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Combination

The polynomia...

Question

The polynomials $a x^{3} + 3 x^{2} - 3$ and $2 x^{3} - 5 x + a$ when divided by (x -4) leaves remainders $R_{1},$ & $R_{2}$ respectively then value of 'a' if $2 R_{1} - R_{2} = 0.$

- \frac{18}{127}

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\frac{18}{127}

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\frac{17}{127}

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- \frac{17}{127}

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Solution

The correct option is C $\frac{18}{127}$
By Remainder Theorem,
$R_{1} = a (4)^{3} + 3 (4)^{2} - 3 = 64 a + 45$ ...(i)
$R_{2} = 2 (4)^{3} - 5 (4) + a = 128 - 20 + a = 108 + a$ ...(ii)
Given: $2 R_{1} - R_{2} = 0$
$∴ 2 (64 a + 45) - (108 + a) = 0$ (from (i) and (ii))
$\Rightarrow 128 a + 90 - 108 - a = 0$
$\Rightarrow 127 a = 18$
$a = \frac{18}{127}$
Option B is correct.

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