Paper Reference(s) 6663A/01 Pearson Edexcel International Advanced Level

Core Mathematics C1

Advanced Subsidiary

Monday 13 January 2014 ( Morning Time: 1 hour 30 minutes

Materials required for examination Items included with question papers Mathematical Formulae (Pink) Nil

Calculators may NOT be used in this examination.

Instructions • Use black ink or ball-point pen. • If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used. • Fill in the boxes at the top of this page with your name, centre number and candidate number. • Answer all questions and ensure that your answers to parts of questions are clearly labelled. • Answer the questions in the spaces provided – there may be more space than you need. • You should show sufficient working to make your methods clear. Answers without working may not gain full credit.

Information • The total mark for this paper is 75. • The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.

Advice • Read each question carefully before you start to answer it. • Try to answer every question. • Check your answers if you have time at the end.

1. Simplify fully

(a) [pic]
(1)
(b) [pic]
(3)

2. [pic], x > 0

(a) Find [pic], giving each term in its simplest form.
(3)
(b) Find [pic], giving each term in its simplest form.
(2)

3. Solve the simultaneous equations

x – 2y – 1 = 0

x2 + 4y2 – 10x + 9 = 0
(7)

4.
[pic]
Figure 1

Figure 1 shows a sketch of a curve with equation y = f(x).

The curve crosses the y-axis at (0, 3) and has a minimum at P (4, 2).

On separate diagrams, sketch the curve with equation

(a) y = f(x + 4),
(2)
(b) y = 2f(x).
(2)
On each diagram, show clearly the coordinates of the minimum point and any point of intersection with the y-axis.

5. Given that for all positive integers n,

[pic]
(a) find the value of [pic]
(2)
(b) find the value of a6
(3)

6.
[pic]
Figure 2

The straight line l1 has equation 2y = 3x + 7.

The line l1 crosses the y-axis at the point A as shown in Figure 2.

(a) (i) State the gradient of l1.

(ii) Write down the coordinates of the point A.
(2)

Another straight line l2 intersects l1 at the point B (1, 5) and crosses the x-axis at the point C, as shown in Figure 2.

Given that 0
(3)
(b) Find the set of possible values of k.
(4)

9. A curve with equation y = f(x) passes through the point (3, 6). Given that

[pic] = (x – 2)(3x + 4)

(a) use integration to find f(x). Give your answer as a polynomial in its simplest form.
(5)
(b) Show that f(x) ≡ (x – 2)2(x + p), where p is a positive constant. State the value of p.
(3)
(c) Sketch the graph of y = f(x), showing the coordinates of any points where the curve touches or crosses the coordinate axes.
(4)

10. The curve C has equation y = x3 – 2x2 – x + 3.

The point P, which lies on C, has coordinates (2, 1).

(a) Show that an equation of the tangent to C at the point P is y = 3x – 5.
(5)

The point Q also lies on C.

Given that the tangent to C at Q is parallel to the tangent to C at P,

(b) find the coordinates of the point Q.
(5)

TOTAL FOR PAPER: 75 MARKS
END