CBSE Class 12 Mathematics Continuity and Differentiability VBQs Set 03

Short Answer Questions–I

Question. Examine the continuity at the indicated points. \( f(x) = |x| + |x - 1| \) at \( x = 1 \)
Answer: Discontinuous

Question. Find \( k \) if \( f(x) \) is continuous at \( x = 0 \). \( f(x) = \begin{cases} \frac{\sin x}{x} + \cos x, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases} \).
Answer: \( k = 2 \)

Question. Find the value of \( c \) in Rolle’s theorem for the function \( f(x) = x^3 - 3x \) in \( [-3, 0] \).
Answer: \( c = -1 \)

Question. If \( f(x) = |\cos x - \sin x| \), find \( f'(\pi/6) \).
Answer: \( -\frac{1}{2}(1 + \sqrt{3}) \)

Question. If \( y = 5 \cos x - 3 \sin x \), prove that \( \frac{d^2 y}{dx^2} + y = 0 \).
Answer: Since \( y = 5 \cos x - 3 \sin x \), then \( \frac{dy}{dx} = -5 \sin x - 3 \cos x \). Differentiating again, \( \frac{d^2 y}{dx^2} = -5 \cos x + 3 \sin x = -(5 \cos x - 3 \sin x) = -y \). Therefore, \( \frac{d^2 y}{dx^2} + y = 0 \).

Short Answer Questions–II

Question. Show that the function ‘f ’ defined by \( f(x) = \begin{cases} 3x - 2, & 0 < x \le 1 \\ 2x^2 - x, & 1 < x \le 2 \\ 5x - 4, & x > 2 \end{cases} \) is continuous at \( x = 2 \), but not differentiable.
Answer: Not differentiable

Question. Show that the function \( f(x) = |x - 1| + |x + 1| \), for all \( x \in R \), but is not differentiable at the points \( x = -1 \) and \( x = 1 \).
Answer: The function \( f(x) \) can be rewritten as a piecewise function. At \( x = 1 \), LHD = 0 and RHD = 2. Since LHD \( \neq \) RHD, it is not differentiable. Similarly for \( x = -1 \).

Question. If \( y = x^3 (\cos x)^x + \sin^{-1} \sqrt{x} \), find \( \frac{dy}{dx} \).
Answer: \( x^3 (\cos x)^x [\frac{3}{x} - x \tan x + \log(\cos x)] + \frac{1}{2\sqrt{x-x^2}} \)

Question. Differentiate \( \tan^{-1} \left[ \frac{\sqrt{1+x^2} + \sqrt{1-x^2}}{\sqrt{1+x^2} - \sqrt{1-x^2}} \right] \) with respect to \( \cos^{-1} x^2 \).
Answer: \( -\frac{1}{2} \)

Question. Differentiate \( \tan^{-1} \left[ \frac{\sqrt{1+x^2} - 1}{x} \right] \) with respect to \( x \).
Answer: \( \frac{1}{2(1 + x^2)} \)

Question. Differentiate \( \tan^{-1} \left[ \frac{x}{\sqrt{1-x^2}} \right] \) with respect to \( \sin^{-1}(2x\sqrt{1-x^2}) \).
Answer: \( \frac{1}{2} \)

Question. If \( y = \cos^{-1} \left( \frac{2^{x+1}}{1 + 4^x} \right) \), then find \( \frac{dy}{dx} \).
Answer: \( \frac{-2^{x+1} \cdot \log_e 2}{1 + 4^x} \)

Question. If \( y = e^x (\sin x + \cos x) \), then show that \( \frac{d^2 y}{dx^2} - 2\frac{dy}{dx} + 2y = 0 \).
Answer: Differentiating \( y \), we get \( \frac{dy}{dx} = e^x(\cos x - \sin x) + e^x(\sin x + \cos x) = 2e^x \cos x \). Differentiating again, \( \frac{d^2 y}{dx^2} = 2e^x \cos x - 2e^x \sin x \). Substituting into the equation yields 0.

Question. Verify Lagrange’s Mean Value Theorem for the following function: \( f(x) = x^2 + 2x + 3 \), for \( [4, 6] \).
Answer: \( f(x) \) is polynomial, hence continuous and differentiable. \( f'(c) = \frac{f(6) - f(4)}{6 - 4} \Rightarrow 2c + 2 = \frac{51 - 27}{2} = 12 \Rightarrow c = 5 \in (4, 6) \). Verified.

Question. If \( y = \sqrt{x^2+1} - \log \left( \frac{1}{x} + \sqrt{1 + \frac{1}{x^2}} \right) \), then find \( \frac{dy}{dx} \).
Answer: \( \frac{\sqrt{x^2+1}}{x} \)

Question. Discuss the differentiability of the function \( f(x) = \begin{cases} 2x - 1, & x < \frac{1}{2} \\ 3 - 6x, & x \ge \frac{1}{2} \end{cases} \) at \( x = \frac{1}{2} \).
Answer: Not differentiable

Question. For what value of \( k \) is the following function continuous at \( x = -\frac{\pi}{6} \)? \( f(x) = \begin{cases} \frac{\sqrt{3}\sin x + \cos x}{x + \frac{\pi}{6}}, & x \neq -\frac{\pi}{6} \\ k, & x = -\frac{\pi}{6} \end{cases} \).
Answer: \( k = 2 \)

Question. If \( x\sqrt{1+y} + y\sqrt{1+x} = 0, -1 < x < 1, x \neq y \), then prove that \( \frac{dy}{dx} = -\frac{1}{(1 + x)^2} \).
Answer: Squaring and simplifying the given equation gives \( y = -\frac{x}{1+x} \). Differentiating with respect to \( x \) yields \( \frac{dy}{dx} = -\frac{1}{(1+x)^2} \).

Question. Differentiate the following function with respect to \( x \): \( (x)^{\cos x} + (\sin x)^{\tan x} \).
Answer: \( x^{\cos x} [\frac{\cos x}{x} - \sin x \log x] + (\sin x)^{\tan x} [1 + \sec^2 x \log \sin x] \)

Question. If \( y = \sin^{-1} (6x \sqrt{1-9x^2}), -\frac{1}{3\sqrt{2}} < x < \frac{1}{3\sqrt{2}} \), then find \( \frac{dy}{dx} \).
Answer: \( \frac{6}{\sqrt{1-9x^2}} \)

Question. If \( y = e^{a \cos^{-1} x}, -1 \le x \le 1 \), show that \( (1 - x^2) \frac{d^2 y}{dx^2} - x \frac{dy}{dx} - a^2 y = 0 \).
Answer: Differentiating \( y \), \( y' = e^{a \cos^{-1} x} \cdot \frac{-a}{\sqrt{1-x^2}} \). Rearranging and differentiating again leads to the required differential equation.

Question. If \( x^m y^n = (x + y)^{m + n} \), prove that \( \frac{d^2 y}{dx^2} = 0 \).
Answer: Taking log and differentiating, we find \( \frac{dy}{dx} = \frac{y}{x} \). Differentiating again, \( \frac{d^2 y}{dx^2} = \frac{x y' - y}{x^2} = \frac{x(y/x) - y}{x^2} = 0 \).

Question. If \( y = \log \tan (\frac{\pi}{4} + \frac{x}{2}) \), show that \( \frac{dy}{dx} = \sec x \). Also find the value of \( \frac{d^2 y}{dx^2} \) at \( x = \frac{\pi}{4} \).
Answer: \( \sqrt{2} \)

Question. If \( x = \tan (\frac{1}{a} \log y) \), show that: \( (1 + x^2) \frac{d^2 y}{dx^2} + (2x - a) \frac{dy}{dx} = 0 \).
Answer: Given \( \frac{1}{a} \log y = \tan^{-1} x \Rightarrow y = e^{a \tan^{-1} x} \). Differentiating twice yields the result.

Question. If \( xy = e^{(x - y)} \), then show that \( \frac{dy}{dx} = \frac{y(x - 1)}{x(y + 1)} \).
Answer: Taking log on both sides, \( \log x + \log y = x - y \). Differentiating w.r.t. \( x \) and rearranging terms gives the result.

Question. If \( y = \tan^{-1} \left[ \frac{\sqrt{1+x^2} + \sqrt{1-x^2}}{\sqrt{1+x^2} - \sqrt{1-x^2}} \right], x^2 \le 1 \), then find \( \frac{dy}{dx} \).
Answer: \( -\frac{x}{\sqrt{1-x^4}} \). [Hint: At first simplify by multiplying with \( \sqrt{1+x^2} + \sqrt{1-x^2} \). Then let \( x^2 = \sin \theta \).]

Question. If \( (ax + b)e^{y/x} = x \), then show that \( x^3 \frac{d^2 y}{dx^2} = (x \frac{dy}{dx} - y)^2 \).
Answer: Hint: \( e^{y/x} = \frac{x}{ax + b} \Rightarrow \frac{y}{x} = \log(\frac{x}{ax + b}) \Rightarrow y = x \log(\frac{x}{ax + b}) \). Differentiate twice to solve.

Question. If \( f(x) = \sqrt{x^2+1}; g(x) = \frac{x+1}{x^2+1} \) and \( h(x) = 2x - 3 \), then find \( f'[h'\{g'(x)\}] \).
Answer: \( \frac{2\sqrt{5}}{5} \). [Hint: At first find \( f'(x), g'(x) \) and \( h'(x) \) and then find \( f'[h'\{g'(x)\}] = f'[h'\{\frac{-x^2-2x+1}{(x^2+1)^2}\}] \).]

Question. Let \( f(x) = x - |x - x^2|, x \in [-1, 1] \). Find the point of discontinuity, (if any), of this function on \( [-1, 1] \).
Answer: No point of discontinuity.

Question. If \( \frac{x}{x - y} = \log \frac{a}{x - y} \), then prove that \( \frac{dy}{dx} = 2 - \frac{x}{y} \).
Answer: Hint: \( \frac{x}{x - y} = \log a - \log (x - y) \) then differentiate.

Question. Let \( y = (\log x)^x + x^{x \cos x} \), then find \( \frac{dy}{dx} \).
Answer: \( (\log x)^x \{ \frac{1}{\log x} + \log(\log x) \} + x^{x \cos x} \{ \cos x + \cos x \log x - x \sin x \log x \} \)

Question. If \( e^y (x + 1) = 1 \), then show that \( \frac{d^2 y}{dx^2} = (\frac{dy}{dx})^2 \).
Answer: Differentiating \( e^y(x+1)=1 \) gives \( \frac{dy}{dx} = -\frac{1}{x+1} \). Differentiating again, \( \frac{d^2 y}{dx^2} = \frac{1}{(x+1)^2} = (\frac{dy}{dx})^2 \).

Question. If \( x = a(\theta - \sin \theta), y = a(1 + \cos \theta) \), then find \( \frac{d^2 y}{dx^2} \).
Answer: \( -\frac{1}{a} \text{cosec}^4(\frac{\theta}{2}) \cdot \frac{1}{2} \text{ or } -\frac{1}{2a \sin^4(\theta/2)} \)

Question. If \( y = 2\cos(\log x) + 3\sin(\log x) \), prove that \( x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + y = 0 \).
Answer: Differentiating twice and substituting the derivatives into the expression gives zero.

Question. Show that the function \( f \) given by \( f(x) = \begin{cases} \frac{e^{1/x} - 1}{e^{1/x} + 1}, & \text{if } x \neq 0 \\ -1, & \text{if } x = 0 \end{cases} \) is discontinuous at \( x = 0 \).
Answer: LHL at \( x=0 \) is -1, but RHL at \( x=0 \) is 1. Since LHL \( \neq \) RHL, it is discontinuous.

Question. Find \( \frac{dy}{dx} \) if \( y = \sin^{-1} \left[ \frac{6x - 4\sqrt{1-4x^2}}{5} \right] \).
Answer: \( \frac{2}{\sqrt{1-4x^2}} \)

Question. Differentiate \( (\sin 2x)^x + \sin^{-1} \sqrt{3x} \) with respect to \( x \).
Answer: \( (\sin 2x)^x [2x \cot 2x + \log(\sin 2x)] + \frac{3}{2\sqrt{3x-9x^2}} \)

Question. Differentiate \( \tan^{-1} \left( \frac{\sqrt{1+x^2}-1}{x} \right) \) w.r.t. \( \sin^{-1} \frac{2x}{1+x^2} \), if \( x \in (-1, 1) \).
Answer: \( \frac{1}{4} \)

Question. If \( x = a(\cos 2t + 2t \sin 2t) \) and \( y = a(\sin 2t - 2t \cos 2t) \), then find \( \frac{d^2 y}{dx^2} \).
Answer: \( \frac{\sec^3 2t}{4at} \)

Question. Find the values of \( a \) and \( b \), if the function \( f \) is defined by \( f(x) = \begin{cases} x^2 + 3x + a, & x \le 1 \\ bx + 2, & x > 1 \end{cases} \) is differentiable at \( x = 1 \).
Answer: \( a = 3, b = 5 \)