If `f(m) = underset(i=0)overset(m)sum({:(30),(30-i):})({:(20),(m-i):})` where `({:(p),(q):})= .^(p)C_(q)`, then

If f(m) = sum_(i=0)_(m) ({:(30),(30-i):})({:(20),(m-i):}) where ({:(p),(q):})= ""^(p)C_(q) , then

The sum sum_(i=0)^m ((10),(i))((20),(m-1)) , where ((p),(q))=0 if p lt q , is maximum when m is equal to (A) 5 (B) 10 (C) 15 (D) 20

A=[{:(l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)),(l_(3),m_(3),n_(3)):}] and B=[{:(p_(1),q_(1),r_(1)),(p_(2),q_(2),r_(2)),(p_(3),q_(3),r_(3)):}] Where p_(i), q_(i),r_(i) are the co-factors of the elements l_(i), m_(i), n_(i) for i=1,2,3 . If (l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)) and (l_(3),m_(3),n_(3)) are the direction cosines of three mutually perpendicular lines then (p_(1),q_(1), r_(1)),(p_(2),q_(2),r_(2)) and (p_(3),q_(),r_(3)) are

sum_(m=1)^(n)(sum_(k=1)^(m)(sum_(p=k)^(m)"^(n)C_(m)*^(m)C_(p)*^(p)C_(k)))=

The distance between the point ( acosalpha,asinalpha) and ( acosbeta,asinbeta) is (a) ( b ) (c) acos( d )(( e )alpha-beta)/( f )2( g ) (h) (i) (j) (b) ( k ) (l)2acos( m )(( n )alpha-beta)/( o )2( p ) (q) (r) (s) (c) ( d ) (e)2as in (f)(( g )alpha-beta)/( h )2( i ) (j)( k ) (l) (d) ( m ) (n) asin( o )(( p )alpha-beta)/( q )2( r ) (s) (t) (u)

A curve is such that the mid-point of the portion of the tangent intercepted between the point where the tangent is drawn and the point where the tangent meets the y-axis lies on the line y=xdot If the curve passes through (1,0), then the curve is (a) ( b ) (c)2y=( d ) x^(( e )2( f ))( g )-x (h) (i) (b) ( j ) (k) y=( l ) x^(( m )2( n ))( o )-x (p) (q) (c) ( d ) (e) y=x-( f ) x^(( g )2( h ))( i ) (j) (k) (d) ( l ) (m) y=2(( n ) (o) x-( p ) x^(( q )2( r ))( s ) (t))( u ) (v)

If z=((sqrt(3)+i)^(17))/((1-i)^(50)) , then find a m p(z)dot

IfI(m , n)=int_0^1x^(m-1)(1-x)^(n-1)dx ,(m , n in I ,m ,ngeq0),t h e n I(m , n)=int_0^oo(x^(m-1))/((1+x)^(m-n))dx I(m , n)=int_0^oo(x^(m-1))/((1+x)^(m+n))dx I(m , n)=int_0^oo(x^(n-1))/((1+x)^(m+n))dx I(m , n)=int_0^oo(x^n)/((1+x)^(m+n))dx