Which of the following could be a sketch of the function $y = \frac{d}{dx}(x \log x)$?
- (a)Figure (a)
- (b)Figure (b)
- (c)Figure (c)
- (d)Figure (d)
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Continuity, Differentiability and Differentiation - ISC Class 12 Mathematics Questions with Answers, Page 2
72 past-paper questions on Continuity, Differentiability and Differentiation from ISC Class 12 Mathematics papers (2027-2017), newest first, in full. Questions 21-40 are on this page, 20 to a page. Tap "Show answer" under a question to see its answer.
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Let $f(x) = \begin{cases} a + \sin^{-1}(x+b), & x \ge 1 \\ x, & x < 1 \end{cases}$, and $f'(1)$ exists.
(a)
The statement “$f(x)$ is continuous at $x = 1$” is true. Justify.
(b)
Hence, find a relation between $a$ and $b$.
(c)
Find $f'(x)$.
(d)
Hence, find the values of $a$ and $b$.
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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 following statements:
Statement I: $h'(2024) = 0$.
Statement II: $h(2) = h\left(\frac{1}{2}\right)$
Which of the statements given above is/are correct?
- (a)I only.
- (b)II only
- (c)Both I and II
- (d)Neither I nor II.
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Statement I: $f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases}$ is continuous at $x = 0$ but $f'(x)$ is not continuous at $x = 0$.
Statement II: The derivative of a continuous function need not be a continuous function.
- (a)Both (I) and (II) are correct and (II) is the correct explanation of (I).
- (b)Both (I) and (II) are correct and (II) is not the correct explanation of (I).
- (c)(I) is correct but (II) is incorrect.
- (d)(II) is correct but (I) is incorrect.
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The function represented by the given graph is not differentiable at which of the following points:
- (a)$-1, 0, 1$
- (b)$-1, 1$
- (c)$0$
- (d)$1$
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The derivative of $\tan^{-1}\left(\frac{\cos x - \sin x}{\cos x + \sin x}\right)$ with respect to $x$, where $x \in \left(0, \frac{\pi}{2}\right)$ is:
- (a)$\frac{\pi}{4}$
- (b)1
- (c)$-\frac{\pi}{4}$
- (d)-1
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Assertion: If $y = \sin^{-1}(x\sqrt{x})$, then $\frac{dy}{dx} = \frac{3\sqrt{x}}{2\sqrt{1-x^3}}$.
Reason: $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}, |x| \le 1$.
Assertion (A): If $y = \sin^{-1}(x\sqrt{x})$, then $\frac{dy}{dx} = \frac{3\sqrt{x}}{2\sqrt{1-x^3}}$.
Reason (R): $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}, |x| \le 1$.
- (a)Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
- (b)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of A.
- (c)Assertion (A) is true, but Reason (R) is false.
- (d)Assertion (A) is false, but Reason (R) is true.
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If $x = \sin t, y = \sin pt$, prove that $(1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + p^2 y = 0$
Given: $x = \sin t, y = \sin pt$
To show: $(1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + p^2 y = 0$
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If $x = \tan\left(\frac{1}{a}\log y\right)$ then show that $(1 + x^2)\frac{d^2y}{dx^2} + (2x - a)\frac{dy}{dx} = 0$
Given: $x = \tan\left(\frac{1}{a} \log y\right)$
To show: $(1 + x^2)\frac{d^2 y}{dx^2} + (2x - a)\frac{dy}{dx} = 0$
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If $x^y = e^{x-y}$, prove that $\frac{dy}{dx} = \frac{\log x}{(1+\log x)^2}$
Given: $x^y = e^{x-y}$
To show: $\frac{dy}{dx} = \frac{\log x}{(1+\log x)^2}$
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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 discontinuity of $f(x)$, is / are:
- (a)$1$
- (b)$3$
- (c)$2$
- (d)$0$
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Find the derivative of $y = \log x + \frac{1}{x}$ with respect to $x$.
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If $y = 3\cos(\log x) + 4\sin(\log x)$, show that $x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + y = 0$
Given: $y = 3\cos(\log x) + 4\sin(\log x)$
To show: $x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + y = 0$
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Determine the value of $k$ for which the following function is continuous at $x = 3$.
$f(x) = \begin{cases} \frac{(x + 3)^2 - 36}{x - 3} & ; x \neq 3 \\ k & ; x = 3 \end{cases}$
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The graph of the function $f$ is shown below.
Of the following options, at what values of $x$ is the function $f$ NOT differentiable?
- (a)At $x = 0$ and $x = 2$
- (b)At $x = 1$ and $x = 3$
- (c)At $x = -1$ and $x = 1$
- (d)At $x = -1.5$ and $x = 1.5$
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If $y = (x + \sqrt{1 + x^2})^n$, then prove that $(1 + x^2) \frac{d^2y}{dx^2} + x \frac{dy}{dx} = n^2 y$.
Given: $y = (x + \sqrt{1 + x^2})^n$
To show: $(1 + x^2) \frac{d^2y}{dx^2} + x \frac{dy}{dx} = n^2 y$
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If $y = e^{ax} \cos bx$, then prove that $\frac{d^2y}{dx^2} - 2a\frac{dy}{dx} + (a^2 + b^2)y = 0$
Given: $y = e^{ax} \cos bx$
To show: $\frac{d^2y}{dx^2} - 2a\frac{dy}{dx} + (a^2 + b^2)y = 0$
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If $f(x) = \frac{4-x^2}{4x-x^3}$ then the function is:
- (a)discontinuous at only one point.
- (b)discontinuous at exactly two points.
- (c)discontinuous at exactly three points.
- (d)discontinuous at exactly four points.
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If the following function is continuous at $x = 2$ then the value of $k$ will be:
$f(x) = \begin{cases} 2x + 1, & \text{if } x < 2 \\ k, & \text{if } x = 2 \\ 3x - 1, & \text{if } x > 2 \end{cases}$
- (a)$2$
- (b)$3$
- (c)$5$
- (d)$-1$
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The derivative of $\log x$ with respect to $\frac{1}{x}$ is:
- (a)$\frac{1}{x}$
- (b)$-\frac{1}{x^3}$
- (c)$-\frac{1}{x}$
- (d)$-x$
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