RS² ACADEMY

PUNJABI BAGH

CLASS 12 - MATHEMATICS
CONTINUITY AND DIFFERENTIABILITY
Time Allowed: 3 hours Maximum Marks: 240

Section A
1. State whether each of the following statement is True or False:
a) Rolle's theorem is applicable for the function f(x) = |x - 1| in [0, 2].
π
b) If f(x) = |cos x|, then f is everywhere continuous but not differentiable at x = (2n + 1) ,n∈
2

Z.
c) The function f(x) = sin x cos x is a continuous function.
d) The function f(x) = |x| + |x - 1| is continuous at x = 0 as well as at x = 1.
e) If f.g is continuous at x = a, then f and g are separately continuous at x = a.
f) The function f(x) = |x| + |x - 1| is continuous at x = 0 but not at x = 1.
g) A continuous function can have some points where limit does not exist.
h) Trigonometric and inverse - trigonometric functions are differentiable in their respective
domain.
i) Rolle's theorem ensures that there is at least one point on the curve y = f(x) at which tangent
is parallel to x-axis.
j) |sin x| is a differentiable function for every value of x.
2. Fill in the blanks with suitable words
1
a) The number of points at which the function f (x) =
log |x|
is discontinuous is ________.
b) For the curve √−
x + √y = 1,
dy 1
at ( 4 ,
1
) is ________.
dx 4

c) Derivative of x2 w.r.t. x3 is ________.

d) An example of a function which is continuous everywhere but fails to be differentiable

exactly at two points is ________.
e) The value of c in Mean value theorem for the function f(x) = x(x - 2), x ∈ [1, 2] is ________.
f) The derivative of log10x w.r.t. x is ________.

g) ax + 1, if x ≥ 1
If f (x) = { is continuous, then a should be equal to _______.
x + 2, if x < 1
π
h) If f(x) = |cos x - sin x|, then f = ________.
′
( )
3

i) The set of points where the function f given by f(x) = |2x - 1| sin x is differentiable is
________.
π
j) If f(x) = |cos x|, then f ′ ( ) = ________.
4

3. MCQs
x −x
dy
a) If y =
e −e
, then is equal to
e + e−x
x
dx

a) 1 + y2 b) None of these

c) 1 - y2 d) y2 + 1

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BY NAVEEN SIR
 b) Differential coefficient of a function f(g(x)) w.r.t. the function g (x) is

a) f‘(g (x)) b) None of these

′

c) f (g(x))
d) f ‘(g (x)) g’ (x)
g ′ (x)

c) where
d −1 1
(cos x) = −
dx √1−x2

a) −1 ⩽ x ⩽ 1 b) -1 < x < 1

c) −1 ⩽ x < 1 d) −1 < x ⩽ 1

−−−− −
d) If y
dy
and then is equal to (0 < x < 1)
−1 −1
= sin x z = cos √1 − x2 ,
dz

−−−− −
a) cos
−1
x b) √1 − x2

c) 1 d)
1

√1−x2

e) If y = aemx + be-mx, then y is equal to

2

a) my1 b) -m2y

c) m2y d) None of these

f) is equal to
d −1
(tan (sec x + tan x)
dx

a) −
1
b) 1

2 2

c) d) None of these
1

2 sec x(sec x+tan x)

g) If xp yq = (x +y)p+q, then dy
is equal to
dx

a) x

y
b) None of these
y y
c) x+y
d) x

h) The derivative of f(x) = | x | at x = 0 is

a) 1 b) – 1

c) All of these d) None of these

i) If f (x)= x2g(x) and g (x) is twice differentiable then f’(x) is equal to

a) 2 g’’(x) b) None of these

c) 2
x g
′′ ′
(x) + 2 x g (x) + 2 g(x) d) x g
2 ′′ ′
(x) + 4 x g (x) + 2 g(x)

j) If x sin (a + y) = sin y, then

dy

dx
is equal to
2

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

sin(a+y)
sin a

c) sin a
2
d) sin(a+y)

sin (a+y) sin a

4. SUBJECTIVE QUESTIONS
1
a) ⎧
x sin ,x ≠ 0
Show that the function f defined by f (x) = ⎨ x is continuous at x = 0.
⎩
0, x = 0

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BY NAVEEN SIR
 b) Find the value of the constant k so that the function f defined below is continuous at x = 0,

⎧ 1 − cos 4x
,x ≠ 0
Where f (x) = ⎨ 8x
2

⎩
k, x = 0

c) Differentiate sin2x w.r.t ecosx

d) Discuss the continuity of the function: f (x) = sin x. cos x

e) Find dy
, y = tan
−1
(
sin x
)
dx 1+cos x

f) Differentiate the function with respect to x : sin(x

2
+ 5)

g) Determine the value of 'k' for which the following function is continuous at x = 3 : f(x) =
2
(x+3) −36
,x ≠ 3
{ x−3 .
k ,x = 3

h)
kx
, if x < 0
Determine the value of the constant 'k' so that the function f(x) = { |x|
is
3, if x ≥ 0

continuous at x= 0.
−1

i) Differentiate the following function with respect to x: esin x

j) Find
dy sin(ax+b)

dx
if y =
cos(cx+d)

Section B
5. Show that the function defined by g(x) = x − [x] is discontinuous at all integral points. Here [x]
denotes the greatest integer less than or equal to x.
dy
6. Find dx
if x2 + xy + y
2
= 100

7. If x and y are connected parametrically by the equation x = sin t, y = cos 2t, without eliminating the
dy
parameter, find dx
.

8. Verify Mean Value Theorem if f(x) = x2 - 4x - 3 in the interval [a, b] where a = 1 and b = 4.
dy
9. Find if sin .
2
y + cos xy = π
dx

10. Find the second-order derivative of the function exsin 5x

dy 2x
11. Find if y
−1
, = cos ( ) , −1 < x < 1
dx 1+x2

12. Differentiate the log(cos ex) w.r.t. x.

−−−−−−
13. Differentiate the following function with respect to x : √cot(x2 )
dy
14. Find if xy + y 2 = tan x + y
dx

15. SOLVE THE FOLLOWING QUESTIONS

2

a) If y = 5 cos x - 3 sin x, prove that d y

+ y = 0
2
dx

b) Differentiate the function with respect to x : cos x3 sin

2 5
(x )
2
c)
2

If ey(x + 1) = 1, show that

d y dy
= ( )
dx2 dx

dy
d) Find if 2x + 3y = sin x .
dx

e) Find the value of c in Rolle's theorem for the function f(x) = x3 - 3x in [-√–
3 , 0].

f) If f : [−5, 5] → R is a differentiable function and if f'(x) does not vanish anywhere, then
prove that f (−5) ≠ f (5)

g) Find the second-order derivative of the function sin(log x)

2

h) If y = 500e7x + 600e-7x show that d y

= 49y .
2
dx

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BY NAVEEN SIR
 i) If x and y are connected parametrically by the equation
dy
x = cos θ − cos 2θ, y = sin θ − sin 2θ , without eliminating the parameter, find .
dx
dy
j) Find if ax + by 2 = cos y .
dx

Section C
16. Write the derivative of sin x with respect to cos x.
2
t d y π
17. If x = a(cos t + log tan , y = a sin t, then evaluate
) 2
at t = .
2 dx 3
2
−1 d y dy
18. If y = e , then show that (1 − x .
m sin x 2 2
) − x − m y = 0
dx2 dx

19. Show that the function f(x)=lx+ 1l + lx - 1l , for all x ∈ R, is not differentiable at the points x = - 1 and x
= 1.
dy
20. Find , If y = (cos x)x + (sin x)1/x.
dx
−−−−− − −−−−− dy
21. If y = sin−1 {x√1 − x − √x √1 − x2 } and 0< x < 1, then find .
dx

22. If y = a sin x + b cos x, then prove that y2 +( = a2 + b2.

dy
)
dx

dy y x
23. Find dx
, if (cos x) = (cos y)

24. Show that the function f (x) = |x − 3| , x ∈ R, is continuous but not differentiable at x = 3.
x+1 x

25. Differentiate the following with respect to x: sin .

−1 2 ⋅3
[ x
]
1+(36)
n
−−−−− d
2
y dy
26. If y = (x + √1 + x2 ) , then show that (1 + x2 ) + x = n y
2
.
dx2 dx
2
dy sin (a+y)
27. If x sin(a + y) + sin a cos(a + y) = 0, then prove that = .
dx sin a
sin x
⎧ + cos x, x > 0
⎪
⎪ x

28. Show that the function f(x) defined by f(x) = ⎨ 2, x = 0 is continuous at x = 0.

⎪
⎩
⎪ 4(1− √1−x)
, x < 0
x

29. Find whether the following function is differentiable at x = 1 and x = 2 or not,

⎧ x , x < 1
⎪

f (x) = ⎨ 2− x , 1 ⩽ x ⩽ 2
⎩
⎪ 2
−2 + 3x − x , x > 2

⎧ |x| + 3, x ≤ −3
⎪
30. Find all points of discontinuity of f where f is defined as follows,f(x) = ⎨ −2x, − 3 < x < 3
⎩
⎪
6x + 2, x ≥ 3
2
λ (x − 2x) , if x ≤ 0
31. For what values of λ is the function f(x) = { is continuous at x = 0?
4x + 1, if x > 0

32. Differentiate the following function with respect to x: cosx cos2x cos3x.
√1+x2 √1−x2
, x2 ≤ 1, then find
dy
33. If y = tan−1 ( 2
−
2
)
dx
.
√1+x √1−x

√1+sin x + √1−sin x
34. Differentiate the function cot−1 [ w.r.t. x.
π
], 0 < x <
√1+sin x − √1−sin x 2

2
2x−3√1−x
35. If y = cos-1[
dy
] , then find
√13 dx

Section D
1

⎧ e x − 1
⎪
if x ≠ 0
36. Show that the function f given by f (x) = ⎨
1
is discontinuous at x = 0.
e x + 1
⎩
⎪
0, if x = 0

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BY NAVEEN SIR
 dy
37. Find dx
if y x y
+ x + x
x
= a
b

3/2
2
dy
[1+ ( ) ]
dx

38. If (x − a)2 + (y − b)
2
= c
2
Prove is a constant independent of a & b.
d 2y

d x2

39. If y = 3cos(logx) + 4sin(logx). Show that x2y2 + xy1 + y = 0x2 y2 + xy1 + y = 0

−−−−−−−−−−−−−−−−−−−−−−−−−− −
−−−−−−−− −−
−−−−
−−−
−−−
−− −− −−−
−−
−−
− dy cos x
40. If y = √sin x + √sin x + √sin x+. . . . . +∞ prove that dx
=
2y−1
1
x
41. Differentiate the function (x cos x) + (x sin x) x
w.r.t. x.
2
x + 3x + p, if x ⩽ 1
42. Find the values of p and q so that f (x) = { is differentiable at x = 1.
qx + 2, if x > 1
3
1− sin x π
⎧
⎪ , if x <
⎪
⎪ 3 cos 2 x 2
⎪
π
43. Find the values of p and q for which f (x) is continuous at x = .
π
= ⎨ p, if x =
2
2
⎪
⎪ q(1−sin x)
⎪
⎩
⎪
π
2
, if x >
2
(π−2x)
−−− − −
−1
−−− − − dy −y
44. If x = √a
sin t
, y = √a
cos −1 t
show that =
x
dx

45. Discuss the continuity of the function

⎧ −2 if x ⩽ −1

f (x) = ⎨ 2x, if −1 < x ⩽ 1

⎩
2 if x > 1
−−−−−−−−−−−−−−−−−−−−−−− −
−−−−−−−−−−
−−−−
−−−
−−
−−
− −
− −− dy
46. If y = √cos x + √cos x + √cos x+. . . . . Prove that (1 − 2y) = sin x
dx

−−−− −−−−− dy
47. If x√1 − y + y √1 + x prove that
1
= 0 = −
dx 2
(1+x)
−−−−
−−−−− −−−− − dy
2
1−y
48. If √1 − x2 + √1 − y
2
= a (x − y) , prove that = √
dx 1−x2

−−−−− −−−−− dy
49. If y √x2 2
+ 1 − log(√x + 1 − x) = 0 prove that (x2 + 1) + xy + 1 = 0
dx

dy
50. If x = a (θ - sin θ) and y = a (1 + cos θ), then find at θ = π
.
dx 3

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BY NAVEEN SIR