Year 12 Trial HSC Mathematics 2006 Essay

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Words: 1139
Pages: 5
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Question 1 Marks

(a) Evaluate correct to 3 significant figures 2

(b) Solve the equation 2

(c) Express in simplest surd form 2

(d) Find the primitive of 2

(e) The angle is subtended at the centre of a circle, by an arc of length 10cm. 2 The centre of a circle has a radius of 15cm. Find the value of to the nearest minute.

(f) Factorise fully 2

Question 2 – (Start a new page)

(a) Differentiate, with respect to x

(i) 2

(ii) 2

(b) Evaluate 2

(c) Find 2

(d) Find p, if the roots of are real 2

(e) Find the equation of the parabola which has its vertex at (0, 2) and its 2 directrix is given by y=5

Marks

Question 3 – (Start a new page)

(a) The coordinates of the points A, B and C on the number plane below are (2, 0) (6, 2) and (6, -1) respectively. AB is parallel to CD.

(Not to scale)

Copy the diagram onto your answer paper

(i) Find the exact length of AB 2

(ii) Find the gradient of AB 1

(iii) Show that the equation of the line through C, parallel to AB is 1

(iv) Find the coordinates of point D (where DC cuts the y-axis) 1

(v) Find the perpendicular distance from A to the line DC 2

(vi) Find the length of DC 1

(vii) Hence find the exact area of the quadrilateral ABCD 2

(b) Solve the equation 2 sin x = for 2

Question 4 – (Start a new page)

(a) For the function

(i) Find the coordinates of the stationary point(s) and their nature 4

(ii) Find any points of inflexion 2

Question 4 continues next page

Marks (b)

(Not to scale)

The graphs of and intersect at the point A and the point B (5, 9)

(i) Show that the point A lies on the y-axis 2

(ii) Find the area of the shaded region bounded by the graphs of 4 and

Question 5 – (Start a new page)

(a) For what value of x is the tangent to the curve parallel to the line 3

(b)

(Not to scale)

In the above diagram , AB=6cm, AC=4cm, CE=8cm and BC=5cm

(i) Copy the diagram onto your answer page

(ii) Prove ∆ ABC is similar to ∆ AED 2

(iii) Find the length of BD 2

Question 5 continues next page

Marks

(c) Organisers of a music festival issued 750 tickets in the first year of the festival.

The number of tickets issued increased by 150 each year after that.

(i) What was the total number of tickets issued in the first fifteen years? 2

(ii) In which year of the festival did the number of tickets issued first 3 exceed 5000?

Question 6 – (Start a new page)

(a) Consider the function

(i) Find the first derivative 1

(ii) Find the stationary point and its nature 4

(b) The curve y = sec x for 0 is rotated around the x-axis 3 Find the exact volume of the solid generated.

(c) (i) State the period of y=3Sin2x 1

(ii) Draw the graph, neatly, of y=3Sin2x in the domain 0 2

(iii) How many solutions are there if 3Sin2x = -1 1

Question 7 – (Start a new page)

(a) In the diagram below ABCD is a parallelogram and BE=EF. AD is produced to F.

(i) Prove that DEF is congruent to CEB 3

(ii) Hence prove that DE = 2

Question 7 continues next page

Marks

(b) Copy the diagram of y = f (x) below neatly onto your answer paper. 4 Then sketch y = f(x) below it, given that it passes through (0, 0). Ensure that both graphs are clear and neat.

(c) The following table lists three values of a function.

x
2.0
3.0
4.0
f(x)
1.7
9.0
4.3

Use these three function values to estimate by: 3 (i) Simpson’s Rule

(ii) The Trapezoidal Rule

Question 8 – (Start a new page)

(a) Consider the series, …… where 0

(i) Show that a limiting sum exists for this series 2

(ii) Find the limiting sum, expressing it in simplest form 2

(b) Express in the form 3

Question 8 continues next page
Marks

(c) ABC is an equilateral triangle with sides of length 6cm. An arc, centre A, and radius 3cm, cuts AB and AC at X and Y respectively. This is
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