NTA JEE Main 1st February 2023 Shift 2

Let $$f : R - \{0, 1\} \to R$$ be a function such that $$f(x) + f\left(\frac{1}{1-x}\right) = 1 + x$$. Then $$f(2)$$ is equal to:

If $$y(x) = x^x, x > 0$$, then $$y''(2) - 2y'(2)$$ is equal to:

The sum of the absolute maximum and minimum values of the function $$f(x) = |x^2 - 5x + 6| - 3x + 2$$ in the interval $$[-1, 3]$$ is equal to:

The area of the region given by $$\{(x, y) : xy \leq 8, 1 \leq y \leq x^2\}$$ is:

Let $$\alpha x = \exp(x^{\beta}y^{\gamma})$$ be the solution of the differential equation $$2x^2 y dy - (1 - xy^2)dx = 0$$, $$x \gt 0$$, $$y(2) = \sqrt{\log_e 2}$$. Then $$\alpha + \beta - \gamma$$ equals:

Let $$\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$$ and $$\vec{b} = \hat{i} + 3\hat{j} + 5\hat{k}$$ be two vectors. Then which one of the following statements is TRUE?

Two dice are thrown independently. Let $$A$$ be the event that the number appeared on the 1st die is less than the number appeared on the 2nd die, $$B$$ be the event that the number appeared on the 1st die is even and that on the second die is odd, and $$C$$ be the event that the number appeared on the 1st die is odd and that on the 2nd is even. Then

Number of integral solutions to the equation $$x + y + z = 21$$, where $$x \geq 1, y \geq 3, z \geq 4$$, is equal to ______.

The total number of six digit numbers, formed using the digits $$4, 5, 9$$ only and divisible by $$6$$, is ______.

The sum of the common terms of the following three arithmetic progressions.
$$3, 7, 11, 15, \ldots, 399$$
$$2, 5, 8, 11, \ldots, 359$$ and
$$2, 7, 12, 17, \ldots, 197$$, is equal to ______.