If `[(a^x),(a^(-x))][(1,2)]=[(p,a^(- 2)),(q,log_2 2)],(a >0)` then `a^(p-q)=`

If Sin^(-1)(2p)/(1+p^(2)) - Cos^(-1)((1-q^(2))/(1+q^(2))) = Tan^(-1) (2x)/(1-x^(2)) , then prove that x = (p-q)/(1+pq)

If the roots of the equation (1-q+(p^(2))/(2))x^(2)+p(1+q)x+q(q-1)+(p^(2))/(2)=0 are equal then show that p^(2)=4q

If (x,2p+q)=(y,p+2q) then p-q=_______

If the roots of the equation (1-q+p^(2)/2)x^(2) + p(1+q)x+q(q-1) +p^(2)/2=0 are equal, then show that p^(2) = 4q

If lim_(x rarr0)(x+ln(sqrt(x^(2)+1)-x))/(x^(3))=(p)/(q) where p and q are coprimes then find the value of |p^(2)-q^(2)|

If {:(2x+y,x+y),(p-q,p+q):} ={:(1,1),(0,0):} then (x,y,p,q) equals

Let f(x)={(0 , x "is irrational"),(2/(2q^(3)-q^(2)+q+sin^(2)q+5) , if x=p/q ("rational")):} (where HCF (p,q)=1,p,q,gt0 ) and f(x) is defined AAxgt0 then which of the following is/are incorrect?

If log_(a)x = p and log_(b)x^(2) =q, then log_(x)sqrt(ab) is equal to (where, a,b, x in R^(+)-{1})- A) 1/p+1/q , B) 1/(2p)+1/q , C) 1/p+1/(2q) , D) 1/(2p)+1/(2q)