Test 5
1. (a)
Encik Syukri pays RM𝑥 income tax every month for his tax assessment in the year 2015. Table 1
shows the sum and the sum of squares of 𝑥.
i. Find the standard deviation of his income tax.
ii. Starting from the year 2016, Encik Syukri needs to pay an additional of RM2 500 every month because of the
improvement performance of his business. Find the mean and the standard deviation of his income tax for his tax
assessment in the year 2016.
(b) A set of data consists of 2, 3, 6 and 9.
i. Determine the mean and the standard deviation of the data.
ii. Two numbers, 𝛼 and 𝛽, are to be added to this set of data, such that the mean is
increased by 1 and the variance is increased by 2.5. Find the value of 𝛼 and 𝛽.
2. (a) Diagram 1 shows the curve 𝑦 = 𝑥 2 − 6𝑥 + 10 passing through the point P
and intersects the curve 𝑦 = 5𝑥 2 at point Q. The line PR is perpendicular
to the x-axis. The curve 𝑦 = 𝑥 2 − 6𝑥 + 10 has a minimum point at P.
i. Find the coordinates of P and Q.
ii. Calculate the area of shaded region.
iii. Calculate the volume generated, in terms of 𝜋, when the region bounded by the curve
𝑦 = 5𝑥 2 , the x-axis and the straight line 𝑥 = 1 is revolved through 360° about the x-axis.
(b) Given that 𝑓(𝑥) = (𝑥 − 5)√2𝑥 + 5 and 𝑓 ′ (𝑥) =
𝑘𝑥
√2𝑥+5
Diagram 1
where k is a constant. Find the value of k.
(c) Syed threw a ball such that the height, s m, if the ball from the ground at time t seconds is given by the equation
𝑠 = −4.9𝑡 2 + 18𝑡 + 1.5. Determine whether the ball could reach a
height of 15m from the ground. Justify your answer.
3. (a) Diagram 2 shows the location of a wrecked ship R at the ocean
using coordinate system with respect to a control tower O. Given the
2
equation of straight line OR is 𝑦 = 𝑥 and is perpendicular to straight
3
line PR. Find
i. The equation of straight line PR,
ii. The coordinates of the wrecked ship R,
Diagram 2
iii. the area in units2, of the triangle bounded by the control tower O,
the wrecked ship R and the control tower P.
iv. As a safety precaution, floating barriers are set 150 units around the wrecked ship R. Find the
equation of the floating barriers.
(b) Diagram 3 shows part of the graph of a straight line obtained by plotting 𝑦√𝑥 against x for the
positive values of variable x and y.
i. Find y in terms of x.
ii. Given point C (9, −8) lies on the graph of 𝑦√𝑥 against x, find the value of y corresponding
to the point C.
4. (a) The third, fourth and fifth term of geometric progression are 𝑥 2 + 1, 5𝑥 and 20 where 𝑥 > 0. Find
i. The value of x
ii. The eighth term of the geometric progression.
1
(b) The first three terms of an arithmetic progression are 15, 13 and 12. Find
2
i. The first negative term
ii. The sum of all positive terms.
(c) Find the value of
i. lim
𝑥 2 −9
𝑥→3 𝑥−3
ii. lim
1
𝑛→1 𝑛+7
iii. lim
5+3𝑟
𝑟→∞ 2+5𝑟
5. (a) Diagram 4 shows the graph of a quadratic function 𝑓(𝑥) = 𝑥 2 − 𝑥 + 𝑛𝑥 + 6, where n is a constant. Find
i. The roots of 𝑓(𝑥)
ii. The value of n
(b) The number of computer chips produced by a machine is given by 𝑛 = 6(5𝑡 ) + (5𝑡+1 ) + 2(5𝑡−1 ), where t is the operation
time, in hours, of the machine. Determine the time used by the machine to produce 7125 computer chips.
(c) Find the dimensions of a rectangle with perimeter 1000 meters so that the area of the rectangle is a maximum.
6. (a) A particle moves along a straight line and passes through a fixed point O. Its velocity, v ms-1, is
given by 𝑣 = 𝑡2 − 7𝑡 + 𝑝, where t is the time, in seconds, after passing through O. Diagram 5 shows
the velocity-time graph of the motion of the particle. It is given that the curve 𝑣 = 𝑡2 − 7𝑡 + 𝑝 has a
minimum value of q.
i. Find the initial velocity,
ii. Find the value of q and explain the meaning of the value of q,
iii. Find the difference in acceleration, in ms-2, of the particle at t = 2 and t = 3,
iv. Calculate the total distance, in m, travelled by the particle in the first 5 seconds.
(b) The National Zoo is having a special promotion on one Sunday. A school plans to organize a trip to the National Zoo. The
trip was participated by x boys and y girls. The number of students participating the trip is based on the following
constraints:
I
The total number of the participants in the trip is not more than 80.
II
The number of girls is more than the number of boys by at most 20.
III
The number of boys is not more than three times the number of girls.
i. Write three inequalities, other than 𝑥 ≥ 0 and 𝑦 ≥ 0, which satisfies all above constraints.
ii. Using a scale of 2 cm to 10 units on both axes, construct and shade the region R which satisfy all the constraints above.
iii. By using your graph in 6(b)(ii), find
1. the maximum number of boy participants if the number of girl participants is 15,
2. the maximum cost for the trip if the cost for a boy is RM30 and the cost for a girl is RM25.
7. (a) Diagram 6 shows a trapezium ABCD. Vector ⃗⃗⃗⃗⃗
𝐴𝐷 is parallel to vector ⃗⃗⃗⃗⃗
𝐵𝐶 . It is given that
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
𝐴𝐵 = 𝑥̰ , 𝐴𝐷 = 𝑦̰ and BF:FD = 2 : 3.
i. Express the following vectors in terms of 𝑥̰ and 𝑦̰.
⃗⃗⃗⃗⃗
1. ⃗⃗⃗⃗⃗⃗
𝐵𝐷
2. 𝐴𝐹
⃗⃗⃗⃗⃗ = 𝑚𝐴𝐶
⃗⃗⃗⃗⃗ and 5𝐵𝐶
⃗⃗⃗⃗⃗ = 𝑛 𝐴𝐷
⃗⃗⃗⃗⃗ . Express vector ⃗⃗⃗⃗⃗
ii. Given that 𝐴𝐹
𝐷𝐶 in terms of
1. 𝑚, 𝑥̰ and 𝑦̰
2. n, 𝑥̰ and 𝑦̰
iii. Hence, find the values of m and n.
(b) The blood pH value could be measured by Henderson-Hasselbalch equation, which is 𝑝𝐻 = 6.1 + log 𝐵 − log 𝐶 . At
which, the B represents bicarbonate concentration and C represents concentration of carbonic acid in the blood. If the
bicarbonate concentration in the blood at pH 7.2 is 25 mol per liter, find the concentration of carbonic acid in the blood.
8. (a) Given that function 𝑓(𝑥) = 𝑥 − 3 and 𝑔(𝑥) = √𝑥 , 𝑥 ≥ 0.
i. Find the value of k such that 𝑓𝑔(𝑘) = 6.
ii. Given function h maps 𝑥 → √𝑥 − 3 , 𝑥 ≥ 3. Express h in terms of f and 𝑔 .
(b) A gardener has the task of digging an area of 800m2. He digs an area of 10m2 on the first day.
On the successive day, he digs an area of 1.2 times the area that he dug on the previous day until
the day when the task is completed. Find the number of days needed to complete the task.
(c) Diagram 7 shows a major segment of a circular manhole cover with center O and radius of
30cm. Given that the cover is hinged at chord AC which is 34cm in length. Use 𝜋 = 3.142, find
i. The length, in cm, of the arc ABC,
ii. The surface area, in cm2, of the manhole cover.
9. Diagram 8 shows a quadrilateral ABCD such that ∠ABC is acute.
(a) Calculate (i) ∠𝐴𝐵𝐶 (ii) ∠𝐴𝐷𝐶 (iii) the area, in cm2, of quadrilateral ABCD.
(b) A triangle 𝐴′𝐵′𝐶′ has the same measurements as those given for triangle ABC i.e.
𝐴′𝐶′ = 12.3cm, 𝐶′𝐵′ = 9.5cm, and ∠𝐵′𝐴′𝐶′ = 40.5°, but which is different in shape to
triangle ABC.
i. Sketch the triangle 𝐴′𝐵′𝐶′
ii. State the size of ∠𝐴′ 𝐵′ 𝐶 ′
10. (a) The sum of first four terms of a geometric progression is 32
i. the first term of the progression
(c) Solve the equation
= 80.
25
√5𝑥 2
81
3
and its common ratio is . Find
ii. the sum to infinity of the progression
(b) Given log 𝑥 3 = 𝑝 and log 𝑥 5 = 𝑞, express log √𝑥
(3𝑥+1 )(5𝑥 )
16
in terms of p and q.
5
Answer
1ai) sd=40 ii) mean=5000, sd=40
3
2
7ai)1)−𝑥̰ + 𝑦̰ 2) 𝑥̰ + 𝑦̰
5
5
bi) mean=5, sd=2.74 ii)5,11
3
2
1
ii1) 𝑥̰ + ( − 1)𝑦̰
2ai)P(3,1), Q(1,5) aii) 6
iii)5𝜋 b)3
5𝑚
5𝑚
3
𝑛
2) 𝑥̰ + ( − 1)𝑦̰
c)can reach, discriminant>0 or reach at
5
t=2.62,1.05
iii)m=3/5 ; n=2
3
b) 1.986 mol/liter
3ai)𝑦 = − 𝑥 + 26 ii)R(12,8) iii)52
2
2
2
8ai) 81 ii) h(x)=gf(x) b)n=16
iv) 𝑥 + 𝑦 – 24𝑥 – 16𝑦 – 22292 = 0
ci) 152.37 ii)2705.8
10
8
bi) 𝑦 = − 2√𝑥 ii)−


'
3
√𝑥
9ai) 57.23 or 57 14
4ai) 2 ii)160 bi)-1.5 ii)82.5
 '
ii) 106 4
ci) 6 ii)1/8 iii)3/5
2
5ai) 2,3 ii)-4 b)4 c)250 x 250
iii) 82.38 cm
1
6ai)10 ii)min q=-9/4 iii)2 iv)13
6
bi)𝑥 + 𝑦 ≤ 80 ; 𝑦 − 𝑥 ≤ 20 ; 𝑥 ≤ 3𝑦
c1) xmax = 45 c2)max pt(60,20) ; max cost=RM2300
bi)
C'
A'
B
B'


bii) 122.77 or 122 46
10ai) 15 ii)37.5
b) 4 + 𝑞 − 8𝑝
c) 1.212
'
1. (a) Solve the equation (3𝑥+1 )(5𝑥 ) = 80.
(b) Given log 9 2𝑦 = 1 − log 3 𝑥, express y in terms of x.
√5𝑥 2
(c) Given log 𝑥 3 = 𝑝 and log 𝑥 5 = 𝑞, express log √𝑥 81 in terms of p and q.
2. (a) The first three terms of a progression are 2, x, 98. Find the value of x if the progression is
i. an arithmetic progression
ii. a geometric progression
16
3
(b) The sum of first four terms of a geometric progression is 32 25 and its common ratio is 5 . Find
i. the first term of the progression
ii. the sum to infinity of the progression
(c) Find the sum of the positive terms of the arithmetic progression 62, 58, 54…
3. (a) The gradient function of a curve is 4𝑥 − 5 and the curve passes through the point (−2, 4). Find the
equation of the curve.
(b) The straight line 8𝑥 + 2𝑦 − 15 = 0 intersects the coordinate axes at A and B. Find the equation of
the straight line that is perpendicular to AB and passes through the midpoint AB.
12𝑥
(c) Given 𝑦 = 𝑥−12 and x is decreasing at the rate of 2 units per second. Find the rate of change of y
when x = 4.
4. (a) Find the turning points of the graph of function 𝑦 = 3𝑥 2 − 𝑥 3 . Determine the nature of each of the
turning points.
(b) Sketch the graph of the function.
(c) The area bounded by the curve and the x-axis is rotated through 360° about x-axis. Find the volume,
in terms of 𝜋, of the solid generated.
Answer
9
1a) 1.212 b)𝑦 = 2𝑥 2 c)4 + 𝑞 − 8𝑝
2ai)x=50 ii)x= +- 14
bi) 15 ii)37.5 c)512
3a) 𝑦 = 2𝑥 2 − 5𝑥 − 14
b) 64𝑦 = 16𝑥 + 225 c)+4.5unit/s
4a) min pt (0,0) ; max pt (2,4)
c) 20.83𝜋