C12 June 2015 Model

BY: A.HARISON

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MUSAEUS COLLEGE

Core Mathematics C12 Advanced Subsidiary

Thursday 02 April 2015 Morning Time: 2 hours 30 minutes

3rd Term

Total Marks

[email protected] Musaeus College, Colombo 07 Leave

blank 1. Given that > 0 and

8 log 2 + log2 = 6

(a) Show that log2 is either 2 or 4.

(b) Hence find both values of x satisfying the above equation.

(5)

(2)


blank 2. Some values of function f, are given in the table below

(a) Use the trapezium rule to estimate

0.50 to 2 decimal places.

(b) Use your answer to part (a) to estimate a value for

0.50 + 2

(4)

(3)

x

0

0.1

0.2

0.3

0.4

0.5

f(x)

0.1

0.23

0.45

0.52

0.44

0.21


blank 3. If

= 5 and = when = .

(a) Show that,

= 2

(b) State the x coordinates of the turning points of the curve = 2 and find the corresponding coordinates.

(c) Make a sketch of the curve = 2 and mark on it

(i) The coordinate axes, (ii) The coordinates of the points where the curve meets the axes.

(d) State the range of values of x for which the gradient of the curve

= 2 is negative.

(5)

(2)

(4)

(3)


blank 4. (a) Calculate the value of the coefficient of in the expansion of ( )8

(b) Using a suitable substitution, find the value of the coefficient of 5 in the expansion of

8

(4)

(3)


blank 5. The first two terms of an arithmetic progression are log2 and log2 .

(a) Show that the common difference of the arithmetic progression is 2.

(b) Given that the sum of the progression is log2 + , find the number of terms in the progression.

(2)

(3)


blank 6. (a) Write down the coordinates of mid-point M of the line joining A(0,1) and B(6,5).

(b) Show that the line + 5 = passes through M and is perpendicular to AB.

(c) Calculate the coordinates of the center of the circle which passes through A, B and the origin O.

(2)

(2)

(3)


blank 7. The series += is geometric series with common ratio k. where 1.

(a) By finding the common ratio, show that the series += is a geometric series

(b) Show that += = 1 1

(4)

(4)


blank 8.

In above Figure, the points O and C are the centres of the circles C1 and C2 respectively. The circle C1 has radius 16 cm and the circle C2 has radius 4 cm. The circles touch at the point D, as shown in the figure. Horizontal line AB is common tangent to circles.

(a) calculate the size of the acute angle AOD, giving your answer in radians to 3 significant figures.

(b) Find the length of AB.

(c) Find the area of Trapezium OABC.

(d) Find the shaded area.

(2)

(1)

(1)

(4)

C

4cm

O

16cm

D

B A

C1

C2


blank 9. The expression + 2 + has a remainder of when it is divided by + and it has a remainder of 66 when it is divided by

(a) Find the values of A and B.

(b) Find the coordinates of turning point.

(4)

(2)


blank 10. The curve C has equation

= 4, 4 94 a) In the space below, sketch the curve C b) Write down the exact coordinates of the points at which C meets the coordinate

axes.

c) Find the range of k, where = has three real roots. Here k is a constant.

(2)

(3)

(3)


blank 11. The curve C has equation = + the line l1, is the normal to the curve C at the point P (1,5).

a) Show that l1 has equation = . Point, , + 4 lies on the curve C. b) Find the values for t when tangent to the curve at Q which is parallel to the above

normal.

c) Hence, find the equations of the two tangents to the curve which are parallel to this normal

(4)

(2)

(5)


blank 11. (a) In ABC the sides AB and AC are of lengths 9cm and 10cm respectively and < = 60. If = , use the cosine rule to show that, 2 9 9 =

b) Find the length of BC to the nearest mm

(4)

(2)


blank 11. A ball is dropped on to a horizontal plane and rebounds successively. The

height above the plane reached by the ball after the first impact with the plane is 4m. After each impact the ball rises to a height which is of the height reached after the previous impact. Calculate the total vertical distance travelled by the ball from the first impact until

a) The fourth impact, b) The eight impact, c) It comes to rest

(3)

(3)

(4)


blank 14. Given that x and y are acute angles,

+ = , sin =

(a) Calculate x and y in terms of . (b) Show that tan = + . (c) Solve for ,

tan tan + =

(4)

(2)

(4)


blank 15.

Figure shows a sketch of part of curve C with equation = (x-2) (8-x) a) Find b) Find the coordinates of its maximum point. c) Find the area enclosed by the curve and x-axis. d) Show that the normal at the point, divides this area in the ratio 91:125

(2)

(2)

(2)

(5)

Time: 2 hours 30 minutes

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C12 June 2015 Model