x ≤ x1
implies f(x) - f(a) = (x - a) f '(t)
[since f(a) = 0]

QUESTION: 4

The volume of ellipsoide  is

Solution:
QUESTION: 5

If f(x+y) = f (x) . f(y) for all x and y. Suppose that f(3) = 3 and f'(0) = 11 then , f'(3) is equal to

Solution:

Given that f'(0) = 11
implies 
implies 
implies 
[Since f(3 + 0) = f(3) - f(0)
=> f(3) = f(3) • (0)]
implies f(0) = 1 Now we have

QUESTION: 6

The area bounded by the curve y = ψ(x), x-axis and the lines x = l , x = m(l <m ) is given by

Solution:
QUESTION: 7

If f(x) and g(x) are real number function defined on and , , then which of following is correct?

Solution:

Here,
cos f(x) - sin (g(x))


using eq. (ii) in eq. (i), then we obtain cos (f (x)) - sin (g (x)) > 0
or 

QUESTION: 8

The volume of an object expressed in spherical coordinates is given by 
The value of the integral is

Solution:
QUESTION: 9

 dx dy is equal to

Solution:
QUESTION: 10

Using the transformation x + y = u, y = v. The value of Jacobian (J) for the integral  is

Solution:
QUESTION: 11

Which of the following function is not called the Euler’s integral of the first kind?

Solution:

Euler’s integral of the first kind is nothing but Beta function. So, here only is not the definition of Beta function.

QUESTION: 12

Consider the following conditions
 (i) f(x) is well defined at x = a
(ii) must exist.
(iii) f(x) is continuous.
(iv) f(x) ≠ 0 at x = a
Q. which of these conditions are necessary for a function f(x) to be derivable at a point x = a of its domain?

Solution:

The necessary condition for a function f(x) to be derivable at a point x = a of its domain.
(i) f(x) is well defined at x = a
(ii) must exist.
(iv) f (x) is continuous.

QUESTION: 13

For (x, y) ∈ R2, let

Solution:

Let us approaches (0, 0) along the line y = mx which passes through origin. Put y = mx, we get

which depends on m. Hence,  does not exists, Therefore f(x, y) is discontinuous at (0,0).
Now, 
and 

Hence, f(x,y) is discontinuous at (0, 0). But both the partial derivative fx and fy exists at origin.

QUESTION: 14

Let 

Then,

Solution:

Let us suppose (x, y) approaches (0, 0) along the line y = mx. Which is a line through the origin. Put y = mx and allows x —> 0, we get

which depends on m, therefore the limit of f(x, y) at (0, 0) does not exists. Hence, f(x, y) is discontinuous at origin.
Now,
since fy exists at origin.

QUESTION: 15

Let f : R2 —> R be defined by

Then, the directional derivative direction of f at (0 , 0) in the direction o f the vector is

Solution:

we are given that u = and a = (0,0). we have directional derivative at a in the direction u as 

QUESTION: 16

L et f : R2 —> R be defined by

Then,

Solution:

We have,
But f(0 , 0) = 0
Hence,

implies f(x, y) is continuous at (0, 0) Now,
where A = 0, B = 0 which does not depends on h and k and

Hence, y(x, y) is continuous as well as differentiable at (0, 0).

QUESTION: 17

The inverse of function f(x) = x3 + 2 is ____________ 

Solution:

To find the inverse of the function equate f(x) then find the value of x in terms of y such that f -1 (y) = x.

QUESTION: 18

Let f : R2 —> R be such that and exist at all points. Then,

Solution:
QUESTION: 19

Let if possible,


 

Solution:
QUESTION: 20

Let f: R2 → R be defined by f(x, y)

Then the value of at the point (0, 0) is

Solution:

Here, f(x, y)