Preliminary Mathematics

Transcription

1 NORTH SYDNEY GIRLS HIGH SCHOOL 010 YEARLY EXAMINATION Preliminary Mathematics General Instructions Reading Time 5 minutes Working Time hours Write using black or blue pen Board-approved calculators may be used All necessary working should be shown in every question Total Marks 84 Attempt Questions 1 7 All questions are of equal value At the end of the eamination, place your solution booklets in order and put this question paper on top. Submit ONE bundle. The bundle will be separated for marking so please ensure your name is written on each solution booklet. Student Name: Teacher: 1 a b 3ab 3c 4 5 6a 6bc 6d 7abc 7d Totals P P3 P4 P5 P6 P7 /6 /3 /9 /1 /1 /1 /1 /7 /31 /6 /7 /13 /5 /5 /10 /7 / /9 /1 /1 /1 /1 /1 /1 /1 /84

2 Total Marks 84 Attempt Questions 1 7 All questions are of equal value Begin each question in a SEPARATE writing booklet. Etra writing booklets are available. Question 1 (1 Marks) (a) Evaluate sin 63 tan14 to two significant figures. (b) Epress y without negative indices if 3 5 = a b and y = ab. (c) Solve (3 )(4 + ) > 0. (d) Simplify (e) Rationalise the denominator of (f) Solve 7 = 3. Question (1 Marks) Start a new booklet. (a) Differentiate with respect to : (i) (ii) (iii) 4 ( + 3) (iv) 1 +. (b) (i) Find the equation of the tangent to 3 y = at the point (1, 4). 3 (ii) For what values of are the tangents parallel to the line y = 4 + 3? 010 Preliminary Mathematics Eamination North Sydney Girls High School

3 Question 3 (1 Marks) Start a new booklet. (a) Sketch on a number plane the region defined by 0 + y 1. (b) A function is defined as f( ) = 4. (i) Find, in simplest form: (α) f ( ) 1 (β) [ f ( )]. (ii) Determine if f ( ) is even, odd or neither. 1 (c) ABCD is a parallelogram. F and E are two points on BC and DC respectively such that BF =, DE = and ADF = BEC = α. FD meets BE at G such that DG = 3 and BG = 6 cm. A B 6 cm 3 G F D α E α C (i) Prove that GDE = GBF. (ii) Prove that GDE is similar to GBF. (iii) Hence find the value of. 010 Preliminary Mathematics Eamination North Sydney Girls High School 3

4 Question 4 (1 Marks) Start a new booklet. Below is a diagram showing a quadrilateral ABCD, where AB CD. y B ( 1, 6) NOT TO SCALE A ( 4, 4) C (, 5) D (, y) (a) Find the gradient of the line AB. 1 (b) Find the eact value of the distance AB. (c) Show that the equation of the line CD is 3y+ 11= 0. (d) Find the perpendicular distance from the line CD to the point A. (e) The distance between C and D is to be 117 units (i) Show that ( ) + 5 = (ii) Find the coordinates of point D, assuming < 0. 3 (f) Find the area of the quadrilateral ABCD Preliminary Mathematics Eamination North Sydney Girls High School 4

5 Question 5 (1 Marks) Start a new booklet. (a) Simplify cosec (90 A)cos(90 A). (b) Find β in the domain 90 < β < 70 if sin β =. 1 (c) (i) Find the angle, in degrees and minutes, that a line with gradient.5 makes with the positive -ais. (ii) Find the acute angle that the same line makes with the y-ais. 1 (d) A section of rainforest is to be scoured in the search for new species. The shape is shown below. The bearing of landmark A from landmark O is 48ºT and is 4 km in distance. The distance from landmark A to B is 40 km and from landmark B to O is 35 km. (i) Find the size of AOB. (ii) Hence or otherwise calculate the area of this section of the rainforest. 1 (iii) What is the bearing of landmark O from landmark B? 010 Preliminary Mathematics Eamination North Sydney Girls High School 5

6 Question 6 (1 Marks) Start a new booklet. (a) For the curve y = a + b + c where a, b and c are constants, it is given that dy y = 1 and = 1 when = 1. Find the relationship between a and c. d (b) (i) Sketch on the same number plane the functions y = and y = (ii) Use your answer to (i) to solve > + 3. (c) State the domain and range of f ( ) 1 =. (d) In the diagram below, BA DF and EF BC. It is also given that BE = 5, AF = 14, DC = 56, BD = 4, EA = and FC = y. 5 E 7 A 14 F y NOT TO SCALE B 4 D 56 C C Find and y, giving reasons Preliminary Mathematics Eamination North Sydney Girls High School 6

7 Question 7 (1 Marks) Start a new booklet. (a) The line + 3y 13= 0is translated 5 units up and parallel to the line. Find the equation of this transformation. (b) Prove the identity cosθ sinθ sinθ cosθ 1 tanθ + 1 cotθ = +. (c) Evaluate 3 lim (d) The diagram below shows the graph of y = f (). y y = f () (i) For which values of is f ( ) > 0? 1 (ii) Eplain what happens to f ( ) for large positive values of. 1 (iii) Sketch the graph y = f ( ). 3 End of Paper 010 Preliminary Mathematics Eamination North Sydney Girls High School 7

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