1
JEE Main 2024 (Online) 30th January Morning Shift
+4
-1

If $$f(x)=\left|\begin{array}{ccc} 2 \cos ^4 x & 2 \sin ^4 x & 3+\sin ^2 2 x \\ 3+2 \cos ^4 x & 2 \sin ^4 x & \sin ^2 2 x \\ 2 \cos ^4 x & 3+2 \sin ^4 x & \sin ^2 2 x \end{array}\right|,$$ then $$\frac{1}{5} f^{\prime}(0)=$$ is equal to :

A
2
B
1
C
0
D
6
2
JEE Main 2024 (Online) 29th January Evening Shift
+4
-1

$$\text { Let } y=\log _e\left(\frac{1-x^2}{1+x^2}\right),-1 < x<1 \text {. Then at } x=\frac{1}{2} \text {, the value of } 225\left(y^{\prime}-y^{\prime \prime}\right) \text { is equal to }$$

A
732
B
736
C
742
D
746
3
JEE Main 2024 (Online) 29th January Morning Shift
+4
-1

Suppose $$f(x)=\frac{\left(2^x+2^{-x}\right) \tan x \sqrt{\tan ^{-1}\left(x^2-x+1\right)}}{\left(7 x^2+3 x+1\right)^3}$$. Then the value of $$f^{\prime}(0)$$ is equal to

A
$$\pi$$
B
$$\sqrt{\pi}$$
C
0
D
$$\frac{\pi}{2}$$
4
JEE Main 2023 (Online) 13th April Morning Shift
+4
-1

For the differentiable function $$f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$$, let $$3 f(x)+2 f\left(\frac{1}{x}\right)=\frac{1}{x}-10$$, then $$\left|f(3)+f^{\prime}\left(\frac{1}{4}\right)\right|$$ is equal to

A
13
B
$$\frac{29}{5}$$
C
$$\frac{33}{5}$$
D
7
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