CAT 2023 Question Paper | Quant Slot 2

CAT 2023 Quant was dominated by Algebra followed by Arithmetic. 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 7 : Let \(k\) be the largest integer such that the equation \((x-1)^2+2 k x+11=0\) has no real roots. If \(y\) is a positive real number, then the least possible value of \(\frac{k}{4 y}+9 y\) is

Video Explanation

Explanatory Answer

\( ( x - 1 ) ^ { 2 } + 2 k x + 11 = 0 \)
\( x ^ { 2 } - 2 x + 1 + 2 k x + 11 = 0 \)
\( x ^ { 2 } + 2 ( k - 1 ) x + 12 = 0 \)
Since the equation above has no real roots, the discriminant of the equation should be negative.
\( b ^ { 2 } - 4 a c < 0 \)
\( b ^ { 2 } < 4 a c \)
\( ( 2 ( k - 1 ) ) ^ { 2 } < 4 \cdot 1 \cdot 12 \)
\( 4 ( k - 1 ) ^ { 2 } < 4 \cdot 12 \)
\( ( k - 1 ) ^ { 2 } < 12 \)
\( - \sqrt { 12 } < k - 1 < \sqrt { 12 } \)
\( - 3.46 \leq k - 1 \leq 3.46 \)
\( - 2.46 \leq k \leq 4.46 \)
The largest integral value that k can take is 4.
Now, we need to minimize \( \frac { k } { 4 y } + 9 y \) where k takes the largest integral value and y is positive…
\( \frac { k } { 4 y } + 9 y = \frac { 4 } { 4 y } + 9 y = \frac { 1 } { y } + 9 y \)
\( \frac { 1 } { y } \& 9 y \) are both positive real numbers, therefore, their A.M is greater than equal to their G.M.
\( A M \left( \frac { 1 } { y } , 9 y \right) \geq G M \left( \frac { 1 } { y } , 9 y \right) \)
\( \frac { \frac { 1 } { y } + 9 y } { 2 } \geq \sqrt { \frac { 1 } { y } ( 9 y ) } \)
\( \frac { \frac { 1 } { y } + 9 y } { 2 } \geq 3 \)
\( \frac { 1 } { y } + 9 y \geq 6 \)
\( \therefore \left( \frac { k } { k y } + 9 y \right) = 6 \)

The question is " Let \(k\) be the largest integer such that the equation \((x-1)^2+2 k x+11=0\) has no real roots. If \(y\) is a positive real number, then the least possible value of \(\frac{k}{4 y}+9 y\) is "

Hence, the answer is '6'