NTA JEE Main 29th July 2022 Shift 2

The domain of the function $$f(x) = \sin^{-1}\left(\frac{x^2-3x+2}{x^2+2x+7}\right)$$ is

Let the function $$f(x) = \begin{cases} \frac{\log_e(1+5x) - \log_e(1+\alpha x)}{x} & \text{if } x \neq 0 \\ 10 & \text{if } x = 0 \end{cases}$$ be continuous at $$x = 0$$. Then $$\alpha$$ is equal to

For $$I(x) = \int \frac{\sec^2 x - 2022}{\sin^{2022} x} dx$$, if $$I\left(\frac{\pi}{4}\right) = 2^{1011}$$, then

If $$[t]$$ denotes the greatest integer $$\leq t$$, then the value of $$\int_0^1 [2x - |3x^2 - 5x + 2| + 1] dx$$ is

If the solution curve of the differential equation $$\frac{dy}{dx} = \frac{x+y-2}{x-y}$$ passes through the point (2, 1) and (k+1, 2), k > 0, then

Let $$y = y(x)$$ be the solution curve of the differential equation $$\frac{dy}{dx} + \frac{2x^2+11x+13}{x^3+6x^2+11x+6}y = \frac{x+3}{x+1}$$, $$x > -1$$, which passes through the point (0, 1). Then $$y(1)$$ is equal to

If $$\langle 2, 3, 9 \rangle$$, $$\langle 5, 2, 1 \rangle$$, $$\langle 1, \lambda, 8 \rangle$$ and $$\langle \lambda, 2, 3 \rangle$$ are coplanar, then the product of all possible values of $$\lambda$$ is

Let $$\vec{a}, \vec{b}, \vec{c}$$ be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and $$(\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c}) + (\vec{b} \times \vec{c}) \cdot (\vec{c} \times \vec{a}) + (\vec{c} \times \vec{a}) \cdot (\vec{a} \times \vec{b}) = 168$$ then $$|\vec{a}| + |\vec{b}| + |\vec{c}|$$ is equal to

Let Q be the foot of perpendicular drawn from the point P(1, 2, 3) to the plane $$x + 2y + z = 14$$. If R is a point on the plane such that $$\angle PRQ = 60°$$, then the area of $$\Delta PQR$$ is equal to

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is