HSC Commerce (English Medium) 11th Standard - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

If ω is a complex cube root of unity, show that `(("a" + "b"omega + "c"omega^2))/("c" + "a"omega + "b"omega^2) = omega^2`.

If ω is a complex cube root of unity, find the value of `omega + 1/omega`

If ω is a complex cube root of unity, find the value of ω² + ω³ + ω⁴.

If ω is a complex cube root of unity, find the value of (1 + ω²)³

If ω is a complex cube root of unity, find the value of (1 - ω - ω²)³ + (1 - ω + ω²)³

If `omega` is a complex cube root of unity, find the value of `(1 + omega)(1 + omega^2)(1 + omega^4)(1 + omega^8)`

If α and β are the complex cube roots of unity, show that α² + β² + αβ = 0.

If x = a + b, y = αa + βb and z = aβ + bα, where α and β are the complex cube roots of unity, show that xyz = a³ + b³.

If ω is a complex cube root of unity, then prove the following: (ω² + ω - 1)³ = – 8

If ω is a complex cube root of unity, then prove the following:  (a + b) + (aω + bω²) + (aω² + bω) = 0.

Find A.M. of two positive numbers whose G.M. and H.M. are 4 and `16/5`. 

Find H.M. of two positive numbers whose A.M. and G.M. are `15/2` and 6.

Find G.M. of two positive numbers whose A.M. and H.M. are 75 and 48.

Find two numbers whose A.M. exceeds their G.M. by `1/2` and their H.M. by `25/26`.

Find two numbers whose A.M. exceeds G.M. by 7 and their H.M. by `63/5`.

Simplify the following and express in the form a + ib.

`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`

If w is a complex cube root of unity, show that `((a+bw+cw^2))/(c+aw+bw^2) = w^2`

If w is a complex cube root of unity, show that

`((a + bw + cw^2)) /( c + aw + bw^2 )= w^2`

If w is a complex cube root of unity, show that, `((a + bw + cw^2))/(c + aw + bw^2) = w^2`

If w is a complex cube root of unity, show that `((a + bw +cw^2))/(c +aw + bw^2) = w^2`