A Level Further Maths Core Pure

A Level Further Maths Core Pure Form of complex numbers ️a + bi Real and imaginary parts of z = a + bi ️Re(z) = a, Im(z) = b When two complex numbers are equal, what is true ️Their real parts are equal and their imaginary parts are equal i² ️-1 i³ ️-i i⁴ ️1 Determinant for two real roots (distinct roots) ️b² - 4ac 0 Determinant for one real root (repeated roots) ️b² - 4ac = 0 Determinant for no real roots (two distinct complex roots) ️b² - 4ac 0 Complex conjugate for z = a + bi ️z* = a - bi Product zz* is always ️Real Expansion of (z - α)(z - β) = 0 ️z² - (α + β)z + αβ = 0 Possible Roots in a Cubic Equation ️Three real roots or one real root and a complex conjugate pair of roots Possible Roots in a Quartic Equation ️Four real roots or two real roots and a complex conjugate pair of roots or two sets of conjugate pairs x-axis on an argand diagram ️Real axis y-axis on an argand diagram ️Imaginary axis Modulus of a complex number |z| ️Distance from the origin to that number (on an Argand diagram) How do you calculate the modulus of a complex number |z|? ️√(x² + y²) Argument of a complex number (arg z) ️Angle between the positive real axis and the line joining that number to the origin (on an Argand diagram) Modulus-argument Form ️z = r(cosθ + i sinθ) | z₁ x z₂ | = ️| z₁ | | z₂ | arg(z₁ x z₂) = ️arg z₁ + arg z₂ |(z₁/z₂)| = ️(| z₁ |)/(| z₂ |) arg(z₁/z₂) = ️arg z₁ - arg z₂ Centre of Circle |z - (a + ib)| = r ️a, b Radius of Circle |z - (a + ib)| = r ️r Cartesian Form ️Equation given in terms of coordinates relative to the x, y and z axes What is the number above the sigma when summing ️The number to end summing at What is the number below the sigma when summing ️The number to start summing fro