NTA JEE Main 27th July 2022 Shift 2

If $$\alpha, \beta$$ are the roots of the equation $$x^2 - (5 + 3^{\sqrt{\log_3 5}} - 5^{\sqrt{\log_5 3}})x + 3(3^{(\log_3 5)^{1/3}} - 5^{(\log_5 3)^{2/3}} - 1) = 0$$ then the equation, whose roots are $$\alpha + \dfrac{1}{\beta}$$ and $$\beta + \dfrac{1}{\alpha}$$, is

Let S be the set of all $$(\alpha, \beta)$$, $$\pi < \alpha, \beta < 2\pi$$, for which the complex number $$\frac{1-i\sin\alpha}{1+2i\sin\alpha}$$ is purely imaginary and $$\frac{1+i\cos\beta}{1-2i\cos\beta}$$ is purely real. Let $$Z_{\alpha\beta} = \sin 2\alpha + i\cos 2\beta, (\alpha,\beta) \in S$$. Then $$\sum_{(\alpha,\beta)\in S}(iZ_{\alpha\beta} + \frac{1}{i\bar{Z}_{\alpha\beta}})$$ is equal to

Let the sum of an infinite G.P., whose first term is $$a$$ and the common ratio is $$r$$, be 5. Let the sum of its first five terms be $$\frac{98}{25}$$. Then the sum of the first 21 terms of an A.P., whose first term is $$10ar$$, $$n^{th}$$ term is $$a_n$$ and the common difference is $$10ar^2$$, is equal to

Let $$S = \left\{\theta \in \left(0, \frac{\pi}{2}\right) : \sum_{m=1}^{9} \sec\left(\theta + (m-1)\frac{\pi}{6}\right) \sec\left(\theta + \frac{m\pi}{6}\right) = -\frac{8}{\sqrt{3}}\right\}$$. Then

The equations of the sides AB, BC and CA of a triangle ABC are $$2x + y = 0$$, $$x + py = 39$$ and $$x - y = 3$$ respectively and P(2,3) is its circumcentre. Then which of the following is NOT true

A circle $$C_1$$ passes through the origin O and has diameter 4 on the positive x-axis. The line $$y = 2x$$ gives a chord OA of circle $$C_1$$. Let $$C_2$$ be the circle with OA as a diameter. If the tangent to $$C_2$$ at the point A meets the x-axis at P and y-axis at Q, then $$QA : AP$$ is equal to

If the length of the latus rectum of a parabola, whose focus is $$(a, a)$$ and the tangent at its vertex is $$x + y = a$$, is 16, then $$|a|$$ is equal to

If the truth value of the statement $$(P \wedge (\sim R)) \to ((\sim R) \wedge Q)$$ is F, then the truth value of which of the following is F?

The angle of elevation of the top P of a vertical tower PQ of height 10 from a point A on the horizontal ground is $$45^\circ$$. Let R be a point on AQ and from a point B, vertically above R, the angle of elevation of P is $$60^\circ$$. If $$\angle BAQ = 30^\circ$$, $$AB = d$$ and the area of the trapezium PQRB is $$\alpha$$, then the ordered pair $$(d, \alpha)$$ is

Let $$A = \begin{pmatrix} 4 & -2 \\ \alpha & \beta \end{pmatrix}$$. If $$A^2 + \gamma A + 18I = O$$, then $$\det(A)$$ is equal to