Continuity and Differentiability

Previous Years' CBSE Board Questions

5.2 Continuity

MCQ

1 The function <math>f(x) = [x]</math>, where <math>[x]</math> denotes the greatest integer less than or equal to x, is continuous at

1. <math>x = 1</math> (b) <math>x = 1.5</math> (c) <math>x = -2</math> (d) <math>x = 4</math>

(2023)

If the function <math>f(x) = \begin{cases} 3x-8, & \text{if } x \le 5 \\ 2k, & \text{if } x > 5 \end{cases}</math> is continuous, then the value of k is

1. 2/7 (b) 7/2 (c) 3/7 (d) 4/7 (Term I, 2021-22) Ap

The function <math>f(x) = |x|</math> is

1. (c) continuous everywhere, but differentiable everywhere except at <math>x = 0</math>.
2. nowhere. (2023)

1. The function <math>f(x) = [x]</math>, where <math>[x]</math> is the greatest integer function that is less than or equal to x, is continuous at
2. 4 (b) -2 (c) 1.5 (d) 1 (Term I, 2021-22)

V5A (1 mark)

The value of <math>\lambda</math> so that the function f defined by <math display="block">f(x) = \begin{cases} \lambda x, & \text{if } x \leq \pi \\ \cos x, & \text{if } x > \pi \end{cases}</math> is continuous at <math>x = \pi</math> is _____ (2020) Ap

1. Determine the value of the constant 'k' so that the function <math>f(x) = \begin{cases} \frac{kx}{|x|}, & \text{if } x < 0 \\ 3, & \text{if } x \ge 0 \end{cases}</math> is continuous at <math>x = 0</math>. 14. The function <math>f(x) = \begin{cases} x^2 & \text{for } x < 1 \\ 2 - x & \text{for } x \ge 1 \end{cases}</math>

1. Determine the value of 'k' for which the following function is continuous at <math>x = 3</math>. <math display="block">f(x) = \begin{cases} \frac{(x+3)^2 - 36}{x-3} &, & x \neq 3 \\ k &, & x = 3 \end{cases}</math> (Al 2017) (Ap)

(2 marks)

1. Find the value(s) of <math>\lambda</math>, if the function <math display="block">f(x) = \begin{cases} \frac{\sin^2 \lambda x}{x^2}, & \text{if } x \neq 0 \\ x \neq 0 \end{cases} \text{ is continuous at } x = 0. \tag{2023}</math>

1. Find the relationship between a and b so that the function f defined by <math>f(x) = \begin{cases} ax+1 & \text{if } x \le 3 \\ bx+3 & \text{if } x > 3 \end{cases}</math> is continuous at <math>x = 3</math>. (2021C) Ap

(4 marks)

Find the values of p and q, for which <math display="block">\frac{1-\sin^3 x}{3\cos^2 x}, \quad \text{if } x < \pi/2</math> <math>f(x) = </math> if <math>x = \pi/2</math> is continuous at <math>x = \pi/2</math>. <math>q(1-\sin x)</math> if <math>x > \pi/2</math> (Delhi 2016) (Ap)

Find the value of the constant k so that the function

<math>f(x)=</math> f, defined below, is continuous at <math>x = 0</math>, where <math>(1-\cos 4x)</math> if <math>x \neq 0</math> (AI 2014C) Ap

if <math>x = 0</math>

5.3 Differentiability

MCQ

The derivative of x2x w.r.t. x is

1. x^2x-1 (b) <math>2x^{2x}\log x</math>
2. <math>2x^{2x}(1 + \log x)</math> (d) <math>2x^{2x}(1 - \log x)</math> (2023)

13. If <math>y^2(2-x) = x^3</math>, then <math>\left(\frac{dy}{dx}\right)_{(1,1)}</math> is equal to

(a) 2 (b) -2 (c) 3 (d) -3/2

(Term I, 2021-22) (Ap)

(a) not differentiable at x = 1

(b) differentiable at x = 1

1. (d) neither continuous nor differentiable at <math>x = 1</math> (Term I, 2021-22) Ev

1. If <math>\sec^{-1}\left(\frac{1+x}{1-y}\right) = a</math>, then <math>\frac{dy}{dx}</math> is equal to (a) <math>\frac{x-1}{y-1}</math> (b) <math>\frac{x-1}{y+1}</math> (c) <math>\frac{y-1}{x+1}</math> (d) <math>\frac{y+1}{y-1}</math> (2020C) Ap

VSA. (1 mark)

16. If <math>y = \tan^{-1} x + \cot^{-1} x</math>, <math>x \in R</math>, then <math>\frac{dy}{dx}</math> is equal to (2020)

If cos (xy) = k, where k is a constant and xy ≠ nπ, n ∈ Z,

then <math>\frac{dy}{dx}</math> is equal to _____. (2020) Ev

18. Differentiate <math>\sin^2(\sqrt{x})</math> with respect to x. (2020) (Ap)

Let f(x) = x|x|, for all x ∈ R check its differentiability at

<math>x = 0</math>. (2020) (EV)

20. If <math>y = f(x^2)</math> and <math>f'(x) = e^{\sqrt{x}}</math>, then find <math>\frac{dy}{dx}</math>. (2020) Ev

21. If <math>f(x) = x+1</math>, find <math>\frac{d}{dx}(f\circ f)(x)</math>.

(Delhi 2019) (U)

SAI (2 marks)

22. If <math>(x^2 + y^2)^2 = xy</math>, then find <math>\frac{dy}{dx}</math>.

(2023)

1. If <math>f(x) = \begin{cases} x^2, & \text{if } x \ge 1 \\ x, & \text{if } x < 1 \end{cases}</math>, then show that f is not differentiable at <math>x = 1</math>. (2023)
2. Check the differentiability of <math>f(x) = |x - 3|</math> at <math>x = 3</math>. (2021C) EV
3. If <math>y = \sqrt{a + \sqrt{a + x}}</math>, then find <math>\frac{dy}{dx}</math>. (2020C)

(AI 2015C) (AD

1. Differentiate <math>tan^{-1}\left(\frac{1+\cos x}{\sin x}\right)</math> with respect to x. (2018)
2. Find <math>\frac{dy}{dx}</math> at <math>x = 1</math>, <math>y = \frac{\pi}{4}</math> if <math>\sin^2 y + \cos xy = K</math>. (Delhi 2017) Ev

LAI (4 marks)

Here, question 28(i) to (iii) is a case study based question of 4 marks.

Let f(x) be a real valued function. Then its Left Hand Derivative (L.H.D.):

<math display="block">Lf'(a) = \lim_{h \to 0} \frac{f(a-h) - f(a)}{-h}</math>

Right Hand Derivative (R.H.D.):

<math display="block">Rf'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}</math>

Also, a function f(x) is said to be differentiable at <math>x = a</math> if its L.H.D. and R.H.D. at <math>x = a</math> exist and both are equal.

For the function <math>f(x) =\begin{cases} |x-3|, x \ge 1 \\ \frac{x^2}{4} - \frac{3x}{2} + \frac{13}{4}, x < 1 \end{cases}</math>

answer the following questions: (Term I, 2021-22) Ev

1. What is R.H.D. of f(x) at <math>x = 1</math>?
2. What is L.H.D. of f(x) at <math>x = 1</math>?
3. If <math>y = \log(\cos e^x)</math>, then find <math>\frac{dy}{dx}</math>. (NCERT, Al 2019) LAI (4 marks)

1. Check if the function f(x) is differentiable at <math>x = 1</math>. OR
2. Find f'(2) and f'(-1).

(2020) EV

Find the values of a and b, if the function f defined by <math display="block">f(x) = \begin{cases} x^2 + 3x + a, & x \le 1 \\ bx + 2, & x > 1 \end{cases}</math> is differentiable at <math>x = 1</math>. (Foreign 2016) Ap

1. If <math>y = \tan^{-1} \left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) x^2 \le 1</math>

then find <math>\frac{dy}{dx}</math>. (NCERT Exemplar, Delhi 2015) <math>\frac{dy}{dx} = \frac{\cos^{-1}x}{(1-x^2)^{3/2}}</math>. (Delay 31. If <math>f(x) = \sqrt{x^2 + 1}</math>; <math>g(x) = \frac{x+1}{x^2 + 1}</math> and <math>h(x) = 2x - 3</math>, then find <math>\frac{dy}{dx} = \frac{\cos^{-1}x}{(1-x^2)^{3/2}</math> f'[h'(g'(x))]. (AI 2015) Ap

Show that the function f(x) = |x - 1| + |x + 1|, for all x ∈ R,

is not differentiable at the points <math>x = -1</math> and <math>x = 1</math>.

(AI 2015) EV

Find whether the following function is differentiable

<math display="block">f(x) = \begin{cases} x, & x < 1 \\ 2 - x, & 1 \le x \le 2 \\ -2 + 3x - x^2, & x > 2 \end{cases}</math> at <math>x = 1</math> and <math>x = 2</math> or not. (Foreign 2015) 🙀

34. For what value of <math>\lambda</math> the function defined by

<math display="block">f(x) = \begin{cases} \lambda(x^2 + 2), & \text{if } x \le 0 \\ 4x + 6, & \text{if } x > 0 \end{cases}</math> is continuous at <math>x = 0</math>? Hence

check the differentiability of f(x) at <math>x = 0</math>.

1. If <math>cosy = xcos(a + y)</math>, where <math>cos a \neq \pm 1</math>, prove that <math display="block">\frac{dy}{dx} = \frac{\cos^2(a+y)}{\sin a}.</math> (Foreign 2014) (EV)
2. If <math>y = \sin^{-1} \left\{ x \sqrt{1-x} - \sqrt{x} \sqrt{1-x^2} \right\}</math> and <math>0 < x < 1</math>, then find <math>\frac{dy}{dy}</math>. (AI 2014C) Ap

5.4 Exponential and Logarithmic

Functions

MCQ

1. excotex 37. If <math>y = \log (\sin e^x)</math>, then <math>\frac{dy}{dx}</math> is (a) <math>\cot e^x</math> (1) (b) cosec e^x (d) ex cosec ex (2023)

38. If <math>y = \tan^{-1}(e^{2x})</math>, then <math>\frac{dy}{dx}</math> is equal to

(a) <math>\frac{2e^{2x}}{1+e^{4x}}</math> (b) <math>\frac{1}{1+e^{4x}}</math> (c) <math>\frac{2}{e^{2x}+e^{-2x}}</math> (d) <math>\frac{1}{e^{2x}-e^{-2x}}</math>

VSA. (1 mark)

1. If <math>y = e^{x^2 \cos x} + (\cos x)^x</math>, then find <math>\frac{dy}{dx}</math>.
2. If <math>\log(x^2+y^2)=2\tan^{-1}\left(\frac{y}{y}\right)</math>, show that <math>\frac{dy}{dx}=\frac{x+y}{x-y}</math>. (Delhi 2019) Ev
3. If <math>y = \frac{x\cos^{-1}x}{\sqrt{1-x^2}} - \log \sqrt{1-x^2}</math>, then prove that (Delhi 2015C) (Ap)

(Foreign 2014) 📝