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Maths-

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Question

Let $alpha$ , and $beta$ be the roots of $x to the power of 2 end exponent minus x plus p equals 0$ and $gamma$ and $delta$ be the root of $x to the power of 2 end exponent minus 4 x plus q equals 0$ . If $alpha comma blank beta comma blank a n d blank gamma comma blank delta$ are in G.P., then the integral values of $p$ and $q$ , respectively, are

$- 2, - 32$
$- 2, 3$
$- 6, 3$
$- 6, - 32$

The correct answer is: $negative 2 comma blank minus 32$

$alpha comma blank beta$ are the roots of $x to the power of 2 end exponent minus x plus p equals 0$ . Hence,
$alpha plus beta equals 1$
$alpha beta equals p$
$gamma comma blank delta$ are the roots of $x to the power of 2 end exponent minus 4 x d plus q equals 0$ . Hence,
$therefore gamma plus delta equals 4$ (3)
$alpha comma blank beta comma blank gamma comma blank delta$ are in G.P. Let $alpha equals a comma blank beta equals a r comma blank gamma equals a r to the power of 2 end exponent comma delta equals a r to the power of 3 end exponent$ , Substituting these values in Eqs. (1), (2), (3) and (4), we get
$alpha plus a r equals 1$ (5)
$a to the power of 2 end exponent r equals p$ (6)
$a r to the power of 2 end exponent plus a r to the power of 3 end exponent equals 4$ (7)
$a to the power of 2 end exponent r to the power of 5 end exponent equals q$ (8)
Dividing (7) by (5), we get
$fraction numerator a r to the power of 2 end exponent open parentheses 1 plus r close parentheses over denominator a open parentheses 1 plus r close parentheses end fraction equals fraction numerator 4 over denominator 1 end fraction blank rightwards double arrow blank r to the power of 2 end exponent equals 4 blank rightwards double arrow r equals 2 comma blank minus 2$
$open parentheses 5 close parentheses rightwards double arrow blank a equals fraction numerator 1 over denominator 1 plus r end fraction equals fraction numerator 1 over denominator 1 plus 2 end fraction blank o r blank fraction numerator 1 over denominator 1 minus 2 end fraction equals fraction numerator 1 over denominator 3 end fraction blank o r minus 1$
As $p$ is an integer (given), $r$ is also an integer (2 or $negative 2$ ). Therefore, from (6), $a not equal to 1 divided by 3$ . Hence, $a equals negative 1$ and $r equals negative 2$
$therefore p equals open parentheses negative 1 close parentheses to the power of 2 end exponent cross times open parentheses negative 2 close parentheses equals negative 2$
$q equals open parentheses negative 1 close parentheses to the power of 2 end exponent cross times open parentheses negative 2 close parentheses to the power of 5 end exponent equals negative 32$

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