CAIE P3 2017 June — Question 6 8 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2017
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex Numbers Argand & Loci
TypeRegion shading with multiple inequalities
DifficultyStandard +0.3 Part (i) is a standard application of complex conjugate roots and polynomial coefficients - routine for P3 students. Part (ii) requires identifying a circle and perpendicular bisector locus, then shading the intersection - straightforward geometric interpretation with no novel insight needed. Both parts are textbook-style exercises with clear methods.
Spec4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines
Throughout this question the use of a calculator is not permitted. The complex number \(2 - \mathrm{i}\) is denoted by \(u\).
  1. It is given that \(u\) is a root of the equation \(x^3 + ax^2 - 3x + b = 0\), where the constants \(a\) and \(b\) are real. Find the values of \(a\) and \(b\). [4]
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(|z - u| < 1\) and \(|z| < |z + \mathrm{i}|\). [4]
Question 6:
AnswerMarks
6(i)EITHER:
x=2+i) x2 x3
AnswerMarks
Substitute x = 2 – i (or in the equation and attempt expansions of and(M1
Equate real and/or imaginary parts to zeroM1
Obtain a = – 2A1
Obtain b = 10A1)
OR1:
x2 x3
AnswerMarks
Substitute x = 2 – i in the equation and attempt expansions of and(M1
x=2+i
AnswerMarks
Substitute in the equation and add/subtract the two equationsM1
Obtain a = – 2A1
Obtain b = 10A1)
OR2:
( )
Factorise to obtain ( x−2+i )( x−2−i )( x− p )  = x 2 −4x+5 ( x− p )  
AnswerMarks
 (M1
Compare coefficientsM1
Obtain a = – 2A1
Obtain b = 10A1)
OR3:
AnswerMarks
Obtain the quadratic factor (x 2 −4x+5)(M1
Use algebraic division to obtain a real linear factor of the form x− p and set the remainder
AnswerMarks
equal to zeroM1
Obtain a = – 2A1
Obtain b = 10A1)
OR4:
AnswerMarks Guidance
Use αβ=5 and α+β=4 in αβ+βγ+γα=−3(M1
Solve for γ and use in αβγ=−b and/or α+β+γ=−aM1
Obtain a = – 2A1
Obtain b = 10A1)
QuestionAnswer Marks
OR5:
AnswerMarks
Factorise as (x–- (2-i))(x2 + ex + g) and compare coefficients to form an equation in a and b(M1
Equate real and/or imaginary parts to zeroM1
Obtain a = – 2A1
Obtain b = 10A1)
Total:4
AnswerMarks Guidance
6(ii)Show a circle with centre 2− iin a relatively correct position B1
Show a circle with radius 1 and centre not at the originB1
Show the perpendicular bisector of the line segment joining 0 to – iB1
Shade the correct regionB1
Total:4 
Question 6:
--- 6(i) ---
6(i) | EITHER:
x=2+i) x2 x3
Substitute x = 2 – i (or in the equation and attempt expansions of and | (M1
Equate real and/or imaginary parts to zero | M1
Obtain a = – 2 | A1
Obtain b = 10 | A1)
OR1:
x2 x3
Substitute x = 2 – i in the equation and attempt expansions of and | (M1
x=2+i
Substitute in the equation and add/subtract the two equations | M1
Obtain a = – 2 | A1
Obtain b = 10 | A1)
OR2:
( )
Factorise to obtain ( x−2+i )( x−2−i )( x− p )  = x 2 −4x+5 ( x− p )  
  | (M1
Compare coefficients | M1
Obtain a = – 2 | A1
Obtain b = 10 | A1)
OR3:
Obtain the quadratic factor (x 2 −4x+5) | (M1
Use algebraic division to obtain a real linear factor of the form x− p and set the remainder
equal to zero | M1
Obtain a = – 2 | A1
Obtain b = 10 | A1)
OR4:
Use αβ=5 and α+β=4 in αβ+βγ+γα=−3 | (M1
Solve for γ and use in αβγ=−b and/or α+β+γ=−a | M1
Obtain a = – 2 | A1
Obtain b = 10 | A1)
Question | Answer | Marks
OR5:
Factorise as (x–- (2-i))(x2 + ex + g) and compare coefficients to form an equation in a and b | (M1
Equate real and/or imaginary parts to zero | M1
Obtain a = – 2 | A1
Obtain b = 10 | A1)
Total: | 4
--- 6(ii) ---
6(ii) | Show a circle with centre 2− iin a relatively correct position | B1
Show a circle with radius 1 and centre not at the origin | B1
Show the perpendicular bisector of the line segment joining 0 to – i | B1
Shade the correct region | B1
Total: | 4
Throughout this question the use of a calculator is not permitted.

The complex number $2 - \mathrm{i}$ is denoted by $u$.

\begin{enumerate}[label=(\roman*)]
\item It is given that $u$ is a root of the equation $x^3 + ax^2 - 3x + b = 0$, where the constants $a$ and $b$ are real. Find the values of $a$ and $b$. [4]

\item On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying both the inequalities $|z - u| < 1$ and $|z| < |z + \mathrm{i}|$. [4]
\end{enumerate}

\hfill \mbox{\textit{CAIE P3 2017 Q6 [8]}}
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