Public Assessment of the HKDSE Mathematics Examination

Transcription

1 Public Assessment of the HKDSE Mathematics Examination. Exam Format (a) The examination consists of one paper. (b) All questions are conventional questions. (c) The duration is hours and 30 minutes. Section Marks Number of Questions Other Details A 50 8 B Section A consists of short questions while Section B consists of long questions. Knowledge of the subject matter in the Compulsory Part together with the Foundation Part and the Non- Foundation Part of Secondary 3 Mathematics Curriculum is assumed.. Standards-referenced Reporting The HKDSE makes use of standards-referenced reporting, which means candidates levels of performance will be reported with reference to a set of standards as defined by cut scores on the variable or scale for a given subject. The following diagram represents the set of standards for a given subject: Cut scores U Variable/ scale Within the context of the HKDSE there will be five cut scores, which will be used to distinguish five levels of performance ( 5), with 5 being the highest. The Level 5 candidates with the best performance will have their results annotated with the symbols and the next top group with the symbol. A performance below the threshold cut score for Level will be labelled as Unclassified (U). II

2 Exam Strategies A. Time Allocation Section Suggested Time Allocation Approximate Time per Question A 80 minutes 6 0 minutes B 60 minutes 0 minutes In general, spend 7 minutes for every 5 marks. Allow 0 minutes for final checking. B. Answering Skills Skip the questions that you do not have confidence on. Go back to the skipped questions after you have finished the others. Show your formulas and steps rather than just writing down the answers. In case you do not get the correct answer, you can get marks for the correct methods used. Make sure your numerical answers are either exact or correct to 4 decimal places, unless otherwise specified. Give intermediate results correct to more significant figures or decimal places to avoid accumulated errors in the final answers. Make sure you give a unit, if any, to each answer. In answering questions involving curve sketching, Always use a pencil. Label significant items like graph titles and axis names. Show rough figures of the correct shapes and intercepts. In proving mathematical results, you should put down your workings in details. Pay attention to question wordings. Write down You can just write down the answer without showing your working steps. Hence You have to use the results obtained earlier to get the answer. III

3 Sample Paper Analysis The analysis is based on the Mathematics (Extended Part) Module sample paper issued by the Hong Kong Examination and Assessment Authority in 009. The paper consists of Section A (0 questions) and Section B (4 questions). Section A Section B. Binomial Theorem / /. Mathematical Induction 0(a) / 3. More about Trigonometric Functions 5, 0(b) / 4. Limits / 5. Differentiation / (a), (b), (c) 6. Applications of Differentiation, 6 / 7. Indefinite Integration 3, 4, 8(a) / 8. Definite Integration / 3 9. Applications of Definite Integration 8(b) (d) 0. Determinants / /. Matrices 0(a), 0(b). Systems of Linear Equations 7 / 3. Vectors / 4 4. Applications of Vectors 9 / IV

4 Useful Formulas. Binomial Theorem (a) Summation Notation n T ( i) T ( ) + T ( ) T ( n) i (b) Binomial Theorem (i) ( a + b) n n n n n n n 0 + C r a r b n n Cn b n C r a b r 0 C a + C a b + C a b +... n n - r r (ii) General term Cr a b, where n and r are integers.. More about Trigonometric Functions (a) (i) p rad. 80 (ii) Arc length rq, where q is measured in radians (iii) Area of a sector r θ rs, where s is the arc length. (b) Trigonometric Functions of General Angles Let P(x, y) be a point on the terminal side of an angle of rotation q. Then, sinθ y r cscθ r y cosθ x r secθ r x where r x + y tanθ y x cotθ x y, (c) Relationship between Trigonometric Functions (i) cscθ sinθ (iii) cotθ tanθ (ii) secθ cosθ sinθ (iv) tanθ cosθ cosθ (v) cotθ (vi) sin θ + cos θ sinθ (vii) + tan θ sec θ (viii) + cot θ csc θ (d) Compound Angle Formulas (i) sin(a + B) sin Acos B + cos Asin B (ii) sin(a - B) sin Acos B - cos Asin B (iii) cos(a + B) cos Acos B - sin Asin B (iv) cos(a - B) cos Acos B + sin Asin B tan A + tan B (v) tan( A + B) tan A tan B tan A tan B (vi) tan( A B) + tan A tan B (e) Double Angle Formulas (i) sin A sin Acos A (ii) cos A cos A sin A cos A sin A tan A (iii) tan A tan A (f) Product-to-sum Formulas: (i) sin A cos B [sin( A + B) + sin( A B)] (ii) cos Asin B [sin( A + B) sin( A B)] (iii) cos A cos B [cos( A + B) + cos( A B)] (iv) sin Asin B [cos( A + B) cos( A B)] (g) Sum-to-product Formulas: x + y x y (i) sin x + sin y sin cos x + y x y (ii) sin x sin y cos sin x + y x y (iii) cos x + cos y cos cos x + y x y (iv) cos x cos y sin sin 3. Limits (a) Limit of a Function Suppose lim f ( x ) and lim g ( x ) exist. x a x a (i) lim k k, k is a constant. x a (ii) lim kf ( x) k lim f ( x), k is a constant. x a x a (iii) lim[ f ( x) ± g( x)] lim f ( x) ± lim g( x) x a x a x a (iv) lim f ( x) g( x) lim f ( x) lim g( x) x a x a x a f (v) lim ( x ) lim f ( x) x a x a g( x) lim g( x) x a V

5 FORMULAS FOR REFERENCE A + B A B sin (A ± B) sin A cos B ± cos A sin B sin A + sin B sin cos cos (A ± B) cos A cos B ± A + B A B sin A sin B sin A sin B cos sin tan (A ± B) tan A ± tan B A + B A B cos A + cos B cos cos tan A tan B A + B A B sin A cos B sin (A + B) + sin (A - B) cos A cos B sin sin cos A cos B cos (A + B) + cos (A - B) sin A sin B cos (A - B) - cos (A + B) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Section A (50 marks) 3. Find d dx ( x + x) from first principles. (4 marks). Given that e x + ln y + 7xy, find dy dx when x 0 and y. (4 marks) 3. The slope at any point (x, y) of a curve is given by dy dx tangent to the curve, find the equation of the curve. 4 x ( x + ). If the straight line y 6 is a (4 marks) 4. If n is a positive integer and the coefficient of x in the expansion of ( x ) n + ( + x) n is 04, find the value(s) of n. (4 marks) M MOCK - 0 Hong Kong Educational Publishing Co.

6 Section B (50 marks). (a) Prove, by mathematical induction, that n(n + ) is divisible by 3 for all positive integers n. (3 marks) (b) Hence prove, by mathematical induction, that (n - n)(n - n + 4) is divisible by for all positive integers n >. (5 marks). Figure 3 In Figure 3, OPQ is an equilateral triangle with OP. M is the mid-point of PQ. A divides OP in the ratio : and B divides OQ in the ratio :. Let OP and OQ be p and q respectively. (a) Find p q. ( marks) (b) Hence find AB and express your answer in surd form if necessary. (3 marks) (c) By considering AB OM, find BCM and determine whether BCMQ is a cyclic quadrilateral. (3 marks) Go on to the next page M MOCK Hong Kong Educational Publishing Co.

7 Mathematics: Mock Exam Papers Extended Part Module Solution Guide (a) M M MM \ M 3M A M A (b) M ( M 3I) ( M 3I) M I (by (a)) M ( M 3I) ( M 3I) M I M ( M 3I) M M A (6) In (a), we have M 3-3M I. If we can rewrite it in the form MN I or NM I, then the matrix N should be the inverse of M. 9. Let (p, q) be the coordinates of A. The slope of the tangent to C dy : 3x + 4 dx dy 3p + 4 dx ( p, q) The equation of the tangent: y q ( 3p + 4)( x p) 3 y ( 3p + 4) x 3p 4 p + q...() Since the x-intercept of the tangent is, the tangent passes through (, 0). Substituting (, 0) into (), 0 3p + 4-3p 3-4p + q M i.e., q 3p 3-3p + 4p () Since A lies on C, q p 3 + 4p...(3) M () - (3): ( 3p 3p + 4 p 4) ( p + 4 p) 3 p 3p 4 0 ( p )( p + p + ) 0 M p or p + p + 0 (rejected) Substituting p into (3), q 3 + 4() 6 \ A (, 6 ) A (6) A M We can first show that p - is a factor of p 3-3p - 4 by using the factor theorem, then factorize p 3-3p - 4 by long division. 5 0 Hong Kong Educational Publishing Co.

8 Mock Exam 0 Alternative Solution Let the line x a cuts the x-axis at D. Area Area of sector OAB - Area of DOAD ( ) ( AOB) ( OA)( OD)sin AOB π ( ) 4 π ( ) sin 4 π 4 ( π ) 8 M M + A A (7) 0. (a) OM : MR : \ OM OR 3 + a b 3 a + b A 3 3 \ The candidate just gives a statement related to an undefined variable R, which is very confusing. Candidates should clearly define all variables when using mathematical expressions to formulate an statement. Let AM : MP : r and BM : MQ : s. Considering DOAP, ( )( OP) + ( r)( OA) OM + r b + ra + r Considering DOBQ, ( )( OQ) + ( s)( OB) OM + s a + sb + s A The candidate can use the correct method to find the ratio AM : MP with details and uses the section formula to find OM. Use the ratios AM : MP and BM : MQ to find OM. \ b + ra a + sb + r + s r s a + b a + b + r ( + r) ( + s) + s M 0 Hong Kong Educational Publishing Co.