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_(m)^(i=0) ({:(10),(i):})({:(20),(m-i):}) , (where ({:(p),(q):})= 0 , if p lt q ) is maximum when 'm' is

The sum sum_(i=0)^m ((10)c,(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

If f(m)=sum_(i=0)^m(30(^) 30-i)(20(^)m-i)w h e r e(p q)=^p C_q ,t h e n (a) maximum value of f(m)i s^(50)C_(25) (b) f(0)+f(1)+...f(50)=2^(50) (c) f(m) is always divisible by 50(1lt=mlt=49) (d)The value of sum_(m=0)^(50)(f(m))^2=^(100)C_(50)

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)))=

Ph-underset((P))underset(OH)underset(|)(CH)-overset(O)overset(||)C-Hunderset(H_(2)O)overset(HO^(-))rarrQ.P and Q are isomers. Identify Q

(i) P("hydrocarbon")underset(H_(2)O)overset(O_(3)//Zn)rarr , , +CH_(2) = O , (ii) Q(C_(6)H_(10))underset(H_(2)O)overset(O_(3)//Zn)rarr Hexane- 1,6 -dial Write the structure of P and Q .

The sum of an angle and one third of its supplementary angle is 90^0dot The measure of the angle is (a) ( b ) (c) (d) (e) 135^(( f )0( g ))( h ) (i) (j) (b) ( k ) (l) (m) (n) 120^(( o )0( p ))( q ) (r) (s) (c) ( d ) (e) (f) (g) 60^(( h )0( i ))( j ) (k) (l) (d) ( m ) (n) (o) (p) 45^(( q )0( r ))( s ) (t) (u)