NTA JEE Main 11th April 2023 Shift 1 - Mathematics

The number of integral solution $$x$$ of $$\log_{x + \frac{7}{2}}\left(\frac{x-7}{2x-3}\right)^2 \geq 0$$ is

Let $$w_1$$ be the point obtained by the rotation of $$z_1 = 5 + 4i$$ about the origin through a right angle in the anticlockwise direction, and $$w_2$$ be the point obtained by the rotation of $$z_2 = 3 + 5i$$ about the origin through a right angle in the clockwise direction. Then the principal argument of $$w_1 - w_2$$ is equal to

The number of triplets $$(x, y, z)$$ where $$x, y, z$$ are distinct non negative integers satisfying $$x + y + z = 15$$, is

Let $$x_1, x_2, \ldots, x_{100}$$ be in an arithmetic progression, with $$x_1 = 2$$ and their mean equal to 200. If $$y_i = ix_i - i$$, $$1 \leq i \leq 100$$, then the mean of $$y_1, y_2, \ldots, y_{100}$$ is

The number of elements in the set $$S = \{\theta \in [0, 2\pi]: 3\cos^4\theta - 5\cos^2\theta - 2\sin^6\theta + 2 = 0\}$$ is

Consider ellipses $$E_k: kx^2 + k^2y^2 = 1$$, $$k = 1, 2, \ldots, 20$$. Let $$C_k$$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $$E_k$$. If $$r_k$$ is the radius of the circle $$C_k$$, then the value of $$\sum_{k=1}^{20} \frac{1}{r_k^2}$$ is

Let R be a rectangle given by the lines $$x = 0$$, $$x = 2$$, $$y = 0$$ and $$y = 5$$. Let $$A(\alpha, 0)$$ and $$B(0, \beta)$$, $$\alpha \in (0, 2)$$ and $$\beta \in (0, 5)$$, be such that the line segment AB divides the area of the rectangle R in the ratio 4:1. Then, the mid-point of AB lies on a

Let sets A and B have 5 elements each. Let the mean of the elements in sets A and B be 5 and 8 respectively and the variance of the elements in sets A and B be 12 and 20 respectively. A new set C of 10 elements is formed by subtracting 3 from each element of A and adding 2 to each element of B. Then the sum of the mean and variance of the elements of C is

An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 men and only five men got medals in all the three events, then how many received medals in exactly two of three events?

Let A be a 2 $$\times$$ 2 matrix with real entries such that $$A' = \alpha A + 1$$, where $$\alpha \in \mathbb{R} - \{-1, 1\}$$. If det$$(A^2 - A) = 4$$, the sum of all possible values of $$\alpha$$ is equal to

Let $$f(x) = x^2 - [x] + |-x + [x]|$$, where $$x \in \mathbb{R}$$ and $$[t]$$ denotes the greatest integer less than or equal to $$t$$. Then, $$f$$ is

Let $$f: [2, 4] \to \mathbb{R}$$ be a differentiable function such that $$x\log_e xf'(x) + \log_e xf(x) + f(x) \geq 1$$, $$x \in [2, 4]$$ with $$f(2) = \frac{1}{2}$$ and $$f(4) = \frac{1}{2}$$.
Consider the following two statements:
(A) $$f(x) \leq 1$$, for all $$x \in [2, 4]$$
(B) $$f(x) \geq 1/8$$, for all $$x \in [2, 4]$$
Then,

The value of the integral $$\int_{-\log_e 2}^{\log_e 2} e^x \log_e e^x + \sqrt{1 + e^{2x}} \, dx$$ is equal to

Area of the region $$\{(x, y): x^2 + (y-2)^2 \leq 4, x^2 \geq 2y\}$$ is

Let $$y = y(x)$$ be a solution curve of the differential equation, $$(1 - x^2y^2)dx = ydx + xdy$$. If the line $$x = 1$$ intersects the curve $$y = y(x)$$ at $$y = 2$$ and the line $$x = 2$$ intersects the curve $$y = y(x)$$ at $$y = \alpha$$, then a value of $$\alpha$$ is

For any vector $$\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$$, with $$10a_i < 1$$, $$i = 1, 2, 3$$, consider the following statements:
$$A: \max(a_1, a_2, a_3) \leq \vec{a}$$
$$B: |\vec{a}| \leq 3\max a_1, a_2, a_3$$

Let $$\vec{a}$$ be a non-zero vector parallel to the line of intersection of the two planes described by $$\hat{i} + \hat{j}, \hat{i} + \hat{k}$$ and $$\hat{i} - \hat{j}, \hat{j} - \hat{k}$$. If $$\theta$$ is the angle between the vector $$\vec{a}$$ and the vector $$\vec{b} = 2\hat{i} - 2\hat{j} + \hat{k}$$ and $$\vec{a} \cdot \vec{b} = 6$$, then the ordered pair $$(\theta, |\vec{a} \times \vec{b}|)$$ is equal to

Let $$(\alpha, \beta, \gamma)$$ be the image of point $$P(2, 3, 5)$$ in the plane $$2x + y - 3z = 6$$. Then $$\alpha + \beta + \gamma$$ is equal to

If the equation of the plane that contains the point $$(-2, 3, 5)$$ and is perpendicular to each of the planes $$2x + 4y + 5z = 8$$ and $$3x - 2y + 3z = 5$$ is $$\alpha x + \beta y + \gamma z + 97 = 0$$ then $$\alpha + \beta + \gamma =$$

Let $$S = M = a_{ij}$$, $$a_{ij} \in \{0, 1, 2\}$$, $$1 \leq i, j \leq 2$$ be a sample space and $$A = \{M \in S: M \text{ is invertible}\}$$ be an even. Then $$P(A)$$ is equal to

If $$a$$ and $$b$$ are the roots of the equation $$x^2 - 7x - 1 = 0$$, then the value of $$\frac{a^{21} + b^{21} + a^{17} + b^{17}}{a^{19} + b^{19}}$$ is equal to _______

In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is _______

Let $$S = 109 + \frac{108}{5} + \frac{107}{5^2} + \ldots + \frac{2}{5^{107}} + \frac{1}{5^{108}}$$. Then the value of $$16S - (25)^{-54}$$ is equal to _______

The number of integral terms in the expansion of $$\left(3^{\frac{1}{2}} + 5^{\frac{1}{4}}\right)^{680}$$ is equal to _______

The mean of the coefficients of $$x, x^2, \ldots, x^7$$ in the binomial expression of $$(2 + x)^9$$ is _______

Let $$H_n: \frac{x^2}{1+n} - \frac{y^2}{3+n} = 1$$, $$n \in \mathbb{N}$$. Let $$k$$ be the smallest even value of $$n$$ such that the eccentricity of $$H_k$$ is a rational number. If $$l$$ is the length of the latus rectum of $$H_k$$, then $$21l$$ is equal to _______

The number of ordered triplets of the truth values of $$p, q$$ and $$r$$ such that the truth value of the statement $$p \vee q \wedge p \vee r \Rightarrow q \vee r$$ is True, is equal to _______

Let $$A = \begin{pmatrix} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{pmatrix}$$, where $$a, c \in \mathbb{R}$$. If $$A^3 = A$$ and the positive value of $$a$$ belongs to the interval $$(n-1, n]$$, where $$n \in \mathbb{N}$$, then $$n$$ is equal to _______.

For $$m, n > 0$$, let $$\alpha(m, n) = \int_0^2 t^m(1 + 3t)^n dt$$. If $$11\alpha(10, 6) + 18\alpha(11, 5) = p \cdot 14^6$$, then $$p$$ is equal to _______

Let a line $$L$$ pass through the origin and be perpendicular to the lines
$$L_1: \vec{r} = (\hat{i} - 11\hat{j} - 7\hat{k}) + \lambda(\hat{i} + 2\hat{j} + 3\hat{k})$$, $$\lambda \in \mathbb{R}$$ and
$$L_2: \vec{r} = (-\hat{i} + \hat{k}) + \mu(2\hat{i} + 2\hat{j} + \hat{k})$$, $$\mu \in \mathbb{R}$$. If $$P$$ is the point of intersection of $$L$$ and $$L_1$$, and $$Q(\alpha, \beta, \gamma)$$ is the foot of perpendicular from $$P$$ on $$L_2$$, then $$9\alpha + \beta + \gamma$$ is equal to _______