Question 3:
3
6- 3
PMT
4
11 (i) Write x2(cid:1)7x(cid:2)6 in the form (x(cid:1)a)2(cid:2)b. [3]
(ii) State the coordinates of the minimum point on the graph of y (cid:3) x2(cid:1)7x(cid:2)6. [2]
(iii) Find the coordinates of the points where the graph of y (cid:3) x2(cid:1)7x(cid:2)6 crosses the axes and
sketch the graph. [5]
(iv) Show that the graphs of y (cid:3) x2(cid:1)7x(cid:2)6 and y (cid:3) x2(cid:1)3x(cid:2)4 intersect only once. Find the
x-coordinate of the point of intersection. [3]
12 (i) Sketch the graph of y (cid:3) x(x(cid:1)3)2. [3]
(ii) Show that the equationx(x(cid:1)3)2 (cid:3) 2 can be expressed as x3(cid:1)6x2(cid:2)9x(cid:1)2 (cid:3) 0. [2]
(iii) Show that x (cid:3) 2 is one root of this equation and find the other two roots, expressing your
answers in surd form.
Show the location of these roots on your sketch graph in part (i). [8]
4751 January 2006
OXFORD CAMBRIDGE AND RSA EXAMINATIONS
Advanced Subsidiary General Certificate of Education
Advanced General Certificate of Education
4751
MEI STRUCTURED MATHEMATICS
Introduction to Advanced Mathematics (C1)
Tuesday 6 JUNE 2006 Afternoon 1hour30minutes
Additional materials:
8 page answer booklet
Graph paper
MEIExamination Formulae and Tables (MF2)
TIME 1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
• Write your name, centre number and candidate number in the spaces provided on the answer
booklet.
• Answer all the questions.
• There is an insertfor use in Question 13.
• You are not permitted to use a calculator in this paper.
• Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.
• You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
• The total number of marks for this paper is 72.
WARNING
You are not allowed to use
a calculator in this paper
This question paper consists of 4 printed pages and an insert.
HN/4
©OCR 2006 [H/102/2647] Registered Charity 106696 [Turn over
Section A(36 marks)
1 The volume of a cone is given by the formula V (cid:4) 1 pr2h. Make r the subject of this formula.
3
[3]
2 One root of the equation x3(cid:2)ax2(cid:2)7 (cid:4) 0is x (cid:4)(cid:1)2. Find the value of a. [2]
3 Aline has equation 3x(cid:2)2y (cid:4) 6.Find the equation of the line parallel to this which passes through
the point (2, 10). [3]
4 In each of the following cases choose one of the statements
P fi Q P ¤ Q P ‹ Q
to describe the complete relationship between Pand Q.
(i) P: x2(cid:2)x(cid:1)2 (cid:4) 0
Q: x (cid:4) 1 [1]
(ii) P: y3 (cid:5) 1
Q: y (cid:5) 1 [1]
5 Find the coordinates of the point of intersection of the lines y (cid:4) 3x(cid:2)1and x(cid:2)3y (cid:4) 6. [3]
6 Solve the inequality x2(cid:2)2x (cid:3) 3. [4]
7 (i) Simplify 6 2¥5 3- 24. [2]
(ii) Express (2-3 5)2 in the form a+b 5, where a and b are integers. [3]
8 Calculate 6C .
3
Find the coefficient of x3 in the expansion of (1(cid:1)2x)6. [4]
9 Simplify the following.
1
162
(i) [2]
3
814
12(a3b2c)4
(ii) [3]
4a2c6
4751 June 2006
1
162
3
814
PMT
3
10 Find the coordinates of the points of intersection of the circle x2(cid:2)y2 (cid:4) 25 and the line y (cid:4) 3x.
Give your answers in surd form. [5]
Section B (36 marks)
11 A(9, 8), B(5, 0) and C(3, 1) are three points.
(i) Show that AB and BC are perpendicular. [3]
(ii) Find the equation of the circle with AC as diameter. You need not simplify your answer.
Show that B lies on this circle. [6]
(iii) BD is a diameter of the circle. Find the coordinates of D. [3]
12 You are given that f(x) (cid:4) x3(cid:2)9x2(cid:2)20x(cid:2)12.
(i) Show that x (cid:4)(cid:1)2 is a root of f(x) (cid:4) 0. [2]
(ii) Divide f(x) by x(cid:2)6. [2]
(iii) Express f(x) in fully factorised form. [2]
(iv) Sketch the graph of y (cid:4) f(x). [3]
(v) Solve the equation f(x) (cid:4) 12. [3]
[Question 13 is printed overleaf.]
4751 June 2006
PMT
4
13 Answer the whole of this question on the insert provided.
1
The insert shows the graph of y (cid:4) , x (cid:6) 0.
x
1
(i) Use the graph to find approximate roots of the equation (cid:4) 2x(cid:2)3, showing your method
x
clearly. [3]
1
(ii) Rearrange the equation (cid:4) 2x(cid:2)3to form a quadratic equation. Solve the resulting equation,
x
p± q
leaving your answers in the form . [5]
r
1
(iii) Draw the graph of y (cid:4) (cid:2)2, x (cid:6) 0, on the grid used for part (i). [2]
x
1
(iv) Write down the values of x which satisfy the equation (cid:2)2 (cid:4) 2x(cid:2)3. [2]
x
4751 June 2006
Candidate
Candidate Name Centre Number Number
This insert consists of 2 printed pages.
HN/2
©OCR 2006 [H/102/2647] Registered Charity 1066969 [Turn over
13 (i) and (iii)
y
5
4
3
2
1
–5 –4 –3 –2 –1 0 1 2 3 4 5 x
–1
–2
–3
–4
–5
....................................................................................................................................................
(ii) ....................................................................................................................................................
....................................................................................................................................................
....................................................................................................................................................
....................................................................................................................................................
....................................................................................................................................................
....................................................................................................................................................
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(iv) ....................................................................................................................................................
4751Insert June2006
5
4
3
2
1
3
10 Simplify (m2(cid:2)1)2 (cid:1) (m2 (cid:1) 1)2, showing your method.
Hence, given the right-angled triangle in Fig. 10, express pin terms of m, simplifying your answer.
[4]
2
m +1
2
m -1
p
Fig. 10
Section B (36 marks)
11 There is an insert for use in this question.
1
The graph of y (cid:4) x(cid:2) is shown on the insert. The lowest point on one branch is (1, 2). The
x
highest point on the other branch is ((cid:1)1, (cid:1)2).
(i) Use the graph to solve the following equations, showing your method clearly.
1
(A) x(cid:2) (cid:4) 4 [2]
x
1
(B) 2x(cid:2) (cid:4) 4 [4]
x
(ii) The equation (x (cid:1) 1)2(cid:2)y2 (cid:4) 4 represents a circle. Find in exact form the coordinates of the
points of intersection of this circle with the y-axis. [2]
(iii) State the radius and the coordinates of the centre of this circle.
Explain how these can be used to deduce from the graph that this circle touches one branch
1
of the curve y (cid:4) x(cid:2) but does not intersect with the other. [4]
x
3
10 The triangle shown in Fig. 10 has height (x(cid:3)1)cm and base (2x (cid:2) 3)cm. Its area is 9cm2.
Not to
x + 1 scale
2x – 3
Fig. 10
(i) Show that 2x2 (cid:2) x (cid:2) 21 (cid:1) 0. [2]
(ii) By factorising, solve the equation 2x2 (cid:2) x (cid:2) 21 (cid:1) 0. Hence find the height and base of the
triangle. [3]
Section B (36 marks)
11
A(3, 7)
Not to
T
scale
C(1, 3)
Fig. 11
Acircle has centre C(1, 3) and passes through the point A(3, 7) as shown in Fig. 11.
(i) Show that the equation of the tangent at Ais x(cid:3)2y (cid:1) 17. [4]
(ii) The line with equation y (cid:1) 2x (cid:2) 9 intersects this tangent at the point T.
Find the coordinates of T. [3]
(iii) The equation of the circle is (x (cid:2) 1)2(cid:3)(y (cid:2) 3)2 (cid:1) 20.
Show that the line with equation y (cid:1) 2x (cid:2) 9 is a tangent to the circle. Give the coordinates of
the point where this tangent touches the circle. [5]
[Turn over
© OCR 2007 4751/01 June 07
x + 1
2x – 3
2x – 3
x + | 1