NYGH 2016-S3EOY-IM1 with ans

Class Register Number Name 南洋女子中学校 NANYANG GIRLS' HIGH SCHOOL End-of-Year Examination 2016 Secondary Three INTEGRATED MATHEMATICS PAPER 1 2 hours Thursday 6 October 2016 0845 - 1045 READ THESE INSTRUCTIONS FIRST INSTRUCTIONS TO CANDIDATES 1. Answer questions 1 – 12 before attempting the Bonus Question. 2. Answer the whole of question 12 on the INSERT provided. 3. Write your answers and working on the separate answer paper provided. 4. Write your name, register number and class on all the work you hand in. At the end of the examination, fasten all your work securely together. 5. Omission of essential working will result in loss of marks. 6. Write in dark blue or black pen on both sides of the paper. 7. You may use an HB pencil for any diagrams or graphs. 8. Do not use staples, paper clips, glue or correction fluid. 9. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. INFORMATION FOR CANDIDATES 1. The number of marks is given in brackets [ ] at the end of each question or part question. 2. The total number of marks for this paper is 80. 3. The use of an approved scientific calculator is expected, where appropriate. 4. You are reminded of the need for clear presentation in your answers. Setter: E.Liow This document consists of 8 printed pages. NANYANG GIRLS' HIGH SCHOOL [ Turn over

2 Mathematical Formulae TRIGONOMETRY Formulae for ∆ABC a2 = b2 + c2 − 2bc cos A ∆ = ab sin C

3 (i) y x O y x O (ii) (i) y x O y x O (ii) 1. It is given that y is directly proportional to ,3x and 10=y when 2=x . (i) Find an equation connecting x and y. [2] (ii) Calculate the value of x when y = 80. [1] 2. (a) The sketch of the graphs nxy= are shown below. Write down a possible value of n for each of the graphs. [2] (b) Write down a possible equation for each of the graphs. [2] 3. The following equations (i) )1)(( 2xaxy −+= , (ii) 02 =−− bxayyx , may be represented by a straight line in the form cmXY += , where a, b, m and c are constants while X and Y are functions of x and/or y. Copy the following table and complete it with an expression each for Y, X, m and c. [4] Y X m c )1)(( 2xaxy −+= x 02 =−− bxayyx b [Turn over

4 4. In a school fun fair, a booth sold three types of items: keychains, notebooks and pens. The table illustrates the number of items sold during different sessions of the day at the booth. Keychains Notebooks Pens Session 1 20 35 48 Session 2 30 50 54 Session 3 54 72 60 The information in the table above can be represented by a matrix P           = 60 54 48 72 50 35 54 30 20 . The price for each item was as follows: Keychain: $1 Notebook: $3 Pen: $2 (a) (i) Write down a matrix Q such that the elements of the matrix product PQ give the total takings for each session. Find the product PQ. [2] (ii) 10% of the takings will be donated to the school’s Pocket Money Fund. Write down a row matrix T such that the element of the matrix product TPQ gives the total amount of donation from the booth. [1] (b) It was later discovered that the number of each type of items sold in Session 3 was wrongly recorded to be 20% more than the actual number. Write down a row matrix R such that the elements of the matrix product RP give the correct number of each type of items sold in Session 3. Find the product RP. [2]

5 5. In the diagram, ADC is a straight line. ∠ABD = 90°, AB = 8 cm, BD = 6 cm, and DC = 9.5 cm. (a) Without the use of calculators, find the value of (i) BDC∠cos , [2] (ii) BDC∠sin . [1] Hence, calculate the area of triangle BDC. [1] (b) Find the shortest length from B to AC. [2] (c) Calculate the length of BC. [2] 6. Given that A =       − 41 23 , B =       − − 11 22 and I =       10 01 . (i) Find the matrix C such that A + C = B + 3I. [2] (ii) Find A-1. Hence, find the matrix D such that AD = B2. [4] 7. The solution of this question by accurate drawing will not be accepted. The diagram shows a triangle ABC, where A, B and C are (3, −1), (−3, 3) and (−6, 2) respectively. (i) Find the area of triangle ABC. [2] (ii) Point X lies on the line segment AB such that BX:XA = 2:1. Find the coordinates of point X. [2] y x O C(−6, 2) B(−3, 3) A(3, −1) 8 cm 6 cm 9.5 cm A B C D [Turn over

6 8. The diagram shows a triangular field ABC, where B is 36 m from C on a bearing of 095°, ∠ABC = 80° and AC = 44 m. (i) Calculate how far C is west of B. [2] (ii) Find the bearing of A from B. [2] (iii) A flagpole CT of height 3.2 m stands at C. An air drone is flying from A, towards the flagpole CT, at an angle of elevation of x° from A. Find the minimum whole number x such that the air drone will not crash onto the flagpole. [2] 9. The points A, B, C, D, E and F lie on the circumference of a circle with centre O. AD and FB are diameters of the circle. The lines GHAI, HFK and KDJ are tangents to the circle. It is given that °=∠ 108BOD , °=∠ 34CBD and °=∠ 36DBO . (a) Find (i) DAF∠ , [1] (ii) CDJ∠ , [1] (iii) FED∠ . [1] (b) Giving your reasons, find DCB∠ . [2] (c) (i) Prove that AOFH is a cyclic quadrilateral. [2] (ii) Hence, find GHF∠ . State your reasons clearly. [2] (d) Given that the length of chord DB = 6 cm, calculate the length of the diameter BF. [2] G H A I B C J D E K F O 108° 36° 34° North North 95° 80° A B C T 3.2 m 44 m 36 m

7 10. The diagram shows part of a straight line l obtained by plotting x y 1+ against x. (i) Line AB is perpendicular to line l. A and B are (1, 2) and (5, 4) respectively. Show that the equation of line l is 1142 2 −+−= xxy , when y is expressed in terms of x. [4] (ii) If (2, k) lies on line l, find the value of k. [2] (iii) Find the vertical-intercept of the new line formed when line l is plotted as a graph of 1 17 + − x xy against x . [2] 11. The solution of this question by accurate drawing will not be accepted. In the diagram, the line segments AC and BD intersect at point B (4, 6). It is given that A, C and D are (0, 8), (10, 3) and (6, 10) respectively. (a) Find the equation of the line parallel to AD and passing through B. [2] (b) Show that AB is perpendicular to BD. [3] (c) If AXCD is a parallelogram, state the coordinates of X. [2] (d) (i) Find the centre-radius form of the equation of the circle which passes through A, B and D. [3] (ii) Write down the coordinates of Y such that BY is a diameter of the circle in (d)(i). [1] y x O C B A D O x y 1+ x A(1, 2) line l (2, k) B(5, 4) [Turn over

8 12. Answer the whole of this question on the INSERT provided. The graph of      −= xxy 2 1 2 1 is partially shown on the INSERT. (a) By drawing a tangent, find the gradient of the graph at the point where 8.0−=x . [3] (b) Find the x-coordinate of a point on the curve at which the gradient is −1. [1] (c) By drawing suitable straight lines on the graph, find (i) the solution to the equation 41 2 =− xx , [3] (ii) the range of values of x for which 016 23 <−+ xx and 0≠x . [3] Bonus Let ABC be an equilateral triangle and M be a point inside the triangle. Let D, E, F be points on BC, CA and AB respectively such that MD, ME and MF are perpendicular to BC, CA and AB respectively. G