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Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability


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38 Questions MCQ Test Mathematics For JEE | Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability

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Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 1

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 1

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 2

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Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 3

Let f (x) = 4 and f ' (x) = 4.  Then  given by

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 3

Apply L H Rule

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 4

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 4


Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 5

denotes greatest integer less than or equal to x)

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 5

Since  does not exist, hence the required limit
does not exist.

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 6

If f(1) = 1, f 1 (1) = 2, then 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 6

 form using L’ Hospital’s rule

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 7

f is defined in [-5, 5] as f(x) = x if x is rational = –x if x is irrational. Then

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 7

Let a is a rational number other than 0, in [–5, 5], then

[As in the immediate neighbourhood of a rational number, we find irrational numbers]
∴ f (x) is not continuous at any rational number

If a is irrational number, then


∴ f (x) is not continuous at any irrational number clearly

∴ f (x) is continuous at x = 0

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 8

f(x) and g(x) are two differentiable functions on [0, 2] such that f ''( x) - g '' (x)= 0, f ' (1) =2g'(1)=4 f(2) =3g(2) = 9 then f(x)–g(x) at x = 3/2 is

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 9

 and f (5) = 2,f '(0)= 3 , then f ' (5) is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 9

f (x + y) = f (x) * f (y)
Differentiate with respect to x, treating y as constant
f ' (x + y) = f ' (x) f (y)
Putting x = 0 and y = x, we get f '(x)= f '(0) f (x) ;
⇒ f ' (5) = 3 f (5) = 3 × 2 = 6.

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 10

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 10

The given expression can be written as

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 11

the value of k is

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Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 12

The value of 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 12

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 13

Let f (a) = g (a) = k  and their nth derivatives

fn (a) , gn (a) exist and are not equal for some n. Further if

then the value of k is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 13

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 14

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Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 15

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Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 16

then the values of a and b, are

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Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 17

 is continuous

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Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 18

equals

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⇒ Given limit is equal to value of integral

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 19

Let a and b be the distinct roots of ax2 + bx +c = 0 , then 

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Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 20

Suppose f(x) is differentiable at x = 1 and 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 20

As function is differentiable so it is continuous as it is given that   and hence f (1) = 0

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 21

Let f  be differentiable for all x. If f (1) = – 2 and f '(x) > 2 for x ∈ [1, 6], then

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 21

Applying Lagrange’s mean value theorem

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 22

If f is a real valued differentiable function satisfying

|f (x) – f (y) | < ( x -y)2 , x, y ∈ R and f (0) = 0, then f (1) equals

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 22


Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 23

Let f : R → R be a function defined by f (x) = min {x + 1,x+ 1} ,Then which of the following is true ?

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 23


Hence, f (x) is differentiable everywhere for all x ∈ R.

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 24

The function f : R /{0} → R given by

can be made continuous at x = 0 by defining f (0) as

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 24


∴ using, L'Hospital rule


Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 25

Then which one of the following is true?

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 25



Let this finite number be l

∴ f is not differentiable at x = 1

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 26

Let f : R →R be  a positive incr easing function with 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 26

f(x) is a  positive increasing function


By Sandwich Theorem.

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 27

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Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 28

The values of p and q for which the function 

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Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 29

Let f : R → [0, ∞) be such that    exists and 

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Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 30

If f : R → R is a function defined by f (x) = [x]  where [x] denotes the greatest integer function, then f is .

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 30


Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 31

Consider the function, f (x) = |x – 2|+ |x – 5|, x ∈ R.

Statement-1 : f '(4) = 0

Statement-2 : f is continuous in [2,5], differentiable in (2,5) and f (2) = f (5).

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 31





∴ statement-2 is also true and a correct explanation for statement 1.

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 32

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 32

Multiply and divide by x in the given expression, we get


 

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 33

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Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 34

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 34

Multiply and divide by x in the given expression, we get

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 35

If the function 

 is differentiable, then the value of k + m is :

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 35

Since g (x) is differentiable, it will be continuous at x = 3


Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 36

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 36

g (x) = f (f (x))
In the neighbourhood of x = 0,
(x) =  | log2 – sin x| = (log 2 – sin x)
∴ g (x) = |log 2 – sin| log 2 – sin x || = (log 2 – sin(log 2 – sin x))
∴ g (x) is differentiable and g'(x) = – cos(log 2 – sin x) (– cos x)
⇒ g'(0) = cos (log 2)

Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 37

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Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 38

then log p is equal to :

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Limits, Continuity and Differentiability - Question 38

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