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SPM Practice 2 (Linear Law) – Question 1

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Question 1 (10 marks):
Use a graph to answer this question.
Table 1 shows the values of two variables, x and y, obtained from an experiment. A straight line will be obtained when a graph of y 2 x  against  1 x is plotted.


(a) Based on Table 1, construct a table for the values of 1 x  and  y 2 x .  

( b ) Plot  y 2 x  against  1 x , using a scale of 2 cm to 0.1 unit on the  1 x -axis   and 2cm to 2 units on the  y 2 x -axis.   Hence, draw the line of best fit.

(c) Using the graph in 1(b)
(i) find the value of y when x = 2.7,
(ii) express y in terms of x.


Solution:
(a)


(b)


(c)(i)
When x=2.7,  1 x =0.37 From graph, y 2 x =5.2 y 2 2.7 =5.2 y=3.75



(c)(ii)
Form graph, y-intercept, c = –4 gradient, m= 16−( −4 ) 0.8−0 =25 Y=mX+c y 2 x =25( 1 x )−4 y= 25−4x


SPM Practice 3 (Linear Law) – Question 6

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Question 6
The table below shows the values of two variables, x and y, obtained from an experiment. The variables x and y are related by the equation a y = b x + 1 , where k and p are constants.


(a) Based on the table above, construct a table for the values of 1 x and 1 y . Plot 1 y against 1 x , using a scale of  2 cm to 0.1 unit on the 1 x - axis and  2 cm to 0.2 unit on the 1 y - axis. Hence, draw the line of best fit.
(b) Use the graph from  (b)  to find the value of
(i)  a,
(ii)  b.


Solution
Step 1 : Construct a table consisting X and Y.




Step 2 : Plot a graph of Y against X, using the scale given and draw a line of best fit

Steps to draw line of best fit - Click here




Step 3 : Calculate the gradient, m, and the Y-intercept, c, from the graph




Step 4 : Rewrite the original equation given and reduce it to linear form

Step 5 : Compare with the values of m and c obtained, find the values of the unknown required

SPM Practice 3 (Linear Law) – Question 5

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Question 5
The following table shows the corresponding values of two variables, x and y, that are related by the equation y = p k x , where p and k are constants.


(a) Plot log 10 y against x  .  Hence, draw the line of best fit
(b) Use your graph in (a) to find the values of p and k.


Solution
Step 1 : Construct a table consisting X and Y.



Step 2 : Plot a graph of Y against X, using the scale given and draw a line of best fit

For steps to draw line of best fit - Click here



Step 3 : Calculate the gradient, m, and the Y-intercept, c, from the graph


Step 4 : Rewrite the original equation given and reduce it to linear form


Step 5 : Compare with the values of m and c obtained, find the values of the unknown required

SPM Practice 2 (Linear Law) – Question 4

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Question 4
The table below shows the corresponding values of two variables, x and y, that are related by the equation y = q x + p q x , where p and q are constants.


One of the values of y is incorrectly recorded.
(a) Using scale of 2 cm to 5 units on the both axis, plot the graph of xy against x 2  .  Hence, draw the line of best fit

(b) Use your graph in (a) to answer the following questions:
(i) State the values of y which is incorrectly recorded and determine its actual value.
(ii) Find the value of p and of q.

Solution
Step 1 : Construct a table consisting X and Y.


Step 2 : Plot a graph of Y against X, using the scale given and draw a line of best fit


Steps to draw line of best fit - Click here

(b) (i) State the values of y which is incorrectly recorded and determine its actual value.


Step 3 : Calculate the gradient, m, and the Y-intercept, c, from the graph

Step 4 : Rewrite the original equation given and reduce it to linear form

Step 5 : Compare with the values of m and c obtained, find the values of the unknown required

SPM Practice (Long Questions)

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Question 9 (12 marks):
(a) Diagram 7.1 shows K (5, 1) drawn on a Cartesian plane.
Diagram  7.1

Transformation T is a translation  ( −3  4 )
Transformation P is a reflection in the line y = 2.
State the coordinates of the images of point K under each of the following transformations:
(i) T2,
(ii) TP.


(b) Diagram 7.2 shows two pentagons KLMNP and QRSTU, drawn on a Cartesian plane.

Diagram  7.2

(i) Pentagon QRSTU  is the image of pentagon KLMNP under the combined transformation WV.
Describe in full the transformation:
(a) V,
(b) W.

(ii) It is given that pentagon QRSTU represents a region with an area of 90 m2.
Calculate the area, in m2, of pentagon KLMNP.


Solution:
(a)



(i) TT = K(5, 1) → T1 → K’(2, 5) ) → T2 → K’’(–1, 9)
(ii) TP =  K(5, 1) → P → K’(5, 3) → T → K’’(2, 7)


(b)


(b)(i)(a)
V: 90o rotation in clockwise direction at point K(3, 7).

(b)(i)(b)
W: Enlargement at point (1, 6) with scale factor 3.

(b)(ii)
Area of QRSTU = (scale factor)2 × area of object
Area of QRSTU = 32 × Area of KLMNP
90 m2 = 9 × Area of KLMNP
Area of KLMNP = 90/9 = 10 m2


6.3 SPM Practice (Long Questions)

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Question 12 (5 marks):
Diagram 7 shows a distance-time graph for the journey of a car from Kuala Lumpur to Ipoh.


(a) State the duration of time, in minutes, the driver stopped and rest at Tapah.
(b) Calculate the speed, in kmh-1, of the car from Kuala Lumpur to Tapah.
(c) Calculate the average speed, in kmh-1, of the car for whole journey.


Solution:
(a)
1400 – 1345 = 15 minutes
The driver stopped and rest at Tapah for 15 minutes.


(b)
Speed= Distance Time Speed= 152 km 1 3 4  h ← 1345−1200 =1 h 45 mins Speed=86.86  kmh −1 Speed of the car from Kuala Lumpur to Tapah=86.86  kmh −1


(c)
Average speed= Total distance Time taken                   = 205 km 3 1 3  h ← 1520−1200 =3 h 20 mins Average speed=61.5  kmh −1 Average speed of the car for the whole journey=61.5  kmh −1


6.3 SPM Practice (Long Questions)

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Question 11 (6 marks):
Diagram 6 shows the distance-time graph of Ursula, Janet and Maria in a 100 m race.

Diagram 6



(a) Who won the race?

(b) During the race, Ursula slipped and fell over. After that, she continued her run.
State the duration, in seconds, before Ursula continued her run.

(c) During the race, Janet was injured and she stopped running.
State Janet's distance, in m, from the finishing line when she stopped running.

(d) Calculate the average speed, in ms-1, of Ursula.


Solution:
(a)
Maria won the race: 100 km in 16 seconds.

(b)
Duration before Ursula continued her run from slipping and falling over
= 18 seconds – 9 seconds
= 9 seconds

(c)
Janet's distance from the finishing line when she stopped running
= 100 m – 70 m
= 30 m

(d)
Average speed= Distance Time                   = 100 m 20 s                   =5  ms −1 Average speed of Ursula=5  ms −1

7.5 SPM Practice (Long Questions)

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Question 10 (5 marks):
The table below shows some members of Red Crescent Society and St John Ambulance Society that are instructed to do an outdoor tasks at various places during the Flag Day.


Two members from the societies were dropped off random at those places.
(a) List all the possible outcomes of the event in this sample space.
You also may use the letters such as A for Amy and so on.

(b) By listing down all the possible outcomes of the event, find the probability that
(i) a boy and a girl were dropped off at a certain place.
(ii) both members that were dropped off at a certain place are from the same society.


Solution:
(a)
S = {(A, J), (A, N), (A, F), (A, C), (A, M), (J, A), (J, N), (J, F), (J, C), (J, M), (N, A), (N, J), (N, F), (N, C), (N, M), (F, A), (F, J), (F, N), (F, C), (F, M), (C, A), (C, J), (C, N), (C, F), (C, M),(M, A), (M, J), (M, N), (M, F), (M, C)}

(b)(i)
{(A, F), (A, C), (A, M), (J, F), (J, C), (J, M), (N, F), (N, C), (N, M), (F, A), (F, J), (F, N), (C, A), (C, J), (C, N), (M, A), (M, J), (M, N)}
Probability = 18 30 = 3 5

(b)(ii)
{(A, F), (A, C), (F, A), (F, C), (C, A), (C, F), (J, N), (J, M), (N, J), (N, M), (M, J), (M, N)}
Probability = 12 30 = 2 5

7.5 SPM Practice (Long Questions)

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Question 9 (6 marks):
Diagram 10.1 shows a disc with four equal sectors and a fixed pointer. Each sector is labelled with water heater, oven, television and iron respectively. Diagram 10.2 shows a box which contains three cash vouchers, RM10, RM20 and RM50.



A lucky customer in a supermarket is given a chance to spin the disc once and then draw a cash voucher from the box.
(a) List the sample space for the combination of prizes that can be won.

(b) By listing down all the possible outcomes of the event, find the probability that

(i) the customer wins a television or cash voucher worth RM50,

(ii) the customer does not wins the water heater and the cash voucher worth RM20.


Solution:
(a)
S = {(Water heater, RM10), (Water heater, RM20), (Water heater, RM50), (Oven, RM10), (Oven, RM20), (Oven, RM50), (Television, RM10), (Television, RM20), (Television, RM50), (Iron, RM10), (Iron, RM20), (Iron, RM50)}

(b)(i)
{( Water heater, RM50), (Oven, RM50), (Iron, RM50), (Television, RM10), (Television, RM20), (Television, RM50)}
Probability = 6 12 = 1 2

(b)(ii)
Probability =1−P( Water heater, RM20 ) =1− 1 12 = 11 12

3.4 SPM Practice (Long Questions)

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Question 9 (3 marks):
The Venn diagram in the answer space shows set P, set Q and set R such that the universal set ξ = P U Q U R.
On the diagram in the answer space, shade the set
(a) P’,
(b) (P ∩ Q) U R

Answer:



Solution:
(a)



(b)





Question 10 (3 marks):
(a) It is given that set E = {perfect square numbers} and set F = {9, 16, 25}.
Complete the Venn diagram in the answer space to show the relationship between set E and set F.

Answer:


(b) The Venn diagram in Diagram 1 shows the sets X, set Y and set Z.
The universal set, ξ = X U Y U Z.

Diagram 1

State the relationship represented by the shaded region between sets X, set Y and set Z.



Solution:
(a)


(b)


Hence, relationship represented by the shaded region between sets X, set Y and set Z are (X ∩ Y) ∪ Z.