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 the equation x^(3) +px +q =0 has three real roots then show that 4p^(3)+ 27q^(2) lt 0 .

If one root of the equation x^2+px+q=0 is 2+sqrt3 , then values of p and q is:

5pq(p^2 - q^2) -: 2p(p+q)

Number of solutions of the equation cos^(4) 2x+2 sin^(2) 2x =17 (cos x + sin x)^(8), 0 lt x lt 2 pi is

Differentiate the following functions : sin^P x cos ^q x

If cos x + sin x = a , (- (pi)/(2) lt x lt - (pi)/(4)) , then cos 2 x is equal to a. a^(2) b. a sqrt((2-a)) c. a sqrt((2+a)) d. asqrt((2-a^(2)))

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) .

ABCD is a trapezium such that AB and CD are parallel and BC bot CD . If angleADB = theta, BC = p and CD = q , then AB is equal to (a) ((p^(2)+q^(2))sin theta)/(p cos theta +q sin theta) (b) (p ^(2) + q ^(2)cos theta)/(p cos theta +q sin theta) (c) (p ^(2)+ q^(2))/(p^(2)cos theta +q^(2) sin theta) (d) ((p^(2) +q^(2))sin theta)/((p cos theta + sin theta)^(2))