Question

Please do 2,4, and 10

d s and y when n the Argand dilagram, the origin is the numbers representing the other vertices Spress 2 + /S and 1 -12 in potar torm and a 60 tex of the triangle is the point 3 -NS. Find the comps centre ot an epuailateral trang 2 Express the result in th e form a jo and in espoment form. nd the fnith roots of -3+3 in polar form and in esponential torm Express S +/12 in polar form and i (S 712), giving the results in the form a jh and in the form re ence evaat thepinpal vlue o ine the fourth roots of-16, giving the results in the form fifth roots of -1, giving the results in polar form. Express the 7 Find the roo principal root in the form re ermine the roots of the equation x+64-O in the form a io, where a and b are real. 2 mine the three cube roots of 2, giving the result in modulus, ent form. Express the principal root in the form a + jb that the equation 1 has one real root and two other roots which are not real, and that, if one of the non-real roots is denoted by , the other is then u2. Mark on the Argand diagram the points which represent three roots and show that they are the vertices of an equilateral triangle. 10 o Show 1 Determine the fifth roots of (2 -j5), giving the results in modulus argument form. Express the principal root in the form a + jb and in the form re 12 Solve the equation 2 + 2(1 +)z+2-0, giving each result in the form 13 Express el /2 in the form a + jb. TA Obtain the expansion of sin 78 in powers of sine 15 Express sinx as a series of terms which are cosines of angles that are a +jb, with a and b correct to 2 places of decimals multiples of x. If Z = χ +/y, wherex and y are real, show that the locus|2+2-2 is a 16 circle and determine its centre and radius. lfz#x+ly, show that the locus arg-r-b and radius. is a circle. Find its Centre 17

Answer 1

(not scale, approximate diagram) 3 + i squareroot 3 is polar form (r e^- theta) = squareroot (3)^2 + (squareroot 3)^2 e i theta (where Tan theta = squareroot 3/3 theta = 30 degree) = squareroot 12 e^i 30 degree = 2 squareroot 3 e^- (70 degree) To get other 2 points we need to shift the current position to + 120 degree and -120 degree respectively to get (x_1, y_1) alpha (x_2, y_2) x_1 + i y_1 = 2 squareroot 3 e^i (30 degree + 120 degree) = 2 squareroot 3 e^i (150 degree) = 2 squareroot 3 (cos 150 degree + i sin 150 degree) = 2 squareroot 3 (-squareroot 3/2 + i/2) x_1 + j y_1 = -3 + i squareroot 3 \r\n Therefore x_2 + iy_2 = 2 squareroot 3 e^i (30 degree - 120 degree) = 2 squareroot 3 e^i (-90 degree)) = 2 squareroot 3 (0 - i) x_2 + i y_2