1.
A laboratory test was carried out on a patient’s intestine. Table 1 shows the number of bacteria found in the patient’s intestine.
The bacteria continue to multiple themselves in the same number
Time (minute)
0
20
40
60
pattern for every 20 minutes. Find
Number of bacteria M 5 250 10 500 21 000 42 000
a. The number of bacteria M in the patient’s intestine after 3
Number of bacteria N
5
15
45
135
hours. [𝑇10 = 5250(2)9 = 2 688 000]
b. The time, in minute, when the number of bacteria N exceeds
the number of bacteria M.
[5(3)𝑛−1 > 5250(2)𝑛−1 , n=19 , time = 380minutes]
2.
Diagram 1 shows parts of the curve 𝑦 = (3𝑥+1)2 , which passes through 𝑆(−1,2).
8
a.
b.
Find the equation of the normal to the curve at point S. [6y+x-11=0]
A region is bonded by the curve, the x-axis and the straight line 𝑥 = −2 and 𝑥 = −3.
i. Find the area of the region. [1/5]
ii. The region is revolved through The region is revolved through 360° about the x-axis. Find the
volume generated, in terms of 𝜋. [0.043𝜋]
3.
A square has sides of 6cm. The midpoints of its sides are joined to form an inscribed square and this
process is continued as shown in Diagram 2. Find
a. The area of third square. [9cm2]
b. The sum of the perimeter of squares if this process is continued infinitely. [81.94]
4.
Given that 𝑎̰ = 2𝑖̰ − 3𝑗̰, 𝑏̰ = 𝑖̰ + 2𝑗̰ and 𝑐̰ = 3𝑖̰ − 𝑗̰.
a. Find the relationship between m and n such that
i. 𝑚𝑎̰ − 𝑛𝑏̰ is parallel to 𝑐̰ . [m=-n]
ii. 𝑚𝑎̰ − 𝑛𝑏̰ is perpendicular to the x-axis. [2m-n=0]
⃗⃗⃗⃗⃗ = 2𝑎̰ + 3𝑐̰ and 𝑂𝑄
⃗⃗⃗⃗⃗⃗ = 𝑏̰ , find
b. Given 𝑂𝑃
i. The coordinates of P, [P(13, -9)]
1
ii. The unit vector in the direction of ⃗⃗⃗⃗⃗
𝑃𝑄 . [
(−12𝑖̰ + 11𝑗̰)]
√265
5.
Diagram 3 shows part of the curve 𝑦 = 𝑥 3 − 𝑥. The curve passes through the point
𝑃(2,6).
a. Find the equation of tangent to the curve at point P. [y=11x-16]
b. Given the tangent to the curve at point Q is parallel to the tangent at P, find the coordinates
of Q. [Q(-2,-6)]
c. The tangent at P meets the normal of Q at the point A. The tangent at Q meets the normal
of P at the point B. State the shape of the figure APBQ. [rectangle]
6.
Diagram 4 shows OAB is a sector of a circle with centre O, radius 5cm and the length of arc AB
is 6.5cm. OPQ is another sector with the same centre O, radius 6cm. It is given that OP
intersects arc AB at M where the ratio of the length of arc AM to the length of arc MB is 2 : 1
and MN is perpendicular to OAQ, Calculate
a. ∠𝐴𝑂𝐵 and ∠𝑃𝑂𝑄 in radians, [1.3 rad, 0.8667rad]
b. The perimeter, in cm, of the shaded region, [12.77cm]
c. The area, in cm2, of the shaded region. [9.432]
7.
Diagram 5 shows part of the curve 𝑦 2 = 𝑥 − 4. The tangent to the curve at point 𝐴(5,1)
intersects the x-axis at point B. Find
a. The equation of the straight line AB, [2y=x-3]
b. The area of the shaded region, [1/3]
c. The volume generated, in terms of 𝜋, when the area bounded by the curve and the straight
line 𝑥 = 6 is revolved 180° about the x-axis. [2𝜋]
8.
Solution by scale drawing is not accepted. Diagram 6 shows a parallelogram ABCD. The
equation of BC is 2𝑦 = 𝑥 + 10 and X is the point on BC such that AX is perpendicular to BC
and BC = 5BX. Find
a. The equation of AX, [y=-2x+18]
b. The coordinates of X and C, [X(5.2,7.6) ; C(18,14)]
c. The area of the parallelogram ABCD. [112unit2]
9.
A particle moves along a straight line and passes through a fixed point O. Its velocity, v ms-1, is given by 𝑣 = 3𝑘𝑡 2 − 2ℎ𝑡, where
h and k are constants and t is the time, in seconds, after passing through O. When t = 3s, the particle stops instantaneously 1m on
the left of O. Assuming motion to the right is positive,
a. Find the value of h and k, [h=1/3 ; k=2/27]
b. Find the time when the velocity of the particle is minimum, [t=1.5s]
c. Sketch the displacement-time graph of the motion of the particle for 0 ≤ 𝑡 ≤ 6.
10. A straight path is made by laying paving slabs end to end. The slabs are laid in order of size and their
lengths are in arithmetic progression. Given that the shortest slab is 0.79 metres long, the longest slab is
1.24 meters long and the difference in lengths of adjoining slabs is 0.025 metres.
a. Calculate the number of slabs used, [n=19]
b. Calculate the length of the path, [19.285m]
c. The costs of the slabs are also in arithmetic progression and vary from RM2.16 for the smallest to RM3.24 for the largest.
Find the total cost of the slabs in the path. [RM51.30]
11. Diagram 7 shows part of the curve 𝑦 2 = 4𝑥. The point P is on the x-axis and the point Q is
on the curve. PQ is parallel to the y-axis and the length is p unit.
a. Given that R is the point (2,0). Express the area, A of the triangle PQR in terms of p
and hence show that
b.
𝑑𝐴
𝑑𝑝
=
3𝑝2
8
− 1. [𝐴 =
𝑝3
8
− 𝑝]
The point P moves along the x-axis and the point Q moves along the curve in such a
way that remains PQ parallel to the y-axis and p increases at the rate of 0.2 units per
second. Find
i. The rate at which A is increasing at the instant when p = 6 unit, [2.5]
ii. An expression for the approximate value of A when 𝑝 = (6 + 𝑘) units, where k is
small. [12.5k]
12. The population of the villages at the beginning of the year 2000 was 240. The population
increased so that, after a period of n years, the new population was 240(1.06)n. Find
a. The population of the villages at the beginning of 2020, [770]
b. The year in which the population of the villages first reached 2500, [2040]
c. The value of q for which log10 𝑞 = 1 + log10 2 − 2 log10 5. [q=4/5]
13. Diagram 8 shows a regular hexagon, ABCDEFG with centre G.
a. Express
i. ⃗⃗⃗⃗⃗
𝑂𝐺 in terms of 𝑎̰ ,𝑏̰ and 𝑐̰ . [a-b+c]
ii. ⃗⃗⃗⃗⃗
𝐶𝐷 in terms of 𝑎̰ , 𝑏̰ and 𝑐̰ . [a-2b+c]
b. The position vectors of points P, Q and R relative to an origin O are 𝑝̰ + 𝑞̰ ,
2𝑝̰ + 3𝑞̰ and 4𝑝̰ − 𝑞̰ respectively.
i. Given that PQRS is a parallelogram, find, in terms of 𝑝̰ and 𝑞̰, the position
vector of S. [3p-3q]
ii. Given that T is a point where position vector relative to O is 5𝑝̰, shows that
OPTR is a parallelogram. [PT // OR and |PT|=|OR| ]
14. a. Find the coordinates of the intersections of the curve 𝑦 = (1 + 𝑥)(1 − 𝑥) and the x-axis. [(1,0);(-1,0)]
b. Calculate the area of the region bounded by the curve and the x-axis. [4/3 units2]
c. Calculate the volume generated, in terms of 𝜋, when this region is bounded by the curve is rotated through 360° about the xaxis. [14𝜋/15]
15. Diagram 9 shows the quadrilateral OABC. The coordinates of A is (4, 8) and the length of OA is √80
units.
a. AB is perpendicular to OA and B lies on the y-axis. Find the equation of AB and the coordinates
of B. [2y+x=20 ; B(0,10)]
b. C is the intersection point of the line that is parallel to 𝑦 + 3𝑥 = 5 and bisector of AB. Calculate
i. The coordinates of C, [C(-1,3)]
ii. The area of quadrilateral OABC. [25units2]
16. A particle moves along a straight line, where t is the time, in seconds, sfter passing through O. Its
velocity, v ms-1, is given by 𝑣 = 𝑡 2 − 6𝑡 + 5. It is given that the particle stops instantaneously at A
and at B. Find
a. The distance AB, [32/3]
b. The total distance travelled by the particle in the first 5 seconds after passing through O. [13cm]
c. Given that C is the point at which the particle has zero acceleration, determine whether C is nearer to O or to B. [OC=3;
BC=16/3 ; C nearer]