HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time)

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1 N E W S O U T H W A L E S HIGHER SCHOOL CERTIFICATE EXAMINATION 996 MATHEMATICS /3 UNIT (COMMON) Time allowed Three hours (Plus minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL questions are of equal value. All necessary working should be shown in every question. may be deducted for careless or badly arranged work. Standard integrals are printed on page. Board-approved calculators may be used. Answer each question in a separate Writing Booklet. You may ask for etra Writing Booklets if you need them.

2 QUESTION. Use a separate Writing Booklet. (a) Find the value of 3 correct to two significant figures. (b) Factorise + 3. (c) Rationalise the denominator of 4 +. (d) Simplify 3. 4 (e) Find a primitive function of 6 3. (f) Graph the solution of + 3 on a number line.

3 3 QUESTION. Use a separate Writing Booklet. y A O + y + 8 = 0 l The line l is shown in the diagram. It has equation + y+ 8= 0 and cuts the ais at A. The line k has equation y = + 6, and is not shown on the diagram. Copy or trace the diagram into your Writing Booklet. (a) Find the coordinates of A. (b) Eplain why k is parallel to l. (c) Draw the graph of k on your diagram, indicating where it cuts the aes. (d) Shade the region + y+ 8 0 on your diagram. (e) Write down a pair of inequalities which define the region between k and l. (f) Show that P(8, ) lies on k. (g) Find the perpendicular distance from P to l. (h) Q(4, 6) lies on l. Show that Q is the point on l which is closest to P.

4 4 QUESTION 3. Use a separate Writing Booklet. (a) Differentiate: e cos. (b) A layer of plastic cuts out % of the light and lets through the remaining 8%. 3 Show that two layers of the plastic let through 7 % of the light. How many layers of the plastic are required to cut out at least 90% of the light? (c) Find sec 6 d. (d) Evaluate e 3 d. 3

5 QUESTION 4. Use a separate Writing Booklet. (a) y 3 y = + O 3 The region which lies between the ais and the line y = + from = 0 to = 3 is rotated about the ais to form a solid. Find the volume of the solid. (b) Let f ( )=. 9 Copy the following table of values into your Writing Booklet and supply the missing values f( ) Use these si values of the function and the trapezoidal rule to find the approimate value of 0 d. Draw the graph of + y = and shade the region whose area is represented by the integral 0 d. (iv) (v) Use your answer to part to eplain why the eact value of the integral is. 4π Use your answers to part and part (iv) to find an approimate value for π.

6 6 QUESTION. Use a separate Writing Booklet. (a) The graph of y f Find f( ). = ( ) passes through the point (, 3), and f ( )= 3. (b) Solve the equation u u = 0, correct to three decimal places. (c) y = e y y = e 6 O The diagram shows the graphs of y = e and y = e. Find the area between the curves from = to =. Leave your answer in terms of e. Show that the curves intersect when e e = 0. Use the results of part (b) with u e = to show that the coordinate of the point of intersection of the curves is approimately (d) For all in the domain 0 4, a function g ( ) satisfies g ( ) < 0 and g ( ) < 0. Sketch a possible graph of y = g( ) in this domain.

7 7 QUESTION 6. Use a separate Writing Booklet. (a) Lee takes some medicine. The amount, M, of medicine present in Lee s bloodstream t hours later is given by M = 4t t 3, for 0 t 3. Sketch the curve M = 4t t for 0 t 3 showing any stationary points. 7 At what time is the amount of medicine in Lee s bloodstream a maimum? When is the amount present increasing most rapidly? (b) 7 mm 3 mm 7 mm A venetian blind consists of twenty-five slats, each 3 mm thick. When the blind is down, the gap between the top slat and the top of the blind is 7 mm and the gap between adjacent slats is also 7 mm, as shown in the first diagram. When the blind is up, all the slats are stacked at the top with no gaps, as shown in the second diagram. (iv) Show that when the blind is raised, the bottom slat rises 67 mm. How far does the net slat rise? Eplain briefly why the distances the slats rise form an arithmetic sequence. Find the sum of all the distances that the slats rise when the blind is raised.

8 8 QUESTION 7. Use a separate Writing Booklet. (a) Sketch the graph of y = cos for 0 π. On the same set of aes, sketch the graph of y = cos for 0 π. Find the eact values of the coordinates of the points where the graph of y = cos crosses the ais in the domain 0 π. (b) D C 7 NOT TO SCALE P A B ABCD is a quadrilateral. The diagonals AC and BD intersect at P. AD = BC and AC = BD. Copy the diagram into your Writing Booklet. (iv) Show that triangles ABC and ABD are congruent. Show that triangle ABP is isosceles. Hence show that triangle CDP is isosceles. Show that AB is parallel to CD.

9 9 QUESTION 8. Use a separate Writing Booklet. (a) Students studying at least one of the languages, French and Japanese, attend a meeting. Of the 8 students present, 8 study French and study Japanese. 4 What is the probability that a randomly chosen student studies French? What is the probability that two randomly chosen students both study French? What is the probability that a randomly chosen student studies both languages? (b) T 8 NOT TO SCALE cm P 6 cm Q 4 cm S R y cm U PQRS is a rectangle with PQ = 6 cm and QR = 4 cm. T and U lie on the lines SP and SR respectively, so that T, Q, and U are collinear, as shown in the diagram. Let PT = cm and RU = y cm. Show that triangles TPQ and QRU are similar. Show that y = 4. (iv) Show that the area, A, of triangle TSU is given by 48 A = Find the height and base of the triangle TSU with minimum area. Justify your answer.

10 0 QUESTION 9. Use a separate Writing Booklet. Two particles P and Q start moving along the ais at time t = 0 and never meet. Particle P is initially at = 4 and its velocity v at time t is given by v = t + 4. ( ) The position of particle Q is given by = + 3log e t +. The diagram shows the graph of = + 3log t +. e ( ) =+3 log e ( t + ) O t (a) Find an epression for the position of P at time t. (b) Copy the diagram into your Writing Booklet and, on the same set of aes, draw the graph of the function found in part (a). (c) P and Q are joined by an elastic string and M is the midpoint of the string. Show that the position of M at time t is given by = t + 4t + 3log ( t + )+. [ ] e (d) Find the time at which the acceleration of M is zero. 3 (e) Find the minimum distance between P and Q. 4

11 QUESTION 0. Use a separate Writing Booklet. (a) On the same set of aes, accurately draw the graphs of y = sin and y = 3 for 0 π. Find the gradient of the tangent to y = sin at the origin. For what values of m does the equation sin = m have a solution in the domain 0 < < π? (b) P 300 m α O 00 m Q NOT TO SCALE The diagram shows a circle centre O with points P and Q on the circle. The angle POQ = α, the length of the chord PQ is 00 metres, and the length of the arc PQ is 300 metres as shown. α Show that sinα = 3. Use your graphs from part (a) to find an approimate value for α. Hence find the size of POQ, and also find the radius of the circle. (c) A and B are two points 00 metres apart. For what values of l is it possible to find a circular arc AB of length l metres? Justify your answer. You may use the results from parts (a) and (b).

12 n d STANDARD INTEGRALS n+ =, n ; 0, if n< 0 n + d = ln, > 0 e a d a e a =, a 0 cosa d = sin a, a 0 a sin a d = cos a, a 0 a sec a d = tan a, a 0 a sec a tan a d = sec a, a a 0 a d = a tan, 0 + a a a d = sin, a> 0, a< < a a ( ) > > d = ln + a, a a ( ) d = ln + + a + a NOTE : ln = log, >0 e 0 Board of Studies NSW 996