Sets

$f\left(A,B\right)=\frac{1-sinB+sinA-sinAsinB+cosAcosB}{cosA\left(1-sinB\right)}$

$\begin{array}{l}\Rightarrow f\left(A,B\right)\left(cosA-cosAsinB-cosB+sinAcosB\right)\\ =2sinA-2sinB\end{array}$

$\Rightarrow f\left(A,B\right)=\frac{2\left(sinA-sinB\right)}{cosA-cosB+sin\left(A-B\right)}=2g\left(A,B\right)$

$n(A\cup B\cup C)+n(A\cup B\cup C{)}^{\mathrm{\text{'}}}=50$

$n(\cup )=50$

$n(A\cup B\cup C{)}^{\mathrm{\text{'}}}=n(\cup )-n(A\cup B\cup C)\Rightarrow 50-29\Rightarrow 21$

$n(A\cup B\cup C)=n\left(A\right)+n\left(B\right)+n\left(C\right)-n(A\cap B)-n(B\cap C)-n(C\cap A)-n(A\cap B\cap C)$

$\Rightarrow \mathrm{}10+15+20-5-6-10-5\Rightarrow 29$

$n(A\cup B\cup C{)}^{\mathrm{\text{'}}}=21$

A = {1,2,3,4,5} B = {0,1,2,3,4}
No. of elements common in A&B = 4.
∴ No. of elements common in A * B and B * A = 4 * 4 = 16

Out of numbers $\frac{1}{2},\frac{2}{4},\frac{3}{6},\frac{1}{3},\frac{2}{6},\frac{2}{3},\frac{4}{6}$ will result only 3 distinct rational numbers.

Total numbers =   

2a² + 3b² = 35
a=0, b≠I
a=±1, b²=11 ⇒ b≠I
a=±2, b=9 ⇒ b=3 or -3
a=±3, b²=17/3 ⇒ b≠I
a=±4, b²=1 ⇒ b=±1

Sum of elements $A\cap \left(B\cup C\right)=274*400$  

In set B numbers of the form 9k + 2 are {101, 109, .992}

  $\therefore sum=\frac{100}{2}\left(101+992\right)=\frac{100*1093}{2}......\left(i\right)$         

Another possible number is 9k + 5 forms are {104, .995}

  $\therefore sum=\frac{100}{2}\left(104+995\right)=\frac{100}{2}*1099.........\left(ii\right)$

$\therefore Total=\frac{100}{2}*\left[1093+1099\right]=100*1096=274*4*100=274*400$           

$\therefore$ possible value of $\mathcal{l}$ = 5

A= {1,2,3,4,5.100}
B= {4,7,10,13,16,19.}
C= {2,4,6,8,10.}
B−C= {7,13,19, .97}
A∩ (B−C)= {7,13,19, .97}
sum of elements=832

2x=tan (π/9)+tan (7π/18)
=sin (π/9+7π/18) / cos (π/9)cos (7π/18)
=sin (π/2) / cos (π/9)cos (7π/18)
=1 / cos (π/9)cos (7π/18)
=1 / cos (π/9)sin (π/2−7π/18)
=1 / cos (π/9)sin (π/9)
⇒x=1 / 2cos (π/9)sin (π/9)
=1 / sin (2π/9)=cosec (2π/9)

Again 2y=tan (π/9)+tan (5π/18)
⇒2y=sin (π/9+5π/18) / cos (π/9)cos (5π/18)
=sin (7π/18) / sin (π/2−π/9)sin (π/2−5π/18)
=sin (7π/18) / sin (7π/18)sin (4π/18) = cosec (2π/9)
⇒|x−2y|=0