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Let w, z be complex numbers. Show that |wz|=|w||z|, and using the fact that x=|x|e^{arg(x)i}, show further that arg(wz)=arg(w)+arg(z) where |.| is the absolute value and arg(.) is the angle (in polar coordinates). Hence, find all solutions to x^n=1 .

Put w=a+bi, z=c+di where a, b, c, d are real, and i 2 =-1. Then |wz| 2 =|(a+bi)(c+di)| 2 =|ac-bd+(ad+cb)i| 2 =(ac) 2 +(bd) 2 -2bcd+(ad) 2 +(cb) 2 +2abcd=(ac) 2 +(bd) 2 +(ad) 2 +(cb) 2 =(a 2 +b 2 )(c 2 +d 2 )...
JK
Answered by Jihoon K. • Maths tutor
8386 Views

Express 9^(3x+)1 in the form 3^y giving y in the form of ax+b where a and b are constants.

Fistly show that: 9 3 x +1 =3 y To find x we need to get the left hand side of the equation to be 3 m (3 x +1) where 3 m =9. I know that 3 2 =9 (but if you don't you can use m=log 3 (9)=2) This gets us: 3 2(...
JG
Answered by Judy G. • Maths tutor
6184 Views

What is the Product Rule?

The product rule is used when differentiating two functions that are multiplied by eachother. The formula for the product rule is: U (dv/dx) + V (du/dx) where 'dv/dx' is the differential of the function, V. ...
AW
Answered by Abbie W. • Maths tutor
4210 Views

A curve has parametric equations x=t(t-1), y=4t/(1-t). The point S on the curve has parameter t=-1. Show that the tangent to the curve at S has equation x+3y+4=0.

To anwser this question we need to find a linear equation of the form y=mx+c which we can rearrange to give the desired equation. Firstly, we must find the gradient at the point S given by dy/dx. Using the p...
MF
Answered by Marcus F. • Maths tutor
8445 Views

Find the value of the discriminant of x2 + 6x + 11

For a quadratic equation in the form ax^2+bx+c=0 the determinant is given by b^2-4ac and is used to help identify the types of roots of the equation. In this case, the determinant is (6)^2-4(1)(11) which is ...
EC
Answered by Emily C. • Maths tutor
9728 Views