英国高考A-LEVEL(CIE考试局)数学Mathematics 2014年 AS A2 考试大纲
2014年AMC_12真题 (B)

(C)
75 2
(D) 40
(E)
300 7
4. Susie pays for 4 muffins and 3 bananas. Calvin spends twice as much paying for 2 muffins and 16 bananas. A muffin is how many times as expensive as a banana?
15. When p 6 , the number e p is an integer. What is the largest power of 2 that is k 1k ln k a factor of e p ? (A) 212 (B) 214 (C) 216 (D) 218 (E) 220
r and s such that the line through Q with slope m does not intersect P if and only if r < m < s. What is r + s? (A)1 (B)26 (C)40 (D)52 (E)80
18. The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is bad if it is not true that for every n from 1 to 15 one can find a subset of the numbers that appear consecutively on the circle that sum to n. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
Mathematics - Further CIE A-LEVEL 进阶数学2013考试大纲

1.3 Cambridge Advanced International Certificate of Education (AICE)
Cambridge AICE is the group award of Cambridge International Advanced Supplementary Level and Advanced Level (AS Level and A Level). Cambridge AICE involves the selection of subjects from three curriculum groups – Mathematics and Science; Languages; Arts and Humanities. An A Level counts as a double-credit qualification and an AS Level as a single-credit qualification within the Cambridge AICE award framework. To be considered for an AICE Diploma, a candidate must earn the equivalent of six credits by passing a combination of examinations at either double credit or single credit, with at least one course coming from each of the three curriculum areas. The examinations are administered in May/June and October/November sessions each year. Further Mathematics (9231) falls into Group A, Mathematics and Sciences. Learn more about AICE at /qualifications/academic/uppersec/aice.
CIE通识培训

评估选择案例:
第一年学习数、理、化、经济,参加AS考试 第二年学习数、理、化,参加A2考试 此考生最后获得3.5个A LEVEL成绩证书(证书样本p21,63)
ICE证书介绍(IGCSE)
课程选择
Group I: Chinese、English、Korean
Group II:Economics、Geography、Global Perspectives
Group III: Physics、Chemistry、Biology Group IV:Additional Mathematics、Mathematics Group V:Accounting、Art and Design、Business Studies、Computer studies 选科要求:从Group I 中选两科;从Group II,III,IV、V四组中各选一科;另外再 从五组中任选一科---共计选修七科。
Over 450 US universities accept Cambridge International AS and A Levels, including all Ivy League and Ivy Plus universities. These universities include: Brown University, Columbia University, Cornell University, Dartmouth College, Harvard University, Massachusetts Institute of Technology (MIT), Princeton University, Stanford University, the University of Pennsylvania and Yale University.
英国A-level数学教材内容汇总

A-Leve l数学(Mathe matl cs)由四亍郃分姐成.换业数学・C ore Mathe matlcs h 力学数学t M ech an les Mathemati csx 轨计數H (Stali^tl cs Mathematitsy 决第數学Decision M ath&mati c& □选择学耳数学(Mathematics)^生,際了孩心数学心时Mathemahcs:^必修的基础数学之尔学生还需^据自己将来的犬学报读若业选择茸中T磯学『力Mechanics Mathematics},统计数学:Statistics f.fathemstics}.决董数学Decision Mathematics]・50将采读工程删]字主.可追力学数学谢xhanlcs);读社会科学觀金融经桥类的.可选:比计數字(Slatistlcs):僂计算机嗽件类的.町选: 决策数韋Decision Maltieinalics^Core Mathematicsl (AS/A2) ------ 核心数学11. Algebra and fun ctio ns --- 代数和函数2. Quadratic functions ----- 二次函数3. Equati ons and in equalities --- 等式和不等式4. Sketchi ng curves ----- 画图(草图)5. Coordinate geometry in the (x, y) plane--------- 平面坐标系中的坐标几何6. Seque nces and series——数歹U7. Differe ntiation ------ 微分8. In tegrati on --- 积分Core Mathematics2 (AS/A2) ----- 核心数学21. Algebra and fun ctio ns --- 代数和函数2. The sine and cos ine rule ---- 正弦和余弦定理3. Expo nen tials and logarithm ----- 指数和对数4. Coordinate geometry in the (x, y) plane--------- 平面坐标系中的坐标几何5. The bi no mial expa nsion --- 二项展开式6. Radia n measure and its applicati on --- 弧度制及其应用7. Geometric seque nces and series ---- 等比数歹U8. Graphs of trig ono metric functions ----- 三角函数的图形9. Differe ntiation ------ 微分10. Trigonometric identities and simple equations ------ 三角恒等式和简单的三角等式11. I ntegration ---- 积分Core Mathematics3 (AS/A2) ----- 核心数学31. Algebra fractio ns ------ 分式代数2. Functions ------ 函数3. The expo nen tial and log fun ctio ns --- 指数函数和对数函数4. Numerical method ----- 数值法5. Tran sform ing graph of functions ---- 函数的图形变换6. Trigon ometry ------- 三角7. Further trig ono metric and their applicati ons ---- 高级三角恒等式及其应用8. Differe ntiation ------ 微分Core Mathematics4 (AS/A2) ----- 核心数学41. Partial fractio ns ---- 部分分式2. Coordinate geometry in the (x, y) plane--------- 平面坐标系中的坐标几何3. The bi no mial expa nsion --- 二项展开式4. Differe ntiation ------ 微分5. Vectors ----- 向量6. In tegrati on --- 积分A-Level :核心数学 Core Maths ,力学数学,统计数学,决策数学 1 2 3 4 5 6 7oio14 14 1520 24 252b3S丽48b4Core Mathematics1 (AS/A2 ) ----- 核心数学 1 8. In tegrati on ------ 积分 每章内容:SketclSketching < \4.1 4.2 4.3 4.44.5Quadratic functions 2A2.2 2.3 2.4 2.5 2.6 1 Algebra and futictlons1.1 1.2 L3 1用 IS L6 L7 L8 Summary of key poinisPlotting the s^phs of Solvingquadratic eqi 】“ 沪 巧 Completingthe: Solving quddratiuSolving quadrate t ions by “Sketching Z> JSummary of kfy 尸為Equations 匚 M Solving sinSolving simultaneob.Using substISolving linGi 『 in 何亦It 華& Solving quadratic^^^ii^sj.jtions by elimination 屈tion* by substitutinn f equation Is linear and the other is quadraticSimplifying an expression by collecting like termsThe laws Qf indicesExpanding an expression Factorhing an expressionFactorising a quadr^k expressionThv las\s of indices for dll rational exponents The use and nianipulation of it rdsRationalising the iknonnridtor of a fraction ivhen 才二dw: XCxsE by ractor i sa ti映>;肯『 c equtijLArby comgTfctjng th 」square .' 'J u iht^m uh Quadratic formulae ; ^ncc^is liiKar rq 3.13.2 33 3J 3.5Suiiunary of 匕叮心疋试he 护ph$ of cubic functions Interpr^lW^yaphs nf cubk fuiKtioiuSketch inutile reciprocal function JK ■ttivinicr sect ion points of o[ functions to solv< equations of the triinsfbr mat ions f(x + ⑷ dnd 冃工-川 舉effect of the transforiiiations fiux) and'Fftrfotming transformations on the sketches of curves詁ry of key pointsAlgebra and fun ctio ns ----- 代数和函数 Quadratic functions ----- 二次函数Equati ons and in equalities --- 等式和不等式 Sketchi ng curves --- 画图(草图)Coordinate geometry in the (x , y ) plane -------- 平面坐标系中的坐标几何 Seque nces and series ——数列Differe ntiation ------ 微分5 Coordinate geonwtry in the (x9 y) plane 6S5.1 The equation of a straight line in the form y = nix + c or ax + 如 + c = 0 655.2 The gradient of a straight line 605.3 The equation of a straight line of the form y - y严ifi(x - 心) 7&5.4 The formula for finding the equation of a straight line5.5 The conditions for two straight lines to be parallel or perpendicular ' 75Summary of key points6 Sequences and series6.1 Introduction to sequences6.2 The nth term of a sequence 0 836.3 Sequences generated by a recurrence relationship i \ // ()856.4 Arithmetic sequences 二886.5 Arithmetic series \C/^ \ °°6.6 The sum to n of an arithmetic series 936.7 Using X notation 」97Summary of key points ' 〃丿) 101 7 Differentiation (//. 1027」The derivative of f(x) as the thiCpn^kto tft^ graph y = f(x) 102105109113114115116117121122122124125126128130Core Mathematics2 (AS/A2 ) 核心数学 21. Algebra and fun ctio ns ---- 代数和函数2. The sine and cos ine rule ---- 正弦和余弦定理3. Expo nen tials and logarithm ---- 指数和对数4. Coordinate geometry in the (x , y ) plane -------- 平面坐标系中的坐标几何5. The bi no mial expa nsion --- 二项展开式6. Radia n measure and its applicati on --- 弧度制及其应用7. Geometric seque nces and series --- 等比数歹 U8. Graphs of trig ono metric functions ---- 三角函数的图形9. Differe ntiation ------ 微分 10. Trigonometric identities and simple equations ------ 三角恒等式和简单的三角等式 11. I ntegration ---- 积分 每章内容: Aigcbrj dEid luiKtions 1J1.2 13 1.4 Simplifying algebraic fractions by division Dividing apolynomial by (x i p)Factorising a polynomial using the Factor TheoremUsing the Remainder Theorem Summary of k (?y pointsThe sint : and cosine ruleUsing the sine rule to find missing sides Using the sine nde find unF^wn angles The rule andfinding two w* Using the cosine ruEc ia Fin# Using the cosine rule tc a Using the sine tl «Calculaikng the area 2A 2.2 23 2.4 2.5 2.6 2.7 f or a nih^F Eo切 Mck ssing an^ic^ L . ■ #4RI le 3 nr< !'『 庶耳竝遁 Theo re mot^Jy^ngle us)闵jExponctuiah an<r^ogaMh * 3J王2 玉33.43.5 3.6Summary of key pointsCk Coordinate in the (x, y\ plant4.1 The 4.2 The ciiibi Suniinjjy of key polrt 115 10 131718 18 21 23 24 27 30 32 36 37 37 39 4() 41 43 45 4ti 49 49 57 60 68 70 70 72 73 75 79tnsTh<bfunctk 严 Writing ns as a Calculating *丄耳 to Laws of JogarithmS Solvi ng equations 汐 a' - b Changing the mt ni ot A line M 峥曲亡two points on a line 4*3 The equatitJiiif a circle Summiiry of fr r/ points iriomTal expansion s triangle X Combinntions and factorialUsing (:) m the binomial expansion5-4 Expanding (d + bxY r using the binomial expansion Summary of key point*11o Kaaian measure ana its applications Using radians to measure angles The length ofthe arc of a circle The area of a sector of a circle The area of a segment of a circle Geometric sequences Geometric progressions and the nth term Usinggeometric sequences to solve problems The sum of a geometric seriesThe sum to infinity of a geometric series Graphs of trigonometric functionsSine, cosine andtangent(unctionsThe values of trigonomef/ functions in the Exact values and surds f Graphs of sine 0f cos J J 、 Simple transformants oDifferentiation9.1 Increasing s ・9.2 Stationarymaximun 、, minipjum and points of inflexion 9.3Using f^rninjf points to Summar 1 “ 亠 inisTrigonom^/ Jidentitie】0.161 6.2 6.3 6.4Summary of key pointsGeometric sequences and series 7.1 7.2 7.3 7.4 7.5Summary of key points8.1 8.2 8.3 8.4 &5Summary of key poE ;' le equations titles ometrical equations e formsin(nd + a), cos(n0 + a) and tan(n0 + a) = k ig?nometrical equationsSimple trigoSolving simj SolvingeqySolving qudIntegratio11.1 11.2 11.3 Are n.4 94 94 9598 100 103 109 110 110 114 117 118 121 127 141 141 146 149 151 156157 157 159 161 164 169 17710129 129 131135 1406 93 ms10.210.310.4Summary ote integrationa curvea curve that gives negative values n a straight line and a curve rapezium Rule of key points11Core Mathematics3 (AS/A2 ) 核心数学 31 2 3 4 5 6 7 8101Ki.S1 12每章内容:7.5 i'hc racloi tbrinuiai'Alxvbrdit Iriiciions1.1 1.2 1.3I'rigonomctry64 6.2AjipJying a corn^ixiatj Sketching trar^8 Differentiation8.1 B.2 8.3 K 4 8.58.68.71281301311322 E r unctioi-i^2,1 2.2 2-3 2.4 2.5Differentiating ti&ing the chain rule Differ ent tatlng using the product rult Differs nt latL ng using the quotient rukr I if fere nt iat j ng the exponential function Finding the differential of the logarithmic function. _Differentiating 5in x(C~Di fferenti ati ng cos xDifferent is tin^ t^n xDifferenliatkng further trigonometrLcaJ functions[differentiating functians formed by combining frigon 九丁贰#乎卜 cxprtncniiaL logfkritl-imLc and polynomial fLinctior^ ;Simplify algebraic fractions by LUI 1{.C IL UI ^ 口 Multiply dix jdici^frjLiujii->Adding and subtracting algvbrd k frautionsI nx alxvbriiit fr*ittic.jri^ jind tiir rcn )i»iii<lvr Ltit.-c.i^int <ln y 4-cn U>iTi 耳$ [JCX JJ€»TLOkl t k J I fLIIlC'Floriiir^^of <> gr^jj^/ica11 y 「_ ___2 =」cth^Js^lcrlind approximate root 萤 of 陶仟彳Tran 露Fermi 订呂 graf^/of fui^ctiini^5-1 Sketch! tig graphs ot 1^hhockx!^^ 4^u net ion y 一 lf(x)l 5u2 Sketching g^r^phs y = f(lxl) (A p olvin^a mcxluliis mictions to sketch erv«?« 什fih 订花JJ CFIM H 11 台 Mlielling lhe co-ordinates ofgiven H, cosecant 仇 and cotaingEfU 丹 f ^tant 也 cosecant 优 and cotangent 甘 xpressicmsj proving iclentiti^ iind solving equations, usingMapping diingrarns and 耳of opaeiiitions ( JFunc-tions <irid functioii notatk>tiRange, mapping diagrams,, graphs and definitionsJUsing composite functions #*f 丿Finding and using inverse-The exponential and log f u net ion s°3/1 Introdticing exponent ial ・ rtions of lhe I'omCj^ . h 心 3・立 Graphs of exponential旷卞:」^ 前m 扌匸卩 存占;逆"二tlxfn 琴, 严U^irig 护 ^Eidinwu b©■—主亠二亠亠」一■■Numerical method? ” 4.1 Finding approx if 4.2 U^ing ilerati algebraic iiicthi l ^irs'lw^nnd approximate rt>ots The fijnr/Q?6?/I TieSimplifying £ sec 他 cowO?R and cot Hidcnlitles l 十 lan 2^ = $2H and 1 + cot-^ = cosec 2IJs.iriglmerse trigcinometricai Uinclions and their graphs7 ..Further tngimonietrk identities and theif applies HonsSt/LMrig addition trigoiionietrical lormulac二Using double an^lc trigoiiDmctrical farmulae7?T Solving equdtiom and proving Idcntltiics using doubk iirigle foniiuLie ^^7 4 Usin^ the fonii a cos b sin B lin striving trigonotnetrical piobiennAlgebra fractions ------ 分式代数 Functions ----- 函数Transforming graph of functions -------函数的图形变换 Trigonometry ----- 三角Further trigonometric and their applications ------ 高级三角恒等式及其应用 Differentiation ------ 微分The exponential and log functions Numerical method ------ 数值法指数函数和对数函数Core Mathematics3 (AS/A2 ) 核心数学4Core Mathematics3 (AS/A2 ) 核心数学 51 2 3 4 556 vector37074 11B4 2110ft6J1111126139 SI6264 fi2 «2 AIXJUL L l||\ ULHJK2A3.2 33 Exam style paperFormulae you need to know List of symbols and notation AnswersIndexof two vectors n of a straightAdding and subtracting algehraic fractionsPartial fractions with two linear factors in the denominatorPartial fractions with ttnee or mor^ linear factors in th<? denominator Partial tract ions with repeated linear factors in the denominator Improper fractions into partial fractions1 Partial fractions L:y 1 ■jitrgrating £t^ndard Junctions Integrating using the reverse chain rule Using trigonometric identities in integrationUsing partial fractions to Integrate expressionsUsing standard patterns to integrdle expre^iorr liitvgraUon by subtjtiti.ition Integration by parts Numericalintegration Integration to find ateas and volumes 1Using integration to solvedifferential equations Difkrtntiai rquatjom in context2 Cootdinate geometry in the (x, y) 2」Parametric equations used toParametric equations used to dtiine the uxirdin^tes ot a Using paranictrkequ 訓 UKndinate 驴oimtr* Converting paramet^. jitions into cartesian 世qiut 档才 Finding the itrea ^iidche airve given by pannr 严旷 ^quations3 Fhe binomial ex3,i UMII^ VtXUMl IU UtSUilW J-^JJJLS I ;In 2 or 3 dimensions 二,二二 55Cartesian toniponeidi Gf a \yytor in 2dimensionsCartesian components ol in 3 dimensio%7^; Extending 2 /悸幺?冲results io ]he seal;| The vect*[nUT^clrnjfetraighi line vector 戸理逖石kFx linesJo between two straight Using partial fracti>#w$ Kjtw tiiv ■binamiai expanjy^f \、 Different la Uon4.1 Differentki(I nti ;ons givenpararnetricaifrf/4 2 Diffenyitiating^uationwhich arc implicitO43 Diffett»y^a!ing the function a 1 4.4{垃 tSftitiibn and rates of change4.5 唏蛙少他rtrntjai equations 5 VecS^ ?<^54,Ve?tor d^fmitipns 4nd vector ^^iiAgrams r 、§,2 Vector arithmetic and the unit vectorThe binomial expulsion a - positive integral index Using the binomidexpand + l^x)"\ j ' 6. In tegrati on ------ 积分 每章内容:The bi no mial expa nsion --- 二项展开式 Differe ntiation ------微分 Vectors ----- 向量Partial fractio ns ---- 部分分式 Coordin ate geometry in the ( x , y ) pla ne 平面坐标系中的坐标几何。
a-level课程数学

a-level课程数学在A-level课程体系中,A-Level数学主要分为数学和进阶数学两个。
一、数学(Mathematics)数学由四个部分组成:1.核心数学(CoreMathematics)是必修的基础数学,有四个单元:C1-C4。
2.力学数学(MechanicsMathematics)有五个单元:M1-M5。
在中国,力学包含在物理中,而在国外,更多的是包含在数学中。
当然物理老师也可以教力学和数学,有时候甚至更好。
3.统计数学(StatisticsMathematics)有四个单元:S1-S4。
4.决策数学(DecisionMathematics)有二个单元:D1和D2。
选择了A-Level数学的学生,除了核心数学是(Core Mathematics)是必修的基础数学之外,学生还要根据自己将来在大学可能会报读的专业选择其中一门数学:力学数学(MechanicsMathematics)或统计数学(Statistics Mathematics)或决策数学(Decision Mathematics)。
而到底选择哪个数学,则要和将来打算报读的专业联系起来:比如以后工科专业的学生可以选择力学和数学;读社会科学或经济金融的任选统计数学;阅读计算机软件的可选决策数学。
二、进阶数学(Further Pure Mathematics)进阶数学分为:FP1、FP2和FP3三个单元。
如此算来,A-Level数学一共有十八个单元,也就是说就A-Level数学一个大科你最多是可以考3个A-Level成绩,拿三个A或A*。
但是这样选的学生凤毛麟角,因为力学和统计越到后面的学习越难,而且没有一个大学会这样要求。
当然学的越多越好,所以通常最多的是15个(2个半),13-14也是很普遍的。
一般是六个单元算一个A-Level数学,如C1、C2、C3、C4、M1、S1这六个组合在一起,如果每个得分都大于等于80分(即480分及以上),就获得一个A-Level数学的A。
IGCSE课程和A-LEVEL课程详解

IGCSE 和 A-LEVEL 课程介绍
1 英国教育体系概况
IGCSE 课程和 A-LEVEL 课程都是英国的教育体系,英国是一个有悠久教育传统的国家。 它的教育体系经过几百年的沿革,相当的完善和复杂,且具有非常大的灵活性。总体来说分 为三个阶段:
义务教育(Compulsory Education),延续教育(Further Education)和高等教育(Higher Education)。这里我们只讲义务教育中的小学教育和中学教育。
课程比GCSE课程稍难一些。GCSE和IGCSE课程一共有四个考试局:AQA,OCR,EDEXCEL,CIE,其
中英国本土学校基本选择前两个考试局,这两个考试局在四个考试局里面也较简单。后两个
考试局除英国以外的学校选择的比较多。在中国,一些纯的国际学校选择EDEXCEL考试局的
居多,而一些普高的国际部等选择CIE的居多。CIE是这几个考试局里公认的最难的。但不论
所属的英国国家考试局派选专家在英国当地统一阅卷。 考试成绩
高 A-Level 的成绩分为 A*,A、B、C、D、E,U 七个等级,A 以上为最优,E 为通过,U 为
2014年A-Level进阶数学纯数学试卷部分试题赏析

学习,凭借考试成绩申请大学.在英国,大多数的中学开设相当
广泛的A—Level课程,多达100多门,真正满足了学生的个性 化发展.在国内的各剑桥高中,考虑到中国学生的学习特点,更 由于师资的限制,各中心实际开设的课程要少很多,通常为英 语、数学、进阶数学、物理、化学、计算机、生物、会计、商业学、经 济学等课程. 2.A—Level进阶数学课程
年的考试大纲为例介绍A—Level进阶数学的数学教学. 1.评价方法 进阶数学需要参加两次考试,分别为纯数学和应用数学, 其中应用数学又包含统计和机械学,每门考试均为3小时,每 张试卷10一11个题目,全部为解答题,总分为100分,考试结束 后,剑桥考试中心根据全球考生的分数划分A、口、C、D、E、u六 个等级,E是通过,u为不通过,下表就是2014年5月/6月进阶 数学考试对每一个部分的评分等级.
224700
崔
辉
①当曰=∥时,△;4(口+1)2—4(口2—1)<O,解得o<一1.
②当曰={o}时,由两个根为。及韦达定={4}时,由两个根为4及韦达定理得
{;翟:二11)6≯解得口不存在.
④当B={0,4}时,由韦达定理得
nⅣ=一
错解的同学认为肘与Ⅳ无公共元素.
Zsln—_
:壶姚。s如川蛐扣=善cos÷㈦啪
zsm—二一 s¨l—写
例2
曲线C的参数方程为z=e‘一4t+3,),=8er,0≤
(1)求曲线C的长度(结果以e的形式表示).(2)曲线c 围绕并轴旋转一圈,求所得图形的表面积(结果以e和可的形式 表示). 赏析 本题主要考查曲线的参数方程,曲线长度公式,旋 转曲面的表面积公式以及定积分的相关知识.
F、l _、Ir }∥匕㈠F =、l_、l 0丁
2017年英国高考A-LEVEL(CIE考试局)数学MathematicsFeb_March 20179709_m17_qp_52

*1694469822*Cambridge International ExaminationsCambridge International Advanced LevelCANDIDATENAMECENTRENUMBERCANDIDATENUMBERMATHEMATICS9709/52 Paper5Mechanics2(M2)February/March20171hour15minutes Candidates answer on the Question Paper.Additional Materials:List of Formulae(MF9)This document consists of13printed pages and3blankpages.JC1703_9709_52/RP©UCLES2017[Turn over1A small ball is projected with speed15m s−1at an angle of60 above the horizontal.Find the distance from the point of projection of the ball at the instant when it is travelling horizontally.[5] 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cylindrical container is open at the top.The curved surface and the circular base of the container are both made from the same thin uniform material.The container has radius0.2m and height0.9m.(i)Show that the centre of mass of the container is0.405m from the base.[3]........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................The container is placed with its base on a rough inclined plane.The container is in equilibrium on the point of slipping down the plane and also on the point of toppling.(ii)Find the coefficient of friction between the container and the plane.[3] ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................©UCLES20179709/52/F/M/17[Turn over3A particle P is projected with speed20m s−1at an angle of60 below the horizontal,from a point O which is30m above horizontal ground.(i)Calculate the time taken by P to reach the ground.[3]................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................©UCLES20179709/52/F/M/17(ii)Calculate the speed and direction of motion of P immediately before it reaches the ground.[4] ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................©UCLES20179709/52/F/M/17[Turn over4A B CD0.75m0.6m 0.9m The diagram shows a uniform lamina ABCD with AB =0.75m,AD =0.6m and BC =0.9m.Angle BAD =angle ABC =90 .(i)Show that the distance of the centre of mass of the lamina from AB is 0.38m,and find the distanceof the centre of mass from BC .[5]........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................©UCLES 20179709/52/F/M/17................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................The lamina is freely suspended at B and hangs in equilibrium.(ii)Find the angle between BC and the vertical.[2] ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................©UCLES20179709/52/F/M/17[Turn over5PQrad s Two particles P and Q have masses 0.4kg and m kg respectively.P is attached to a fixed point A by a light inextensible string of length 0.5m which is inclined at an angle of 60 to the vertical.P and Q are joined to each other by a light inextensible vertical string.Q is attached to a fixed point B ,which is vertically below A ,by a light inextensible string.The string BQ is taut and horizontal.The particles rotate in horizontal circles about an axis through A and B with constant angular speed rad s −1(see diagram).The tension in the string joining P and Q is 1.5N.(i)Find the tension in the string AP and the value of .[4]................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................©UCLES 20179709/52/F/M/17................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................(ii)Find m and the tension in the string BQ.[3] ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................©UCLES20179709/52/F/M/17[Turn over。
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Introduction
3.1 Aims 3.2 Assessment objectives
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Cambridge International AS and A Level Mathematics 9709
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