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

Let $$g: \mathbf{R} \rightarrow \mathbf{R}$$ be a non constant twice differentiable function such that $$\mathrm{g}^{\prime}\left(\frac{1}{2}\right)=\mathrm{g}^{\prime}\left(\frac{3}{2}\right)$$. If a real valued function $$f$$ is defined as $$f(x)=\frac{1}{2}[g(x)+g(2-x)]$$, then

A
$$f^{\prime \prime}(x)=0$$ for atleast two $$x$$ in $$(0,2)$$
B
$$f^{\prime}\left(\frac{3}{2}\right)+f^{\prime}\left(\frac{1}{2}\right)=1$$
C
$$f^{\prime \prime}(x)=0$$ for no $$x$$ in $$(0,1)$$
D
$$f^{\prime \prime}(x)=0$$ for exactly one $$x$$ in $$(0,1)$$
2
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
3
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
4
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
$$\frac{\pi}{2}$$
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