NTA JEE Main 25th January 2023 Shift 1

Let $$z_1 = 2 + 3i$$ and $$z_2 = 3 + 4i$$. The set $$S = \{z \in \mathbb{C} : |z - z_1|^2 - |z - z_2|^2 = |z_1 - z_2|^2\}$$ represents a

If $$a_r$$ is the coefficient of $$x^{10-r}$$ in the Binomial expansion of $$(1+x)^{10}$$, then $$\sum_{r=1}^{10} r^3 \left(\frac{a_r}{a_{r-1}}\right)^2$$ is equal to

The points of intersection of the line $$ax + by = 0$$, ($$a \neq b$$) and the circle $$x^2 + y^2 - 2x = 0$$ are $$A(\alpha, 0)$$ and $$B(1, \beta)$$. The image of the circle with $$AB$$ as a diameter in the line $$x + y + 2 = 0$$ is:

The distance of the point $$(6, -2\sqrt{2})$$ from the common tangent $$y = mx + c$$, $$m > 0$$, of the curves $$x = 2y^2$$ and $$x = 1 + y^2$$ is

The value of $$\lim_{n \to \infty} \frac{1+2-3+4+5-6+\ldots+(3n-2)+(3n-1)-3n}{\sqrt{2n^4+4n+3} - \sqrt{n^4+5n+4}}$$ is

The statement $$(p \wedge (\sim q)) \Rightarrow (p \Rightarrow (\sim q))$$ is

The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2, then their new variance is equal to:

Let $$x, y, z > 1$$ and $$A = \begin{bmatrix} 1 & \log_x y & \log_x z \\ \log_y x & 2 & \log_y z \\ \log_z x & \log_z y & 3 \end{bmatrix}$$. Then $$|adj(adj A^2)|$$ is equal to

Let $$S_1$$ and $$S_2$$ be respectively the sets of all $$a \in \mathbb{R} - \{0\}$$ for which the system of linear equations
$$ax + 2ay - 3az = 1$$
$$(2a+1)x + (2a+3)y + (a+1)z = 2$$
$$(3a+5)x + (a+5)y + (a+2)z = 3$$
has unique solution and infinitely many solutions. Then

Let $$f : (0,1) \to \mathbb{R}$$ be a function defined by $$f(x) = \frac{1}{1-e^{-x}}$$, and $$g(x) = (f(-x) - f(x))$$. Consider two statements
(I) $$g$$ is an increasing function in $$(0, 1)$$
(II) $$g$$ is one-one in $$(0, 1)$$
Then,