NTA JEE Main 24th June 2022 Shift 2

Let $$S = \{z \in \mathbb{C} : |z - 3| \leq 1$$ and $$z(4 + 3i) + \bar{z}(4 - 3i) \leq 24\}$$. If $$\alpha + i\beta$$ is the point in $$S$$ which is closest to $$4i$$, then $$25(\alpha + \beta)$$ is equal to ______.

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits $$1, 2, 3, 4, 5, 7$$ and $$9$$ is ______.

The remainder on dividing $$1 + 3 + 3^2 + 3^3 + \ldots + 3^{2021}$$ by $$50$$ is ______.

Let a circle $$C : (x - h)^2 + (y - k)^2 = r^2, k > 0$$, touch the $$x$$-axis at $$(1, 0)$$. If the line $$x + y = 0$$ intersects the circle $$C$$ at $$P$$ and $$Q$$ such that the length of the chord $$PQ$$ is $$2$$, then the value of $$h + k + r$$ is equal to ______.

Let $$P_1$$ be a parabola with vertex $$(3, 2)$$ and focus $$(4, 4)$$ and $$P_2$$ be its mirror image with respect to the line $$x + 2y = 6$$. Then the directrix of $$P_2$$ is $$x + 2y =$$ ______.

Let the hyperbola $$H : \frac{x^2}{a^2} - y^2 = 1$$ and the ellipse $$E : 3x^2 + 4y^2 = 12$$ be such that the length of latus rectum of $$H$$ is equal to the length of latus rectum of $$E$$. If $$e_H$$ and $$e_E$$ are the eccentricities of $$H$$ and $$E$$ respectively, then the value of $$12(e_H^2 + e_E^2)$$ is equal to ______.

The sum of all the elements of the set $$\{\alpha \in \{1, 2, \ldots, 100\} : HCF(\alpha, 24) = 1\}$$ is ______.

Let $$S = \left\{\begin{pmatrix} -1 & a \\ 0 & b \end{pmatrix} ; a, b \in \{1, 2, 3, \ldots 100\}\right\}$$ and let $$T_n = \{A \in S : A^{n(n+1)} = I\}$$. Then the number of elements in $$\bigcap_{n=1}^{100} T_n$$ is ______.

The area (in sq. units) of the region enclosed between the parabola $$y^2 = 2x$$ and the line $$x + y = 4$$ is ______.

In an examination, there are $$10$$ true-false type questions. Out of $$10$$, a student can guess the answer of $$4$$ questions correctly with probability $$\frac{3}{4}$$ and the remaining $$6$$ questions correctly with probability $$\frac{1}{4}$$. If the probability that the student guesses the answers of exactly $$8$$ questions correctly out of $$10$$ is $$\frac{27k}{4^{10}}$$, then $$k$$ is equal to ______.