Important Knowledge and Formulas

Transcription

1 DSE Pre-Training Mathematics (Compulsory Part) Section A Important Knowledge and Formulas Junior Secondary (involve knowledge in Non-foundation Part) 1. Estimation, Approximation and Errors (a) (b) Absolute error estimated value exact value Maximum absolute error largest possible uncertainty of an estimation or a measurement (c) Maximum absolute error Relative error Measured value or Absolute error Exact value (d) Percentage error Relative error 100%. Percentages (a) Percentage change New value Original Original value value 100% (b) (i) New value Original value (1 Percentage increase) (c) (ii) New value Original value (1 Percentage decrease) Profit and loss Selling price Cost price Percentage change 100% Cost price If the percentage change 0, then there is a profit. If the percentage change 0, then there is a loss. (d) Selling price Cost price (1 Profit percentage) or Cost price (1 Loss percentage) (e) (f) (g) Marked price Selling price Discount percentage 100% Marked price Selling price Marked price (1 Discount percentage) Let P be the principal, r% be the interest rate per period, n be the number of periods, I be the interest and A be the total amount. (i) Simple interest (1) I P r% n () A P I (ii) Compound interest n n (1) A P (1 r%) () I P (1 r%) P (h) Let n be the number of periods (i) Growth New value Original value (1 Growth rate) n (ii) Depreciation New value Original value (1 Depreciation rate) n II

2 Useful Theorems and Definitions (Geometry) Useful Theorems and Definitions (Geometry) Junior Secondary (involve knowledge in Non-foundation Part) A. Angles and Parallel Lines 1. Angles related to intersecting lines AOB is a straight line. a + b = 180 a + b + c + d = 0 a = b (adj. s on st. line) ( s at a pt.) (vert. opp. s). Parallel lines (a) If AB // CD, then a = b b = c c + d = 180 (corr. s, AB // CD) (alt. s, AB // CD) (int. s, AB // CD) (b) (i) If a = b, then AB // CD. (ii) If b = c, then AB // CD. (iii) If c + d = 180, then AB // CD. (corr. s equal) (alt. s equal) (int. s supp.) B. Triangles 1. Angles of a triangle (a) a + b + c = 180 ( sum of D) (b) d = a + b (ext. of D). Special triangles (a) Isosceles triangle (i) If AB = AC, then b = c. (base s, isos. D) (ii) If b = c, then AB = AC. (sides opp. eq. s) (iii) If AB = AC and any one of the following conditions is satisfied, then the other two are also satisfied. AD BC BAD = CAD BD = CD (property of isos. D) XI

3 Essential Calculator Programs Essential Calculator Programs We will introduce some useful built-in and user-defined programs for fx-0fh II and fx-0p II. Command Keys P1 (or P, P, P)... 1 (or,, ) A... ALPHA A B... ALPHA B C... ALPHA C D... ALPHA D X... ALPHA X Y... ALPHA Y M... ALPHA M M+... M+?... SHIFT P-CMD 1... SHIFT P-CMD :... SHIFT P-CMD... SHIFT P-CMD... SHIFT P-CMD =... SHIFT P-CMD... SHIFT P-CMD >... SHIFT P-CMD <... SHIFT P-CMD... SHIFT P-CMD... SHIFT P-CMD Goto... SHIFT P-CMD Lbl... SHIFT P-CMD 1 1 p r... SHIFT EXP SHIFT ANS 1 Special Command Keys Press and to move the cursor. Press DEL to delete the number, variable or function at the position of the blinking cursor. Press SHIFT DEL to insert number, variable or function. The cursor becomes. Press SHIFT DEL to change the insert cursor back to a normal cursor. XIX

4 DSE Pre-Training Mathematics (Compulsory Part) Section A Question Distribution Paper 1 Mock Practice The following table shows the corresponding topics of the questions in Paper 1 Mock Practice (MP) Laws of Indices Percentages Estimation and Error Formulas and Polynomials More about Polynomials Equations Functions and Graphs Rate, Ratio and Variations Sequences Inequalities Symmetry, Transformation of Figures and -D Figures Straight Lines and Rectilinear Figures Basic Properties of Circles Mensuration Coordinates and Equations of Straight Lines Locus and Equations of Circles Trigonometry Probability Statistics Topic MP 1 1, 8 7 Junior Secondary Senior Secondary MP MP MP MP MP MP 7 MP 8 MP 9 MP ,,,,, , , XXIV

5 DSE Pre-Training Mathematics (Compulsory Part) Section A The table below shows the related topics and the corresponding questionss in Paper 1 Mock Practice (MP) 1 for each Skill. Topic Skill MP 1 MP MP MP MP Laws of Indices Skill Skill Formulas and Polynomials Skill Skill Estimation and Error Skill Skill Equations Skill7 Skill8 Rate, Ratio and Variations Skill9 Formulas and Polynomials Skill10 Percentages Skill11 Skill1 Skill1 9 7 Straight Lines and Rectilinear Figures Skill1 7 Skill1 9 7 Skill1 9 7 Skill17 8 Skill Mensuration Skill19 8 Skill0 8 8 Skill Skill Coordinates and Equations of Straight Lines Skill Skill Skill Probability Skill Skill7 Skill8 7 8 Statistics Skill9 7 8 Skill0 8 Skill1 7

6 Pre-Training Course 1 Skill1 Use laws of indices to simplify expressions Example Simplify the following expressions and express your answer with positive indices. (a) (x y ) (b) (x y ) (c) x x y Solution: (a) (x y ) = x y = x 9 y Instant Practice Simplify the following expressions and express your answer with positive indices. (a) (a b ) (b) (a b ) (c) a b b (b) (x y ) x = x y = y 0 x (c) x y y x = y = x Skill Perform change of subject Example (a) Make x the subject of the formula x y. (b) Make k the subject of the formula 7h + k = h. (c) Make s the subject of the formula t 1. s t (d) Make m the subject of the formula 7m + pm =. (e) Make p the subject of the formula pq = (p + t). Solution: (a) x y (b) 7h + k = h k = h x y 0 h k x 0 y Instant Practice (a) Make g the subject of the formula g. f (b) Make b the subject of the formula a 9b = a. (c) Make R the subject of the formula S. R S 1 (d) Make x the subject of the formula 8 + kx x = 7. (e) Make r the subject of the formula (p r) = p(q + r). (c) t 1 s t s t t 1 t s t 1 (d) 7m + pm = (7 + p)m = m 7 p (e) pq = (p + t) pq = p + t pq p = t (q )p = t t p = q

7 Paper 1 Mock Practicee 1 Name: Class: Date: 11 a b 1. Simplify b 9 and express your answer with positive indices. Marks: /00 ( marks) _. Make k the subject of the formula h 7k. k ( marks). (a) Round down 987. to 1 significant figure. (b) Round up 987. to 1 decimal place. (c) Round off 987. to the nearest integer. ( marks) 1

8 DSE Pre-Training Mathematics (Compulsory Part) Section A Name: Class: Date: Answer all questions (1 mark for each question) Marks: / (x x 1)(x 1) = A. (x 1). B. x x + x + 1. C. x x x 1. D. x x x 1. (x ) x = A. x. B. x 1. C. 7x.. D. 7x 1 9 (s + t) = A. ( s + t)( + s t). B. ( s t)( + s + t). C. (9 s + t)(9 + s t). D. (9 s t)(9 + s + t). Which of the following is an identity/ are identities? I. 9x 1 = 0 II. 9x 1 = (x ) III. 9x 1 = (x + )(x ) A. II only B. III only C. I and II only D. I and III only. If x + y = 10 and 7x + yy = 7, then x = A. 7. B.. C.. D. 7.. There are 80 audiences in a theater. If the number of male audiences is 0% more than that of female audiences, then the number of male audiences is A. 0. B. 0. C. 90. D If the volume of a balloon is decreased by 0% and then increased by 0%, find the percentage change in the volume of the balloon. A. 70% B. 0% C. 0% D. 10% 8. The width, the length and the height of a rectangular block are measured as cm, cm and 7 cm correct to the nearest cm respectively. Let v cm be the actual volume of the block. Find the range of values of v. A v 18. B v 18. C. 19. v 10. D. 19. v 10. 0

9 Paper 1 Mock Practicee Name: Class: Date: Marks: / 8x 1. (a) Solve the inequality 17 x. 8x (b) Find all integers satisfying both the inequalities 17 x and x ( marks). It is given that g( (x) is the sum of two parts, one part varies as x and the Suppose that g() ) = 0 and g() = 1. Find g(7). other part is a constant. ( marks) _ 7

10 DSE Pre-Training Mathematics (Compulsory Part) Section A Name: Class: Date: Answer all questions (1 mark for each question) Marks: / Let a be a constant. Solve the equation (x + a) = 9a. A. x = a B. x = a C. x = a or x = aa D. x a or x = aa The figure shows the graph of y = ax x b, where a and b are constants. Which of the following is true? y y = ax x b x O. It is given that b partly varies directly as a and partly varies inversely as a. When a = 1, b = 1; when a =, b = 1. When a = 0., b = A.. B. 8. C. 1. D. 0.. [sin (70 ) 1][cos ( ) + 1] = A. sin. B. cos. C. sin. D. cos. A. a > 0 and b > 0 B. a > 0 and b < 0 C. a < 0 and b > 0 D. a < 0 and b < 0. In the figure, ABCD is a circle. BD is a diameter and intersects AC at E. If AD // BC and CED = 108, then ABE = D. The solution of x 7 > 1 or < + x is A. x <. B. x >. C. x <. D. x >. A.. A 108 E B C B.. C. 8. D

11 DSE Pre-Training Mathematics (Compulsory Part) Section A Cross-Topic questions in DSE Exam (Section A) Paper , 10, , , Practice Paper Sample Paper Paper, 8, 8, 7 1, 8 The table below shows the strand(s) and related topics of each of the following cross-topic questions. Note that some of them are modified from the DSE Exam questions. Question Strand 1 Geometry Geometry Geometry Geometry Algebra Geometry Data 7 Handling Algebra / 8 Geometry Trigonometric Ratios Senior Secondary: Basic Properties of Circles Trigonometric Ratios Mensuration Senior Secondary: Basic Properties of Circles Coordinate Geometry Symmetry and Transformation Coordinate Geometry Symmetry and Transformation Trigonometric Ratios Percentages Senior Secondary: Quadratic Equations in One Unknown Senior Secondary: Senior Secondary: DSE Exam Related Topics Reference Straight Lines and Rectilinear Figures Trigonometric Ratios Equations of Straight Lines Statistical Diagrams Simple Idea of Probability Coordinate Geometry Quadratic Equations in One Unknown 01 I 7 01 I 01 I 9 110

12 Cross-Topic Questions Pre-Training 7. The frequency distribution table and the cumulative frequency distribution table below show the distribution of the ages of the visitors to a theme park. Age Frequency Age less than Cumulative frequency p m 1 0 q n 1 0 r (a) Find p, q and r. (b) A visitor with age less than 0. or greater than 0. can buy the ticket with half price. If a visitor is randomly selected, find the probability that he / she CANNOT buy a ticket with half price. ( marks) 11

13 Pre-DSE Assessmentt 1 Name: Class: Date: 70 Marks: /7 Unless otherwise specified, numerical answers should be either exact or correct to significant figures. Section A(1) ( marks) 8 ( m n ) 1. Simplify ( m n) and express your answer with positive indices. ( marks). Make a the subject of the formula 1 x a y b. ( marks) 1

14 Pre-DSE Assessmentt Name: Class: Date: 0 Marks: / Answer all questions (1 mark for each question) 1. (a + a + a)(b b) = A. a + b. B. ab. C. a b. D. a b. x x 1. x x = 8 A. 7x. B. x. C. x. D.. x. 1x z 1xyz 1y z = A. z (x y)(x y). B. z (x y)(x + y). C. z (x + y)(x y). D. z (x + y)(x y).. Let m and n be constants. If m(x x) n(x + x) x + x, then m = A.. C.. B. D.... Express 00 as a decimal correct to significant figures. A B C D If < s < and < t < 7, then the range of values of s t is A. s 1. t B. s. t 7 C. 7 s. t D. s 1. 7 t Let k be a constant. Solve (x + k)(x k + ) = ( x + k). A. x = k B. x = k 1 C. x = k or k 1 D. x = k or k the equation The base of a parallelogram is increased from 1 cm to cm but its height is decreasedd by 0% %. Find the percentage change in the area of the parallelogram. A. An increase of 8.% B. 0% C. A decrease of 8.% D. A decrease of % 1