
	Trending ▼ ICSE CBSE 10th ISC CBSE 12th CTET GATE UGC NET Vestibulares ResFinder

NSW HSC 2007 : MATHEMATICS

16 pages, 65 questions, 0 questions with responses, 0 total responses,

	nsw_hsc	+Fave Message
	Profile Timeline Uploads Q & A

Home > nsw_hsc >

Instantly get Model Answers to questions on this ResPaper. Try now!
NEW ResPaper Exclusive!

Formatting page ...

2007 H I G H E R S C H O O L C E R T I F I C AT E E X A M I N AT I O N Mathematics General Instructions Reading time 5 minutes Working time 3 hours Write using black or blue pen Board-approved calculators may be used A table of standard integrals is provided at the back of this paper All necessary working should be shown in every question 212 Total marks 120 Attempt Questions 1 10 All questions are of equal value Total marks 120 Attempt Questions 1 10 All questions are of equal value Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. Marks Question 1 (12 marks) Use a SEPARATE writing booklet. (a) Evaluate 2 + 5 correct to two decimal places. (b) Solve 2x 5 > 3 and graph the solution on a number line. (c) Rationalise the denominator of 1 3 1 . 2 2 2 (d) Find the limiting sum of the geometric series 2 33 3 + + + . 4 16 64 (e) Factorise 2x2 + 5x 12. 2 (f) Find the equation of the line that passes through the point ( 1, 3) and is perpendicular to 2x + y + 4 = 0 . 2 2 Marks Question 2 (12 marks) Use a SEPARATE writing booklet. (a) Differentiate with respect to x : (i) 2 x 2 ex + 1 (ii) 2 (i) Find (1 + 3 ) d x . cos x 2 (ii) (b) (1 + tan x)10. 8 . Evaluate dx 1 x 2 4 (c) 3 The point P ( , 0) lies on the curve y = x sin x . Find the equation of the tangent to the curve at P . 3 3 Marks Question 3 (12 marks) Use a SEPARATE writing booklet. y (a) B (12, 16) NOT TO SCALE C (2, 11) A (10, 5) O x In the diagram, A, B and C are the points (10, 5), (12, 16) and (2, 11) respectively. Copy or trace this diagram into your writing booklet. (i) 1 (ii) Find the midpoint of AC . 1 (iii) Show that OB AC. 2 (iv) Find the midpoint of OB and hence explain why OABC is a rhombus. 2 (v) (b) Find the distance AC . Hence, or otherwise, find the area of OABC . 1 Heather decides to swim every day to improve her fitness level. On the first day she swims 750 metres, and on each day after that she swims 100 metres more than the previous day. That is, she swims 850 metres on the second day, 950 metres on the third day and so on. (i) Write down a formula for the distance she swims on the nth day. 1 (ii) How far does she swim on the 10th day? 1 (iii) What is the total distance she swims in the first 10 days? 1 (iv) After how many days does the total distance she has swum equal the width of the English Channel, a distance of 34 kilometres? 2 4 Marks Question 4 (12 marks) Use a SEPARATE writing booklet. 2 sin x = 1 for 0 x 2 . (a) Solve (b) Two ordinary dice are rolled. The score is the sum of the numbers on the top faces. 2 (i) What is the probability that the score is 10? 2 (ii) What is the probability that the score is not 10? 1 (c) A O NOT TO SCALE C B An advertising logo is formed from two circles, which intersect as shown in the diagram. The circles intersect at A and B and have centres at O and C . The radius of the circle centred at O is 1 metre and the radius of the circle centred at C is 3 metres. The length of OC is 2 metres. . 2 (i) Use Pythagoras theorem to show that OAC = (ii) Find ACO and AOC . 2 (iii) Find the area of the quadrilateral AOBC. 1 (iv) Find the area of the major sector ACB. 1 (v) Find the total area of the logo (the sum of all the shaded areas). 2 5 1 Marks Question 5 (12 marks) Use a SEPARATE writing booklet. (a) A B E NOT TO SCALE F C D In the diagram, ABCDE is a regular pentagon. The diagonals AC and BD intersect at F. Copy or trace this diagram into your writing booklet. (i) 1 (ii) Find the size of BAC. Give reasons for your answer. 2 (iii) (b) Show that the size of ABC is 108 . By considering the sizes of angles, show that ABF is isosceles. 2 A particle is moving on the x-axis and is initially at the origin. Its velocity, v metres per second, at time t seconds is given by v= 2t 16 + t 2 . (i) What is the initial velocity of the particle? 1 (ii) Find an expression for the acceleration of the particle. 2 (iii) Find the time when the acceleration of the particle is zero. 1 (iv) Find the position of the particle when t = 4 . 3 6 Marks Question 6 (12 marks) Use a SEPARATE writing booklet. (a) 2 Solve the following equation for x : 2e2x e x = 0. (b) Let ( x ) = x 4 4 x 3 . (i) Find the coordinates of the points where the curve crosses the axes. 2 (ii) Find the coordinates of the stationary points and determine their nature. 4 (iii) Find the coordinates of the points of inflexion. 1 (iv) Sketch the graph of y = ( x ) , indicating clearly the intercepts, stationary points and points of inflexion. 3 End of Question 6 Please turn over 7 Marks Question 7 (12 marks) Use a SEPARATE writing booklet. (a) (i) Find the coordinates of the focus, S , of the parabola y = x 2 + 4. (ii) The graphs of y = x 2 + 4 and the line y = x + k have only one point of intersection, P . Show that the x-coordinate of P satisfies 2 1 x2 x + 4 k = 0. (iii) Using the discriminant, or otherwise, find the value of k. 1 (iv) Find the coordinates of P . 2 (v) Show that SP is parallel to the directrix of the parabola. 1 (b) y A y = sin x O 2 y = 3 cos x x B The diagram shows the graphs of y = 3 cos x and y = sin x . The first two points of intersection to the right of the y-axis are labelled A and B. 3 cos x = sin x to find the x-coordinates of A and B. (i) Solve the equation (ii) Find the area of the shaded region in the diagram. 8 2 3 Marks Question 8 (12 marks) Use a SEPARATE writing booklet. (a) One model for the number of mobile phones in use worldwide is the exponential growth model, N = Ae kt, where N is the estimate for the number of mobile phones in use (in millions), and t is the time in years after 1 January 2008. (i) It is estimated that at the start of 2009, when t = 1, there will be 1600 million mobile phones in use, while at the start of 2010, when t = 2, there will be 2600 million. Find A and k. 3 (ii) According to the model, during which month and year will the number of mobile phones in use first exceed 4000 million? 2 (b) E 27 q D 8 p A B NOT TO SCALE C In the diagram, AE is parallel to BD, AE = 27, CD = 8, BD = p, BE = q and ABE, BCD and BDE are equal. Copy or trace this diagram into your writing booklet. (i) Prove that ABE BCD . 2 (ii) Prove that EDB BCD . 2 (iii) Show that 8, p, q, 27 are the first four terms of a geometric series. 1 (iv) Hence find the values of p and q. 2 9 Marks Question 9 (12 marks) Use a SEPARATE writing booklet. (a) 3 y NOT TO SCALE O 1 x The shaded region in the diagram is bounded by the curve y = x2 + 1, the x-axis, and the lines x = 0 and x = 1. Find the volume of the solid of revolution formed when the shaded region is rotated about the x-axis. (b) A pack of 52 cards consists of four suits with 13 cards in each suit. (i) One card is drawn from the pack and kept on the table. A second card is drawn and placed beside it on the table. What is the probability that the second card is from a different suit to the first? 1 (ii) The two cards are replaced and the pack shuffled. Four cards are chosen from the pack and placed side by side on the table. What is the probability that these four cards are all from different suits? 2 Question 9 continues on page 11 10 Marks Question 9 (continued) (c) Mr and Mrs Caine each decide to invest some money each year to help pay for their son s university education. The parents choose different investment strategies. (i) Mr Caine makes 18 yearly contributions of $1000 into an investment fund. He makes his first contribution on the day his son is born, and his final contribution on his son s seventeenth birthday. His investment earns 6% compound interest per annum. 3 Find the total value of Mr Caine s investment on his son s eighteenth birthday. (ii) Mrs Caine makes her contributions into another fund. She contributes $1000 on the day of her son s birth, and increases her annual contribution by 6% each year. Her investment also earns 6% compound interest per annum. 2 Find the total value of Mrs Caine s investment on her son s third birthday (just before she makes her fourth contribution). (iii) Mrs Caine also makes her final contribution on her son s seventeenth birthday. Find the total value of Mrs Caine s investment on her son s eighteenth birthday. End of Question 9 11 1 Marks Question 10 (12 marks) Use a SEPARATE writing booklet. (a) dx , of the dt object, as a function of time, t. The coordinates of the points shown on the graph An object is moving on the x-axis. The graph shows the velocity, are A(2, 1) , B (4, 5) , C (5, 0) and D (6, 5) . The velocity is constant for t 6. dx dt B 5 1 O 5 NOT TO SCALE A C 2 6 t 4 D (i) Using Simpson s rule, estimate the distance travelled between t = 0 and t = 4. 2 (ii) The object is initially at the origin. During which time(s) is the displacement of the object decreasing? 1 (iii) Estimate the time at which the object returns to the origin. Justify your answer. 2 (iv) Sketch the displacement, x, as a function of time. 2 Question 10 continues on page 13 12 Marks Question 10 (continued) (b) The noise level, N, at a distance d metres from a single sound source of loudness L is given by the formula N= L 2 . d Two sound sources, of loudness L1 and L 2 are placed m metres apart. L1 x P m L2 The point P lies on the line between the sound sources and is x metres from the sound source with loudness L1 . (i) Write down a formula for the sum of the noise levels at P in terms of x . 1 (ii) There is a point on the line between the sound sources where the sum of the noise levels is a minimum. 4 Find an expression for x in terms of m, L1 and L 2 if P is chosen to be this point. End of paper 13 BLANK PAGE 14 BLANK PAGE 15 STANDARD INTEGRALS x n dx = 1 dx x = ln x , x > 0 e ax dx = 1 ax e , a 0 a cos ax d x = 1 sin ax , a 0 a sin ax d x 1 = cos ax , a 0 a sec 2 ax d x = 1 tan ax , a 0 a sec ax tan ax d x = 1 sec ax , a 0 a 1 a +x 2 2 1 1 x a 1 x tan 1 , a 0 a a dx x = sin 1 , a > 0 , a < x < a a dx a2 x 2 2 = dx 1 n +1 x , n 1; x 0 , if n < 0 n +1 = ln x + x 2 a 2 , x > a > 0 dx = ln x + x 2 + a 2 2 1 2 x 2 + a ( ) ( ) NOTE : ln x = loge x , x > 0 16 Board of Studies NSW 2007

Formatting page ...

Additional Info : New South Wales Higher School Certificate Mathematics 2007
Tags : new south wales higher school certificate mathematics 2007, nsw hsc online mathematics, nsw hsc maths, nsw hsc mathematics syllabus, nsw hsc maths model exam papers, mathematics sample papers, mathematics course, nsw hsc maths solved paper., australia new south wales, nsw hsc online, nsw hsc past papers, nsw hsc papers, nsw hsc syllabus, nsw board of studies, higher school certificate new south wales, nsw australia, hsc syllabus, nsw hsc exams, nsw hsc question papers, nsw hsc solved question papers, nsw hsc previous exam papers, nsw university.

nsw_hsc chat

Horizontal lines at:
Guest Horizontal lines at:
AutoRM Data:
Box geometries: