NTA JEE Main 29th January 2023 Shift 2 - Mathematics

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is:

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is

Let $$K$$ be the sum of the coefficients of the odd powers of $$x$$ in the expansion of $$(1+x)^{99}$$. Let $$a$$ be the middle term in the expansion of $$\left(2 + \frac{1}{\sqrt{2}}\right)^{200}$$. If $$\frac{^{200}C_{99}K}{a} = \frac{2^l m}{n}$$, where $$m$$ and $$n$$ are odd numbers, then the ordered pair $$(l, n)$$ is equal to:

The set of all values of $$\lambda$$ for which the equation $$\cos^2 2x - 2\sin^4 x - 2\cos^2 x = \lambda$$

If the tangent at a point P on the parabola $$y^2 = 3x$$ is parallel to the line $$x + 2y = 1$$ and the tangents at the points Q and R on the ellipse $$\frac{x^2}{4} + \frac{y^2}{1} = 1$$ are perpendicular to the line $$x - y = 2$$, then the area of the triangle $$PQR$$ is:

The statement $$B \Rightarrow ((\neg A) \vee B)$$ is not equivalent to:

Let $$R$$ be a relation defined on $$\mathbb{N}$$ as $$a R b$$ is $$2a + 3b$$ is a multiple of $$5, a, b \in \mathbb{N}$$. Then $$R$$ is

The set of all values of $$t \in \mathbb{R}$$, for which the matrix $$\begin{bmatrix} e^t & e^{-t}(\sin t - 2\cos t) & e^{-t}(-2\sin t - \cos t) \\ e^t & e^{-t}(2\sin t + \cos t) & e^{-t}(\sin t - 2\cos t) \\ e^t & e^{-t}\cos t & e^{-t}\sin t \end{bmatrix}$$ is invertible, is

Let $$f$$ and $$g$$ be twice differentiable functions on $$R$$ such that
$$f''(x) = g''(x) + 6x$$
$$f'(1) = 4g'(1) - 3 = 9$$
$$f(2) = 3, g(2) = 12$$
Then which of the following is NOT true?

The value of the integral $$\int_1^2 \left(\frac{t^4+1}{t^6+1}\right) dt$$ is:

The value of the integral $$\int_{1/2}^{2} \frac{\tan^{-1}x}{x} dx$$ is equal to

The area of the region $$A = \{(x,y) : |\cos x - \sin x| \leq y \leq \sin x, 0 \leq x \leq \frac{\pi}{2}\}$$

Let $$y = y(x)$$ be the solution of the differential equation $$x \log_e x \frac{dy}{dx} + y = x^2 \log_e x$$, $$(x > 1)$$. If $$y(2) = 2$$, then $$y(e)$$ is equal to

If $$\vec{a} = \hat{i} + 2\hat{k}$$, $$\vec{b} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{c} = 7\hat{i} - 3\hat{j} + 4\hat{k}$$, $$\vec{r} \times \vec{b} + \vec{b} \times \vec{c} = \vec{0}$$ and $$\vec{r} \cdot \vec{a} = 0$$ then $$\vec{r} \cdot \vec{c}$$ is equal to:

Let $$\vec{a} = 4\hat{i} + 3\hat{j}$$ and $$\vec{b} = 3\hat{i} - 4\hat{j} + 5\hat{k}$$ and $$\vec{c}$$ is a vector such that $$\vec{c} \cdot (\vec{a} \times \vec{b}) + 25 = 0$$, $$\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = 4$$ and projection of $$\vec{c}$$ on $$\vec{a}$$ is $$1$$, then the projection of $$\vec{c}$$ on $$\vec{b}$$ equals:

Shortest distance between the lines $$\frac{x-1}{2} = \frac{y+8}{-7} = \frac{z-4}{5}$$ and $$\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-6}{-3}$$ is:

The plane $$2x - y + z = 4$$ intersects the line segment joining the points $$A(a, -2, 4)$$ and $$B(2, b, -3)$$ at the point $$C$$ in the ratio $$2 : 1$$ and the distance of the point $$C$$ from the origin is $$\sqrt{5}$$. If $$ab < 0$$ and $$P$$ is the point $$(a-b, b, 2b-a)$$ then $$CP^2$$ is equal to:

If the lines $$\frac{x-1}{1} = \frac{y-2}{2} = \frac{z+3}{1}$$ and $$\frac{x-a}{2} = \frac{y+2}{3} = \frac{z-3}{1}$$ intersects at the point $$P$$, then the distance of the point $$P$$ from the plane $$z = a$$ is:

Let $$S = \{w_1, w_2, \ldots\}$$ be the sample space associated to a random experiment. Let $$P(w_n) = \frac{P(w_{n-1})}{2}$$, $$n \geq 2$$. Let $$A = \{2k + 3l; k, l \in \mathbb{N}\}$$ and $$B = \{w_n; n \in A\}$$. Then $$P(B)$$ is equal to

Let $$\alpha_1, \alpha_2, \ldots, \alpha_7$$ be the roots of the equation $$x^7 + 3x^5 - 13x^3 - 15x = 0$$ and $$|\alpha_1| \geq |\alpha_2| \geq \ldots \geq |\alpha_7|$$. Then, $$\alpha_1\alpha_2 - \alpha_3\alpha_4 + \alpha_5\alpha_6$$ is equal to ______.

Let $$\alpha = 8 - 14i$$, $$A = \left\{z \in \mathbb{C} : \frac{\alpha\bar{z} - \bar{\alpha}z}{z^2 - (\bar{z})^2 - 112i} = 1\right\}$$ and $$B = \{z \in \mathbb{C} : |z + 3i| = 4\}$$. Then, $$\sum_{z \in A \cap B} (Re\ z - Im\ z)$$ is equal to ______.

The total number of 4-digit numbers whose greatest common divisor with $$54$$ is $$2$$, is

Let $$a_1 = b_1 = 1$$ and $$a_n = a_{n-1} + (n-1)$$, $$b_n = b_{n-1} + a_{n-1}$$, $$\forall n \geq 2$$. If $$S = \sum_{n=1}^{10} \left(\frac{b_n}{2^n}\right)$$ and $$T = \sum_{n=1}^{8} \frac{n}{2^{n-1}}$$ then $$2^7(2S - T)$$ is equal to

Let $$\{a_k\}$$ and $$\{b_k\}$$, $$k \in \mathbb{N}$$, be two G.P.s with common ratio $$r_1$$ and $$r_2$$ respectively such that $$a_1 = b_1 = 4$$ and $$r_1 < r_2$$. Let $$c_k = a_k + b_k$$, $$k \in \mathbb{N}$$. If $$c_2 = 5$$ and $$c_3 = \frac{13}{4}$$ then $$\sum_{k=1}^{\infty} c_k - (12a_6 + 8b_4)$$ is equal to

A circle with centre $$(2, 3)$$ and radius $$4$$ intersects the line $$x + y = 3$$ at the points $$P$$ and $$Q$$. If the tangents at $$P$$ and $$Q$$ intersect at the point $$S(\alpha, \beta)$$, then $$4\alpha - 7\beta$$ is equal to

A triangle is formed by the tangents at the point $$(2, 2)$$ on the curves $$y^2 = 2x$$ and $$x^2 + y^2 = 4x$$, and the line $$x + y + 2 = 0$$. If $$r$$ is the radius of its circumcircle, then $$r^2$$ is equal to

Let $$X = \{11, 12, 13, \ldots, 40, 41\}$$ and $$Y = \{61, 62, 63, \ldots, 90, 91\}$$ be the two sets of observations. If $$\bar{x}$$ and $$\bar{y}$$ are their respective means and $$\sigma^2$$ is the variance of all the observations in $$X \cup Y$$, then $$|\bar{x} + \bar{y} - \sigma^2|$$ is equal to

Let A be a symmetric matrix such that $$|A| = 2$$ and $$\begin{bmatrix} 2 & 1 \\ 3 & \frac{3}{2} \end{bmatrix} A = \begin{bmatrix} 1 & 2 \\ \alpha & \beta \end{bmatrix}$$. If the sum of the diagonal elements of A is $$s$$, then $$\frac{\beta s}{\alpha^2}$$ is equal to ______.

Consider a function $$f : \mathbb{N} \to \mathbb{R}$$, satisfying $$f(1) + 2f(2) + 3f(3) + \ldots + xf(x) = x(x+1)f(x)$$; $$x \geq 2$$ with $$f(1) = 1$$. Then $$\frac{1}{f(2022)} + \frac{1}{f(2028)}$$ is equal to

If the equation of the normal to the curve $$y = \frac{x-a}{(x+b)(x-2)}$$ at the point $$(1, -3)$$ is $$x - 4y = 13$$ then the value of $$a + b$$ is equal to ______.