JEE (Advanced) 2023 Paper-2

Let $$A_1, A_2, A_3, \ldots, A_8$$ be the vertices of a regular octagon that lie on a circle of radius 2. Let P be a point on the circle and let $$PA_i$$ denote the distance between the points P and $$A_i$$ for $$i = 1, 2, \ldots, 8$$. If P varies over the circle, then the maximum value of the product $$PA_1 \cdot PA_2 \cdots PA_8$$ is

Let $$R = \left\{\begin{pmatrix} a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0 \end{pmatrix} : a, b, c, d \in \{0, 3, 5, 7, 11, 13, 17, 19\}\right\}$$. Then the number of invertible matrices in R is

Let $$C_1$$ be the circle of radius 1 with center at the origin. Let $$C_2$$ be the circle of radius $$r$$ with center at the point $$A = (4, 1)$$, where $$1 < r < 3$$. Two distinct common tangents PQ and ST of $$C_1$$ and $$C_2$$ are drawn. The tangent PQ touches $$C_1$$ at P and $$C_2$$ at Q. The tangent ST touches $$C_1$$ at S and $$C_2$$ at T. Mid points of the line segments PQ and ST are joined to form a line which meets the x-axis at a point B. If $$AB = \sqrt{5}$$, then the value of $$r^2$$ is

Let a be the area of the triangle ABC. Then the value of $$(64a)^2$$ is

Then the inradius of the triangle ABC is

Let $$p_i$$ be the probability that a randomly chosen point has $$i$$ many friends, $$i = 0, 1, 2, 3, 4$$. Let $$X$$ be a random variable such that for $$i = 0, 1, 2, 3, 4$$, the probability $$P(X = i) = p_i$$. Then the value of $$7E(X)$$ is

Two distinct points are chosen randomly out of the points $$A_1, A_2, \ldots, A_{49}$$. Let $$p$$ be the probability that they are friends. Then the value of $$7p$$ is

An electric dipole is formed by two charges $$+q$$ and $$-q$$ located in xy-plane at (0, 2) mm and (0, -2) mm, respectively, as shown in the figure. The electric potential at point P(100, 100) mm due to the dipole is $$V_0$$. The charges $$+q$$ and $$-q$$ are then moved to the points (-1, 2) mm and (1, -2) mm, respectively. What is the value of electric potential at P due to the new dipole?

Young's modulus of elasticity $$Y$$ is expressed in terms of three derived quantities, namely, the gravitational constant $$G$$, Planck's constant $$h$$ and the speed of light c, as $$Y = c^\alpha h^\beta G^\gamma$$. Which of the following is the correct option?

A particle of mass m is moving in the $$xy$$-plane such that its velocity at a point $$(x, y)$$ is given as $$\vec{v} = \alpha(y\hat{x} + 2x\hat{y})$$, where $$\alpha$$ is a non-zero constant. What is the force $$\vec{F}$$ acting on the particle?