NTA JEE Main 27th July 2022 Shift 2

The domain of the function $$f(x) = \sin^{-1}[2x^2 - 3] + \log_2\left(\log_{\frac{1}{2}}(x^2 - 5x + 5)\right)$$, where $$[t]$$ is the greatest integer function, is

If for $$p \neq q \neq 0$$, the function $$f(x) = \frac{7\sqrt[p]{729+x} - 3}{\sqrt[3]{729+qx} - 9}$$ is continuous at $$x = 0$$, then

Let $$f(x) = 2 + |x| - |x-1| + |x+1|$$, $$x \in \mathbb{R}$$. Consider
$$(S_1): f'(-3/2) + f'(-1/2) + f'(1/2) + f'(3/2) = 2$$
$$(S_2): \int_{-2}^{2} f(x) dx = 12$$
Then,

$$\int_0^2 \left|2x^2 - 3x + \left[x - \frac{1}{2}\right]\right| dx$$, where $$[t]$$ is the greatest integer function, is equal to

The area of the region enclosed by $$y \leq 4x^2$$, $$x^2 \leq 9y$$ and $$y \leq 4$$, is equal to

Consider a curve $$y = y(x)$$ in the first quadrant as shown in the figure. Let the area $$A_1$$ is twice the area $$A_2$$. Then the normal to the curve perpendicular to the line $$2x - 12y = 15$$ does NOT pass through the point

If the length of the perpendicular drawn from the point $$P(a, 4, 2)$$, $$a > 0$$ on the line $$\frac{x+1}{2} = \frac{y-3}{3} = \frac{z-1}{-1}$$ is $$2\sqrt{6}$$ units and Q$$(\alpha_1, \alpha_2, \alpha_3)$$ is the image of the point P in this line, then $$a + \sum_{i=1}^{3} \alpha_i$$ is equal to

If the line of intersection of the planes $$ax + by = 3$$ and $$ax + by + cz = 0$$, $$a > 0$$ makes an angle $$30^\circ$$ with the plane $$y - z + 2 = 0$$, then the direction cosines of the line are

Let X have a binomial distribution $$B(n, p)$$ such that the sum and the product of the mean and variance of X are 24 and 128 respectively. If $$P(X > n-3) = \frac{k}{2^n}$$, then $$k$$ is equal to

A six faced die is biased such that $$3 \times P(\text{a prime number}) = 6 \times P(\text{a composite number}) = 2 \times P(1)$$. Let X be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of X is