CUET Mathematics: CalculusContinuity & Differentiability. Free, no login required.

Let $e^y(x+1) = 1$. Then which of the following are TRUE?

(A) $\frac{d^2y}{dx^2} = -\frac{1}{(x+1)^2}$

(B) $\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2$

(C) $\left.\frac{d^2y}{dx^2}\right|_{x=0} = -1$

(D) $\left.\frac{d^2y}{dx^2}\right|_{x=0} = 1$

(E) $\left.\frac{d^2y}{dx^2}\right|_{x=1} = \frac{1}{4}$

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Let $f(x)=\begin{cases}\dfrac{|x|}{x},&x\ne0\\1,&x=0\end{cases}$ and $g(x)=\begin{cases}x\sin\left(\dfrac{1}{x}\right),&x\ne0\\0,&x=0\end{cases}$

Then at the origin, which one of the following is true?

If $x = \frac{1-t}{1+t}$ and $y = \frac{3t}{1+t}$, then $\frac{d^2y}{dx^2}$ is equal to

If $y = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0 < x < 1$, then $\frac{dy}{dx}$ is equal to

If $f(x) = x^3 e^{-x}$, then the value of $f''(1)$ is equal to

If $x^2 - y^2 = 1$, then which of the following is correct?

(A) $(x^2 - 1)\left(\frac{dy}{dx}\right)^2 = x^2$

(B) $(x^2 - 1)\left(\frac{d^2y}{dx^2}\right)^2 = x^2$

(C) $(x^2 - 1)^3\left(\frac{d^2y}{dx^2}\right)^2 = x^2$

(D) $(x^2 - 1)^3\left(\frac{d^2y}{dx^2}\right)^2 = 1$

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If $f(x) = x^3\log_e x$, Then $f''(e^2)$ is equal to

Match List-I with List-II

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If $y = 3^x + e^x + x^x + x^3$, then the value of $\frac{dy}{dx}$ at $x = 3$ is

The function $f(x) = [x]$, where $[x]$ denotes the greatest integer function, is continuous at $x =$

(A) 2.9

(B) 5

(C) -3

(D) 6.5

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If $y = \sqrt{2024x + 2025}$, then which of the following is correct?

Match List-I with List-II

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If $t=e^{2 x}$ and $y=\log _{e} t^{2}$, then $\frac{d^{2} y}{d x^{2}}$ is :

The second order derivative of which of the following functions is $5^{\mathrm{x}}$ ?

Let $[x]$ denote the greatest integer function. Then match List-I with List-II:

If $f(x)$, defined by $f(x)=\left\{\begin{array}{lll}k x+1 & \text { if } & x \leq \pi \\ \cos x & \text { if } & x>\pi\end{array}\right.$ is continuous at $x=\pi$, then the value of $k$ is :

If $\mathrm{e}^{\mathrm{y}}=\mathrm{x}^{\mathrm{x}}$, then which of the following is true ?

If $y = x^x$, $\frac{dy}{dx}$ will be:

If $y = \sin^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ then $\frac{dy}{dx} =$

Points of discontinuity of the greatest integer function $f(x) = [x]$, where $[x]$ denotes integer less than or equal to $x$, are

If $y = x^{(x \sin x)}$ then $\frac{dy}{dx} = ?$

If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$, then which of the following can be a discontinuous function ?

If $f(x) = \begin{cases} ax^2 + b, & x < -1 \\ bx^2 + ax + 4, & x \geq -1 \end{cases}$ is everywhere differentiable, then :

Match List - I with List - II.

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Let $y = \log_e \left(\frac{a + b \sin x}{a - b \sin x}\right)$, then value of $\frac{dy}{dx}$ is :

Match List - I with List - II.

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The given function $f(x) = [x]$ is discontinuous at :

If $x = a\left(t - \frac{1}{t}\right)$, $y = b\left(t + \frac{1}{t}\right)$, then $\frac{dy}{dx} =$

Which of the following statements are correct ?

(A) If $f : R \to R$ then $f(x) = |x|$ is continuous everywhere.

(B) If $f : R \to R$ then $f(x) = |x|$ is continuous everywhere but not differentiable at $x = 0$.

(C) Let $f : R - \{0\} \to R$ then $f(x) = \frac{1}{x}$ is continuous everywhere.

(D) Let $f : R \to R$ then $f(x) = |x - 1| + |x - 2|$ is continuous everywhere but not differentiable at exactly 2 points.

(E) If $f : R \to R$ then $f(x) = \cot x$ is continuous everywhere.

Choose the correct answer from the options given below :