The cubic equation x³ +bx² +cx+d = 0, where b, c and d are constants, has roots α, β and γ. It is given that α+β + γ = 2, α² + β² + γ² = 3, α⁴ + β⁴ + γ⁴ = 5.

By finding a cubic equation whose roots are α, β and γ, solve the set of simultaneous equations α + β + γ = −1, α² + …

It is given that the equation x³ – 21x² + kx – 216 = 0, where k is a constant, has real roots a, ar and ar⁻¹.

The equation x³ − x + 1 = 0 has roots α, β, γ.

The curves C₁ and C₂ have polar equations C₁: r = 1/√2, for 0 < θ < 2π, C₂: r = √(sin10), for 0 ≤ θ≤ π. Find the pola…

A curve has equation x² – 6xy + 25y² = 16. Show that dy/dx = 0 at the point (3, 1). By finding the value of d²y/dx² a…

Let In = ∫₀^π xⁿ sin x dx, where n is a non-negative integer. Show that In = n(π)ⁿ⁻¹ – n(n − 1)In-2, for n ≥ 2. Find …