1
JEE Main 2024 (Online) 5th April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If $$y(\theta)=\frac{2 \cos \theta+\cos 2 \theta}{\cos 3 \theta+4 \cos 2 \theta+5 \cos \theta+2}$$, then at $$\theta=\frac{\pi}{2}, y^{\prime \prime}+y^{\prime}+y$$ is equal to :

A
$$\frac{1}{2}$$
B
1
C
$$\frac{3}{2}$$
D
2
2
JEE Main 2024 (Online) 4th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$f(x)=x^5+2 \mathrm{e}^{x / 4}$$ for all $$x \in \mathbf{R}$$. Consider a function $$g(x)$$ such that $$(g \circ f)(x)=x$$ for all $$x \in \mathbf{R}$$. Then the value of $$8 g^{\prime}(2)$$ is :

A
4
B
2
C
16
D
8
3
JEE Main 2024 (Online) 30th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$$ be a function satisfying $$f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$$ for all $$x, y, f(y) \neq 0$$. If $$f^{\prime}(1)=2024$$, then

A
$$x f^{\prime}(x)+2024 f(x)=0$$
B
$$x f^{\prime}(x)-2023 f(x)=0$$
C
$$x f^{\prime}(x)-2024 f(x)=0$$
D
$$x f^{\prime}(x)+f(x)=2024$$
4
JEE Main 2024 (Online) 30th January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

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 $$(0,1)$$
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