NTA JEE Main 30th January 2023 Shift 1 - Mathematics

If the solution of the equation $$\log_{\cos x} \cot x + 4\log_{\sin x} \tan x = 1$$, $$x \in (0, \frac{\pi}{2})$$ is $$\sin^{-1}\frac{\alpha + \sqrt{\beta}}{2}$$, where $$\alpha, \beta$$ are integers, then $$\alpha + \beta$$ is equal to:

If $$a_n = \frac{-2}{4n^2 - 16n + 15}$$, then $$a_1 + a_2 + \ldots + a_{25}$$ is equal to:

If the coefficient of $$x^{15}$$ in the expansion of $$\left(ax^3 + \frac{1}{bx^{\frac{1}{2}}}\right)^{15}$$ is equal to the coefficient of $$x^{-15}$$ in the expansion of $$\left(ax^{\frac{1}{3}} - \frac{1}{bx^3}\right)^{15}$$, where $$a$$ and $$b$$ are positive real numbers, then for each such ordered pair $$a, b$$:

The coefficient of $$x^{301}$$ in $$1 + x^{500} + x \cdot 1 + x^{499} + x^2 \cdot 1 + x^{498} + \ldots + x^{500}$$ is:

If $$\tan 15° + \frac{1}{\tan 75°} + \frac{1}{\tan 105°} + \tan 195° = 2a$$, then the value of $$a + \frac{1}{a}$$ is:

Let $$y = x + 2$$, $$4y = 3x + 6$$ and $$3y = 4x + 1$$ be three tangent lines to the circle $$(x - h)^2 + (y - k)^2 = r^2$$. Then $$h + k$$ is equal to:

If $$P(h, k)$$ be point on the parabola $$x = 4y^2$$, which is nearest to the point $$Q(0, 33)$$, then the distance of $$P$$ from the directrix of the parabola $$y^2 = 4(x + y)$$ is equal to:

Among the statements:
S1: $$p \vee q \Rightarrow r \Leftrightarrow p \Rightarrow r$$
S2: $$p \vee q \Rightarrow r \Leftrightarrow p \Rightarrow r \vee q \Rightarrow r$$

The minimum number of elements that must be added to the relation $$R = \{(a, b), (b, c)\}$$ on the set $$\{a, b, c\}$$ so that it becomes symmetric and transitive is:

Let $$A = \begin{pmatrix} m & n \\ p & q \end{pmatrix}$$, $$d = |A| \neq 0$$ and $$A - d(\text{Adj } A) = 0$$. Then

Let the system of linear equations
$$x + y + kz = 2$$
$$2x + 3y - z = 1$$
$$3x + 4y + 2z = k$$
have infinitely many solutions. Then the system
$$(k+1)x + (2k-1)y = 7$$
$$(2k+1)x + (k+5)y = 10$$ has:

Suppose $$f: R \to (0, \infty)$$ be a differentiable function such that $$5f(x+y) = f(x) \cdot f(y), \forall x, y \in R$$. If $$f(3) = 320$$, then $$\sum_{n=0}^{5} f(n)$$ is equal to:

The number of points on the curve $$y = 54x^5 - 135x^4 - 70x^3 + 180x^2 + 210x$$ at which the normal lines are parallel to $$x + 90y + 2 = 0$$ is:

If $$[t]$$ denotes the greatest integer $$\leq 1$$, then the value of $$\frac{3e-1}{e} \int_1^2 x^2 e^{x+x^3} dx$$ is:

Let the solution curve $$y = y(x)$$ of the differential equation $$\frac{dy}{dx} - \frac{3x^5\tan^{-1}x^3}{1+x^{6\cdot 7}} y = 2x \exp\frac{x^3 - \tan^{-1}x^3}{(1+x)^7}$$ pass through the origin. Then $$y(1)$$ is equal to:

If $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ are three non-zero vectors and $$\hat{n}$$ is a unit vector perpendicular to $$\vec{c}$$ such that $$\vec{a} = \alpha\vec{b} - \hat{n}$$, $$\alpha \neq 0$$ and $$\vec{b} \cdot \vec{c} = 12$$, then $$\vec{c} \times \vec{a} \times \vec{b}$$ is equal to:

The line $$l_1$$ passes through the point $$(2, 6, 2)$$ and is perpendicular to the plane $$2x + y - 2z = 10$$. Then the shortest distance between the line $$l_1$$ and the line $$\frac{x+1}{2} = \frac{y+4}{-3} = \frac{z}{2}$$ is:

Let a unit vector $$\vec{OP}$$ make angle $$\alpha, \beta, \gamma$$ with the positive directions of the co-ordinate axes OX, OY, OZ respectively, where $$\beta \in (0, \frac{\pi}{2})$$. $$\vec{OP}$$ is perpendicular to the plane through points $$(1, 2, 3)$$, $$(2, 3, 4)$$ and $$(1, 5, 7)$$, then which one is true?

If an unbiased die, marked with $$-2, -1, 0, 1, 2, 3$$ on its faces is thrown five times, then the probability that the product of the outcomes is positive, is:

A straight line cuts off the intercepts OA = a and OB = b on the positive directions of x-axis and y-axis respectively. If the perpendicular from origin O to this line makes an angle of $$\frac{\pi}{6}$$ with positive direction of y-axis and the area of $$\triangle OAB$$ is $$\frac{98}{3}\sqrt{3}$$, then $$a^2 - b^2$$ is equal to:

Let $$z = 1 + i$$ and $$z_1 = \frac{1 + i\bar{z}}{\bar{z}(1-z) + \frac{1}{z}}$$. Then $$\frac{12}{\pi} \arg z_1$$ is equal to

Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to

$$\sum_{n=0}^{\infty} \frac{n^3((2n)!) + (2n-1)(n!)}{(n!)(2n)!} = ae + \frac{b}{e} + c$$ where $$a, b, c \in \mathbb{Z}$$ and $$e = \sum_{n=0}^{\infty} \frac{1}{n!}$$. Then $$a^2 - b + c$$ is equal to ______.

The mean and variance of 7 observations are 8 and 16 respectively. If one observation 14 is omitted, $$a$$ and $$b$$ are respectively mean and variance of remaining 6 observation, then $$a + 3b - 5$$ is equal to ______.

Let $$S = \{1, 2, 3, 4, 5, 6\}$$. Then the number of one-one functions $$f: S \to P(S)$$, where $$P(S)$$ denote the power set of $$S$$, such that $$f(n) \subset f(m)$$ where $$n < m$$ is

Let $$f^1(x) = \frac{3x+2}{2x+3}$$, $$x \in R - \{-\frac{3}{2}\}$$. For $$n \geq 2$$, define $$f^n x = f^1 \circ f^{n-1}(x)$$. If $$f^5 x = \frac{ax+b}{bx+a}$$, $$\gcd(a,b) = 1$$, then $$a + b$$ is equal to ______.

$$\lim_{x \to 0} \frac{48}{x^4} \int_0^x \frac{t^3}{t^6+1} dt$$ is equal to

Let $$\alpha$$ be the area of the larger region bounded by the curve $$y^2 = 8x$$ and the lines $$y = x$$ and $$x = 2$$, which lies in the first quadrant. Then the value of $$3\alpha$$ is equal to

If the equation of the plane passing through the point $$(1, 1, 2)$$ and perpendicular to the line $$x - 3y + 2z - 1 = 0 = 4x - y + z$$ is $$Ax + By + Cz = 1$$, then $$140(C - B + A)$$ is equal to

If $$\lambda_1 < \lambda_2$$ are two values of $$\lambda$$ such that the angle between the planes $$P_1: \vec{r} \cdot (3\hat{i} - 5\hat{j} + \hat{k}) = 7$$ and $$P_2: \vec{r} \cdot (\lambda\hat{i} + \hat{j} - 3\hat{k}) = 9$$ is $$\sin^{-1}\frac{2\sqrt{6}}{5}$$, then the square of the length of perpendicular from the point $$(38\lambda_1, 10\lambda_2, 2)$$ to the plane $$P_1$$ is