**SPM Mathematics (Model Test Paper)**

**Section**

*A* (52 marks)

Answer

**all**questions in this section.**1.**The Venn diagram in the answer space shows sets

*E*,

*F*and

*G*such that the universal set, $\xi =E\cup F\cup G.$

On the diagrams in the answer space, shade

$\begin{array}{l}(a)\text{}E\text{}\text{'}\cap G\text{'}\\ (b)\text{}\left(E\cap F\right)\text{'}\text{}\cup G\text{'}\end{array}$

[3 marks]

Answer*:*

**2.**Calculate the value of

**and of**

*x***that satisfy the following simultaneous linear equations:**

*y**$\begin{array}{l}x+6y=12\\ \frac{2}{3}x+2y=6\end{array}$*

[4 marks]

**3.**

Diagram 1

Diagram 1 shows a square

*ABCD*and a right-angled triangle*PQR*with equal area.Based on the information, find the value of

*x*.Hence, find the perimeter, in cm, of the square

*ABCD*. [4 marks]

**4 (a)**For each of the following statements, determine whether the statement is true or false.

(i) 35 is multiple of 3 and 5

(ii) 7 is a factor of 42 or 16 is a multiple of 6.

(b) Write down two implications based on the following sentence:

*p*

^{3 }= –8 if and only if

*p*= –2

(c) Make a general conclusion by induction for a list of numbers 2, 11, 26, 47, ……. which follows the following pattern.

2 = 3(1)

^{2}– 111 = 3(2)

^{2}– 126 = 3(3)

^{2}– 147 = 3(4)

^{2}– 1……………

[6 marks]

[6 marks]

**5.**Diagram 2 in the answer space shows a right prism. Trapezium

*ABGF*is the uniform cross section of the prism.

(a) On Diagram 2, mark the angle between the plane

*ADE*and the plane*ADCB*.(b) Calculate the angle between the plane

*ADE*and the plane*ADCB*.*[3 marks]*

Answer:

(

*a*)Diagram 2

**6.**Diagram 3 shows a right cylinder with a diameter of (

*y*+ 4 ) cm.

Given the volume of the cylinder is 269.5 cm

^{3}and by using find the value of it’s radius. [4 marks]

**7.**Diagram 4 shows a straight line

*ST*and a straight line

*PQ*drawn on a Cartesian plane.

*ST*is parallel to

*PQ*. Given that equation of the straight line

*ST*is 2

*y*= 8

*x*+ 3.

Diagram 4

Find,

(a) the equation of the straight line

*PQ*,(b) the

*x*-intercept of the straight line*PQ*. [ 5 marks ]

**8.**Diagram 5, shows a quadrant

*KLM*with centre

*M*and sector

*JMN*with centre

*J*.

Diagram 5

Using $\pi =\frac{22}{7}$ , calculate

(a) the perimeter, in cm, of the whole diagram,

(b) the area, in cm

^{2}, of the shaded region. [6 marks]

**9.**Diagram 6 shows the speed-time graph for the movement of two particles,

*J*and

*K*, for a period of

*t*s. The graph

*ABCD*represents the movement of

*J*and the graph

*AE*represents the movement of

*K*. Both particles start at the same point and move along the same route.

Diagram 6

(a) State the uniform speed, in

*ms*^{-1}, of particle*J*.(b) Calculate the rate of change of speed, in

*ms*^{-2}, of particle*J*for the first 13 s.(c) At

*t*s, the difference between the distance travelled by*J*and*K*is 169 m. Calculate the value of*t*.[ 6 marks ]

**10.**Diagram 7.1 shows three cards labelled with letters in bag A and three numbered cards in bag B.

Diagram 7.1

A card is picked at random from bag A and then a card is picked at random from bag B.

(a) Diagram 7.2 in the answer space shows the incomplete possible outcomes of the event. Complete the possible outcomes in Diagram 7.2.

(b) Using the complete possible outcomes in 10(a), find the probability that

(i) a card labelled

*Q*and the card with odd number are picked,

(ii) a card labelled

*P*or the card with a number which is multiple of 3 are picked.

*[5 marks]*

Answer:

(a)

Diagram 7.2

**11.**It is given that matrix

*M*is a 2 × 2 matrix such that

$M\left(\begin{array}{l}-2\text{}1\\ 1\text{}3\end{array}\right)=\left(\begin{array}{l}\text{10}\\ \text{01}\end{array}\right)$

(a) Find matrix

*M*.(b) Write the following simultaneous linear equations as matrix equations:

–2

*x*+*y*= 10*x*+ 3*y*= 9Hence, using matrix method, calculate the value of

*x*and of*y*. [6 marks]