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

The area of the region

$$\left\{ {\matrix{ {(x,y):0 \le x \le {9 \over 4},} & {0 \le y \le 1,} & {x \ge 3y,} & {x + y \ge 2} \cr } } \right\}$$ is

Consider three sets E₁ = {1, 2, 3}, F₁ = {1, 3, 4} and G₁ = {2, 3, 4, 5}. Two elements are chosen at random, without replacement, from the set E₁, and let S₁ denote the set of these chosen elements. Let E₂ = E₁ $$-$$ S₁ and F₂ = F₁ $$\cup$$ S₁. Now two elements are chosen at random, without replacement, from the set F₂ and let S₂ denote the set of these chosen elements.

Let G₂ = G₁ $$\cup$$ S₂. Finally, two elements are chosen at random, without replacement, from the set G₂ and let S₃ denote the set of these chosen elements.

Let E₃ = E₂ $$\cup$$ S₃. Given that E₁ = E₃, let p be the conditional probability of the event S₁ = {1, 2}. Then the value of p is

Let $\theta_1, \theta_2, \ldots, \theta_{10}$ be positive valued angles (in radian) such that $\theta_1+\theta_2+\cdots+\theta_{10}=2 \pi$. Define the complex numbers $z_1=e^{i \theta_1}, z_k=z_{k-1} e^{i \theta_k}$ for $k=2,3, \ldots, 10$, where $i=\sqrt{-1}$. Consider the statements $P$ and $Q$ given below:

$$P:\left| {{z_2} - {z_1}} \right| + \left| {{z_3} - {z_2}} \right| + ..... + \left| {{z_{10}} - {z_9}} \right| + \left| {{z_1} - {z_{10}}} \right| \le 2\pi $$

$$Q:\left| {z_2^2 - z_1^2} \right| + \left| {z_3^2 - z_2^2} \right| + .... + \left| {z_{10}^2 - z_9^2} \right| + \left| {z_1^2 - z_{10}^2} \right| \le 4\pi $$

Then,

Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, ......, 100}. Let p₁ be the probability that the maximum of chosen numbers is at least 81 and p₂ be the probability that the minimum of chosen numbers is at most 40.

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

Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, ......, 100}. Let p₁ be the probability that the maximum of chosen numbers is at least 81 and p₂ be the probability that the minimum of chosen numbers is at most 40.

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

Let $$\alpha$$, $$\beta$$ and $$\gamma$$ be real numbers such that the system of linear equations

x + 2y + 3z = $$\alpha$$

4x + 5y + 6z = $$\beta$$

7x + 8y + 9z = $$\gamma $$ $$-$$ 1

is consistent. Let | M | represent the determinant of the matrix

$$M = \left[ {\matrix{ \alpha & 2 & \gamma \cr \beta & 1 & 0 \cr { - 1} & 0 & 1 \cr } } \right]$$

Let P be the plane containing all those ($$\alpha$$, $$\beta$$, $$\gamma$$) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.

The value of | M | is _________.

Let $$\alpha$$, $$\beta$$ and $$\gamma$$ be real numbers such that the system of linear equations

x + 2y + 3z = $$\alpha$$

4x + 5y + 6z = $$\beta$$

7x + 8y + 9z = $$\gamma $$ $$-$$ 1

is consistent. Let | M | represent the determinant of the matrix

$$M = \left[ {\matrix{ \alpha & 2 & \gamma \cr \beta & 1 & 0 \cr { - 1} & 0 & 1 \cr } } \right]$$

Let P be the plane containing all those ($$\alpha$$, $$\beta$$, $$\gamma$$) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.

The value of D is _________.

Consider the lines L₁ and L₂ defined by

$${L_1}:x\sqrt 2 + y - 1 = 0$$ and $${L_2}:x\sqrt 2 - y + 1 = 0$$

For a fixed constant $$\lambda$$, let C be the locus of a point P such that the product of the distance of P from L₁ and the distance of P from L₂ is $$\lambda$$². The line y = 2x + 1 meets C at two points R and S, where the distance between R and S is $$\sqrt {270} $$. Let the perpendicular bisector of RS meet C at two distinct points R' and S'. Let D be the square of the distance between R' and S'.

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

Consider the lines L₁ and L₂ defined by

$${L_1}:x\sqrt 2 + y - 1 = 0$$ and $${L_2}:x\sqrt 2 - y + 1 = 0$$

For a fixed constant $$\lambda$$, let C be the locus of a point P such that the product of the distance of P from L₁ and the distance of P from L₂ is $$\lambda$$². The line y = 2x + 1 meets C at two points R and S, where the distance between R and S is $$\sqrt {270} $$. Let the perpendicular bisector of RS meet C at two distinct points R' and S'. Let D be the square of the distance between R' and S'.

The value of D is __________.

For any 3 $$\times$$ 3 matrix M, let | M | denote the determinant of M. Let

$$E = \left[ {\matrix{ 1 & 2 & 3 \cr 2 & 3 & 4 \cr 8 & {13} & {18} \cr } } \right]$$, $$P = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 0 & 1 \cr 0 & 1 & 0 \cr } } \right]$$ and $$F = \left[ {\matrix{ 1 & 3 & 2 \cr 8 & {18} & {13} \cr 2 & 4 & 3 \cr } } \right]$$

If Q is a nonsingular matrix of order 3 $$\times$$ 3, then which of the following statements is(are) TRUE?

Let f : R $$\to$$ R be defined by $$f(x) = {{{x^2} - 3x - 6} \over {{x^2} + 2x + 4}}$$

Then which of the following statements is (are) TRUE?

Let E, F and G be three events having probabilities $$P(E) = {1 \over 8}$$, $$P(F) = {1 \over 6}$$ and $$P(G) = {1 \over 4}$$, and let P (E $$\cap$$ F $$\cap$$ G) = $${1 \over {10}}$$. For any event H, if H^c denotes the complement, then which of the following statements is (are) TRUE?

For any 3 $$\times$$ 3 matrix M, let |M| denote the determinant of M. Let I be the 3 $$\times$$ 3 identity matrix. Let E and F be two 3 $$\times$$ 3 matrices such that (I $$-$$ EF) is invertible. If G = (I $$-$$ EF)^$$-$$1, then which of the following statements is (are) TRUE?

For any positive integer n, let S_n : (0, $$\infty$$) $$\to$$ R be defined by $${S_n}(x) = \sum\nolimits_{k = 1}^n {{{\cot }^{ - 1}}\left( {{{1 + k(k + 1){x^2}} \over x}} \right)} $$, where for any x $$\in$$ R, $${\cot ^{ - 1}}(x) \in (0,\pi )$$ and $${\tan ^{ - 1}}(x) \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$$. Then which of the following statements is (are) TRUE?

For any complex number w = c + id, let $$\arg (w) \in ( - \pi ,\pi ]$$, where $$i = \sqrt { - 1} $$. Let $$\alpha$$ and $$\beta$$ be real numbers such that for all complex numbers z = x + iy satisfying $$\arg \left( {{{z + \alpha } \over {z + \beta }}} \right) = {\pi \over 4}$$, the ordered pair (x, y) lies on the circle $${x^2} + {y^2} + 5x - 3y + 4 = 0$$, Then which of the following statements is (are) TRUE?

For x $$\in$$ R, the number of real roots of the equation $$3{x^2} - 4\left| {{x^2} - 1} \right| + x - 1 = 0$$ is ________.

In a triangle ABC, let AB = $$\sqrt {23} $$, BC = 3 and CA = 4. Then the value of $${{\cot A + \cot C} \over {\cot B}}$$ is _________.

Let $$\overrightarrow u $$, $$\overrightarrow v $$ and $$\overrightarrow w $$ be vectors in three-dimensional space, where $$\overrightarrow u $$ and $$\overrightarrow v $$ are unit vectors which are not perpendicular to each other and $$\overrightarrow u $$ . $$\overrightarrow w $$ = 1, $$\overrightarrow v $$ . $$\overrightarrow w $$ = 1, $$\overrightarrow w $$ . $$\overrightarrow w $$ = 4

If the volume of the paralleopiped, whose adjacent sides are represented by the vectors, $$\overrightarrow u $$, $$\overrightarrow v $$ and $$\overrightarrow w $$, is $$\sqrt 2 $$, then the value of $$\left| {3\overrightarrow u + 5\overrightarrow v } \right|$$ is ___________.