Show that
 

where a, b, c are in arithmetic progression.

Prove that
 

where p is a scalar.

Prove that
 

Prove that
 

Prove that
 

Prove that

Prove that
 

Prove that  when x, y, z are different and

Prove that
 

Prove that
 

Prove that
 

Prove that
  

Prove that

Prove that
 

Show that

is divisible by λ³ and find the other factors.

Prove that

Solve the equation
 

Find the value of the expression

1.(2−ω)(2−+ 2.(3−ω)(3−+ … (n−1).(n−ω)(n−

where ω is an imaginary cube root of unity.

a) n(n−1)(+3n+4)

b) n(n+1)(+3n+4)

c) n(n−1)(+n+1)

d) n(n+1)(+n+1)

Find the coefficient of x¹⁶ in the expansion of (x² – 2x)¹⁰.

a) 2980

b) 3160

c) 3360

d) 3580

Find the coefficient of x^r in the expansion of
 

a)  

b)

c)

d)

Unless  is a positive integer there will be no term containing x^r (common to all options)

Find the coefficient of x¹⁸ in (ax⁴ – bx)⁹

a) 84a³b³

b) 84a⁶b³

c) 84a³b⁶

d) 84a⁶b⁶

Find the middle term of  

a) 216

b) 252

c) 278

d) 312

Find the two middle terms of
 

a)

b)

c)

d)