XI test 26/27

1) If p= a+ b+ c, q= a+ wb+ w²c, r= w²b+ wc where w is a nonreal cube root of 1, show that p³+ q³+ r³ - 3pqr= 27abc.

2) Solve: x¹⁾³ + (2x -3)¹⁾³ = {12(x -1)}¹⁾³.

3) In an arithmetic progression of n terms (n is even) the two middle terms are p- q, p+ q respectively. Show that the sum of the squares of all the terms of the progression is n[p² + (n² -1)q²/3].

4) Show that 3[sin⁴(3π/2 - x) + sin⁴(3π+ x)] - 2[sin⁶(π/2+ x) + sin⁶(5π - x)] is independent of x.

.5) If 3ˣ - 3ˣ⁻² = 8, find the value of xˣ.

6) If cosα + cosβ = cos(3π/7) and sinα + sinβ= sin(3π/7) find cos²{(α -β)/2}.