First slide

Theory of equations

Question

Let $f (x) = {ax}^{3} + 5 x^{2} - bx + 1$ .If f (x) when divided by 2x +1 leaves 5 as remainder, and f (x) is divisible by 3x -1, then

Moderate

A

$a = 26, b = 10$
B

$a = 6, b = - 4$
C

$a = 26, b = 12$
D

$a = - 6, b = 4$

Solution

we have,
$\begin{aligned} f (- \frac{1}{2}) = 5 and f (\frac{1}{3}) = 0 \\ \Rightarrow & - \frac{a}{8} + \frac{5}{4} + \frac{b}{2} + 1 = 5 and \frac{a}{27} + \frac{5}{9} - \frac{b}{3} + 1 = 0 \\ \Rightarrow & - a + 10 + 4 b + 8 = 40 and a - 9 b + 42 = 0 \\ \Rightarrow & - a + 4 b = 22 and a - 9 b + 42 = 0 \\ \Rightarrow & a = - 6, b = 4 . \end{aligned}$

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