Exam week’s coming, so here it is.

###### Section A (35 marks)

(1) Simplify and express your answer with positive indices. (3 marks)

*Solution*

(2) Make the subject of the formula . (3 marks)

*Solution*

(3) Simplify . (3 marks)

*Solution*

(4) Factorize

(a) .

(b) .

(c) . (4 marks)

*Solution*

(a)

(b)

(c)

(5) In a recreation club, there are 180 members and the number of male members is 40% more than the number of female members. Find the difference of the number of male members and the number of female members. (4 marks)

*Solution*

Let be the number of female members.

(6) Consider the compound inequality

(a) Solve (*).

(b) Write down the greatest negative integer satisfying (*). (4 marks)

*Solution*

(a)

(b)

(7) In a polar coordinate system, is the pole. The polar coordinate of the points and are and respectively.

(a) Find .

(b) Find the perimeter of .

(c) Write down the number of folds of rotational symmetry of . (4 marks)

*Solution*

(a)

(b) Obviously the triangle is isosceles and hence and hence is equilateral. Therefore the perimeter is 36.

(c) 3

(8) It is given that is the sum of two parts, one part varies as and the other part varies as . Suppose that and .

(a) Find .

(b) Solve the equation . (5 marks)

*Solution*

(a) Let for some non-zero real numbers and .

(2) – (1) gives

(b)

(9) The frequency distribution table and the cumulative frequency distribution table below show the distribution of the heights of the plants in a garden.

(a) Find , , and .

(b) If a plant is randomly selected from the garden, find the probability that the height of the selected plant is less than 1.25 m but not less than 0.65 m.

*Solution*

(a)

(b)

(10) The coordinates of the points and are and respectively. Let be a moving point in the rectangular coordinate plane such that is equidistant from and . Denote the locus of by .

(a) Find the equation of . (2 marks)

(b) intersects the *x*-axis and *y*-axis at and respectively. Denote the origin by . Let be the circle which passes through , , and . Someone claims that the circumference of exceeds 30. Is the claim correct? Explain your answer. (3 marks)

*Solution*

(a)

(b) Let , then , hence

Let , then , hence

Since is a right triangle where , hence is the diameter of the circle .

Therefore, the claim is correct.

(11) An inverted right circular conical vessel contains some milk. The vessel is held vertically. The depth of milk in the vessel is . Peter pours of milk into the vessel without overflowing. He now finds that the depth of milk in the vessel is .

(a) Express the final volume of milk in the vessel in terms of . (3 marks)

(b) Peter claims that the final area of the wet curved surface of the vessel is at least . Do you agree? Explain your answer. (3 marks)

*Solution*

(a) Let be the original volume of milk.

(b)

Therefore, disagree.

(12) The bar chart below shows the distribution of the ages of the children in a group, where and . The median of the ages of the children in the group is 7.5.

(a) Find and . (3 marks)

(b) Four more children now join the group. It is found that the ages of these four children are all different and the range of the ages of the children in the group remains unchanged. Find

(i) the greatest possible median of the ages of the children in the group.

(ii) the least possible mean of the ages of the children in the group. (4 marks)

*Solution*

(a) Since the median age is 7.5, hence

Now, .

Since

(b) (i) To maximize the median, we add age 7, 8, 9, 10. Therefore the new median will be 8.

(b) (ii) To minimize the mean, we add age 6, 7, 8, 9.

Case ,

Case ,

(13) In figure 1, is a triangle. , , and are points lying on such that , and .

(a) Prove that . (2 marks)

(b) Suppose that , and .

(i) Find .

(ii) Is a right-angled triangle? Explain your answer. (5 marks)

*Solution*

(a)

(b) (i)

Hence, is isosceles.

(b) (ii)

By Pyth. thm,

Therefore, is right-angled.

(14) Let , where , , and are constants. When is divided by and when is divided by , the two remainders are equal. It is given that , where , , and are constants.

(a) Find , , and . (5 marks)

(b) How many real roots does the equation have? Explain your answer. (5 marks)

*Solution*

(a)

By matching the coefficient,

(b)

For no real root.

For two real roots.

Therefore, has two real roots.

###### Section B (35 marks)

(15) If 4 boys and 5 girls randomly form a queue, find the probability that no boys are next to each other in the queue. (3 marks)

*Solution*

Consider this : _ G _ G _ G _ G _ G _

There are 6 places to put 4 B’s so there are 6P4 ways to place the boys. Moreover, there are 5! ways to permute those 5 G’s. Therefore,

(16) In a test, the mean of the distribution of the scores of a class of students is 61 marks. The standard scores of Albert and Mary are -2.6 and 1.4 respectively. Albert gets 22 marks. A student claims that the range of the distribution is at most 59 marks. Is the claim correct? Explain your answer. (3 marks)

*Solution*

So, the standard deviation is 15. Now, let’s calculate what Mary gets on the test.

, which is greater than 59. Therefore the claim is false.

(17) The 1st term and the 38th term of an arithmetic sequence are 666 and 555 respectively. Find

(a) the common difference of the sequence. (2 marks)

(b) the greatest value of such that the sum of the first term of the sequence is positive. (3 marks)

*Solution*

(a)

(b)

Since is positive integer, we can divide both side by without switching the sign.

Therefore, the greatest such is 444.

(18) Let .

(a) Using the method of completing the square, find the coordinates of the vertex of the graph of . (2 marks)

(b) The graph of is obtained by translating the graph of vertically. If the graph of touches the *x*-axis, find . (2 marks)

(c) Under a transformation, is changed to . Describe the geometric meaning of the transformation. (2 marks)

*Solution*

(a)

Therefore, the vertex is .

(b) Basically translate the graph of 13 units upward.

(c) Since

Therefore, the transformation is reflecting the graph of f(x) horizontally along the *y*-axis.

(19) Figure 2 shows a geometric model in the form of tetrahedron. It is given that .

(a) Find and . (4 marks)

(b) A craftsman claims that the angle between and the face is . Do you agree? Explain your answer. (2 marks)

*Solution*

(a) By Sine Law,

By Cosine Law, .

(b) By Sine law,

Therefore, do not agree.

(20) is an obtuse-angled triangle. Denote the in-centre and the circumcentre of by and respectively. It is given that , , and are collinear.

(a) Prove that . (3 marks)

(b) A rectangular coordinate system is introduced so that the coordinates of and are and respectively while the *y*-coordinate of is 19. Let be the circle which passes through , , and .

(i) Find the equation of .

(ii) Let and be two tangents to such that the slope of each tangent is and the *y*-intercept of is greater than that of . cuts the *x*-axis and *y*-axis at and respectively while cuts the *x*-axis and the *y*-axis at and respectively. Someone claims that the area of the trapezium exceeds 17000. Is the claim correct? Explain your answer. (9 marks)

*Solution*

(a) Since , , and are collinear, we can draw a line passing through all three points and meet the line segment at . Since is incenter, is angle bisector of , hence . Since is circumcenter, then is perpendicular bisector of and so . obviously by common sides. Hence, by ASA. Therefore the corresponding sides of the two congruent triangles are congruent and so .

(b) (i) Let . Since ,

Let circle

Substitute into circle C yields . Hence,

3(1)-19(2):

Therefore, the equation of circle C is

(b) (ii) Since and have slope of , They both have the equation of the form:

They are both tangents to the circle hence:

Since they are tangents to the circle,

Let

Let

Therefore, the claim is correct.