Have a query? Ask a Tutor!!

Access to best educators and exclusive paper discussions

Offsite Campus Turito Championship

Can’t find what you’re looking for?

Our tutors are from top schools around the world, and they are waiting for you 24/7, anytime, anywhere.

Homework- One to One Solutions
Understand concepts to solve better
Get your doubts solved instantly

Maths-

General

Easy

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 $r equals negative 2$
$therefore p equals open parentheses negative 1 close parentheses to the power of 2 end exponent cross times open parentheses negative 2 close parentheses equals negative 2$
$q equals open parentheses negative 1 close parentheses to the power of 2 end exponent cross times open parentheses negative 2 close parentheses to the power of 5 end exponent equals negative 32$

Related Questions to study

If $x comma blank y comma$ and $z$ are in G.P., and $x plus 3 comma blank y plus 3$ , and $z plus 3$ are in H.P., then

If $x comma blank y comma$ and $z$ are in G.P., and $x plus 3 comma blank y plus 3$ , and $z plus 3$ are in H.P., then

Three numbers form an increasing G.P. If the middle number is doubled, then the new numbers are in A.P. The common ratio of the G.P. is

Three numbers form an increasing G.P. If the middle number is doubled, then the new numbers are in A.P. The common ratio of the G.P. is

If $H subscript n end subscript equals 1 plus fraction numerator 1 over denominator 2 end fraction plus horizontal ellipsis plus fraction numerator 1 over denominator n end fraction$ , then value of $S subscript n end subscript equals 1 plus fraction numerator 3 over denominator 2 end fraction plus fraction numerator 5 over denominator 3 end fraction plus horizontal ellipsis plus fraction numerator 99 over denominator 50 end fraction$ is

If $H subscript n end subscript equals 1 plus fraction numerator 1 over denominator 2 end fraction plus horizontal ellipsis plus fraction numerator 1 over denominator n end fraction$ , then value of $S subscript n end subscript equals 1 plus fraction numerator 3 over denominator 2 end fraction plus fraction numerator 5 over denominator 3 end fraction plus horizontal ellipsis plus fraction numerator 99 over denominator 50 end fraction$ is

parallel

Let $T subscript r end subscript$ and $S subscript r end subscript$ be the $r to the power of t h end exponent$ term and sum up to $r to the power of t h end exponent$ term of a series respectively. If for an odd number $n comma blank S subscript n end subscript equals n$ and $T subscript n end subscript equals fraction numerator T subscript n minus 1 end subscript over denominator n to the power of 2 end exponent end fraction$ , then $T subscript m end subscript$ ( $m$ being even) is

Let $T subscript r end subscript$ and $S subscript r end subscript$ be the $r to the power of t h end exponent$ term and sum up to $r to the power of t h end exponent$ term of a series respectively. If for an odd number $n comma blank S subscript n end subscript equals n$ and $T subscript n end subscript equals fraction numerator T subscript n minus 1 end subscript over denominator n to the power of 2 end exponent end fraction$ , then $T subscript m end subscript$ ( $m$ being even) is

If $not stretchy sum from r equals 1 to n of r to the power of 4 end exponent equals l open parentheses n close parentheses comma blank$ then $blank not stretchy sum from r equals 1 to n of open parentheses 2 r minus 1 close parentheses to the power of 4 end exponent$ is equal to

If $not stretchy sum from r equals 1 to n of r to the power of 4 end exponent equals l open parentheses n close parentheses comma blank$ then $blank not stretchy sum from r equals 1 to n of open parentheses 2 r minus 1 close parentheses to the power of 4 end exponent$ is equal to

If $a subscript 1 end subscript comma a subscript 2 end subscript comma a subscript 3 end subscript comma horizontal ellipsis$ are in A.P., then $a subscript p end subscript comma blank a subscript q end subscript comma blank a subscript r end subscript$ are in A.P. if $p comma q comma r$ are in

If $a subscript 1 end subscript comma a subscript 2 end subscript comma a subscript 3 end subscript comma horizontal ellipsis$ are in A.P., then $a subscript p end subscript comma blank a subscript q end subscript comma blank a subscript r end subscript$ are in A.P. if $p comma q comma r$ are in

parallel

If $1 minus fraction numerator 1 over denominator 3 end fraction plus fraction numerator 1 over denominator 5 end fraction minus fraction numerator 1 over denominator 7 end fraction plus fraction numerator 1 over denominator 9 end fraction minus fraction numerator 1 over denominator 11 end fraction plus horizontal ellipsis equals fraction numerator pi over denominator 4 end fraction$ , then value of $fraction numerator 1 over denominator 1 cross times 3 end fraction plus fraction numerator 1 over denominator 5 cross times 7 end fraction plus fraction numerator 1 over denominator 9 cross times 11 end fraction plus horizontal ellipsis$ is

If $1 minus fraction numerator 1 over denominator 3 end fraction plus fraction numerator 1 over denominator 5 end fraction minus fraction numerator 1 over denominator 7 end fraction plus fraction numerator 1 over denominator 9 end fraction minus fraction numerator 1 over denominator 11 end fraction plus horizontal ellipsis equals fraction numerator pi over denominator 4 end fraction$ , then value of $fraction numerator 1 over denominator 1 cross times 3 end fraction plus fraction numerator 1 over denominator 5 cross times 7 end fraction plus fraction numerator 1 over denominator 9 cross times 11 end fraction plus horizontal ellipsis$ is

If $a subscript 1 end subscript comma a subscript 2 end subscript comma a subscript 3 end subscript left parenthesis a subscript 1 end subscript greater than 0 right parenthesis$ are three successive terms of a G.P. with common ratio $r$ , the value of $r$ for which $a subscript 3 end subscript greater than 4 a subscript 2 end subscript minus 3 a subscript 1 end subscript$ holds is given by

If $a subscript 1 end subscript comma a subscript 2 end subscript comma a subscript 3 end subscript left parenthesis a subscript 1 end subscript greater than 0 right parenthesis$ are three successive terms of a G.P. with common ratio $r$ , the value of $r$ for which $a subscript 3 end subscript greater than 4 a subscript 2 end subscript minus 3 a subscript 1 end subscript$ holds is given by

If the sum to infinity of the series $1 plus 2 r plus 3 r to the power of 2 end exponent plus 4 r to the power of 3 end exponent plus midline horizontal ellipsis$ is 9/4, then value of $r$ is

If the sum to infinity of the series $1 plus 2 r plus 3 r to the power of 2 end exponent plus 4 r to the power of 3 end exponent plus midline horizontal ellipsis$ is 9/4, then value of $r$ is

parallel

If $a comma blank b comma blank c$ are in A.P., then $fraction numerator a over denominator b c end fraction comma fraction numerator 1 over denominator c end fraction comma fraction numerator 2 over denominator b end fraction$ will be in

If $a comma blank b comma blank c$ are in A.P., then $fraction numerator a over denominator b c end fraction comma fraction numerator 1 over denominator c end fraction comma fraction numerator 2 over denominator b end fraction$ will be in

BiologyGeneral

Rubisco has the active site that binds to

Rubisco has the active site that binds to

parallel

In dark reaction, regeneration of RuBP needs

In dark reaction, regeneration of RuBP needs

ATPase has

ATPase has

Find out the reason that creates an important difference between $C subscript 3 end subscript$ and $C subscript 4 end subscript$ -plant

Find out the reason that creates an important difference between $C subscript 3 end subscript$ and $C subscript 4 end subscript$ -plant

parallel

card img

With Turito Academy.

card img

With Turito Foundation.

card img

Get an Expert Advice From Turito.

Turito Academy

card img

With Turito Academy.

Test Prep

card img

With Turito Foundation.