# CAT 2021 Question Paper | Quant Slot 2

###### CAT Previous Year Paper | CAT Quant Questions | Question 13

CAT 2021 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 13 : For all real values of x, the range of the function f(x) = $$frac{x^{2} + 2x + 4}{2x^{2} + 4x + 9}$ is 1. [ $\frac{3}{7}$ , $\frac{8}{9}$ ) 2. [ $\frac{4}{9}$ , $\frac{8}{9}$ ] 3. [ $\frac{3}{7}$ , $\frac{1}{2}$ ) 4.$ $$frac{3}{7}$ , $\frac{1}{2}$ ) ## 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 We have a function f$x) = $$frac{x^{2} + 2x + 4}{2x^{2} + 4x + 9}$ We can complete the squares in the numerator as f$x) = $$frac{x^{2} + 2x + 1 + 3}{2x^{2} + 4x + 9}$ f$x) = $$frac{$x + 1$_{2} + 3}{2x^{2} + 4x + 9})
Now we try to get (x + 1)2in the denominator as well
f(x) = $$frac{$x + 1$_{2} + 3}{2(x + 1)^{2} + 7})
From here we can take out (x+1)2 + 3 from both, the numerator and the denominator and put it as ‘k’
Replacing (x+1)2 + 3 with k we get
f(x) = $$frac{k}{2k + 1}$ The minimum value of k = 3, as the square part is always positive and another positive number is added to the equation. So the fraction will have the minimum value when k = 3 f$x) = $$frac{3}{2$3$ + 1})= $$frac{3}{7}$ And, the maximum value when k approaches its maximum value. We can observe that the fraction has a 2k + 1 in the denominator and so it will always be one more than twice the numerator. As the value of k increases the value of the addition of one gets less and less of a value to the overall denominator. The fraction approaches 1/2 , but never 1/2 itself. Hence the range becomes [$\frac{3}{7}$,$\frac{1}{2}$) The question is " For all real values of x, the range of the function f$x) = $$frac{x^{2} + 2x + 4}{2x^{2} + 4x + 9}$ is " ##### Hence, the answer is '[ $\frac{3}{7}$ , $\frac{1}{2}$ )' Choice C is the correct answer. ###### Prepare for CAT 2023 with 2IIM's Daily Preparation Schedule ###### Know all about CAT Exam Syllabus and what to expect in CAT ###### Best CAT Online Coaching Try upto 40 hours for free Learn from the best! ###### Already have an Account? ###### CAT Coaching in ChennaiCAT 2023 Classroom Batches Starting Now! @Gopalapuram ###### Best CAT Coaching in Chennai Introductory offer of 5000/- Attend a Demo Class ###### Best Indore IPM & Rohtak IPM CoachingSignup and sample 9 full classes for free. Register now! ##### 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
WhatsApp: WhatsApp Now
Email: info@2iim.com