Continuity, Differentiability and Differentiation - ISC Class 12 Mathematics Questions with Answers
72 past-paper questions on Continuity, Differentiability and Differentiation from ISC Class 12 Mathematics papers (2027-2017), newest first. Open one to see its answer.
- Observe the graph given below: State the value(s) of $x$ where the function has removable discontinuity. State the value(s) of… 2027 · 2 marks · Short answer
- Determine the values of constants $p$ and $q$ such that the function… 2027 · 2 marks · Short answer
- If $x = 3z + 1$ and $y = f(x)$, then prove that $9\frac{d^2y}{dx^2} = \frac{d^2y}{dz^2}$. 2027 · 3 marks · Short answer
- $f(x) = \begin{cases} 1 + x, & x \le 2 \\ 5 - x, & x > 2 \end{cases}$ at $x = 2$ is not differentiable. 2026 · 1 mark · Assertion-reason
- If $y = (A + Bx)e^{-2x}$, prove that: $\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 4y = 0$ (Application) 2026 · 4 marks · Derivation
- If $y = x^3 \log\left(\frac{1}{x}\right)$, then prove that $x\frac{d^2 y}{dx^2} - 2\frac{dy}{dx} + 3x^2 = 0$. 2026 · 4 marks · Derivation
- If $y = x + \tan x$, show that $\cos^2 x \frac{d^2 y}{dx^2} - 2y + 2x = 0$. 2026 · 4 marks · Derivation
- Differentiate $y = x^x$ with respect to $x$. 2026 · 1 mark · Short answer
- Statement 1: The sum, difference and product of two continuous functions are also continuous. Statement 2: The function $f(x)$ is… 2026 · 1 mark · MCQ
- Consider the function $f$ given by $f(x) = \log x$, $x > 0$, then the function $f$ is: 2026 · 1 mark · MCQ
- Consider the graph given below of the function $y = x^{2/3}$ and answer the question that follows: Statement 1: The function is… 2026 · 1 mark · MCQ
- If $x = e^{\frac{x}{y}}$, then prove that $\frac{dy}{dx} = \frac{x-y}{x \log x}$ (Understanding) Find the values of $'a'$ for… 2026 · 2 marks · Derivation
- If $x = e^{y + e^{y + e^{y + \dots \infty}}}$, $x > 0$ then find $\frac{dy}{dx}$. 2026 · 1 mark · Short answer
- If $x^y = y^x$, then find $\frac{dy}{dx}$. [Understanding] 2025 · 2 marks · Short answer
- If $x = a\cos\theta, y = a\sin\theta$, then $\frac{dy}{dx}$ at $\theta = \frac{\pi}{4}$ will be: 2025 · 1 mark · MCQ
- The function defined by $f(x) = \sin^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ is differentiable at $x = 0$. Is this statement true… 2025 · 2 marks · Short answer
- Differentiate $\sin^{-1}\left(\frac{2^{x+1}\cdot 3^x}{1 + (36)^x}\right)$ with respect to $x$. 2025 · 2 marks · Short answer
- If $y = (x + \sqrt{a^2 + x^2})^m$, prove that $(a^2 + x^2)\frac{d^2y}{dx^2} + x\frac{dy}{dx} - m^2y = 0$. [Analysis] 2025 · 4 marks · Derivation
- The figure given below shows the graph of a function $y = f(x)$. What is the derivative of the function? 2025 · 1 mark · Short answer
- The adjacent figure is the graph of function $y = f(x)$. Give answers to the following questions. Name the type of discontinuity… 2025 · 2 marks · Short answer
- Which of the following could be a sketch of the function $y = \frac{d}{dx}(x \log x)$? 2025 · 1 mark · MCQ
- Let $f(x) = \begin{cases} a + \sin^{-1}(x+b), & x \ge 1 \\ x, & x < 1 \end{cases}$, and $f'(1)$ exists. The statement “$f(x)$ is… 2025 · 6 marks · Long answer
- Let $f(x)$ be a function such that $f'(x) = g(x)$ and $f''(x) = -f(x)$. Let $h(x) = \{f(x)\}^2 + \{g(x)\}^2$. Then, consider the… 2025 · 1 mark · MCQ
- Statement I: $f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases}$… 2025 · 1 mark · MCQ
- The function represented by the given graph is not differentiable at which of the following points: 2025 · 1 mark · MCQ
- The derivative of $\tan^{-1}\left(\frac{\cos x - \sin x}{\cos x + \sin x}\right)$ with respect to $x$, where… 2025 · 1 mark · MCQ
- If $y = \sin^{-1}(x\sqrt{x})$, then $\frac{dy}{dx} = \frac{3\sqrt{x}}{2\sqrt{1-x^3}}$. Assertion (A): If… 2025 · 1 mark · Assertion-reason
- If $x = \sin t, y = \sin pt$, prove that $(1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + p^2 y = 0$ 2025 · 4 marks · Derivation
- If $x = \tan\left(\frac{1}{a}\log y\right)$ then show that $(1 + x^2)\frac{d^2y}{dx^2} + (2x - a)\frac{dy}{dx} = 0$ 2025 · 4 marks · Derivation
- If $x^y = e^{x-y}$, prove that $\frac{dy}{dx} = \frac{\log x}{(1+\log x)^2}$ 2025 · 2 marks · Derivation
- If $f(x) = \begin{cases} x+2 & x < 0 \\ -x^2-2 & 0 \le x < 1 \\ x & x \ge 1 \end{cases}$ then the number of point(s) of… 2025 · 1 mark · MCQ
- Find the derivative of $y = \log x + \frac{1}{x}$ with respect to $x$. 2024 · 1 mark · Short answer
- If $y = 3\cos(\log x) + 4\sin(\log x)$, show that $x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + y = 0$ 2024 · 4 marks · Derivation
- Determine the value of $k$ for which the following function is continuous at $x = 3$… 2024 · 2 marks · Numerical
- The graph of the function $f$ is shown below. Of the following options, at what values of $x$ is the function $f$ NOT… 2024 · 1 mark · MCQ
- If $y = (x + \sqrt{1 + x^2})^n$, then prove that $(1 + x^2) \frac{d^2y}{dx^2} + x \frac{dy}{dx} = n^2 y$. 2023 · 4 marks · Derivation
- If $y = e^{ax} \cos bx$, then prove that $\frac{d^2y}{dx^2} - 2a\frac{dy}{dx} + (a^2 + b^2)y = 0$ 2023 · 4 marks · Derivation
- If $f(x) = \frac{4-x^2}{4x-x^3}$ then the function is: 2023 · 1 mark · MCQ
- If the following function is continuous at $x = 2$ then the value of $k$ will be… 2023 · 1 mark · MCQ
- The derivative of $\log x$ with respect to $\frac{1}{x}$ is: 2023 · 1 mark · MCQ
- If $y = \sqrt{\sin x + y}$, then find $\frac{dy}{dx}$. 2023 · 2 marks · Short answer
- If $\sin^{-1} x + \sin^{-1} y = \frac{\pi}{2}$, then $\frac{dy}{dx}$ is equal to 2023 · 1 mark · MCQ
- The second derivative of $y = x^3 - 5x^2 + x$ is: 2022 · 2 marks · MCQ
- The value of $\lim_{x \to 0} \frac{\log(1+x)}{x}$ is equal to: 2022 · 2 marks · MCQ
- If $\sin^{-1} x + \sin^{-1} y = \frac{\pi}{2}$, then $\frac{dy}{dx}$ is equal to: 2022 · 2 marks · MCQ
- If $y = t^2$ and $t = x + 3$ then $\frac{dy}{dx}$ is equal to: 2022 · 2 marks · MCQ
- What will be the derivative of $\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ with respect to… 2022 · 2 marks · MCQ
- The set of points, where the function $f(x) = x |x|$ is differentiable in: 2022 · 2 marks · MCQ
- If $x = \tan\left(\frac{1}{a} \log y\right)$, prove that $(1 + x^2)\frac{d^2y}{dx^2} + (2x - a)\frac{dy}{dx} = 0$ 2021 · 4 marks · Derivation
- Differentiate the function with reference to $x$: $f(x) = \cos^{-1}\left(\sqrt{\frac{1 - \cos x}{2}}\right)$ 2021 · 2 marks · Short answer
- Using L’Hospital’s Rule, evaluate: $\lim_{x \to 0} \frac{8^x - 4^x}{4x}$ 2021 · 2 marks · Short answer
- Find $\frac{dy}{dx}$, if $x = at^2$ and $y = 2at$. 2021 · 2 marks · Short answer
- Find $\frac{dy}{dx}$ if $x^3 + y^3 = 3axy$. 2020 · 2 marks · Short answer
- Using L’Hospital’s rule, evaluate: $\lim_{x \to 0} \frac{xe^x - \log(1 + x)}{x^2}$. 2020 · 4 marks · Numerical
- If $y = e^{m\sin^{-1} x}$, prove that $(1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} = m^2 y$. 2020 · 4 marks · Derivation
- Verify Rolle’s theorem for the function, $f(x) = -1 + \cos x$ in the interval $[0, 2\pi]$. 2020 · 4 marks · Short answer
- Using L'Hospital's Rule, evaluate: $\lim_{x \to 0} \frac{8^x - 4^x}{4x}$ 2019 · 2 marks · Short answer
- Show that the function $f(x) = |x - 4|$, $x \in \mathbb{R}$ is continuous, but not differentiable at $x = 4$. 2019 · 4 marks · Derivation
- $f(x) = \frac{x^2 - 9}{x - 3}$ is not defined at $x = 3$. What value should be assigned to $f(3)$ for continuity of $f(x)$ at… 2019 · 2 marks · Short answer
- Verify the Lagrange's mean value theorem for the function: $f(x) = x + \frac{1}{x}$ in the interval $[1, 3]$ 2019 · 4 marks · Short answer
- If $y = e^{\sin^{-1} x}$ and $z = e^{-\cos^{-1} x}$, prove that $\frac{dy}{dz} = e^{\frac{\pi}{2}}$ 2019 · 4 marks · Derivation
- Verify Rolle’s Theorem for the following function: $f(x) = e^x \sin x, x \in [0, \pi]$ 2018 · 4 marks · Short answer
- Verify Rolle’s theorem for the following function: $f(x) = e^{-x} \sin x$ on $[0, \pi]$. 2018 · 4 marks · Short answer
- Find the value of constant $k$ so that the function $f(x)$ defined as… 2018 · 2 marks · Numerical
- Find $\frac{dy}{dx}$, if $x = at^2$ and $y = 2at$. 2018 · 2 marks · Short answer
- Prove that the function $f(x) = |x - 1|, x \in \mathbb{R}$, is continuous at $x = 1$ but not differentiable. 2018 · 4 marks · Derivation
- If $x = \tan\left(\frac{1}{a} \log y\right)$, prove that $(1 + x^2) \frac{d^2y}{dx^2} + (2x - a) \frac{dy}{dx} = 0$. 2018 · 4 marks · Derivation
- Show that the function $f(x) = \begin{cases} x^2, & x \le 1 \\ \frac{1}{x}, & x > 1 \end{cases}$ is continuous at $x = 1$ but not… 2018 · 4 marks · Derivation
- If $y = e^{a \cos^{-1} x}$, where $-1 \leq x \leq 1$, then show that: $(1 - x^2) y_2 - xy_1 - a^2y = 0$ 2018 · 4 marks · Derivation
- Verify Lagrange’s mean value theorem for the function: $f(x) = x(1 - \log x)$ and find the value of ‘c’ in the interval $[1, 2]$. 2017 · 5 marks · Numerical
- Using L’Hospital’s Rule, evaluate: $\lim_{x \to \pi/2} \left(x \tan x - \frac{\pi}{2} \sec x\right)$ 2017 · 3 marks · Short answer
- If $y = \cos(\sin x)$, show that: $\frac{d^2y}{dx^2} + \tan x \frac{dy}{dx} + y \cos^2 x = 0$ 2017 · 5 marks · Derivation
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