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If p and q are the roots of the equation x2 - 7x - 13 = 0, which of these equations has 4p + 3 and 4q + 3 as roots? (Difficult)
- a)x2 - 34x - 115 = 0
- b)x2 - 28x - 58 = 0
- c)x2 - 28x - 133 = 0
- d)x2 - 22x + 58 = 0
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
If p and q are the roots of the equation x2 - 7x - 13 = 0, which of t...
The equation with 4p and 4q as its roots is
⇒ x2 - 28x - 208 = 0
The equation with 4p+3 and 4q+3 as its roots is:
(x - 3)2 - 28(x - 3) - 208 = 0
⇒ x2 - 6x + 9 - 28x + 84 -208 = 0
⇒ x2 - 34x - 115 = 0
Hence, the correct option is (a).
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If p and q are the roots of the equation x2 - 7x - 13 = 0, which of t...
Finding the Roots p and q
To begin, we need to find the roots p and q of the quadratic equation:
x² - 7x - 13 = 0.
Using the quadratic formula, we find:
- p + q = 7 (sum of roots)
- pq = -13 (product of roots)
Calculating the roots:
- p = (7 + √(7² + 4 * 13)) / 2
- q = (7 - √(7² + 4 * 13)) / 2
This gives us specific numerical values for p and q.
Transforming the Roots
Next, we want to transform the roots p and q into new roots 4p + 3 and 4q + 3.
- New roots:
- 4p + 3
- 4q + 3
To find the sum and product of these new roots:
Sum of New Roots
- Sum = (4p + 3) + (4q + 3) = 4(p + q) + 6 = 4(7) + 6 = 28 + 6 = 34.
Product of New Roots
- Product = (4p + 3)(4q + 3) = 16pq + 12(p + q) + 9 = 16(-13) + 12(7) + 9
- Simplifying gives: -208 + 84 + 9 = -115.
New Quadratic Equation
Now that we have the sum and product of the new roots, we can write the quadratic equation:
x² - (sum)x + (product) = 0
=> x² - 34x - 115 = 0.
Conclusion
Thus, the equation that has roots 4p + 3 and 4q + 3 is:
Option A: x² - 34x - 115 = 0.
To begin, we need to find the roots p and q of the quadratic equation:
x² - 7x - 13 = 0.
Using the quadratic formula, we find:
- p + q = 7 (sum of roots)
- pq = -13 (product of roots)
Calculating the roots:
- p = (7 + √(7² + 4 * 13)) / 2
- q = (7 - √(7² + 4 * 13)) / 2
This gives us specific numerical values for p and q.
Transforming the Roots
Next, we want to transform the roots p and q into new roots 4p + 3 and 4q + 3.
- New roots:
- 4p + 3
- 4q + 3
To find the sum and product of these new roots:
Sum of New Roots
- Sum = (4p + 3) + (4q + 3) = 4(p + q) + 6 = 4(7) + 6 = 28 + 6 = 34.
Product of New Roots
- Product = (4p + 3)(4q + 3) = 16pq + 12(p + q) + 9 = 16(-13) + 12(7) + 9
- Simplifying gives: -208 + 84 + 9 = -115.
New Quadratic Equation
Now that we have the sum and product of the new roots, we can write the quadratic equation:
x² - (sum)x + (product) = 0
=> x² - 34x - 115 = 0.
Conclusion
Thus, the equation that has roots 4p + 3 and 4q + 3 is:
Option A: x² - 34x - 115 = 0.
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If p and q are the roots of the equation x2 - 7x - 13 = 0, which of these equations has 4p + 3 and 4q + 3 as roots? (Difficult)a)x2 - 34x - 115 = 0b)x2 - 28x - 58 = 0c)x2 - 28x - 133 = 0d)x2 - 22x + 58 = 0Correct answer is option 'A'. Can you explain this answer?
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If p and q are the roots of the equation x2 - 7x - 13 = 0, which of these equations has 4p + 3 and 4q + 3 as roots? (Difficult)a)x2 - 34x - 115 = 0b)x2 - 28x - 58 = 0c)x2 - 28x - 133 = 0d)x2 - 22x + 58 = 0Correct answer is option 'A'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about If p and q are the roots of the equation x2 - 7x - 13 = 0, which of these equations has 4p + 3 and 4q + 3 as roots? (Difficult)a)x2 - 34x - 115 = 0b)x2 - 28x - 58 = 0c)x2 - 28x - 133 = 0d)x2 - 22x + 58 = 0Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If p and q are the roots of the equation x2 - 7x - 13 = 0, which of these equations has 4p + 3 and 4q + 3 as roots? (Difficult)a)x2 - 34x - 115 = 0b)x2 - 28x - 58 = 0c)x2 - 28x - 133 = 0d)x2 - 22x + 58 = 0Correct answer is option 'A'. Can you explain this answer?.
If p and q are the roots of the equation x2 - 7x - 13 = 0, which of these equations has 4p + 3 and 4q + 3 as roots? (Difficult)a)x2 - 34x - 115 = 0b)x2 - 28x - 58 = 0c)x2 - 28x - 133 = 0d)x2 - 22x + 58 = 0Correct answer is option 'A'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about If p and q are the roots of the equation x2 - 7x - 13 = 0, which of these equations has 4p + 3 and 4q + 3 as roots? (Difficult)a)x2 - 34x - 115 = 0b)x2 - 28x - 58 = 0c)x2 - 28x - 133 = 0d)x2 - 22x + 58 = 0Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If p and q are the roots of the equation x2 - 7x - 13 = 0, which of these equations has 4p + 3 and 4q + 3 as roots? (Difficult)a)x2 - 34x - 115 = 0b)x2 - 28x - 58 = 0c)x2 - 28x - 133 = 0d)x2 - 22x + 58 = 0Correct answer is option 'A'. Can you explain this answer?.
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