Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle $$\alpha$$ with the positive x-axis and the equations of its diagonals are $(\sqrt{3}+1)x+(\sqrt{3}-1)y=0$ and $(\sqrt{3}-1)x-(\sqrt{3}+1)y+8\sqrt{3}=0$. Then $a$2 is equal to :

A line passing through the point P($a$, 0) makes an acute angle $$\alpha$$ with the positive x-axis. Let this line be rotated about the point P through an angle $\frac{\alpha }{2}$ in the clockwise direction. If in the new position, the slope of the line is $2-\sqrt{3}$ and its distance from the origin is $\frac{1}{\sqrt{2}}$, then the value of $3a^2{\tan }^2\alpha -2\sqrt{3}$ is :

Let ABC be the triangle such that the equations of lines AB and AC be $3 y-x=2$ and $x+y=2$, respectively, and the points B and C lie on $x$-axis. If P is the orthocentre of the triangle ABC , then the area of the triangle PBC is equal to

If the orthocenter of the triangle formed by the lines y = x + 1, y = 4x - 8 and y = mx + c is at (3, -1), then m - c is :

Let the area of the triangle formed by a straight line $\mathrm{L}:x+\mathrm{b}y+\mathrm{c}=0$ with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line L makes an angle of $45^{\circ }$ with the positive $x$-axis, then the value of ${\mathrm{b}}^2+{\mathrm{c}}^2$ is :

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines ${\mathrm{L}}_1:2x+y+6=0$ and ${\mathrm{L}}_2:4x+2y-p=0,p>0$, at the points A and B , respectively. If $AB=\frac{9}{\sqrt{2}}$ and the foot of the perpendicular from the point $A$ on the line $L_2$ is $M$, then $\frac{AM}{BM}$ is equal to

Let the three sides of a triangle are on the lines $4 x-7 y+10=0, x+y=5$ and $7 x+4 y=15$. Then the distance of its orthocentre from the orthocentre of the tringle formed by the lines $x=0, y=0$ and $x+y=1$ is

Consider the lines $x(3\lambda +1)+y(7\lambda +2)=17\lambda +5,\lambda$ being a parameter, all passing through a point P. One of these lines (say $L$ ) is farthest from the origin. If the distance of $L$ from the point $(3,6)$ is $d$, then the value of $d^2$ is

The least value of n for which the number of integral terms in the Binomial expansion of $(\sqrt[3]{7}+\sqrt[12]{11}{)}^n$ is 183, is :

The remainder, when $7^{103}$ is divided by 23, is equal to:

Let $\alpha ,\beta ,\gamma$ and $\delta$ be the coefficients of $x^7,x^5,x^3$ and $x$ respectively in the expansion of $$\begin{array}{rl}& {\left(x+\sqrt{x^3-1}\right)}^5+{\left(x-\sqrt{x^3-1}\right)}^5,x>1\text{. If }u\text{ and }v\text{ satisfy the equations }\\ & \\ & \alpha u+\beta v=18,\\ & \\ & \gamma u+\delta v=20,\end{array}$$ then $\mathrm{u}+\mathrm{v}$ equals :

Let the coefficients of three consecutive terms $T_r$, $T_{r+1}$ and $T_{r+2}$ in the binomial expansion of $(a+b{)}^{12}$ be in a G.P. and let $p$ be the number of all possible values of $r$. Let $q$ be the sum of all rational terms in the binomial expansion of $(\sqrt[4]{3}+\sqrt[3]{4}{)}^{12}$. Then $p + q$ is equal to:

If in the expansion of $(1+x{)}^{\mathrm{p}}(1-x{)}^{\mathrm{q}}$, the coefficients of $x$ and $x^2$ are 1 and -2 , respectively, then ${\mathrm{p}}^2+{\mathrm{q}}^2$ is equal to :

For some $\mathrm{n}\ne 10$, let the coefficients of the 5 th, 6 th and 7 th terms in the binomial expansion of $(1+\mathrm{x}{)}^{\mathrm{n}+4}$ be in A.P. Then the largest coefficient in the expansion of $(1+\mathrm{x}{)}^{\mathrm{n}+4}$ is:

Suppose $A$ and $B$ are the coefficients of $30^{\text{th }}$ and $12^{\text{th }}$ terms respectively in the binomial expansion of $(1+x{)}^{2\mathrm{n}-1}$. If $2\mathrm{A}=5\mathrm{B}$, then n is equal to:

The number of integral terms in the expansion of ${\left(5^{\frac{1}{2}}+7^{\frac{1}{8}}\right)}^{1016}$ is:

The largest $\mathrm{n}\in \mathbf{N}$ such that $3^{\mathrm{n}}$ divides 50 ! is :

The term independent of $x$ in the expansion of ${\left(\frac{(x+1)}{\left(x^{2/3}+1-x^{1/3}\right)}-\frac{(x-1)}{\left(x-x^{1/2}\right)}\right)}^{10},x>1$, is :

The remainder when ${\left((64{)}^{(64)}\right)}^{(64)}$ is divided by 7 is equal to