This is a question from Polynomails that appeared in the CAT 2018 Question Paper. There are essentially two ways to solve the question. Try it yourself before looking at the video explanation where both the ways to solve this questionare explained.

To know about the exam's blueprint, visit **2IIM's CAT Blueprint**.

Question 14 : The smallest integer n such that n^{3} - 11n^{2} + 32n - 28 ＞ 0 is [TITA]

Limited Seats Available - Register Now!

We have to find the smallest integer n such that n^{3} - 11n^{2} + 32n - 28 > 0

For this we can assume some values for n, such that n = 10 , 9 , 8 ,.. so on and can find the smallest integer

n^{3} - 11n^{2} + 32n - 28 > 0

Let us now assume that n = 10

1000 – 1100 + 320 - 28 > 0

192 > 0 Hence this n = 10 works out

Similarly n = 9 , 8 also works so we can try out with n = 8 and if it works we can try for lower numbers otherwise we can try with 9

n^{3} - 11n^{2} + 32n - 28 > 0

512 – 704 + 256 – 28 > 0

36 > 0

Since n = 8 works lets check for n = 7

343 – 539 + 224 – 28 = 0

Hence the smallest integer was found to be 8

The other way of thinking about it can be by substituting it with smaller numbers and checking if something can factorize it

So \\frac{P(x)}{x-a}) where remainder = 0

Let's substitute n = 1 with n^{3} - 11n^{2} + 32n - 28

1 – 11 + 32 – 28 = -6 this doesn’t work and now we can substitute n = 2 and check it out

8 – 44 – 64 – 28 = 0

So this number n^{3} - 11n^{2} + 32n - 28 is a multiple of n-2

(n-2)(n^{2} – 9n + 14)

(n – 2)(n - 2)(n - 7) > 0

This will be greater than 0 when n = 7 and when n is between 2 and 7 it will be negative

When n = 1 ,

1 - 11 + 32 - 28 = -6 so this doesn’t work

Hence the smallest integer which will work is n = 8

The question is **"The smallest integer n such that n ^{3} - 11n^{2} + 32n - 28 ＞ 0 is [TITA]"**

Copyrights © All Rights Reserved by 2IIM.com - A Fermat Education Initiative.

Privacy Policy | Terms & Conditions

CAT^{®} (Common Admission Test) is a registered trademark of the Indian Institutes of Management. This website is not endorsed or approved by IIMs.

2IIM Online CAT Coaching

A Fermat Education Initiative,

58/16, Indira Gandhi Street,

Kaveri Rangan Nagar, Saligramam, Chennai 600 093

**Mobile:** (91) 99626 48484 / 94459 38484

**WhatsApp:** WhatsApp Now

**Email: **info@2iim.com