If the equations `x^2+p x+q=0a n dx^2+p^(prime)x+q^(prime)=0` have a common root, then it must be equal to a.`(p^(prime)-p ^(prime) q)/(q-q^(prime))` b. `(q-q ')/(p^(prime)-p)` c. `(p^(prime)-p)/(q-q^(prime))` d. `(p q^(prime)-p^(prime) q)/(p-p^(prime))`

If the equation x^(2)+px+q=0 and x^(2)+p'x+q'=0 have a common root show that it must be equal to (pq'-p'q)/(q-q') or (q-q')/(p'-p)

If alpha,beta are the roots of x^2-p x+q=0a n dalpha^(prime),beta' are the roots of x^2-p^(prime)x+q^(prime)=0, then the value of (alpha-alpha^(prime))^2+(beta-alpha^(prime))^2+(alpha-beta^(prime))^2+(beta-beta^(prime))^2 is 2{p^2-2q+p^('2)-2q^(prime)-p p '} 2{p^2-2q+p^('2)-2q^(prime)-q q '} 2{p^2-2q-p^('2)-2q^(prime)-p p '} 2{p^2-2q-p^('2)-2q^(prime)-q q '}

If the quadratic equation x^(2)-px+q=0 where p, q are the prime numbers has integer solutions, then minimum value of p^(2)-q^(2) is equal to _________

Consider the family of all circles whose centers lie on the straight line y=x . If this family of circles is represented by the differential equation P y^(primeprime)+Q y^(prime)+1=0, where P ,Q are functions of x , y and y^(prime)(h e r ey^(prime)=(dy)/(dx),y^=(d^2y)/(dx^2)), then which of the following statements is (are) true? (a)P=y+x (b)P=y-x (c)P+Q=1-x+y+y^(prime)+(y^(prime))^2 (d)P-Q=x+y-y^(prime)-(y^(prime))^2

(v) If p and q are prime factors of a number a then their product p xx q is also a factor of a.