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Edexcel IALAS Level

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t Use black ink or ball-point pen.t Fill in the boxes at the top of this page with your name, centre number and candidate number.t Answer all questions.t Without sufficient working, correct answers may be awarded no marks.t Answer the questions in the spaces provided

there may be more space than you need.

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t The total mark for this paper is 100.t 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

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

Dimuth SandaruwanB.Sc(special)

Calculators may be used.

Time: 2 hours & 30 minutes

Mathematics C12Sample Paper

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01 The equation x2 + 4px + 9 = 0 has unequal real roots. Find the set of possible values of p.(4)

f(x) = 3x2 + 6x + 7

Given that f(x) can be written in the form A(x + B)2 + C, where A, B and C are rational numbers,

(a) find the value of A, the value of B and the value of C.(3)

(b) Hence, or otherwise, find

(i) the value of x for which 1f (x)

is a maximum,

(ii) the maximum value of 1

f (x).

(2)

02

Figure 1

Figure 1 shows the sector, AOB of a circle with centre O and radius 8 cm. A circle of radius 2 cm touches the lines OA and OB and the arc AB. Angle AOB is 2 radians,

0 <

.4

(a) Find, to 4 significant figures, the value of (3)

(b) Find, to 3 significant figures, the area of the region shaded in Figure 1.(3)

2

2 cm

Diagram NOT accurately drawn

A

O

B8 cm

03 Solve the equations

(a) logx 1024 = 5(2)

(b) log5 (6y + 11) = 3(3)

(c) 2log3 t + logt 9 = 5(6)

10

6

Figure 1

Figure 1 shows triangle ABC with AB = 10 cm, BC = 6 cm and BAC = 28. The point D lies on AC such that BD = 6 cm.

(a) Find, to the nearest 0.1, the size of DBC.(4)

(b) Find, to 3 significant figures, the length of AD.(3)

(c) Find, to 3 significant figures, the area of triangle ABC.(3)

Diagram NOTaccurately drawn

A

10 cm

B

6 cm

CD

6 cm

28

n4 (a) Show that r3( 4 =) n n(3 5 )2r =1 (3)

(b) Hence, or otherwise, evaluate r 3 4( )11

50

r =

(2)n

Given that 3r 4( ) = 186r =1

(c) find the value of n.(3)

5 A particle P moves along the x-axis. At time t seconds (t 0) the velocity, v m/s, of P is given by v = 5cos2t. Find

(a) the least value of t for which P is instantaneously at rest,(2)

(b) the magnitude of the maximum acceleration of P.(3)

When t = 0, P is at the point (2, 0).

(c) Find the distance of P from the origin when P first comes to instantaneous rest.(4)

8 Solve, for 0 , giving each solution to 3 significant figures, (a) 5 sin 1 = 0

(3)

(b) tan 2 0.43 + = (4)

(c) 4 sin2 7 cos = 2(4)

9 The sum Sn of the first n terms of an arithmetic series is given by Sn = n(2n + 3). The first term of the series is a.

(a) Show that a = 5(2)

(b) Find the common difference of the series.(3)

(c) Find the 12th term of the series.(2)

Given that 1 + Sp + 4 = 2 Sp

(d) find the value of p.(4)

7 The line l passes through the points with coordinates (1, 6) and (3, 2).

(a) Show that an equation of l is y + 2x = 8(3)

The curve C has equation xy = 8

(b) Show that l is a tangent to C.(3)

Given that l is the tangent to C at the point A,

(c) find the coordinates of A.(2)

(d) Find an equation, with integer coefficients, of the normal to C at A.(3)

10 Solve the equations

(a) logx 1024 = 5(2)

(b) log5 (6y + 11) = 3(3)

(c) 2log3 t + logt 9 = 5(6)

11 f(x) = x3 + px2 + qx + 6 p, q

Given that f(x) = (x 1)(x 3) (x + r)

(a) find the value of r.(1)

Hence, or otherwise,

(b) find the value of p and the value of q.(3)

Figure 2

Figure 2 shows the curve C with equation y = f(x) which crosses the x-axis at the points with coordinates (3, 0) and (1, 0) and at the point A. The point B on C has x-coordinate 2

(c) Find an equation of the tangent to C at B.(5)

(d) Show that the tangent at B passes through A.(2)

(e) Use calculus to find the area of the finite region bounded by C and the tangent at B.(5)

Diagram NOTaccurately drawn

A

y

C

B

x1 3O

12 The point C with coordinates (2, 1) is the centre of a circle which passes through the point A with coordinates (3, 3).

(a) Find the radius of the circle.(2)

The line AB is a diameter of the circle.

(b) Find the coordinates of B.(2)

The points D with coordinates (0, 2) and E with coordinates (4, 0) lie on the circle.

(c) Show that DE is a diameter of the circle.(2)

The point P has coordinates (x, y).

(d) Find an expression, in terms of x and y, for the length of CP.(2)

Given that the point P lies on the circle,

(e) show that x2 + y2 4x 2y = 0(2)

13 A solid paperweight in the shape of a cuboid has volume 15 cm3. The paperweight has a rectangular base of length 5x cm and width x cm and a height of h cm. The total surface area of the paperweight is A cm2.

(a) Show that A = 10x2 + 36

(3)x

(b) Find, to 3 significant figures, the value of x for which A is a minimum, justifying that this value of x gives a minimum value of A.

(6)

(c) Find, to 3 significant figures, the minimum value of A.(2)

14 The third and fifth terms of a geometric series S are 48 and 768 respectively. Find

(a) the two possible values of the common ratio of S,(3)

(b) the first term of S.(1)

Given that the sum of the first 5 terms of S is 615

(c) find the sum of the first 9 terms of S.(4)

Another geometric series T has the same first term as S. The common ratio of T is 1r

where r is one of the values obtained in part (a). The nth term of T is tn Given that t2 > t3

(d) find the common ratio of T.(1)

The sum of the first n terms of T is Tn

(e) Writing down all the numbers on your calculator display, find T9(2)

The sum to infinity of T is T

Given that T Tn > 0.002

(f) find the greatest value of n.(5)

End

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