2001 HG2, 2006 HI6, 2010 HI1
|
|
- Ruby Clarissa Bryant
- 5 years ago
- Views:
Transcription
1 - Individual ( 8 ) x, y 0 ( ) Group Individual Events I * see the remark Find the value of the unit digit of (Reference: 996 HG0) (mod 0) 5 (mod 0) ( ) + + ( ) (mod 0) (mod 0) 9 (mod 0) I Given that a, b and c are positive even integers which satisfy the equation a + b + c 0. I How many solutions does the equation have? Reference: 00 HG, 006 HI6, 00 HI Let a p, b q, c r, where p, q, r are positive integers. a + b + c 0 (p + q + r) 0 p + q + r 006 The question is equivalent to find the number of ways to put 006 identical balls into different boxes, and each box must contain at least one ball. Align the 006 balls in a row. There are 005 gaps between these balls. Put sticks into three of these gaps, so as to divide the balls into groups. The following diagrams show one possible division. The three boxes contain balls, 00 balls and ball. p, q 00, r. The number of ways is equivalent to the number of choosing gaps as sticks from 005 gaps The number of ways is In Figure, ABD is a square. The coordinates of B and D are (5, ) and (, ) respectively. If A(a, b) lies in the first quadrant, find the value of a + b. Mid-point of BD M(, ) MB MA (a ) + (b ) (5 ) + (+ ) 0 a + b a b 8 0 () b MA MB a 5 b a () Sub. () into (): a + (a ) a (a ) 8 0 5a 0a 5 0 a a 0 (a )(a + ) 0 a or (rejected) b () 5 a + b D O y A Figure 圖圖 B x Page
2 I4 ( ) Method mbd 5 BDA 45 tan 45 mad b a + m + tan 45 BD mbd tan 45 ; mad + ( ) b a + b 9 a + b a () Mid-point of BD M(, ) b a () (similar to method ) Solving () and () gives a, b 5 a + b 8 Method by Mr. Jimmy Pang from Lee Shing Pik ollege Mid-point of BD M(, ) Translate the coordinate system by x x, y y The new coordinate of M is M (, ) (0, 0) The new coordinate of B is B (5, ) (4, ) Rotate B about M in anticlockwise direction through 90 The new coordinate of A (, 4) Translate the coordinate system by x x +, y y + The old coordinate of A ( +, 4 + ) (, 5) (a, b) a + b 8 D(-, ) D' ' M(, )M'(0,0) O y y' A A' B(5, -) B' Find the number of places of the number 0 5. (Reference: 98 FG0., 99 HI7) The number of places I5 Given that log4 N , find the value of N. (Reference: 994 HI) 9 7 I6 I7 log4 N (sum to infinity of a geometric series, a, r.) N 4 8 Given that a and b are distinct prime numbers, a 9a + m 0 and b 9b + m 0. Find a b the value of +. (Reference: 996 HG8, 996FG7., 00 FG4.4, 005 FG.) b a a and b are prime distinct roots of x 9x + m 0 a + b sum of roots 9 (odd) a and b are prime number and all prime number except, are odd. a, b 7 (or a 7, b ) a b ( 8 ) b a Given that a, b and c are positive numbers, and a + b + c 9. Suppose the maximum value among a + b, a + c and b + c is P, find the minimum value of P. WLOG assume that a + b P, a + c P, c + a P. 8 (a + b + c) (a + b) + (b + c) + (c + a) P 6 P The minimum value of P is 6. x' x Page
3 I8 If the quadratic equation (k 4)x (4k + 4)x has two distinct positive integral roots, find the value(s) of k. learly k 4 0; otherwise, the equation cannot have two real roots. Let the roots be α, β. (4k + 4) 4(48) (k 4) [(7k + ) 48k + 9] (k + 8k + 96) [(k + 4)] 4k α k [ ( k + 4) ] ( 4) 7k + + k + 4 8k k 6, β. k 4 k 4 k k 4 k + For positive integral roots, k is a positive factor of 8 and k + is a positive factor of 6. k,, 4, 8 and k +,,, 6 k, 4, 6, 0 and k, 0,, 4 k 4 only Method provided by Mr. Jimmy Pang from Po Leung Kuk Lee Shing Pik ollege The quadratic equation can be factorised as: [(k )x 8][(k + )x 6] k and k x or k k + By similar argument as before, for positive integral root, k 4 only. I9 Given that x, y are positive integers and x > y, solve x 89 + y. x y (x y)(x + xy + y ) and both and 99 are primes. x + xy + y (x y) + xy x y (x y) + xy xy x y 89 + xy 79. (rejected) 99 + xy xy (rejected) xy (rejected) x, y Page
4 I0 In figure, AE 4, EB 7, A 9 and BD D 0. Find the value of BF. Reference: 005 HI5, 009 HG8 AB 4 + 7, B AB + B A AB 90 (converse, Pythagoras theorem) Let BF a, BF θ, ABF 90 θ Area of BEF + area of BF area of BE a sin θ + a 7cos θ A E 圖圖 Figure F B D 0a sin θ + 7a cos θ 40 () Area of BDF + area of ABF area of ABD 0 a cosθ + a 0sin θ a cos θ + 0a sin θ 0 () () (): 5 a cos θ 80 a cos θ 8 () () (): 50 a sin θ 0 a sin θ 5 (4) () + (4) : BF a ( ) Method AB 90 (similar to method ) A 0 Regard B as the origin, B as the x-axis, BA as the y-axis, then d 0i, c 0i, e 7j, a j Suppose F divides AD in the ratio p and p. 5 Also, F divides E in the ratio t : t. f p d + ( p) a 0pi + ( p)j () f t c + ( t) e 0ti + 7( t)j () ompare coefficients: 0p 0t and ( p) 7( t) p t () and ( p) t (4) () + (4): 5t + t 5 0 E 5 t p F 9 BF f 0( 5 )i + 7( 5 )j 8i + )j B - p - t 0D 0 Page 4
5 Method AB 90 (converse, Pythagoras theorem, similar to method ) Regard B as the origin, B as the x-axis, BA as the y-axis, then Equation of AD: x + y 0 (); equation of E: x + y 0 7 () 5 y x () (): y ; () (): 5 4 x 8 BF x + y ( ) Method 4 AB 90 (similar to method ) Regard B as the origin, B as the x-axis, BA as the y-axis, then A(0, ), E(0, 7), (0, 0), D(0, 0). Let AF : FD r : s Apply Menelaus theorem on ABD with EF. AE B DF EB D FA 4 0 s s 7 0 r r 4 By the point of division formula, F, 5 5 8, 5 BF A(0, ) r E(0,7) B(0,0) F s D(0,0) (0,0) Method 5 (Provided by Mr. Lee hun Yu, James from St. Paul s o-educational ollege) Area of AB 0 0 Area of ABD : area of AD : Area of BDF : area of DF Area of ABF : area of AF : Area of EB : area of EA : Area of FEB : area of FEA Area of FB : area of FA : Area of ABF : area of AF : area of BF : : Area of BF Distance from F to B 4. 0 Area of ABF Distance from F to AB BF (Pythagoras theorem) Page 5
6 Group Events G Given that x, y and z are three consecutive positive integers, and integer. Find the value of x + y + z. x y, z y + y z x z x y y + y + y + y + y + y x x y y z z y y y + ( ) ( ) ( y + y y + + y y ) + ( y ) y ( y ) ( y ) y( y + ) y + y + y + y y + y y + y 6 y 6y ( y ) y( y + ) ( y ) y ( y +) y learly y does not divide y, so y + and y are factors of 6. y y, y + x + y + z 6 y y, but y + 4 which is not a factor of 6, rejected. y or 6 are similarly rejected. Method x y, z y + y z x z x y y + y + y + y + y + y x x y y z z y y y + y y + is an integer y > (); is an integer y () y y + Solving () and () gives y, x, z x + y + z 6 G Given that x is a real number and x + ( 5 x) x G G4 If 5 x, the equation is equivalent to y z x z x y is an x x y y z z 0. Find the value of x. x x x x 0 x 5 x 0 4x 0x + 5 4x x (4)(07) < 0 no real solution, rejected If 5 < x, then the equation becomes x 0 + x 5 x x 0 5 x 07 Evaluate (Answer can be expressed in index form.) ( 005 ) 005 ( + ) Evaluate (Answer can be expressed in surd form.) Page 6
7 G5 Find the minimum value of x + y 0x 6y Reference: 999 HG7, 00 HI, 08 HI x + y 0x 6y (x 5) + (y ) G6 In Figure, AB is an isosceles triangle. Suppose AB A. If D is a point on B produced such that DAB 90 and D, find the length of B. Let AB θ AB (base, isos. ) AD 80 θ (adj. s on st. line) BD sec θ B cos θ BD sec θ cos θ + cos θ 6 0 ( cos θ )(4 cos θ + ) 0 cos θ or (rejected) 4 B cos θ 6 Method Draw AE BD. A ABE AE (R.H.S.) Let BE x E. x x 6 cos B x + x + x + x 7 0 (x 8)(x + 9) 0 B x E x D x 8 B x 6 G7 Given that a x b y c z 0 w and + +, where a, b, c are positive integers (a b c) x y z w and x, y, z, w are real numbers, find the value of a + b + c. log a x log b y log c z log 0 w x log a y log b z log c w log 0 log a logb logc log abc w x y z wlog0 wlog0 wlog0 wlog0 abc 0 a and b (otherwise x log a y log b z log c w log 0 0 w log 0 w 0) a, b, c 5 a + b + c 0 G8 Given that the roots of the equation x + px + q 0 are integers and q > 0. If p + q 60, find the value of q. Let the roots be α and β. α + β p () αβ q > 0 () p + q 60 (α + β) + αβ 60 α β( α) 6 (α )(β ) 6, which is a prime α, β 6 or α, β 6 When α, β 6 α 0, β 60 αβ q 0 (contradicting to the given condition q > 0, rejected) α, β 6 Remark the original question is: Given that the roots of the equation x + px + q 0 are integers and p, q > 0. If p + q 60, find the value of q. α + β p < 0 (), αβ q > 0 () α < 0 and β < 0 and (α )(β ) 6 α, β 6 α 0, β 60 αβ q 0 (rejected), no solution Page 7
8 α, β 6, q αβ 4 G9 Evaluate sin + sin + sin + + sin 59 + sin 60. (Reference 00 FG.) sin + sin 89 sin + cos sin + sin 88 sin + cos... sin 44 + sin 46 sin 44 + cos 44 sin + sin + + sin 89 (sin + sin 89 ) + + (sin 44 + sin 46 ) + sin sin + sin + sin + + sin 59 + sin sin sin G0 In a gathering, originally each guest will shake hands with every other guest, but Steven only shakes hands with people whom he knows. If the total number of handshakes in the gathering is 60, how many people in the gathering does Steven know? (Note: when two persons shake hands with each other, the total number of handshakes will be one (not two).) Suppose there are n persons and Steven knows m persons (where n > m). If everyone shakes hands with each other, then the total number of hand-shaking n In this case, Steven shakes hands with n persons. However, he had made only m hand-shaking. Method n < 60 n By trial and error, n m Method n + n n m 59 + n n ( ) m 60 ( ) 0 55 < m < n 0 ( 8 + n n ) < n n n 8 0 and n n 8 > 0 (n.5) and (n 0.5) 8.5 > n n n > ( n )( ) 0 and ( )( ) 0 ( n ) and ( n < or n > ) < n < 8. 5, 0.5 < < n.5 For integral value, n ( ) + m 60 m 5 Page 8
9 Geometrical onstruction. In the space provided, construct an equilateral triangle AB with sides equal to the length of MN below. () 作綫段 AB, 使得 AB MN () 以 A 為圓心,AB 為半徑作一弧 ; 以 B 為圓心,BA 為半徑作一弧 ; 兩弧相交於 () 連接 A 及 B AB 為等邊三角形 A B. As shown in Figure, construct a circle inside the triangle AB, so that AB, B and A are tangents to the circle. Reference: 009 HS, 04 H, 09 H () 作 AB 的角平分綫 A () 作 AB 的角平分綫 P 為兩條角平分綫的交點 T () 連接 AP 4 4 S (4) 分別過 P 作垂直綫至 B A 及 AB,R S T 為 5 對應的垂足 P BPR BPT (A.A.S.) PR PS (A.A.S.) B R PT PR PS ( 全等三角形的對應邊 ) (5) 若 P 至 B 的垂足為 R, 以 P 為圓心,PR 為半徑作一圓, 此圓內切於三角形的三 邊, 稱為內切圓 (inscribed circle) ( 切綫 半徑的逆定理 ). Figure shows a triangle PQR. onstruct a line MN parallel to QR so that (i) M and N lie on PQ and PR respectively; and (ii) the area of PMN the area of PQR. 首先, 我們計算 MN 和 QR 的關係 : QPR MPN ( 公共角 ) PQR PMN (QR // MN, 對應角 ) PRQ PNM (QR // MN, 對應角 ) PQR ~ PMN ( 等角 ) PMN 的的的 PQR 的的的 PM PQ PM PQ PM PQ 作圖步驟 : () 利用垂直平分綫, 找 PQ 之中點 O () 以 O 為圓心 OP OQ 為半徑, 向外作一半圓, 與剛才的垂直平分綫相交於 F PFQ 90 ( 半圓上的圓周角 ) PFQ 為一個直角等腰三角形 QPF 45 PQ PF PQ sin 45 () PQ 以 P 為圓心,PF 為半徑, 作一圓弧, 交 PQ 於 M PM (4) 自 M 作一綫段平行於 QR, 交 PR 於 N, 則 PMN 平分 PQR 的面積 F M Q P M Q O P N N R R Page 9
10 Percentage of correct questions 4.48%.58%.66% % 5 0.7% 6 0.8% 7 4.9% 8.08% % 0 4.6% 57.49% 5.% 8.06% % % 6 8.6% % % % 0 4.5% Page 0
Answers: ( HKMO Heat Events) Created by: Mr. Francis Hung Last updated: 23 November see the remark
9 50450 8 4 5 8-9 04 6 Individual 6 (= 8 ) 7 6 8 4 9 x =, y = 0 (= 8 = 8.64) 4 4 5 5-6 07 + 006 4 50 5 0 Group 6 6 7 0 8 *4 9 80 0 5 see the remark Individual Events I Find the value of the unit digit
More informationPage 1
nswers: (008-09 HKMO Heat Events) reated by: Mr. Francis Hung Last updated: 8 ugust 08 08-09 Individual 00 980 007 008 0 7 9 8 7 9 0 00 (= 8.) Spare 0 9 7 Spare 08-09 Group 8 7 8 9 0 0 (=.) Individual
More informationAnswers: ( HKMO Heat Events) Created by: Mr. Francis Hung Last updated: 15 December 2017
Answers: (0- HKMO Het Events) reted y: Mr. Frncis Hung Lst updted: 5 Decemer 07 - Individul - Group Individul Events 6 80 0 4 5 5 0 6 4 7 8 5 9 9 0 9 609 4 808 5 0 6 6 7 6 8 0 9 67 0 0 I Simplify 94 0.
More information= lim(x + 1) lim x 1 x 1 (x 2 + 1) 2 (for the latter let y = x2 + 1) lim
1061 微乙 01-05 班期中考解答和評分標準 1. (10%) (x + 1)( (a) 求 x+1 9). x 1 x 1 tan (π(x )) (b) 求. x (x ) x (a) (5 points) Method without L Hospital rule: (x + 1)( x+1 9) = (x + 1) x+1 x 1 x 1 x 1 x 1 (x + 1) (for the
More informationNumbers and Fundamental Arithmetic
1 Numbers and Fundamental Arithmetic Hey! Let s order some pizzas for a party! How many pizzas should we order? There will be 1 people in the party. Each people will enjoy 3 slices of pizza. Each pizza
More informationS Group Events G1 a 47 G2 a *2
Answers: (003-0 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 9 September 07 Individual Events I a 6 I P I3 a I a IS P 8 b + Q 6 b 9 b Q 8 c 7 R 6 c d S 3 d c 6 R 7 8 d *33 3 see the remark
More informationChapter 22 Lecture. Essential University Physics Richard Wolfson 2 nd Edition. Electric Potential 電位 Pearson Education, Inc.
Chapter 22 Lecture Essential University Physics Richard Wolfson 2 nd Edition Electric Potential 電位 Slide 22-1 In this lecture you ll learn 簡介 The concept of electric potential difference 電位差 Including
More informationChap. 4 Force System Resultants
Chap. 4 Force System Resultants Chapter Outline Moment of a Force Scalar Formation Cross Product Moment of Force Vector Formulation Principle of Moments Moment of a Force about a Specified xis Moment of
More informationIndividual Events 1 I2 x 0 I3 a. Group Events. G8 V 1 G9 A 9 G10 a 4 4 B
Answers: (99-95 HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 I a Individual Events I x 0 I3 a I r 3 I5 a b 3 y 3 b 8 s b c 3 z c t 5 c d w d 0 u d 6 3 6 G6 a 5 G7 a Group Events
More information0 0 = 1 0 = 0 1 = = 1 1 = 0 0 = 1
0 0 = 1 0 = 0 1 = 0 1 1 = 1 1 = 0 0 = 1 : = {0, 1} : 3 (,, ) = + (,, ) = + + (, ) = + (,,, ) = ( + )( + ) + ( + )( + ) + = + = = + + = + = ( + ) + = + ( + ) () = () ( + ) = + + = ( + )( + ) + = = + 0
More informationFrequency Response (Bode Plot) with MATLAB
Frequency Response (Bode Plot) with MATLAB 黃馨儀 216/6/15 適應性光子實驗室 常用功能選單 File 選單上第一個指令 New 有三個選項 : M-file Figure Model 開啟一個新的檔案 (*.m) 用以編輯 MATLAB 程式 開始一個新的圖檔 開啟一個新的 simulink 檔案 Help MATLAB Help 查詢相關函式 MATLAB
More information生物統計教育訓練 - 課程. Introduction to equivalence, superior, inferior studies in RCT 謝宗成副教授慈濟大學醫學科學研究所. TEL: ext 2015
生物統計教育訓練 - 課程 Introduction to equivalence, superior, inferior studies in RCT 謝宗成副教授慈濟大學醫學科學研究所 tchsieh@mail.tcu.edu.tw TEL: 03-8565301 ext 2015 1 Randomized controlled trial Two arms trial Test treatment
More informationAlgorithms and Complexity
Algorithms and Complexity 2.1 ALGORITHMS( 演算法 ) Def: An algorithm is a finite set of precise instructions for performing a computation or for solving a problem The word algorithm algorithm comes from the
More informationAdvanced Engineering Mathematics 長榮大學科工系 105 級
工程數學 Advanced Engineering Mathematics 長榮大學科工系 5 級 姓名 : 學號 : 工程數學 I 目錄 Part I: Ordinary Differential Equations (ODE / 常微分方程式 ) Chapter First-Order Differential Equations ( 一階 ODE) 3 Chapter Second-Order
More informationMT EDUCARE LTD. SUMMATIVE ASSESSMENT Roll No. Code No. 31/1
CBSE - X MT EDUCARE LTD. SUMMATIVE ASSESSMENT - 03-4 Roll No. Code No. 3/ Series RLH Please check that this question paper contains 6 printed pages. Code number given on the right hand side of the question
More informationpseudo-code-2012.docx 2013/5/9
Pseudo-code 偽代碼 & Flow charts 流程圖 : Sum Bubble sort 1 Prime factors of Magic square Total & Average Bubble sort 2 Factors of Zodiac (simple) Quadratic equation Train fare 1+2+...+n
More informationFinite Interval( 有限區間 ) open interval ( a, closed interval [ ab, ] = { xa x b} half open( or half closed) interval. Infinite Interval( 無限區間 )
Finite Interval( 有限區間 ) open interval ( a, b) { a< < b} closed interval [ ab, ] { a b} hal open( or hal closed) interval ( ab, ] { a< b} [ ab, ) { a < b} Ininite Interval( 無限區間 ) [ a, ) { a < } (, b] {
More informationBOARD QUESTION PAPER : MARCH 2016 GEOMETRY
BOARD QUESTION PAPER : MARCH 016 GEOMETRY Time : Hours Total Marks : 40 Note: (i) Solve All questions. Draw diagram wherever necessary. (ii) Use of calculator is not allowed. (iii) Diagram is essential
More informationMultiple sequence alignment (MSA)
Multiple sequence alignment (MSA) From pairwise to multiple A T _ A T C A... A _ C A T _ A... A T _ G C G _... A _ C G T _ A... A T C A C _ A... _ T C G A G A... Relationship of sequences (Tree) NODE
More informationCh.9 Liquids and Solids
Ch.9 Liquids and Solids 9.1. Liquid-Vapor Equilibrium 1. Vapor Pressure. Vapor Pressure versus Temperature 3. Boiling Temperature. Critical Temperature and Pressure 9.. Phase Diagram 1. Sublimation. Melting
More informationtan θ(t) = 5 [3 points] And, we are given that d [1 points] Therefore, the velocity of the plane is dx [4 points] (km/min.) [2 points] (The other way)
1051 微甲 06-10 班期中考解答和評分標準 1. (10%) A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is π/3, this angle is decreasing
More informationTriangles. Example: In the given figure, S and T are points on PQ and PR respectively of PQR such that ST QR. Determine the length of PR.
Triangles Two geometric figures having the same shape and size are said to be congruent figures. Two geometric figures having the same shape, but not necessarily the same size, are called similar figures.
More informationTopic 2 [312 marks] The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths
Topic 2 [312 marks] 1 The rectangle ABCD is inscribed in a circle Sides [AD] and [AB] have lengths [12 marks] 3 cm and (\9\) cm respectively E is a point on side [AB] such that AE is 3 cm Side [DE] is
More informationANSWER KEY & SOLUTIONS
PRE-HALFYEARLY ASSESSMENT- [P-H-A MATHS SYLLABUS] ANSWER KEY & SOLUTIONS General Instructions:. The question paper comprises of four sections, A, B, C & D.. All questions are compulsory. 3. Section A Q
More informationDESIGN OF THE QUESTION PAPER Mathematics Class X
SET-I DESIGN OF THE QUESTION PAPER Mathematics Class X Time : 3 Hours Maximum Marks : 80 Weightage and the distribution of marks over different dimensions of the question shall be as follows: (A) Weightage
More informationBasic Properties of Circles (II) 圓的基本特性 ( 二 )
Basic Properties of Circles (II) 圓的基本特性 ( 二 ) Exercises( 練習 ) 1. In the figure, AB is a diameter of the circle, DC is the tangent to the circle at D and BAD = 3. If ABC is a straight line, find x. 在圖中,AB
More information2008 Euclid Contest. Solutions. Canadian Mathematics Competition. Tuesday, April 15, c 2008 Centre for Education in Mathematics and Computing
Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario 008 Euclid Contest Tuesday, April 5, 008 Solutions c 008
More information邏輯設計 Hw#6 請於 6/13( 五 ) 下課前繳交
邏輯設計 Hw#6 請於 6/3( 五 ) 下課前繳交 . A sequential circuit with two D flip-flops A and B, two inputs X and Y, and one output Z is specified by the following input equations: D A = X A + XY D B = X A + XB Z = XB
More information2, find the value of a.
Answers: (99-9 HKMO Final Events) reated by: Mr. Francis Hung Last updated: 7 December 05 Individual Events I a I a 6 I a 4 I4 a 8 I5 a 0 b b 60 b 4 b 9 b c c 00 c 50 c 4 c 57 d d 50 d 500 d 54 d 7 Group
More informationCHAPTER 4. Thermochemistry ( 熱化學是熱力學的一支, 在化學反應或相變化過程中發生的能量吸收或釋出, 若以吸放熱的形式表現, 即為熱化學研究的對象 ) Chap. 4 Thermochemistry
CHAPTER 4 Thermochemistry ( 熱化學是熱力學的一支, 在化學反應或相變化過程中發生的能量吸收或釋出, 若以吸放熱的形式表現, 即為熱化學研究的對象 ) 1 2 4.1 Energy Stored in Chemical Bonds Is Released or Taken Up in Chemical Reactions 3 I 2.2 Food and Energy reserves
More informationCBSE MATHEMATICS (SET-2)_2019
CBSE 09 MATHEMATICS (SET-) (Solutions). OC AB (AB is tangent to the smaller circle) In OBC a b CB CB a b CB a b AB CB (Perpendicular from the centre bisects the chord) AB a b. In PQS PQ 4 (By Pythagoras
More information9.1 (10.1) Parametric Curves ( 參數曲線 )
9.1 (10.1) Parametric Curves ( 參數曲線 ) [Ex]Sketch and identify the curve defined by parametric equations x= 6 t, y = t, t 4 (a) Sketch the curve by using the parametric equations to plot points: (b) Eliminate
More informationChapter 20 Cell Division Summary
Chapter 20 Cell Division Summary Bk3 Ch20 Cell Division/1 Table 1: The concept of cell (Section 20.1) A repeated process in which a cell divides many times to make new cells Cell Responsible for growth,
More informationUNIT-8 SIMILAR TRIANGLES Geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art. 1. ABC is a right-angled triangle, right-angled
More information11 is the same as their sum, find the value of S.
Answers: (998-99 HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Individual Events I P 4 I a 8 I a 6 I4 a I5 a IS a Q 8 b 0 b 7 b b spare b 770 R c c c c 0 c 57 S 0 d 000 d 990 d
More informationLocating a point: Introduction the coordinates system. In mathematics and daily life we often need to describe the position of an object.
Precaution: This chapter requires some knowledge in Elementar Geometr. Coordinates Sstem Locating a point: Introduction the coordinates sstem In mathematics and dail life we often need to describe the
More informationSOLUTIONS FOR IMO 2005 PROBLEMS
SOLUTIONS FOR IMO 005 PROBLEMS AMIR JAFARI AND KASRA RAFI Problem. Six points are chosen on the sides of an equilateral triangle AB: A, A on B; B, B on A;, on AB. These points are the vertices of a convex
More informationPre-Regional Mathematical Olympiad Solution 2017
Pre-Regional Mathematical Olympiad Solution 07 Time:.5 hours. Maximum Marks: 50 [Each Question carries 5 marks]. How many positive integers less than 000 have the property that the sum of the digits of
More information6 CHAPTER. Triangles. A plane figure bounded by three line segments is called a triangle.
6 CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius A plane figure bounded by three line segments is called a triangle We denote a triangle by the symbol In fig ABC has
More informationcomplicated calculations, similar to HKDSE difficult types ** Please answer any 30 questions ** Level 1 In the figure, sin θ =
Level : single concept, simple calculation, easier than HKDSE types Level : one or two concept, some calculations, similar to HKDSE easy types Level 3: involving high level, logical and abstract thinking
More informationChapter 6. Series-Parallel Circuits ISU EE. C.Y. Lee
Chapter 6 Series-Parallel Circuits Objectives Identify series-parallel relationships Analyze series-parallel circuits Determine the loading effect of a voltmeter on a circuit Analyze a Wheatstone bridge
More informationMATHEMATICS. Time allowed : 3 hours Maximum Marks : 100 QUESTION PAPER CODE 30/1/1 SECTION - A
MATHEMATICS Time allowed : 3 hours Maximum Marks : 100 GENERAL INSTRUCTIONS : 1. All questions are compulsory 2. The question paper consists of 30 questions divided into four sections - A, B, C and D.
More informationCBSE CLASS X MATH -SOLUTION Therefore, 0.6, 0.25 and 0.3 are greater than or equal to 0 and less than or equal to 1.
CBSE CLASS X MATH -SOLUTION 011 Q1 The probability of an event is always greater than or equal to zero and less than or equal to one. Here, 3 5 = 0.6 5% = 5 100 = 0.5 Therefore, 0.6, 0.5 and 0.3 are greater
More informationCCE PR Revised & Un-Revised
D CCE PR Revised & Un-Revised 560 00 KARNATAKA SECONDARY EDUCATION EXAMINATION BOARD, MALLESWARAM, BANGALORE 560 00 08 S.S.L.C. EXAMINATION, JUNE, 08 :. 06. 08 ] MODEL ANSWERS : 8-K Date :. 06. 08 ] CODE
More informationWork Energy And Power 功, 能量及功率
p. 1 Work Energy And Power 功, 能量及功率 黃河壺口瀑布 p. 2 甚麼是 能量? p. 3 常力所作的功 ( Work Done by a Constant Force ) p. 4 F F θ F cosθ s 要有出力才有 功 勞 造成位移才有 功 勞 W = F cos θ s ( Joule, a scalar ) = F s or F Δx F : force,
More information2018 LEHIGH UNIVERSITY HIGH SCHOOL MATH CONTEST
08 LEHIGH UNIVERSITY HIGH SCHOOL MATH CONTEST. A right triangle has hypotenuse 9 and one leg. What is the length of the other leg?. Don is /3 of the way through his run. After running another / mile, he
More informationCBSE CLASS-10 MARCH 2018
CBSE CLASS-10 MARCH 2018 MATHEMATICS Time : 2.30 hrs QUESTION & ANSWER Marks : 80 General Instructions : i. All questions are compulsory ii. This question paper consists of 30 questions divided into four
More information37th United States of America Mathematical Olympiad
37th United States of America Mathematical Olympiad 1. Prove that for each positive integer n, there are pairwise relatively prime integers k 0, k 1,..., k n, all strictly greater than 1, such that k 0
More information台灣大學開放式課程 有機化學乙 蔡蘊明教授 本著作除另有註明, 作者皆為蔡蘊明教授, 所有內容皆採用創用 CC 姓名標示 - 非商業使用 - 相同方式分享 3.0 台灣授權條款釋出
台灣大學開放式課程 有機化學乙 蔡蘊明教授 本著作除另有註明, 作者皆為蔡蘊明教授, 所有內容皆採用創用 姓名標示 - 非商業使用 - 相同方式分享 3.0 台灣授權條款釋出 hapter S Stereochemistry ( 立體化學 ): chiral molecules ( 掌性分子 ) Isomerism constitutional isomers butane isobutane 分子式相同但鍵結方式不同
More information1983 FG8.1, 1991 HG9, 1996 HG9
nswers: (1- HKMO Heat Events) reated by: Mr. Francis Hung Last updated: 6 February 017 - Individual 1 11 70 6 1160 7 11 8 80 1 10 1 km 6 11-1 Group 6 7 7 6 8 70 10 Individual Events I1 X is a point on
More information授課大綱 課號課程名稱選別開課系級學分 結果預視
授課大綱 課號課程名稱選別開課系級學分 B06303A 流體力學 Fluid Mechanics 必 結果預視 課程介紹 (Course Description): 機械工程學系 三甲 3 在流體力學第一課的學生可能會問 : 什麼是流體力學? 為什麼我必須研究它? 我為什麼要研究它? 流體力學有哪些應用? 流體包括液體和氣體 流體力學涉及靜止和運動時流體的行為 對流體力學的基本原理和概念的了解和理解對分析任何工程系統至關重要,
More informationKARNATAKA SECONDARY EDUCATION EXAMINATION BOARD, MALLESWARAM, BANGALORE G È.G È.G È.. Æ fioê, d È 2018 S. S. L. C. EXAMINATION, JUNE, 2018
CCE RR REVISED & UN-REVISED O %lo ÆË v ÃO y Æ fio» flms ÿ,» fl Ê«fiÀ M, ÊMV fl 560 00 KARNATAKA SECONDARY EDUCATION EXAMINATION BOARD, MALLESWARAM, BANGALORE 560 00 G È.G È.G È.. Æ fioê, d È 08 S. S. L.
More informationMathematics. Sample Question Paper. Class 10th. (Detailed Solutions) Mathematics Class X. 2. Given, equa tion is 4 5 x 5x
Sample Question Paper (Detailed Solutions Matematics lass 0t 4 Matematics lass X. Let p( a 6 a be divisible by ( a, if p( a 0. Ten, p( a a a( a 6 a a a 6 a 6 a 0 Hence, remainder is (6 a.. Given, equa
More informationMaharashtra State Board Class X Mathematics - Geometry Board Paper 2016 Solution
Maharashtra State Board Class X Mathematics - Geometry Board Paper 016 Solution 1. i. ΔDEF ΔMNK (given) A( DEF) DE A( MNK) MN A( DEF) 5 5 A( MNK) 6 6...(Areas of similar triangles) ii. ΔABC is 0-60 -90
More informationEXPERMENT 9. To determination of Quinine by fluorescence spectroscopy. Introduction
EXPERMENT 9 To determination of Quinine by fluorescence spectroscopy Introduction Many chemical compounds can be excited by electromagnetic radication from normally a singlet ground state S o to upper
More information2 13b + 37 = 54, 13b 37 = 16, no solution
Answers: (999-00 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 6 February 07 Individual Events SI P 6 I P 5 I P 6 I P I P I5 P Q 7 Q 8 Q 8 Q Q Q R R 7 R R 996 R R S 990 S 6 S S 666 S S
More informationPaper: 02 Class-X-Math: Summative Assessment - I
1 P a g e Paper: 02 Class-X-Math: Summative Assessment - I Total marks of the paper: 90 Total time of the paper: 3.5 hrs Questions: 1] The relation connecting the measures of central tendencies is [Marks:1]
More informationDigital Image Processing
Dgtal Iage Processg Chater 08 Iage Coresso Dec. 30, 00 Istructor:Lh-Je Kau( 高立人 ) Deartet of Electroc Egeerg Natoal Tae Uversty of Techology /35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology
More informationVectors. Vectors ( 向量 ) Representation of Vectors. Special Vectors. Equal vectors. Chapter 16
Vectors ( 向量 ) Chapter 16 2D Vectors A vector is a line which has both magnitde and direction. For example, in a weather report yo may hear a statement like the wind is blowing at 25 knots ( 海浬 ) in the
More information2016 SEC 4 ADDITIONAL MATHEMATICS CW & HW
FEB EXAM 06 SEC 4 ADDITIONAL MATHEMATICS CW & HW Find the values of k for which the line y 6 is a tangent to the curve k 7 y. Find also the coordinates of the point at which this tangent touches the curve.
More informationReview exercise 2. 1 The equation of the line is: = 5 a The gradient of l1 is 3. y y x x. So the gradient of l2 is. The equation of line l2 is: y =
Review exercise The equation of the line is: y y x x y y x x y 8 x+ 6 8 + y 8 x+ 6 y x x + y 0 y ( ) ( x 9) y+ ( x 9) y+ x 9 x y 0 a, b, c Using points A and B: y y x x y y x x y x 0 k 0 y x k ky k x a
More informationKARNATAKA SECONDARY EDUCATION EXAMINATION BOARD, MALLESWARAM, BANGALORE G È.G È.G È.. Æ fioê, d È 2018 S. S. L. C. EXAMINATION, JUNE, 2018
CCE PR REVISED & UN-REVISED O %lo ÆË v ÃO y Æ fio» flms ÿ,» fl Ê«fiÀ M, ÊMV fl 560 00 KARNATAKA SECONDARY EDUCATION EXAMINATION BOARD, MALLESWARAM, BANGALORE 560 00 G È.G È.G È.. Æ fioê, d È 08 S. S. L.
More informationIt is known that the length of the tangents drawn from an external point to a circle is equal.
CBSE -MATHS-SET 1-2014 Q1. The first three terms of an AP are 3y-1, 3y+5 and 5y+1, respectively. We need to find the value of y. We know that if a, b and c are in AP, then: b a = c b 2b = a + c 2 (3y+5)
More information5.5 Using Entropy to Calculate the Natural Direction of a Process in an Isolated System
5.5 Using Entropy to Calculate the Natural Direction of a Process in an Isolated System 熵可以用來預測自發改變方向 我們現在回到 5.1 節引入兩個過程 第一個過程是關於金屬棒在溫度梯度下的自然變化方向 試問, 在系統達平衡狀態時, 梯度變大或更小? 為了模擬這過程, 考慮如圖 5.5 的模型, 一孤立的複合系統受
More informationANSWER KEY MATHS P-SA- 1st (FULL SA-1 SYLLABUS) Std. X
ANSWER KEY MATHS P-SA- 1st (FULL SA-1 SYLLABUS) Std. X General Instructions: 1. The question aer comrises of four sections, A, B, C & D.. All questions are comulsory. 3. Section A Q 1 to 4 are 1 mark each
More information1 dx (5%) andˆ x dx converges. x2 +1 a
微甲 - 班期末考解答和評分標準. (%) (a) (7%) Find the indefinite integrals of secθ dθ.) d (5%) and + d (%). (You may use the integral formula + (b) (%) Find the value of the constant a for which the improper integral
More informationCAREER POINT. PRMO EXAM-2017 (Paper & Solution) Sum of number should be 21
PRMO EXAM-07 (Paper & Solution) Q. How many positive integers less than 000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3? Sum
More information2003 AIME Given that ((3!)!)! = k n!, where k and n are positive integers and n is as large as possible, find k + n.
003 AIME 1 Given that ((3!)!)! = k n!, where k and n are positive integers and n is as large 3! as possible, find k + n One hundred concentric circles with radii 1,, 3,, 100 are drawn in a plane The interior
More informationSAMPLE QUESTION PAPER Class-X ( ) Mathematics. Time allowed: 3 Hours Max. Marks: 80
SAMPLE QUESTION PAPER Class-X (017 18) Mathematics Time allowed: 3 Hours Max. Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided
More information大原利明 算法点竄指南 点竄術 算 額 絵馬堂
算額 大原利明 算法点竄指南 点竄術 算 額 絵馬 絵 馬 絵馬堂 甲 一 乙 二 丙 三 丁 四 戊 五 己 六 庚 七 辛 八 壬 九 癸 十 十五 二十 百 千 甲 乙 丙 丁 傍書法 関孝和 点竄術 松永良弼 甲 甲 甲 甲 甲 乙 甲 甲 乙 甲 乙 甲 乙 甲 乙 丙 丁 戊 a + b 2 c 甲 a a 二乙 2 b b 2 小 c c SOLVING SANGAKU 5 3.1.
More informationCBSE QUESTION PAPER CLASS-X MATHS
CBSE QUESTION PAPER CLASS-X MATHS SECTION - A Question 1:If sin α = 1 2, then the value of 4 cos3 α 3 cos α is (a)0 (b)1 (c) 1 (d)2 Question 2: If cos 2θ = sin(θ 12 ), where2θ and (θ 12 ) are both acute
More informationCBSE CLASS-10 MARCH 2018
CBSE CLASS-10 MARCH 2018 MATHEMATICS Time : 2.30 hrs QUESTION Marks : 80 General Instructions : i. All questions are compulsory ii. This question paper consists of 30 questions divided into four sections
More informationThe CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca Euclid Contest. Wednesday, April 15, 2015
The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca 015 Euclid Contest Wednesday, April 15, 015 (in North America and South America) Thursday, April 16, 015 (outside of North America
More informationSample Question Paper Mathematics First Term (SA - I) Class X. Time: 3 to 3 ½ hours
Sample Question Paper Mathematics First Term (SA - I) Class X Time: 3 to 3 ½ hours M.M.:90 General Instructions (i) All questions are compulsory. (ii) The question paper consists of 34 questions divided
More information14 th Annual Harvard-MIT Mathematics Tournament Saturday 12 February 2011
14 th Annual Harvard-MIT Mathematics Tournament Saturday 1 February 011 1. Let a, b, and c be positive real numbers. etermine the largest total number of real roots that the following three polynomials
More informationInternational Mathematical Olympiad. Preliminary Selection Contest 2004 Hong Kong. Outline of Solutions 3 N
International Mathematical Olympiad Preliminary Selection Contest 004 Hong Kong Outline of Solutions Answers:. 8. 0. N 4. 49894 5. 6. 004! 40 400 7. 6 8. 7 9. 5 0. 0. 007. 8066. π + 4. 5 5. 60 6. 475 7.
More informationPRMO _ (SOLUTIONS) [ 1 ]
PRMO 07-8_0-08-07 (SOLUTIONS) [ ] PRMO 07-8 : QUESTIONS & SOLUTIONS. How many positive integers less than 000 have the property that the sum of the digits of each such number is divisible by 7 and the
More informationMECHANICS OF MATERIALS
CHAPTER 2 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Lecture Notes: J. Walt Oler Texas Tech University Stress and Strain Axial Loading 2.1 An Introduction
More informationNEW YORK CITY INTERSCHOLASTIC MATHEMATICS LEAGUE Senior A Division CONTEST NUMBER 1
Senior A Division CONTEST NUMBER 1 PART I FALL 2011 CONTEST 1 TIME: 10 MINUTES F11A1 Larry selects a 13-digit number while David selects a 10-digit number. Let be the number of digits in the product of
More informationLinear Regression. Applied Linear Regression Models (Kutner, Nachtsheim, Neter, Li) hsuhl (NUK) SDA Regression 1 / 34
Linear Regression 許湘伶 Applied Linear Regression Models (Kutner, Nachtsheim, Neter, Li) hsuhl (NUK) SDA Regression 1 / 34 Regression analysis is a statistical methodology that utilizes the relation between
More informationCBSE Sample Question Paper 1 ( )
CBSE Sample Question Paper (07-8 Time: Hours Maximum Marks: 80 General Instructions: (i All questions are compulsory. (ii The question paper consists of 0 questions divided into four sections A, B, C and
More informationMT - MATHEMATICS (71) GEOMETRY - PRELIM II - PAPER - 6 (E)
04 00 Seat No. MT - MTHEMTIS (7) GEOMETRY - PRELIM II - (E) Time : Hours (Pages 3) Max. Marks : 40 Note : ll questions are compulsory. Use of calculator is not allowed. Q.. Solve NY FIVE of the following
More informationOlimpiada Matemática de Centroamérica y el Caribe
Olimpiada Matemática de Centroamérica y el Caribe 1st Centromerican 1999 Problem A1 A, B, C, D, E each has a unique piece of news. They make a series of phone calls to each other. In each call, the caller
More informationDISCUSSION CLASS OF DAX IS ON 22ND MARCH, TIME : 9-12 BRING ALL YOUR DOUBTS [STRAIGHT OBJECTIVE TYPE]
DISCUSSION CLASS OF DAX IS ON ND MARCH, TIME : 9- BRING ALL YOUR DOUBTS [STRAIGHT OBJECTIVE TYPE] Q. Let y = cos x (cos x cos x). Then y is (A) 0 only when x 0 (B) 0 for all real x (C) 0 for all real x
More information在雲層閃光放電之前就開始提前釋放出離子是非常重要的因素 所有 FOREND 放電式避雷針都有離子加速裝置支援離子產生器 在產品設計時, 為增加電場更大範圍, 使用電極支援大氣離子化,
FOREND E.S.E 提前放電式避雷針 FOREND Petex E.S.E 提前放電式避雷針由 3 個部分組成 : 空中末端突針 離子產生器和屋頂連接管 空中末端突針由不鏽鋼製造, 有適合的直徑, 可以抵抗強大的雷擊電流 離子產生器位於不鏽鋼針體內部特別的位置, 以特別的樹脂密封, 隔絕外部環境的影響 在暴風雨閃電期間, 大氣中所引起的電場增加, 離子產生器開始活化以及產生離子到周圍的空氣中
More informationOBJECTIVE TEST. Answer all questions C. N3, D. N3, Simplify Express the square root of in 4
. In a particular year, the exchange rate of Naira (N) varies directly with the Dollar ($). If N is equivalent to $8, find the Naira equivalent of $6. A. N8976 B. N049 C. N40. D. N.7. If log = x, log =
More informationApTutorGroup. SAT II Chemistry Guides: Test Basics Scoring, Timing, Number of Questions Points Minutes Questions (Multiple Choice)
SAT II Chemistry Guides: Test Basics Scoring, Timing, Number of Questions Points Minutes Questions 200-800 60 85 (Multiple Choice) PART A ----------------------------------------------------------------
More informationWrite your Name, Registration Number, Test Centre, Test Code and the Number of this booklet in the appropriate places on the answersheet.
2009 Booklet No. Test Code : SIA Forenoon Questions : 30 Time : 2 hours Write your Name, Registration Number, Test Centre, Test Code and the Number of this booklet in the appropriate places on the answersheet.
More informationKENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32
KENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32 SAMPLE PAPER 05 (2018-19) SUBJECT: MATHEMATICS(041) BLUE PRINT : CLASS X Unit Chapter VSA (1 mark) SA I (2 marks) SA II (3 marks) LA (4 marks) Total Unit
More informationMaharashtra Board Class X Mathematics - Geometry Board Paper 2014 Solution. Time: 2 hours Total Marks: 40
Maharashtra Board Class X Mathematics - Geometry Board Paper 04 Solution Time: hours Total Marks: 40 Note: - () All questions are compulsory. () Use of calculator is not allowed.. i. Ratio of the areas
More information國立中正大學八十一學年度應用數學研究所 碩士班研究生招生考試試題
國立中正大學八十一學年度應用數學研究所 碩士班研究生招生考試試題 基礎數學 I.(2%) Test for convergence or divergence of the following infinite series cos( π (a) ) sin( π n (b) ) n n=1 n n=1 n 1 1 (c) (p > 1) (d) n=2 n(log n) p n,m=1 n 2 +
More informationMathematics CLASS : X. Time: 3hrs Max. Marks: 90. 2) If a, 2 are three consecutive terms of an A.P., then the value of a.
1 SAMPLE PAPER 4 (SAII) MR AMIT. KV NANGALBHUR Mathematics CLASS : X Time: 3hrs Max. Marks: 90 General Instruction:- 1. All questions are Compulsory. The question paper consists of 34 questions divided
More information11 th Philippine Mathematical Olympiad Questions, Answers, and Hints
view.php3 (JPEG Image, 840x888 pixels) - Scaled (71%) https://mail.ateneo.net/horde/imp/view.php3?mailbox=inbox&inde... 1 of 1 11/5/2008 5:02 PM 11 th Philippine Mathematical Olympiad Questions, Answers,
More informationFILL THE ANSWER HERE
HOM ASSIGNMNT # 0 STRAIGHT OBJCTIV TYP. If A, B & C are matrices of order such that A =, B = 9, C =, then (AC) is equal to - (A) 8 6. The length of the sub-tangent to the curve y = (A) 8 0 0 8 ( ) 5 5
More informationCBSE QUESTION PAPER CLASS-X MATHS
CBSE QUESTION PAPER CLASS-X MATHS SECTION - A Question 1: In figure, AB = 5 3 cm, DC = 4cm, BD = 3cm, then tan θ is (a) (b) (c) (d) 1 3 2 3 4 3 5 3 Question 2: In figure, what values of x will make DE
More informationEdexcel New GCE A Level Maths workbook Circle.
Edexcel New GCE A Level Maths workbook Circle. Edited by: K V Kumaran kumarmaths.weebly.com 1 Finding the Midpoint of a Line To work out the midpoint of line we need to find the halfway point Midpoint
More informationTRIANGLES CHAPTER 7. (A) Main Concepts and Results. (B) Multiple Choice Questions
CHAPTER 7 TRIANGLES (A) Main Concepts and Results Triangles and their parts, Congruence of triangles, Congruence and correspondence of vertices, Criteria for Congruence of triangles: (i) SAS (ii) ASA (iii)
More information( )( ) PR PQ = QR. Mathematics Class X TOPPER SAMPLE PAPER-1 SOLUTIONS. HCF x LCM = Product of the 2 numbers 126 x LCM = 252 x 378
Mathematics Class X TOPPER SAMPLE PAPER- SOLUTIONS Ans HCF x LCM Product of the numbers 6 x LCM 5 x 378 LCM 756 ( Mark) Ans The zeroes are, 4 p( x) x + x 4 x 3x 4 ( Mark) Ans3 For intersecting lines: a
More informationPRMO Solution
PRMO Solution 0.08.07. How many positive integers less than 000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3?. Suppose a, b
More informationMATHEMATICS. 61. If letters of the word KUBER are written in all possible orders and arranged as in a dictionary, then rank of the word KUBER will be:
MATHEMATICS 61. If letters of the word KUBER are written in all possible orders and arranged as in a dictionary, then rank of the word KUBER will be: (A) 67 (B) 68 (C) 65 (D) 69 : Alphabetical order of
More information