NURTURE WORKSHOP

ADDITIONAL MATHEMATICS SPMCHECK LISTS

3472/2 (Paper 2)

Working and AnswersProvided by

………………………….Hj. Busrah Bin Md.SehSMK Jasin, Melaka.April 2014

Topics Page

1. Simultaneous Equations 2

2. Quadratic Function (QF) & Quadratic Equation (QE)

3

3. Progression (AP & GP) 5

4. Statistics 7

5. Application on Differentiation 9

6. Coordinates Geometry 10

7. Sketching The Trigonometric Function Graph 11

8. Linear Law 15

9. Circular Measures 17

10. Vectors 19

11. Application on Integration 21

12. Probability Distributions 23

13. Index Numbers 24

14. Solution Of Triangles 26

15. Motion Along a Straight Line 28

16. Linear Programming 30

1. Solve the following simultaneous equations

3x + y = 1 , 5x2 + y2 + 4xy – 5 = 0. SPM 2012 / (5 Marks)

2. Solve the simultaneous equations x + 2y = 3 and 2x2 – y2 = 2. Give your answers correct to three decimal places.

SMKJ 2013 / (6 Marks)

2. A quadratic equation x2 + 4(3x + k) = 0, where k is a constant has roots p and 2p.

a. Find the value of p and of k.b. Hence, form the quadratic equation which has the roots p – 1 and p + 1

SPM 2012 / (8 marks)

3. It is given that the quadratic function f(x) = 2x2 – 6x + 5

a. By using completing the square, express f(x) in the form of f(x) = a(x + p)2 + q

b. Sketch the graph for f(x) = 2x2 – 6x + 5 such that

c. Find the range of values of x for SMKJ 2013 / (8 marks)

8cm 6cm 4cm

4. Diagram shows the arrangement of cylinders having the same radius, 3 cm. The height of the first cylinder is 4 cm and the height of each subsequent cylinder increases by 2 cm.

(Volume of cylinder = )

a. Calculate the volume, in cm3, of the 17th cylinder, in term of b. Given the total volume of the first n cylinder is 1620 cm3, find the value of n.

SPM 2010 / (6 marks)

5. Table shows the marks obtained by 40 candidates in a test.Marks Number

ofcandidate

s

10 – 19

4

20 – 29

5

30 – 39

13

40 – 49

10

50 – 59

8

a. State the modal class.b. Without using an ogive, calculate

i. the median ii. the variance of the marks.

Iii. the interquartel range of the marks

(7 marks)

6. A curve has a gradient function kx – 6, where k is a constant. Given that the minimum point of

the curve is (3, – 5), find

i. the value of kii. the equation of the curve.

SPM 2012 / (6 marks)

7. Diagram shows the triangle AOB where 0 is the origin. Point C lies on the of straight line AB

a. Calculate the area, in unit2 , of triangle AOB

b. Given that AC : CB = 3 : 2, find the coordinates of C

c. A point P moves such that PA = 2PB. Find the equation of the locus of P

C

A(-3, 4)

B(6, -2)

0 x

y

8. a. Sketch the graph of

b. Hence, using the same axes, sketch a suitable straight line to find the number of

solution to the equation

State the number of solutions.SPM 2009 / (6 marks)

9. a. Sketch the graph of

b. Hence, using the same axes, sketch a suitable straight line to find the number of solution

to the equation State the number of solutions.

Melaka 2012 / (7 marks)

10. Table shows the values of two variables, x and y, obtained from an experiment. The variables x and y are related by the equation , where p and k are

constants.

a. Plot log y against (x + 1), using a scale of 2 cm to 1 unit on (x + 1)-axis and 2 cm to 0.2 unit on the log y-axis. Hence, draw the line of best fit.

b. Use the graph in (a) to find the value ofi. pii. k SPM 2006 / (10 marks)

x 1 2 3 4 5 6

y 4.0 5.7 8.7 13.2 20.0 28.8

11. In diagram, AOBC is a semicircle with centre 0 and radius 4 cm. APD is a sector of a circle with centre P and radius 6 cm. It is given that OC is perpendicular to AOB.

Calculatea. the value of θ, in radians.b. the perimeter, in cm, of the shaded regionc. the area, in cm2, of the shaded region

SPM 2011/ (10 marks)

A

C

D

P0B

θ

12. Diagram shows a parallelogram ABCD. Point P on the straight line AB and point Q lies on the straight line DC. The straight line AQ is extended to the point R such that AQ = 2QR. It is given that AP : PB = 3 : 1, DQ : QC = 3 : 1 , = 6u and = v

a. Express, in term of u and vi.

ii. Hence, show that the points P, C and R are collinear

b. It is given that u = 3 i and v = 2 i + 5 j.i. Express in term of i and j

ii. Find the unit vector in the direction of SPM 2011 / (10 marks)

RD Q

C

A P B

13. Diagram shows part of the curve y = f(x) which passes through Q(1, 3). The straight line PQ is parallel to the x-axis. The curve has a gradient function of 2x.

Finda. the equation of the curveb. the area of the shaded regionc. the volume of revolution, in term of π, when the shaded region is rotated

through 3600 about the y-axis.

SPM 2011 / (10 marks)

y

0x

Q(1, 3)

y = f(x)

P

14. a. In a survey carried out in a particular district, It is found that three out of five families own national car. If 10 families are chosen at random from the district, calculate the probability

that at least 8 families own a national car.

b. The diameters of limes from a farm have a normal distribution with mean 3.2 cm and standard deviation of 1.5 cm.

Calculatei. the probability that a lime chosen at random from this farm has a diameter

of more than 3.9 cm

ii. the value of d if 12% of the limes have diameters less than d cm

SPM 2012 / (10 marks)

15. Table shows the prices, price indices and percentage expenditure of four ingredients P, Q, R and S, used in the making of a kind of food.

Ingredient

Price (RM) per kg Price index in the year 2007 based on the

year 2005

Percentage expenditure

(%)2005 2007

P 4.00 5.00 x 16

Q 3.00 y 150 12

R 8.00 10.00 125 48

S z 3.00 120 24

a. Find the values of x, y and zb. Calculate the composite index for the cost of making the food in the year 2007

based on the year 2005.c. The cost of making a packet of the food in the year 2005 was RM 50.00.

Calculate the corresponding cost in the year 2007.d. The cost of all the ingredients increases by 15% from the ear 2007 to the year

2009. Find he composite index for the year 2009 based on the year 2005.

SPM 2011 (10 marks)

16. In Diagram shows a quadrilateral ABCD such that ABC is acute.

a. Calculate

i. < ABCii. < ADCiii. the area, in cm2, of quadrilateral ABCD

b. A triangle A’B’C’ has the same measurements as those given triangle ABC, that is, A’C’ = 12.3 cm, C’B’ = 9.5 cm and B’A’C’ = 40.50.

i. Sketch the triangle A’B’C’ii. State the size of A’B’C’

SPM 2004 / (10 marks)

40.50

D

A

C

9.8 cm

B

5.2 cm

12.3 cm

9.5 cm

17. A particle moves along a straight line from a fixed point P. Its velocity, V ms-1, is given

by V = 2t(6 – t), where t is time, in seconds, after leaving the point P.

Finda. the maximum velocity of the particleb. the distance traveled during the third second,c. the value of t when the particle passes the point P again.d. the time between leaving P and when the particle reverses its

direction of motion.SPM 2004 / (10 marks)

18. Sebuah pusat latihan menawarkan dua kursus, A dan B. Bilangan peserta (participants) kursus A ialah x dan bilangan peserta kursus B ialah y. Pengambilan peserta adalah berdasarkan kekangan berikut.

I The maximum number of participants is 80II The number of participants for course B is at least 10

III The number of participants for course B ia at most times the

number of participants for course A

a. Write three inequalities, other than x ≥ 0 and y ≥ 0, which satisfy all the above constraints.

b. Using a scale of 2 cm to 10 participants on both axes, construct and shade the region R which satisfies all the above constraints.

c. Using the graph, find

i. the minimum number of participants for course Aii. the maximum total fees collected per month if the monthly fees per

participant for course A is RM 300 and for course B is RM 400.

SPM 2010 / (10 marks)