NTA JEE Main 28th July 2022 Shift 1

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