EdExcel Further Pure 2

Transcription

1 EdExcel Further Pure 2 Complex Numbers Section : Loci in the Argand diagram Multiple Choice Test Questions 1 are about the following loci: P: z i = 2 Q: z i = z R: arg( z i) = S: z i = 2 z 1) Which of the loci above represent a circle? (a) P and Q (c) P only (b) Q (d) P and S 2) Which of the loci above represent a straight line (or part of a straight line)? (a) Q, R and S (c) R only (b) Q and R (d) Q only ) Which of the loci above are satisfied by the point + i? (a) none of them (c) R (b) all of them (d) P, Q and S ) The set of points for which z 2 + i = is (a) a circle, centre -2 + i, radius (b) a circle, centre 2 i, radius 2 (c) a circle, centre 2 i, radius (d) a circle, centre -2 + i, radius 2 MEI, 22/07/08 1/7

2 EdExcel FP2 Complex nos Section MC test solutions 5) Im Re The shaded area in the Argand diagram represents the points z for which (a) z 2 + i 2 (b) z 2 + i < 2 (c) z + 2 i < 2 (d) z + 2 i 2 6) Im 1+2i Re The bold half-line in the diagram above shows the set of points z for which (a) arg( z 1 2i) = (b) arg( z 1 2i) = (c) arg( z i) = (d) arg( z i) = MEI, 22/07/08 2/7

3 EdExcel FP2 Complex nos Section MC test solutions 7) Im -1+i Re The shaded area in the diagram above shows the set of points z for which 2 (a) arg( z + 1 i) (b) 2 (c) 0 arg( z 1+ i) (d) 2 arg( z 1+ i) 2 0 arg( z + 1 i) 8) The sketch, on an Argand diagram, of the locus of points satisfying z = z+ 1 i is (a) the line joining the points (, 0) and (-1, ) (b) the line joining the points (, 0) and (1, -) (c) the line joining the points (-1, 0) and (1, 2) (d) the line joining the points (-, 0) and (1, -) 9) A transformation is given by w= z 2+ i. Under this transformation, the circle z 1 = 2 is transformed into the circle with (a) centre 1 + i, radius 6 (b) centre 1 + i, radius 1.5 (c) centre -1 + i, radius 6 (d) centre 1 + i, radius 1.5 MEI, 22/07/08 /7

4 EdExcel FP2 Complex nos Section MC test solutions 2+ iz 10) A transformation is given by w =. 1 z Under this transformation, the circle z = 1 is transformed into (a) the perpendicular bisector of the points (-2, 0) and (0, 1) (b) the perpendicular bisector of the points (2, 0) and (0, -1) (c) a circle, centre (2, -1), radius 1 (d) a circle, centre (-2, 1), radius 1 MEI, 22/07/08 /7

5 EdExcel FP2 Complex nos Section MC test solutions Solutions to Multiple Choice Test 1) The correct answer is (d) For P, z represents all points whose distance from the point + i is 2. This means that P is a circle. For Q, z represents all points whose distance from the point + i is equal to its distance from the origin. This means that Q is the perpendicular bisector of a line joining the origin and the point + i. For R, z represents all points for which a line from the point to + i makes an angle of with the real axis. This is the half-line from the point + i in the direction making an angle of with the real axis. For S, writing z = x + iy and squaring gives ( x ) + ( y ) = ( x + y ) x 6x + 9+ y 8y + 16 = x + y 2 2 x + 6x + y + 8y = 25 This is the equation of a circle. Therefore P and S both represent circles. 2) The correct answer is (b) From the solution to question 1, Q is a straight line since it is a perpendicular bisector, and R is a half- line so is part of a straight line. ) The correct answer is (a) The point + i is the centre of the circle represented by P, so it does not lie on the circle itself. For both Q and S, for the point + i the value of z i is zero, but the value of z is 5, so + i cannot lie on either of these loci. R represents a line starting from + i, but the point + i is not part of the locus since the argument of zero is not defined. Therefore the point does not lie on any of the loci. MEI, 22/07/08 5/7

6 EdExcel FP2 Complex nos Section MC test solutions ) The correct answer is (c) z 2+ i = z (2 i) = This means that the distance of the point z from the point 2 i is always. Therefore the set of points is a circle, centre 2 i, radius. 5) The correct answer is (b) The shaded area is the area inside the circle with centre (2, -1) and radius 2. So this is the locus of a point z whose distance from the point 2 i is always less than 2. This is the set of points given by z (2 i) < 2 z 2+ i < 2 6) The correct answer is (a) The line starts from the point 1 + 2i and goes in the direction The set of points is therefore arg( z (1 + 2i)) = arg( z 1 2i) =. 7) The correct answer is (d) The lines bounding the region both start from the point -1 + i and have 2 directions 0 and respectively. These lines are given by arg( z + 1 i) = 0 and So the shaded region is given by 0 arg( z + 1 i) 2 arg( z + 1 i) = 2. MEI, 22/07/08 6/7

7 EdExcel FP2 Complex nos Section MC test solutions 8) The correct answer is (c) z = z + 1 i z = z ( 1+ i) This means that the distance of the point representing z from the point representing (the point (, 0)) is equal to the distance of the point representing z from the point representing -1 + i (the point (-1, )). The locus of z is therefore the perpendicular bisector of the points (, 0) and (-1, ). The midpoint of (, 0) and (-1, ) is (1, 2), so the locus passes through (1, 0). The gradient of the line joining (, 0) and (-1, ) is 1 =, so the gradient of the perpendicular bisector is 1. The equation of the perpendicular bisector is therefore y 2 = x 1 y = x + 1 The locus is therefore the line joining any two points satisfying y = x + 1, such as (-1, 0) and (1, 2). 9) The correct answer is (a) w = z 2+ i w + 2 i = z 1 ( w + 2 i) = z The circle z 1 = 2 is transformed into 1 ( w + 2 i) 1 = 2 ( w + 2 i) = 6 w 1 i = 6 This is a circle, centre 1 + i, radius 6. 10) The correct answer is (b) 2+ iz w = 1 z w wz = 2+ iz w 2 = z( w + i) w 2 z = w + i The circle z = 1 is transformed into w 2 = 1 w + i w 2 = w + i This is the perpendicular bisector of the points (2, 0) and (0, -1). MEI, 22/07/08 7/7