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If Every Pair from among the equations x^2 px qr =0 , x^2 qx rp = 0 and x^2 rx pq = 0 has a common root, then the sum of the three common roots is? OPTIONS:- a) 2(pqr) b) pqr c) -(pqr) d) pqr?
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If Every Pair from among the equations x^2 px qr =0 , x^2 qx rp = 0 an...
Explanation:
Introduction:
We are given three equations:
- x^2 px qr = 0
- x^2 qx rp = 0
- x^2 rx pq = 0
And we are asked to find the sum of the three common roots.
Solution:
Let's assume that the common roots of the first two equations are x1 and x2, the common roots of the second and third equations are x2 and x3, and the common roots of the first and third equations are x1 and x3.
Common root of first two equations:
For the first two equations, we have:
- x^2 px qr = 0
- x^2 qx rp = 0
If x1 is a common root of the first two equations, then it satisfies both equations, and we get:
- x1^2 px qr = 0
- x1^2 qx rp = 0
Dividing the second equation by x1, we get:
- x1 qx rp = 0
Since x1 is not equal to 0 (otherwise it wouldn't be a root), we can divide the first equation by x1^2, and we get:
- px qr = 0
So, either p or q or r must be 0. Without loss of generality, let's assume that p = 0. Then the first equation becomes:
- x^2 qr = 0
And the only root of this equation is x = 0. Therefore, x1 = 0.
Common root of second and third equations:
For the second and third equations, we have:
- x^2 qx rp = 0
- x^2 rx pq = 0
If x2 is a common root of the second and third equations, then it satisfies both equations, and we get:
- x2^2 qx rp = 0
- x2^2 rx pq = 0
Dividing the first equation by x2 and the second equation by x2, we get:
- x2 qx rp = 0
- x2 rx pq = 0
Since x2 is not equal to 0 (otherwise it wouldn't be a root), we can divide the first equation by x2, and we get:
- qx rp = 0
So, either q or r must be 0. Without loss of generality, let's assume that q = 0. Then the first equation becomes:
- x^2 rp = 0
And the only root of this equation is x = 0. Therefore, x2 = 0.
Common root of first and third equations:
For the first and third equations, we have:
- x^2 px qr = 0
- x^2 rx pq = 0
If x3 is a common root of the first and third equations, then it satisfies both equations, and we get:
- x3^2 px qr = 0
- x3^2 rx pq = 0
Dividing the first equation by x3 and the second equation by x3, we get:
- x3 px qr = 0
- x3 rx pq = 0
Since x3 is not
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If Every Pair from among the equations x^2 px qr =0 , x^2 qx rp = 0 and x^2 rx pq = 0 has a common root, then the sum of the three common roots is? OPTIONS:- a) 2(pqr) b) pqr c) -(pqr) d) pqr?
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