JEE (Advanced) 2021 Paper-1

Consider a triangle $$\triangle$$ whose two sides lie on the x-axis and the line $$x + y + 1 = 0$$. If the orthocenter of $$\triangle$$ is (1,1), then the equation of the circle passing through the vertices of the triangle $$\triangle$$ is

The area of the region
$$\left\{(x, y) : 0 \leq x \leq \frac{9}{4}, 0 \leq y \leq 1, x \geq 3y, x + y \geq 2\right\}$$
is

Consider three sets $$E_1 = \left\{1, 2, 3 \right\}, F_1 = \left\{1, 3, 4 \right\}$$ and $$G_1 = \left\{2, 3, 4, 5 \right\}$$. Two elements are chosen at random, without replacement, from the set $$E_1$$, and let $$S_1$$ denote the set of these chosen elements. Let $$E_2 = E_1 − S_1$$ and $$F_2 = F_1 \cup S_1$$. Now two elements are chosen at random, without replacement, from the set $$F_2$$ and let $$S_2$$ denote the set of these chosen elements.

Let $$G_2 = G_1 \cup S_2$$. Finally, two elements are chosen at random, without replacement, from the set $$G_2$$ and let $$S_3$$ denote the set of these chosen elements. Let $$E_3 = E_2 \cup S_3$$. Given that $$E_1 = E_3$$, let p be the conditional probability of the event $$S_1 = \left\{1, 2 \right\}$$. Then the value of p is

Let $$\theta_1, \theta_2, ......., \theta_{10}$$ be positive valued angles (in radian) such that $$\theta_1 + \theta_2 + ..... + \theta_{10} = 2 \pi$$. Define the complex numbers $$Z_1 = e^{i \theta_1}, Z_k = Z_{k - 1}e^{i \theta_k}$$ for $$k = 2, 3, ...., 10,$$ where $$i = \sqrt{-1}$$. Consider the statements P and Q given below:
$$P : \mid Z_2 - Z_1 \mid + \mid Z_3 - Z_2 \mid + .... + \mid Z_{10} - Z_9 \mid + \mid Z_1 - Z_{10} \mid \leq 2 \pi$$
$$Q : \mid Z_2^2 - Z_1^2 \mid + \mid Z_3^2 - Z_2^2 \mid + .... + \mid Z_{10}^2 - Z_9^2 \mid + \mid Z_1^2 - Z_{10}^2 \mid \leq 4 \pi$$
Then

The value of $$\frac{625}{4} p_1$$ is .........

The value of $$\frac{125}{4} p_2$$ is .........

The value of $$\mid M \mid$$ is ..........

The value of D is .........

The value of $$\lambda^2$$ is .........

The value of D is ..........