NTA JEE Main 25th June 2022 Shift 1

Let a circle $$C$$ in complex plane pass through the points $$z_1 = 3 + 4i, z_2 = 4 + 3i$$ and $$z_3 = 5i$$. If $$z \neq z_1$$ is a point on $$C$$ such that the line through $$z$$ and $$z_1$$ is perpendicular to the line through $$z_2$$ and $$z_3$$, then $$\arg z$$ is equal to

If $$\frac{1}{2 \cdot 3^{10}} + \frac{1}{2^2 \cdot 3^9} + \cdots + \frac{1}{2^{10} \cdot 3} = \frac{K}{2^{10} \cdot 3^{10}}$$, then the remainder when $$K$$ is divided by $$6$$ is

Let a circle $$C$$ touch the lines $$L_1: 4x - 3y + K_1 = 0$$ and $$L_2: 4x - 3y + K_2 = 0$$, $$K_1, K_2 \in R$$. If a line passing through the centre of the circle $$C$$ intersects $$L_1$$ at $$(-1, 2)$$ and $$L_2$$ at $$(3, -6)$$, then the equation of the circle $$C$$ is

If $$y = m_1 x + c_1$$ and $$y = m_2 x + c_2$$, $$m_1 \neq m_2$$ are two common tangents of circle $$x^2 + y^2 = 2$$ and parabola $$y^2 = x$$, then the value of $$8|m_1 m_2|$$ is equal to

Let $$x = 2t, y = \frac{t^2}{3}$$ be a conic. Let $$S$$ be the focus and $$B$$ be the point on the axis of the conic such that $$SA \perp BA$$, where $$A$$ is any point on the conic. If $$k$$ is the ordinate of the centroid of the $$\triangle SAB$$, then $$\lim_{t \to 1} k$$ is equal to

Let $$f(x)$$ be a polynomial function such that $$f(x) + f'(x) + f''(x) = x^5 + 64$$. Then, the value of $$\lim_{x \to 1} \frac{f(x)}{x - 1}$$ is equal to

Consider the following two propositions :
$$P_1 : \sim p \to \sim q$$
$$P_2 : p \wedge \sim q \wedge \sim p \vee q$$
If the proposition $$p \to \sim p \vee q$$ is evaluated as FALSE, then

Let $$a, b$$ and $$c$$ be the length of sides of a triangle $$ABC$$ such that $$\frac{a+b}{7} = \frac{b+c}{8} = \frac{c+a}{9}$$. If $$r$$ and $$R$$ are the radius of incircle and radius of circumcircle of the triangle $$ABC$$, respectively, then the value of $$\frac{R}{r}$$ is equal to

Let $$A = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix}$$. If $$M$$ and $$N$$ are two matrices given by $$M = \sum_{k=1}^{10} A^{2k}$$ and $$N = \sum_{k=1}^{10} A^{2k-1}$$ then $$MN^2$$ is

Let $$A$$ be a $$3 \times 3$$ real matrix such that $$A\begin{pmatrix}1\\1\\0\end{pmatrix} = \begin{pmatrix}1\\1\\1\end{pmatrix}; A\begin{pmatrix}0\\0\\1\end{pmatrix} = \begin{pmatrix}-1\\0\\1\end{pmatrix}$$. If $$X = \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}^T$$ and $$I$$ is an identity matrix of order $$3$$, then the system $$(A - 2I)X = \begin{pmatrix}4\\1\\1\end{pmatrix}$$ has