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What is the positive value of m for which the roots of the equation 12x2 + mx + 5 = 0 are in the ratio 3 ∶ 2?
- a)5 √10
- b)5 √10/12
- c)5/12
- d)12/5
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
What is the positive value of m for which the roots of the equation 12...
Let the roots be 3t and 2t respectively
Product of the roots = c/a
3t × 2t = 5/12
t2 = 5/72
∴ t = ± √5/(6√2)
Sum of roots = -b/a
3t + 2t = -m/12
-60t = m
∴ m = -60 × ± √5/(6√2) = ± 5√10
Only positive value of m is required
∴ m = 5√10
Most Upvoted Answer
What is the positive value of m for which the roots of the equation 12...
Let the roots of the equation be 3k and 2k, where k is a constant.
By Vieta's formulas, we know that the sum of the roots is equal to -b/a and the product of the roots is equal to c/a.
The sum of the roots is 3k + 2k = 5k.
The product of the roots is (3k)(2k) = 6k^2.
We can set up the following equations:
5k = -m/12 (equation 1)
6k^2 = 5/12 (equation 2)
From equation 2, we can solve for k:
6k^2 = 5/12
k^2 = 5/72
k = ±√(5/72)
k = ±(√5)/(6√2)
k = ±(√5)/(6√2) * (√2/√2)
k = ±(√10)/(12)
Since m is positive, we can take the positive value of k:
k = (√10)/(12)
Now we can substitute this value of k into equation 1 to solve for m:
5k = -m/12
5(√10)/(12) = -m/12
√10/12 = -m/12
m = -√10
However, since we are looking for the positive value of m, the positive value of m for which the roots of the equation are in the ratio 3:2 is:
m = √10
Therefore, the positive value of m is √10.
By Vieta's formulas, we know that the sum of the roots is equal to -b/a and the product of the roots is equal to c/a.
The sum of the roots is 3k + 2k = 5k.
The product of the roots is (3k)(2k) = 6k^2.
We can set up the following equations:
5k = -m/12 (equation 1)
6k^2 = 5/12 (equation 2)
From equation 2, we can solve for k:
6k^2 = 5/12
k^2 = 5/72
k = ±√(5/72)
k = ±(√5)/(6√2)
k = ±(√5)/(6√2) * (√2/√2)
k = ±(√10)/(12)
Since m is positive, we can take the positive value of k:
k = (√10)/(12)
Now we can substitute this value of k into equation 1 to solve for m:
5k = -m/12
5(√10)/(12) = -m/12
√10/12 = -m/12
m = -√10
However, since we are looking for the positive value of m, the positive value of m for which the roots of the equation are in the ratio 3:2 is:
m = √10
Therefore, the positive value of m is √10.
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What is the positive value of m for which the roots of the equation 12x2 + mx + 5 = 0 are in the ratio 3 ∶ 2?a)5 √10b)5 √10/12c)5/12d)12/5Correct answer is option 'A'. Can you explain this answer?
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