OCR MEI FP1 2006 June — Question 2 5 marks

Exam BoardOCR MEI
ModuleFP1 (Further Pure Mathematics 1)
Year2006
SessionJune
Marks5
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Mark schemeDownload PDF ↗
TopicPolynomial Division & Manipulation
TypePolynomial Identity Matching
DifficultyEasy -1.2 This is a straightforward algebraic manipulation requiring expansion and coefficient comparison. While it's a Further Maths question, it involves only routine polynomial algebra with no problem-solving insight needed—students simply expand the right side and match coefficients systematically. The 5 marks reflect the four unknowns plus working, but the task is purely mechanical.
Spec1.02j Manipulate polynomials: expanding, factorising, division, factor theorem
Find the values of \(A\), \(B\), \(C\) and \(D\) in the identity $$2x^3 - 3x^2 + x - 2 \equiv (x + 2)(Ax^2 + Bx + C) + D.$$ [5]
Question 2:
2
Section A(36 marks)
Ê1 0ˆ
1 (i) State the transformation represented by the matrix . [1]
Ë0 -1¯
(ii) Write down the 2 ¥ 2 matrix for rotation through 90° anticlockwise about the origin. [1]
(iii) Find the 2 ¥ 2 matrix for rotation through 90° anticlockwise about the origin, followed by
reflection in the x-axis. [2]
2 Find the values of A, B, C and D in the identity
2x3(cid:3)3x2(cid:1)x(cid:3)2 (cid:2) (x(cid:1)2)(Ax2(cid:1)Bx(cid:1)C)(cid:1)D. [5]
3 The cubic equation z3(cid:1)4z2(cid:3)3z(cid:1)1 (cid:4) 0 has roots a, b and g .
(i) Write down the values ofa (cid:1) b (cid:1) g,ab (cid:1) bg (cid:1) ga and abg . [3]
(ii) Show that a2(cid:1) b2(cid:1) g2 (cid:4) 22. [3]
4 Indicate, on separate Argand diagrams,
(i) the set of points z for which(cid:1) z(cid:3)(3(cid:3)j) (cid:1) (cid:6) 3, [3]
(ii) the set of points z for which 1 (cid:5) (cid:1) z(cid:3)(3(cid:3)j) (cid:1) (cid:6) 3, [2]
(iii) the set of points z for whicharg (z(cid:3)(3(cid:3)j)) (cid:4) 1 p . [3]
4
Ê-1 2ˆ
5 (i) The matrix S= represents a transformation.
Ë-3 4¯
(A) Show that the point (1, 1) is invariant under this transformation. [1]
(B) Calculate S–1. [2]
(C) Verify that (1, 1) is also invariant under the transformation represented by S–1. [1]
(ii) Part (i) may be generalised as follows.
If (x, y) is an invariant point under a transformation represented by the non-singular
matrix T, it is also invariant under the transformation represented by T–1.
Êxˆ Êxˆ
Starting with T = , or otherwise, prove this result. [2]
Ëy¯ Ëy¯
6 Prove by induction that 3(cid:1)6(cid:1)12(cid:1)…(cid:1)3(cid:2)2n(cid:3)1 (cid:4) 3(2n(cid:3)1) for all positive integers n. [7]
Question 2:
2
Section A(36 marks)
Ê1 0ˆ
1 (i) State the transformation represented by the matrix . [1]
Ë0 -1¯
(ii) Write down the 2 ¥ 2 matrix for rotation through 90° anticlockwise about the origin. [1]
(iii) Find the 2 ¥ 2 matrix for rotation through 90° anticlockwise about the origin, followed by
reflection in the x-axis. [2]
2 Find the values of A, B, C and D in the identity
2x3(cid:3)3x2(cid:1)x(cid:3)2 (cid:2) (x(cid:1)2)(Ax2(cid:1)Bx(cid:1)C)(cid:1)D. [5]
3 The cubic equation z3(cid:1)4z2(cid:3)3z(cid:1)1 (cid:4) 0 has roots a, b and g .
(i) Write down the values ofa (cid:1) b (cid:1) g,ab (cid:1) bg (cid:1) ga and abg . [3]
(ii) Show that a2(cid:1) b2(cid:1) g2 (cid:4) 22. [3]
4 Indicate, on separate Argand diagrams,
(i) the set of points z for which(cid:1) z(cid:3)(3(cid:3)j) (cid:1) (cid:6) 3, [3]
(ii) the set of points z for which 1 (cid:5) (cid:1) z(cid:3)(3(cid:3)j) (cid:1) (cid:6) 3, [2]
(iii) the set of points z for whicharg (z(cid:3)(3(cid:3)j)) (cid:4) 1 p . [3]
4
Ê-1 2ˆ
5 (i) The matrix S= represents a transformation.
Ë-3 4¯
(A) Show that the point (1, 1) is invariant under this transformation. [1]
(B) Calculate S–1. [2]
(C) Verify that (1, 1) is also invariant under the transformation represented by S–1. [1]
(ii) Part (i) may be generalised as follows.
If (x, y) is an invariant point under a transformation represented by the non-singular
matrix T, it is also invariant under the transformation represented by T–1.
Êxˆ Êxˆ
Starting with T = , or otherwise, prove this result. [2]
Ëy¯ Ëy¯
6 Prove by induction that 3(cid:1)6(cid:1)12(cid:1)…(cid:1)3(cid:2)2n(cid:3)1 (cid:4) 3(2n(cid:3)1) for all positive integers n. [7]
Find the values of $A$, $B$, $C$ and $D$ in the identity
$$2x^3 - 3x^2 + x - 2 \equiv (x + 2)(Ax^2 + Bx + C) + D.$$ [5]

\hfill \mbox{\textit{OCR MEI FP1 2006 Q2 [5]}}
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