NEW SYLLABUS MATHEMATICS

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1 M8 NW SYLLUS MTHMTIS MULTIPL HOI PRTI S - Problems NM: LSS: LSS NO.: M Leo.F. Leung Hong Kong am & Research

2 ONTNTS Section Level Topics Questions ngle etween a Line and a Plane 8 ngle etween Two Lines -, 9-0 ngle etween Two Planes 6 Relationship between ngle and Length 6-7, -, 7 Mensuration 8 ngle etween Two Lines 9-0, ngle etween Two Planes -, 8 Relationship between ngle and Length, -0, -, -7, 0 Projection 9, earing,, Mensuration 9 Hong Kong am & Research

3 STION Level :. The figure shows a cube FGH. sin G =.. H.... F G... The figure shows a cube FGH. Find tanθ.. H... F θ G. In the figure, FGH is a cube and M is the mid-point of. Find cosθ.. H.. 7 F θ G. 7 9 M Hong Kong am & Research

4 . The figure shows a cube FGH. Which of the following angles is/are equal to 60? I. HG II. F H III. GF IV. HF. IV only F G. I and III only. II and III only. II and IV only. The figure shows a cuboid FGH. and intersect at T. Which of the following pairs of angles are equal? I. T and TH H II. TG and FTH III. G and HF. III only. I and II only F G. II and III only T. I, II and III 6. The figure shows a cuboid FGH. G = H F G Hong Kong am & Research

5 7. Refer to Question 6. Find cos G The figure shows a cuboid FGH, where is a square. M is the mid-point of GH. Let θ be the angle that the line M makes with the plane FGH, =, =. Find cosθ. H M G. F In the figure,, and O lie in a horizontal plane; is vertically above O. Which of the following angles is/are equal to 90? I. O II. O III. O. I only. III only O. II and III only. I, II and III Hong Kong am & Research

6 6 0. The figure shows a rectangular block. Find θ θ The figure shows a rectangular block FGH. M is the mid-point of. Find MF.. cm. 0 cm. cm H 9cm G cm. 0 cm M F cm. The figure shows a rectangular plane F which inclines to the horizontal plane, where and F are vertically above and respectively. M is a point on. rrange α, β and γ in ascending order of magnitude.. α, β, γ F. β, α, γ. β, γ, α. γ, α, β α M β γ Hong Kong am & Research

7 . In the figure, O, O and O are three mutually perpendicular planes. Let O = m, O = 60 and O =. =. m.. 0 m.. m.. 0 m. 60 O m 7. The figure shows a right pyramid V with a square base. Let =, VH =. V =... V H. In the figure, O is a horizontal plane. is vertically above O. Find O O 60 8 Hong Kong am & Research

8 8 6. The figure shows a right pyramid V with a rectangular base in which V is its height. Let = a, = b, V = c and the angle between the planes V and be θ. Find sinθ..... b + c c c a + b c b + c c a + b + c V 7. In the figure, is a horizontal plane. Find..... psin y cos p tan y cos p tan sin y p tan y tan p y 8. The figure shows a right triangular prism F. Find the area of F...8 F Hong Kong am & Research

9 LVL 9 Section : 9. The figure shows a cube. Find θ θ 0. The figure shows a cube FGH. and F intersect at I. Find the angle between the lines H and HI. H..... I G. 6.9 F. The figure shows a cuboid FGH. Find tanθ F θ 60 H G Hong Kong am & Research

10 0. H θ F G y N z M The figure shows a cuboid FGH. M and N are the mid-points of and FG respectively. Let MHN =θ, GH =, G = y and = z. Find cosθ z + y + z + z + y + z + z z + y z z + z + y + z + z. The figure shows a cuboid FGH. Let θ be the angle between G and. Which of the following is/are true? I. cosθ= 8 H II. cos θ = III. cos θ = IV. cos θ = θ F 0 G. I only. III only. I and III only. I and IV only Hong Kong am & Research

11 . The figure shows a right triangular prism. Find θ θ. In the figure, OZ is perpendicular to the plane OXY. Let OZ = h and XY = 7h. Find XOY Z h O Y 0 7h X 6. The figure shows two identical rectangular planes and F in which = and =. The angle between the two planes is 60. Let the angle between and be θ. Find cosθ. F θ Hong Kong am & Research

12 7. In the figure, O is a horizontal plane, is vertically above O and M is the mid-point of. Find θ O θ 0 M 8. Refer to Question 7. If N is a point on such that ON, find NO In the figure, O, O and O are mutually perpendicular. Find tanθ θ O 60 Hong Kong am & Research

13 0. The figure shows a right triangular prism F with =. Let = 6, = and F =. Find G F G. In the figure, is a square in a horizontal plane and V is perpendicular to the plane. Let =, V = y and θ be the angle between the planes V and. tanθ=. y.. y +. V. y +.. y +.. The figure shows a right pyramid V with a square base in which = and V =. Find the angle between the planes V and V.. V Hong Kong am & Research

14 . Find the angle between the planes V and V of a regular tetrahedron V In the figure, O is vertical. Let O = 70, O = 0 and =φ. Find φ O N 70 0 φ. In the figure, is a trapezoidal plane inclined at an angle of 0 to the horizontal. Let = 0 and F = 8. Find F 0 0 Hong Kong am & Research

15 6. In the figure, F is a right triangular prism. Let =, = 6, =, F = and G : G = :. Find cosθ... G θ.. 6 F 7. In the figure, is a horizontal plane and V is a vertical rod. Let V = 60, V =α, = 0, = and = 7. Find tan α.. V α 8. In the figure,,, = and =. Let = a, =θ and =φ. =. a cos φ.. a cos φ tanθ. a. a cos φsinθ.. a cos φ + cosθ. φ θ Hong Kong am & Research

16 6 9. In the figure, is a plane inclined at an angle of φ to the horizontal with //. Let =θ, = and =. tanθ=. sin φ.... tan φ. 0 sin φ. 0 tan φ. θ φ 0. The figure shows a right triangular prism. = The figure shows a right triangular prism F. M and N are the mid-points of F and respectively. Find MN... F M N Hong Kong am & Research

17 . In the figure,, and are three mutually perpendicular planes. Find In the figure, is vertical. Let = φ, =φ and =α. Find sin α.. 0 tan φ. tan φ+tan φ. tan φ tanφ + N φ α. tan φ+tan φ φ. In the figure, is a right-angled triangle and is a rectangle in a vertical plane. Let =θ, = 60 and =. Find sinθ.. sin cos 60. cos sin 60. tan cos 60. tan tan 60 θ 60 Hong Kong am & Research

18 8. a α β h 0 60 F G The figure shows a right triangular prism F. press a in terms of h, α and β. h. sin α+sin β... h h h sin α + sin β + cos α+β cos α+β 6. In the figure, is a horizontal plane with = and = 0. TS is a vertical pole of height h such that S = 90. Let θ be the angle between the lines S and T. If sinθ=, find h h S T θ 0 Hong Kong am & Research

19 7. In the figure, V, and are mutually perpendicular, where V = =. Let M and N be the mid-points of V and V respectively. Find MN.. 0 V.. 60 N. 90 M 9 8. The figure shows a pyramid O with a square base of side. O is perpendicular to the base. Let O = and θ be the angle between the planes O and O. cosθ=. 0. O In the figure, is a plane inclined at an angle of 0 to the horizontal, where is a rectangle. The overhead sun casts a shadow F of the plane on the horizontal ground. Find the ratio of the area of the shadow F and the area of the plane.. :. :. : F. : Hong Kong am & Research

20 0 0. In the figure, is a piece of triangular sheet placed on a horizontal plane. The sheet is folded along such that = 60 and is vertically above. M is the mid-point of and N is the foot of the perpendicular from to. Let = and MN = 0. Find M... 0 M 60. N.. sun rays 0 m m 8m N In the figure, a vertical trapezoidal wall runs along the north direction on the horizontal ground. If the sun bears N60 with an angle of elevation 60, find the area of the shadow of the wall on the horizontal ground.. 8 m. 0 m 8. m. 0 m Hong Kong am & Research

21 M8 - Problems Section Level. Let one side of the cube be G G sin G G. Let one side of the cube be G G G cosg G 0 G tan. Let M MH, H cos M. II, IV: F and HF are equilateral. II: GT FT, T HT, G FH III: G F, H, G HF M M cos y z M M cos M 0 69 z y z HONG KONG XM & RSRH 9

22 Level 9. Let P sin Let one side of the cube be I HI I HI cos tan 0, F tan H tan 60 F F H F H tan 0 tan 60 HONG KONG XM & RSRH

23 . 8. h OX h tan 0 h OY h tan cosxoy h XOY 0 h h h 6. 60, is equilateral 7. cos Let O h 7h 9 68 h h O h, O h tan0 tan OM h h h M h h tan O h O 0 tan O OM h h h h h h cos 0 h h ON Osin0 tan NO h NO O tan, O tan O O O O h tan 60 O O tan 60 tan O O 0. Let M be the mid-point of M G 6 Similarly, M 7 cosg M MG G G G G G G 7 7 HONG KONG XM & RSRH

24 8. RHS cos a cos a cos 9. Let F be a point on such that F is a rectangle F sin F sin tan F sin 0. sin 60 a 8 a tan 60 b 8 b a c tan 0 a 8 c 8 8 b c 8. Let P be the foot of perpendicular from M to tan 60 N N PN MN P MP 6.8 N PN tan 60 7 sin sin0 sin0 sin HONG KONG XM & RSRH

25 Let V 0 S VM S T h S h M h sin T Similarly, N h h MN VM h h V 9 h h 6h 96 MN h 8 h 9 or h 9 rejected MN 8. Let M be the mid-point on O and N on O such that MN // OM 90 OMN O 90 MN MN O O M N MN MN 60 M O OM N O coso N N N coso cos MN M N MN M 7 0 HONG KONG XM & RSRH 7

26 M NW SYLLUS MTHMTIS S MULTIPL HOI PRTI More bout Statistics NM: LSS: LSS NO.: M Leo.F. Leung Hong Kong am & Research

27 ONTNTS Section Level Topics Questions Problems involving Variances - Standard Scores -8 Normal istributions 9- Problems involving Variances -0 Standard Scores - Normal istributions -8 Hong Kong am & Research

28 STION For a normal distribution, the following are assumed hereafter: 68% of the data lie within one standard deviation 9% of the data lie within two standard deviations 99% of the data lie within three standard deviations Level :. The graph shows the frequency curves of three normal distributions P, Q and R, each having the same frequency. Which distribution has the greatest variance and which has the smallest? f Greatest Smallest. P Q. P R. Q R. R P. The variance of the three numbers a d, a, a d is d. d.. d. Hong Kong am & Research

29 . The figures show the histograms of three frequency distributions I, II and III. rrange their variances in ascending order of magnitude. I. II. III.. I, II, III. II, I, III. II, III, I. III, II, I Hong Kong am & Research

30 . The graph shows the frequency curves of two normal distributions and. f Which of the following is/are true? I. Mode of > Mode of II. Inter-quartile range of < Inter-quartile range of III. Variance of < Variance of. I only. II only. I and II only. II and III only. May of age 6 years 0 months is a student of Secondary of a school. The mean and standard deviation of the age distribution of the form are as follows: Mean Standard deviation Secondary 6 years months months Find the standard score of the age of May Hong Kong am & Research

31 6 6. girl gets 70 marks in an nglish test with mean 60. If her standard score is, find the standard deviation In a test, the mean mark is 60 and the standard deviation is. If Mary got a standard score of 0. 6, find her mark Jacky, ndy, Leo, Jerry weigh 0 kg, 60 kg, 70 kg and 80 kg respectively. Jacky is years old, ndy is 0 years old, Leo is years old, Jerry is 0 years old. The mean weights and standard deviations of people at their age groups are shown below: ge 0 0 Mean weight kg 6 7 Standard deviation kg 8 etermine who is relatively over-weight.. Jack. ndy. Leo. Jerry Hong Kong am & Research

32 9- The I.Q. of children in a certain school are found to be normally distributed. The mean and standard deviation of the I.Q. of children are 00 and respectively Find the percentage of children that have an I.Q. between 8 and.. %. 0%. 68%. 8% 0. Find the percentage of children that have an I.Q. more than 8.. 0%. 68%. 8%. 9%. Find the percentage of children that have an I.Q. less than 70.. %..%. %. 6%. The marks of an eamination are found to be normally distributed with a mean of 00 and a standard deviation of. student will get a merit certificate if he/she scores above. What is the probability that a student taking this eamination receives a merit certificate? Hong Kong am & Research

33 8 Level :. In the figure, P and Q are curves showing the distribution of heights of students in two schools, each having the same number of students. Which of the following is/are true? f I. Variance of P > Variance of Q II. Mean of P > Mean of Q III. Median of P > Median of Q. I only. I and II only. II and III only. I, II and III. The variance of a distribution of test scores is v. If marks are added to each datum of the distribution, what is the variance of the new distribution?. v. v. v. v. The variance of a distribution of test scores is v. If each datum of the distribution is divided by, what is the variance of the new distribution?. v. v 6. v. 6 v Hong Kong am & Research

34 6. Given that,,, } and p q, p q,, p }, { n { n q where each of the data set contains n elements, and p, q are constants. Let v and v be the variances of and respectively. Which of the following must be true?. v v. v pv. v p v. v pv q 9 7. The figure shows the cumulative frequency curves of three distributions I, II and III. rrange the three distributions in the order of their variances, from the smallest to the largest. I. II. III.. I, II, III. I, III, II. II, I, III. II, III, I Hong Kong am & Research

35 0 8. Given two sets of numbers { a, a, a } and { b, b, b }, where a b. m and m are respectively the means of the two sets, and v and v are respectively their variances. Which of the following must be true?. m m and. m and. m v v m v v m and v v. m m and v v 9. {,,, 6, 8} and {,,, 7, 9} are two sets of numbers. Which of the following is/are true? I. The two sets of numbers have the same range. II. The two sets of numbers have the same variance. III. The two sets of numbers have the same mean.. I only. II only. I and II only. I and III only 0. is the mean of the set of numbers { abcde,,,, }. Which of the following about the two sets of numbers { abcde,,,, } and { abc,,, de,, } is/are true? I. The two sets of numbers have the same mean. II. The two sets of numbers have the same range. III. The two sets of numbers have the same variance.. I only. III only. I and II only. II and III only Hong Kong am & Research

36 - In a mathematics test given to a class, the mean and standard deviation were 6 and 8 respectively. The table below shows the marks of four students in the class: Student Mark In a second mathematics test given to the same class, the mean and standard deviation were and 6 respectively. The table below shows the marks of those four students: Student Mark Of the students, which students improved most?....,. Of the students, which student has/have a consistent performance in both tests?...., - The weights of 00 children are normally distributed with a mean of kg and a standard deviation of kg.. Find the number of children whose weights lie between kg and 9 kg Find the number of children whose weights lie between 9 kg and 9 kg Hong Kong am & Research

37 . The quality of a product is measured using an inde. It follows a normal distribution with mean 0 and standard deviation 0. The top 6% of products are classified grade, the bottom 6% grade and the others grade. Find the range of inde of grade products To qualify as a contestant in a swimming event, a swimmer has to be in the top 6% of all entrants. The swimming times are normally distributed, with a mean of minute 0 seconds and a standard deviation of seconds. What is the qualifying time for the event?. minute seconds. minute 7 seconds. minute seconds. minute 6 seconds 7. The lengths of metal wires produced in a factory follow a normal distribution with a mean of 80 cm. If.% of the wires are shorter than 78. cm, the standard deviation is. 0. cm cm.. cm... cm. 8. The capacities of the soft-drink cans produced in a factory follow a normal distribution with a mean of 0 cm. If 8.% of soft-drink cans have capacities between 8 cm and cm, find the standard deviation.. 0. cm. cm.. cm. cm Hong Kong am & Research

38 M More bout Statistics Section Level. R is most dispersed, P is least dispersed R has the greatest variance, P has the smallest variance. a d a a d Mean a a d a Variance a a a d a d. II is least dispersed, III is most dispersed i.e. Variance of II < Variance of I < Variance of III.. Standard score of the age of May 6 years0 months 6 years months months. 6. Let s be the standard deviation s s I, II: From the figure: Mode of < Mode of 7. i.e. I is not true Let her mark be Inter-quartile range of < Inter-quartile range of i.e. II is true III. is more dispersed than Variance of < Variance of i.e. III is true HONG KONG XM & RSRH 8

39 . student will get a merit certificate if he/she scores above = student will get a merit certificate if he/she scores above s Required probability 0% % 6% 0. 6 Level. I. P is more dispersed than Q P has the larger variance i.e. Variance of P > Variance of Q i.e. I is true II, III: From the figure: Mean of P < Mean of Q Median of P < Median of Q i.e. II, III are not true HONG KONG XM & RSRH 0

40 6. HONG KONG XM & RSRH },,, { n Let be the mean of the distribution n n n v n },,, { q p q p q p n Mean of n q p q p q p n q p n qn p n n n v p n p n q p q p q p q p q p q p v ] [ ] [ ] [ ] [ 7. I. The frequency curve for I: II. The frequency curve for II: III. The frequency curve for III: From the frequency curves, III is most dispersed, I is least dispersed Variance of I < Variance of II < Variance of III

41 0. I. Mean of the st set e d c b a e d c b a Mean of the nd set e d c b a 6 6 I is true II. a,b,c,d,e a,b,c,d,e Min Ma thest set of Range is the mean, it cannot be the ma. or min. of e d c b a,,,,, New range = Original range II is true III. Variance of the st set e d c b a Variance of the nd set 6 6 e d c b a e d c b a learly, the variance of the sets are not the same III is not true HONG KONG XM & RSRH

42 .. % of children whose weights lie between kg and 9 kg = Percentage of data lying between s and s = 68%.% = 8.% Number of children whose weights lie between kg and 9 kg 00 8.% 6 The middle % of product are classified grade The range of inde of grade products is: s s % of children whose weights lie between 9 kg and 9 kg = Percentage of data lying between s and s = 9% % = 97% Number of children whose weights lie between 9 kg and 9 kg 00 97% 88 6% 0% % i.e. the swimming time should be lying below s Qualifying time for the event 70 s 67s min 7s HONG KONG XM & RSRH 6