1
MHT CET 2023 10th May Evening Shift
+2
-0

Let $$f$$ be a differentiable function such that $$\mathrm{f}(1)=2$$ and $$\mathrm{f}^{\prime}(x)=\mathrm{f}(x)$$, for all $$x \in \mathrm{R}$$. If $$\mathrm{h}(x)=\mathrm{f}(\mathrm{f}(x))$$, then $$\mathrm{h}^{\prime}(1)$$ is equal to

A
$$4 \mathrm{e}^2$$
B
$$4 \mathrm{e}$$
C
$$2 \mathrm{e}$$
D
$$2 \mathrm{e}^2$$
2
MHT CET 2023 10th May Evening Shift
+2
-0

If $$y$$ is a function of $$x$$ and $$\log (x+y)=2 x y$$, then $$\frac{d y}{d x}$$ at $$x=0$$ is

A
0
B
$$-$$1
C
1
D
2
3
MHT CET 2023 10th May Evening Shift
+2
-0

If $$x=3 \tan \mathrm{t}$$ and $$y=3 \sec \mathrm{t}$$, then the value of $$\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}$$ at $$\mathrm{t}=\frac{\pi}{4}$$ is

A
$$\frac{-1}{6 \sqrt{2}}$$
B
$$\frac{1}{6 \sqrt{2}}$$
C
$$\frac{1}{3 \sqrt{2}}$$
D
$$\frac{3}{2 \sqrt{2}}$$
4
MHT CET 2023 10th May Evening Shift
+2
-0

If $$y=\tan ^{-1}\left(\frac{\log \left(\frac{\mathrm{e}}{x^2}\right)}{\log \left(e x^2\right)}\right)+\tan ^{-1}\left(\frac{4+2 \log x}{1-8 \log x}\right)$$, then $$\frac{\mathrm{d} y}{\mathrm{~d} x}$$ is

A
0
B
$$\frac{1}{2}$$
C
$$\frac{1}{4}$$
D
1
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