The equation `x^(2) + px +q =0` has two roots ` alpha, beta` then
Choose the correct answer :
(1) The equation ` qx^(2) +(2q -p^(2))x +q` has roots
` (1) alpha /beta , beta/alpha ` (2) `1/alpha ,1/beta`
` (3) alpha^(2), beta^(2)` (4) None of these
(2) The equation whose roots are `1/alpha , 1/beta ` is
`(1) px^(2) + qx + 1 =0` (2) `qx^(2) +px +1 =0`
(3)` x^(2) + qx + p =0 `
(4) `qx^(2) +x+ p =0`
3. the values of p and q when roots are -1,2
(1) p = -1 , q = -2 (2) p =-1 , q =2
(3) p=1 , q = -2 (4) None of these

Explanation

The equation ` x^(2) + px + q =0` has two roots a,b .
so , `alpha + beta = -p , alpha ,beta = q `
(1) Answrt (1)
Now `alpha/beta + beta/alpha (alpha^(2) +beta^(2))/(alphabeta) = ((alpha+beta)^(2) -2alpha beta)/(alphabeta)`
` = (p^(2) -2q)/q`
Product or roots = ` alpha/beta. beta/alpha =1`
So , the equation is
`x^(2) - ((p^(2) -2q)/q) x + 1 =0`
`Rightarrow qx^(2) + (2q -p^2) x+q=0`
Hence the roots of given equation is ` alpha/beta, beta/alpha`
2. (Answer (2))
Sum of roots `= 1/alpha + 1/beta = (alpha +beta)/(alpha beta) = - p/q`
Product of roots `= 1/alpha .1/beta =1/q`
Hence, the required quadratic equation is ` x^(2)` - (sum of roots) x + product of roots =0
` Rightarrow x^(2) + (px)/q + 1/q =0`
` Rightarrow qx^(2) + px + 1=0`
3. Answer (1) given roots are -1,2
so, sum of root s = -1 + 2 = + 1 = -p
` Rightarrow p = =-1 `
Product of roots = -1.2 =-2 =q
` Rightarrow q=-2 `