NTA JEE Main 29th January 2023 Shift 1

Let $$\lambda \neq 0$$ be a real number. Let $$\alpha, \beta$$ be the roots of the equation $$14x^2 - 31x + 3\lambda = 0$$ and $$\alpha, \gamma$$ be the roots of the equation $$35x^2 - 53x + 4\lambda = 0$$. Then $$\frac{3\alpha}{\beta}$$ and $$\frac{4\alpha}{\gamma}$$ are the roots of the equation:

For two non-zero complex numbers $$z_1$$ and $$z_2$$, if $$\text{Re}(z_1 z_2) = 0$$ and $$\text{Re}(z_1 + z_2) = 0$$, then which of the following are possible?
(A) Im $$(z_1) > 0$$ and Im $$(z_2) > 0$$
(B) Im $$(z_1) < 0$$ and Im $$(z_2) > 0$$
(C) Im $$(z_1) > 0$$ and Im $$(z_2) < 0$$
(D) Im $$(z_1) < 0$$ and Im $$(z_2) < 0$$
Choose the correct answer from the options given below:

Let $$f(\theta) = 3\left(\sin^4\left(\frac{3\pi}{2} - \theta\right) + \sin^4(3\pi + \theta)\right) - 2\left(1 - \sin^2 2\theta\right)$$ and $$S = \left\{\theta \in [0, \pi] : f'(\theta) = -\frac{\sqrt{3}}{2}\right\}$$. If $$4\beta = \sum_{\theta \in S} \theta$$ then $$f(\beta)$$ is equal to

A light ray emits from the origin making angle $$30°$$ with the positive $$x$$-axis. After getting reflected by the line $$x + y = 1$$, if this ray intersects x-axis at Q, then the abscissa of Q is

Let $$B$$ and $$C$$ be the two points on the line $$y + x = 0$$ such that $$B$$ and $$C$$ are symmetric with respect to the origin. Suppose $$A$$ is a point on $$y - 2x = 2$$ such that $$\triangle ABC$$ is an equilateral triangle. Then, the area of the $$\triangle ABC$$ is

Let the tangents at the points $$A(4, -11)$$ and $$B(8, -5)$$ on the circle $$x^2 + y^2 - 3x + 10y - 15 = 0$$, intersect at the point $$C$$. Then the radius of the circle, whose centre is $$C$$ and the line joining $$A$$ and $$B$$ is its tangent, is equal to

Let $$x = 2$$ be a root of the equation $$x^2 + px + q = 0$$ and $$f(x) = \begin{cases} \frac{1-\cos(x^2-4px+q^2+8q+16)}{(x-2p)^4}, & x \neq 2p \\ 0, & x = 2p \end{cases}$$. Then $$\lim_{x \to 2p^+} [f(x)]$$
where $$[.]$$ denotes greatest integer function, is

If $$p, q$$ and $$r$$ are three propositions, then which of the following combination of truth values of $$p$$, $$q$$ and $$r$$ makes the logical expression $$\{(p \vee q) \wedge ((\neg p) \vee r)\} \to ((\neg q) \vee r)$$ false?

Let $$\alpha$$ and $$\beta$$ be real numbers. Consider a $$3 \times 3$$ matrix $$A$$ such that $$A^2 = 3A + \alpha I$$. If $$A^4 = 21A + \beta I$$, then

Consider the following system of equations
$$\alpha x + 2y + z = 1$$
$$2\alpha x + 3y + z = 1$$
$$3x + \alpha y + 2z = \beta$$
For some $$\alpha, \beta \in \mathbb{R}$$. Then which of the following is NOT correct.