#### Online Mock Tests

#### Chapters

Chapter 2: Inverse Trigonometric Functions

Chapter 3: Matrices

Chapter 4: Determinants

Chapter 5: Continuity And Differentiability

Chapter 6: Application Of Derivatives

Chapter 7: Integrals

Chapter 8: Application Of Integrals

Chapter 9: Differential Equations

Chapter 10: Vector Algebra

Chapter 11: Three Dimensional Geometry

Chapter 12: Linear Programming

Chapter 13: Probability

## Chapter 5: Continuity And Differentiability

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 5 Continuity And Differentiability Solved Examples [Pages 91 - 107]

#### Short Answer

Find the value of the constant k so that the function f defined below is continuous at x = 0, where f(x) = `{{:((1 - cos4x)/(8x^2)",", x ≠ 0),("k"",", x = 0):}`

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

If f(x) = `{{:((x^3 + x^2 - 16x + 20)/(x - 2)^2",", x ≠ 2),("k"",", x = 2):}` is continuous at x = 2, find the value of k.

Show that the function f defined by f(x) = `{{:(x sin 1/x",", x ≠ 0),(0",", x = 0):}` is continuous at x = 0.

Given f(x) = `1/(x - 1)`. Find the points of discontinuity of the composite function y = f[f(x)]

Let f(x) = x|x|, for all x ∈ R. Discuss the derivability of f(x) at x = 0

Differentiate `sqrt(tansqrt(x))` w.r.t. x

If y = tan(x + y), find `("d"y)/("d"x)`

If e^{x} + e^{y} = e^{x+y} , prove that `("d"y)/("d"x) = -"e"^(y - x)`

Find `("d"y)/("d"x)`, if y = `tan^-1 ((3x - x^3)/(1 - 3x^2)), -1/sqrt(3) < x < 1/sqrt(3)`

If y = `sin^-1 {xsqrt(1 - x) - sqrt(x) sqrt(1 - x^2)}` and 0 < x < 1, then find `("d"y)/(dx)`

If x = a sec^{3}θ and y = a tan^{3}θ, find `("d"y)/("d"x)` at θ = `pi/3`

If x^{y} = e^{x–y}, prove that `("d"y)/("d"x) = logx/(1 + logx)^2`

If y = tanx + secx, prove that `("d"^2y)/("d"x^2) = cosx/(1 - sinx)^2`

If f(x) = |cos x|, find f'`((3pi)/4)`

If f(x) = |cos x – sinx|, find `"f'"(pi/6)`

Verify Rolle’s theorem for the function, f(x) = sin 2x in `[0, pi/2]`.

Verify mean value theorem for the function f(x) = (x – 3)(x – 6)(x – 9) in [3, 5].

#### Long Answer

If f(x) = `(sqrt(2) cos x - 1)/(cot x - 1), x ≠ pi/4` find the value of `"f"(pi/4)` so that f (x) becomes continuous at x = `pi/4`

Show that the function f given by f(x) = `{{:(("e"^(1/x) - 1)/("e"^(1/x) + 1)",", "if" x ≠ 0),(0",", "if" x = 0):}` is discontinuous at x = 0.

Let f(x) = `{{:((1 - cos 4x)/x^2",", "if" x < 0),("a"",", "if" x = 0),(sqrt(x)/(sqrt(16) + sqrt(x) - 4)",", "if" x > 0):}`. For what value of a, f is continuous at x = 0?

Examine the differentiability of the function f defined by

f(x) = `{{:(2x + 3",", "if" -3 ≤ x < - 2),(x + 1",", "if" -2 ≤ x < 0),(x + 2",", "if" 0 ≤ x ≤ 1):}`

Differentiate `tan^-1 (sqrt(1 - x^2)/x)` with respect to`cos^-1(2xsqrt(1 - x^2))`, where `x ∈ (1/sqrt(2), 1)`

#### Objective Type Questions from 24 to 25

The function f(x) = `{{:(sinx/x + cosx",", "if" x ≠ 0),("k"",", "if" x = 0):}` is continuous at x = 0, then the value of k is ______.

3

2

1

1.5

The function f(x) = [x], where [x] denotes the greatest integer function, is continuous at ______.

4

– 2

1

1.5

The number of points at which the function f(x) = `1/(x - [x])` is not continuous is ______.

1

2

3

None of these

The function given by f (x) = tanx is discontinuous on the set ______.

`{"n"pi: "n" ∈ "Z"}`

`{2"n"pi: "n" ∈ "Z"}`

`{(2"n" + 1) pi/2 : "n" ∈ "Z"}`

`{("n"pi)/2 : "n" ∈ "Z"}`

Let f(x)= |cosx|. Then, ______.

f is everywhere differentiable

f is everywhere continuous but not differentiable at n = nπ, n ∈ Z

f is everywhere continuous but not differentiable at x = `(2"n" + 1) pi/2, "n" ∈ "Z"`

None of these

The function f(x) = |x| + |x – 1| is ______.

Continuous at x = 0 as well as at x = 1

Continuous at x = 1 but not at x = 0

Discontinuous at x = 0 as well as at x = 1

Continuous at x = 0 but not at x = 1

The value of k which makes the function defined by f(x) = `{{:(sin 1/x",", "if" x ≠ 0),("k"",", "if" x = 0):}`, continuous at x = 0 is ______.

8

1

–1

None of these

The set of points where the functions f given by f(x) = |x – 3| cosx is differentiable is ______.

R

R – {3}

`(0, oo)`

None of these

Differential coefficient of sec (tan^{–1}x) w.r.t. x is ______.

`x/sqrt(1 + x^2)`

`x/(1 + x^2)`

`xsqrt(1 + x^2)`

`1/sqrt(1 + x^2)`

If u = `sin^-1 ((2x)/(1 + x^2))` and v = `tan^-1 ((2x)/(1 - x^2))`, then `"du"/"dv"` is ______.

`1/2`

x

`(1 - x^2)/(1 + x^2)`

1

The value of c in Rolle’s Theorem for the function f(x) = e x sinx, x ∈ π [0, π] is ______.

`pi/6`

`pi/4`

`pi/2`

`(3pi)/4`

The value of c in Mean value theorem for the function f(x) = x(x – 2), x ∈ [1, 2] is ______.

`3/2`

`2/3`

`1/2`

`3/2`

#### Match the column

COLUMN-I |
COLUMN-II |

(A) If a function f(x) = `{((sin3x)/x, "if" x = 0),("k"/2",", "if" x = 0):}` is continuous at x = 0, then k is equal to |
(a) |x| |

(B) Every continuous function is differentiable |
(b) True |

(C) An example of a function which is continuouseverywhere but not differentiable at exactly one point |
(c) 6 |

(D) The identity function i.e. f (x) = x ∀ ∈x R is a continuous function |
(d) False |

#### Fill in the blanks 37 to 41

The number of points at which the function f(x) = `1/(log|x|)` is discontinuous is ______.

If f(x) = `{{:("a"x + 1, "if" x ≥ 1),(x + 2, "if" x < 1):}` is continuous, then a should be equal to ______.

The derivative of log_{10}x w.r.t. x is ______.

If y = `sec^-1 ((sqrt(x) + 1)/(sqrt(x + 1))) + sin^-1((sqrt(x) - 1)/(sqrt(x) + 1))`, then `"dy"/"dx"` is equal to ______.

The derivative of sin x w.r.t. cos x is ______.

#### State whether the following is True or False: 42 to 46

For continuity, at x = a, each of `lim_(x -> "a"^+) "f"(x)` and `lim_(x -> "a"^-) "f"(x)` is equal to f(a).

True

False

y = |x – 1| is a continuous function.

True

False

A continuous function can have some points where limit does not exist.

True

False

|sinx| is a differentiable function for every value of x.

True

False

cos |x| is differentiable everywhere.

True

False

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 5 Continuity And Differentiability Exercise [Pages 107 - 116]

#### Short Answer

Examine the continuity of the function f(x) = x^{3} + 2x^{2} – 1 at x = 1

#### Find which of the functions in 2 to 10 is continuous or discontinuous at the indicated points:

f(x) = `{{:(3x + 5",", "if" x ≥ 2),(x^2",", "if" x < 2):}` at x = 2

f(x) = `{{:((1 - cos 2x)/x^2",", "if" x ≠ 0),(5",", "if" x = 0):}` at x = 0

f(x) = `{{:((2x^2 - 3x - 2)/(x - 2)",", "if" x ≠ 2),(5",", "if" x = 2):}` at x = 2

f(x) = `{{:(|x - 4|/(2(x - 4))",", "if" x ≠ 4),(0",", "if" x = 4):}` at x = 4

f(x) = `{{:(|x|cos 1/x",", "if" x ≠ 0),(0",", "if" x = 0):}` at x = 0

f(x) = `{{:(|x - "a"| sin 1/(x - "a")",", "if" x ≠ 0),(0",", "if" x = "a"):}` at x = a

f(x) = `{{:(("e"^(1/x))/(1 + "e"^(1/x))",", "if" x ≠ 0),(0",", "if" x = 0):}` at x = 0

f(x) = `{{:(x^2/2",", "if" 0 ≤ x ≤ 1),(2x^2 - 3x + 3/2",", "if" 1 < x ≤ 2):}` at x = 1

f(x) = |x| + |x − 1| at x = 1

#### Find the value of k in the 11 to 14 so that the function f is continuous at the indicated point:

f(x) = `{{:(3x - 8",", "if" x ≤ 5),(2"k"",", "if" x > 5):}` at x = 5

f(x) = `{{:((2^(x + 2) - 16)/(4^x - 16)",", "if" x ≠ 2),("k"",", "if" x = 2):}` at x = 2

f(x) = `{{:((sqrt(1 + "k"x) - sqrt(1 - "k"x))/x",", "if" -1 ≤ x < 0),((2x + 1)/(x - 1)",", "if" 0 ≤ x ≤ 1):}` at x = 0

f(x) = `{{:((1 - cos "k"x)/(xsinx)",", "if" x ≠ 0),(1/2",", "if" x = 0):}` at x = 0

Prove that the function f defined by

f(x) = `{{:(x/(|x| + 2x^2)",", x ≠ 0),("k", x = 0):}`

remains discontinuous at x = 0, regardless the choice of k.

Find the values of a and b such that the function f defined by

f(x) = `{{:((x - 4)/(|x - 4|) + "a"",", "if" x < 4),("a" + "b"",", "if" x = 4),((x - 4)/(|x - 4|) + "b"",", "if" x > 4):}`

is a continuous function at x = 4.

Given the function f(x) = `1/(x + 2)`. Find the points of discontinuity of the composite function y = f(f(x))

Find all points of discontinuity of the function f(t) = `1/("t"^2 + "t" - 2)`, where t = `1/(x - 1)`

Show that the function f(x) = |sin x + cos x| is continuous at x = π.

Examine the differentiability of f, where f is defined by

f(x) = `{{:(x[x]",", "if" 0 ≤ x < 2),((x - 1)x",", "if" 2 ≤ x < 3):}` at x = 2

Examine the differentiability of f, where f is defined by

f(x) = `{{:(x^2 sin 1/x",", "if" x ≠ 0),(0",", "if" x = 0):}` at x = 0

Examine the differentiability of f, where f is defined by

f(x) = `{{:(1 + x",", "if" x ≤ 2),(5 - x",", "if" x > 2):}` at x = 2

Show that f(x) = |x – 5| is continuous but not differentiable at x = 5.

A function f: R → R satisfies the equation f( x + y) = f(x) f(y) for all x, y ∈ R, f(x) ≠ 0. Suppose that the function is differentiable at x = 0 and f′(0) = 2. Prove that f′(x) = 2f(x).

#### Differentiate the following w.r.t. x 25 to 43:

`2^(cos^(2_x)`

`8^x/x^8`

`log (x + sqrt(x^2 + "a"))`

`log [log(logx^5)]`

`sin sqrt(x) + cos^2 sqrt(x)`

sin^{n} (ax^{2} + bx + c)

`cos(tan sqrt(x + 1))`

sinx^{2} + sin^{2}x + sin^{2}(x^{2})

`sin^-1 1/sqrt(x + 1)`

(sin x)^{cosx}

sin^{m}x . cos^{n}x

(x + 1)^{2}(x + 2)^{3}(x + 3)^{4}

`cos^-1 ((sinx + cosx)/sqrt(2)), (-pi)/4 < x < pi/4`

`tan^-1 (sqrt((1 - cosx)/(1 + cosx))), - pi/4 < x < pi/4`

`tan^-1 (secx + tanx), - pi/2 < x < pi/2`

`tan^-1 (("a"cosx - "b"sinx)/("b"cosx - "a"sinx)), - pi/2 < x < pi/2` and `"a"/"b" tan x > -1`

`sec^-1 (1/(4x^3 - 3x)), 0 < x < 1/sqrt(2)`

`tan^-1 ((3"a"^2x - x^3)/("a"^3 - 3"a"x^2)), (-1)/sqrt(3) < x/"a" < 1/sqrt(3)`

`tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/(sqrt(1 + x^2) - sqrt(1 - x^2))), -1 < x < 1, x ≠ 0`

#### Find dy/dx of the functions expressed in parametric form in 44 to 48.

x = `"t" + 1/"t"`, y = `"t" - 1/"t"`

x = `"e"^theta (theta + 1/theta)`, y= `"e"^-theta (theta - 1/theta)`

x = 3cosθ – 2cos^{3}θ, y = 3sinθ – 2sin^{3}θ

sin x = `(2"t")/(1 + "t"^2)`, tan y = `(2"t")/(1 - "t"^2)`

x = `(1 + log "t")/"t"^2`, y = `(3 + 2 log "t")/"t"`

If x = e^{cos2t} and y = e^{sin2t}, prove that `"dy"/"dx" = (-y log x)/(xlogy)`

If x = asin2t (1 + cos2t) and y = b cos2t (1–cos2t), show that `("dy"/"dx")_("at t" = pi/4) = "b"/"a"`

If x = 3sint – sin 3t, y = 3cost – cos 3t, find `"dy"/"dx"` at t = `pi/3`

Differentiate `x/sinx` w.r.t. sin x

Differentiate `tan^-1 ((sqrt(1 + x^2) - 1)/x)` w.r.t. tan^{–1}x, when x ≠ 0

#### Find dy/dx when x and y are connected by the relation given in 54 to 57

`sin xy + x/y` = x^{2} – y

sec(x + y) = xy

tan^{–1}(x^{2} + y^{2}) = a

(x^{2} + y^{2})^{2} = xy

If ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0, then show that `"dy"/"dx" * "dx"/"dy"` = 1

If x = `"e"^(x/y)`, prove that `"dy"/"dx" = (x - y)/(xlogx)`

If y^{x} = e^{y – x}, prove that `"dy"/"dx" = (1 + log y)^2/logy`

If y = `(cos x)^((cos x)^((cosx)....oo)`, show that `"dy"/"dx" = (y^2 tanx)/(y log cos x - 1)`

If x sin (a + y) + sin a cos (a + y) = 0, prove that `"dy"/"dx" = (sin^2("a" + y))/sin"a"`

If `sqrt(1 - x^2) + sqrt(1 - y^2) = "a"(x - y)`, proe that `"dy"/"dx" = sqrt((1 - y^2)/(1 - x^2)`

If y = tan^{–1}x, find `("d"^2y)/("dx"^2)` in terms of y alone.

#### Verify the Rolle’s theorem for the functions in 65 to 69.

f(x) = x(x – 1)^{2} in [0, 1]

f(x) = `sin^4x + cos^4x` in `[0, pi/2]`

f(x) = log(x^{2} + 2) – log3 in [–1, 1]

f(x) = `x(x + 3)e^((–x)/2)` in [–3, 0]

f(x) = `sqrt(4 - x^2)` in [– 2, 2]

Discuss the applicability of Rolle’s theorem on the function given by f(x) = `{{:(x^2 + 1",", "if" 0 ≤ x ≤ 1),(3 - x",", "if" 1 ≤ x ≤ 2):}`

Find the points on the curve y = (cosx – 1) in [0, 2π], where the tangent is parallel to x-axis

Using Rolle’s theorem, find the point on the curve y = x(x – 4), x ∈ [0, 4], where the tangent is parallel to x-axis

#### Verify mean value theorem for the functions given 73 to 76

f(x) = `1/(4x - 1)` in [1, 4]

f(x) = x^{3} – 2x^{2} – x + 3 in [0, 1]

f(x) = sinx – sin2x in [0, π]

f(x) = `sqrt(25 - x^2)` in [1, 5]

Find a point on the curve y = (x – 3)^{2}, where the tangent is parallel to the chord joining the points (3, 0) and (4, 1)

Using mean value theorem, prove that there is a point on the curve y = 2x^{2} – 5x + 3 between the points A(1, 0) and B(2, 1), where tangent is parallel to the chord AB. Also, find that point

#### Long Answer

Find the values of p and q so that f(x) = `{{:(x^2 + 3x + "p"",", "if" x ≤ 1),("q"x + 2",", "if" x > 1):}` is differentiable at x = 1

If x^{m} . y^{n} = (x + y)^{m+n}, prove that `"dy"/"dx" = y/x`

If x^{m} . y^{n} = (x + y)^{m+n}, prove that `("d"^2"y")/("dx"^2)` = 0

If x = sint and y = sin pt, prove that `(1 - x^2) ("d"^2"y")/("dx"^2) - x "dy"/"dx" + "p"^2y` = 0

Find `"dy"/"dx"`, if y = `x^tanx + sqrt((x^2 + 1)/2)`

#### Objective Type Questions from 83 to 96

If f(x) = 2x and g(x) = `x^2/2 + 1`, then which of the following can be a discontinuous function ______.

f(x) + g(x)

f(x) – g(x)

f(x) . g(x)

`("g"(x))/("f"(x))`

The function f(x) = `(4 - x^2)/(4x - x^3)` is ______.

Discontinuous at only one point

Discontinuous at exactly two points

Discontinuous at exactly three points

None of these

The set of points where the function f given by f(x) = |2x − 1| sinx is differentiable is ______.

R

`"R" - {1/2}`

`(0, oo)`

None of these

The function f(x) = cot x is discontinuous on the set ______.

{x = nπ : n ∈ Z}

{x = 2nπ : n ∈ Z}

`{x = (2"n" + 1)pi/2 ; "n" ∈ "Z"}`

`{x = ("n"pi)/2 ; "n" ∈ "Z"}`

The function f(x) = `"e"^|x|` is ______.

Continuous everywhere but not differentiable at x = 0

Continuous and differentiable everywhere

Not continuous at x = 0

None of these

If f(x) = `x^2 sin 1/x` where x ≠ 0, then the value of the function f at x = 0, so that the function is continuous at x = 0, is ______.

0

– 1

1

None of these

If f(x) = `{{:("m"x + 1",", "if" x ≤ pi/2),(sin x + "n"",", "If" x > pi/2):}`, is continuous at x = `pi/2`, then ______.

m = 1, n = 0

m = `("n"pi)/2 + 1`

n = `("m"pi)/2`

m = n = `pi/2`

Let f(x) = |sin x|. Then ______.

f is everywhere differentiable

f is everywhere continuous but not differentiable at x = nπ, n ∈ Z

f is everywhere continuous but not differentiable at x = `(2"n" + 1) pi/2`, n ∈ Z

None of these

If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.

`(4x^3)/(1 - x^4)`

`(-4x)/(1 - x^4)`

`1/(4 - x^4)`

`(-4x^3)/(1 - x^4)`

If y = `sqrt(sinx + y)`, then `"dy"/"dx"` is equal to ______.

`cos/(2y - 1)`

`cosx/(1 - 2y)`

`sinx/(1 - 2y)`

`sinx/(2y - 1)`

The derivative of cos^{–1}(2x^{2} – 1) w.r.t. cos^{–1}x is ______.

2

`(-1)/(2sqrt(1 - x^2)`

`2/x`

1 – x

^{2}

If x = t^{2}, y = t^{3}, then `("d"^2"y")/("dx"^2)` is ______.

`3/2`

`3/(4"t")`

`3/(2"t")`

`3/4`

The value of c in Rolle’s theorem for the function f(x) = x^{3} – 3x in the interval `[0, sqrt(3)]` is ______.

1

– 1

`3/2`

`1/3`

For the function f(x) = `x + 1/x`, x ∈ [1, 3], the value of c for mean value theorem is ______.

1

`sqrt(3)`

2

None of these

#### Fill in the blanks 97 to 101:

An example of a function which is continuous everywhere but fails to be differentiable exactly at two points is ______.

Derivative of x^{2} w.r.t. x^{3} is ______.

If f(x) = |cosx|, then `"f'"(pi/4)` = ______.

If f(x) = |cosx – sinx| , then `"f'"(pi/4)` = ______.

For the curve `sqrt(x) + sqrt(y)` = 1, `"dy"/"dx"` at `(1/4, 1/4)` is ______.

#### State whether the following is True or False: 102 to 106

Rolle’s theorem is applicable for the function f(x) = |x – 1| in [0, 2].

True

False

If f is continuous on its domain D, then |f| is also continuous on D.

True

False

The composition of two continuous function is a continuous function.

True

False

Trigonometric and inverse-trigonometric functions are differentiable in their respective domain.

True

False

If f.g is continuous at x = a, then f and g are separately continuous at x = a.

True

False

## Chapter 5: Continuity And Differentiability

## NCERT solutions for Mathematics Exemplar Class 12 chapter 5 - Continuity And Differentiability

NCERT solutions for Mathematics Exemplar Class 12 chapter 5 (Continuity And Differentiability) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 12 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics Exemplar Class 12 chapter 5 Continuity And Differentiability are Higher Order Derivative, Algebra of Continuous Functions, Derivative - Exponential and Log, Concept of Differentiability, Proof Derivative X^n Sin Cos Tan, Infinite Series, Continuous Function of Point, Mean Value Theorem, Second Order Derivative, Derivatives of Functions in Parametric Forms, Logarithmic Differentiation, Exponential and Logarithmic Functions, Derivatives of Implicit Functions, Derivatives of Inverse Trigonometric Functions, Derivatives of Composite Functions - Chain Rule, Concept of Continuity.

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