NTA JEE Main 27th July 2022 Shift 1 - Mathematics

Let the minimum value $$v_0$$ of $$v=|z|^{2} + |z-3|^{2} + |z-6i|^{2}, z ∈ \mathbb{C} $$ s attained at $$z=z_{0}$$. Then $$|2z_0^2- \overline{z}_0^3+3|^{2}+v_0^2 $$ is equal to

Suppose $$a_1, a_2, \ldots, a_n$$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms to the sum of first nine terms of the progression is $$5:12$$and $$110 < a_{15} < 120$$, then the sum of the first ten terms of the progression is equal to 

The remainder when $$(2021)^{2022} + (2022)^{2021}$$ is divided by $$7$$ is

Let $$A(1, 1)$$, $$B(-4, 3)$$, $$C(-2, -5)$$ be vertices of a triangle $$ABC$$, $$P$$ be a point on side $$BC$$, and $$\Delta_1$$ and $$\Delta_2$$ be the areas of triangle $$APB$$ and $$ABC$$ respectively. If $$\Delta_1 : \Delta_2 = 4 : 7$$, then the area enclosed by the lines $$AP$$, $$AC$$ and the $$x$$-axis is

If the circle $$x^2 + y^2 - 2gx + 6y - 19c = 0$$, $$g, c \in \mathbb{R}$$ passes through the point $$(6, 1)$$ and its centre lies on the line $$x - 2cy = 8$$, then the length of intercept made by the circle on $$x$$-axis is

Let $$P(a, b)$$ be a point on the parabola $$y^2 = 8x$$ such that the tangent at $$P$$ passes through the centre of the circle $$x^2 + y^2 - 10x - 14y + 65 = 0$$. Let $$A$$ be the product of all possible values of $$a$$ and $$B$$ be the product of all possible values of $$b$$. Then the value of $$A + B$$ is equal to

Let $$f : \mathbb{R} \to \mathbb{R}$$ be a function defined as $$f(x) = a \sin\left(\frac{\pi[x]}{2}\right) + [2 - x]$$, $$a \in \mathbb{R}$$, where $$[t]$$ is the greatest integer less than or equal to $$t$$. If $$\lim_{x \to -1} f(x)$$ exists, then the value of $$\int_0^4 f(x) \, dx$$ is equal to

$$(p \wedge r) \Leftrightarrow (p \wedge (\sim q))$$ is equivalent to $$(\sim p)$$ when $$r$$ is

Let a vertical tower $$AB$$ of height $$2h$$ stands on a horizontal ground. Let from a point $$P$$ on the ground a man can see upto height $$h$$ of the tower with an angle of elevation $$2\alpha$$. When from $$P$$, he moves a distance $$d$$ in the direction of $$\overrightarrow{AP}$$, he can see the top of the tower with an angle of elevation $$\alpha$$. If $$d = \sqrt{7}h$$, then $$\tan \alpha$$ is equal to

Let $$R_1$$ and $$R_2$$ be two relations defined on $$\mathbb{R}$$ by $$a R_1 b \Leftrightarrow ab \geq 0$$ and $$a R_2 b \Leftrightarrow a \geq b$$, then

Let $$A = \begin{pmatrix} 1 & 2 \\ -2 & -5 \end{pmatrix}$$. Let $$\alpha, \beta \in \mathbb{R}$$ be such that $$\alpha A^2 + \beta A = 2I$$. Then $$\alpha + \beta$$ is equal to

Let $$f, g : \mathbb{N} - \{1\} \to \mathbb{N}$$ be functions defined by $$f(a) = \alpha$$, where $$\alpha$$ is the maximum of the powers of those primes $$p$$ such that $$p^\alpha$$ divides $$a$$, and $$g(a) = a + 1$$, for all $$a \in \mathbb{N} - \{1\}$$. Then, the function $$f + g$$ is

Let a function $$f : \mathbb{R} \to \mathbb{R}$$ be defined as:

$$f(x) = \begin{cases} \int_0^x (5 - |t - 3|) \, dt, & x > 4 \\ x^2 + bx, & x \leq 4 \end{cases}$$

where $$b \in \mathbb{R}$$. If $$f$$ is continuous at $$x = 4$$, then which of the following statements is NOT true?

$$I = \int_{\pi/4}^{\pi/3} \left(\frac{8 \sin x - \sin 2x}{x}\right) dx$$. Then

The area of the smaller region enclosed by the curves $$y^2 = 8x + 4$$ and $$x^2 + y^2 + 4\sqrt{3}x - 4 = 0$$ is equal to

Let $$y = y_1(x)$$ and $$y = y_2(x)$$ be two distinct solutions of the differential equation $$\frac{dy}{dx} = x + y$$, with $$y_1(0) = 0$$ and $$y_2(0) = 1$$ respectively. Then, the number of points of intersection of $$y = y_1(x)$$ and $$y = y_2(x)$$ is

Let $$\vec{a} = \alpha \hat{i} + \hat{j} + \beta \hat{k}$$ and $$\vec{b} = 3\hat{i} - 5\hat{j} + 4\hat{k}$$ be two vectors, such that $$\vec{a} \times \vec{b} = -\hat{i} + 9\hat{j} + 12\hat{k}$$. Then the projection of $$\vec{b} - 2\vec{a}$$ on $$\vec{b} + \vec{a}$$ is equal to

Let $$\vec{a} = 2\hat{i} - \hat{j} + 5\hat{k}$$ and $$\vec{b} = \alpha \hat{i} + \beta \hat{j} + 2\hat{k}$$. If $$\left(\left(\vec{a} \times \vec{b}\right) \times \hat{i}\right) \cdot \hat{k} = \frac{23}{2}$$, then $$\left|\vec{b} \times 2\hat{j}\right|$$ is equal to

If the plane $$P$$ passes through the intersection of two mutually perpendicular planes $$2x + ky - 5z = 1$$ and $$3kx - ky + z = 5$$, $$k < 3$$ and intercepts a unit length on positive $$x$$-axis, then the intercept made by the plane $$P$$ on the $$y$$-axis is

Let $$S$$ be the sample space of all five digit numbers. If $$p$$ is the probability that a randomly selected number from $$S$$, is a multiple of $$7$$ but not divisible by $$5$$, then $$9p$$ is equal to

Let $$S = \{z \in \mathbb{C} : \bar{z}^2 + \bar{z} = 0\}$$. Then $$\sum_{z \in S} (\text{Re}(z) + \text{Im}(z))$$ is equal to______.

Let $$f(x) = 2x^2 - x - 1$$ and $$S = \{n \in \mathbb{Z} : |f(n)| \leq 800\}$$. Then, the value of $$\sum_{n \in S} f(n)$$ is equal to

If the length of the latus rectum of the ellipse $$x^2 + 4y^2 + 2x + 8y - \lambda = 0$$ is $$4$$, and $$l$$ is the length of its major axis, then $$\lambda + l$$ is equal to_______.

An ellipse $$E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ passes through the vertices of the hyperbola $$H : \frac{x^2}{49} - \frac{y^2}{64} = -1$$. Let the major and minor axes of the ellipse $$E$$ coincide with the transverse and conjugate axes of the hyperbola $$H$$. Let the product of the eccentricities of $$E$$ and $$H$$ be $$\frac{1}{2}$$. If $$l$$ is the length of the latus rectum of the ellipse $$E$$, then the value of $$113l$$ is equal to______.

The mean and variance of $$10$$ observations were calculated as $$15$$ and $$15$$ respectively by a student who took by mistake $$25$$ instead of $$15$$ for one observation. Then, the correct standard deviation is______.

Let $$S$$ be the set containing all $$3 \times 3$$ matrices with entries from $$\{-1, 0, 1\}$$. The total number of matrices $$A \in S$$ such that the sum of all the diagonal elements of $$A^T A$$ is $$6$$ is

For $$k \in \mathbb{R}$$, let the solutions of the equation $$\cos\left(\sin^{-1}\left(x \cot\left(\tan^{-1}\left(\cos(\sin^{-1} x)\right)\right)\right)\right) = k$$, $$0 < |x| < \frac{1}{\sqrt{2}}$$ be $$\alpha$$ and $$\beta$$, where the inverse trigonometric functions take only principal values. If the solutions of the equation $$x^2 - bx - 5 = 0$$ are $$\frac{1}{\alpha^2} + \frac{1}{\beta^2}$$ and $$\frac{\alpha}{\beta}$$, then $$\frac{b}{k^2}$$ is equal to

Let $$M$$ and $$N$$ be the number of points on the curve $$y^5 - 9xy + 2x = 0$$, where the tangents to the curve are parallel to $$x$$-axis and $$y$$-axis, respectively. Then the value of $$M + N$$ equals

Let $$y = y(x)$$ be the solution curve of the differential equation $$\sin(2x^2) \log_e(\tan x^2) \, dy + \left(4xy - 4\sqrt{2}x \sin\left(x^2 - \frac{\pi}{4}\right)\right) dx = 0$$, $$0 < x < \sqrt{\frac{\pi}{2}}$$, which passes through the point $$\left(\sqrt{\frac{\pi}{6}}, 1\right)$$. Then $$\left|y\left(\sqrt{\frac{\pi}{3}}\right)\right|$$ is equal to

Let the line $$\dfrac{x-3}{7} = \dfrac{y-2}{-1} = \dfrac{z-3}{-4}$$ intersect the plane containing the lines $$\dfrac{x-4}{1} = \dfrac{y+1}{-2} = \dfrac{z}{1}$$ and $$4ax - y + 5z - 7a = 0 = 2x - 5y - z - 3$$, $$a \in \mathbb{R}$$ at the point $$P(\alpha, \beta, \gamma)$$. Then the value of $$\alpha + \beta + \gamma$$  equals ______.