JEE MAIN 2022 Previous Year Questions (PYQs) – Page 8 of 31

The value of the integral $\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {(1 + {e^x})({{\sin }^6}x + {{\cos }^6}x)}}} $ is equal to

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with :

Let $S_{1} = \{ z_{1} \in \mathbb{C} : |z_{1} - 3| = \tfrac{1}{2} \}$ and $S_{2} = \{ z_{2} \in \mathbb{C} : |z_{2} - |z_{2} + 1|| = |z_{2} + |z_{2} - 1|| \}$. Then, for $z_{1} \in S_{1}$ and $z_{2} \in S_{2}$, the least value of $|z_{2} - z_{1}|$ is :

Let the maximum area of the triangle that can be inscribed in the ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 4} = 1,\,a > 2$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be $6\sqrt 3 $. Then the eccentricity of the ellipse is :

If the minimum value of $f(x)=\dfrac{5x^{2}}{2}+\dfrac{\alpha}{x^{5}},; x>0,$ is $14$, then the value of $\alpha$ is equal to:

Let the area of the triangle with vertices $A(1,\alpha)$, $B(\alpha,0)$ and $C(0,\alpha)$ be $4$ sq. units. If the points $(\alpha,-\alpha)$, $(-\alpha,\alpha)$ and $(\alpha^2,\beta)$ are collinear, then $\beta$ is equal to:

$ \text{Let } \alpha, \beta \text{ and } \gamma \text{ be three positive real numbers. Let } f(x) = \alpha x^{5} + \beta x^{3} + \gamma x,; x \in \mathbb{R} \text{ and } g : \mathbb{R} \to \mathbb{R} \text{ be such that } g(f(x)) = x \text{ for all } x \in \mathbb{R}. \text{ If } a_{1}, a_{2}, a_{3}, \ldots, a_{n} \text{ be in arithmetic progression with mean zero, then the value of } f!\left(g!\left(\frac{1}{n}\sum_{i=1}^{n} f(a_{i})\right)\right) \text{ is equal to:}$

The number of distinct real roots of the equation

x⁷ $-$ 7x $-$ 2 = 0 is

The minimum value of the twice differentiable function $f(x)=\int_{0}^{x} e^{,x-t},f'(t),dt-(x^{2}-x+1)e^{x},; x\in\mathbb{R}$, is: