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Test: Functions Of One,Two Or Three Real Variables - 4 - Mathematics MCQ


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20 Questions MCQ Test Topic-wise Tests & Solved Examples for Mathematics - Test: Functions Of One,Two Or Three Real Variables - 4

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Test: Functions Of One,Two Or Three Real Variables - 4 - Question 1

Convert the set x in roster form if set x contains the positive prime number, which divides 72. 

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables - 4 - Question 1

2 and 3 are the divisors of 72, which are prime. So, the roster form of set x is (2, 3}.

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 2

Let f(x + y) = f(x)f(y) for all x and y and f(5) = - 2 and f'(0) = 3. What is the value of f'(5)?

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables - 4 - Question 2

f'(0) = 3
implies 
or 
[Since f(5 + 0) = f(5) • f(0)
implies f(5) = f(5) - f(0)
implies f(0) = 1]
Now, we have

[by eq. (1)]
Hence, f'(5) = -6 

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Test: Functions Of One,Two Or Three Real Variables - 4 - Question 3

Let l  = { 1 } ∪ { 2 }  For x ∈ R, let φ(x) = dis (x, l) = ln f{| x —y | : y ∈ l} then

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables - 4 - Question 3

Given that,

= ln f{|x -.y| : y ∈ 1}
where l = { l } ∪ { 2 }
= ln f{| x -l| ,| x -2 | } 
the graph of the φ(x) is given by.

Clearly, the graph of the function have sharp edges at x = 1,3/2 and 2. Therefore, f(x) is not differentiable at x
= 1, 3/2 and 2.
Hence, φ(x) is continuous on R but not
differentiable at x = 1, 3/2 and 2

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 4

Let f(x) = 

then f is

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables - 4 - Question 4


At x =1/2

Since 
therefore, f(x) is not continuous x =1/2

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 5

If the function defined as

Then, the function f (x) has

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables - 4 - Question 5


So, the function is not continuous at the point x = 1. The discontinuity is of the first kind and can be removed by defining function as f (x) = – 1.

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 6

Let

then,

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 7

Let

then,

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 8

Let

then,

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 9

Let

then,

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 10

Let

then,

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 11

Write the roaster form for A = {x : x2 – 5x + 6 = 0}
 

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables - 4 - Question 11

Concept: Tabular form / Roaster form: In this method, a set is described by listing all the elements, separated by commas, within the braces {}.

Example: A = {2, 3, 5} is a set of first three prime numbers. Set –builder form: In this method, all the elements of the set possess a single common property, which is being enlisted. Example: B = {x : 6 ≤ x ∈ N ≤ 12}

Calculation: Given: A = {x : x2 – 5x + 6 = 0} x2 – 5x + 6 = 0 ⇒ x2 – 2x -3x + 6 = 0 ⇒ x(x - 2) – 3(x - 2) = 0 ⇒ (x - 3)(x - 2) = 0 ⇒ x = 3 or 2 As we know that, roaster form of a set is described by listing all the elements, separated by commas, within the braces {}.

Hence the required roaster form of set B = {2, 3}

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 12

If we expand sinx by Taylor’s series about  then a1, a7, a4, a3 are

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables - 4 - Question 12

Given function f(x) = sin x



the value of a2, a7 a4, a3 in (i) are

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 13

If f(x) is twice differentiable and | f(x) | < α, f'(x)| < β, in the range x > α, then which of the following is correct?

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables - 4 - Question 13

Let x > a and n a positive number then



or 
Now, for maxima or minima of f(h), we have

or 
and 
Hence, the least value of φ(h)

= 2√AB
thus, 

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 14

The function f(x) = | x + 2 | is not differentiable at a point

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables - 4 - Question 14

Given that

Clearly, the rule of the function is changing at x = 0, so we shall test the differentiability of f(x) only at the point x = -2 obviously, being polynomial, at all other points the function is differentiable.


Since Lf '(- 2) ≠ R'f(- 2) , therefore , f(x) is not differentiable at x = - 2 .

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 15

Using Rolle’s theorem, the equation. = 0 has atleast one root between 0 and 1, if

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables - 4 - Question 15

Consider the function f defined by
Since , f(x) is a polynomial, it is continuous and differentiablex. Consequently f(x) is continuous in the closed interval [0, 1] and differentiable in the open interval [0,1] Also , f(0) = 0 and

i.e. , f(0) = f(1)
Thus, all the three conditions of Rolle’s theorem are satisfied
Hence, there is atleast one value of x in the open interval [0, 1]
where f'(x) = 0
i.e., cos xn + a1.xn-1 + . . . + an-1 + a= 0 
 

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 16

A function f : R-->R satisfies the e q .f(x +y) = f(x).f(y), and f(x) ≠ 0,  . If f(x) is differentiate at 0 a n d f'(0 ) = 2 then f'(x) is equal to

Detailed Solution for Test: Functions Of One,Two Or Three Real Variables - 4 - Question 16

We have

Now f(0 + 0)= f(0).f(0)

[Since f(x)  0 ,]
from eq. (ii) we get
Gives 

 

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 17

Which one of the following is true?

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 18

Let

then,

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 19

Let

then,

Test: Functions Of One,Two Or Three Real Variables - 4 - Question 20

Let

then,

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