# CAT 2023 Question Paper | Quant Slot 2

###### CAT Previous Year Paper | CAT Quant Questions | Question 7

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 $$frac{k}{4 y}+9 y$ is ## Best CAT Online Coaching Try upto 40 hours for free Learn from the best! #### 2IIM : Best Online CAT Coaching. ### Video Explanation ## Best CAT Coaching in Chennai #### CAT Coaching in Chennai - CAT 2022Limited Seats Available - Register Now! ### Explanatory Answer $$ 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' ###### Best CAT Online Coaching Try upto 40 hours for free Learn from the best! ###### Prepare for CAT 2024 with 2IIM's Daily Preparation Schedule ###### Know all about CAT Exam Syllabus and what to expect in CAT ###### Already have an Account? ###### CAT Coaching in ChennaiCAT 2024 Classroom Batches Starting Now! @Gopalapuram and @Anna nagar ###### Best CAT Coaching in Chennai Attend a Demo Class ###### Best Indore IPM & Rohtak IPM CoachingSignup and sample 9 full classes for free. Register now! ## CAT Questions | CAT Quantitative Aptitude ## CAT Questions | Verbal Ability for CAT ##### Where is 2IIM located? 2IIM Online CAT Coaching A Fermat Education Initiative, 58/16, Indira Gandhi Street, Kaveri Rangan Nagar, Saligramam, Chennai 600 093 ##### How to reach 2IIM? Mobile:$91) 99626 48484 / 94459 38484
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