# CAT 2023 Question Paper | Quant Slot 3

###### CAT Previous Year Paper | CAT Quant Questions | Question 5

CAT 2023 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 5 : A quadratic equation $x^2+b x+c=0$ has two real roots. If the difference between the reciprocals of the roots is $$frac{1}{3}$, and the sum of the reciprocals of the squares of the roots is $$frac{5}{9}$, then the largest possible value of $$b+c$$ is

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let $$alpha $& \beta$ be the roots of $x ^ { 2 } + b x + c = 0$, $\frac { 1 } { \alpha } - \frac { 1 } { \beta } = \frac { 1 } { 3 }$ $\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } = \frac { 5 } { 9 }$ $\left$ $frac { 1 } { \alpha } - \frac { 1 } { \beta } \right$ ^ { 2 } = $frac { 1 } { 9 }$ $\frac { 4 } { 9 } = \frac { 2 } { \alpha \beta }$ $\alpha \beta = 9 / 2$ $\frac { 1 } { \alpha ^ { 2 } + \beta ^ { 2 } } = \frac { \alpha ^ { 2 } + \beta ^ { 2 } } { \alpha ^ { 2 } \beta ^ { 2 } } = \frac {$ $alpha + \beta$ ^ { 2 } - 2 $alpha \beta } {$ $alpha \beta$ ^ { 2 } } = $frac { 5 } { 9 }$ $$ $alpha + \beta$ ^ { 2 } = $frac { 5 } { 9 } \left$ $frac { 9 } { 2 } \right$ $left$ $frac { 9 } { 2 } \right$ + 2 $left$ $frac { 9 } { 2 } \right$$ $$ $alpha + $beta$ ^ { 2 } = $frac { 45 } { 4 } + 9 = 20 + \frac { 1 } { 4 } = 20.25$ $\alpha + \beta = \pm 4.5$ $\alpha + \beta = \frac { - b } { 1 }$ $b = -$ $alpha + \beta$$ $$alpha \beta = \frac { c } { 1 }$ $c = \alpha \beta = 4.5$ $b + c = -$ $alpha + \beta$ + $alpha$ to maximize $$ b + c$$, we pick $( $alpha + $beta$ = - 4.5$ $b + c = -$ - 4.5 ) + 4.5 = 9$
So, the largest possible value of (b + c) is 9.

The question is " A quadratic equation $x^2+b x+c=0$ has two real roots. If the difference between the reciprocals of the roots is $$frac{1}{3}$, and the sum of the reciprocals of the squares of the roots is $$frac{5}{9}$, then the largest possible value of $$b+c$$ is "

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