NTA JEE Mains 1st feb 2023 Shift 2

Let $$P(x_0, y_0)$$ be the point on the hyperbola $$3x^2 - 4y^2 = 36$$, which is nearest to the line $$3x + 2y = 1$$. Then $$\sqrt{2}(y_0 - x_0)$$ is equal to :

Which of the following statements is a tautology?

Let $$9 = x_1 < x_2 < \ldots < x_7$$ be in an A.P. with common difference $$d$$. If the standard deviation of $$x_1, x_2, \ldots, x_7$$ is 4 and the mean is $$\bar{x}$$, then $$\bar{x} + x_6$$ is equal to :

Let $$P(S)$$ denote the power set of $$S = \{1, 2, 3, \ldots, 10\}$$. Define the relations $$R_1$$ and $$R_2$$ on $$P(S)$$ as $$AR_1B$$ if $$(A \cap B^c) \cup (B \cap A^c) = \phi$$ and $$AR_2 B$$ if $$A \cup B^c = B \cup A^c, \forall A, B \in P(S)$$. Then :

If $$A = \frac{1}{2}\begin{bmatrix} 1 & \sqrt{3} \\ -\sqrt{3} & 1 \end{bmatrix}$$ then,

For the system of linear equations $$ax + y + z = 1$$, $$x + ay + z = 1$$, $$x + y + az = \beta$$, which one of the following statements is NOT correct?

Let $$S = \left\{x \in R : 0 < x < 1 \text{ and } 2\tan^{-1}\left(\frac{1-x}{1+x}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\}$$. If $$n(S)$$ denotes the number of elements in $$S$$ then :

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 :