[1 Mark Question Answer]

Question 11 Mark

Find the value of a , if $(x - a)$ is a factor of $x^3 - a^2x + x + 2$.

Answer

Let $f(x) = x^2- a^2x + x + 2$
Put $x - a = 0$
$\therefore x = a$
$f(a) = a^3 - a^2·a + a + 2$
$0 = a^3 - a^3 + a + 2$
or
$a + 2 = 0$
$\therefore a = -2$

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Question 21 Mark

Find the value of $p$ if the division of $px^3 + 9x^2 + 4x - 10 by (x + 3)$ leaves the remainder $5$.

Answer

Here, $P(-3) = 5$
$\Rightarrow p(-3)^3 + 9(-3)^2 + 4(-3) - 10 = 5$
$\Rightarrow -27p + 81 - 12 - 10 = 5$
$\Rightarrow -27p = -54$
$\Rightarrow p = 2$

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Question 31 Mark

When $x^3 + 3x^2– kx + 4$ is divided by $(x – 2)$, the remainder is $k$. Find the value of $k$.

Answer

Here, $P(2) = k$
$\Rightarrow 2^3 + 3(2)^2 - k(2) + 4 = k$
$\Rightarrow 8 + 12 - 2k + 4 = k$
$\Rightarrow 3k = 24$
$\Rightarrow k = 8$

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Question 41 Mark

If $x – 2$ is a factor of $2x^3 - x^2 - px - 2$.
Find the value of $p$

Answer

Given expression is $2x^3 - x^2- px - 2$ and $x - 2$ is the factor.
$x - 2 = 0, x = 2$ in expression
$2(2)^3 - (2)^2 - p(2) - 2 = 0$
$16 - 4 - 2p - 2 = 0$
$10 - 2p = 0$
$p = 5$

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Question 51 Mark

Use remainder theorem and find the remainder when the polynomial $g(x) = x^3 + x^2 – 2x + 1$ is divided by $x – 3$.

Answer

By the remainder theorem, required remainder
$8^{(3)} = (3)^3 + (3)^2 - 2 x 3 + 1$
$= 27 + 9 - 6 + 1$
$= 31.$

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Mathematics [1 Mark Question Answer]

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