NTA JEE Main 1st February 2023 Shift 2

The number of integral values of $$k$$, for which one root of the equation $$2x^2 - 8x + k = 0$$ lies in the interval $$(1, 2)$$ and its other root lies in the interval $$(2, 3)$$, is:

Let $$a, b$$ be two real numbers such that $$ab < 0$$. If the complex number $$\frac{1+ai}{b+i}$$ is of unit modulus and $$a + ib$$ lies on the circle $$|z - 1| = |2z|$$, then a possible value of $$\frac{1+[a]}{4b}$$, where $$[t]$$ is greatest integer function, is:

The sum $$\sum_{n=1}^{\infty} \frac{2n^2+3n+4}{(2n)!}$$ is equal to:

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: