NTA JEE Main 28th July 2022 Shift 1 - Mathematics

Let $$S_1 = \{z_1 \in \mathbb{C} : |z_1 - 3| = \frac{1}{2}\}$$ and $$S_2 = \{z_2 \in \mathbb{C} : |z_2 - |z_2 + 1|| = |z_2 + |z_2 - 1||\}$$. Then, for $$z_1 \in S_1$$ and $$z_2 \in S_2$$, the least value of $$|z_2 - z_1|$$ is

If the minimum value of $$f(x) = \frac{5x^2}{2} + \frac{\alpha}{x^5}, x \gt 0$$, is 14, then the value of $$\alpha$$ is equal to

Consider the sequence $$a_1, a_2, a_3, \ldots$$ such that $$a_1 = 1, a_2 = 2$$ and $$a_{n+2} = \frac{2}{a_{n+1}} + a_n$$ for $$n = 1, 2, 3, \ldots$$. If $$\frac{a_1 + \frac{1}{a_2}}{a_3} \cdot \frac{a_2 + \frac{1}{a_3}}{a_4} \cdot \frac{a_3 + \frac{1}{a_4}}{a_5} \cdots \frac{a_{30} + \frac{1}{a_{31}}}{a_{32}} = 2^\alpha \cdot {}^{61}C_{31}$$ then $$\alpha$$ is equal to

The remainder when $$7^{2022} + 3^{2022}$$ is divided by 5 is

For $$t \in (0, 2\pi)$$, if ABC is an equilateral triangle with vertices $$A(\sin t, -\cos t)$$, $$B(\cos t, \sin t)$$ and $$C(a, b)$$ such that its orthocentre lies on a circle with centre $$(1, \frac{1}{3})$$, then $$a^2 - b^2$$ is equal to

Let C be the centre of the circle $$x^2 + y^2 - x + 2y = \frac{11}{4}$$ and P be a point on the circle. A line passes through the point C, makes an angle of $$\frac{\pi}{4}$$ with the line CP and intersects the circle at the points Q and R. Then the area of the triangle PQR (in unit$$^2$$) is

If the tangents drawn at the points P and Q on the parabola $$y^2 = 2x - 3$$ intersect at the point $$R(0, 1)$$, then the orthocentre of the triangle PQR is

Let the operations $$*, \odot \in \{\wedge, \vee\}$$. If $$(p * q) \odot (p \odot \sim q)$$ is a tautology, then the ordered pair $$(*, \odot)$$ is

For $$\alpha \in \mathbb{N}$$, consider a relation R on $$\mathbb{N}$$ given by $$R = \{(x, y) : 3x + \alpha y$$ is a multiple of 7$$\}$$. The relation R is an equivalence relation if and only if

Let the matrix $$A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ and the matrix $$B_0 = A^{49} + 2A^{98}$$. If $$B_n = \text{Adj}(B_{n-1})$$ for all $$n \geq 1$$, then $$\det(B_4)$$ is equal to

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $$\cos^{-1}(x) - 2\sin^{-1}(x) = \cos^{-1}(2x)$$ is equal to

Let $$\alpha, \beta$$ and $$\gamma$$ be three positive real numbers. Let $$f(x) = \alpha x^5 + \beta x^3 + \gamma x$$, $$x \in \mathbb{R}$$ and $$g: \mathbb{R} \to \mathbb{R}$$ be such that $$g(f(x)) = x$$ for all $$x \in \mathbb{R}$$. If $$a_1, a_2, a_3, \ldots, a_n$$ be in arithmetic progression with mean zero, then the value of $$f\left(g\left(\frac{1}{n}\sum_{i=1}^{n} f(a_i)\right)\right)$$ is equal to

Considering only the principal values of the inverse trigonometric functions, the domain of the function $$f(x) = \cos^{-1}\left(\frac{x^2 - 4x + 2}{x^2 + 3}\right)$$ is

The minimum value of the twice differentiable function $$f(x) = \int_0^x e^{x-t} f'(t) dt - (x^2 - x + 1)e^x$$, $$x \in \mathbb{R}$$, is

Let the solution curve of the differential equation $$x dy = (\sqrt{x^2 + y^2} + y) dx$$, $$x > 0$$, intersect the line $$x = 1$$ at $$y = 0$$ and the line $$x = 2$$ at $$y = \alpha$$. Then the value of $$\alpha$$ is

If $$y = y(x)$$, $$x \in \left(0, \frac{\pi}{2}\right)$$ be the solution curve of the differential equation $$\sin^2(2x)\frac{dy}{dx} + (8\sin^2(2x) + 2\sin(4x))y = 2e^{-4x}(2\sin(2x) + \cos(2x))$$, with $$y\left(\frac{\pi}{4}\right) = e^{-\pi}$$, then $$y\left(\frac{\pi}{6}\right)$$ is equal to

Let the vectors $$\vec{a} = (1+t)\hat{i} + (1-t)\hat{j} + \hat{k}$$, $$\vec{b} = (1-t)\hat{i} + (1+t)\hat{j} + 2\hat{k}$$ and $$\vec{c} = t\hat{i} - t\hat{j} + \hat{k}$$, $$t \in \mathbb{R}$$ be such that for $$\alpha, \beta, \gamma \in \mathbb{R}$$, $$\alpha\vec{a} + \beta\vec{b} + \gamma\vec{c} = \vec{0} \Rightarrow \alpha = \beta = \gamma = 0$$. Then, the set of all values of $$t$$ is

Let a vector $$\vec{a}$$ has magnitude 9. Let a vector $$\vec{b}$$ be such that for every $$(x, y) \in \mathbb{R} \times \mathbb{R} - \{(0,0)\}$$, the vector $$x\vec{a} + y\vec{b}$$ is perpendicular to the vector $$6y\vec{a} - 18x\vec{b}$$. Then the value of $$|\vec{a} \times \vec{b}|$$ is equal to

The foot of the perpendicular from a point on the circle $$x^2 + y^2 = 1, z = 0$$ to the plane $$2x + 3y + z = 6$$ lies on which one of the following curves?

Out of 60% female and 40% male candidates appearing in an exam, 60% candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, that the chosen candidate is a female, is

The sum of all real values of $$x$$ for which $$\frac{3x^2 - 9x + 17}{x^2 + 3x + 10} = \frac{5x^2 - 7x + 19}{3x^2 + 5x + 12}$$ is equal to

Let S be the set of all passwords which are six to eight characters long, where each character is either an alphabet from {A, B, C, D, E} or a number from {1, 2, 3, 4, 5} with the repetition of characters allowed. If the number of passwords in S whose at least one character is a number from {1, 2, 3, 4, 5} is $$\alpha \times 5^6$$, then $$\alpha$$ is equal to

For $$p, q \in \mathbb{R}$$, consider the real valued function $$f(x) = (x - p)^2 - q$$, $$x \in \mathbb{R}$$ and $$q > 0$$. Let $$a_1, a_2, a_3$$ and $$a_4$$ be in an arithmetic progression with mean $$p$$ and positive common difference. If $$|f(a_i)| = 500$$ for all $$i = 1, 2, 3, 4$$, then the absolute difference between the roots of $$f(x) = 0$$ is

Let $$x_1, x_2, x_3, \ldots, x_{20}$$ be in geometric progression with $$x_1 = 3$$ and the common ratio $$\frac{1}{2}$$. A new data is constructed replacing each $$x_i$$ by $$(x_i - i)^2$$. If $$\bar{x}$$ is the mean of new data, then the greatest integer less than or equal to $$\bar{x}$$ is

For the hyperbola $$H: x^2 - y^2 = 1$$ and the ellipse $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a > b > 0$$, let the
(1) eccentricity of E be reciprocal of the eccentricity of H, and
(2) the line $$y = \sqrt{\frac{5}{2}}x + K$$ be a common tangent of E and H.
Then $$4(a^2 + b^2)$$ is equal to

$$\lim_{x \to 0} \left(\frac{(x+2\cos x)^3 + 2(x+2\cos x)^2 + 3\sin(x+2\cos x)}{(x+2)^3 + 2(x+2)^2 + 3\sin(x+2)}\right)^{\frac{100}{x}}$$ is equal to

Let $$A = \begin{pmatrix} 1 & -1 \\ 2 & \alpha \end{pmatrix}$$ and $$B = \begin{pmatrix} \beta & 1 \\ 1 & 0 \end{pmatrix}$$, $$\alpha, \beta \in \mathbb{R}$$. Let $$\alpha_1$$ be the value of $$\alpha$$ which satisfies $$(A + B)^2 = A^2 + \begin{pmatrix} 2 & 2 \\ 2 & 2 \end{pmatrix}$$ and $$\alpha_2$$ be the value of $$\alpha$$ which satisfies $$(A + B)^2 = B^2$$. Then $$|\alpha_1 - \alpha_2|$$ is equal to

Let $$f: [0, 1] \to \mathbb{R}$$ be a twice differentiable function in (0, 1) such that $$f(0) = 3$$ and $$f(1) = 5$$. If the line $$y = 2x + 3$$ intersects the graph of $$f$$ at only two distinct points in (0, 1), then the least number of points $$x \in (0, 1)$$, at which $$f''(x) = 0$$, is

If $$\int_0^{\sqrt{3}} \frac{15x^3}{\sqrt{(1+x^2)} + \sqrt{(1+x^2)^3}} dx = \alpha\sqrt{2} + \beta\sqrt{3}$$, where $$\alpha, \beta$$ are integers, then $$\alpha + \beta$$ is equal to

Let $$P(-2, -1, 1)$$ and $$Q\left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)$$ be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are $$\alpha, -1, \beta$$, where both $$\alpha$$ and $$\beta$$ are integers of minimum absolute values, then $$\alpha^2 + \beta^2$$ is equal to