OCR MEI FP1 2007 June — Question 3 5 marks

Exam BoardOCR MEI
ModuleFP1 (Further Pure Mathematics 1)
Year2007
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicFactor & Remainder Theorem
TypePolynomial identity or expansion
DifficultyEasy -1.2 This is a straightforward algebraic identity question requiring expansion and coefficient comparison. While it's from FP1, the technique is mechanical: expand the right side, equate coefficients of powers of x, and solve the resulting simple equations. No problem-solving insight needed, just careful algebraic manipulation.
Spec1.02j Manipulate polynomials: expanding, factorising, division, factor theorem
Find the values of the constants \(A\), \(B\), \(C\) and \(D\) in the identity $$x^3 - 4 \equiv (x - 1)(Ax^2 + Bx + C) + D.$$ [5]
Question 3:
3
1 1 1
6 (i) Show that (cid:2) (cid:3) . [2]
r(cid:1)2 r(cid:1)3 (r(cid:1)2)(r(cid:1)3)
1 1 1 1
(ii) Hence use the method of differences to find (cid:1) (cid:1) (cid:1)…(cid:1) . [4]
3 (cid:4) 4 4 (cid:4) 5 5 (cid:4) 6 52 (cid:4) 53
(cid:2)n 3n(cid:2)1
7 Prove by induction that 3r(cid:2)1 (cid:3) . [6]
r(cid:3)1 2
Section B (36 marks)
x2 (cid:2) 4
8 Acurve has equation y (cid:3) .
(x (cid:2) 3)(x(cid:1)1)(x (cid:2) 1)
(i) Write down the coordinates of the points where the curve crosses the axes. [3]
(ii) Write down the equations of the three vertical asymptotes and the one horizontal asymptote.
[4]
(iii) Determine whether the curve approaches the horizontal asymptote from above or below for
(A) large positive values of x,
(B) large negative values of x. [3]
(iv) Sketch the curve. [4]
9 The cubic equation x3(cid:1)Ax2(cid:1)Bx(cid:1)15 (cid:3) 0, where A and B are real numbers, has a root
x (cid:3) 1(cid:1)2j.
(i) Write down the other complex root. [1]
(ii) Explain why the equation must have a real root. [1]
(iii) Find the value of the real root and the values of A and B. [9]
[Question 10 is printed overleaf.]
[Turn over
© OCR 2007 4755/01 June 07
Question 3:
3
1 1 1
6 (i) Show that (cid:2) (cid:3) . [2]
r(cid:1)2 r(cid:1)3 (r(cid:1)2)(r(cid:1)3)
1 1 1 1
(ii) Hence use the method of differences to find (cid:1) (cid:1) (cid:1)…(cid:1) . [4]
3 (cid:4) 4 4 (cid:4) 5 5 (cid:4) 6 52 (cid:4) 53
(cid:2)n 3n(cid:2)1
7 Prove by induction that 3r(cid:2)1 (cid:3) . [6]
r(cid:3)1 2
Section B (36 marks)
x2 (cid:2) 4
8 Acurve has equation y (cid:3) .
(x (cid:2) 3)(x(cid:1)1)(x (cid:2) 1)
(i) Write down the coordinates of the points where the curve crosses the axes. [3]
(ii) Write down the equations of the three vertical asymptotes and the one horizontal asymptote.
[4]
(iii) Determine whether the curve approaches the horizontal asymptote from above or below for
(A) large positive values of x,
(B) large negative values of x. [3]
(iv) Sketch the curve. [4]
9 The cubic equation x3(cid:1)Ax2(cid:1)Bx(cid:1)15 (cid:3) 0, where A and B are real numbers, has a root
x (cid:3) 1(cid:1)2j.
(i) Write down the other complex root. [1]
(ii) Explain why the equation must have a real root. [1]
(iii) Find the value of the real root and the values of A and B. [9]
[Question 10 is printed overleaf.]
[Turn over
© OCR 2007 4755/01 June 07
Find the values of the constants $A$, $B$, $C$ and $D$ in the identity
$$x^3 - 4 \equiv (x - 1)(Ax^2 + Bx + C) + D.$$ [5]

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