1. A laboratory test was carried out on a patientβs intestine. Table 1 shows the number of bacteria found in the patientβs 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 patientβs intestine after 3 Number of bacteria N 5 15 45 135 hours. [π10 = 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)πβ1 > 5250(2)πβ1 , n=19 , time = 380minutes] 2. Diagram 1 shows parts of the curve π¦ = (3π₯+1)2 , which passes through π(β1,2). 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 π₯ = β2 and π₯ = β3. 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 πΜ° = 2πΜ° β 3πΜ°, πΜ° = πΜ° + 2πΜ° and πΜ° = 3πΜ° β πΜ°. a. Find the relationship between m and n such that i. ππΜ° β ππΜ° is parallel to πΜ° . [m=-n] ii. ππΜ° β ππΜ° is perpendicular to the x-axis. [2m-n=0] βββββ = 2πΜ° + 3πΜ° and ππ ββββββ = πΜ° , find b. Given ππ i. The coordinates of P, [P(13, -9)] 1 ii. The unit vector in the direction of βββββ ππ . [ (β12πΜ° + 11πΜ°)] β265 5. Diagram 3 shows part of the curve π¦ = π₯ 3 β π₯. The curve passes through the point π(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. β π΄ππ΅ and β πππ 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 π¦ 2 = π₯ β 4. The tangent to the curve at point π΄(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 π₯ = 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 2π¦ = π₯ + 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 π£ = 3ππ‘ 2 β 2βπ‘, 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 β€ π‘ β€ 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 π¦ 2 = 4π₯. 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. ππ΄ ππ = 3π2 8 β 1. [π΄ = π3 8 β π] 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 π = (6 + π) 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 π = 1 + log10 2 β 2 log10 5. [q=4/5] 13. Diagram 8 shows a regular hexagon, ABCDEFG with centre G. a. Express i. βββββ ππΊ in terms of πΜ° ,πΜ° and πΜ° . [a-b+c] ii. βββββ πΆπ· in terms of πΜ° , πΜ° and πΜ° . [a-2b+c] b. The position vectors of points P, Q and R relative to an origin O are πΜ° + πΜ° , 2πΜ° + 3πΜ° and 4πΜ° β πΜ° respectively. i. Given that PQRS is a parallelogram, find, in terms of πΜ° and πΜ°, the position vector of S. [3p-3q] ii. Given that T is a point where position vector relative to O is 5πΜ°, shows that OPTR is a parallelogram. [PT // OR and |PT|=|OR| ] 14. a. Find the coordinates of the intersections of the curve π¦ = (1 + π₯)(1 β π₯) 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 β80 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 π¦ + 3π₯ = 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 π£ = π‘ 2 β 6π‘ + 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]