2012/021) If the expression E= a(sin^6t + cos^6t) - b(sin^4t + cos^4t) + 1 vanishes for all values of θ, then (a + b) .

sin ^ 6 t + cos^ 6 t= ( sin ^2 t+ cos^2 t) ^3 - 3 sin ^2 t cos^ 2 t( sin ^2 t + cos^2 t) = 1 - 3 sin ^2 t cos ^2 t

sin ^4 t + cos^ 4t = (sin ^2 t + cos ^2 t)^2 - 2 sin^2 t cos^2 t

a(sin^6t + cos^6t) - b(sin^4t + cos^4t) + 1
= a(1- 3 sin^2 t cos^2 t) - b(1- 2 sin^2 t cos ^2 t) + 1
= ( a + 1 - b) - sin^2 t cos ^2)(-3a + 2b)

as it is zero for any t so

a + 1 = b
- 3a + 2b = 0
or -3a + 2(a+1) = 0
a= 2 and b = 3 hence a+ b = 5