Out of 100 students, 15 passed in English, 12 passed in Mathematics, 8 in Science, 4 in English and Science, 4 in all the three.Find how many students passed in Mathematics and Science but not in English?

4

(a)

If \(cos\left( \alpha +\beta \right) =\frac { 4 }{ 5 } ,\quad sin(\alpha -\beta )=\frac { 5 }{ 13 } \) and \(\alpha , \beta \) lie between 0 and \(\frac { \pi }{ 4 } \) then prove that \(tan 2\alpha =\frac { 56 }{ 33 } .\)

4

(a)

If \(cos \left( \theta +\phi \right) = m\quad cos\quad \left( \theta -\phi \right) \)then find the value of \(\frac { 1-m }{ 1+m } cot\quad \phi .\)

4

(a)

If z is a complex number such that \(|z-1|=|z+1|\)  then show that Re(z) = 0.

4

(a)

If (1+i)(1+2i)(1+3i)...(1+ni)=(x+iy), then show that 2.5.10....(1+n²)=x²+y²

4

(a)