NTA JEE Main 29th July 2022 Shift 1 - Mathematics

If $$z = 2 + 3i$$, then $$z^5 + \bar{z}^5$$ is equal to:

If $$\frac{1}{(20-a)(40-a)} + \frac{1}{(40-a)(60-a)} + \ldots + \frac{1}{(180-a)(200-a)} = \frac{1}{256}$$, then the maximum value of $$a$$ is

Let the circumcentre of a triangle with vertices $$A(a, 3)$$, $$B(b, 5)$$ and $$C(a, b)$$, $$ab > 0$$ be $$P(1, 1)$$. If the line AP intersects the line BC at the point $$Q(k_1, k_2)$$, then $$k_1 + k_2$$ is equal to

Let a line L pass through the point of intersection of the lines $$bx + 10y - 8 = 0$$ and $$2x - 3y = 0$$, $$b \in \mathbb{R} - \{\frac{4}{3}\}$$. If the line L also passes through the point (1, 1) and touches the circle $$17(x^2 + y^2) = 16$$, then the eccentricity of the ellipse $$\frac{x^2}{5} + \frac{y^2}{b^2} = 1$$ is

Let the focal chord of the parabola $$P: y^2 = 4x$$ along the line $$L: y = mx + c, m > 0$$ meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola $$H: x^2 - y^2 = 4$$. If O is the vertex of P and F is the focus of H on the positive x-axis, then the area of the quadrilateral OMFN is

If $$\lim_{x \to 0} \frac{\alpha e^x + \beta e^{-x} + \gamma \sin x}{x \sin^2 x} = \frac{2}{3}$$, where $$\alpha, \beta, \gamma \in \mathbb{R}$$, then which of the following is NOT correct?

The statement $$(p \wedge q) \Rightarrow (p \wedge r)$$ is equivalent to

The angle of elevation of the top of a tower from a point A due north of it is $$\alpha$$ and from a point B at a distance of 9 units due west of A is $$\cos^{-1}\left(\frac{3}{\sqrt{13}}\right)$$. If the distance of the point B from the tower is 15 units, then $$\cot\alpha$$ is equal to

Let R be a relation from the set $$\{1, 2, 3, \ldots, 60\}$$ to itself such that $$R = \{(a, b) : b = pq$$, where $$p, q \geq 3$$ are prime numbers$$\}$$. Then, the number of elements in R is

Let A and B be two $$3 \times 3$$ non-zero real matrices such that AB is a zero matrix. Then

The number of points, where the function $$f: \mathbb{R} \to \mathbb{R}$$, $$f(x) = |x - 1|\cos|x - 2|\sin|x - 1| + (x - 3)|x^2 - 5x + 4|$$, is NOT differentiable, is

Let $$f(x) = 3(x^2 - 2)^3 + 4$$, $$x \in \mathbb{R}$$. Then which of the following statements are true?
P: $$x = 0$$ is a point of local minima of f
Q: $$x = \sqrt{2}$$ is a point of inflection of f
R: $$f'$$ is increasing for $$x > \sqrt{2}$$

The integral $$\int_0^{\pi/2} \frac{1}{3 + 2\sin x + \cos x} dx$$ is equal to:

If $$f(\alpha) = \int_1^\alpha \frac{\log_{10} t}{1+t} dt$$, $$\alpha > 0$$, then $$f(e^3) + f(e^{-3})$$ is equal to

The area of the region $$\{(x, y) : |x - 1| \leq y \leq \sqrt{5 - x^2}\}$$ is equal to

Let the solution curve $$y = y(x)$$ of the differential equation $$(1 + e^{2x})\left(\frac{dy}{dx} + y\right) = 1$$ pass through the point $$\left(0, \frac{\pi}{2}\right)$$. Then, $$\lim_{x \to \infty} e^x y(x)$$ is equal to

Let $$\vec{a} = 3\hat{i} + \hat{j}$$ and $$\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$$. Let $$\vec{c}$$ be a vector satisfying $$\vec{a} \times (\vec{b} \times \vec{c}) = \vec{b} + \lambda\vec{c}$$. If $$\vec{b}$$ and $$\vec{c}$$ are non-parallel, then the value of $$\lambda$$ is

Let $$\hat{a}$$ and $$\hat{b}$$ be two unit vectors such that the angle between them is $$\frac{\pi}{4}$$. If $$\theta$$ is the angle between the vectors $$(\hat{a} + \hat{b})$$ and $$(\hat{a} + 2\hat{b} + 2(\hat{a} \times \hat{b}))$$ then the value of $$164\cos^2\theta$$ is equal to

If the foot of the perpendicular from the point $$A(-1, 4, 3)$$ on the plane $$P: 2x + my + nz = 4$$, is $$\left(-2, \frac{7}{2}, \frac{3}{2}\right)$$, then the distance of the point A from the plane P, measured parallel to a line with direction ratios 3, -1, -4, is equal to

Let $$S = \{1, 2, 3, \ldots, 2022\}$$. Then the probability, that a randomly chosen number n from the set S such that $$HCF(n, 2022) = 1$$, is

Let $$S = \{4, 6, 9\}$$ and $$T = \{9, 10, 11, \ldots, 1000\}$$. If $$A = \{a_1 + a_2 + \ldots + a_k : k \in \mathbb{N}, a_1, a_2, a_3, \ldots, a_k \in S\}$$, then the sum of all the elements in the set $$T - A$$ is equal to _______

Let $$a_1, a_2, a_3, \ldots$$ be an A.P. If $$\sum_{r=1}^{\infty} \frac{a_r}{2^r} = 4$$, then $$4a_2$$ is equal to ______

If $$\frac{1}{2 \times 3 \times 4} + \frac{1}{3 \times 4 \times 5} + \frac{1}{4 \times 5 \times 6} + \ldots + \frac{1}{100 \times 101 \times 102} = \frac{k}{101}$$, then $$34k$$ is equal to _______

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $$\left(\sqrt[4]{2} + \frac{1}{\sqrt[4]{3}}\right)^n$$, in the increasing powers of $$\frac{1}{\sqrt[4]{3}}$$ be $$\sqrt[4]{6} : 1$$. If the sixth term from the beginning is $$\frac{\alpha}{\sqrt[4]{3}}$$, then $$\alpha$$ is equal to _______

Let $$S = \{\theta \in (0, 2\pi) : 7\cos^2\theta - 3\sin^2\theta - 2\cos^2(2\theta) = 2\}$$. Then the sum of roots of all the equations $$x^2 - 2(\tan^2\theta + \cot^2\theta)x + 6\sin^2\theta = 0$$, $$\theta \in S$$, is _______

Let the mirror image of a circle $$c_1: x^2 + y^2 - 2x - 6y + \alpha = 0$$ in line $$y = x + 1$$ be $$c_2: 5x^2 + 5y^2 + 10gx + 10fy + 38 = 0$$. If r is the radius of circle $$c_2$$, then $$\alpha + 6r^2$$ is equal to ______

Let the mean and the variance of 20 observations $$x_1, x_2, \ldots, x_{20}$$ be 15 and 9, respectively. For $$\alpha \in \mathbb{R}$$, if the mean of $$(x_1 + \alpha)^2, (x_2 + \alpha)^2, \ldots, (x_{20} + \alpha)^2$$ is 178, then the square of the maximum value of $$\alpha$$ is equal to _______

The number of matrices of order $$3 \times 3$$, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is _______

Let p and p+2 be prime numbers and let $$\Delta = \begin{vmatrix} p! & (p+1)! & (p+2)! \\ (p+1)! & (p+2)! & (p+3)! \\ (p+2)! & (p+3)! & (p+4)! \end{vmatrix}$$
Then the sum of the maximum values of $$\alpha$$ and $$\beta$$, such that $$p^\alpha$$ and $$(p+2)^\beta$$ divide $$\Delta$$, is _______

Let a line with direction ratios $$a, -4a, -7$$ be perpendicular to the lines with direction ratios $$3, -1, 2b$$ and $$b, a, -2$$. If the point of intersection of the line $$\frac{x+1}{a^2+b^2} = \frac{y-2}{a^2-b^2} = \frac{z}{1}$$ and the plane $$x - y + z = 0$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + \gamma$$ is equal to ________