NTA JEE Main 25th January 2023 Shift 1 - Mathematics

Let $$z_1 = 2 + 3i$$ and $$z_2 = 3 + 4i$$. The set $$S = \{z \in \mathbb{C} : |z - z_1|^2 - |z - z_2|^2 = |z_1 - z_2|^2\}$$ represents a

If $$a_r$$ is the coefficient of $$x^{10-r}$$ in the Binomial expansion of $$(1+x)^{10}$$, then $$\sum_{r=1}^{10} r^3 \left(\frac{a_r}{a_{r-1}}\right)^2$$ is equal to

The points of intersection of the line $$ax + by = 0$$, ($$a \neq b$$) and the circle $$x^2 + y^2 - 2x = 0$$ are $$A(\alpha, 0)$$ and $$B(1, \beta)$$. The image of the circle with $$AB$$ as a diameter in the line $$x + y + 2 = 0$$ is:

The distance of the point $$(6, -2\sqrt{2})$$ from the common tangent $$y = mx + c$$, $$m > 0$$, of the curves $$x = 2y^2$$ and $$x = 1 + y^2$$ is

The value of $$\lim_{n \to \infty} \frac{1+2-3+4+5-6+\ldots+(3n-2)+(3n-1)-3n}{\sqrt{2n^4+4n+3} - \sqrt{n^4+5n+4}}$$ is

The statement $$(p \wedge (\sim q)) \Rightarrow (p \Rightarrow (\sim q))$$ is

The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2, then their new variance is equal to:

Let $$x, y, z > 1$$ and $$A = \begin{bmatrix} 1 & \log_x y & \log_x z \\ \log_y x & 2 & \log_y z \\ \log_z x & \log_z y & 3 \end{bmatrix}$$. Then $$|adj(adj A^2)|$$ is equal to

Let $$S_1$$ and $$S_2$$ be respectively the sets of all $$a \in \mathbb{R} - \{0\}$$ for which the system of linear equations
$$ax + 2ay - 3az = 1$$
$$(2a+1)x + (2a+3)y + (a+1)z = 2$$
$$(3a+5)x + (a+5)y + (a+2)z = 3$$
has unique solution and infinitely many solutions. Then

Let $$f : (0,1) \to \mathbb{R}$$ be a function defined by $$f(x) = \frac{1}{1-e^{-x}}$$, and $$g(x) = (f(-x) - f(x))$$. Consider two statements
(I) $$g$$ is an increasing function in $$(0, 1)$$
(II) $$g$$ is one-one in $$(0, 1)$$
Then,

Let $$y(x) = (1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$$. Then $$y' - y''$$ at $$x = -1$$ is equal to

Let $$x = 2$$ be a local minima of the function $$f(x) = 2x^4 - 18x^2 + 8x + 12$$, $$x \in (-4, 4)$$. If $$M$$ is local maximum value of the function $$f$$ in $$(-4, 4)$$, then $$M =$$

The minimum value of the function $$f(x) = \int_0^2 e^{|x-t|} dt$$ is

Let $$y = y(x)$$ be the solution curve of the differential equation $$\frac{dy}{dx} = \frac{y}{x}(1 - xy^2(1 + \log_e x))$$, $$x \gt 0$$, $$y(1) = 3$$. Then $$\frac{y^2(x)}{9}$$ is equal to:

The distance of the point $$P(4, 6, -2)$$ from the line passing through the point $$(-3, 2, 3)$$ and parallel to a line with direction ratios $$3, 3, -1$$ is equal to:

Let M be the maximum value of the product of two positive integers when their sum is 66. Let the sample space $$S = \{x \in \mathbb{Z} : x(66-x) \geq \frac{5}{9}M\}$$ and the event $$A = \{x \in S : x$$ is a multiple of 3$$\}$$. Then P(A) is equal to

Let $$S = \{\alpha : \log_2(9^{2\alpha-4} + 13) - \log_2(\frac{5}{2} \cdot 3^{2\alpha-4} + 1) = 2\}$$. Then the maximum value of $$\beta$$ for which the equation $$x^2 - 2(\sum_{\alpha \in s} \alpha)^2 x + \sum_{\alpha \in s}(\alpha+1)^2\beta = 0$$ has real roots, is _____.

Let $$x$$ and $$y$$ be distinct integers where $$1 \leq x \leq 25$$ and $$1 \leq y \leq 25$$. Then, the number of ways of choosing $$x$$ and $$y$$, such that $$x + y$$ is divisible by 5, is _____.

Let $$S = \{1, 2, 3, 5, 7, 10, 11\}$$. The number of non-empty subsets of $$S$$ that have the sum of all elements a multiple of 3, is _____.

Let $$A_1, A_2, A_3$$ be the three A.P. with the same common difference $$d$$ and having their first terms as $$A, A+1, A+2$$, respectively. Let $$a, b, c$$ be the 7$$^{th}$$, 9$$^{th}$$, 17$$^{th}$$ terms of $$A_1$$, $$A_2$$, $$A_3$$, respectively such that $$\begin{vmatrix} a & 7 & 1 \\ 2b & 17 & 1 \\ c & 17 & 1 \end{vmatrix} + 70 = 0$$. If $$a = 29$$, then the sum of first 20 terms of an AP whose first term is $$c - a - b$$ and common difference is $$\frac{d}{12}$$, is equal to _____.

The constant term in the expansion of $$\left(2x + \frac{1}{x^7} + 3x^2\right)^5$$ is _____.

The vertices of a hyperbola H are $$(\pm 6, 0)$$ and its eccentricity is $$\frac{\sqrt{5}}{2}$$. Let N be the normal to H at a point in the first quadrant and parallel to the line $$\sqrt{2}x + y = 2\sqrt{2}$$. If $$d$$ is the length of the line segment of N between H and the y-axis then $$d^2$$ is equal to _____.

If the sum of all the solutions of $$\tan^{-1}\left(\frac{2x}{1-x^2}\right) + \cot^{-1}\left(\frac{1-x^2}{2x}\right) = \frac{\pi}{3}$$, $$-1 < x < 1$$, $$x \neq 0$$, is $$\alpha - \frac{4}{\sqrt{3}}$$, then $$\alpha$$ is equal to _____.

For some $$a, b, c \in \mathbb{N}$$, let $$f(x) = ax - 3$$ and $$g(x) = x^b + c$$, $$x \in \mathbb{R}$$. If $$(f \circ g)^{-1}(x) = \left(\frac{x-7}{2}\right)^{1/3}$$, then $$(f \circ g)(ac) + (g \circ f)(b)$$ is equal to _____.

Let $$f(x) = \int \frac{2x}{(x^2+1)(x^2+3)} dx$$. If $$f(3) = \frac{1}{2}(\log_e 5 - \log_e 6)$$, then $$f(4)$$ is equal to

If the area enclosed by the parabolas $$P_1: 2y = 5x^2$$ and $$P_2: x^2 - y + 6 = 0$$ is equal to the area enclosed by $$P_1$$ and $$y = \alpha x$$, $$\alpha > 0$$, then $$\alpha^3$$ is equal to _____.

Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three non zero vectors such that $$\vec{b} \cdot \vec{c} = 0$$ and $$\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\vec{b} - \vec{c}}{2}$$. If $$\vec{d}$$ be a vector such that $$\vec{b} \cdot \vec{d} = \vec{a} \cdot \vec{b}$$, then $$(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d})$$ is equal to

The vector $$\vec{a} = -\hat{i} + 2\hat{j} + \hat{k}$$ is rotated through a right angle, passing through the y-axis in its way and the resulting vector is $$\vec{b}$$. Then the projection of $$3\vec{a} + \sqrt{2}\vec{b}$$ on $$\vec{c} = 5\hat{i} + 4\hat{j} + 3\hat{k}$$ is

Consider the lines $$L_1$$ and $$L_2$$ given by
$$L_1: \frac{x-1}{2} = \frac{y-3}{1} = \frac{z-2}{2}$$
$$L_2: \frac{x-2}{1} = \frac{y-2}{2} = \frac{z-3}{3}$$
A line $$L_3$$ having direction ratios $$1, -1, -2$$, intersects $$L_1$$ and $$L_2$$ at the points $$P$$ and $$Q$$ respectively. Then the length of line segment $$PQ$$ is

Let the equation of the plane passing through the line $$x - 2y - z - 5 = 0 = x + y + 3z - 5$$ and parallel to the line $$x + y + 2z - 7 = 0 = 2x + 3y + z - 2$$ be $$ax + by + cz = 65$$. Then the distance of the point $$(a, b, c)$$ from the plane $$2x + 2y - z + 16 = 0$$ is _____.