Differentiation · Mathematics · AP EAPCET

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MCQ (Single Correct Answer)

1
If $y=\sinh ^{-1}\left(\frac{1-x}{1+x}\right)$, then $\frac{d y}{d x}$ is equal to
AP EAPCET 2024 - 23th May Morning Shift
2
If $y=(x-1)(x+2)\left(x^2+5\right)\left(x^4+8\right)$, then $\lim _{x \rightarrow-1}\left(\frac{d y}{d x}\right)$ is equal to
AP EAPCET 2024 - 23th May Morning Shift
3
If $y=\left(\tan ^{-1} 2 x\right)^2+\left(\cot ^{-1} 2 x\right)^2$, then $\left(1+4 x^2\right)^2 y^{\prime \prime}-16$ is equal to
AP EAPCET 2024 - 23th May Morning Shift
4
If $y=\tan ^{-1} \frac{x}{1+2 x^2}+\tan ^{-1} \frac{x}{1+6 x^2}+\tan ^{-1} \frac{x}{1+12 x^2}$, then $\left(\frac{d y}{d x}\right)_{x=\frac{1}{2}}$ is equal to
AP EAPCET 2024 - 22th May Evening Shift
5

If $f(x)=5 \cos ^3 x-3 \sin ^2 x$ and $g(x)=4 \sin ^3 x+\cos ^2 x$, then the derivative of $f(x)$ with respect to $g(x)$ is

AP EAPCET 2024 - 22th May Evening Shift
6
If $y=1+x+x^2+x^3+\ldots \ldots \infty$ and $|x|<1$, then $y^{\prime \prime}$ is equal to
AP EAPCET 2024 - 22th May Evening Shift
7

    If $y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+\ldots \infty}}}$, then the value of $\frac{d^2 y}{d x^2}$ at the point $(\pi, 1)$ is

AP EAPCET 2024 - 22th May Morning Shift
8
64. If $f(0)=0, f^{\prime}(0)=3$, then the derivative of $y=f(f(f(f(f(x)))))$ at $x=0$ is
AP EAPCET 2024 - 22th May Morning Shift
9
If $\frac{d}{d x}\left(\frac{1+x^2+x^4}{1+x+x^2}\right)=a x+b$, then $(a, b)=$
AP EAPCET 2024 - 21th May Evening Shift
10
The rate of change of $x^{\sin x}$ with respect to $(\sin x)^x$ is
AP EAPCET 2024 - 21th May Morning Shift
11
If $y=\frac{\alpha x+\beta}{\gamma \alpha+\delta}$, then $2 y_1 y_3=$
AP EAPCET 2024 - 21th May Morning Shift
12
Which one of the following is false ?
AP EAPCET 2024 - 21th May Morning Shift
13
If $y=t^2+t^3$ and $x=t-t^4$, then $\frac{d^2 y}{d x^2}$ at $t=1$ is
AP EAPCET 2024 - 20th May Evening Shift
14
If $y=\tan (\log x)$, then $\frac{d^2 y}{d x^2}=$
AP EAPCET 2024 - 20th May Morning Shift
15
For $x<0, \frac{d}{d x}\left[|x|^x\right]=$
AP EAPCET 2024 - 20th May Morning Shift
16
If $y=x-x^2$, then the rate of change of $y^2$ with respect to $x^2$ at $x=2$ is
AP EAPCET 2024 - 20th May Morning Shift
17
If $y=f(x)$ is a thrice differentiable function and a bijection, then $\frac{d^2 x}{d y^2}\left(\frac{d y}{d x}\right)^3+\frac{d^2 y}{d x^2}=$
AP EAPCET 2024 - 19th May Evening Shift
18
If $y=\tan ^{-1}\left(\frac{2-3 \sin x}{3-2 \sin x}\right)$, then $\frac{d y}{d x}=$
AP EAPCET 2024 - 18th May Morning Shift
19
If $x=3\left[\sin t-\log \left(\cot \frac{t}{2}\right)\right]$ and $y=6\left[\cos t+\log \left(\operatorname{tin} \frac{t}{2}\right)\right]$ then $\frac{d y}{d x}=$
AP EAPCET 2024 - 18th May Morning Shift
20
The length of the tangent drawn at the point $P\left(\frac{\pi}{4}\right)$ on the curve $x^{2 / 3}+y^{2 / 3}=2^{2 / 3}$ is
AP EAPCET 2024 - 18th May Morning Shift
21

Assertion (A) $$\frac{d}{d x}\left(\frac{x^2 \sin x}{\log x}\right)=\frac{x^2 \sin x}{\log x}\left(\cot x+\frac{2}{x}-\frac{1}{x \log x}\right)$$

Reason (R) $$\frac{d}{d x}\left(\frac{u v}{w}\right)=\frac{u v}{w}\left[\frac{u^{\prime}}{u}+\frac{v^{\prime}}{v}+\frac{w^{\prime}}{w}\right]$$

AP EAPCET 2022 - 5th July Morning Shift
22

If $$x=f(\theta)$$ and $$y=g(\theta)$$, then $$\frac{d^2 y}{d x^2}=$$

AP EAPCET 2022 - 5th July Morning Shift
23

$$y=x^3-a x^2+48 x+7$$ is an increasing function for all real values of $$x$$, then $$a$$ lies in the interval

AP EAPCET 2022 - 5th July Morning Shift
24

If $$x \neq 0$$ and $$f(x)$$ satisfies $$8 f(x)+6 f(1 / x) =x+5$$, then $$\frac{d}{d x}\left(x^2 f(x)\right)$$ at $$x=1$$ is

AP EAPCET 2022 - 4th July Evening Shift
25

If $$f(x)=\cot ^{-1}\left(\frac{x^x+x^{-x}}{2}\right)$$, then $$f^{\prime}(1)=$$

AP EAPCET 2022 - 4th July Morning Shift
26

If $$f(x)=2x^2+3x-5$$, then the value of $$f'(0)+3f'(-1)$$ is equal to

AP EAPCET 2021 - 20th August Morning Shift
27

If $$y=\left(1+\frac{1}{x}\right)\left(1+\frac{2}{x}\right)\left(1+\frac{3}{x}\right) \ldots\left(1+\frac{n}{x}\right)$$ and $$x \neq 0$$. When $$x=-1, \frac{d y}{d x}$$ is equal to

AP EAPCET 2021 - 20th August Morning Shift
28

If $$\log \left(\sqrt{1+x^2}-x\right)=y\left(\sqrt{1+x^2}\right)$$, then $$\left(1+x^2\right) \frac{d y}{d x}+x y$$ is equal to

AP EAPCET 2021 - 19th August Evening Shift
29

If $$y=e^{x^2+e^{x^2+e^{x^2+\cdots \infty}}}$$, then $$\frac{d y}{d x}$$ is equal to

AP EAPCET 2021 - 19th August Evening Shift
30

$$\frac{d}{d x}\left[\tan ^{-1}\left(\frac{\cos x}{1+\sin x}\right)\right]$$ is equal to

AP EAPCET 2021 - 19th August Evening Shift
31

If $$x^2+y^2=1$$, then

AP EAPCET 2021 - 19th August Evening Shift
32

If $$y=x+\frac{1}{x}$$, then which among the following holds?

AP EAPCET 2021 - 19th August Morning Shift
33

If $$3 \sin x y+4 \cos x y=5$$, then $$\frac{d y}{d x}$$ is equal to

AP EAPCET 2021 - 19th August Morning Shift
34

$$f(x)=\sqrt{x^2+1}: g(x)=\frac{x+1}{x^2+1}: h(x)=2 x-3$$, then the value of $$f^{\prime}\left[h^{\prime}\left(g^{\prime}(x)\right)\right]$$ is equal to

AP EAPCET 2021 - 19th August Morning Shift
35

For which value(s) of $$a$$ $$f(x)=-x^3+4 a x^2+2 x-5$$ is decreasing for every $$x$$ ?

AP EAPCET 2021 - 19th August Morning Shift
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