Evaluate: `(lim)_(x->0)sin8xcot3x`. If it is of the form `p/q`, then `(p+q)=`?

If log_(b)a=3,log_(b)=-4. If the value of x satisfying theequation a^(3x)=c^(x)-1 is expressed in the form (p)/(q), where p and q are relatively prime then p+q is

If the value of lim_(x->0)( ((3/ x)+1)/((3/x)-1))^(1/x) can be expressed in the form of e^(p/q), where p and q are relative prime then find the value of (p+q) .

If alpha,beta are the roots of the equation x^(2)+px-r=0 and (alpha)/(3),3 beta are the roots of the equation x^(2)+qx-r=0, then r equals (i)((3)/(8))(p-3q)(3p+q)(ii)((3)/(8))(3p-q)(3p+q)(iii)((3)/(64))(q-3p)(p-3p)(iv)((3)/(64))(p-q)(q-3p)

Let P(x) and Q(x) be two quadratic polynomials such that minimum value of P(x) is equal to twice of maximum value of Q(x) and Q(x)>0AA x in(1,3) and Q(x)<0AA x in(-oo ,1)uu(3, oo) If sum of the roots of the equation P(x) =0 is 8 and P(3)-P(4)=1 and distance between the vertex of y=P(x) and vertex of y=Q(x) is 2sqrt(2) The polynomial P(x) is:

Let P(x) and Q(x) be two quadratic polynomials such that minimum value of P(x) is equal to twice of maximum value of Q(x) and Q(x)>0AA x in(1, 3) and Q (x)<0AA x in(-oo, 1)uu(3, oo) . If sum of the roots of the equation P(x)=0 is 8 and P(3)-P(4)=1 and distance between the vertex of y=P(x) and vertex of y=Q(x) is 2sqrt(2) The polynomial P(x) is:

If the sum of square of roots of equation x^(2)+(p+iq)x+3i=0 is 8, then find p and q,where p and q are real.

If p and q are the roots of x^(2)+px+q=0 then find p.

The value of ("lim")_(xvec1)(p/(1-x^p)-q/(1-x^q)),p ,q , in N , equal (a)(p+q)/2 (b) (p q)/2 (c) (p-q)/2 (d) sqrt(p/q)