1
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
$$\frac{3}{2 \sqrt{2}}$$
2
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
3
MHT CET 2023 10th May Morning Shift
+2
-0

If $$y=\cos ^{-1}\left(\frac{\mathrm{a}^2}{\sqrt{x^4+\mathrm{a}^4}}\right)$$, then $$\frac{\mathrm{d} y}{\mathrm{~d} x}$$ is

A
$$\frac{2 a^2 x}{x^4+a^4}$$
B
$$\frac{2 a^2 x^2}{\sqrt{x^4+a^4}}$$
C
$$\frac{a^4 x^4}{x^4+a^4}$$
D
$$\frac{a^4 x^2}{2 \sqrt{x^4+a^4}}$$
4
MHT CET 2023 10th May Morning Shift
+2
-0

For $$x>1$$, if $$(2 x)^{2 y}=4 \mathrm{e}^{2 x-2 y}$$, then $$(1+\log 2 x)^2 \frac{\mathrm{d} y}{\mathrm{~d} x}$$ is equal to

A
$$\frac{x \log 2 x+\log 2}{x}$$
B
$$\frac{x \log 2 x-\log 2}{x}$$
C
$$x \log 2 x$$
D
$$\log 2 x$$
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