NTA JEE Main 30th January 2023 Shift 1

If the solution of the equation $$\log_{\cos x} \cot x + 4\log_{\sin x} \tan x = 1$$, $$x \in (0, \frac{\pi}{2})$$ is $$\sin^{-1}\frac{\alpha + \sqrt{\beta}}{2}$$, where $$\alpha, \beta$$ are integers, then $$\alpha + \beta$$ is equal to:

If $$a_n = \frac{-2}{4n^2 - 16n + 15}$$, then $$a_1 + a_2 + \ldots + a_{25}$$ is equal to:

If the coefficient of $$x^{15}$$ in the expansion of $$\left(ax^3 + \frac{1}{bx^{\frac{1}{2}}}\right)^{15}$$ is equal to the coefficient of $$x^{-15}$$ in the expansion of $$\left(ax^{\frac{1}{3}} - \frac{1}{bx^3}\right)^{15}$$, where $$a$$ and $$b$$ are positive real numbers, then for each such ordered pair $$a, b$$:

The coefficient of $$x^{301}$$ in $$1 + x^{500} + x \cdot 1 + x^{499} + x^2 \cdot 1 + x^{498} + \ldots + x^{500}$$ is:

If $$\tan 15° + \frac{1}{\tan 75°} + \frac{1}{\tan 105°} + \tan 195° = 2a$$, then the value of $$a + \frac{1}{a}$$ is:

Let $$y = x + 2$$, $$4y = 3x + 6$$ and $$3y = 4x + 1$$ be three tangent lines to the circle $$(x - h)^2 + (y - k)^2 = r^2$$. Then $$h + k$$ is equal to:

If $$P(h, k)$$ be point on the parabola $$x = 4y^2$$, which is nearest to the point $$Q(0, 33)$$, then the distance of $$P$$ from the directrix of the parabola $$y^2 = 4(x + y)$$ is equal to:

Among the statements:
S1: $$p \vee q \Rightarrow r \Leftrightarrow p \Rightarrow r$$
S2: $$p \vee q \Rightarrow r \Leftrightarrow p \Rightarrow r \vee q \Rightarrow r$$

The minimum number of elements that must be added to the relation $$R = \{(a, b), (b, c)\}$$ on the set $$\{a, b, c\}$$ so that it becomes symmetric and transitive is:

Let $$A = \begin{pmatrix} m & n \\ p & q \end{pmatrix}$$, $$d = |A| \neq 0$$ and $$A - d(\text{Adj } A) = 0$$. Then