Back to Mathematics Part I
Class 12 Mathematics

Chapter 5 Solutions — Continuity and Differentiability

Open Solutions PDFReads in your browser→Also seeTextbook page→
Get the full experience in the app
Free practice, offline PDFs & solutions.
Download on the App StoreGet it on Google Play
Already installed? Open in the app
Solutions

Overview

Step-by-step NCERT solutions for Continuity and Differentiability (Chapter 5, CBSE Class 12 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Continuity and Differentiability textbook chapter.

Solved

What these solutions cover

All 137 questions in Continuity and Differentiability are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise 5.1

  1. Prove that the function f(x) = 5x - 3 is continuous at x = 0, at x = -3 and at x = 5.
  2. Examine the continuity of the function f(x) = 2x² - 1 at x = 3.
  3. Examine the following functions for continuity.
    • (a) f(x) = x - 5
    • (b) f(x) = 1/(x - 5), x ≠ 5
    • (c) f(x) = (x² - 25)/(x + 5), x ≠ -5
    • (d) f(x) = |x - 5|
  4. Prove that the function f(x) = x^n is continuous at x = n, where n is a positive integer.
  5. Is the function f defined by f(x) = x, if x <= 1 f(x) = 5, if x > 1 continuous at x = 0? At x = 1? At x = 2?
  6. Find all points of discontinuity of f, where f is defined by f(x) = 2x + 3, if x <= 2 f(x) = 2x - 3, if x > 2
  7. Find all points of discontinuity of f, where f is defined by f(x) = |x| + 3, if x <= -3 f(x) = -2x, if -3 < x < 3 f(x) = 6x + 2, if x >= 3
  8. Find all points of discontinuity of f, where f is defined by f(x) = |x|/x, if x ≠ 0 f(x) = 0, if x = 0
  9. Find all points of discontinuity of f, where f is defined by f(x) = x/|x|, if x < 0 f(x) = -1, if x >= 0
  10. Find all points of discontinuity of f, where f is defined by f(x) = x + 1, if x >= 1 f(x) = x² + 1, if x < 1
  11. Find all points of discontinuity of f, where f is defined by f(x) = x³ - 3, if x <= 2 f(x) = x² + 1, if x > 2
  12. Find all points of discontinuity of f, where f is defined by f(x) = x¹⁰ - 1, if x <= 1 f(x) = x², if x > 1
  13. Is the function defined by f(x) = x + 5, if x <= 1 f(x) = x - 5, if x > 1 a continuous function?
  14. Discuss the continuity of the function f, where f is defined by f(x) = 3, if 0 <= x <= 1 f(x) = 4, if 1 < x < 3 f(x) = 5, if 3 <= x <= 10
  15. Discuss the continuity of the function f, where f is defined by f(x) = 2x, if x < 0 f(x) = 0, if 0 <= x <= 1 f(x) = 4x, if x > 1
  16. Discuss the continuity of the function f, where f is defined by f(x) = -2, if x <= -1 f(x) = 2x, if -1 < x <= 1 f(x) = 2, if x > 1
  17. Find the relationship between a and b so that the function f defined by f(x) = ax + 1, if x <= 3 f(x) = bx + 3, if x > 3 is continuous at x = 3.
  18. For what value of λ is the function defined by f(x) = λ(x² - 2x), if x <= 0 f(x) = 4x + 1, if x > 0 continuous at x = 0? What about continuity at x = 1?
  19. 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.
  20. Is the function defined by f(x) = x² - sin x + 5 continuous at x = π?
  21. 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
  22. Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
  23. Find all points of discontinuity of f, where f(x) = sin x / x, if x < 0 f(x) = x + 1, if x >= 0
  24. Determine if f defined by f(x) = x² sin(1/x), if x ≠ 0 f(x) = 0, if x = 0 is a continuous function?
  25. Examine the continuity of f, where f is defined by f(x) = sin x - cos x, if x ≠ 0 f(x) = -1, if x = 0
  26. Find the value of k so that the function f is continuous at x = π/2: f(x) = k cos x / (π - 2x), if x ≠ π/2 f(x) = 3, if x = π/2
  27. Find the value of k so that the function f is continuous at x = 2: f(x) = kx², if x <= 2 f(x) = 3, if x > 2
  28. Find the value of k so that the function f is continuous at x = π: f(x) = kx + 1, if x <= π f(x) = cos x, if x > π
  29. Find the value of k so that the function f is continuous at x = 5: f(x) = kx + 1, if x <= 5 f(x) = 3x - 5, if x > 5
  30. Find the values of a and b such that the function defined by f(x) = 5, if x <= 2 f(x) = ax + b, if 2 < x < 10 f(x) = 21, if x >= 10 is a continuous function.
  31. Show that the function defined by f(x) = cos(x²) is a continuous function.
  32. Show that the function defined by f(x) = |cos x| is a continuous function.
  33. Examine that sin |x| is a continuous function.
  34. Find all the points of discontinuity of f defined by f(x) = |x| - |x + 1|.

Exercise 5.2

  1. Differentiate sin(x² + 5) with respect to x.
  2. Differentiate cos(sin x) with respect to x.
  3. Differentiate sin(ax + b) with respect to x.
  4. Differentiate sec(tan(√x)) with respect to x.
  5. Differentiate sin(ax + b) / cos(cx + d) with respect to x.
  6. Differentiate cos(x³) · sin²(x⁵) with respect to x.
  7. Differentiate 2 cot(x²) with respect to x.
  8. Differentiate cos(√x) with respect to x.
  9. Prove that the function f given by f(x) = |x - 1|, x in R, is not differentiable at x = 1.
  10. Prove that the greatest integer function defined by f(x) = [x], 0 < x < 3, is not differentiable at x = 1 and x = 2.

Exercise 5.3

  1. Find dy/dx if 2x + 3y = sin x.
  2. Find dy/dx if 2x + 3y = sin y.
  3. Find dy/dx if ax + by² = cos y.
  4. Find dy/dx if xy + y² = tan x + y.
  5. Find dy/dx if x² + xy + y² = 100.
  6. Find dy/dx if x³ + x²y + xy² + y³ = 81.
  7. Find dy/dx if sin² y + cos(xy) = κ.
  8. Find dy/dx if sin² x + cos² y = 1.
  9. Find dy/dx if y = sin^{-1}(2x / (1 + x²)).
  10. Find dy/dx if y = tan^{-1}((3x - x³) / (1 - 3x²)), -1/√3 < x < 1/√3.
  11. Find dy/dx if y = cos^{-1}((1 - x²) / (1 + x²)), 0 < x < 1.
  12. Find dy/dx if y = sin^{-1}((1 - x²) / (1 + x²)), 0 < x < 1.
  13. Find dy/dx if y = cos^{-1}(2x / (1 + x²)), -1 < x < 1.
  14. Find dy/dx if y = sin^{-1}(2x√(1 - x²)), -1/√2 < x < 1/√2.
  15. Find dy/dx if y = sec^{-1}(1 / (2x² - 1)), 0 < x < 1/√2.

Exercise 5.4

  1. Differentiate e^x / sin x with respect to x.
  2. Differentiate e^{sin^{-1} x} with respect to x.
  3. Differentiate e^{x³} with respect to x.
  4. Differentiate sin(tan^{-1} e^{-x}) with respect to x.
  5. Differentiate log(cos e^x) with respect to x.
  6. Differentiate e^x + e^{x²} + e^{x³} + e^{x⁴} + e^{x⁵} with respect to x.
  7. Differentiate √(e^x) for x > 0 with respect to x.
  8. Differentiate log(log x), x > 1, with respect to x.
  9. Differentiate cos x / log x, x > 0, with respect to x.
  10. Differentiate cos(log x + e^x), x > 0, with respect to x.

Exercise 5.5

  1. Differentiate cos x · cos 2x · cos 3x with respect to x.
  2. Differentiate √[(x-1)(x-2) / ((x-3)(x-4)(x-5))] with respect to x.
  3. Differentiate (log x)^{cos x} with respect to x.
  4. Differentiate x^x - 2^{sin x} with respect to x.
  5. Differentiate (x+3)² · (x+4)³ · (x+5)⁴ with respect to x.
  6. Differentiate (x + 1/x)^x + x^{1+1/x} with respect to x.
  7. Differentiate (log x)^x + x^{log x} with respect to x.
  8. Differentiate (sin x)^x + sin^{-1}(√x) with respect to x.
  9. Differentiate x^{sin x} + (sin x)^{cos x} with respect to x.
  10. Differentiate x^x cos x + (x²⁺1)/(x²⁻1) with respect to x.
  11. Differentiate (x cos x)^x + (x sin x)^{1/x} with respect to x.
  12. Find dy/dx if x^y + y^x = 1.
  13. Find dy/dx if y^x = x^y.
  14. Find dy/dx if (cos x)^y = (cos y)^x.
  15. Find dy/dx if x^y = e^{x-y}.
  16. Find the derivative of f(x) = (1+x)(1+x²)(1+x⁴)(1+x⁸) and hence find f'(1).
  17. Differentiate (x² - 5x + 8)(x³ + 7x + 9) in three ways:
    • (i) by product rule,
    • (ii) by expanding,
    • (iii) by logarithmic differentiation. Do they all give the same answer?
  18. If u, v and w are functions of x, show that d(uvw)/dx = (du/dx)vw + u(dv/dx)w + uv(dw/dx) in two ways: first by repeated application of product rule, second by logarithmic differentiation.

Exercise 5.6

  1. If x and y are connected parametrically by x = 2at², y = at⁴, find dy/dx without eliminating the parameter.
  2. If x and y are connected parametrically by x = a cos θ, y = b cos θ, find dy/dx without eliminating the parameter.
  3. If x and y are connected parametrically by x = sin t, y = cos 2t, find dy/dx without eliminating the parameter.
  4. If x and y are connected parametrically by x = 4t, y = 4/t, find dy/dx without eliminating the parameter.
  5. If x and y are connected parametrically by x = cos θ - cos 2θ, y = sin θ - sin 2θ, find dy/dx without eliminating the parameter.
  6. If x and y are connected parametrically by x = a(θ - sin θ), y = a(1 + cos θ), find dy/dx without eliminating the parameter.
  7. If x and y are connected parametrically by x = sin³ t / √(cos 2t), y = cos³ t / √(cos 2t), find dy/dx without eliminating the parameter.
  8. If x and y are connected parametrically by x = a(cos t + log tan(t/2)), y = a sin t, find dy/dx without eliminating the parameter.
  9. If x and y are connected parametrically by x = a sec θ, y = b tan θ, find dy/dx without eliminating the parameter.
  10. If x and y are connected parametrically by x = a(cos θ + θ sin θ), y = a(sin θ - θ cos θ), find dy/dx without eliminating the parameter.
  11. If x = a^{sin^{-1} t}, y = a^{cos^{-1} t}, show that dy/dx = -y/x.

Exercise 5.7

  1. Find the second order derivative of y = x² + 3x + 2.
  2. Find the second order derivative of y = x^{20}.
  3. Find the second order derivative of y = x cos x.
  4. Find the second order derivative of y = log x.
  5. Find the second order derivative of y = x³ log x.
  6. Find the second order derivative of y = e^x sin 5x.
  7. Find the second order derivative of y = e^{6x} cos 3x.
  8. Find the second order derivative of y = tan^{-1} x.
  9. Find the second order derivative of y = log(log x).
  10. Find the second order derivative of y = sin(log x).
  11. If y = 5 cos x - 3 sin x, prove that d²y/dx² + y = 0.
  12. If y = cos^{-1} x, find d²y/dx² in terms of y alone.
  13. If y = 3 cos(log x) + 4 sin(log x), show that x² y_2 + x y_1 + y = 0.
  14. If y = Ae^{mx} + Be^{nx}, show that d²y/dx² - (m+n) dy/dx + mny = 0.
  15. If y = 500e^{7x} + 600e^{-7x}, show that d²y/dx² = 49y.
  16. If e^y(x+1) = 1, show that d²y/dx² = (dy/dx)².
  17. If y = (tan^{-1} x)², show that (x²⁺1)² y_2 + 2x(x²⁺1) y_1 = 2.

Miscellaneous Exercise

  1. Differentiate (3x² - 9x + 5)⁹ with respect to x.
  2. Differentiate sin³ x + cos⁶ x with respect to x.
  3. Differentiate (5x)^{3 cos 2x} with respect to x.
  4. Differentiate sin^{-1}(x√x), 0 ≤ x ≤ 1, with respect to x.
  5. Differentiate (cos^{-1}(x/2)) / √(2x+7), -2 < x < 2, with respect to x.
  6. Differentiate cot^{-1}[(√(1+sin x) + √(1-sin x)) / (√(1+sin x) - √(1-sin x))], 0 < x < π/2, with respect to x.
  7. Differentiate (log x)^{log x}, x > 1, with respect to x.
  8. Differentiate cos(a cos x + b sin x), for some constant a and b, with respect to x.
  9. Differentiate (sin x - cos x)^{sin x - cos x}, π/4 < x < 3π/4, with respect to x.
  10. Differentiate x^x + x^a + a^x + a^a, for some fixed a > 0 and x > 0, with respect to x.
  11. Differentiate x^{x²⁻3} + (x-3)^{x²}, for x > 3, with respect to x.
  12. Find dy/dx if y = 12(1 - cos t), x = 10(t - sin t), -π/2 < t < π/2.
  13. Find dy/dx if y = sin^{-1} x + sin^{-1}(√(1-x²)), 0 < x < 1.
  14. If x√(1+y) + y√(1+x) = 0, for -1 < x < 1, prove that dy/dx = -1/(1+x)².
  15. If (x-a)² + (y-b)² = c², for some c > 0, prove that [(1 + (dy/dx)²)^{3/2}] / (d²y/dx²) is a constant independent of a and b.
  16. If cos y = x cos(a+y), with cos a ≠ ±1, prove that dy/dx = cos²(a+y) / sin a.
  17. If x = a(cos t + t sin t) and y = a(sin t - t cos t), find d²y/dx².
  18. If f(x) = |x|³, show that f''(x) exists for all real x and find it.
  19. Using the fact that sin(A+B) = sin A cos B + cos A sin B and differentiation, obtain the sum formula for cosines.
  20. Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.
  21. 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|.
  22. If y = e^{a cos^{-1} x}, -1 ≤ x ≤ 1, show that (1-x²) d²y/dx² - x dy/dx - a² y = 0.
Sample

Worked examples

3 questions from Continuity and Differentiability, solved step by step.

Exercise 5.1

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

f(x) = 5x - 3 is a polynomial, so it is continuous at every real number. We verify at the three given points.

At x = 0: f(0) = -3. lim_{x→0} (5x - 3) = -3 = f(0). Continuous at x = 0.

At x = -3: f(-3) = -18. lim_{x→-3} (5x - 3) = -18 = f(-3). Continuous at x = -3.

At x = 5: f(5) = 22. lim_{x→5} (5x - 3) = 22 = f(5). Continuous at x = 5.

Answer: f is continuous at x = 0, x = -3, and x = 5.

Exercise 5.2

Differentiate sec(tan(√x)) with respect to x.

Let u = √x, v = tan u, f = sec v.

d/dx[sec(tan(√x))] = sec(tan(√x)) · tan(tan(√x)) · d/dx[tan(√x)]

= sec(tan(√x)) · tan(tan(√x)) · sec²(√x) · d/dx(√x)

= sec(tan(√x)) · tan(tan(√x)) · sec²(√x) · 1/(2√x).

Answer: sec(tan √x) · tan(tan √x) · sec²(√x) / (2√x).

Exercise 5.3

Find dy/dx if xy + y² = tan x + y.

Differentiating w.r.t. x:

y + x dy/dx + 2y dy/dx = sec² x + dy/dx.

dy/dx (x + 2y - 1) = sec² x - y.

dy/dx = (sec² x - y) / (x + 2y - 1).

Answer: dy/dx = (sec² x - y) / (x + 2y - 1).

All 137 questions in Continuity and Differentiability are solved step by step in the Solutions PDF above.

Keep solving

More solutions in Mathematics Part I

01Relations and Functions02Inverse Trigonometric Functions03Matrices04Determinants06Application of Derivatives
Explore

More NCERT Solutions for Class 12

Read the Continuity and Differentiability textbook chapter / PDF, or browse all CBSE Class 12 Mathematics solutions.

← PreviousCh 4: DeterminantsNext →Ch 6: Application of Derivatives

Solve offline with notes, solutions & mock tests

CBSE Prepmaster — free on iOS & Android

Get the App