Question

Given that `alpha,gamma` are roots of the equation `A x^2-4x+1=0,a n dbeta,delta` the roots of the equation of `B x^2-6x+1=0,` such that `alpha,beta,gamma,a n ddelta` are in H.P., then a.`A=3` b. `A=4` `B=2` d. `B=8`
A. A = 3
B. A = 4
C. B = 2
D. B = 8

Answer 1

Correct Answer - 1,4
Since ` alpha , beta , gamma, delta ` are in H.P. , ` 1//alpha , 1//beta, 1//gamma, 1//delta` are in A.P. and
the may be taken as ` a - 3d , a - d , a + d, a + 3 d` . Replacing x
by `1//x`, we get the equation whose roots are ` 1//alpha , `1//beta, 1//gamma, `//sigma` .
Therefore , equation `x^(2) - 6x + B = 0 ` has roots a - d , a + 3d Sum of
the roots is
` 2 (a - d) = 4, 2(a + b) = 6`
`therefore a = 5 //2 2, d = 1//2 `
Product of the roots is
` (a - 3d ) (a + d) = A = 3 `
` (a - d) (a + 3d) = B = 8` .