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ASSIGNMENT OF CONTINUITY AND DIFFERENTIABILITY

Class 12 - Mathematics

2 2

1. If sin −1
(
x −y
) = log a then
dy
is equal to [1]
2 2
x +y dx

2 2 y
a) x −y
b) x
2 2
x +y

2 2

c) x

y
d) x +y

2 2
x −y

2. f: [-2n, 2a] → R is an odd function such that the left hand derivative at x = a is zero and f(x) = f(2a - x) ∀x ∈ (a, [1]
2a). Then its left hand derivative at x = -a is

a) does not exist b) 1

c) 0 d) a

d [1]
2

If y = xx find
d y
3.
dx
2
fie
a) 0 b) xx{(1 - log x)2 + 1
}
x

c) xx{(1 + log x)2 + 1

} d) xx{(1 + log x)2 - 1
}
pli
x x

4. If y = f(
3x+4
) and f'(x) = tan x2 then
dy
is equal to [1]
5x+6 dx

a) -2 tan ( b) tan x2
3x+4 1
Sim

) ×
2
5x+3
(5x+3)

c) f( 3 tan x +4

2
) d) -2 tan ( 3x+4

5x+6
) ×
1

2
5 tan x +6 (5x+6)

1/x
e −1
, x ≠ 0 [1]
5. The function f(x) = { e
1/x
+1

0, x = 0

a) is not continuous at x = 0, but can be made b) is continuous at x = 0

continuous at x = 0

c) is not continuous at x = 0 d) is continuous at x = 1

dy
6. If x sin (a + y) = sin y, then is equal to [1]
dx

a) sin a

sin(a+y)
b) sin (a+y)

sin a

c) d)
sin a sin(a+y)

2
sin (a+y) sin a

7. If √−
x + √y =
−
−
√a, then (
d y

2
) is equal to [1]
dx
x=a

a) 1

2a
b) a

c) 1

4a
d) 1

∣tan(
∣
π
+ x)∣
∣,
x
x ≠ 0 [1]
8. Let f(x) = { 4
then the value of k such that f(x) holds continuity at x = 0 is
k, x = 0

a) e2 b) 1

e
2

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 c) e d) e-2
ax+b
9. If y = 2
, then (2xy1 + y) y3 = [1]
x +c

a) 3 (xy1 + y2) y2 b) 3 (xy2 + y1) y2

c) 4 (xy1 + y2) y2 d) 3 (xy2 + y1) y1

10. If y = log xx, then the value of

dy
is [1]
dx

a) xx(1 + log x) b)
e
log
x

c) log( d) log (ex)

x
)
e

11. If ex + ey = ex+y, prove that

dy
= −e
y−x
. [1]
dx

12. Differentiate the function with respect to x: tan −1

(
a+x
). [1]
1−ax

2
x −3x+2
, if x ≠ 1 [1]
13. Determine the value of the constant k so that the function f(x) = {
x−1
is continuous at x =
k , if x = 1

1
2

14. If y = |x - x2|, then find

d y
[1]
2
dx

15. Show that f(x) = [x] is not continuous at x = n, where n is an integer. [1]

16. If y = sin-1x + cos-1x, find

dy
.
d [1]
3
dx
fie
17. Differentiate e x
w.r.t. x. [1]
18. If f(x) = |log x|, x > 0, find f'(1/e) and f'(e). [1]
2

If y = ea cos-1 x, – 1 ≤ x ≤ 1, show that (1 - x2) - a2y = 0. [2]

pli
d y dy
19. 2
− x
dx dx
2
d y [2]
20. If x = a (θ - sin θ ), y = a (1 - cos θ ), a > 0, then find 2
at θ = π

3
.
dx
[2]
Sim

−−−−− dy
21. If y = sin
−1
x + sin
−1
√1 − x2 , find dx
at x ∈ (0, 1).
−
22. Differentiate the function sin −1
(x√x ) , 0 ≤ x ≤ 1 w.r.t. to x. [2]
−1

23. Differentiate e tan √x

w.r.t. x. [2]
24. Differentiate xe √sin x
w.r.t. x. [2]
25. Differentiate w.r.t. x: tan-1 ( cos x
) . [2]
1+sin x

26. Examine the continuity of the function f(x) = 2x2 - 1 at x = 3. [2]

27. If xy = yx, find

dy
. [2]
dx

28. Find which of the functions is continuous or discontinuous at the indicated points: [2]
2
2x −3x−2
, if x ≠ 2
f(x) = {
x−2
at x = 2
5, if x = 2

⎧
⎪
1, if x ≤ 3 [3]
29. Find the values of a and b so that the function f given by f(x) = ⎨ ax + b, if 3 < x < 5 is continuous at x = 3
⎩
⎪
7, if x ≥ 5

and x = 5
⎧
√1+kx −√1−kx
, if − 1 ≤ x < 0
[3]
30. Find the value of k, for which f(x) = ⎨ x
is continuous at x = 0.
⎩
if 0 ≤
2x+1
, x < 1
x−1

⎧ 1 − cos kx
⎪
⎪ , if x ≠ 0
[3]
31. Find the value of k so that the function f is continuous at the indicated point: f(x) = ⎨
x sin x
1
⎪
⎩
⎪ , if x = 0
2

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 at x = 0
If y = xcot x + [3]
2
dy
32. 2x −3

2
x +x+2
, find dx
.
2n

33. Differentiate the function with respect to x: cos −1

(
1−x

2n
), 0 < x < ∞ . [3]
1+x

34. If x = sec θ - cos θ and y = secnθ - cosnθ , prove that (x2 + 4)(
dy
) = n2(y2 + 4) [3]
dx

dy y log x
35. If x = e
cos 2t
and y = e
sin 2t
, then prove that dx
= −
x log y
[3]

36. If xy . yx = 1, Prove that

dy
= −
y(y+x log y)
. [3]
dx x(y log x+x)

37. Differentiate the function with respect to x: tan −1

{
x
}, −1 < x < 1 . [3]
2
1+√1−x

38. Find
dy
when y = xx + x1/x [3]
dx

39. Differentiate tan −1

(
x
) with respect to sin −1
−−−−−
2
(2x√1 − x ) , if − 1
< x <
1
. [5]
√1−x2 √2 √2

40. Find all the points of discontinuity of f defined by f(x) = |x| – |x + 1|. [5]
−−−−−−−−−−−−−−−−−−−−−−− −
−−−−−−−− −−− −−− − − −
−−− −−−− − −
[5]
dy
41. If y = √cos x + √cos x + √cos x+. . . . . Prove that (1 − 2y) dx
= sin x

dy
42. Find dx
if x = a (cos θ + θ sin θ) and y = a (sin θ − θ cos θ) [5]
43. Read the following text carefully and answer the questions that follow: [4]
The function f(x) will be discontinuous at x = a if f(x) has
Discontinuity of first kind : lim f(a − h) and lim f(a + h) both exist but are not equal. If is also known as

irremovable discontinuity.
h→0 h→0

d
fie
Discontinuity of second kind : If none of the limits lim f(a − h) and lim f(a + h) exist.
h→0 h→0

Removable discontinuity: lim f(a − h) and lim f(a + h) both exist and equal but not equal to f(a).
pli
h→0 h→0

2
x −9
, for x ≠ 3
i. What can you tell me about the behavior of the function f(x) = { x−3
, at x = 3 irremovable
4, for x = 3
Sim

or removable discontinuity? (1)

x + 2, if x ≤ 4
ii. What can you tell me about the behavior of the function f(x) = { at x = 4 irremovable or
x + 4, if x > 4

removable discontinuity? (1)

2

for x ≠
x −4
, 2
iii. What can you tell me about the behavior of the function f(x) = { x−2
, at x = 2 irremovable or
5, for x = 2

removable discontinuity? (2)

OR
x−|x|
, x ≠ 0
What can you tell me about the behavior of the function f(x) = { x
, at x = 0 irremovable or
2, x = 0

removable discontinuity? (2)

44. Read the following text carefully and answer the questions that follow: [4]
Mansi started to read the notes on the topic 'differentiability' which she has prepared in the class of mathematics.
She wanted to solve the questions based on this topic, which teacher gave as home work. She has written
following matter in her notes:
Let f(x) be a real valued function, then its Left Hand Derivative (LHD) is:
f(a−h)−f(a)
′
Lf (a) = lim h→0
−h

Right Hand Derivative (RHD) is:

f(a+h)−f(a)
′
Rf (a) = lim
h
h→0

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 Also, a function f(x) is said to be differentiable at x = a if its LHD and RHD at x = a exist and are equal.
|x − 3|, x ≥ 1
For the function, f(x) = {
x
2
3x 13
− + , x < 1
4 2 4

i. Find the value f'(-1). (1)

ii. Find the value of f'(2). (1)
iii. Check the differentiability of function at x = 1. (2)
OR
Check the differentiability of the given function at x = 1. (2)
45. Read the following text carefully and answer the questions that follow: [4]
If a relation between x and y is such that y cannot be expressed in terms of x, then y is called implicit function of
x.

Assume a function, y = 6x2 - 11ey

This function can be rewritten as

y + 11ey = 6x2
But it is not possible to completely separate and represent it as a function of y. This type of function is known as
an implicit function.
To differentiate an implicit function, we consider y as a function of x and then we use the chain rule to
differentiate any term consisting of y.
d
fie
Now to differentiate the above function, we differentiate directly w.r.t. x the entire function. This step basically
indicates the use of chain rule.
y 2
d(6x )
pli
dy d(11e )
i.e., + =
dx dx dx
dy dy
⇒ + 11e
y
= 12x
dx dx
dy
⇒ (1 + 11e )
y
= 12x
Sim

dx
dy 12x
⇒ =
y
dx (1+11e )

i. If x3 + x2y + xy2 + y3 = 81, then find

dy

dx
. (1)
ii. Find the slope of the tangent to the curve y = x2 + 6y2 + xy. (1)

if sin2y + cos xy = K. (2)

dy
iii. Find at x = 1, y = π

dx 4

OR
If y = (√−
dy
x ) , then find . (2)
y

dx

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