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

Mathematics Textbook for Class 12

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NCERT Mathematics Class 12

Mathematics Textbook for Class 12 - Shaalaa.com

Chapter 5: Continuity and Differentiability

Chapter 5: Continuity and Differentiability solutions [Pages 159 - 161]

Q 1 | Page 159

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

Q 1.3 | Page 159

Examine the following functions for continuity.

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

Q 2 | Page 159

Examine the continuity of the function f (x) = 2x2 – 1 at x = 3.

Q 3.1 | Page 159

Examine the following functions for continuity. 

f (x) = x – 5

 

Q 3.2 | Page 159

Examine the following functions for continuity

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

Q 3.4 | Page 159

Examine the following functions for continuity

f(x) = | x – 5|

Q 4 | Page 159

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

Q 5 | Page 159

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?

Q 6 | Page 159

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

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

Q 7 | Page 159

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):}`

Q 8 | Page 159

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

Q 9 | Page 159

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

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

Q 10 | Page 159

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):}`

Q 11 | Page 159

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):}`

Q 12 | Page 159

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):}`

Q 13 | Page 159

Is the function defined by

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

Q 14 | Page 160

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):}`

Q 15 | Page 160

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):}`

Q 16 | Page 160

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):}`

Q 17 | Page 160

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 = 3.

Q 18 | Page 160

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?

Q 19 | Page 160

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.

Q 20 | Page 160

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

Q 21 | Page 160

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

Q 22 | Page 160

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

Q 23 | Page 160

Find the points of discontinuity of f, where

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

Q 24 | Page 160

Determine if f defined by

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

Q 25 | Page 161

Examine the continuity of f, where f is defined by

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

Q 26 | Page 161

Find the values of 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` 

Q 27 | Page 161

Find the values of 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`

 

Q 28 | Page 161

Find the values of 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`

Q 29 | Page 161

Find the values of 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`

Q 30 | Page 161

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.

Q 31 | Page 161

Show that the function defined by f (x) = cos (x2) is a continuous function.

Q 32 | Page 161

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

 
Q 33 | Page 161

Examine sin |x| is a continuous function.

 
Q 34 | Page 161

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

 

Chapter 5: Continuity and Differentiability solutions [Page 166]

Q 1 | Page 166

Differentiate the functions with respect to x.

sin (x2 + 5)

Q 2 | Page 166

Differentiate the functions with respect to x.

cos (sin x)

Q 3 | Page 166

Differentiate the functions with respect to x.

sin (ax + b)

Q 4 | Page 166

Differentiate the functions with respect to x.

`sec(tan (sqrtx))`

Q 5 | Page 166

Differentiate the functions with respect to x.

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

Q 6 | Page 166

Differentiate the functions with respect to x

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

Q 7 | Page 166

Differentiate the functions with respect to x

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

Q 8 | Page 166

Differentiate the functions with respect to x.

`cos (sqrtx)`

Q 10 | Page 166

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

Chapter 5: Continuity and Differentiability solutions [Page 169]

Q 1 | Page 169

Find  `dy/dx`

2x + 3y = sin x

Q 2 | Page 169

Find `dy/dx`

2x + 3y = sin y

Q 3 | Page 169

Find `dy/dx`

ax + by2 = cos y

Q 4 | Page 169

Find `dy/dx`

xy + y2 = tan x + y

Q 5 | Page 169

Find `dx/dy`

x2 + xy + y2 = 100

Q 6 | Page 169

Find `dy/dx`

x3 + x2y + xy2 + y3 = 81

Q 7 | Page 169

Find `dy/dx`

sin2 y + cos xy = Π

Q 8 | Page 169

Find `dy/dx`

sin2 x + cos2 y = 1

Q 9 | Page 169

Find `dy/dx`

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

Q 10 | Page 169

Find `dy/dx`

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

Q 11 | Page 169

Find `dy/dx`

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

Q 12 | Page 169

Find `dy/dx`

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

Q 13 | Page 169

Find `dx/dy`

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

Q 14 | Page 169

Find `dy/dx`

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

Q 15 | Page 169

Find `dy/dx`

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

Chapter 5: Continuity and Differentiability solutions [Pages 147 - 174]

Q 1 | Page 174

Differentiate the following w.r.t. x:

`e^x/sinx`

Q 2 | Page 147

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

Q 3 | Page 174

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

Q 4 | Page 174

Differentiate the following w.r.t. x

sin (tan–1 e–x)

Q 5 | Page 174

Differentiate the following w.r.t. x:

`log(cos e^x)`

Q 6 | Page 174

Differentiate the following w.r.t. x:

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

Q 7 | Page 174

Differentiate the following w.r.t. x:

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

Q 8 | Page 174

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

Q 9 | Page 174

Differentiate the following w.r.t. x

`cos x/log x, x >0`

Q 10 | Page 174

Differentiate the following w.r.t. x:

cos (log x + ex), x > 0

Chapter 5: Continuity and Differentiability solutions [Pages 178 - 179]

Q 1 | Page 178

Differentiate the function with respect to x

cos x . cos 2x . cos 3x

Q 2 | Page 178

Differentiate the function with respect to x.

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

Q 3 | Page 178

Differentiate the function with respect to x.

`(log x)^(cos x)`

Q 4 | Page 178

Differentiate the function with respect to x.

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

Q 5 | Page 178

Differentiate the function with respect to x.

(x + 3)2 . (x + 4)3 . (x + 5)4

Q 6 | Page 178

Differentiate the function with respect to x.

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

Q 7 | Page 178

Differentiate the function with respect to x.

(log x)x + xlog x

Q 8 | Page 178

Differentiate the function with respect to x.

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

Q 9 | Page 178

Differentiate the function with respect to x.

xsin x + (sin x)cos x

Q 10 | Page 178

Differentiate the function with respect to x.

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

Q 11 | Page 178

Differentiate the function with respect to x.

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

Q 12 | Page 178

Find `dy/dx` of function

xy + yx = 1

Q 13 | Page 178

Find `dy/dx` of Function yx = xy

Q 14 | Page 178

Find `dy/dx` of Function

(cos x)y = (cos y)x

Q 15 | Page 178

Find `dy/dx` of function

xy = e(x – y)

Q 16 | Page 178

Find the derivative of the function given by f (x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) and hence find f ′(1).

Q 17 | Page 178

Differentiate (x2 – 5x + 8) (x3 + 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:

Q 18 | Page 179

If uv 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.

 

Chapter 5: Continuity and Differentiability solutions [Page 181]

Q 1 | 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`

Q 3 | Page 181

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

x = sin ty = cos 2t

 

Q 4 | Page 181

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

x = 4t, y = 4/y

Q 5 | Page 181

If 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

Q 5.6 | Page 181

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

x = a cos θy = b cos θ

Q 6 | Page 181

If 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)

Q 7 | Page 181

If 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)`

Q 8 | Page 181

If 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`

Q 9 | Page 181

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

x = a sec θ, y = b tan θ

Q 10 | Page 181

If 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)

Q 11 | Page 181

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

Chapter 5: Continuity and Differentiability solutions [Pages 183 - 184]

Q 1 | Page 183

Find the second order derivatives of the function.

x2 + 3x + 2

Q 3 | Page 183

Find the second order derivatives of the function.

x . cos x

Q 4 | Page 183

Find the second order derivatives of the function.

log x

Q 5 | Page 183

Find the second order derivatives of the function.

x3 log x

Q 6 | Page 183

Find the second order derivatives of the function.

ex sin 5x

Q 7 | Page 183

Find the second order derivatives of the function.

e6x cos 3x

Q 8 | Page 183

Find the second order derivatives of the function.

tan–1 x

Q 9 | Page 183

Find the second order derivatives of the function.

log (log x)

Q 10 | Page 183

Find the second order derivatives of the function.

sin (log x)

Q 11 | Page 183

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

Q 12 | Page 184

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

Q 13 | Page 184

If y = 3 cos (log x) + 4 sin (log x), show that x2 y2 + xy1 + y = 0

Q 14 | Page 184

If y = Aemx + Benx, show that `(d^2y)/dx^2  - (m+ n) (dy)/dx + mny = 0`

Q 15 | Page 184

If y = 500e7x + 600e–7x, show that `(d^2y)/(dx^2) = 49y`

Q 16 | Page 184

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

Q 17 | Page 184

If y = (tan–1 x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2

Q 183 | Page 183

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

 

Chapter 5: Continuity and Differentiability solutions [Page 186]

Q 1 | Page 186

Verify Rolle’s theorem for the function f (x) = x2 + 2x – 8, x ∈ [– 4, 2].

Q 2.1 | Page 186

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]

Q 2.2 | Page 186

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]

Q 2.3 | Page 186

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) = x2 – 1 for x ∈ [1, 2]

Q 3 | Page 186

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

Q 4 | Page 186

Verify Mean Value Theorem, if f (x) = x2 – 4x – 3 in the interval [a, b], where a = 1 and b = 4.

Q 5 | Page 186

Verify Mean Value Theorem, if f (x) = x3 – 5x2 – 3x in the interval [a, b], where a = 1 and b = 3. Find all c ∈ (1, 3) for which f ′(c) = 0.

Q 6 | Page 186

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

Chapter 5: Continuity and Differentiability solutions [Pages 191 - 192]

Q 1 | Page 191

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

Q 2 | Page 191

Differentiate w.r.t. x the function sin3 x + cos6 x

Q 3 | Page 191

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

Q 4 | Page 191

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

Q 5 | Page 191

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

Q 6 | Page 191

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`

Q 7 | Page 191

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

Q 8 | Page 191

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

Q 9 | Page 191

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

Q 10 | Page 191

Differentiate w.r.t. x the function xx + xa + ax + aa, for some fixed a > 0 and x > 0

Q 11 | Page 191

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

Q 12 | Page 191

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

Q 13 | Page 191

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

Q 14 | Page 191

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

Q 15 | Page 191

If (x – a)2 + (y – b)2 = c2, for some c > 0, prove that

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

Q 16 | Page 192

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

Q 17 | Page 192

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

Q 18 | Page 192

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

Q 19 | Page 192

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

Q 20 | Page 192

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

Q 21 | Page 192

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

Q 22 | Page 192

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)|`

Q 23 | Page 192

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 Mathematics Class 12

Mathematics Textbook for Class 12 - Shaalaa.com

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

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

Further, we at Shaalaa.com are providing 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 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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