2 Marks Questions

Question 12 Marks

Find the quadratic polynomial whose sum and product of the zeroes are $\frac{21}{8}$ and $\frac{5}{16}$ respectively.

Answer

Let the polynomial be
$
p(x)=a x^2+b x+c
$
Its given that, Sum of zeroes $=\frac{21}{8}$
$
\Rightarrow \frac{-b}{a}=\frac{21}{8}
$
Assuming $a=1$,
$
\Rightarrow b=-\frac{21}{8}
$
We also know that, Product of zeroes $=\frac{5}{16}$
$
\Rightarrow \frac{c}{a}=\frac{5}{16}
$
Assuming $a=1$,$
\Rightarrow c=\frac{5}{16}
$
Now, $a=1, b=-\frac{21}{8}, c=\frac{5}{16}$
Hence, the required quadratic polynomial
$
=a x^2+b x+c
$
Substituting the values of $a, b$ and $c$ in the above equation, we get,
$
x^2-\frac{21}{8} x+\frac{5}{16}
$
Multiply the equation by 16 .
$
\Rightarrow 16 x^2-42 x+5
$
Hence, the required quadratic polynomial is $16 x^2-42 x+5$

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Question 22 Marks

Find a quadratic polynomial whose zeroes are $2+\sqrt{3}$ and $2-\sqrt{3}$.

Answer

Let the zeroes of a quadratic polynomial be $a =2+\sqrt{3}$ and $b =2-\sqrt{3}$
We know that,
The quadratic polynomial with $a$ and $b$ as zeroes is $x^2-(a+b) x+a b$
Substituting values of $a$ and $b$ in the above equation, we get,
$\Rightarrow x^2-(2+\sqrt{3}+2-\sqrt{3}) x+(2+\sqrt{3})(2-\sqrt{3})$
Applying, $(a+b)(a-b)=a^2-b^2$
$\Rightarrow x^2-4 x+(4-3)$
$\Rightarrow x^2-4 x+1$
Hence, the required quadratic polynomial is
$x^2-4 x+1$

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Question 32 Marks

Divide $6 x^3+13 x^2+x-2$ by $2 x+1$, and find quotient and remainder.

Answer

Use long division to divide $6 x^3+13 x^2+x-2$ by $2 x+1$.

Hence, quotient is $3 x^2+5 x-2$ and remainder is 0 .

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Question 42 Marks

A teacher asked $10$ of his students to write a polynomial in one variable on a paper and then to handover the paper.
The following were the answer given by the students:
$2 x+3,3 x^2+7 x+2,4 x^3+3 x^2+2, x^3+\sqrt{3 x}+7$
$7 x+\sqrt{7}, 5 x^3-7 x+2,2 x^2+3-\frac{5}{x}, 5 x-\frac{1}{2}$
$a x^3+b x^2+c x+d, x+\frac{1}{x}$
Answer the following questions:
$(i)$ How many of the above ten, are not polynomials?
$(ii)$ How many of the above ten, are quadratic polynomials?

Answer

$(i) x^3+\sqrt{3 x}+7$
$(ii) 3 x^2+7 x+2, x+\frac{1}{x}$

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