Mathematics Exam > Mathematics Questions > Let 𝐶 be set of integer pairs (𝑎, 𝑏) for w...
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Let 𝐶 be set of integer pairs (𝑎, 𝑏) for which the three
complex roots 𝑟1
, 𝑟2 and 𝑟3 of the polynomial 𝑝(𝑥) = 𝑥
3 −
2𝑥
2 + 𝑎𝑥 − 𝑏 satisfy 𝑟1
3 + 𝑟2
3 + 𝑟3
3 = 0. Then the
cardinality of 𝐶 is?
complex roots 𝑟1
, 𝑟2 and 𝑟3 of the polynomial 𝑝(𝑥) = 𝑥
3 −
2𝑥
2 + 𝑎𝑥 − 𝑏 satisfy 𝑟1
3 + 𝑟2
3 + 𝑟3
3 = 0. Then the
cardinality of 𝐶 is?
Most Upvoted Answer
Let 𝐶 be set of integer pairs (𝑎, 𝑏) for which the three complex ro...
Understanding the Problem
To solve for the cardinality of the set C containing integer pairs (a, b), we need to analyze the cubic polynomial given by:
- p(x) = x^3 - 2x^2 + ax - b
The roots r1, r2, and r3 satisfy the condition:
- r1^3 + r2^3 + r3^3 = 0
Using Symmetric Sums
We can use the properties of symmetric sums of roots:
- Sum of roots: r1 + r2 + r3 = 2 (from the coefficient of x^2)
- Sum of product of roots taken two at a time: r1r2 + r2r3 + r3r1 = a
- Product of roots: r1r2r3 = b
The identity for the sum of cubes can be expressed as:
- r1^3 + r2^3 + r3^3 = (r1 + r2 + r3)(r1^2 + r2^2 + r3^2 - r1r2 - r2r3 - r3r1)
Given that r1 + r2 + r3 = 2, we can simplify this expression to find conditions on a and b.
Derived Condition
From the condition r1^3 + r2^3 + r3^3 = 0, we can derive:
- (2)(r1^2 + r2^2 + r3^2 - a) = 0
This leads us to find:
- r1^2 + r2^2 + r3^2 = a
Using the identity:
- r1^2 + r2^2 + r3^2 = (r1 + r2 + r3)^2 - 2(r1r2 + r2r3 + r3r1)
We derive:
- 4 - 2a = a
- Hence, a = 4/3, which is not an integer.
Cardinality of Set C
Since a must be an integer and the derived expression for a does not yield integer values, we conclude:
- No integer pairs (a, b) satisfy the condition.
Thus, the cardinality of set C is:
- 0.
To solve for the cardinality of the set C containing integer pairs (a, b), we need to analyze the cubic polynomial given by:
- p(x) = x^3 - 2x^2 + ax - b
The roots r1, r2, and r3 satisfy the condition:
- r1^3 + r2^3 + r3^3 = 0
Using Symmetric Sums
We can use the properties of symmetric sums of roots:
- Sum of roots: r1 + r2 + r3 = 2 (from the coefficient of x^2)
- Sum of product of roots taken two at a time: r1r2 + r2r3 + r3r1 = a
- Product of roots: r1r2r3 = b
The identity for the sum of cubes can be expressed as:
- r1^3 + r2^3 + r3^3 = (r1 + r2 + r3)(r1^2 + r2^2 + r3^2 - r1r2 - r2r3 - r3r1)
Given that r1 + r2 + r3 = 2, we can simplify this expression to find conditions on a and b.
Derived Condition
From the condition r1^3 + r2^3 + r3^3 = 0, we can derive:
- (2)(r1^2 + r2^2 + r3^2 - a) = 0
This leads us to find:
- r1^2 + r2^2 + r3^2 = a
Using the identity:
- r1^2 + r2^2 + r3^2 = (r1 + r2 + r3)^2 - 2(r1r2 + r2r3 + r3r1)
We derive:
- 4 - 2a = a
- Hence, a = 4/3, which is not an integer.
Cardinality of Set C
Since a must be an integer and the derived expression for a does not yield integer values, we conclude:
- No integer pairs (a, b) satisfy the condition.
Thus, the cardinality of set C is:
- 0.
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Let 𝐶 be set of integer pairs (𝑎, 𝑏) for which the three complex roots 𝑟1 , 𝑟2 and 𝑟3 of the polynomial 𝑝(𝑥) = 𝑥 3 − 2𝑥 2 + 𝑎𝑥 − 𝑏 satisfy 𝑟1 3 + 𝑟2 3 + 𝑟3 3 = 0. Then the cardinality of 𝐶 is? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let 𝐶 be set of integer pairs (𝑎, 𝑏) for which the three complex roots 𝑟1 , 𝑟2 and 𝑟3 of the polynomial 𝑝(𝑥) = 𝑥 3 − 2𝑥 2 + 𝑎𝑥 − 𝑏 satisfy 𝑟1 3 + 𝑟2 3 + 𝑟3 3 = 0. Then the cardinality of 𝐶 is? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let 𝐶 be set of integer pairs (𝑎, 𝑏) for which the three complex roots 𝑟1 , 𝑟2 and 𝑟3 of the polynomial 𝑝(𝑥) = 𝑥 3 − 2𝑥 2 + 𝑎𝑥 − 𝑏 satisfy 𝑟1 3 + 𝑟2 3 + 𝑟3 3 = 0. Then the cardinality of 𝐶 is?.
Let 𝐶 be set of integer pairs (𝑎, 𝑏) for which the three complex roots 𝑟1 , 𝑟2 and 𝑟3 of the polynomial 𝑝(𝑥) = 𝑥 3 − 2𝑥 2 + 𝑎𝑥 − 𝑏 satisfy 𝑟1 3 + 𝑟2 3 + 𝑟3 3 = 0. Then the cardinality of 𝐶 is? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let 𝐶 be set of integer pairs (𝑎, 𝑏) for which the three complex roots 𝑟1 , 𝑟2 and 𝑟3 of the polynomial 𝑝(𝑥) = 𝑥 3 − 2𝑥 2 + 𝑎𝑥 − 𝑏 satisfy 𝑟1 3 + 𝑟2 3 + 𝑟3 3 = 0. Then the cardinality of 𝐶 is? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let 𝐶 be set of integer pairs (𝑎, 𝑏) for which the three complex roots 𝑟1 , 𝑟2 and 𝑟3 of the polynomial 𝑝(𝑥) = 𝑥 3 − 2𝑥 2 + 𝑎𝑥 − 𝑏 satisfy 𝑟1 3 + 𝑟2 3 + 𝑟3 3 = 0. Then the cardinality of 𝐶 is?.
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Let 𝐶 be set of integer pairs (𝑎, 𝑏) for which the three complex roots 𝑟1 , 𝑟2 and 𝑟3 of the polynomial 𝑝(𝑥) = 𝑥 3 − 2𝑥 2 + 𝑎𝑥 − 𝑏 satisfy 𝑟1 3 + 𝑟2 3 + 𝑟3 3 = 0. Then the cardinality of 𝐶 is?, a detailed solution for Let 𝐶 be set of integer pairs (𝑎, 𝑏) for which the three complex roots 𝑟1 , 𝑟2 and 𝑟3 of the polynomial 𝑝(𝑥) = 𝑥 3 − 2𝑥 2 + 𝑎𝑥 − 𝑏 satisfy 𝑟1 3 + 𝑟2 3 + 𝑟3 3 = 0. Then the cardinality of 𝐶 is? has been provided alongside types of Let 𝐶 be set of integer pairs (𝑎, 𝑏) for which the three complex roots 𝑟1 , 𝑟2 and 𝑟3 of the polynomial 𝑝(𝑥) = 𝑥 3 − 2𝑥 2 + 𝑎𝑥 − 𝑏 satisfy 𝑟1 3 + 𝑟2 3 + 𝑟3 3 = 0. Then the cardinality of 𝐶 is? theory, EduRev gives you an
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