1
MHT CET 2021 24th September Morning Shift
+2
-0

If $$y=\log _{10} x+\log _x 10+\log _x x+\log _{10} 10$$, then $$\frac{d y}{d x}=$$

A
$$\frac{1}{x \log _e 10}+\frac{1}{x \log _{10} e}$$
B
$$\frac{1}{x \log _e 10}+\frac{\log _e 10}{x\left(\log _{10} e\right)^2}$$
C
$$\frac{1}{x \log _e 10}-\frac{1}{x \log _{10} e}$$
D
$$\frac{1}{x \log _e 10}-\frac{\log _e 10}{x\left(\log _e x\right)^2}$$
2
MHT CET 2021 23rd September Evening Shift
+2
-0

If $$y=x \tan y$$, then $$\frac{d y}{d x}=$$

A
$$\frac{\tan x}{x-y^2}$$
B
$$\frac{y}{x-x^2-y^2}$$
C
$$\frac{\tan x}{x-x^2-y^2}$$
D
$$\frac{\tan y}{y-x}$$
3
MHT CET 2021 23rd September Evening Shift
+2
-0

The derivative of the function $$\cot ^{-1}\left[(\cos 2 x)^{1 / 2}\right]$$ at $$x=\pi / 6$$ is

A
$$\left(\frac{1}{3}\right)^{1 / 2}$$
B
$$\left(\frac{2}{3}\right)^{1 / 2}$$
C
$$\left(\frac{3}{2}\right)^{1 / 2}$$
D
$$(3)^{1 / 2}$$
4
MHT CET 2021 23rd September Evening Shift
+2
-0

For all real $$x$$, the minimum value of the function $$f(x)=\frac{1-x+x^2}{1+x+x^2}$$ is

A
$$\frac{1}{3}$$
B
0
C
3
D
1
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