NTA JEE Main 27th July 2022 Shift 1

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