NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS

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1 NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER 1. There will be oe -hour paper cosistig of 4 questios.. Each questio carries 5 marks. 3. Cadidates will be required to aswer all 4 questios. The detailed syllabus is o the ext page. Nayag Techological Uiversity August 01

2 CONTENT OUTLINE Kowledge of the cotet of the O Level Mathematics syllabus ad of some of the cotet of the O Level Additioal Mathematics syllabus are assumed i the syllabus below ad will ot be tested directly, but it may be required idirectly i respose to questios o other topics. The assumed kowledge for O Level Additioal Mathematics is appeded after this sectio. Topic/Sub-topics Cotet PURE MATHEMATICS 1 Fuctios ad graphs 1.1 Fuctios, iverse fuctios ad composite fuctios Iclude: cocepts of fuctio, domai ad rage use of otatios such as f( x ) = x + 5, f : x a x + 5, f 1 ( x ), fg( x) ad f ( x ) fidig iverse fuctios ad composite fuctios coditios for the existece of iverse fuctios ad composite fuctios domai restrictio to obtai a iverse fuctio relatioship betwee a fuctio ad its iverse as reflectio i the lie y = x Exclude the use of the relatio 1 (fg) = g 1 f 1 1. Graphig techiques Iclude: use of a graphic calculator to graph a give fuctio relatig the followig equatios with their graphs x y ± = 1 b a ax + b y = cx + d ax + bx + c y = dx + e characteristics of graphs such as symmetry, itersectios with the axes, turig poits ad asymptotes determiig the equatios of asymptotes, axes of symmetry, ad restrictios o the possible values of x ad/or y effect of trasformatios o the graph of y = f(x) as represeted by y = a f(x), y = f( x) + a, y = f( x + a) ad y = f(ax), ad combiatios of these trasformatios relatig the graphs of y = f( x), y = f( x ), 1 y = ad y = f( x) to the graph of y = f(x) f( x) simple parametric equatios ad their graphs

3 Topic/Sub-topics Cotet 1.3 Equatios ad iequalities Iclude: f( x) solvig iequalities of the form > 0 where g( x) Sequeces ad series f(x) ad g(x) are quadratic expressios that are either factorisable or always positive solvig iequalities by graphical methods formulatig a equatio or a system of liear equatios from a problem situatio fidig the umerical solutio of equatios (icludig system of liear equatios) usig a graphic calculator.1 Summatio of series Iclude: cocepts of sequece ad series relatioship betwee u (the th term) ad (the sum to terms) sequece give by a formula for the th term sequece geerated by a simple recurrece relatio of the form x = f( x ) + 1 S use of otatio summatio of series by the method of differeces covergece of a series ad the sum to ifiity biomial expasio of ( 1+ x) for ay ratioal coditio for covergece of a biomial series proof by the method of mathematical iductio. Arithmetic ad geometric series Iclude: formula for the th term ad the sum of a fiite arithmetic series formula for the th term ad the sum of a fiite geometric series coditio for covergece of a ifiite geometric series formula for the sum to ifiity of a coverget geometric series solvig practical problems ivolvig arithmetic ad geometric series 3 Vectors 3.1 Vectors i two ad three dimesios Iclude: additio ad subtractio of vectors, multiplicatio of a vector by a scalar, ad their geometrical iterpretatios x x use of otatios such as, y y, x i + yj, z x i + yj + zk, AB, a positio vectors ad displacemet vectors magitude of a vector uit vectors 3

4 Topic/Sub-topics Cotet distace betwee two poits agle betwee a vector ad the x-, y- or z-axis use of the ratio theorem i geometrical applicatios 3. The scalar ad vector products of vectors Iclude: cocepts of scalar product ad vector product of vectors calculatio of the magitude of a vector ad the agle betwee two directios calculatio of the area of triagle or parallelogram geometrical meaigs of a.b ad a b, where b is a uit vector Exclude triple products a.b c ad a b c 3.3 Three-dimesioal geometry Iclude: vector ad cartesia equatios of lies ad plaes fidig the distace from a poit to a lie or to a plae fidig the agle betwee two lies, betwee a lie ad a plae, or betwee two plaes relatioships betwee (i) two lies (coplaar or skew) (ii) a lie ad a plae (iii) two plaes (iv) three plaes fidig the itersectios of lies ad plaes 4 Complex umbers Exclude: fidig the shortest distace betwee two skew lies fidig a equatio for the commo perpedicular to two skew lies 4.1 Complex umbers expressed i cartesia form Iclude: extesio of the umber system from real umbers to complex umbers complex roots of quadratic equatios four operatios of complex umbers expressed i the form ( x + iy) equatig real parts ad imagiary parts cojugate roots of a polyomial equatio with real coefficiets 4

5 Topic/Sub-topics 4. Complex umbers expressed i polar form Cotet Iclude: complex umbers expressed i the form iθ r (cosθ + isiθ ) or r e, where r > 0 ad π < θ Ğ π calculatio of modulus ( r ) ad argumet (θ ) of a complex umber multiplicatio ad divisio of two complex umbers expressed i polar form represetatio of complex umbers i the Argad diagram geometrical effects of cojugatig a complex umber ad of addig, subtractig, multiplyig, dividig two complex umbers loci such as z c Y r, z a = z b ad arg( z a) = α use of de Moivre s theorem to fid the powers ad th roots of a complex umber Exclude: loci such as z a = k z b, where k 1 ad arg( z a) arg( z b) = α properties ad geometrical represetatio of the th roots of uity use of de Moivre s theorem to derive trigoometric idetities 5 Calculus 5.1 Differetiatio Iclude: graphical iterpretatio of (i) f ( x ) > 0, f ( x ) = 0 ad f ( x ) < 0 (ii) f ( x ) > 0 ad f ( x ) < 0 relatig the graph of y = f ( x) to the graph of y = f(x) differetiatio of simple fuctios defied implicitly or parametrically fidig the umerical value of a derivative at a give poit usig a graphic calculator fidig equatios of tagets ad ormals to curves solvig practical problems ivolvig differetiatio Exclude: fidig o-statioary poits of iflexio problems ivolvig small icremets ad approximatio 5

6 Topic/Sub-topics Cotet 5. Maclauri s series Iclude: derivatio of the first few terms of the series x expasio of ( 1+ x), e, si x, l( 1+ x), ad other simple fuctios fidig the first few terms of the series expasios of sums ad products of fuctios, e.g. e x cosx, usig stadard series summatio of ifiite series i terms of stadard series 1 si x x, cos x 1 x, ta x x cocepts of covergece ad approximatio Exclude derivatio of the geeral term of the series 5.3 Itegratio techiques Iclude: itegratio of f ( x) f( x) si x, cos x, ta x 1 1 1,, a + x a x a x ad itegratio by a give substitutio itegratio by parts 1 x a Exclude reductio formulae 5.4 Defiite itegrals Iclude: cocept of defiite itegral as a limit of sum defiite itegral as the area uder a curve evaluatio of defiite itegrals fidig the area of a regio bouded by a curve ad lies parallel to the coordiate axes, betwee a curve ad a lie, or betwee two curves area below the x-axis fidig the area uder a curve defied parametrically fidig the volume of revolutio about the x- or y-axis fidig the umerical value of a defiite itegral usig a graphic calculator Exclude approximatio of area uder a curve usig the trapezium rule 6

7 Topic/Sub-topics Cotet 5.5 Differetial equatios Iclude: solvig differetial equatios of the forms STATISTICS 6 Permutatios, combiatios ad probability d y = f d x d y = f d x d d x y ( x) ( y ) = f ( x) formulatig a differetial equatio from a problem situatio use of a family of solutio curves to represet the geeral solutio of a differetial equatio use of a iitial coditio to fid a particular solutio iterpretatio of a solutio i terms of the problem situatio 6.1 Permutatios ad combiatios Iclude: additio ad multiplicatio priciples for coutig cocepts of permutatio (! or P r ) ad combiatio ( C r ) arragemets of objects i a lie or i a circle cases ivolvig repetitio ad restrictio 6. Probability Iclude: additio ad multiplicatio of probabilities mutually exclusive evets ad idepedet evets use of tables of outcomes, Ve diagrams, ad tree diagrams to calculate probabilities calculatio of coditioal probabilities i simple cases use of: P( A ) = 1 P( A) P( A B) = P( A) + P( B) P( A B) P( A B) P( A B) = P( B) 7

8 Topic/Sub-topics Cotet 7 Biomial, Poisso ad ormal distributios 7.1 Biomial ad Poisso distributios Iclude: cocepts of biomial distributio B(, p) ad Poisso distributio Po(µ ) ; use of B(, p) ad Po(µ ) as probability models use of mea ad variace of biomial ad Poisso distributios (without proof) solvig problems ivolvig biomial ad Poisso variables additive property of the Poisso distributio Poisso approximatio to biomial Exclude calculatio of mea ad variace for other probability distributios 7. Normal distributio Iclude: cocept of a ormal distributio ad its mea ad variace; use of N( µ, σ ) as a probability model stadard ormal distributio fidig the value of P( X < x1) give the values of x 1, µ, σ use of the symmetry of the ormal distributio fidig a relatioship betwee x 1, µ, σ give the value of P( X < x1) solvig problems ivolvig ormal variables solvig problems ivolvig the use of E( ax + b) ad Var( ax + b) solvig problems ivolvig the use of E( ax + by ) ad Var( ax + by ), where X ad Y are idepedet ormal approximatio to biomial ormal approximatio to Poisso 8 Samplig ad hypothesis testig Exclude: fidig probability desity fuctios ad distributio fuctios calculatio of E(X ) ad Var(X ) from other probability desity fuctios 8.1 Samplig Iclude: cocepts of populatio ad sample radom, stratified, systematic ad quota samples advatages ad disadvatages of the various samplig methods distributio of sample meas from a ormal populatio use of the Cetral Limit Theorem to treat sample meas as havig ormal distributio whe the sample size is sufficietly large 8

9 Topic/Sub-topics Cotet calculatio of ubiased estimates of the populatio mea ad variace from a sample solvig problems ivolvig the samplig distributio 8. Hypothesis testig Iclude: cocepts of ull ad alterative hypotheses, test statistic, level of sigificace ad p-value tests for a populatio mea based o: * a sample from a ormal populatio of kow variace * a sample from a ormal populatio of ukow variace * a large sample from ay populatio 1-tail ad -tail tests use of t-test 9 Correlatio ad Regressio Exclude testig the differece betwee two populatio meas 9.1 Correlatio coefficiet ad liear regressio Iclude: cocepts of scatter diagram, correlatio coefficiet ad liear regressio calculatio ad iterpretatio of the product momet correlatio coefficiet ad of the equatio of the least squares regressio lie cocepts of iterpolatio ad extrapolatio use of a square, reciprocal or logarithmic trasformatio to achieve liearity Exclude: derivatio of formulae hypothesis tests 9

10 ASSUMED KNOWLEDGE Cotet from O Level Additioal Mathematics A ALGEBRA 1 Equatios ad iequalities coditios for a quadratic equatio to have: (i) two real roots (ii) two equal roots (iii) o real roots coditios for ax + bx + c to be always positive (or always egative) Polyomials multiplicatio ad divisio of polyomials use of remaider ad factor theorems 3 Idices ad surds four operatios o surds ratioalisig the deomiator 4 Simultaeous equatios i two ukows solvig simultaeous equatios with at least oe liear equatio, by substitutio express a pair of liear equatios i matrix form ad solvig the equatios by iverse matrix method 5 Expoetial ad logarithmic fuctios x x fuctios a, e, log a x, l x ad their graphs laws of logarithms x equivalece of y = a ad x = loga y chage of base of logarithms fuctio x ad graph of f(x ), where f(x ) is liear, quadratic or trigoometric solvig simple equatios ivolvig expoetial ad logarithmic fuctios 6 Partial fractios Iclude cases where the deomiator is of the form ( ax + b)( cx + d) ( ax + b)( cx + d) ( ax + b)( x + c ) 10

11 Cotet from O Level Additioal Mathematics B GEOMETRY AND TRIGONOMETRY 7 Coordiate geometry i two dimesios graphs of equatios * y = ax, where is a simple ratioal umber * y = kx coordiate geometry of the circle with the equatio i the form ( x a) + ( y b) = r or x + y + gx + fy + c = 0 8 Trigoometry six trigoometric fuctios, ad pricipal values of the iverses of sie, cosie ad taget trigoometric equatios ad idetities (see List of Formulae) expressio of a cosθ + b siθ i the forms R si( θ ± α ) ad R cos( θ ± α ) C CALCULUS 9 Differetiatio ad itegratio derivative of f ( x) as the gradiet of the taget to the graph of y = f( x) at a poit derivative as rate of chage x derivatives of x for ay ratioal, si x, cos x, ta x, e ad l x, together with costat multiples, sums ad differeces derivatives of composite fuctios derivatives of products ad quotiets of fuctios icreasig ad decreasig fuctios statioary poits (maximum ad miimum turig poits ad poits of iflexio) use of secod derivative test to discrimiate betwee maxima ad miima coected rates of chage maxima ad miima problems itegratio as the reverse of differetiatio x itegratio of x for ay ratioal, e, si x, cos x, sec x ad their costat multiples, sums ad differeces ax+b itegratio of ( ax + b) for ay ratioal, si( ax + b), cos( ax + b) ad e 10 11

12 MATHEMATICAL NOTATION 1. Set Notatio is a elemet of is ot a elemet of {x 1, x, } the set with elemets x 1, x, {x: } (A) the set of all x such that the umber of elemets i set A the empty set uiversal set A the complemet of the set A Z Z + the set of itegers, {0, ±1, ±, ±3, } the set of positive itegers, {1,, 3, } Q the set of ratioal umbers Q + the set of positive ratioal umbers, {x Q: x > 0} Q + 0 R the set of positive ratioal umbers ad zero, {x Q: x ğ 0} the set of real umbers R + the set of positive real umbers, {x R: x > 0} R + 0 R ` the set of positive real umbers ad zero, {x R: x ğ 0} the real tuples the set of complex umbers is a subset of is a proper subset of is ot a subset of is ot a proper subset of uio itersectio [a, b] the closed iterval {x R: a Ğ x Ğ b} [a, b) the iterval {x R: a Ğ x < b} (a, b] the iterval {x R: a < x Ğ b} (a, b) the ope iterval {x R: a < x < b} 1

13 . Miscellaeous Symbols = is equal to is ot equal to is idetical to or is cogruet to is approximately equal to is proportioal to < is less tha Y; is less tha or equal to; is ot greater tha > is greater tha [; is greater tha or equal to; is ot less tha ifiity 3. Operatios a + b a b a b, ab, a.b a a b,, a/b b a plus b a mius b a multiplied by b a divided by b a : b a i i= 1 a a the ratio of a to b a 1 + a a the positive square root of the real umber a the modulus of the real umber a! factorial for Z + U {0}, (0! = 1) r the biomial coefficiet! r!( r)!, for, r Z + U {0}, 0 Y r Y ( 1)...( r + 1), for Q, r Z + U {0} r! 13

14 4. Fuctios f f(x) fuctio f the value of the fuctio f at x f: A B f is a fuctio uder which each elemet of set A has a image i set B f: x y the fuctio f maps the elemet x to the elemet y f 1 g o f, gf the iverse of the fuctio f the composite fuctio of f ad g which is defied by (g o f)(x) or gf(x) = g(f(x)) lim f(x) the limit of f(x) as x teds to a x a x ; δ x a icremet of x dy dx d y dx f'(x), f' (x),, f () (x) yd x b a yd x x&, & x&, the derivative of y with respect to x the th derivative of y with respect to x the first, secod, th derivatives of f(x) with respect to x idefiite itegral of y with respect to x the defiite itegral of y with respect to x for values of x betwee a ad b the first, secod, derivatives of x with respect to time 5. Expoetial ad Logarithmic Fuctios e base of atural logarithms e x, exp x expoetial fuctio of x log a x logarithm to the base a of x l x atural logarithm of x lg x logarithm of x to base Circular Fuctios ad Relatios si, cos, ta, cosec, sec, cot si 1, cos 1, ta 1 cosec 1, sec 1, cot 1 } the circular fuctios } the iverse circular fuctios 14

15 7. Complex Numbers i square root of 1 z a complex umber, z = x + iy = r(cos θ + i si θ ), r R + 0 = re iθ, r R + 0 Re z the real part of z, Re (x + iy) = x Im z the imagiary part of z, Im (x + iy) = y z the modulus of z, x + iy = (x + y ), r (cos θ + i siθ) = r arg z the argumet of z, arg(r(cos θ + i si θ )) = θ, π < θ Ğπ z* the complex cojugate of z, (x + iy)* = x iy 8. Matrices M M 1 M T det M a matrix M the iverse of the square matrix M the traspose of the matrix M the determiat of the square matrix M 9. Vectors a the vector a AB the vector represeted i magitude ad directio by the directed lie segmet AB â a uit vector i the directio of the vector a i, j, k uit vectors i the directios of the cartesia coordiate axes a the magitude of a AB a.b apb the magitude of AB the scalar product of a ad b the vector product of a ad b 10. Probability ad Statistics A, B, C, etc. evets A B uio of evets A ad B A B itersectio of the evets A ad B P(A) probability of the evet A A' complemet of the evet A, the evet ot A P(A B) probability of the evet A give the evet B X, Y, R, etc. radom variables x, y, r, etc. value of the radom variables X, Y, R, etc

16 x, x, 1 f, f, 1 p(x) observatios frequecies with which the observatios, x 1, x occur the value of the probability fuctio P(X = x) of the discrete radom variable X p, p probabilities of the values x, x, of the discrete radom variable X 1 1 f(x), g(x) the value of the probability desity fuctio of the cotiuous radom variable X F(x), G(x) E(X) E[g(X)] Var(X) B(, p) the value of the (cumulative) distributio fuctio P(X Y x) of the radom variable X expectatio of the radom variable X expectatio of g(x) variace of the radom variable X biomial distributio, parameters ad p Po(µ) Poisso distributio, mea µ N(µ, σ ) ormal distributio, mea µ ad variace σ µ populatio mea σ σ x s φ Φ ρ r populatio variace populatio stadard deviatio sample mea ubiased estimate of populatio variace from a sample, s 1 1 ( x x) probability desity fuctio of the stadardised ormal variable with distributio N (0, 1) correspodig cumulative distributio fuctio liear product-momet correlatio coefficiet for a populatio liear product-momet correlatio coefficiet for a sample 16

NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS

NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be oe 2-hour paper cosistig of 4 questios.