jee-advanced 2005 Q18

jee-advanced · India · mains Solving quadratics and applications Determining quadratic function from given conditions

18. If
$$\left[ \begin{array} { l l l } 4 a ^ { 2 } & 4 a & 1 \\ 4 b ^ { 2 } & 4 b & 1 \\ 4 c ^ { 2 } & 4 c & 1 \end{array} \right] \left[ \begin{array} { c } f ( - 1 ) \\ f ( 1 ) \\ f ( 2 ) \end{array} \right] = \left[ \begin{array} { l l l } 3 a ^ { 2 } & + & 3 a \\ 3 b ^ { 2 } & + & 3 b \\ 3 c ^ { 2 } & + & 3 c \end{array} \right] ,$$
$f ( x )$ is a quadratic function and its maximum value occurs at a point $V$. $A$ is a point of intersection of $y = f ( x )$ with $x$-axis and point $B$ is such that chord $A B$ subtends a right angle at V . Find the area enclosed by $\mathrm { f } ( \mathrm { x } )$ and chord AB .

If $f ( x )$ is a continuous and differentiable function and $f \left( \frac { 1 } { n } \right) = 0$ for every positive integer $n$, then

18. If

$$\left[ \begin{array} { l l l } 
4 a ^ { 2 } & 4 a & 1 \\
4 b ^ { 2 } & 4 b & 1 \\
4 c ^ { 2 } & 4 c & 1
\end{array} \right] \left[ \begin{array} { c } 
f ( - 1 ) \\
f ( 1 ) \\
f ( 2 )
\end{array} \right] = \left[ \begin{array} { l l l } 
3 a ^ { 2 } & + & 3 a \\
3 b ^ { 2 } & + & 3 b \\
3 c ^ { 2 } & + & 3 c
\end{array} \right] ,$$

$f ( x )$ is a quadratic function and its maximum value occurs at a point $V$. $A$ is a point of intersection of $y = f ( x )$ with $x$-axis and point $B$ is such that chord $A B$ subtends a right angle at V . Find the area enclosed by $\mathrm { f } ( \mathrm { x } )$ and chord AB .

Paper Questions

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