# Mathematic Spm 2006

Topics: Analytic geometry, Cartesian coordinate system, Euclidean geometry Pages: 13 (1786 words) Published: April 15, 2011
SPM(U) 2006 : http://mathsmozac.blogspot.com Section A [52 marks] Answer all questions in this section. 1 The Venn diagram in the answer space shows sets P, Q and R such that the universal set, ξ = P ∪ Q ∪ R . On the diagram in the answer space, shade (a) the set P ∪ R , (b) the set (P ∩ R ) ∪ Q ' . [3 marks] Answer: (a) P Q R

(b)

P R

Q

2

Diagram 1 shows a solid cuboid. A cone is removed from this solid.

12 cm 10 cm 15 cm
DIAGRAM 1

The diameter of the base of the cone is 7 cm and the height of the cone is 9 cm. Calculate 22 ⎞ ⎛ the volume, in cm 3 , of the remaining solid. ⎜ Use π = [3 marks] ⎟ 7 ⎠ ⎝ Answer:

SAH@MOAZC2008

1

SPM(U) 2006 : http://mathsmozac.blogspot.com 3 (a) State whether each of the following statements is true or false. 3 64 = 4 (i) (ii) − 5 > −8 and 0 ⋅ 03 = 3 × 10 −1 . Write down two implications based on the following sentence. ∆ ABC is an equilateral triangle if and only if each of the interior angle of ∆ ABC is 60 o . Complete the premise in the following argument: Premise 1 : ………………………………………………… Premise 2 : 90 o ≤ x ≤ 180 o . Conclusion : sin x o is positive. [5 marks] Answer: (a) (i) ……………………………………………………………….. (ii) ……………………………………………………………….. Implication I : ……………………………………………………………………….. ……………………………………………………………………….. Implication II: ……………………………………………………………………….. ……………………………………………………………………….. Premise 1 : …………………………………………………………………………...

(b)

(c)

(b)

(c) 4

Diagram 2 shows a right prism. The base HJKL is a horizontal rectangle. The right angled triangle NHJ is the uniform cross section of the prism. M

N 8 cm L 6 cm H 12 cm
DIAGRAM 2

K

J

Identify and calculate the angle between the line KN and the plane HLMN. Answer :

[4 marks]

SAH@MOAZC2008

2

SPM(U) 2006 : http://mathsmozac.blogspot.com 5 Calculate the value of d and of e that satisfy the following simultaneous linear equations: [4 marks] 3d − 2e = 9

6d + e = −2

6

In diagram 3, O is the origin and PQRS is a trapezium. PS is parallel to QR. The straight line RS is parallel to the y-axis. The points Q and S lie on the x-axis. y P(− 1, 10 )

S Q O

x

R(4, − 11)
DIAGRAM 3

Find (a) (b)

the equation of the straight line QR, the x-intercept of the straight line QR. [5 marks]

(b)

SAH@MOAZC2008

3

SPM(U) 2006 : http://mathsmozac.blogspot.com 7 Solve the quadratic equation 2 x 2 + 3 x = 15 + 2 x . Answer : [4 marks]

8

Table 1 shows the number of teachers in a two-session school. Session Morning Afternoon Men 6 4 Number of teachers Women 10 8

TABLE 1

Two teachers from the school are chosen at random to attend an assembly of Teacher’s Day at the state level. Calculate the probability that both teachers chosen (a) are men, (b) are from the same session. [5 marks] Answer : (a)

(b)

SAH@MOAZC2008

4

SPM(U) 2006 : http://mathsmozac.blogspot.com 9 In diagram 4, QRS and UT are arcs of two circles, centres P and S respectively. R

Q T

120 o
P U

45 o
S

DIAGRAM 4

It is given that PUS is a straight line, PQ = 21 cm and US = 14 cm. 22 Using π = , calculate 7 (a) the area, in cm 2 , of the shaded region, (b) the perimeter, in cm, of the shaded region. [6 marks] Answer : (a)

(b)

SAH@MOAZC2008

5

SPM(U) 2006 : http://mathsmozac.blogspot.com 10 ⎛3 − 4⎞ It is given that matrix M = ⎜ ⎜1 2 ⎟ . ⎟ ⎝ ⎠ (a) (b) Find the inverse matrix of M. Write the following simultaneous linear equations as matrix equation: 3 x − 4 y = 13 x + 2y = 6 Hence, using matrices, calculate the value of x and of y. [6 marks] Answer : (a)

(b)

SAH@MOAZC2008

6

SPM(U) 2006 : http://mathsmozac.blogspot.com 11 Diagram 5 shows the speed-time graph for the movement of a particle for a period of 30 seconds. Speed ms −1 25

(

)

u

O

16

22
DIAGRAM 5

30

Time (s)

(a) (b) (c)

State the length of time, in s, for which the particle moves with uniform speed. Calculate the rate of change of speed, in m s −2...