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Test: Single Correct MCQs: Limits, Continuity and Differentiability | JEE Advanced

36 Questions MCQ Test Mathematics For JEE | Test: Single Correct MCQs: Limits, Continuity and Differentiability | JEE Advanced

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This mock test of Test: Single Correct MCQs: Limits, Continuity and Differentiability | JEE Advanced for JEE helps you for every JEE entrance exam. This contains 36 Multiple Choice Questions for JEE Test: Single Correct MCQs: Limits, Continuity and Differentiability | JEE Advanced (mcq) to study with solutions a complete question bank. The solved questions answers in this Test: Single Correct MCQs: Limits, Continuity and Differentiability | JEE Advanced quiz give you a good mix of easy questions and tough questions. JEE students definitely take this Test: Single Correct MCQs: Limits, Continuity and Differentiability | JEE Advanced exercise for a better result in the exam. You can find other Test: Single Correct MCQs: Limits, Continuity and Differentiability | JEE Advanced extra questions, long questions & short questions for JEE on EduRev as well by searching above.

QUESTION: 1

A.
0
B.
∞
C.
1
D.
none of these

Solution:

QUESTION: 2

For a real number y, let [y] denotes the greatest integer less than or equal to y : Then the function

A.
discontinuous at some x
B.
continuous at all x, but the derivative f ' (x) does not exist for some x
C.
f ' (x) exists for all x, but the second derivative f ' (x) does not exist for some x
D.
f ' (x) exists for all x

Solution:

By def. [x – π] is an integer whatever be the value of x.
And so p[x – π] is an integral multiple of p.
Consequently tan
And since 1 + [x]2 ≠ 0 for any x, we conclude that f (x) = 0.
Thus f (x) is constant function and so, it is continuous and differentiable any no. of times, that is f '(x), f ''(x), f '''(x),... all exist for every x, their value being 0 at every pt. x. Hence, out of all the alternatives only (d) is correct.

QUESTION: 3

There exist a function f (x), satisfying f (0) = 1, f '(0) = –1, f (x) > 0 for all x, and

A.
f ''(x) > 0 for all x
B.
–1<f "(x) < 0 for all x
C.
– 2 < f "(x) < –1 for all
D.
f " (x) < –2 for all x

Solution:

f (x) = e^–x is one such function.

QUESTION: 4

has the value

A.
1/24
B.
1/5
C.
D.
none of these

Solution:

QUESTION: 5

If f (a) = 2, f ' (a) = 1 , g (a) = -1 , g ' (a) = 2 , then the value of

A.
-5
B.
1/5
C.
5
D.
none of these

Solution:

QUESTION: 6

The function is not defined at x = 0. The value which should be assigned to f at x = 0 so that it is continuous at x = 0, is

A.
a – b
B.
a + b
C.
ln a – ln b
D.
none of these

Solution:

For f (x) to be continuous at x = 0

= a + b

QUESTION: 7

A.
0
B.
C.
1/2
D.
none of these

Solution:

QUESTION: 8

Where [x] denotes the greatest integer less than or equal to x. then

A.
1
B.
0
C.
–1
D.
non e of these

Solution:

The given function can be restated as

QUESTION: 9

Let f : R → R be a differentiable function and f (1) = 4. Then the value of

A.
8 f '(1)
B.
4 f '(1)
C.
2 f '(1)
D.
f '(1)

Solution:

We have f : R → R, a differentiable function and f (1) = 4
NOTE THIS STEP

QUESTION: 10

Let [.] den ote th e greatest integer function and f (x) = [tan ²x], then:

A.
B.
f (x) is continuous at x = 0
C.
f (x) is not differentiable at x = 0
D.
f ' (0) = 1

Solution:

We have f (x) = [tan² x]

tan x is an increasing function for

QUESTION: 11

The function denotes the greatest integer function, is discontinuous at

A.
All x
B.
All integer points
C.
No x
D.
x which is not an integer

Solution:

When x is not an integer, both the functions [x] and cos are continuous.
∴f (x) is continuous on all non integral points.

∴ f is continuous at all integral pts as well.
Thus, f is continuous everywhere.

QUESTION: 12

Solution:

QUESTION: 13

The function f(x) = [x]² – [x²] (where [y] is the greatest integer less than or equal to y), is discontinuous at

A.
all integers
B.
all integers except 0 and 1
C.
all integers except 0
D.
all integers except 1

Solution:

We have f (x) = [x]² – [x²]

At x = 0,

f (1) = [1]² – [1²] = 1 – 1 = 0
∴ L.H.L. = R.H.L. = f (1)
∴ f (x) is continuous at x = 1.
Clearly at other integral pts f (x) is not continuous.

QUESTION: 14

The function f (x) = (x² - 1) | x² - 3x + 2 |+cos (|x|) is NOT differentiable at

A.
–1
B.
0
C.
1
D.
2

Solution:

∴ Given function can be written as

This function is differentiable at all points except possibly at x = 1 and x = 2.

Lf (2) ≠ Rf '(2)
∴ f is not differentiable at x = 2.

QUESTION: 15

A.
2
B.
–2
C.
1/2
D.
–1/2

Solution:

QUESTION: 16

A.
e
B.
e^–1
C.
e^–5
D.
e⁵

Solution:

For x ∈ R

QUESTION: 17

A.
-π
B.
π
C.
π/2
D.
1

Solution:

QUESTION: 18

The left-hand derivative of f(x) = [x] sin(p x) at x = k, k an integer, is

A.
(-1)^k(k - 1)π
B.
(-1)^{k - 1}(k -1)π
C.
(-1)^kkπ
D.
(-1)^{k - 1}kπ

Solution:

QUESTION: 19

Let f : R → R be a function defined by f (x) = max {x, x³}. The set of all points where f (x) is NOT differentiable is

A.
{-1, 1}
B.
{-1, 0}
C.
{0, 1}
D.
{-1, 0, 1}

Solution:

KEY CONCEPT
A continuous function f (x) is not differentiable at x = a if graphically it takes a sharp turn at x = a.
Graph of f (x) = max {x, x³} is as shown with solid lines.

From graph of f (x) at x = – 1, 0, 1, we have sharp turns.
∴ f (x) is not differentiable at x = – 1, 0, 1.

QUESTION: 20

Which of the following functions is differentiable at x = 0?

A.
cos(|x|) + |x|
B.
cos (|x|) - |x|
C.
sin (|x|) + |x|
D.
sin (|x|) - |x|

Solution:

Let us test each of four options.

QUESTION: 21

The domain of the derivative of the function

A.
R – {0}
B.
R – {1}
C.
R – {–1}
D.
R – {–1, 1}

Solution:

The given function is

∴ f (x) is discontinuous at x = – 1
Also we can prove in the same way, that f (x) is discontinuous at x = 1
∴ f ' (x) can not be found for x = ± 1 or domain of f ' (x) = R – {– 1, 1}

QUESTION: 22

The integer n for which is a finite non-zero number is