Hi guys,
My teacher gave our class a big booklet of mixed homework questions to help us prepare for our exams, I am finding some difficult questions for me:
1. Solve z^2+4z-1+12i.
2. The region R of the argand diagram consists of the set of al values of z such that both 0 less than Re(z^2) less than 3 and 0 less than Im(z^2) less than 4 hold true. Draw a sketch of the ragion R, showing coordinates of vertices.
3. binomial expansion of (z-1/z)^5 for which I got z^5+5z^3+10z+10/z+5/z^3+1/z^5 which Im fairly sure is right, then it says express sin^5theta in the form asin5theta+bsin3theta+csintheta, and bit after that says hence solve 16sin^5theta=sin5theta, which im not sure how to do.
I know u use demoivres thing but I just cant algebraically fet the asin5theta.... thing, I keep stuffing up.
4.prove for any polynomial P(x), if alpha is zero of multiplicity 2, then alpha is also a zero of p'(x)
Iirc then u go y=(x-alpha)^2Q(x)+R(x),
then p'(x)=2(x-alpha)Q(x)+R'(x) or something, but yea im sure thats wrong, I remember doing a question like this before but forgot what im missing.And finally it says "Hence facorise polynomial P(x)=x^6-7x^4+8x^2+16.
5. When monic cubic polynomial divided by x^2+x+1, remainder=2x+3, when polynomial divided by x(x+3), remainder 5(x+1), find the polynomial.
6. P and Q two points on ellipse x^2/9+y^2/4=1, parameters theta and theta+pi/2 respectively, find coordinates of point of intersection T of the tangents to the ellipse at points P and Q"
I got x coord. as being (6costh-6sinth)/2, is that right?
My teacher gave our class a big booklet of mixed homework questions to help us prepare for our exams, I am finding some difficult questions for me:
1. Solve z^2+4z-1+12i.
2. The region R of the argand diagram consists of the set of al values of z such that both 0 less than Re(z^2) less than 3 and 0 less than Im(z^2) less than 4 hold true. Draw a sketch of the ragion R, showing coordinates of vertices.
3. binomial expansion of (z-1/z)^5 for which I got z^5+5z^3+10z+10/z+5/z^3+1/z^5 which Im fairly sure is right, then it says express sin^5theta in the form asin5theta+bsin3theta+csintheta, and bit after that says hence solve 16sin^5theta=sin5theta, which im not sure how to do.
I know u use demoivres thing but I just cant algebraically fet the asin5theta.... thing, I keep stuffing up.
4.prove for any polynomial P(x), if alpha is zero of multiplicity 2, then alpha is also a zero of p'(x)
Iirc then u go y=(x-alpha)^2Q(x)+R(x),
then p'(x)=2(x-alpha)Q(x)+R'(x) or something, but yea im sure thats wrong, I remember doing a question like this before but forgot what im missing.And finally it says "Hence facorise polynomial P(x)=x^6-7x^4+8x^2+16.
5. When monic cubic polynomial divided by x^2+x+1, remainder=2x+3, when polynomial divided by x(x+3), remainder 5(x+1), find the polynomial.
6. P and Q two points on ellipse x^2/9+y^2/4=1, parameters theta and theta+pi/2 respectively, find coordinates of point of intersection T of the tangents to the ellipse at points P and Q"
I got x coord. as being (6costh-6sinth)/2, is that right?
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