3A - 2A - A = - Vidyadip

For a fixed constant $\lambda$, let $C$ be the locus of a point $P$ such that the product of the distance of $P$ from $L_1$ and the distance of $P$ from $L_2$ is $\lambda^2$. The line $y=2 x+1$ 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 $R ^{\prime}$ and $S ^{\prime}$. Let $D$ be the square of the distance between $R ^{\prime}$ and $S ^{\prime}$.

($1$) The value of $\lambda^2$ is

($2$) The value of $D$ is

→

The value of $\mathop {\lim }\limits_{n \to \infty } \frac{1}{{1.3}} + \frac{1}{{3.5}} + \frac{1}{{5.7}} + \frac{1}{{7.9}} + ... + \frac{1}{{(2n - 1)(2n + 1)}}$ is equal to

→

If $|x|\, < 1$, then the sum of the series $1 + 2x + 3{x^2} + 4{x^3} + ...........\infty $ will be

→

A contractors has two teams of workers, team $A$ and team B. Team $A$ can complete a project $P$ in $12$ days and team $B$ can complete $P$ in $36$ days. Team $A$ starts working on $P$ and team $B$ joins team $A$ after four days. Team $A$ is withdrawn after another two days and team $B$ is asked to double its efficiency. The number of additional days required for team $B$ to complete $P$ is

→

If $s$ is the sum of an infinite $G.P.$, the first term $a$ then the common ratio $r$ given by

→

The points $A (a , 0) , B (0 , b) , C (c , 0) \& D (0 , d)$ are such that $ac = bd \,\,\&\,\, a, b, c, d$ are all non-zero. Then the points :