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FINAL REVISION EXERCISE SPM 2105Nov. 24, 2015

1) (a) Prove that

tanθ sin 2 θsin 2 θ−tan θ

= tan 2θ.

(b) (i) Sketch the graph of y=|tan 2θ| for 0≤θ≤π .(ii) Hence, using the same axes, draw a suitable straight line to find the number of

solutions for the equation 2|tan 2θ|−2θ

π=6

for 0≤θ≤π . State the number of solutions.

[(b) no of solns = 4]

2) Diagram 1 shows a square with length x cm was cut into four squares shown at stage 2. Then every square was cut into another four squares for the subsequent squares.

Diagram 1

(a) Show that the sum of the perimeters of the squares at every stage forms a geometric progression.

(b) Given the sum of the perimeters of the squares cut at stage 10 is 10240 cm, find the value of x.

(c) Calculate the total numbers of squares cut from stage 5 to stage 10.

[(a) r = 2 (b) x = 5 (c) 349440]

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3) Sands are poured at the rate of 24 π cm3 s−1 to form a vertical cone shape as shown in

Diagram 2. The radius is r cm and the height is h cm.

Diagram 2

Given that r=3

4h

and the volume of cone is V=1

3πr2 h

.(a) (i) Express the volume of cone, V, in terms of r.

(ii) Write an expression for

dVdr .

(b) Hence, calculate(i) the small change in V when r increases from 9 cm to 9.03 cm,

(ii) the rate of change of r when h = 12 cm.

[(a)(i)

49

πr3

(ii)

43

πr2

(b)(i) 3 .24 π (ii)

29

cms−1

]

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4) In Diagram 3, O is the centre of a semicircle PQR with radius 14 cm and OST is a right angled triangle.

Diagram 3

The length of arc PQ is 28 cm and RS = 7 cm.

Find [Use π=3 . 142](a) the value of θ , in radian,

(b) the perimeter, in cm, of the shaded region X,

(c) the area, in cm2 of the shaded region Y.[(a) 1.142 rad (b) 105.45 cm (c) 106.87 cm2]

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5) (a) In a study conducted on a group of graduates, it is found that 65% of them succeeded in gaining employment after graduation.(i) If 2 graduates are chosen at random, find the probability that there are not

more than 10 graduates employed after graduation.

(ii) If the standard deviation of the employment of the graduates is 10.2, calculate the number of graduates who participated in this study.

(b) A school with 2000 students take part in a cross-country event. The cross-country event started at 0800 hours. The time taken for the students to finish the event is normally distributed with a mean of 40 minutes and a variance of 100 minutes2.(i) Find the probability of students who finished the event after 1 hour.

(ii) If 450 students finished the event in less than t minutes, find the value of t.[(a)(i)0.9576 (ii) 457 (b)(i)0.02280 (ii) 32.45]

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6) Diagram 4 shows an ogive that represents the distribution of the lengths of 80 siakap fish reared by a fish breeder.

Diagram 4

(a) Construct the frequency table using class intervals of the same size, based on the information obtained from the ogive.

(b) Based on the frequency table, calculate(i) the median,(ii) the mean,(iii) the varianceof the lengths of the siakap fish.

[(b)(i) 12.69 cm (ii)12.75 cm (iii) 26.19 cm2]