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KEAM 2013 Math Paper

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Question : 13 of 120

Marks: +1, -0

Let α and β be the roots of the equation

px^2+qx+r=0.

If p, q, r are in A.P. and

\alpha+\beta=4,

then αβ is equal to

-9
9
-5
5
-4

Solution:

Given,

(\alpha, \beta)

are the roots of the equation

p x^{2}+q x+r=0

\therefore \;\; \alpha+\beta=-\;\frac{q}{p}

.....(i)

and

\;\; \alpha\beta=\;\frac{r}{p}

....(ii)

Also given,

\alpha+\beta=4

.....(iii)

From Eqs. (i) and (iii), we get

-\;\frac{q}{p}=4 \Rightarrow q=-4 p

.....(iv)

Also,

p, q

and

r

are in

A P

.

\therefore \;\; p+r=2 q

\Rightarrow \;\; p+r=-8 p

[from Eq. (iv)]

\Rightarrow \;\; 9 p+r=0

\Rightarrow \;\; \;\frac{r}{p}=-9

....(v)

From Eqs. (ii) and (v).

\alpha\beta=-9

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