OCR MEI FP1 2007 June — Question 2 3 marks

Exam BoardOCR MEI
ModuleFP1 (Further Pure Mathematics 1)
Year2007
SessionJune
Marks3
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Mark schemeDownload PDF ↗
TopicComplex Numbers Argand & Loci
TypeCircle equations in complex form
DifficultyEasy -1.2 This is a straightforward recall question requiring students to identify and write down the equation of a circle from an Argand diagram. It tests basic knowledge of complex number loci with minimal problem-solving required—students simply need to read the center and radius from the diagram and apply the standard form |z - a| = r.
Spec4.02k Argand diagrams: geometric interpretation
Write down the equation of the locus represented by the circle in the Argand diagram shown in Fig. 2. [3] \includegraphics{figure_2}
Question 2:
2
Section A (36 marks)
Ê2 -1ˆ
1 You are given the matrix M= .
Ë4 3¯
(i) Find the inverse of M. [2]
(ii) A triangle of area 2 square units undergoes the transformation represented by the matrix M.
Find the area of the image of the triangle following this transformation. [1]
2 Write down the equation of the locus represented by the circle in the Argand diagram shown in Fig. 2.
[3]
Im
O 3 Re
–2j
Fig. 2
3 Find the values of the constants A, B, C and D in the identity
x3 (cid:2) 4 (cid:1) (x (cid:2) 1)(Ax2(cid:1)Bx(cid:1)C)(cid:1)D. [5]
4 Two complex numbers, aand b, are given by a (cid:3) 1 (cid:2) 2j and b (cid:3)(cid:2)2 (cid:2) j.
(i) Represent b and its complex conjugate b* on an Argand diagram. [2]
(ii) Express abin the form a(cid:1)bj. [2]
(cid:1)
a b
(iii) Express in the form a(cid:1)bj. [3]
b
5 The roots of the cubic equation x3(cid:1)3x2 (cid:2) 7x(cid:1)1 (cid:3) 0 are a, band g. Find the cubic equation
whose roots are 3a, 3band 3g, expressing your answer in a form with integer coefficients. [6]
3
Question 2:
2
Section A (36 marks)
Ê2 -1ˆ
1 You are given the matrix M= .
Ë4 3¯
(i) Find the inverse of M. [2]
(ii) A triangle of area 2 square units undergoes the transformation represented by the matrix M.
Find the area of the image of the triangle following this transformation. [1]
2 Write down the equation of the locus represented by the circle in the Argand diagram shown in Fig. 2.
[3]
Im
O 3 Re
–2j
Fig. 2
3 Find the values of the constants A, B, C and D in the identity
x3 (cid:2) 4 (cid:1) (x (cid:2) 1)(Ax2(cid:1)Bx(cid:1)C)(cid:1)D. [5]
4 Two complex numbers, aand b, are given by a (cid:3) 1 (cid:2) 2j and b (cid:3)(cid:2)2 (cid:2) j.
(i) Represent b and its complex conjugate b* on an Argand diagram. [2]
(ii) Express abin the form a(cid:1)bj. [2]
(cid:1)
a b
(iii) Express in the form a(cid:1)bj. [3]
b
5 The roots of the cubic equation x3(cid:1)3x2 (cid:2) 7x(cid:1)1 (cid:3) 0 are a, band g. Find the cubic equation
whose roots are 3a, 3band 3g, expressing your answer in a form with integer coefficients. [6]
3
Write down the equation of the locus represented by the circle in the Argand diagram shown in Fig. 2. [3]

\includegraphics{figure_2}

\hfill \mbox{\textit{OCR MEI FP1 2007 Q2 [3]}}
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Q1 3 Q2 3 Q3 5 Q4 7 Q5 6 Q6 6 Q7 6 Q8 14 Q9 11 Q10 11