Simplify `((p^(a))/(p^(b)))^(c)times((p^(b))/(p^(c)))times((p^(c))/(p^(a)))`

The plots of (1)/(X_(A))vs.(1)/(Y_(A)) (where X_(A) and Y_(A) are the mole fraction of liquid A in liquid and vapour phase respectively) is linear with slope and intercept respectively are given as: (a) (P_A^(@))/(P_B^(@)),(P_B^(@)-P_A^(@))/(P_B^(@)) (b) (P_B^(@))/(P_A^(@)),(P_A^(@)-P_B^(@))/(P_A^(@)) (c) (P_B^(@))/(P_A^(@)),(P_B^(@))/(P_B^@-P_A^(@)) (d) P_A^(@)-P_B^(@),(P_A^(@))/(P_B^(@))

For positive real numbers a ,bc such that a+b+c=p , which one holds? (a) (p-a)(p-b)(p-c)lt=8/(27)p^3 (b) (p-a)(p-b)(p-c)geq8a b c (c) (b c)/a+(c a)/b+(a b)/clt=p (d) none of these

If a x+b y+c z=p , then minimum value of x^2+y^2+z^2 is (p/(a+b+c))^2 (b) (p^2)/(a^2+b^2+c^2) (a^2+b^2+c^2)/(p^2) (d) ((a+b+c)/p)^2

If A and B are independent events such that P(A) gt 0, P(B) gt 0 , then (a) 1-P(A//B) (b) 1-P(A //bar(B)) (c) (1- P(A uu B))/(P(B)) (d) (P(A))/(P(B))+(P(bar(A)))/(P(B))=1

If a^(2), b^(2),c^(2) are in A.P prove that (a)/(b+c), (b)/(c+a) ,(c)/(a+b) are in A.P.

If A and B are events such that P(A uu B) = (3)//(4), P(A nn B) = (1)//(4) and P(A^(c)) = (2)//(3) , then find (a) P(A) (b) P(B) (c ) P(A nn B^(c )) (d) P(A^(c ) nn B)

If vec a is perpendicular to vec b and vec r is non-zero vector such that p vec r+( vec rdot vec a) vec b= vec c , then vec r= vec c/p-(( vec adot vec c) vec b)/(p^2) (b) vec a/p-(( vec cdot vec b) vec a)/(p^2) vec a/p-(( vec adot vec b) vec c)/(p^2) (d) vec c/(p^2)-(( vec adot vec c) vec b)/p

The expression log_p.log_p(((p)^(1/p))^(1/p)........n time)^(1/p) where p>= 2 , when simplified is (A)independent of p, but dependent on n (B) independent of n, but dependent on p (C) dependent on both p & n (D) negative

If A and B are two events such that P(A)=0.3 , P(B)=0.25 , P(AnnB)=0.2 , then P(((A^(C ))/(B^(C )))^(C )) is equal to

If A and B are two events such that A!=varphi,\ B=varphi, then a. P(A//B)=(P(AnnB))/(P(B)) b. P(A//B)=P(A)P(B) c. P(A//B)=P(B//A)=1 d. P(A//B)=(P(A))/(P(b))