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Class 12 Mathematics

Chapter 5 Solutions — Continuity and Differentiability

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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.

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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) = 2x^2 - 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) = (x^2 - 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) = x^2 + 1, if x < 1
  11. Find all points of discontinuity of f, where f is defined by f(x) = x^3 - 3, if x <= 2 f(x) = x^2 + 1, if x > 2
  12. Find all points of discontinuity of f, where f is defined by f(x) = x^10 - 1, if x <= 1 f(x) = x^2, 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(x^2 - 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) = x^2 - 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) = x^2 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) = kx^2, 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(x^2) 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(x^2 + 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(x^3) · sin^2(x^5) with respect to x.
  7. Differentiate 2 cot(x^2) 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 + by^2 = cos y.
  4. Find dy/dx if xy + y^2 = tan x + y.
  5. Find dy/dx if x^2 + xy + y^2 = 100.
  6. Find dy/dx if x^3 + x^2y + xy^2 + y^3 = 81.
  7. Find dy/dx if sin^2 y + cos(xy) = κ.
  8. Find dy/dx if sin^2 x + cos^2 y = 1.
  9. Find dy/dx if y = sin^{-1}(2x / (1 + x^2)).
  10. Find dy/dx if y = tan^{-1}((3x - x^3) / (1 - 3x^2)), -1/√3 < x < 1/√3.
  11. Find dy/dx if y = cos^{-1}((1 - x^2) / (1 + x^2)), 0 < x < 1.
  12. Find dy/dx if y = sin^{-1}((1 - x^2) / (1 + x^2)), 0 < x < 1.
  13. Find dy/dx if y = cos^{-1}(2x / (1 + x^2)), -1 < x < 1.
  14. Find dy/dx if y = sin^{-1}(2x√(1 - x^2)), -1/√2 < x < 1/√2.
  15. Find dy/dx if y = sec^{-1}(1 / (2x^2 - 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^{x^3} 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^{x^2} + e^{x^3} + e^{x^4} + e^{x^5} 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 + (x^2+1)/(x^2-1) 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+x^2)(1+x^4)(1+x^8) and hence find f'(1).
  17. Differentiate (x^2 - 5x + 8)(x^3 + 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 = 2at^2, y = at^4, 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 = sin^3 t / √(cos 2t), y = cos^3 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 = x^2 + 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 = x^3 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 d^2y/dx^2 + y = 0.
  12. If y = cos^{-1} x, find d^2y/dx^2 in terms of y alone.
  13. If y = 3 cos(log x) + 4 sin(log x), show that x^2 y_2 + x y_1 + y = 0.
  14. If y = Ae^{mx} + Be^{nx}, show that d^2y/dx^2 - (m+n) dy/dx + mny = 0.
  15. If y = 500e^{7x} + 600e^{-7x}, show that d^2y/dx^2 = 49y.
  16. If e^y(x+1) = 1, show that d^2y/dx^2 = (dy/dx)^2.
  17. If y = (tan^{-1} x)^2, show that (x^2+1)^2 y_2 + 2x(x^2+1) y_1 = 2.

Miscellaneous Exercise

  1. Differentiate (3x^2 - 9x + 5)^9 with respect to x.
  2. Differentiate sin^3 x + cos^6 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^{x^2-3} + (x-3)^{x^2}, 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-x^2)), 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 = 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.
  16. If cos y = x cos(a+y), with cos a ≠ ±1, prove that dy/dx = cos^2(a+y) / sin a.
  17. If x = a(cos t + t sin t) and y = a(sin t - t cos t), find d^2y/dx^2.
  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-x^2) d^2y/dx^2 - x dy/dx - a^2 y = 0.
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01Relations and Functions02Inverse Trigonometric Functions03Matrices04Determinants06Application of Derivatives
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