Penang_jitsin Trial Stpm 2012 Mathst Paper 2(Q&A)

Trial STPM Penang Jit Sin 2012 Mathematics Paper 2

Answer all questions.

1. Solve the differential equation , given that y = 2 when

x = . [4]

2. By using vector method, prove that the two diagonals of a rhombus are perpendicular to each other. [4]

3. WXYZ is a trapezium in which WX is parallel to ZY and ZW = WX. Given that XYZ is twice XZY. Prove that WXYZ is a cyclic quadrilateral. [7]

4. At noon, a ship A is 2 km to the north of ship B. The velocities for A and B are 12 kmh1 to the west and 15 kmh1 to the north-west respectively. When are the ships nearest to each other ? Find the shortest distance between the two ships. [8]

5. The rate at which a substance evaporates is k times the amount of substance which has not yet evaporated, where k is a constant. If the initial amount of substance was A and the amount which has evaporated at time t is x, write down a differential equation involving x, and solve it to give x in term of A, k and t. Sketch the graph

of x against t. Show that the time taken for half the substance to evaporate is . [8]

6. (a) Prove that, for all values of x,cos 3x + cos 5x + cos 7x = cos 5x (4cos2x 1).Hence, solve the equationcos 3x + cos 5x + cos 7x = 0 for 0 x . [8]

(b) Find R (>0) and (0 < < ) such that for all values of ,

.Hence, obtain the value of for which is maximum for 0 x . [6]

7. A clerk produces documents with an average of 0.3 error per page. Find the probability that a document consisting of 15 pages has(a) no error, [2](b) at least one error, [1](c) at least one error on each of five pages and none on the others. [3]

8. Analysis of the results of a certain group of students who had taken examination in both mathematics and physics produced the following information : 75% of the students passed in mathematics, 70% passed in physics and 40% failed in at least one of these subjects.(a) Find the percentage of students who passed in exactly one of the two subjects. [4](b) Of the students who passed in mathematics, find the percentage who also passed in physics.

[3]

9. A certain variety of flower seed is sold in packets containing 1000 seeds. The packet claims that 40% will bloom white and 60% red.If five seeds are planted, estimate the probability that

1

1


(a) exactly three will bloom white, [2](b) at least one will bloom white, [2]If 100 seeds are planted, use approximation to estimate the probability of obtaining between 30 and 45 white flowers. [3]

10. The continuous random variable X has the probability density function f(x) given by

f(x) =

where a and b are constants.(a) If the mean of X is 2.25, find the values of a and b. [3]

(b) Determine the value of . [3]

(c) Find the value of k such that P(X < k) = 0.375. State, with a reason, whether the median m of X is greater than or less than k. [4]

11. The discrete random variable X denotes the number of ‘sixes’ shown when two fair dice are thrown.(a) Tabulate the probability distribution of X. [3](b) Two dice are thrown repeatedly. Find the probability that in 5 throws, the result X = 2 occurs at least once. [3](c) The dice are thrown n times. Find the least value of n such thatP(X = 2 occurs at least once in n throws) > 0.9. [3](d) Use an appropriate approximation to estimate the probability that in 150 throws, the result X = 2 will occur at least twice. [3]

12. The table below shows the weights of 100 students.Weight(kg) 31- 39- 47- 55- 63- 71- 79- 87- 95-Cumulative Relative Frequency

1.00 0.98 0.90 0.76 0.50 0.26 0.10 0.04 0.00

“63-“ means 63 kg and above.(a) Find the mode of the weights of the students. [4](b) Draw a cumulative relative frequency curve for the above distribution. From the curve, find the median and the inter-quartile range of the weights of the students. [9]

2

2


3

3


4

4


5

5


6

6


7

7

Penang_jitsin Trial Stpm 2012 Mathst Paper 2(Q&A)

Documents

Transcript of Penang_jitsin Trial Stpm 2012 Mathst Paper 2(Q&A)