CAT Question > The number of roots common between the two eq...
Download as PDF
Download as PDF
The number of roots common between the two equations x3 + 3x2 + 4x + 5 = 0 and x3+ 2x2 + 7x + 3 = 0 is
- a)0
- b)1
- c)2
- d)3
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
The number of roots common between the two equations x3 + 3x2+ 4x + 5 ...
Subtracting one equation from other we get a quadratic equation x2 - 3x + 2 = 0, which has two roots 1,2, but neither of them satisfies these equations.
Hence, 0 will be the answer.
Related Content
Most Upvoted Answer
The number of roots common between the two equations x3 + 3x2+ 4x + 5 ...
Solution:
To find the number of roots common between the two equations, we need to find their intersection points.
Let the roots of the first equation be a, b, and c. Then, we can write the equation as:
(x-a)(x-b)(x-c) = 0
Expanding this, we get:
x3 - (a+b+c)x2 + (ab+bc+ca)x - abc = 0
Comparing this with the first equation, we get:
a+b+c = 3
ab+bc+ca = 4
abc = -5
Similarly, for the second equation, let the roots be p, q, and r. Then, we can write the equation as:
(x-p)(x-q)(x-r) = 0
Expanding this, we get:
x3 - (p+q+r)x2 + (pq+qr+rp)x - pqr = 0
Comparing this with the second equation, we get:
p+q+r = 2
pq+qr+rp = 7
pqr = -3
To find the common roots, we need to solve these two sets of equations simultaneously.
Adding the two sets of equations, we get:
a+b+c+p+q+r = 5
ab+bc+ca+pq+qr+rp = 11
abc+pqr = -8
Substituting the values of a+b+c and p+q+r from the above equations, we get:
5 - (ab+bc+ca) + 2 - (pq+qr+rp) = 0
Simplifying this, we get:
ab+bc+ca+pq+qr+rp = 7
Substituting this in the equation abc+pqr = -8, we get:
abc = -5
pqr = -3
Since the product of the roots is different for the two equations, they do not have any common roots.
Hence, the correct answer is option A.
To find the number of roots common between the two equations, we need to find their intersection points.
Let the roots of the first equation be a, b, and c. Then, we can write the equation as:
(x-a)(x-b)(x-c) = 0
Expanding this, we get:
x3 - (a+b+c)x2 + (ab+bc+ca)x - abc = 0
Comparing this with the first equation, we get:
a+b+c = 3
ab+bc+ca = 4
abc = -5
Similarly, for the second equation, let the roots be p, q, and r. Then, we can write the equation as:
(x-p)(x-q)(x-r) = 0
Expanding this, we get:
x3 - (p+q+r)x2 + (pq+qr+rp)x - pqr = 0
Comparing this with the second equation, we get:
p+q+r = 2
pq+qr+rp = 7
pqr = -3
To find the common roots, we need to solve these two sets of equations simultaneously.
Adding the two sets of equations, we get:
a+b+c+p+q+r = 5
ab+bc+ca+pq+qr+rp = 11
abc+pqr = -8
Substituting the values of a+b+c and p+q+r from the above equations, we get:
5 - (ab+bc+ca) + 2 - (pq+qr+rp) = 0
Simplifying this, we get:
ab+bc+ca+pq+qr+rp = 7
Substituting this in the equation abc+pqr = -8, we get:
abc = -5
pqr = -3
Since the product of the roots is different for the two equations, they do not have any common roots.
Hence, the correct answer is option A.
Attention CAT Students!
To make sure you are not studying endlessly, EduRev has designed CAT study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in CAT.
|  |
| |
Related Content
| |
| |
Top Courses for CATView all
| 1 Crore+ students have signed up on EduRev. Have you? Download the App |
1 Crore+ students have signed up on EduRev. Have you?
Learn this topic in detail
 |
Practice Test for XAT - 3
100 Ques | 180 Mins
|
Start Test |
 |
More from Related Course
Quantitative Aptitude (Quant)
|
View notes & videos |
 |
View courses related to this question
|
Explore CAT courses |
Similar CAT Doubts
Quick links for CAT exam
The number of roots common between the two equations x3 + 3x2+ 4x + 5 = 0 and x3+ 2x2+ 7x + 3 = 0 isa)0 b)1 c)2 d)3Correct answer is option 'A'. Can you explain this answer?
Question Description
The number of roots common between the two equations x3 + 3x2+ 4x + 5 = 0 and x3+ 2x2+ 7x + 3 = 0 isa)0 b)1 c)2 d)3Correct answer is option 'A'. Can you explain this answer? for CAT 2023 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The number of roots common between the two equations x3 + 3x2+ 4x + 5 = 0 and x3+ 2x2+ 7x + 3 = 0 isa)0 b)1 c)2 d)3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2023 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number of roots common between the two equations x3 + 3x2+ 4x + 5 = 0 and x3+ 2x2+ 7x + 3 = 0 isa)0 b)1 c)2 d)3Correct answer is option 'A'. Can you explain this answer?.
The number of roots common between the two equations x3 + 3x2+ 4x + 5 = 0 and x3+ 2x2+ 7x + 3 = 0 isa)0 b)1 c)2 d)3Correct answer is option 'A'. Can you explain this answer? for CAT 2023 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The number of roots common between the two equations x3 + 3x2+ 4x + 5 = 0 and x3+ 2x2+ 7x + 3 = 0 isa)0 b)1 c)2 d)3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2023 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number of roots common between the two equations x3 + 3x2+ 4x + 5 = 0 and x3+ 2x2+ 7x + 3 = 0 isa)0 b)1 c)2 d)3Correct answer is option 'A'. Can you explain this answer?.
Solutions for The number of roots common between the two equations x3 + 3x2+ 4x + 5 = 0 and x3+ 2x2+ 7x + 3 = 0 isa)0 b)1 c)2 d)3Correct answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
Download more important topics, notes, lectures and mock test series for CAT Exam by signing up for free.
Here you can find the meaning of The number of roots common between the two equations x3 + 3x2+ 4x + 5 = 0 and x3+ 2x2+ 7x + 3 = 0 isa)0 b)1 c)2 d)3Correct answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The number of roots common between the two equations x3 + 3x2+ 4x + 5 = 0 and x3+ 2x2+ 7x + 3 = 0 isa)0 b)1 c)2 d)3Correct answer is option 'A'. Can you explain this answer?, a detailed solution for The number of roots common between the two equations x3 + 3x2+ 4x + 5 = 0 and x3+ 2x2+ 7x + 3 = 0 isa)0 b)1 c)2 d)3Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of The number of roots common between the two equations x3 + 3x2+ 4x + 5 = 0 and x3+ 2x2+ 7x + 3 = 0 isa)0 b)1 c)2 d)3Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The number of roots common between the two equations x3 + 3x2+ 4x + 5 = 0 and x3+ 2x2+ 7x + 3 = 0 isa)0 b)1 c)2 d)3Correct answer is option 'A'. Can you explain this answer? tests, examples and also practice CAT tests.
Do you know? How Toppers prepare for CAT Exam
With help of the best CAT teachers & toppers, We have prepared a guide for student who are
preparing for CAT : 15 Steps to clear CAT Exam
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Download free EduRev App
Track your progress, build streaks, highlight & save important lessons and more!
(Scan QR code)
|
|
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup
