NTA JEE Main 6th April 2023 Shift 1

The sum of all the roots of the equation $$|x^2 - 8x + 15| - 2x + 7 = 0$$ is

The sum of the first 20 terms of the series $$5 + 11 + 19 + 29 + 41 + \ldots$$ is

Let $$a_1, a_2, a_3, \ldots, a_n$$ be n positive consecutive terms of an arithmetic progression. If $$d > 0$$ is its common difference, then $$\lim_{n \to \infty} \sqrt{\dfrac{d}{n}}\dfrac{1}{\sqrt{a_1}+\sqrt{a_2}} + \dfrac{1}{\sqrt{a_2}+\sqrt{a_3}} + \ldots + \dfrac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}$$ is

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $$\sqrt[4]{2} + \dfrac{1}{\sqrt[4]{3}}  ^n$$ is $$\sqrt{6}:1$$, then the third term from the beginning is:

If $$^{2n}C_3 : ^nC_3 = 10:1$$, then the ratio $$n^2+3n : n^2-3n+4$$ is

The straight lines $$l_1$$ and $$l_2$$ pass through the origin and trisect the line segment of the line $$L: 9x + 5y = 45$$ between the axes. If $$m_1$$ and $$m_2$$ are the slopes of the lines $$l_1$$ and $$l_2$$, then the point of intersection of the line $$y = (m_1 + m_2)x$$ with L lies on

Statement $$(P \Rightarrow Q) \wedge (R \Rightarrow Q)$$ is logically equivalent to

The mean and variance of a set of 15 numbers are 12 and 14 respectively. The mean and variance of another set of 15 numbers are 14 and $$\sigma^2$$ respectively. If the variance of all the 30 numbers in the two sets is 13, then $$\sigma^2$$ is equal to

From the top $$A$$ of a vertical wall $$AB$$ of height 30 m, the angles of depression of the top $$P$$ and bottom $$Q$$ of a vertical tower $$PQ$$ are 15° and 60°, respectively. $$B$$ and $$Q$$ are on the same horizontal level. If $$C$$ is a point on $$AB$$ such that $$CB = PQ$$, then the area (in m$$^2$$) of the quadrilateral $$BCPQ$$ is equal to

Let $$A = a_{ij}{}_{2\times 2}$$ where $$a_{ij} \neq 0$$ for all $$i, j$$ and $$A^2 = I$$. Let $$a$$ be the sum of all diagonal elements of $$A$$ and $$b = |A|$$. Then $$3a^2 + 4b^2$$ is equal to