NTA JEE Main 6th April 2023 Shift 1 - Mathematics

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

If the system of equations
$$x + y + az = b$$
$$2x + 5y + 2z = 6$$
$$x + 2y + 3z = 3$$
has infinitely many solutions, then $$2a + 3b$$ is equal to

Let $$5f(x) + 4f\left(\dfrac{1}{x}\right) = \dfrac{1}{x} + 3$$, $$x > 0$$. Then $$18\int_1^2 f(x)\,dx$$ is equal to

Let $$A = \left\{x \in \mathbb{R}: |x+3| + |x+4| \le 3\right\}$$, $$B = \left\{x \in \mathbb{R}: 3^x \sum_{r=1}^{\infty} \dfrac{3^{x-3}}{10^r} < 3^{-3x}\right\}$$, where $$[t]$$ denotes greatest integer function. Then,

If $$2x^y + 3y^x = 20$$, then $$\dfrac{dy}{dx}$$ at (2, 2) is equal to:

Let $$Ix = \int \dfrac{x^x \sec^2 + \tan x}{(x \tan x + 1)^2} dx$$. If $$I(0) = 0$$, then $$I\left(\dfrac{\pi}{4}\right)$$ is equal to

Let the position vectors of the points A, B, C and D be $$5\hat{i} + 5\hat{j} + 2\lambda\hat{k}$$, $$\hat{i} + 2\hat{j} + 3\hat{k}$$, $$-2\hat{i} + \lambda\hat{j} + 4\hat{k}$$ and $$-\hat{i} + 5\hat{j} + 6\hat{k}$$. Let the set $$S = \{\lambda \in \mathbb{R}$$: the points A, B, C and D are coplanar$$\}$$. The $$\sum_{\lambda \in S} (\lambda + 2)^2$$ is equal to

Let $$\vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k}$$, $$\vec{b} = \hat{i} - 2\hat{j} - 2\hat{k}$$ and $$\vec{c} = -\hat{i} + 4\hat{j} + 3\hat{k}$$. If $$\vec{d}$$ is a vector perpendicular to both $$\vec{b}$$ and $$\vec{c}$$, and $$\vec{a} \cdot \vec{d} = 18$$, then $$|\vec{a} \times \vec{d}|^2$$ is equal to

One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is

If the equation of the plane passing through the line of intersection of the planes $$2x - y + z = 3$$, $$4x - 3y + 5z + 9 = 0$$ and parallel to the line $$\dfrac{x+1}{-2} = \dfrac{y+3}{4} = \dfrac{z-2}{5}$$ is $$ax + by + cz + 6 = 0$$, then $$a + b + c$$ is equal to

A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is $$\dfrac{k}{3^{11}}$$, then $$k$$ is equal to

The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is ______.

The coefficient of $$x^{18}$$ in the expansion of $$\left(x^4 - \dfrac{1}{x^3}\right)^{15}$$ is ______.

A circle passing through the point $$P(\alpha, \beta)$$ in the first quadrant touches the two coordinate axes at the points A and B. The point P is above the line AB. The point Q on the line segment AB is the foot of perpendicular from P on AB. If PQ is equal to 11 units, then the value of $$\alpha\beta$$ is ______.

Let the point $$p, p+1$$ lie inside the region $$E = \{x, y: 3-x \le y \le \sqrt{9-x^2}, 0 \le x \le 3\}$$. If the set of all values of $$p$$ is the interval $$(a, b)$$, then $$b^2 + b - a^2$$ is equal to ______.

Let $$A = 1, 2, 3, 4, \ldots, 10$$ and $$B = 0, 1, 2, 3, 4$$. The number of elements in the relation $$R = \{(a, b) \in A \times A: 2a - b^2 + 3a - b \in B\}$$ is ________.

Let $$a \in \mathbb{Z}$$ and $$t$$ be the greatest integer $$\le t$$, then the number of points, where the function $$f(x) = a + 13|\sin x|$$, $$x \in (0, \pi)$$ is not differentiable, is ______.

Let the tangent to the curve $$x^2 + 2x - 4y + 9 = 0$$ at the point P(1, 3) on it meet the y-axis at A. Let the line passing through P and parallel to the line $$x - 3y = 6$$ meet the parabola $$y^2 = 4x$$ at B. If B lies on the line $$2x - 3y = 8$$, then $$AB^2$$ is equal to ______.

If the area of the region $$S = \{(x,y): 2y - y^2 \le x^2 \le 2y, x \ge y\}$$ is equal to $$\dfrac{n+2}{n+1} - \dfrac{\pi}{n-1}$$, then the natural number $$n$$ is equal to ______.

Let $$y = y(x)$$ be a solution of the differential equation $$(x\cos x)dy + (xy\sin x + y\cos x - 1)dx = 0$$, $$0 \lt x \lt \dfrac{\pi}{2}$$. If $$\dfrac{\pi}{3}y\left(\dfrac{\pi}{3}\right) = \sqrt{3}$$, then $$\left|\dfrac{\pi}{6}y''\left(\dfrac{\pi}{6}\right) + 2y'\left(\dfrac{\pi}{6}\right)\right|$$ is equal to ______.

Let the image of the point P(1, 2, 3) in the plane $$2x - y + z = 9$$ be Q. If the coordinates of the point R are (6, 10, 7), then the square of the area of the triangle PQR is ______.