First slide

Theory of equations

Question

If $3 p^{2} = 5 p + 2$ and $3 q^{2} = 5 q + 2$ where $p \neq q$ , then the equation whose roots are $3 p - 2 q$ and $3 q - 2 p$ is

Moderate

A

$3 x^{2} - 5 x - 100 = 0$
B

$5 x^{2} + 3 x + 100 = 0$
C

$5 x^{2} - 5 x + 100 = 0$
D

$5 x^{2} - 3 x - 100 = 0$

Solution

Given roots are $3 p - 2 q and 3 q - 2 p .$
Sum of roots $= (3 p - 2 q) + (3 q - 2 p) = (p + q) = \frac{5}{3}$
Product of roots $= (3 p - 2 q) (3 q - 2 p)$
$= 9 pq - 6 q^{2} - 6 p^{2} + 4 pq = 13 pq - 2 (3 p^{2} + 3 q^{2})$
$= 13 (\frac{- 2}{3}) - 2 (5 p + 2 + 5 q + 2)$
$= 13 (\frac{- 2}{3}) - 2 [5 (\frac{5}{3}) + 4]$
$= \frac{- 26}{3} - 2 [\frac{25}{3} + 4] = \frac{- 100}{3}$
Hence, equation is $3 x^{2} - 5 x - 100 = 0$

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