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. 2. Each questio carries 25 marks. 3. Cadidates will be required to aswer all 4 questios. Electroic Calculators. The use of commo electroic scietific calculators is allowed. The detailed syllabus is o the ext page. Nayag Techological Uiversity September 200
CONTENT OUTLINE Kowledge of the cotet of the Ordiary Level Syllabus (or a equivalet syllabus) is assumed. The use of a scietific calculator, where appropriate, is expected. Topic / Sub-topics PURE MATHEMATICS Fuctios ad graphs. Expoetial ad logarithmic fuctios ad Graphig techiques Cotet Iclude: cocept of fuctio use of otatio such as f( x ) = x 2 + 5 x fuctios e ad l x ad their graphs laws of logarithms equivalece of y = e x ad x = l y use of a graphic calculator to graph a give fuctio characteristics of graphs such as symmetry, itersectios with the axes, turig poits ad asymptotes Exclude: cocepts of domai ad rage the use of otatio f : x a x 2 + 5.2 Equatios ad iequalities Iclude: solvig simultaeous equatios, oe liear ad oe quadratic, by substitutio coditios for a quadratic equatio to have real or equal roots solvig quadratic iequalities 2 coditios for ax + bx + c to be always positive (or always egative) solvig iequalities by graphical methods formulatig a equatio from a problem situatio fidig the umerical solutio of a equatio usig a graphic calculator 2
Topic / Sub-topics Cotet 2 Calculus 2. Differetiatio Iclude: derivative of f(x ) as the gradiet of the taget to the graph of y = f(x) at a poit d y use of stadard otatios f ( x) ad d x x derivatives of x for ay ratioal, e, l x, together with costat multiples, sums ad differeces use of chai rule graphical iterpretatio of f ( x ) > 0, f ( x ) = 0 ad f ( x ) < 0 statioary poits (local maximum ad miimum poits ad poits of iflexio) 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: differetiatio from first priciples derivatives of products ad quotiets of fuctios dy use of = dx dx dy differetiatio of fuctios defied implicitly or parametrically fidig o-statioary poits of iflexio problems ivolvig small icremets ad approximatio relatig the graph of y = f ( x) to the graph of y = f(x) 2.2 Itegratio Iclude: itegratio as the reverse of differetiatio x itegratio of x, for ay ratioal, ad e, together with costat multiples, sums ad differeces ( + ) itegratio of ( ax + b), for ay ratioal, ad e ax b 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 fidig the umerical value of a defiite itegral usig a graphic calculator Exclude: defiite itegral as a limit of sum approximatio of area uder a curve usig the trapezium rule area below the x-axis 3
Topic / Sub-topics Cotet STATISTICS 3 Probability 3. 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 PA ( ) = PA ( ) P( A B) = P( A) + P( B) P( A B ) = P( A B P( AB) ) P( B) 4 Biomial ad ormal distributios 4. Biomial distributio Iclude: kowledge of the biomial expasio of ( a+ b ) for positive iteger use of the otatios! ad r cocept of biomial distributio B( p, ) ad use of B( p, ) as a probability model use of mea ad variace of a biomial distributio (without proof) solvig problems ivolvig biomial variables Exclude calculatio of mea ad variace for other probability distributios 4.2 Normal distributio Iclude: cocept of a ormal distributio ad its mea ad variace; 2 use of N( µ, σ ) as a probability model stadard ormal distributio fidig the value of P( X < x ) give the values of x, µ, σ use of the symmetry of the ormal distributio fidig a relatioship betwee x, µ, σ give the value of P( X < x ) 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 Exclude: fidig probability desity fuctios ad distributio fuctios calculatio of E( X ) ad Var( X ) from other probability desity fuctios 4
Topic / Sub-topics Cotet 5 Samplig ad hypothesis testig 5. 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 calculatio of ubiased estimates of the populatio mea ad variace from a sample solvig problems ivolvig the samplig distributio 5.2 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 large sample from ay populatio -tail ad 2-tail tests Exclude testig the differece betwee two populatio meas 6 Correlatio ad Regressio 6. 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 Exclude: derivatio of formulae hypothesis tests use of a square, reciprocal or logarithmic trasformatio to achieve liearity 5
MATHEMATICAL NOTATION The list which follows summarises the otatio used i NTU Etrace Examiatios i Mathematics. Although primarily directed towards A-Level, the list also applies, where relevat, at AO-Level.. Set Notatio is a elemet of is ot a elemet of {x, x 2, } the set with elemets x, x 2, {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, ±, ±2, ±3, } the set of positive itegers, {, 2, 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} 6
2. 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= a a the ratio of a to b a + a 2 +... + a the positive square root of the real umber a the modulus of the real umber a! factorial for Z + U {0}, (0! = ) r the biomial coefficiet! r!( r)!, for, r Z + U {0}, 0 Y r Y ( )...( r + ), for Q, r Z + U {0} r! 7
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 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 0 6. Circular Fuctios ad Relatios si, cos, ta, cosec, sec, cot si, cos, ta cosec, sec, cot } the circular fuctios } the iverse circular fuctios 8
7. Complex Numbers i square root of 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 2 + y 2 ), r (cos θ + i siθ) = r arg z the argumet of z, arg(r(cos θ + i si θ )) = θ, π < θ Ğ π z* the complex cojugate of z, (x + iy)* = x iy 8. Matrices M M 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 0. 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 9
x, y, r, etc. value of the radom variables X, Y, R, etc. x, x, 2 observatios f, p(x) f, 2 frequecies with which the observatios, x, x 2 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 2 2 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 Ğ 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 2 ) ormal distributio, mea µ ad variace σ 2 µ populatio mea σ 2 σ x s 2 φ Φ ρ r populatio variace populatio stadard deviatio sample mea ubiased estimate of populatio variace from a sample, s 2 ( x x) 2 probability desity fuctio of the stadardised ormal variable with distributio N (0, ) correspodig cumulative distributio fuctio liear product-momet correlatio coefficiet for a populatio liear product-momet correlatio coefficiet for a sample 0