03 Preliminary Course FINAL EXAMINATION Friday, September 6 Mathematics General Instructions o Reading Time 5 minutes. o Working Time 3 hours. o Write using a black pen. o Approved calculators may be used. o All necessary working should be shown for every question. Outcomes to be Assessed: A student: Total marks (00) Section I 0 marks o Attempt Questions 0 o Answer on the multiple choice answer sheet provided o Allow approximately 5 minutes for this section Section II 90 marks o Attempt Questions 6 o Answer in the booklets provided o Begin each question in a new booklet o Allow approximately hours 45 minutes for this section Name: Student Number: Teacher:
P P3 P4 P5 provides reasoning to support conclusions which are appropriate to the context. performs routine arithmetic and algebraic manipulation involving surds, simple rational expressions and trigonometric identities. chooses and applies appropriate arithmetic, algebraic, graphical, trigonometric and geometric techniques. understands the concept of a function and the relationship between a function and its graph.
Section I 0 marks Attempt Questions -0 Allow about 5 minutes for this section Use the multiple choice answer sheet for your responses to Questions 0. What is the value of (A).4 (B).45 (C).46 (D).5 8.8 3.47.79 + 7.75 correct to two significant figures? Which of the following equations has solutions x= and x= 3? (A) (B) (C) (D) x x x x 5x 6= 0 + 5x 6= 0 x 6= 0 + x 6= 0 3 Which statement is incorrect? (A) The diagonals of a rhombus bisect each other (B) The diagonals of a rhombus are equal (C) The diagonals of a rhombus are perpendicular to each other (D) The diagonals of a rhombus bisect the vertex angles 4 x The function f( x) = is: x (A) an even function (B) an odd function (C) neither an even nor odd function (D) a zero function
5 Which graph does not represent a function? (A) (B) (C) (D)
6 Which of the following is a correct expression involving θ in triangle ABC? (A) (B) (C) (D) 5 = 6 + 0 + 6 0 cosθ 6 + 0 5 cosθ = 0 5 5 6 sin sin 5 3' θ = sinθ sin 5 3' = 6 5 7 What is the gradient of the line perpendicular to the line x+ y+ 3= 0? (A) (B) (C) (D) 8 The solution to t (A) 0< t < (B) t < 0 or t > (C) t > > t is: (D) t < or t > 0
9 What is the exact value of cos 40? (A) (B) (C) 3 (D) 3 0 Simplify cos(90 θ ) sin(90 θ) (A) (B) cotθ (C) (D) tanθ tanθ End of Section I
Section II 90 marks Attempt Questions -6 Allow about hours 45 minutes for this section Begin each question in a new booklet All necessary working should be shown All questions are of equal value Question (5 marks). Use a SEPARATE writing booklet. Marks (a) Solve 5x = x+ 0 (b) Factorise: (i) (ii) x 7x 4 + ( x ) 6 (c) The line 6x ky = passes through the point (3, ). Find the value of k. (d) Solve the following pair of simultaneous equations: 3 x+ y = 5x 4y = 4 (e) Find the value of x if 75 + 7 = x 3 Question continues on the next page
Question (continued) Marks (f) ABCD is a parallelogram. CB is produced to E so that CB = BE. 3 Prove AFD EFB
Question (5 marks). Use a SEPARATE writing booklet. Marks (a) Solve 3 5 = x 3 (b) Solve + 3x= x+ 3 (c) What are the values of a and b if 5 = a+ b? + 3 (d) Prove + sinθ tanθ + secθ = cosθ (e) BF is parallel to CG, BC = EC and ABE =. (i) Show that BEC = 68. (ii) Hence, or otherwise, show that CG bisects DCE.
Question 3 (5 marks). Use a SEPARATE writing booklet. Marks (a) In the diagram below the points A, B and C have coordinates (, ), ( 4, 3) and (, 3) respectively. (i) Calculate the exact length of interval BC (ii) Find the gradient of BC (iii) Hence, show that the equation of BC is y = x+ 5 (iv) Find, to the nearest degree, the acute angle between the x-axis and the line BC (v) Find the perpendicular distance between A and the line BC (vi) Find the coordinates of D, in the first quadrant, so that ABCD is a parallelogram (vii) Find the exact area of the parallelogram ABCD (b) The lengths of the sides of triangle ABC are a = 5. cm, b = 7.3 cm and c = 6.7 cm. (i) Explain why BAC is the smallest angle in the triangle. (ii) Calculate the size of the smallest angle in ABC. Give the answer correct to the nearest minute. (iii) Hence find the area of the triangle. Give the answer correct to the nearest square centimetre.
Question 4 (5 marks). Use a SEPARATE writing booklet. Marks (a) In the diagram, AOB and COD are straight lines. AC AB and AB BD. (i) Prove ACO BDO (ii) If CD is 35 cm, find the length of OD (b) A ship leaves port P and travels on a bearing of 04 a distance of 300 km to point Q. It then turns and travels on a bearing of 39 for 00 km to point R. (i) Show that PQR = 45. (ii) What is the distance from R to P? Answer correct to the nearest kilometre. (iii) Find the bearing of R from P? 3 Answer correct to the nearest degree. (c) (i) Show that cos x + 3sin x= sin x (ii) Hence, or otherwise, solve cos x 3sin x for 0 x 360 + =
Question 5 (5 marks). Use a SEPARATE writing booklet. Marks (a) The function y = f( x) is defined as follows: x for x f( x) = for < x< x + for x (i) Evaluate f( ) + f() (ii) Write an expression for f( a + ) (b) Make neat sketches of the following graphs on separate number planes. Mark clearly the essential features of each graph. (i) (ii) ( x 3) ( y 4) 5 y + + = = 4 x (iii) xy = (iv) y = x (c) Solve 7 3x < 3 and graph your solution on a number line. 3 (d) If 4 tanθ = and cosθ > 0, is the value of sinθ positive or negative? 5
Question 6 (5 marks). Use a SEPARATE writing booklet. Marks (a) In the diagram below AO BP CQ. Find the value of x (b) Graph the region represented by the inequalities: x y x + > 5 and 0 (c) Simplify 3 x x x x x x 4 5 4 + 4 + 4 3 (d) Find the domain of y =. 5 x (e) Find the equation of the line through the point of intersection of the lines 3 6x 5y = 3and 4x+ y = and also through the point (, ) (f) If the points ( a, 3), ( a, a ) and ( a 3, a+ ) are collinear, find the 3 value of a. End of Examination Paper
Section I Answer Sheet Student Number: 0 marks Attempt Questions -0 Allow about 5 minutes for this section Use this multiple choice answer sheet for questions 0. Select the alternative A, B, C or D that best answers the question. Fill in the response oval completely. Sample + 4 =? (A) (B) 6 (C) 8 (D) 9 A B C D If you think you have made a mistake, put a cross through the incorrect answer and fill in the new answer. A B C D If you change your mind and have crossed out what you consider to be the correct answer, then indicate this by writing the word correct and drawing an arrow as follows: correct A B C D Completely fill the response oval representing the most correct answer. A B C D A B C D 3 A B C D 4 A B C D 5 A B C D 6 A B C D 7 A B C D 8 A B C D 9 A B C D 0 A B C D
Solutions to Year Mathematics Preliminary Examination 03 Section I Multiple Choice Solutions 8.8 3.47.79 7.75.455....5 (sf) (D) 3 The diagonals of a rhombus are equal (B) ( x )( x3) 0 x x60 (D) x 4 f( x) x x f( x) ( x) ( ) x x f( x) (B) 5 For some x values there are two matching y values 6 5 6 sin sin 53 (C) (C) 7 x y 30 y x3 m m (C) 8 t t t t 0 tt ( ) 0 t 0ort (B) 9 cos 40 cos(8060 ) cos 60 (B) 0 cos(90 ) sin(90 ) sin cos tan (C)
Question (a) 5x x0 4x x 3 Marking Criteria Correct answer (b) (i) x 7x4 (x)( x4) Correct answer (b) (ii) x ( ) 6 ( x4)( x4) ( x)( x6) Correct solution Attempt at difference of two squares Correct expansion x 4x (c) Sub. (3, ) into 6xky : 6(3) k() k 6 k 8 Correct solution Correct substitution of (3, ) Correct answer without justification (d) x y...() 5x4y 4...() () 5 : 5x5y 5...(3) (3) () : 9y 9 y In () : x x 3 Correct solution Correct elimination/substitution method with only one correct value for either x or y Correct attempt at either elimination or substitution method e) 75 7 5 33 3 8 3 643 9 x 9 3 Correct solution x = 9 Substantially correct solution Correct attempt at simplifying the surds
Question (continued) (f) Marking Criteria 3 Correct solution AD BC (equal opposite sides in parallelogram ABCD) CB BE (given) AD EB (both equal BC) AFD BFE (equal vertically opposite angles) DAF EBF (equal alternate angles, AD EC) AFD EFB (SAA) Communication: d) clear setting out and labelling of equations mark f) articulate reasoning, clear argument mark Two correct statements, Correct proof but not fully One correct statement, Correct proof without any justification
Question (a) 3 5 x 3 7 x 7x 46 7x 0 0 x 7 (b) 3x x 3x x or 3xx x 4x3 3 x x 4 Solutions are valid if: 3x 0 i.e. x 3 x is only valid solution Check solutions by substitution into equation (c) 5 55 4 9 7 97 a9andb7 (d) sin tan sec cos LHS tan sec sin cos cos sin cos RHS starting from RHS show it equals LHS Marking Criteria 3 Correct solution Substantially correct solution Correct attempt at solution 3 Correct solution One mistake in the solution Correct attempt at solution 3 Correct solution One mistake in working towards the answer Correctly attempting to rationalise the denominator Correct solution with correct setting out Correct attempt at proof by rewriting expression in terms of sin and/or cos equivalent
Question (continued) (e) (i) Marking Criteria Correct proof CBE 68 (supplement of ABE) CBE BEC 68 (angles opposite equal sides in isosceles CBE) One correct statement, Correct proof, not fully (ii) CBE DCG 68 (equal corresponding angles, BF CG) BEC ECG 68 (equal alternate angles, BF CG) CG bisects DCE ( ECG GCD) Communication: (e) clear, concise and correct setting out of proofs 4 marks Correct proof One correct statement, Correct proof, not fully
Question 3 (a) (i) Marking Criteria Correct answer BC 3 6 45 3 5 units Correct length as an unsimplified surd (ii) 6 m 3 (iii) y3( x) y3x y x5 Correct answer Correct solution (iv) tan tan () 63.434... 63 (nearest deg) Correct solution Correct statement tan (v) axbyc d a b () ( ) 5 9 units 5 Correct solution Correct substitution into correct formula (vi) B A: Right 5, up C D : Right 5, up ( CD BA, CD BA) Correct answer D ( 5,3) (4, 4) One correct coordinate
Question 3 (continued) (vii) A bh 9 3 5 5 7 units Marking Criteria Correct answer (b) (i) BAC is opposite the shortest side Correct reason (ii) 7.3 6.7 5. cos BAC 7.36.7 0.775... BAC cos (0.775...) 43035.9 43 (nearest min) (iii) A bcsin A 7.3 6.7sin 43 0 35 6.785... 7 cm (nearest cm ) Correct solution rounded correctly Correct substitution into cosine rule Substantially correct solution Correct solution A 7.3 6.7sin 43 ' 6.787... 7 cm (nearest cm ) Communication: a) v) showing substitution mark vi) explaining how coordinates are found mark b) ii) showing unrounded answer in degrees/minutes/seconds before rounding mark
Question 4 (a) (i) Marking Criteria Correct solution AB 90 (given) AOC BOD (equal vertically opposite angles) ACO BDO (equiangular) One correct statement, Correct proof, not fully (ii) AC CO BD DO 5 (corresponding sides in similar triangles are in same ratio) 5 OD 35 7 5 cm Correct solution, Correct solution, not fully Correct first statement with reason (b) (i) Correct proof PQT 76 (supplementary cointerior angles on parallel lines) TQR (angles at point add to 360 ) PQR 45 ( TQR PQT ) (ii) RP 300 00 300 00 cos 45 4547.8... RP 4547.8....47... km (nearest km) Correctly finding PQT or TQR, Correct proof, not fully Correct solution Correct substitution in cosine rule
Question 4 (continued) (iii) sin QPR sin 45 00 RP 00sin 45 sin QPR RP 0.665... QPR 4.76... Bearing of Rfrom P 044 47 Marking Criteria 3 Correct solution Correct value of QPR Correct substitution into sine rule (c) (i) cos x 3sin xsin x LHS cos x 3sin ( sin x) 3sin sin x 3sin sin x RHS x x x Correct proof Correct attempt at proof (ii) cos x 3sin x sin x sin x x 90, 70 Correct solution One correct value for x Communication: (a)(i) mark for copying diagram mark for stating vertices in corresponding order mark for sufficient proof i.e. equiangular when two pairs of corresponding angles are equal (ii) mark for correct reason why corresponding pairs of sides are in same ratio mark for correct units for OD (b)(i) marks for each of correct geometric reasons to show angle PQR = 45 o (ii) mark for correct rounding (iii) mark for correct rounding
Question 5 (a) f( ) f() (i) ( ) () Marking Criteria Correct solution Correct attempt at solution (ii) f( a ) a a Correct answer (b) (i) Correct graph showing all essential features Correct graph without all essential features shown (ii) Correct graph showing all essential features Correct graph without all essential features shown
Question 5 (continued) (iii) Marking Criteria Correct graph showing all essential features, including asymptotes Correct graph without all essential features shown (iv) Correct graph showing all essential features, including asymptotes Correct graph without all essential features shown (c) 73x 3 73x3or73x3 3x4 3x0 4 0 x x 3 3 4 0 x 3 3 3 Correct solution and number line graph Correct solution Correct attempt at solution and correct number line graphfrom their working (d) tan is negative in nd and 4th quadrants cos is positive in st and 4th quadrants lies in 4th quadrant where sin is negative Correct attempt at solution Correct number line graph from incorrect solution Correct answer Communication: (a) Showing clear and correct substitution for (i) and (ii) 3 marks (d) Clear setting out of reason for sin being negative mark
Question 6 (a) Marking Criteria Correct proof with full and correct reason x 3 (intercepts on transversals are cut 3 4 in same ratio by a family of parallel lines) 9 x cm 4 Correct first statement, Correct proof, not fully (b) Correct region One correct region (c) 3 x x 4x5 x 4x 4x4 ( x)( x x) ( x5)( x) ( x)( x) 4( x x) x 5 4 3 Correct solution Substantially correct solution Correct attempt at factorisation
Question 6 (continued) (d) y, x5 5 x Domain: 5x 0 x 5 (e) 6x5y3 k(4x y) 0 Sub. (,) : 5 3 k(8 ) 0 40k 0 4 k 0 k 5 6x5y3 (4x y) 0 5 30x5y5 4x y 0 6x6y6 0 x y0 Marking Guidelines Correct answer Correct attempt at finding domain by stating x 5 or x 5 3 Correct solution Correct value of k Correct point of intersection and attempt to find equation of required line Correct form of equation shown by first line and attempt to substitute (, ) Correct attempt to solve equations simultaneously (f) ( a) 3 ( a) 3 ( a) a ( a3) a a5 a 3a 3a3 (3a3)( a5) (3a)( a) 3a 8a53a 7a a 3 3 a 3 Correct solution Substantially correct solution Correct attempt at solution shown by first line of working Communication: Question 6 (4 marks) (b) The graph is drawn neatly with template. Intercepts shown and graph labelled. Mark (d) Clear setting out. For example stating limitations such as x 5. Mark (e) Clear and logical setting out of solution (example: mention of substitution). Mark (f) Clear explanation that equal gradients will prove the points are collinear. Mark