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Two different quadratic equations have a common root. Let the three unique roots of the two equations be A, B and C - all of them are positive integers. If (A + B + C) = 41 and the product of the roots of one of the equations is 35, which of the following options is definitely correct?
  • a)
    The common root is 29.
  • b)
    The smallest among the roots is 1.
  • c)
    One of the roots is 5.
  • d)
    Product of the roots of the other equation is 5.
  • e)
    All of the above are possible, but none are definitely correct.
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Two different quadratic equations have a common root. Let the three u...
It has been given that A+B+C = 41.
Let the common root be B.
All the roots are positive integers.
The product of the roots of one of the equations is 35.
35 can be obtained only in 2 ways - either as 5*7 or 35*1.
A+B+C = 41.
If A and B are 5 and 7 in any order, then C = 41 - 5 - 7 = 29.
If A and B are 35 and 1 in any order, then C = 41 - 35 -1 =5.
As we can see, in either case, 5 is one of the 3 roots.
Therefore, option C is the right answer.
Most Upvoted Answer
Two different quadratic equations have a common root. Let the three u...
Given information:
- Two different quadratic equations have a common root.
- The three unique roots of the two equations are A, B, and C, all of which are positive integers.
- (A + B + C) = 41.
- The product of the roots of one of the equations is 35.

To determine the correct option, we need to analyze the given information.

- Let's assume the two quadratic equations are:
Equation 1: ax^2 + bx + c = 0
Equation 2: px^2 + qx + r = 0

- Since the equations have a common root, this means that the common root satisfies both equations. Let's call the common root 'k'.

- Therefore, substituting 'k' into both equations, we get:
ak^2 + bk + c = 0
pk^2 + qk + r = 0

- Since k is a root of both equations, these equations can be factored as:
a(x - k)(x - m) = 0
p(x - k)(x - n) = 0

- From the first equation, we can infer that the product of the other two roots (m and n) is c/a.
- From the second equation, we can infer that the product of the other two roots (m and n) is r/p.

- Since the product of the roots of one of the equations is 35, we can conclude that either c/a = 35 or r/p = 35.

- Let's consider the options given:
a) The common root is 29.
b) The smallest among the roots is 1.
c) One of the roots is 5.
d) The product of the roots of the other equation is 5.
e) All of the above are possible, but none are definitely correct.

- Let's analyze each option based on the given information:

a) The common root is 29:
- We cannot determine if this is true or false based on the given information.

b) The smallest among the roots is 1:
- We cannot determine if this is true or false based on the given information.

c) One of the roots is 5:
- This option is definitely correct because it is given that all the roots are positive integers. So, one of the roots can be 5.

d) The product of the roots of the other equation is 5:
- We cannot determine if this is true or false based on the given information.

e) All of the above are possible, but none are definitely correct:
- This option is not correct because option c is definitely correct, as explained above.

Therefore, the correct option is c) One of the roots is 5.
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Two different quadratic equations have a common root. Let the three unique roots of the two equations be A, B and C - all of them are positive integers. If (A + B + C) = 41 and the product of the roots of one of the equations is 35, which of the following options is definitely correct?a)The common root is 29.b)The smallest among the roots is 1.c)One of the roots is 5.d)Product of the roots of the other equation is 5.e)All of the above are possible, but none are definitely correct.Correct answer is option 'C'. Can you explain this answer?
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Two different quadratic equations have a common root. Let the three unique roots of the two equations be A, B and C - all of them are positive integers. If (A + B + C) = 41 and the product of the roots of one of the equations is 35, which of the following options is definitely correct?a)The common root is 29.b)The smallest among the roots is 1.c)One of the roots is 5.d)Product of the roots of the other equation is 5.e)All of the above are possible, but none are definitely correct.Correct answer is option 'C'. 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 Two different quadratic equations have a common root. Let the three unique roots of the two equations be A, B and C - all of them are positive integers. If (A + B + C) = 41 and the product of the roots of one of the equations is 35, which of the following options is definitely correct?a)The common root is 29.b)The smallest among the roots is 1.c)One of the roots is 5.d)Product of the roots of the other equation is 5.e)All of the above are possible, but none are definitely correct.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2023 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two different quadratic equations have a common root. Let the three unique roots of the two equations be A, B and C - all of them are positive integers. If (A + B + C) = 41 and the product of the roots of one of the equations is 35, which of the following options is definitely correct?a)The common root is 29.b)The smallest among the roots is 1.c)One of the roots is 5.d)Product of the roots of the other equation is 5.e)All of the above are possible, but none are definitely correct.Correct answer is option 'C'. Can you explain this answer?.
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