#### 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 Class 12 Mathematics Textbook Chapter 5 Continuity and Differentiability [Pages 159 - 161]

Prove that the function f (x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.

Examine the following functions for continuity.

`f(x) = (x^2 - 25)/(x + 5), x != -5`

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

Examine the following functions for continuity.

f (x) = x – 5

Examine the following functions for continuity

`1/(x - 5), x != 5`

Examine the following functions for continuity

f(x) = | x – 5|

Prove that the function `f(x) = x^n` is continuous at *x* = *n*, where *n* is a positive integer

Is the function *f* defined by f(x)= `{(x, if x<=1),(5, if x > 1):}`

continuous at *x* = 0? At *x* = 1? At *x* = 2?

Find all points of discontinuity of *f*, where *f* is defined by

`f(x) = {(2x +3, if zx <=2),(2x - 3, if x > 2):}`

Find all points of discontinuity of *f*, where *f* is defined by `f(x) = {(|x|+3, if x<= -3),(-2x, if -3 < x < 3),(6x + 2, if x >= 3):}`

Find all points of discontinuity of *f*, where *f* is defined by `f(x) = {(|x|/x , if x != 0),(0, if x = 0):}`

Find all points of discontinuity of *f*, where *f* is defined by

`f(x) = {(x/|x|, ","if x < 0),(-1, ","if x >= 0):}`

Find all points of discontinuity of *f*, where *f* is defined by

`f(x) = {(x+1, "," if x >= 1),(x^2 + 1, ","if x < 1):}`

Find all points of discontinuity of f, where f is defined by `f(x) = {(x^3 - 3, if x <= 2),(x^2 + 1, if x > 2):}`

Find all points of discontinuity of *f*, where *f* is defined by `f(x) = {(x^10 - 1, ","if x <= 1),(x^2, ","if x > 1):}`

Is the function defined by

`f(x) = {(x+5, if x <= 1),(x -5, if x > 1):}` a continuous function?

Discuss the continuity of the function *f*, where *f* is defined by

`f(x) = {(3, ","if 0 <= x <= 1),(4, ","if 1 < x < 3),(5, ","if 3 <= x <= 10):}`

Discuss the continuity of the function *f*, where *f* is defined by

`f(x) = {(2x , ","if x < 0),(0, "," if 0 <= x <= 1),(4x, "," if x > 1):}`

Discuss the continuity of the function *f*, where *f* is defined by

`f(x) = {(-2,"," if x <= -1),(2x, "," if -1 < x <= 1),(2, "," if x > 1):}`

Find the relationship between *a* and *b* so that the function *f* defined by `f(x)= {(az + 1, if x<= 3),(bx + 3, if x > 3):}` is continuous at *x *= 3.

For what value of `lambda` is the function defined by

`f(x) = {(lambda(x^2 - 2x), "," if x <= 0),(4x+ 1, "," if x > 0):}` continuous at *x* = 0? What about continuity at *x* = 1?

Show that the function defined by g(x) = x = [x] is discontinuous at all integral point. Here [x] denotes the greatest integer less than or equal to *x*.

Is the function defined by `f(x) = x^2 - sin x + 5` continuous at *x *= π?

Discuss the continuity of the following functions.

(a) *f* (*x*) = sin *x* + cos *x*

(b) *f* (*x*) = sin *x* − cos *x*

(c) *f* (*x*) = sin *x* × cos x

Discuss the continuity of the cosine, cosecant, secant and cotangent functions,

Find the points of discontinuity of *f*, where

`f(x) = {((sinx)/x, "," if x < 0),(x + 1, "," if x >= 0):}`

Determine if *f* defined by

`f(x) = {(x^2 sin 1/x, "," if x != 0),(0, "," if x = 0):}` is a continuous function?

Examine the continuity of *f*, where *f* is defined by

`f(x) = {(sin x - cos x, if x != 0),(-1, "," if x = 0):}`

Find the values of *k *so that the function *f* is continuous at the indicated point.

`f(x) = {((kcosx)/(pi-2x), "," if x != pi/2),(3, "," if x = pi/2):} " at x =" pi/2`

Find the values of *k *so that the function *f* is continuous at the indicated point.

`f(x) = {(kx^2, "," if x<= 2),(3, "," if x > 2):} " at x" = 2`

Find the values of *k *so that the function *f* is continuous at the indicated point.

`f(x) = {(kx +1, if x<= pi),(cos x, if x > pi):} " at x " = pi`

Find the values of *k *so that the function *f* is continuous at the indicated point.

`f(x) = {(kx + 1, "," if x <= 5),(3x - 5, "," if x > 5):} " at x " = 5`

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

`f(x) = {(5, "," if x <= 2),(ax +b, "," if 2 < x < 10),(21, "," if x >= 10):}`

is a continuous function.

Show that the function defined by* f *(*x*) = cos (*x*^{2}) is a continuous function.

Show that the function defined by f(x) = |cos x| is a continuous function.

Examine sin |x| is a continuous function.

Find all the points of discontinuity of *f *defined by `f(x) = |x| - |x + 1|`.

#### NCERT solutions for Class 12 Mathematics Textbook Chapter 5 Continuity and Differentiability [Page 166]

Differentiate the functions with respect to *x*.

sin (x^{2} + 5)

Differentiate the functions with respect to *x*.

cos (sin x)

Differentiate the functions with respect to *x*.

sin (ax + b)

Differentiate the functions with respect to *x*.

`sec(tan (sqrtx))`

Differentiate the functions with respect to *x*.

`(sin (ax + b))/cos (cx + d)`

Differentiate the functions with respect to *x*.

`cos x^3. sin^2 (x^3)`

Differentiate the functions with respect to *x*.

`2sqrt(cot(x^2))`

Differentiate the functions with respect to *x*.

`cos (sqrtx)`

Prove that the function *f *given by `f(x) = |x - 1|, x in R` is notdifferentiable at *x* = 1.

#### NCERT solutions for Class 12 Mathematics Textbook Chapter 5 Continuity and Differentiability [Page 169]

Find `dy/dx`

2x + 3y = sin x

Find `dy/dx`

2x + 3y = sin y

Find `dy/dx`

ax + by^{2} = cos y

Find `dy/dx`

xy + y^{2} = tan x + y

Find `dx/dy`

x^{2} + xy + y2 = 100

Find `dy/dx`

x^{3} + x2y + xy^{2} + y^{3} = 81

Find `dy/dx`

sin^{2} y + cos xy = Π

Find `dy/dx`

sin^{2} x + cos^{2} y = 1

Find `dy/dx`

`y = sin^(-1)((2x)/(1+x^2))`

Find `dy/dx`

`y = tan^(-1) ((3x -x^3)/(1 - 3x^2)), - 1/sqrt3 < x < 1/sqrt3`

Find `dy/dx`

`y = cos^(-1) ((1-x^2)/(1+x^2)), 0 < x < 1`

Find `dy/dx`

`y = sin^(-1) ((1-x^2)/(1+x^2)), 0 < x < 1`

Find `dx/dy`

`y = cos^(-1) ((2x)/(1+x^2)), -1 < x < 1`

Find `dy/dx`

`y = sin^(-1)(2xsqrt(1-x^2)), -1/sqrt2 < x < 1/sqrt2`

Find `dy/dx`

`y = sec^(-1) (1/(2x^2 - 1)), 0 < x < 1/sqrt2`

#### NCERT solutions for Class 12 Mathematics Textbook Chapter 5 Continuity and Differentiability [Pages 147 - 174]

Differentiate the following w.r.t. *x*:

`e^x/sinx`

Differentiate the following w.r.t. x: `e^(sin^(-1) x)`

Differentiate the following w.r.t. *x*: `e^(x^3)`

Differentiate the following w.r.t. *x*:

sin (tan–1 e^{–x})

Differentiate the following w.r.t. *x*:

`log(cos e^x)`

Differentiate the following w.r.t. *x*:

`e^x + e^(x^2) + ....+ e^(x^3)`

Differentiate the following w.r.t. *x*:

`sqrt(e^(sqrtx)), x > 0`

Differentiate the following w.r.t. *x*: log (log x), x > 1

Differentiate the following w.r.t. *x*:

`cos x/log x, x >0`

Differentiate the following w.r.t. *x*:

cos (log x + e^{x}), x > 0

#### NCERT solutions for Class 12 Mathematics Textbook Chapter 5 Continuity and Differentiability [Pages 178 - 179]

Differentiate the function with respect to *x*.

cos x . cos 2x . cos 3x

Differentiate the function with respect to *x*.

`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`

Differentiate the function with respect to *x*.

`(log x)^(cos x)`

Differentiate the function with respect to *x*.

`x^x - 2^(sin x)`

Differentiate the function with respect to *x*.

(x + 3)^{2} . (x + 4)^{3} . (x + 5)^{4}

Differentiate the function with respect to *x*.

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

Differentiate the function with respect to *x*.

(log x)^{x} + xl^{og x}

Differentiate the function with respect to *x*.

`(sin x)^x + sin^(-1) sqrtx`

Differentiate the function with respect to *x*.

x^{sin x} + (sin x)^{cos x}

Differentiate the function with respect to *x*.

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

Differentiate the function with respect to *x*.

`(x cos x)^x + (x sin x)^(1/x)`

Find `dy/dx` of function

x^{y} + y^{x} = 1

Find `dy/dx` of Function y^{x} = x^{y}

Find `dy/dx` of Function

(cos x)^{y} = (cos y)^{x}

Find `dy/dx` of function

xy = e^{(x – y)}

Find the derivative of the function given by f (x) = (1 + x) (1 + x^{2}) (1 + x^{4}) (1 + x^{8}) and hence find f ′(1).

Differentiate (x^{2} – 5x + 8) (x^{3} + 7x + 9) in three ways mentioned

(i) by using product rule

(ii) by expanding the product to obtain a single polynomial.

(iii) by logarithmic differentiation.

Do they all give the same answer?d below:

If *u*, *v* and *w* are functions of *x*, then show that

`d/dx(u.v.w) = (du)/dx v.w+u. (dv)/dx.w + u.v. (dw)/dx`

in two ways-first by repeated application of product rule, second by logarithmic differentiation.

#### NCERT solutions for Class 12 Mathematics Textbook Chapter 5 Continuity and Differentiability [Page 181]

If *x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

`x = 2at^2, y = at^4`

If *x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

*x* = sin *t*, *y* = cos 2*t*

If *x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

x = 4t, y = 4/y

*x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

x = cos θ – cos 2θ, y = sin θ – sin 2θ

*x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

*x* = *a* cos *θ*, *y* = *b* cos *θ*

*x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

x = a (θ – sin θ), y = a (1 + cos θ)

*x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

`x = (sin^3t)/sqrt(cos 2t), y = (cos^3t)/sqrt(cos 2t)`

*x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

`x = a(cos t + log tan t/2), y = a sin t`

*x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

x = a sec θ, y = b tan θ

*x* and *y* are connected parametrically by the equation, without eliminating the parameter, find `dy/dx`

x = a (cos θ + θ sin θ), y = a (sin θ – θ cos θ)

if `x = sqrt(a^(sin^(-1)))`, y = `sqrt(a^(cos^(-1)))` show that `dy/dx = - y/x`

#### NCERT solutions for Class 12 Mathematics Textbook Chapter 5 Continuity and Differentiability [Pages 183 - 184]

Find the second order derivatives of the function.

x^{2} + 3x + 2

Find the second order derivatives of the function.

x . cos x

Find the second order derivatives of the function.

log x

Find the second order derivatives of the function.

x^{3} log x

Find the second order derivatives of the function.

e^{x} sin 5x

Find the second order derivatives of the function.

e^{6x} cos 3x

Find the second order derivatives of the function.

tan^{–1} x

Find the second order derivatives of the function.

log (log x)

Find the second order derivatives of the function.

sin (log x)

If y = 5 cos x – 3 sin x, prove that `(d^2y)/(dx^2) + y = 0`

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

If y = 3 cos (log x) + 4 sin (log x), show that x^{2} y_{2} + xy_{1} + y = 0

If y = Ae^{mx} + Be^{nx}, show that `(d^2y)/dx^2 - (m+ n) (dy)/dx + mny = 0`

If y = 500e^{7x} + 600e^{–7x}, show that `(d^2y)/(dx^2) = 49y`

If e^{y} (x + 1) = 1, show that `(d^2y)/(dx^2) =((dy)/(dx))^2`

If y = (tan^{–1} x)^{2}, show that (x^{2} + 1)^{2} y_{2} + 2x (x^{2} + 1) y_{1} = 2

Find the second order derivatives of the function. `x^20`

#### NCERT solutions for Class 12 Mathematics Textbook Chapter 5 Continuity and Differentiability [Page 186]

Verify Rolle’s theorem for the function f (x) = x^{2} + 2x – 8, x ∈ [– 4, 2].

Examine if Rolle’s Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s Theorem from these examples?

f (x) = [x] for x ∈ [5, 9]

Examine if Rolle’s Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s Theorem from these examples?

f (x) = [x] for x ∈ [– 2, 2]

Examine if Rolle’s Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s Theorem from these examples?

f (x) = x^{2} – 1 for x ∈ [1, 2]

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

Verify Mean Value Theorem, if f (x) = x^{2} – 4x – 3 in the interval [a, b], where a = 1 and b = 4.

Verify Mean Value Theorem, if f (x) = x^{3} – 5x^{2} – 3x in the interval [a, b], where a = 1 and b = 3. Find all c ∈ (1, 3) for which f ′(c) = 0.

Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.

#### NCERT solutions for Class 12 Mathematics Textbook Chapter 5 Continuity and Differentiability [Pages 191 - 192]

Differentiate w.r.t. x the function (3x2 – 9x + 5)9

Differentiate w.r.t. x the function sin^{3} x + cos^{6} x

Differentiate w.r.t. x the function (5x)^{3cos 2x}

Differentiate w.r.t. x the function `sin^(–1)(xsqrtx ), 0 ≤ x ≤ 1`

Differentiate w.r.t. x the function `(cos^(-1) x/2)/sqrt(2x+7), -2 < x < 2`

Differentiate w.r.t. x the function `cot^(-1) [(sqrt(1+sinx) + sqrt(1-sinx))/(sqrt(1+sinx) - sqrt(1-sinx))]`, ` 0 < x < pi/2`

Differentiate w.r.t. x the function (log x)^{log x}, x > 1

Differentiate w.r.t. x the function cos (a cos x + b sin x), for some constant a and b.

Differentiate w.r.t. x the function (sin x – cos x) ^{(sin x – cos x)}, `pi/4 < x < (3pi)/4`

Differentiate w.r.t. x the function x^{x} + x^{a} + a^{x} + a^{a}, for some fixed a > 0 and x > 0

Differentiate w.r.t. x the function `x^(x^2 -3) + (x -3)^(x^2)`, for x > 3

Find `dy/dx` ,if y = 12 (1 – cos t), x = 10 (t – sin t), `-pi/2< t< pi/2`

Find `dy/dx` , if y = sin^{–1} x + sin^{–1} `sqrt(1-x^2)`, 0 < x < 1

if `xsqrt(1+y) + ysqrt(1+x) = 0`, for, −1 < *x* <1, prove that `dy/dx = 1/(1+ x)^2`

If (x – a)^{2} + (y – b)^{2} = c^{2}, for some c > 0, prove that

`[1+ (dy/dx)^2]^(3/2)/((d^2y)/dx^2)` is a constant independent of *a* and *b*.

If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`

If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`

If f (x) = |x|^{3}, show that f ″(x) exists for all real x and find it.

Using mathematical induction prove that `d/(dx) (x^n) = nx^(n -1)` for all positive integers *n*.

Using the fact that sin (*A* + *B*) = sin *A* cos *B* + cos *A* sin *B* and the differentiation, obtain the sum formula for cosines

Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer ?

if y = `[(f(x), g(x), h(x)),(l, m,n),(a,b,c)]`, prove that `dy/dx` =`|(f'(x), g'(x), h'(x)),(l,m, n),(a,b,c)|`

if `y = e^(acos^(-1)x)`, -1 <= x <= 1 show that `(1- x^2) (d^2y)/(dx^2) -x dy/dx - a^2y = 0`

## Chapter 5: Continuity and Differentiability

## NCERT solutions for Class 12 Mathematics Textbook chapter 5 - Continuity and Differentiability

NCERT solutions for Class 12 Mathematics Textbook 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 Class 12 Mathematics Textbook solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 12 Mathematics Textbook 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.

Using NCERT Class 12 solutions Continuity and Differentiability exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer NCERT Textbook Solutions to score more in exam.

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