Practice Question on Continuity , differentiability For IIT-JEE , NDA and Airforce

Continuity , differentiability

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Question no. 1

What is equal to ?

looks_one 0

looks_two 1

looks_3 1/2

looks_4 limit does not exist

Option looks_3 0

Solution :

Question no. 2

Which one of the following is correct in respect pf the function f(x)= |x| + x²

looks_one f(x) is not continuous at x =0

looks_two f(x) is differentiable at x =0

looks_3 f(x) is continuous but not differentiable at x = 0

looks_4 None of these

Option looks_3 f(x) is continuous but not differentiable at x = 0

Solution :
f(x)= |x| + x²
Continuity :

So is is continuous at x = 0
Differentiability :

Question no. 3

What is the set of all points , where the function is differentiable

looks_one (-∞, ∞) only

looks_two (0, ∞) only

looks_3 (-∞,0) ∪ (0, ∞) only

looks_4 (-∞,0) only

Option looks_3 5: 7

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Question no. 4

what is equal to ?

looks_one 1/2

looks_two 1

looks_3 2

looks_4 0

Option looks_4 0

Solution :

Question no. 5

At how many points is the function f(x) = [x] discontinuous ?

looks_one 1

looks_two 2

looks_3 3

looks_4 infinity

Option looks_4 infinity

Solution :
Greater integer function has infinity discontinuity

Question no. 6

In the function , x ≠ ±2 is continuous at x =2 , then what is f(2) equal to ?

looks_one 0

looks_two 1/2

looks_3 1

looks_4 2

Option looks_two 1/2

Solution :

Question no. 7

What is the value of ?

looks_one a - b

looks_two a + b

looks_3

looks_4

Option looks_3

Solution :

Question no. 8

Consider the following statements :
1. f(x) = |x-3| is continuous at x = 0
2. f(x) = |x-3| is differentiate at x =0
which of the statements given above is/are correct ?

looks_one 1 only

looks_two 2 only

looks_3 Both 1 and 2

looks_4 neither 1 nor 2

Option looks_3 Both 1 and 2

Solution :
f(x) = |x-3| = -(x-3) when x ≤ 3
f(x) = |x-3| = (x-3) when x ≥ 3
therefore at x = 0 , f(x) = -(x-3) it is continuous and differentiable .

Question no. 9

The function f(x) = x cosec x is

looks_one Continuous for all values of x

looks_two Discontinuous everywhere

looks_3 Continuous for all x except at x = nπ , where n is an integer

looks_4 Continuous for all x except at x = nπ/2 , where n is an integer

Option looks_3 Continuous for all x except at x = nπ , where n is an integer

Solution :
cosecx is not defined at x = nπ , So it is discontinuous at x = nπ

Question no. 10

consider the following statements
1. Every function has a primitive
2. A primitive of a function is unique
which of the following given above is /are correct ?

looks_one 1 only

looks_two 2 only

looks_3 Both 1 and 2

looks_4 neither 1 nor 2

Option looks_two 2 sq. unit

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Question no. 11

what is the value of ?

looks_one 0

looks_two 1

looks_3 -1

looks_4 The limit does not exist

Option looks_one 0

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Question no. 12

What is the value of ?

looks_one e

looks_two e²

looks_3 e⁴

looks_4 e⁵

Option looks_4 1/3

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Question no. 13

consider the following function f : R → R such that f(x) = x if x ≥ 0 and f(x) = -x² if x<0 , then which one of the following is correct ?

looks_one f(x) is continonus at every x ∈ R

looks_two f(x) is continonus at x = 0 only

looks_3 f(x) is discontinonus at x = 0 only

looks_4 f(x) is discontinonus at at every x ∈ R

Option looks_3 X=2

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Question no. 14

What is equal to ?

looks_one e

looks_two e³

looks_3 e^-9

looks_4 e⁹

Option looks_3 Y=-8

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Question no. 15

Let y(x) = axⁿ and δy denote small change in y. What is the limit of ?

looks_one 0

looks_two 1

looks_3 anx^n-1

looks_4 axⁿlog(ax)

Option looks_one never intersect

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