CAT 2022 Question Paper | Quant Slot 2

CAT 2022 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 9 : Let \(r\) and \(c\) be real numbers. If \(r\) and \(-r\) are roots of \(5 x^3+c x^2-10 x+9=0\), then \(c\) equals

Video Explanation

Explanatory Answer

Since r and (-r) are the roots of the cubic equation \( 5 x ^ { 3 } + c x ^ { 2 } - 10 x + 9 = 0 \)
\( 5 ( r ) ^ { 3 } + c ( r ) ^ { 2 } - 10 ( r ) + 9 = 0 \)
\( 5 ( - r ) ^ { 3 } + c ( - r ) ^ { 2 } - 10 ( - r ) + 9 = 0 \)
\( 5 r ^ { 3 } + c r ^ { 2 } - 10 r + 9 = 0 \)
\( - 5 r ^ { 3 } + c r ^ { 2 } + 10 r + 9 = 0 \)
10 r³ - 20 r = 0
r³ - 2r = 0
r (r² - 2) = 0
r = \( \pm \sqrt { 2 } \)
Assuming the third root of the cubic equation as k, we can reconstruct it as…
\( ( x - \sqrt { 2 } ) ( x + \sqrt { 2 } ) ( x - k ) = 0 \)
\( \left( x ^ { 2 } - 2 \right) ( x - k ) = 0 \)
\( x ^ { 3 } - k x ^ { 2 } - 2 x + 2 k = 0 \)
\( 5 x ^ { 3 } + c x ^ { 2 } - 10 x + 9 = 0 \) can be re-written by making the coefficient of x³ as 1.
\( x ^ { 3 } + \frac { c } { 5 } x ^ { 2 } - 2 x + \frac { 9 } { 5 } = 0 \)
\( x ^ { 3 } + \frac { c } { 5 } x ^ { 2 } - 2 x + \frac { 9 } { 5 } = 0 \) and \( x ^ { 3 } - k x ^ { 2 } - 2 x + 2 k = 0 \) are the same.
\( k = \frac { - c } { 5 } \)
\( 2 k = \frac { 9 } { 5 } \)
This implies, \( c = \frac { - 9 } { 2 } \)

The question is " Let \(r\) and \(c\) be real numbers. If \(r\) and \(-r\) are roots of \(5 x^3+c x^2-10 x+9=0\), then \(c\) equals "

Hence, the answer is '\(-\frac{9}{2}\)'

Choice A is the correct answer.