Junior Secondary. A. Errors 1. Absolute error = Estimated value - Exact value
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1 Junior Secondary A. Errors. Absolute error = Estimated value - Exact value. Maximum absolute error = Scale interval of the measuring tool 3. Maximum absolute error Absolute error Relative error = or Measured value Exact value 4. Percentage error = Relative error 00% B. Percentages New value - Original value. Percentage change = 00% Original value. (a) New value = Original value ( + Percentage increase) (b) New value = Original value ( - Percentage decrease) 3. Profit and loss Selling price - Cost price Percentage change = 00% Cost price If the percentage change > 0, then there is a profit. If the percentage change < 0, then there is a loss. 4. Selling price = Cost price ( + Profit percentage) or Cost price ( - Loss percentage) 5. Marked price - Selling price Discount percentage = 00% Marked price 6. Selling price = Marked price ( - Discount percentage) 7. Let P be the principal, r% be the interest rate per annum, n be the number of years and A be the total amount. (a) Simple interest (i) Interest, I = P r% n (ii) Total amount, A = P + I (b) Compound interest (i) Compounded yearly Total amount A = P ( + r%) n r% Compounded half-yearly A= P + r% Compounded quarterly A= P + 4 r% Compounded monthly A= P + (ii) Compound interest, I = A - P 8. Let n be the number of periods. (a) Growth New value = Original value ( + Growth rate) n (b) Depreciation New value = Original value ( - Depreciation rate) n n 4n n
2 Junior Secondary A. Angles and Parallel Lines. Angles related to intersecting lines AOB is a straight line. a + b = 80 a + b + c + d = 360 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 = 80 (corr. s, AB // CD) (alt. s, AB // CD) (int. s, AB // CD) (b) (i) If a = b, then AB // CD. (corr. s equal) (ii) If b = c, then AB // CD. (iii) If c + d = 80, then AB // CD. (alt. s equal) (int. s supp.) B. Triangles. Angles of a triangle (a) a + b + c = 80 ( 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)
3 We will introduce some useful techniques for using the calculators fx-50fh II and fx-3650 P II to solve problems in the examinations. Command Keys A... ALPHA A B... ALPHA B C... ALPHA C D... ALPHA D X... ALPHA X Y... ALPHA Y M... ALPHA M M+... M+?... Prog... Prog :... Prog 3... Prog 4... Prog =... Prog... Prog >... Prog <... Prog... Prog 3... Prog 4 Goto... Prog Lbl... Prog p r... SHIFT EXP SHIFT ANS 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. In CMPLX mode, arg... SHIFT ( Abs... SHIFT )... SHIFT ( ) 3
4 Tips for Using Calculators 5 0 Conversion between the Rectangular Coordinates and Polar Coordinates The built-in functions Pol( and Rec( of fx-50fh II and fx-3650p II interconvert the rectangular coordinates and polar coordinates. Example : If the rectangular coordinates of a point P are (-3, 3), find the polar coordinates of the point P. Display Key-in sequence SHIFT + ( ) 3, 3 EXE ( = 8) RCL, (The radius vector, r) Y 35 (The polar angle, q) \ Polar coordinates of P = ( 8, 35 ) Note: You may need to use the fact that -θ = θ to convert a negative angle into positive. Example : If the polar coordinates of a point P are (0, 300 ), find the rectangular coordinates of P. Display Key-in sequence SHIFT 0, EXE 5 RCL, (The x-coordinate) Y Instant Practice If the rectangular coordinates of a point P are (- 6 3, -6), find the polar coordinates of the point P. Ans: (, 0 ) Instant Practice If the polar coordinates of a point P are ( 98, 35 ), find the rectangular coordinates of P. Ans: (7, -7) ( = 5 3) (The y-coordinate) \ Rectangular coordinates of P = (5, ) 03 Operations of Complex Numbers After entering the CMPLX mode of fx-50fh II and fx-3650p II, we can manipulate the operations of complex numbers directly. Program Execution Step : MODE SHIFT MODE [CMPLX mode] Step : Input the real part, the imaginary part of the complex numbers and the operations. Press ENG to input the imaginary unit i. Step 3: Press EXE after inputting the expression to obtain the real part of the result and press SHIFT EXE to obtain the imaginary part of the result. If the result is a purely imaginary number, the imaginary part will be displayed immediately after pressing EXE.
5 Chapter VII Coordinate Geometry Topic Year PP SP , , ( Coordinates ) The rectangular coordinates of the point P are (-, ). If P is reflected with respect to the x-axis, then the polar coordinates of its image are A. (, 45 ). B. (, 5 ). C. (4, 45 ). D. (4, 5 ). Answer: D Analysis Note that P lies in quadrant II. After reflection, the image of P lies in quadrant III. Ref. DSE 05 Q3 Solution: The rectangular coordinates of the image of P are (-, - ), which lies in quadrant III. OP = ( ) + ( ) = 4 The polar angle is θ, where Thus, we have θ = 5. θ = tan =. Thus, the required polar coordinates are (4, 5 ). Common Mistake Confuse reflection with respect to the x-axis with reflection with respect to the y-axis. Wrongly calculate the radius r and the polar angle θ. Correct Concept For a point (x, y) in rectangular coordinate system, reflection with respect to the x-axis: (x, -y) reflection with respect to the y-axis: (-x, y) r = x + y and tan θ = y. x Determine the quadrant in which θ lies in by the corresponding rectangular coordinates.
6 Chapter X Algebraic Expressions () 9 Choose the BEST answer for each question. 34. Quadratic Equations and Functions (). If the roots of the equation x - 5x + 3 = 0 are α and β, then α + β = A. 9. B.. C. 5. D. 3.. It is given that α and β are the roots of the equation + x x = ( ) Find +. α β A. -4 B. - C. D It is given that α and β are the roots of the equation x(x - 4) = 0. Find (α - β). A. -40 B. 4 C. 6 D It is given that α and β are the roots of the equation ( x + )( x ) 3 = 0. Find α 3 + β 3. A. 9 B. 6 C. D If the roots of the equation x + x - 6 = 0 are a and b, find a - b. A. -6 B. C. 6 D. 0 CE 003 Q4 6. Let k be a constant. If the difference between the roots of x - 4x - k = 0 is 8, then k = A. -. B. -. C. 6. D.. CE 997 Q30 4α = α 3 7. If a b and, then a + b = 4β = β 3 A. -4. B. -3. C. 3. D. 4. DSE 03 Q35 8. Let k be a constant. If the roots of the equation x - kx - = 0 are α and b, then + = α β A. - k. B. - k. C.. D.. k k DSE PP Q33 9. Let k be a constant. If the roots of the quadratic equation x - kx - 8 = 0 are α and β, then α + αβ + β = A. k. B. k - 8. C. k + 8. D. k + 6. DSE 05 Q34 0. How many roots are there for the equation cos θ + cos θ - = 0 for 0 θ < 360? A. B. 3 C. 4 D. 5 DSE 06 Q38
7 4 MCQ Fast-track Course: Mathematics (Compulsory Part). 4. If the graph of y = log x is translated upwards by unit, it becomes the graph of The figure above shows the graph of y = f(x). If f(x) = 3g(x), which of the following may represent the graph of y = g(x)? A. B. C. A. y = log(x + ). B. y = log(x - ). C. y = log(0x). x D. y = log Let f(x) be a quadratic function. If the coordinates of the vertex of the graph of y = f(x) are (-4, -6), which of the following must be true? A. The roots of f(x) = 0 are integers. B. The roots of f(x) + 4 = 0 are real. C. f(x) + 6 = 0 has a double real root. D. f(x) + 8 = 0 has unreal roots. 6. DSE 0 Q34 D. The figure shows the graph of A. y = + cos x. DSE 07 Q3 3. If the graph of y = x is translated to the left by units, it becomes the graph of A. y = x -. B. y = x +. C. y = x -. D. y = 4( x ). B. y = + cos x. C. y = 3 cos x. D. y = 3 cos x. DSE 0 Q39
8 Chapter IV Plane Geometry 9 Chapter IV Plane Geometry Answers. A. C 3. C 4. B 5. C 6. B 7. D 8. D 9. B 0. B. D. B 3. C 4. B 5. C 6. C 7. A 8. B 9. D 0. B. B. B 3. A 4. C 5. B 6. C 7. A 8. B 9. C 30. C 3. D 3. A 33. A 34. A 35. C 36. A 37. B 38. C 39. B 40. C 4. C 4. C 43. B 44. B 45. A 46. B 47. D 48. B 49. C 50. C 5. A 5. A 53. C 54. D 55. C 56. A 57. D 58. D 5. Angle Properties of Rectilinear Figures. A For I, since x = y, AB // CD (alt. s equal). C y = b (alt. s, // lines) a + y = 80 (int. s, // lines) \ a + b = C x + y + z = 80 ( sum of ) x : y : z = 3 : 4 : 8 4 \ y = 80 = ABD + y = 80 (adj. s on st. line) ABD + 48 = 80 ABD = 3 4. B DAC = ACB (alt. s, AD // BC) ABCD is an isosceles trapezium. \ ABC = DCB = 30 + ACB = 30 + DAC 7 + DAC DAC = 80 (int. s, AD // BC) DAC = 78 DAC = C Draw PU and QR such that AB // CD // PU // QR. BPU = x (alt. s, AB // PU) DPU = y (alt. s, PU // CD) \ x + y = 70 BQR = x (alt. s, AB // QR) DQR = y (alt. s, QR // CD) \ BQD = x + y = ( x + y) = 70 = B For I, let ADE = BDE = BDC = x. ABD = x (alt. s, AB // DC) = BDE \ BE = DE (sides opp. eq. s) Since E is the mid-point of AB, AE = BE = DE. \ BAD = ADE (base s, isos. D) = x Since BAD = ABD = x, DABD is an isosceles triangle. \ I is true. For II, BAD + ABD + ADB = 80 ( sum of D) x + x + x = 80 4x = 80 x = 45 \ ADC = 3x = 3(45 ) = 35, which is an obtuse angle. \ II is not true.
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