NTA JEE Main 26th June 2022 Shift 1

Let $$A = \left\{z \in C : \left|\frac{z+1}{z-1}\right| < 1\right\}$$ and $$B = \left\{z \in C : \arg\left(\frac{z-1}{z+1}\right) = \frac{2\pi}{3}\right\}$$. Then $$A \cap B$$ is

The remainder when $$(2021)^{2023}$$ is divided by $$7$$ is

Let $$R$$ be the point $$(3, 7)$$ and let $$P$$ and $$Q$$ be two points on the line $$x + y = 5$$ such that $$PQR$$ is an equilateral triangle. Then the area of $$\triangle PQR$$ is

Let $$C$$ be a circle passing through the points $$A(2,-1)$$ and $$B(3,4)$$. The line segment $$AB$$ is not a diameter of $$C$$. If $$r$$ is the radius of $$C$$ and its centre lies on the circle $$(x-5)^2 + (y-1)^2 = \frac{13}{2}$$, then $$r^2$$ is equal to

Let the normal at the point $$P$$ on the parabola $$y^2 = 6x$$ pass through the point $$(5, -8)$$. If the tangent at $$P$$ to the parabola intersects its directrix at the point $$Q$$, then the ordinate of the point $$Q$$ is

$$\lim_{x \to \frac{1}{\sqrt{2}}} \frac{\sin(\cos^{-1} x) - x}{1 - \tan(\cos^{-1} x)}$$ is equal to

Let $$\Delta, \nabla \in \{\wedge, \vee\}$$ be such that $$p\nabla q \to ((p\Delta q)\nabla r)$$ is a tautology. Then $$(p\nabla q) \Delta r$$ is logically equivalent to

The mean of the numbers $$a, b, 8, 5, 10$$ is $$6$$ and their variance is $$6.8$$. If $$M$$ is the mean deviation of the numbers about the mean, then $$25M$$ is equal to

Let $$A$$ be a $$3 \times 3$$ invertible matrix. If $$|\text{adj}(24A)| = |\text{adj}(3 \text{ adj}(2A))|$$, then $$|A|^2$$ is equal to

The ordered pair $$(a, b)$$, for which the system of linear equations
$$3x - 2y + z = b$$
$$5x - 8y + 9z = 3$$
$$2x + y + az = -1$$
has no solution, is