1
MHT CET 2023 13th May Morning Shift
+2
-0

Differentiation of $$\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$$ w.r.t. $$\cos ^{-1}\left(\sqrt{\frac{1+\sqrt{1+x^2}}{2 \sqrt{1+x^2}}}\right)$$ is

A
$$\frac{1}{2}$$
B
1
C
2
D
$$\frac{1}{4}$$
2
MHT CET 2023 13th May Morning Shift
+2
-0

If $$y=\tan ^{-1}\left(\frac{4 \sin 2 x}{\cos 2 x-6 \sin ^2 x}\right)$$, then $$\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)$$ at $$x=0$$ is

A
3
B
5
C
8
D
1
3
MHT CET 2023 13th May Morning Shift
+2
-0

Let $$\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$$ be a function such that $$\mathrm{f}(x)=x^3+x^2 \mathrm{f}^{\prime}(1)+x \mathrm{f}^{\prime \prime}(2)+6, x \in \mathrm{R}$$, then $$\mathrm{f}(2)$$ is

A
30
B
$$-$$4
C
$$-$$2
D
8
4
MHT CET 2023 12th May Evening Shift
+2
-0

If $$\mathrm{f}(x)=\sin ^{-1}\left(\frac{2 \log x}{1+(\log x)^2}\right)$$, then $$\mathrm{f}^{\prime}(\mathrm{e})$$ is

A
$$\frac{2}{\mathrm{e}}$$
B
$$\frac{1}{2 \mathrm{e}}$$
C
$$\mathrm{e}$$
D
$$\frac{1}{\mathrm{e}}$$
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