NTA JEE Main 10th April 2023 Shift 1 - Mathematics

Let the complex number $$z = x + iy$$ be such that $$\frac{2z - 3i}{2z + i}$$ is purely imaginary. If $$x + y^2 = 0$$, then $$y^4 + y^2 - y$$ is equal to

Let the first term a and the common ratio r of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to

If the coefficient of $$x^7$$ in $$\left(ax - \frac{1}{bx^2}\right)^{13}$$ and the coefficient of $$x^{-5}$$ in $$\left(ax + \frac{1}{bx^2}\right)^{13}$$ are equal, then $$a^4 b^4$$ is equal to:

$$96 \cos\frac{\pi}{33} \cos\frac{2\pi}{33} \cos\frac{4\pi}{33} \cos\frac{8\pi}{33} \cos\frac{16\pi}{33}$$ is equal to

A line segment $$AB$$ of length $$\lambda$$ moves such that the points $$A$$ and $$B$$ remain on the periphery of a circle of radius $$\lambda$$. Then the locus of the point, that divides the line segment $$AB$$ in the ratio 2:3, is a circle of radius

Let the ellipse $$E: x^2 + 9y^2 = 9$$ intersect the positive $$x$$- and $$y$$-axes at the points $$A$$ and $$B$$ respectively. Let the major axis of $$E$$ be a diameter of the circle $$C$$. Let the line passing through $$A$$ and $$B$$ meet the circle $$C$$ at the point $$P$$. If the area of the triangle with vertices $$A$$, $$P$$ and the origin $$O$$ is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime, then $$m - n$$ is equal to

The negation of the statement $$p \vee q \wedge q \vee \sim r$$ is

If $$A$$ is a $$3 \times 3$$ matrix and $$|A| = 2$$, then $$|3 \text{ adj}(|3A| \cdot A^2)|$$ is equal to

For the system of linear equations
$$2x - y + 3z = 5$$
$$3x + 2y - z = 7$$
$$4x + 5y + \alpha z = \beta$$,
which of the following is NOT correct?

If $$f(x) = \frac{\tan^{-1} x + \log_e 123}{x \log_e 1234 - \tan^{-1} x}$$, $$x > 0$$, then the least value of $$f(f(x)) + f\left(f\left(\frac{4}{x}\right)\right)$$ is

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm$$^2$$) is equal to

If $$Ix = \int e^{\sin^2 x} \cos x \sin 2x \cdot \sin x \, dx$$ and $$I(0) = 1$$, then $$I\left(\frac{\pi}{3}\right)$$ is equal to

Let $$f$$ be a differentiable function such that $$x^2 f(x) - x = 4\int_0^x tf(t) \, dt$$, $$f(1) = \frac{2}{3}$$. Then $$18f(3)$$ is equal to

The slope of tangent at any point $$(x, y)$$ on a curve $$y = y(x)$$ is $$\frac{x^2 + y^2}{2xy}$$, $$x > 0$$. If $$y(2) = 0$$, then a value of $$y(8)$$ is

An arc $$PQ$$ of a circle subtends a right angle at its centre $$O$$. The mid point of the arc $$PQ$$ is $$R$$. If $$\overrightarrow{OP} = \vec{u}$$, $$\overrightarrow{OR} = \vec{v}$$ and $$\overrightarrow{OQ} = \alpha\vec{u} + \beta\vec{v}$$, then $$\alpha$$, $$\beta^2$$, are the roots of the equation

Let $$O$$ be the origin and the position vector of the point $$P$$ be $$-\hat{i} - 2\hat{j} + 3\hat{k}$$. If the position vectors of the points $$A$$, $$B$$ and $$C$$ are $$-2\hat{i} + \hat{j} - 3\hat{k}$$, $$2\hat{i} + 4\hat{j} - 2\hat{k}$$ and $$-4\hat{i} + 2\hat{j} - \hat{k}$$ respectively, then the projection of the vector $$\overrightarrow{OP}$$ on a vector perpendicular to the vectors $$\overrightarrow{AB}$$ and $$\overrightarrow{AC}$$ is

Let two vertices of a triangle $$ABC$$ be $$(2, 4, 6)$$ and $$(0, -2, -5)$$, and its centroid be $$(2, 1, -1)$$. If the image of the third vertex in the plane $$x + 2y + 4z = 11$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha\beta + \beta\gamma + \gamma\alpha$$ is equal to

The shortest distance between the lines $$\frac{x+2}{1} = \frac{y}{-2} = \frac{z-5}{2}$$ and $$\frac{x-4}{1} = \frac{y-1}{2} = \frac{z+3}{0}$$ is

Let $$P$$ be the point of intersection of the line $$\frac{x+3}{3} = \frac{y+2}{1} = \frac{1-z}{2}$$ and the plane $$x + y + z = 2$$. If the distance of the point $$P$$ from the plane $$3x - 4y + 12z = 32$$ is $$q$$, then $$q$$ and $$2q$$ are the roots of the equation

Let $$N$$ denote the sum of the numbers obtained when two dice are rolled. If the probability that $$2^N < N!$$ is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime, $$4m - 3n$$ is equal to

Let $$a$$, $$b$$, $$c$$ be the three distinct positive real numbers such that $$2a^{\log_e a} = bc^{\log_e b}$$ and $$b^{\log_e 2} = a^{\log_e c}$$. Then $$6a + 5bc$$ is equal to _______.

The number of permutations, of the digits 1, 2, 3, ..., 7 without repetition, which neither contain the string 153 nor the string 2467, is _______.

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total numbers of persons, who participated in the tournament, is _______.

The sum of all those terms, of the arithmetic progression 3, 8, 13, ..., 373, which are not divisible by 3, is equal to _______.

The coefficient of $$x^7$$ in $$(1 - x + 2x^3)^{10}$$ is _______.

Let a common tangent to the curves $$y^2 = 4x$$ and $$x - 4^2 + y^2 = 16$$ touch the curves at the points $$P$$ and $$Q$$. Then $$PQ^2$$ is equal to _______.

If the mean of the frequency distribution

The number of elements in the set $$\{n \in \mathbb{Z}: |n^2 - 10n + 19| < 6\}$$ is _______.

Let $$f: [-2, 2] \to \mathbb{R}$$ be defined by $$f(x) = \begin{cases} x[x], & -2 < x < 0 \\ (x - 1)[x], & 0 \leq x \leq 2 \end{cases}$$ where $$[x]$$ denotes the greatest integer function. If $$m$$ and $$n$$ respectively are the number of points in $$(-2, 2)$$ at which $$y = |f(x)|$$ is not continuous and not differentiable, then $$m + n$$ is equal to _______.

Let $$y = px$$ be the parabola passing through the points $$(-1, 0)$$, $$(0, 0)$$, $$(1, 0)$$ and $$(1, 0)$$. If the area of the region $$\{(x, y): (x+1)^2 + (y-1)^2 \leq 1, y \leq px\}$$ is $$A$$, then $$12\pi - 4A$$ is equal to _______.