HKCEE 1993 Mathematics II

Transcription

1 HK 9 Mathematics II If f() = 0, then f(y) = 0 y 0 + y 0 8y 0 y 0 y If s = n [a + (n )d], then d = ( s an) n( n ) ( s an) ( n ) s n( n ) as n a( n ) ( s an) n( n ) Simplify ( + )( + + ) b a a b ab a a b a b a b If + a (b )( ), then a =, b = a =, b = a =, b = a =, b = a =, b = O y + y = 0 Find the greatest value of + y if (, y) is a point lying in the region O (including the boundary) a a b + a b a a a ab b a b b MTHS II

2 7 - y 0 y = c + d y = a + b The diagram shows the graphs of y = a + b and y = c + d The solutions of the equation a + b = c + d are,, 0, 0,, Find the HF and LM of ab c and abc HF LM a a b c abc ab c abc a b c ab c abc a b c abc If and are the roots of the quadratic equation = 0, find the value of + 8 If log(p + q) = log p + log q, then p = q = q p = q q p = q q p = q q p = q If the simultaneous equations y k have only one solution, y find k 9 The epression + k is divisible by ( + ) Find the remainder when it is divided by ( + ) If, a, b, c, are in P, then a + b + c = --MTHS II

3 r h 6 The price of a cylindrical cake of radius r and height h varies directly as the volume If r = cm and h = cm, the price is $0 Find the price when r = cm and h = 6 cm h r $ $880 $0 $6 $ rad cm 7 In the figure, the base of the conical vessel is inscribed in the bottom of the cubical bo If the bo and the conical vessel have the same capacity, find h : r : : 6 : : 8 : Find the perimeter of the sector in the figure h r cm cm 60 cm 6 cm cm The figure shows a solid consisting of a cylinder of height h and a hemisphere of radius r The area of the curved surface of the cylinder is twice that of the hemisphere Find the ratio volume of cylinder : volume of hemisphere : : : : : --MTHS II

4 8 merchant marks his goods % above the cost He allows 0 % discount on the marked price for a cash sale Find the percentage profit the merchant makes for a cash sale % % % % 7% cos cos 9 sin sin 89 The largest value of sin + cos is sin cos tan sin cos 0 cos sin + sin = P 0 ( sin ) ( cos ) (cos sin ) 8 In the figure, =, P = P and P P Find tan In the figure, cos = 7 Find a --MTHS II

5 - 0 o - o + o 7 0 o 0 o o 0 o If the points (, ), (, ) and (7, k) are on the same straight line, then k = In the figure, points,, and are concyclic Find 0 o o o 7 o 0 o (0, 0), (, 0) and (, 6) are the vertices of a triangle P(9, ), Q(6, 6) and R(, 9) are three points Which of the following triangles has/have area(s) greater than the area of? 8 o 7 o In the figure, // and = Find o o 70 o 7 o 76 o 9 I P II Q III R I only II only III only I and II only II and III only circle of radius touches both the positive -ais and the positive y-ais Which of the following is/are true? 6 I Its centre is in the first quadrant II Its centre lies on the line y = 0 III Its centre lies on the line + y = 0 o I only II only III only I and II only I and III only In the figure, is a diameter Find 00 o 0 What is the area of the circle + y 0 + 6y = 0? --MTHS II

6 6 8 Two fair dice are thrown What is the probability of getting a total of or 0? group of n numbers has mean m If the numbers, and 6 are removed from the group, the mean of the remaining n numbers remains unchanged Find m 6 n The figure shows the frequency polygons of two symmetric distributions and with the same mean Which of the following is/are true? I Interquartile range of < Interquartile range of II Standard deviation of > 6 III Standard deviation of Mode of > Mode of I only II only III only I and II only II and III only If 9 + = 6, then = 6 9 If a : b = : and b : c = :, then a b c = a b c 7 Sign of f() From the table, a root of the equation f() = 0 is 7 (correct to sig fig) 7 (correct to sig fig) 77 (correct to sig fig) 7 (correct to sig fig) 8 (correct to sig fig) --MTHS II 6

7 7 Given that the positive numbers p, q, r, s are in GP, which of the following must be true? I kp, kq, kr, ks are in GP, where k is a non-zero constant II a p, a q, a r, a s are in GP, where a is a positive constant III log p, log q, log r, log s are in P 0 If the solution of the inequality a is c, then a =, c = a =, c = a =, c = a =, c = a =, c = I only II only I and II only I and III only I, II and III only 8 y In the figure, is a square and is an equilateral triangle rea of = rea of 9 In the figure, the rectangle has perimeter 6 cm and area cm Find the length of its diagonal cm cm 7 cm 6 cm cm In factorizing the epression a + a b + b, we find that (a b ) is a factor (a + b ) is a factor (a ab b ) is a factor (a ab + b ) is a factor it cannot be factorized Q 8 S O In the figure, the radii of the sectors OPQ and ORS are cm and cm rea of shaded region respectively = rea of sector OPQ R P --MTHS II 7

8 9 6 6 Solve tan + tan = 0 for 0 o < 60 o o, o only o, o only o, 60 o, o, 0 o o, 0 o, o, 00 o o, o, o, o y Which of the following gives the compound interest on $ at 6% pa for one year, compounded monthly? 0 80 o 70 o 60 o 0 o 006 $ $ 0 000(06 ) 006 $ $ $0 000 Originally of the students in a class failed in an eamination fter taking a re-eamination, 0% of the failed students passed Find the total pass percentage of the class 7 - The figure shows the graph of the function y = sin(0 o ) y = sin( + 0 o ) y = cos( + 0 o ) y = sin( 0 o ) y = cos( 0 o ) 6 % % 0% 60% 7 % In the figure, is an equilateral triangle and the radii of the three circles are each equal to Find the perimeter of the triangle ( + tan0 o ) 6( + tan0 o ) --MTHS II 8

9 8 o tan0 6 o tan0 F G In the figure, FGH is a cuboid The diagonal H makes an angle with the base Find tan H 0 II a : b : c = : : III sin : sin : sin = : : I only II only III only I and II only I, II and III only T M In the figure, TP and TQ are tangent to the circle at P and Q respectively if M is a point on the minor arc PQ and PMQ =, then PTQ = P o 80 o 80 o 80 o H O 9 M K b a N c In the figure, O is the centre of the circle touches the circle at N Which of the following is/are correct? In the figure, if arc : arc : arc = : :, which of the following is/are true? I : : = : : I M, N, K, O are concyclic II HN ~ NK III ON = NO I only II only --MTHS II 9

10 III only I and II only I, II and III only H G X In the figure, the three circles touch one another XY is their common tangent The two larger circles are equal If the radius of the smaller circle is cm, find the radii of the larger circles Y In the figure and FGH are two squares and H is an equilateral triangle Find : F F 8 cm 0 cm cm cm 6 cm : : : : : F F F In the figure, a rectangular piece of paper is folded along F so that and coincide If = cm, = 6 cm, find cm cm cm 8 cm cm --MTHS II 0