IIT-JEE 2011 · Paper 1 · Mathematics Q60

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Paragraph for Question Nos. 58 to 60

Let $a$ , $b$ and $c$ be three real numbers satisfying
$\begin{bmatrix}a&b&c\end{bmatrix}\begin{bmatrix}1&9&7\\8&2&7\\7&3&7\end{bmatrix}=\begin{bmatrix}0&0&0\end{bmatrix}\qquad\text{(E)}$

Let $b=6$ , with $a$ and $c$ satisfying (E). If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2+bx+c=0$ , then
$\sum_{n=0}^\infty\left(\frac1\alpha+\frac1\beta\right)^n$
is

\dfrac67

\infty

Answer and worked solution

Official answer: B

Values. $b=6$ gives $k=-1$ : $a=1$ , $c=-7$ ; the quadratic is $x^2+6x-7=0$ .

Sum of reciprocals. $\frac1\alpha+\frac1\beta=-\frac bc=\frac67$ .

Geometric series. $\displaystyle\sum_{n=0}^\infty\left(\frac67\right)^n=\frac1{1-\frac67}=7$ .

(B)

Topics:Geometric progression Roots and coefficients

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IIT-JEE 2011 · Paper 1, Mathematics question 60