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

Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be a function defined by $$f(x)=\frac{x}{\left(1+x^4\right)^{1 / 4}}$$, and $$g(x)=f(f(f(f(x))))$$. Then, $$18 \int_0^{\sqrt{2 \sqrt{5}}} x^2 g(x) d x$$ is equal to

A
36
B
33
C
39
D
42
2
JEE Main 2024 (Online) 30th January Evening Shift
+4
-1

Let $$y=f(x)$$ be a thrice differentiable function in $$(-5,5)$$. Let the tangents to the curve $$y=f(x)$$ at $$(1, f(1))$$ and $$(3, f(3))$$ make angles $$\pi / 6$$ and $$\pi / 4$$, respectively with positive $$x$$-axis. If $$27 \int_\limits1^3\left(\left(f^{\prime}(t)\right)^2+1\right) f^{\prime \prime}(t) d t=\alpha+\beta \sqrt{3}$$ where $$\alpha, \beta$$ are integers, then the value of $$\alpha+\beta$$ equals

A
26
B
$$-$$16
C
36
D
$$-$$14
3
JEE Main 2024 (Online) 30th January Evening Shift
+4
-1

Let $$a$$ and $$b$$ be real constants such that the function $$f$$ defined by $$f(x)=\left\{\begin{array}{ll}x^2+3 x+a & , x \leq 1 \\ b x+2 & , x>1\end{array}\right.$$ be differentiable on $$\mathbb{R}$$. Then, the value of $$\int_\limits{-2}^2 f(x) d x$$ equals

A
21
B
19/6
C
17
D
15/6
4
JEE Main 2024 (Online) 30th January Evening Shift
+4
-1

Let $$\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$$ be defined as $$f(x)=a e^{2 x}+b e^x+c x$$. If $$f(0)=-1, f^{\prime}\left(\log _e 2\right)=21$$ and $$\int_0^{\log _e 4}(f(x)-c x) d x=\frac{39}{2}$$, then the value of $$|a+b+c|$$ equals

A
16
B
12
C
8
D
10
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