Prove
`root(p+q)(x^(p^(2))/(x^(q^(2))))xxroot(q+r)(x^(q^(2))/(x^(r^(2))))xxroot(r+p)(x^(r^(2))/(x^(p^(2))))=1`

Simply (x^(p)/x^(q))^(p+q)div((x^(p+q))/(x^(p-q)))^(p^(2)/(q))

Simply (a^(p)/a^(q))^(p+q-r).(a^(q)/a^(r ))^(q+r-p).(a^(r )/a^(p))^(r+p-q)

Prove that : ((a^q)/(a^r))^p xx((a^r)/(a^p))^qxx((a^p)/(a^q))^r =1

Consider int(x^(3)+3x^(2)+2x+1)/(sqrt(x^(2)+x+1))dx =(ax^(2)+bx+c)sqrt(x^(2)+x+1)+lambda int(dx)/(sqrt(x^(2)+x+1)) Now, match the following lists and then choose the correct code. Codes: {:(,a,b,c,d),((1),q,p,s,r),((2),s,p,q,r),((3),r,q,p,s),((4),q,s,p,r):}

Prove that the points (p,p^(2)),(q,q^(2))and(r,r^(2))(pner) can never be collinear.

If the ratio of the roots of a x ^2+2b x+c=0 is same as the ratio of roots of p x^2+2q x+r=0, then a. (2b)/(a c)=(q^2)/(p r) b. b/(a c)=(q^2)/(p r) c. (b^2)/(a c)=(q^2)/(p r) d. none of these

Prove that the points (p,p^(2)), (q,q^(2))and(r,r^(2))(pneqner) can never be collinear.

Let p and q be real numbers such that p!=0,p^3!=q ,and p^3!=-qdot If alpha and beta are nonzero complex numbers satisfying alpha+beta=-p and alpha^3+beta^3=q , then a quadratic equation having alpha//beta and beta//alpha as its roots is A. (p^3+q)x^2-(p^3+2q)x+(p^3+q)=0 B. (p^3+q)x^2-(p^3-2q)x+(p^3+q)=0 C. (p^3+q)x^2-(5p^3-2q)x+(p^3-q)=0 D. (p^3+q)x^2-(5p^3+2q)x+(p^3+q)=0

Prove that : (v) p^(log_(x)q) = q^(log_(x)p)

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)