1
JEE Main 2024 (Online) 31st January Evening Shift
+4
-1

If the function $$f:(-\infty,-1] \rightarrow(a, b]$$ defined by $$f(x)=e^{x^3-3 x+1}$$ is one - one and onto, then the distance of the point $$P(2 b+4, a+2)$$ from the line $$x+e^{-3} y=4$$ is :

A
$$2 \sqrt{1+e^6}$$
B
$$\sqrt{1+e^6}$$
C
$$3 \sqrt{1+e^6}$$
D
$$4 \sqrt{1+e^6}$$
2
JEE Main 2024 (Online) 31st January Morning Shift
+4
-1

$$\text { If } f(x)=\left|\begin{array}{ccc} x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{array}\right| \text { for all } x \in \mathbb{R} \text {, then } 2 f(0)+f^{\prime}(0) \text { is equal to }$$

A
24
B
18
C
42
D
48
3
JEE Main 2024 (Online) 30th January Evening Shift
+4
-1

Let $$f(x)=(x+3)^2(x-2)^3, x \in[-4,4]$$. If $$M$$ and $$m$$ are the maximum and minimum values of $$f$$, respectively in $$[-4,4]$$, then the value of $$M-m$$ is

A
108
B
392
C
608
D
600
4
JEE Main 2024 (Online) 30th January Morning Shift
+4
-1

The maximum area of a triangle whose one vertex is at $$(0,0)$$ and the other two vertices lie on the curve $$y=-2 x^2+54$$ at points $$(x, y)$$ and $$(-x, y)$$, where $$y>0$$, is :

A
108
B
122
C
88
D
92
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