Prove that the equation ` p cos x - q sin x =r ` admits solution for x only if `-sqrt(p^(2)+q^(2)) lt r lt sqrt(p^(2)+q^(2))`

if p: q :: r , prove that p : r = p^(2): q^(2) .

If the line of regression of Y on X is p_(1)x + q_(1)y + r_(1)= 0 and that of X on Y is p_(2)x + q_(2)y + r_(2)= 0 , prove that (p_(1)q_(2))/(p_(2)q_(1)) le 1

If one root of the equation x^(2)+px+q=0 is 2+sqrt(3) where p, q epsilon I and roots of the equation rx^(2)+x+q=0 are tan 268^(@) and cot 88^(@) then value of p+q+r is :

Let alpha,beta be the roots of the equation x^(2)-px+r=0 and alpha//2,2beta be the roots of the equation x^(2)-qx+r=0 , then the value of r is (1) (2)/(9)(p-q)(2q-p) (2) (2)/(9)(q-p)(2p-q) (3) (2)/(9)(q-2p)(2q-p) (4) (2)/(9)(2p-q)(2q-p)

If the ratio of the roots of the equation lx^2+nx+n=0 is p:q prove that sqrt(p/q) +sqrt(q/p)+sqrt(n/l)=0

If sin A + cos A = p and sec A + cosec A = q , then prove that : q(p^(2) - 1) = 2 p

For p, q in R , the roots of (p^(2) + 2)x^(2) + 2x(p +q) - 2 = 0 are

If the roots of the equation 1/ (x+p) + 1/ (x+q) = 1/r are equal in magnitude but opposite in sign, show that p+q = 2r & that the product of roots is equal to (-1/2)(p^2+q^2) .

If p, q, r each are positive rational number such tlaht p gt q gt r and the quadratic equation (p + q - 2r)x^(2) + (q + r- 2p)x + (r + p - 2q) = 0 has a root in (-1 , 0) then which of the following statement hold good? (A) (r + p)/(q) lt 2 (B) Both roots of given quadratic are rational (C) The equation px^(2) + 2qx + r = 0 has real and distinct roots (D) The equation px^(2) + 2qx + r = 0 has no real roots

Show that: x p q p x q q q xx-p)(x^2+p x-2q^2) .