FOR EDEXCEL
GCE Examinations Advanced Subsidiary
Core Mathematics C2
Paper L Time: 1 hour 30 minutes
Instructions and Information
Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and / or integration. Full marks may be obtained for answers to ALL questions. Mathematical formulae and statistical tables are available. This paper has nine questions.
Advice to Candidates
You must show sufficient working to make your methods clear to an examiner. Answers without working may gain no credit.
Written by Shaun Armstrong
Solomon Press These sheets may be copied for use solely by the purchasers institute.
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1.
A geometric series has first term 75 and second term 15. (a) (b) Find the common ratio of the series. Find the sum to infinity of the series. (2) (2)
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2.
A circle has the equation x2 + y2 + 8x 4y + k = 0, where k is a constant. (a) Find the coordinates of the centre of the circle. (2)
Given that the x-axis is a tangent to the circle, (b) find the value of k. (3)
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3.
A B O C D Figure 1 Figure 1 shows a circle of radius r and centre O in which AD is a diameter. The points B and C lie on the circle such that OB and OC are arcs of circles of radius r with centres A and D respectively. Show that the area of the shaded region OBC is
1 2 r (3 6
3 ).
(6)
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3.
continued
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4.
(a)
Sketch the graphs of y = sin 2x and y = tan
x 2
on the same diagram in the (4)
space provided, for x in the interval 0 x 360. (b) Hence state how many solutions exist to the equation sin 2x = tan
x 2
, (2)
for x in the interval 0 x 360 and give a reason for your answer.
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4.
continued
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5.
(a)
Find the value of a such that loga 27 = 3 + loga 8. (3)
(b)
Solve the equation 2x + 3 = 6x 1, giving your answer to 3 significant figures. (4)
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5.
continued
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6.
(a) (b)
Expand (2 + x)4 in ascending powers of x, simplifying each coefficient. Find the integers A, B and C such that (2 + x)4 + (2 x)4 A + Bx2 + Cx4.
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(4)
(2)
(c)
Find the real values of x for which (2 + x)4 + (2 x)4 = 136. (3)
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6.
continued
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7. (a) (b) (c) (d)
f(x) = 2x3 5x2 + x + 2. Show that (x 2) is a factor of f(x). Fully factorise f(x). Solve the equation f(x) = 0. Find the values of in the interval 0 2 for which 2 sin3 5 sin2 + sin + 2 = 0, giving your answers in terms of . (4) (2) (4) (1)
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7.
continued
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8.
The curve C has the equation y = 3 x 2 2x (a) (b) (c) (d)
1
1 2
, x > 0. (4) (5) (2) (2)
Find the coordinates of the points where C crosses the x-axis. Find the exact coordinates of the stationary point of C. Determine the nature of the stationary point. Sketch the curve C in the space provided.
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8.
continued
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9.
C O Figure 2 Figure 2 shows the curve C with equation y = 3x 4 x + 2 and the tangent to C at the point A. Given that A has x-coordinate 4, (a) show that the tangent to C at A has the equation y = 2x 2. (6) x
The shaded region is bounded by C, the tangent to C at A and the positive coordinate axes. (b) Find the area of the shaded region. (8)
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9.
continued
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9.
continued
END
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