NTA JEE Main 30th January 2023 Shift 2

The number of ways of selecting two numbers $$a$$ and $$b$$, $$a \in \{2, 4, 6, \ldots, 100\}$$ and $$b \in \{1, 3, 5, \ldots, 99\}$$ such that $$2$$ is the remainder when $$a + b$$ is divided by $$23$$ is

Let $$a, b, c > 1$$, $$a^3, b^3$$ and $$c^3$$ be in A.P. and $$\log_a b$$, $$\log_c a$$ and $$\log_b c$$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $$\frac{a+4b+c}{3}$$ and the common difference is $$\frac{a-8b+c}{10}$$ is $$-444$$, then $$abc$$ is equal to

Let $$x = \left(8\sqrt{3} + 13\right)^{13}$$ and $$y = \left(7\sqrt{2} + 9\right)^{9}$$. If $$[t]$$ denotes the greatest integer $$\leq t$$, then

The parabolas: $$ax^2 + 2bx + cy = 0$$ and $$d^2 + 2ex + fy = 0$$ intersect on the line $$y = 1$$. If $$a, b, c, d, e, f$$ are positive real numbers and $$a, b, c$$ are in G.P., then

Let $$A$$ be a point on the $$x$$-axis. Common tangents are drawn from $$A$$ to the curves $$x^2 + y^2 = 8$$ and $$y^2 = 16x$$. If one of these tangents touches the two curves at $$Q$$ and $$R$$, then $$(QR)^2$$ is equal to

Let $$f, g$$ and $$h$$ be the real valued functions defined on $$\mathbb{R}$$ as
$$f(x) = \begin{cases} \frac{x}{|x|}, & x \neq 0 \\ 1, & x = 0 \end{cases}$$, $$g(x) = \begin{cases} \frac{\sin(x+1)}{(x+1)}, & x \neq -1 \\ 1, & x = -1 \end{cases}$$ and $$h(x) = 2[x] - f(x)$$, where $$[x]$$ is the greatest integer $$\leq x$$. Then the value of $$\lim_{x \to 1} g(h(x-1))$$ is

Consider the following statements:
$$P$$: I have fever
$$Q$$: I will not take medicine
$$R$$: I will take rest
The statement "If I have fever, then I will take medicine and I will take rest" is equivalent to:

Let $$S$$ be the set of all values of $$a_1$$ for which the mean deviation about the mean of $$100$$ consecutive positive integers $$a_1, a_2, a_3, \ldots, a_{100}$$ is $$25$$. Then $$S$$ is

If $$P$$ is a $$3 \times 3$$ real matrix such that $$P^T = aP + (a-1)I$$, where $$a > 1$$, then

For $$\alpha, \beta \in \mathbb{R}$$, suppose the system of linear equations
$$x - y + z = 5$$
$$2x + 2y + \alpha z = 8$$
$$3x - y + 4z = \beta$$
has infinitely many solutions. Then $$\alpha$$ and $$\beta$$ are the roots of