OCR Maths FP1 Topic Questions from Papers Complex Numbers Answers PhysicsAndMathsTutor.com
. 1 (i) i Correct real and imaginary parts z* = i 1i Correct conjugate seen or implied Correct real and imaginary parts (iii) 17 + 1 17 i Attempt to use w* Obtain correct answer in any form 7 (Q, June 00). x y =1 and xy = 10 ±( i ) Attempt to equate real and imaginary parts of (x + iy) and 1 0i Obtain each result Eliminate to obtain a quadratic in x or y Solve to obtain x = (±) or y =(±) Obtain correct answers as complex numbers. Show correct process for subtracting fractions (Q, June 00). (i) Circle Centre (0, ) Radius Straight line Through origin with positive slope Sketch(s) showing correct features, each mark independent 0 or 0 +0i and + i ftf Obtain intersections as complex numbers t 7 8. (a) (i) + = = Values stated (Q, June 00) 1. (i) + 1i i 8i 10 +1i 1 (10 + 1i) or + i ft Attempt to multiply correctly Obtain correct answer Multiply numerator & denominator by conjugate Obtain denominator Their part (i) or 10 + 1i derived again / (Q1, Jan 00)
7. (a) (i) 1 1 Obtain correct answer, decimals OK - 0.9 (b) 1 i Using tan 1 b / a, or equivalent trig allow + or - Obtain 0.9 Obtain correct answer Express LHS in Cartesian form & equate real and imaginary parts Obtain x = 1 and y = - Correct answer written as a complex number (c) Sketch of vertical straight line Through ( - 0., 0) 8. (i) 10 (Q7, Jan 00). (i) + i 1 Conjugate seen (Q, June 00). 7 (i) -7i Real part correct Imaginary part correct + i - + 1i iz stated or implied or i = -1 seen Real part correct Imaginary part correct (iii) 1 ( 7i) or equivalent 8 Multiply by conjugate Real part correct Imaginary part correct N.B. Working must be shown.. (i) Circle, Centre O radius Sketch showing correct features (Q, June 00).. 8 (i) Circle, Centre O radius One straight line Through O with +ve slope In 1 st quadrant only 1+i 7 Sketch showing correct features Attempt to find intersections by trig, solving equations or from graph Correct answer stated as complex number (Q, June 00)
. 9 Attempt to equate real and imaginary parts of (x +iy) and 1 x y = 1 and xy = ± ( + i) D +8i Obtain each result Eliminate to obtain a quadratic in x or y Solve to obtain x = ( ± ), or y = ( ± ) 1 Obtain only correct two answers as complex numbers. Expand to obtain r r (Q, Jan 007). 10 (i) Circle Centre (1, -1) Passing through (0, 0) Sketch a concentric circle Inside (i) and touching axes Shade between the circles. (i) 1 Show given answer correctly (Q, Jan 007). 11 (i) 1 Show given answer correctly 1 ± i (iii) 7 Attempt to solve quadratic equation or substitute x + iy and equate real and imaginary parts Obtain answers as complex numbers Obtain correct answers, simplified Correct root on x axis, co-ords. shown Other roots in nd and rd quadrants Correct lengths and angles or coordinates or complex numbers shown. (i) Correct expression for u (Q, Jan 007)
1 EITHER a = b = OR, a = b = Use trig to find an expression for a (or b) Obtain correct answer Attempt to find other value Obtain correct answer a.e.f. (Allow. ) State equations for a and b Attempt to solve these equations Obtain correct answers a.e.f. SR ± scores only Show result true for n = 1 (Q1, June 007) 1 8 (i) Circle, centre (, 0), y-axis a tangent at origin Straight line, through (1, 0) with +ve slope In 1 st quadrant only Inside circle, below line, above x-axis Bft 8 8 Deduce no solutions Sketch showing correct features N.B. treat diagrams asa MR Sketch showing correct region SR: ft for any correct features (Q8, June 007) 10 1 (i) x y =1 and xy =1 9 Attempt to equate real and imaginary parts of (x + iy) and 1+0i Obtain each result Eliminate to obtain a quadratic in x or y ±( + i ) z = 1± 1 + 0i + i, - - i * *dep ft 11 Solve to obtain x =(±) or y =(±) Obtain correct answers as complex numbers Use quadratic formula or complete the square Simplify to this stage Use answers from (i) Obtain correct answers (Q10, June 007)
1 (i) z* = + i 1 +1i PhysicsAndMathsTutor.com Conjugate seen or implied Obtain correct answer i -1 0i ft ft Correct z i or expansion of (z I) seen Real part correct Imaginary part correct (iii) 9 + i 1 8 Multiply by conjugate Numerator correct Denominator correct (Q, Jan 008) 1 (i) + i 8 7 (i) Use det A = 0 Horizontal straight line in quadrants Through (0, ) Straight line Through O with positive slope In 1 st quadrant only State or obtain algebraically that y = Use suitable trigonometry Obtain correct answer a.e.f. decimals OK must be a complex number (Q, Jan 008) 17 (i) Correct modulus 0.97 or.1 o Correct argument, any equivalent form (a) Circle centre A (, ) Through O, allow if centre is (, ) (b) A(, ) Half line with +ve slope Starting at (, 0) Parallel to OA, (implied by correct arg shown) O r (Q, June 008) 18 9 (i) Attempt to equate real and imaginary parts of (x + iy) and + 1i x y = and xy = Obtain both results Eliminate to obtain a quadratic in x or y ±( + i) Solve a term quadratic & obtain x or y Obtain correct answers as complex nos. 1i Correct real and imaginary parts (iii) Attempt to solve a quadratic equation x = ± 1i Obtain correct answers x =± ( ± i) Each pair of correct answers a.e.f. (Q9, June 008)
191 7 + i. 17 Multiply by conjugate of denominator Obtain correct numerator Obtain correct denominator Both diagonals correct (Q1, Jan 009) 10 0 (i) x y =, xy = Attempt to equate real and imaginary parts Obtain both results a.e.f. x 8 x ( 10 + i = 0 x = ± 10, y = ± ± ) z = ± i 10 z ± ( ± i ) = ft Eliminate to obtain quadratic in x or y Solve to obtain x (or y) values Correct values for both x & y obtained a.e.f. Correct answers as complex numbers Solve quadratic in z Obtain correct answers Use results of (i) Obtain correct answers, ft must include root from conjugate (iii) ft 1 Sketch showing roots correctly (iv) ft ft 1 Sketch of straight line, Bisector toα (Q10, Jan 009). 1 (i) 11 9i Obtain correct answers. Correct real and imaginary parts 1 + 1i Correct real and imaginary parts. Either p q 1, pq 8 Both values stated or used (Q, June 009). (i), or o AEF State correct answers (a) (b) ft Circle, centre (, -), through O ft for (, ) only Straight line with +ve slope, through (, -) or their centre Half line only starting at centre (iii) ft ft ft Area above horizontal through a, below (b) Outside circle 11 7. (i) Show that terms cancel (Q, in pairs June 009)
x iy Conjugate known Equate real and imaginary parts x + y = 1 x + y = 9 Obtain both equations, OK with factor of i Solve pair of equations z = + i Obtain correct answer as a complex number S.C. Solving z + iz = 1 + 9i can get max /, not first (Q, Jan 010) 8 (i) Attempt to equate real and imaginary parts of (x + iy) & 1i x y = and xy = Obtain both results, a.e.f Obtain quadratic in x or y Solve to obtain x = ( ± ) or y = ( ± ) ± ( i) Obtain correct answers as complex nos -------------------------------------------------------------------------------------------------------------------------------------- ft Circle with centre at their square root Circle passing through origin ft nd circle centre correct relative to 1 st Circle passing through origin 9 (Q8, Jan 010) (i) + 1i 1 7. o or 1.18 ft ft Correct real and imaginary parts Correct modulus Correct argument 11 7 8 8 i 7 Multiply by conjugate Obtain correct numerator Obtain correct denominator (Q, June 010)
(i) (a) (b) Circle centre (, ), through origin Vertical line, clearly x = ft ft Inside their circle And to right of their line, if vertical (Q, June 010) 10 7 (i) Attempt to equate real and imaginary parts x y = xy = Obtain both results Eliminate to obtain quadratic in x or y Solve to obtain x or y value z = + i Obtain correct answer as a complex no. 1 Obtain given answer correctly (iii) w = ± 11i w = i 11 Attempt to solve quadratic equation Obtain correct answers Choose negative sign Relate required value to conjugate of (i) Obtain correct answer (Q10, June 010). 8(i) 1 +1i Real and imaginary parts correct ------------------------------------------------------------------------------------------------------------------------------------------------ z* seen Multiply by w* 7 1 i 7 7 Obtain correct real part or numerator Obtain correct imaginary part or denom. Sufficient working must be shown (Q, Jan 011)
9 (i) (a) * Vertical line dep Clearly through (, 0 ) (b) Sloping line with +ve slope Through ( 0, -) ft Half line starting on y-axis o shown convincingly ------------------------------------------------------------------------------------------------------------------------------------------------ ft Shaded to left of their (i) (a) ft Shaded below their (i) (b) must be +ve slope ft Shaded above horizontal through their (0, -) NB These marks are independent, but / only for fully correct answer. 8 (Q, Jan 011) 0 (i) a Correct modulus arg a 0,, 1.0 Correct argument Circle Centre ( 1, ) Through origin, centre ( 1, ) and another y intercept Vertical line * Through a or their centre, with +ve gradient D Correct half line 8 (Q, June 011) 1 9 (i) 1 + 0i 1 State correct value ------------------------------------------------------------------------------------------------------------------------------------------------ Use a = ( sum of roots ) a = Obtain correct answer Use b = product of roots b = 11 Obtain correct answer Substitute, expand and equate imag. parts Obtain a = - Equate real parts Obtain b = 11 ------------------------------------------------------------------------------------------------------------------------------------------------ (iii) Attempt to equate real and imaginary parts of (p+iq) & 1 0i or root from p q 1 and pq 1 Obtain both results cao Obtain quadratic in p or q Solve to ob tain p ( ) or q ( ) Obtain correct answers as complex nos Attempt at all roots ( ± i) 7 State other two roots as complex nos 1 (Q9, June 011)
1 a 1 Use formula for modulus a = 1 Obtain correct answer tan a Use formula for argument 0.9 or. or 0.1π FT Obtain correct answer allow 0.9 [] 1 (Q1, Jan 01) Attempt to equate real and imaginary parts x y and xy Obtain both results x x 18 0 or y y 18 0 Eliminate to obtain quadratic in x or y Solve to obtain x or y value x or y Both values correct ( i ) Correct answers as complex numbers [] (Q, Jan 01) Circle Centre (,1) Passing through O and crosses y-axis again Line, with correct slope shown line starting at O 1 Completely correct diagram for both loci Ignore shading [] (Q, Jan 01) MT 1 (i) 1 +11i Real part correct Imaginary part correct [] 1 Multiply by conjugate of denominator or find a pair of simultaneous equations 9i Obtain correct numerator or real part 9 Obtain correct denominator or imaginary part i 1 1 [] (Q1, June 01) 7 (i) Circle, centre (, ) ft Touching x-axis, ft for ( as centre ft Crossing y-axis twice Horizontal line, y intercept [] 7 1 +i 7 + i State correct answers [] 7 (iii) ft Inside circle or above line Completely correct diagram [] (Q7, June 01)
7 (i) z Allow. argz. or 0. Allow -7 o or -0.() [] z* = + i stated or used Obtain two equations from real and imaginary parts a b, b a 8 Obtain correct equations Attempt to solve linear equations a =, b = Obtain correct answers [] (Q, Jan 01) 8 7 (i) (a) Circle Centre O and radius [] 7 (i) (b) Horizontal line (, 1 ) on their line ½ line to left i.e. horizontal [] 7 Shade only inside their circle or above their horizontal line Completely correct diagram [] 8 (i) Obtain correct numerator from addition or partial fractions (Q7, Jan 01) 9 1 Use correct trig expression Obtain correct answer Correct expression for modulus FT Obtain correct answer aef i FT Correct conjugate seen or implied i FT Correct answer [] (i) (Q1, June 01) MT 0 x y 11 and xy Attempt to equate real and imaginary parts of (x + iy) and 11 + 1 Obtain both results cao * Obtain a quadratic in x or y ±( + i) D Solve a term quadratic to obtain a value for x or y Obtain 1 correct answer as complex number Obtain only the other correct answer [] Establish result true for n = 1 or n = (Q, June 01) 1 (i) Use arg (z a) = in equation for l condone missing brackets 1 arg( z i) Obtain correct answer Use z a k in equation for C, k must be real z i Obtain correct answer z i or e.g. x 1 1 [] ( y ) 9 Obtain correct inequality, or answer consistent with sensible (i) arg( z i) or y x, x 0 Each correct single inequality, or answer consistent with sensible (i) [] SC if < used consistently, but otherwise all correct, B (Q, June 01)