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### JEE 2012 Math: ${\lim _{a \to {0^ + }}}\alpha (a)$ & ${\lim _{a \to {0^ + }}}\beta (a)$

Let $\alpha (a)$ and $\beta (a)$ be the roots of the equation

$\left( {\sqrt[3]{{1 + a}} - 1} \right){x^2} + \left( {\sqrt {1 + a} - 1} \right)x + \left( {\sqrt[6]{{1 + a}} - 1} \right) = 0$ where a>-1.

Then ${\lim _{a \to {0^ + }}}\alpha (a)$ and ${\lim _{a \to {0^ + }}}\beta (a)$ are

(A) $-\frac {5}{2}$ and 1
(B) $-\frac {1}{2}$ and -1
(C) $-\frac {7}{2}$ and 2
(D) $-\frac {9}{2}$ and 3

image

Solution

Looking at the options and noticing that while the sum of the limits is same the product of the limits is different and hence the following quick method can be employed:

Consider,

assuming both limits exist

Hence, (B).

### Sum of the coefficients in the expansion of $(x+y)^n$ ....

If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$ $\therefore 2^n = 4096 =2^{12}$ $\Rightarrow n = 12$ Greatest coefficient = ${}^{12}{C_6} = 924$