Categories
I.S.I. and C.M.I. Entrance

Root of Equation- B.Stat. (Hons.) Admission Test 2005 – Objective Problem 2

Try this beautiful problem of Algebra prticularly in cubic equation fromB.Stat. (Hons.) Admission Test 2005. You may use sequential hints to help you solve the problem.

What are we learning ?

Competency in Focus: Root of Equation

This problem from Root of equation for B.Stat. (Hons.) Admission Test 2005 Objective Problem 2  is based on calculating a variable in a given equation.

First look at the knowledge graph:-

calculation of  mean and median- AMC 8 2013 Problem

Next understand the problem

If \(\sqrt{3}+1\) is a root of the equation \(3x^{3}+ax^{2}+bx+12=0\) where a and b are rational numbers, then b is equal to (A) -6 (B) 2 (C) 6 (D) 10
Source of the problem

B.Stat. (Hons.) Admission Test 2005 – Objective problem 2

Key Competency

Algebra (Root of equation)

Difficulty Level
4/10
Suggested Book

Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics

Start with hints 

Do you really need a hint? Try it first!
If the co effeicent of any polynomial equation is rational number and one of the root is a surd or coplex number, then the other root must be the conjugate.
Since our polynomial is of degree 3, there must be three roots of the equation. We already know the two of them so let the third one is $\gamma$.
we can use Vieta’s Theorem, that gives, the product of the roots is \(\frac{c}{a}\) in our case \(\frac{-12}{3} \) =\(-4\). so we can say that $(1+\sqrt{3}) \times(1-\sqrt{3}) \times \gamma=-4$
Now we have calculated the value of gamma. Also we have from Vieta’s Theorem, that sum of products of the roots taken two roots at a time is \(\frac{b}{3}\). So we can write,   $\frac{b}{3}=(1+\sqrt{3}) \times(1-\sqrt{3})+_{\gamma} \times\{(1+\sqrt{3})+(1-\sqrt{3})\}=2$.

I.S.I. & C.M.I. Program

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are: B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.
The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

Learn more

Carpet Strategy in Geometry | Watch and Learn

Here is a video solution for a Problem based on Carpet Strategy in Geometry. This problem is helpful for Math Olympiad, ISI & CMI Entrance, and other math contests. Watch and Learn! Here goes the question… Suppose ABCD is a square and X is a point on BC such that...
Solve weird equation using inequality

Bijection Principle Problem | ISI Entrance TOMATO Obj 22

Here is a video solution for a Problem based on Bijection Principle. This is an Objective question 22 from TOMATO for ISI Entrance. Watch and Learn! Here goes the question… Given that: x+y+z=10, where x, y and z are natural numbers. How many such solutions are...
Solve weird equation using inequality

What is the Area of Quadrilateral? | AMC 12 2018 | Problem 13

Here is a video solution for a Problem based on finding the area of a quadrilateral. This question is from American Mathematics Competition, AMC 12, 2018. Watch and Learn! Here goes the question… Connect the centroids of the four triangles in a square. Can you find...
Solve weird equation using inequality

Solving Weird Equations using Inequality | TOMATO Problem 78

Here is a video solution for ISI Entrance Number Theory Problems based on solving weird equations using Inequality. Watch and Learn! Here goes the question… Solve: 2 \cos ^{2}\left(x^{3}+x\right)=2^{x}+2^{-x} We will recommend you to try the problem yourself. Done?...

AM-GM Inequality Problem | ISI Entrance

Here is a video solution for ISI Entrance Number Theory Problems based on AM-GM Inequality Problem. Watch and Learn! Here goes the question… a, b, c, d are positive real numbers. Prove that: (1+a)(1+b)(1+c)(1+d) <= 16. We will recommend you to try the problem...

Sum of 8 fourth powers | ISI Entrance Problem

Here is a video solution for ISI Entrance Number Theory Problems based on Sum of 8 fourth powers. Watch and Learn! Can you show that the sum of 8 fourth powers of integers never adds up to 1993? How can you solve this fourth-degree diophantine equation? Let’s...

ISI MStat Entrance 2020 Problems and Solutions

Problems and Solutions of ISI MStat Entrance 2020 of Indian Statistical Institute.

ISI Entrance 2020 Problems and Solutions – B.Stat & B.Math

Problems and Solutions of ISI BStat and BMath Entrance 2020 of Indian Statistical Institute.

Testing of Hypothesis | ISI MStat 2016 PSB Problem 9

This is a problem from the ISI MStat Entrance Examination,2016 making us realize the beautiful connection between exponential and geometric distribution and a smooth application of Central Limit Theorem.
Cheena Statistics Logo

ISI MStat PSB 2006 Problem 8 | Bernoullian Beauty

This is a very simple and regular sample problem from ISI MStat PSB 2009 Problem 8. It It is based on testing the nature of the mean of Exponential distribution. Give it a Try it !

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.