Michaelmas Term 2010

Form 5

Paper 2(2 hrs 40 min)

Instructions to Candidates

1. Answer ALL questions in Section I and ANY TWO in Section II.

2. Begin the answer for EACH question on a NEW page.

3. Full marks may not be awarded unless full working or explanation is shown with the answer.

4. Mathematical instruments and silent electronic calculators may be used for this paper.

5. You are advised to use the first 10 minutes of the examination time to read through this paper. Writing may begin during this 10-minute period.

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO

LIST OF FORMULAE

Volume of a prism V = Ah where A is the area of a cross-section and h is the perpendicular length.

Volume of a right pyramid V = ¹∕3 Ah where A is the area of the base and h is the perpendicular height.

Circumference of a circle C = 2πr where r is the radius of the circle.

Area of a circle A = πr2 where r is the radius of the circle.

Area of a trapezium A = ½(a+b)h where a and b are the lengths of the parallel sides and h is the perpendicular distance between the parallel sides.

Roots of quadratic equations If ax 2 + bx + c = 0,

Then x =

Trigonometric ratios Sinθ =

Cosθ =

Tanθ =

Area of triangle Area of Δ = ½bh where b is the length of the base and h is the perpendicular height.

Area of ΔABC = ½ab sinC

Area of ΔABC =

where s =

Sine rule

Cosine rule a2 = b2 + c2 – 2bc CosA

SECTION I

Answer ALL the questions in this section.

ALL working must be clearly shown.

1. (a) Using a calculator, or otherwise, determine the exact value of

(i) (1.5)2 + (2.1)2

(ii)

(iii)

(7 marks)

(b) (i) Write the answer in Part (a) (i) correct to one significant figure.

(ii) Write your answer in Part (a) (ii) in standard form.

(2 marks)

(c) Give the three (3) mathematical laws. (3 marks)

Total 12 marks

2. (a) Simplify

(i) 3m – 2(m + 1) (2 marks)

(ii) (3 marks)

(b) Solve the equation

3 ( x−8 )=2x (3 marks)

(c) Expand and Simplify

25

(3 x−7 )(5 x+10) (3 marks)

Total 11 marks

3. (a) Given that A = and B =

(i) Calculate the matrix product AB. (3 marks)

(ii) If C = and AB = C, calculate the values of a, b, x and y.

(5 marks)

(b) Consumer Arithmetic

Total 14 marks

4. (i) Factorise completely(a) 3a3−12ab2 (2 marks)

(b) 6a−8ay+3b−4 by (2 marks)

(c) 6 x2+7 x−5 (3 marks)

(ii) Indices

5. (a) The figures shown below, not drawn to scale, represent the cross sections of two circular pizzas. Both pizzas are equally thick and contain the same toppings.

Small pizza Medium pizza

Diameter = 15 cm Diameter = 30 cm

(i) Is a medium pizza twice as large as a small pizza?

Use calculations to support your answer. (4 marks)

(ii) A medium pizza is cut into 3 equal parts, and each part is sold for $15.95. A small pizza is sold for $12.95.

Which is the better buy?

Use calculations to support your answer. (3 marks)

(b) In the figure below, BCD=900 ,BAD=370 , DBC=450∧AB=20m.

Find the length ofCD.

(5 marks) Total

12 marks

6. In an examination, Part A was attempted by 70 students, Part B by 50 and Part C by 42. 30 students attempted both parts A and B, 8 attempted both Parts B and C, 28 attempted both Parts A and C, and 3 attempted all three parts.

Answer the following questions using set notation.

(a) How many students attempted Part A but not Parts B and C? (2 marks)

(b) How many students attempted Part B but not Parts A and C? (2 marks)

(c) How many students attempted at least 2 parts? (2 marks)

(d) Draw a suitable Venn diagram to represent all the information given.

(5 marks)

Total 11 marks

7. The table below gives the distribution of marks of a set of Form 5 students.

MarksNumber of students

31 – 40 541 – 50 851 – 60 1061 – 70 1271 – 80 881 – 90 7

91 – 100 5

(a) Estimate the mean mark of the students (3 marks)

(b) Using a cumulative frequency curve, estimate the following marks:

(i) Median (3 marks)

(ii) Semi-Interquartile Range (3 marks)

(c) Calculate the probability that a student chosen at random had a mark of less than 80. (2 marks)

Credit will be given for drawing appropriate lines on your graph to show how the estimates were obtained. Total 11 marks

8. Investigation/Combination (Form 4 paper)

SECTION II

Answer ANY TWO questions in this section.

8. (a) Given that M =

(i) Show that M is a non-singular matrix. (2 marks)

(ii) Write down the inverse of M. (3 marks)

(iii) Write down the 2x2 matrix which is equal to the product MM-1. (1 mark)

(iv) Pre-multiply both sides of the following matrix equation by M-1.

=

Hence solve for x and y. (4 marks)

(b) A bag contains 2 white marbles and 3 blue marbles.

(i) A marble is selected at random from the bag. Calculate the probability that the marble is white. (2 marks)

(ii) If the white marble is NOT replaced in the bag, calculate the probability that the second marble selected from the bag will be blue. (3 marks)

Total 15 marks

8.

(10 marks)

1. (a) A = (3 40 5 ), B = (1 4

x 0), C = (3 y0 4 )

(i) Evaluate the determinant of A.(ii) If |A| = |B|, find x.(iii) Find (a) AC (b) CA(iv) Given that AC = CA, calculate the value of y.

(b) Given that −2 x+3 y=8

4 x+5 y=6

(i) express the above simultaneous equations in the form AX=B

(ii) hence, solve the simultaneous equations for x and y .

The ratio of the prices of two pens of different quality is 3:7. The total cost for 30 of the cheaper pens and 10 of the more expensive pens is $1,568. Given that p dollars represent the cost of one of the cheaper pens, determine:

(i) an algebraic expression in p for the cost of ONE of the more expensive pens (1 marks)

(ii) the value of p (3 marks)

(iii) the cost of ONE of the more expensive pens. (2 marks)