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KINGDAVIDHIGH SCHOOL, LINKSFIELD

MATHEMATICS

GRADE 12 EXAMINATION

APRIL 2008

Total: 150 marks

Reading Time: 10 minutes Writing Time: 3 hours

LO’s Being Assessed / Mark Allocation For Each
LO 1 ; LO 2 / 75
LO 3 ; LO 4 / 75

This paper contains 15 pages excluding the front page and the instructions and the following enclosures: Data sheet

Check that your paper is complete.

NAME:______

SECTION A LO 1 AND LO 2 75 MARKS

QUESTION 1

Solve for x if :

(a) x(x – 1) = 2

(3)

(b) x(x – 1) > 2

(3)

QUESTION 2

Solve for x if

(5)

QUESTION 3

(a) Evaluate without using a calculator:

(i)

(4)

(ii)

(6)

(iii)

(3)

(b) Solve for x if

(5)

QUESTION 4

Express as a single log without using a calculator:

(a)

(2)

(b) +

(2)

(c) + + . . . +

(2)

QUESTION 5

Given that x3 –x2 – x – 15 = 0

(a) Show that x = 1 is not a root of the equation:

(2)

(b) Prove that x = 3 is the only root of the equation.

______

(6)

QUESTION 6

A researcher described the rat population of a large city by the equation

y = 54(2,7183) 0,12 t

where t is the time measured in years since 1995 and y is the number of rats in millions.

(a) What was the rat population in 1995, when the research started?

(1)

(b) What was the rat population in 2007?

(2)

(c) Sketch the graph of the rat population function, clearly showing the

y-intercept and basic shape.

(2)

QUESTION 7

The drawing on the right shows the graphs

of a parabola, f, and two straight lines, g and h.

f , g and h all intersect at A on the y-axis, while

f and g also intersect at D(q ; r), the turning point

of f. g(x) = 2x – 4 and h(x) = mx – 4.

(a) If g is perpendicular to h, write down the

value of m

(1)

(b) The graph of h cuts the x-axis at P(q ;0).

Show that q = 8.

(2)

(c) Determine the value of r.

(2)

(d) Determine the equation of f if r = 20, leaving your answer in the form

y = a(x – t)2 + n

(5)

(e) Explain how f must be moved to obtain the parabola with equation

y = .

(2)

(f) Show that y = also passes through A.

(2)

QUESTION 8

The sketch shows the graph of

y = f(x) which is a shift of the graph

.

(a) Write down the equation of f.

______(2)

(b) Write down the domain of h if

the graph of f is shifted 3 units to

the right and 1 unit down to obtain

y = h(x).

______(1)

(c) Write down the equation of the

horizontal asymptote of h.

______(1)

QUESTION 9

The graphs in the sketch represent

f(x) = acosbx and g(x) = asin(x + )

(a) What is the period of f?

______(1)

(b) What are the values of a and b?

______

______(2)

(c) What is the equation of g?

______(1)

(d) If g is represented by the equation

y = a cos(x + ), write down the value

of .

______(1)

QUESTION 10

The graph of g(x) = logp x is sketched.

(a) Find the value of p

______

______

______(1)

(b) Write down the equation of g –1 in the

form

g –1(x) = ______(1)

(c) Sketch the graph of g –1 on the same

set of axes, showing the co-ordinates

of any 3 points on the graph. (2)

NAME:______

SECTION B – LO 3 AND LO 4 (75 MARKS)

QUESTION 1

D is the midpoint of the line joining A(6 ; 4) and B(10 ; 6). Find the equation of the circle with its centre at C(1 ; 2) which passes through D. (4)

QUESTION 2

The circle defined by the equation x2 + y2 + 2ax + 8y + 1 = 0 has a radius of 8units.

(a)Find the value of “a” if a > 0.(5)

(b)If a = 7, find the length of the tangent from the point T(2 ; 3) to a point

S on the circle, leaving your answer in surd form.(5)

QUESTION 3
QUESTION 4

(a) Prove that = cos A(6)

(b) For what values of A is the identity in Q4(a) undefined?(6)

(c) Without using a calculator, and showing sufficient working to justify your

answers, evaluate the following:

(i)sin2 1 + sin2 89

(1)

(ii)sin2 1 + sin2 2 + sin2 88 + sin2 89

(1)

(iii)sin2 1 + sin2 2+sin2 3 + . . . . + sin287 + sin2 88 + sin2 89

(3)

QUESTION 5

(a) If 2tan + 1 = 0 and  [180 ; 0], find the value of sin2 without

using a calculator.(5)

(b) If cos 42 = p, determine each of the following in terms of p:

(i)cos 762

(1)

(ii)sin2201

(3)

QUESTION 6

(a) (i) Prove that: 4cos2 x – 4sin2 x = 4cos2x(2)

(ii) Hence, solve for x if

4cos2 x = 4sin2 x + 3sin 4x(6)

(b) Determine the general solution of the equation

2 sin (x  30) = cos x(6)

QUESTION 7

In the diagram, BD = x, and

(a) Determine in terms of .(1)

(b) Prove that (5)

(c) Find the area of ADC if

x= 4 units and  = 15(3)