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

## 20 Questions MCQ Test Topic-wise Tests & Solved Examples for 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

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)  