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5B Guide to “O” Level Elementary Mathematics Revision S/N. TOPIC What I need to know… OK () 1. Arithmetic The Number System: Natural, whole, integers, even, odd, prime, rational, irrational, perfect squares & square roots, perfect cubes & cube roots, real & complex Order of Real Numbers: Symbols like =, , , , , Laws of Arithmetic: 1. Commutative Law: a + b = b + a & ab = ba 2. Associative Law: a + (b + c) = (a + b) = c & a(bc) = (ab)c 3. Distributive Law: a(b + c) = ab + ac & a(b - c) = ab – ac Reciprocal of a Number Order of Operations Factors & Multiples & problems involving their applications Directed Numbers [i.e. positive & negative numbers]: Representation of Directed Numbers using the Number Line, Rules of Multiplication & Division for Directed Numbers Fractions: Types of fractions, the four operations of fractions Decimal Places & Significant Figures Estimation & Approximation Standard form: n A x 10 , where 1 A 10 and n is an integer Ratio Proportion: Direct & indirect (inverse) Proportions Average Speed Formula & its application Percentage Personal & Household Finances: Profit & Losses, Discount, Simple Interest Conversion of Currencies Scales & Maps Area Scale Units of Measurements & their Conversions Time 2. Indices The 9 Laws of Indices & their Applications 1. a m a n a m n 2. a m a n a mn 3. a m n a mn 4. a m b m (ab) m m a 5. a m bm , b 0 6. a0 1 b 1 1 1 7. a n n , n a n , a 0 8. a n n a , a 0, n 0 a a a m m 9. a n n n a m , a 0, n 0 3. Algebra Expansion of Algebraic Expressions Factorization of Algebraic Expressions: 1. Extracting of common factors 2. By Grouping 3. “Cross-Method” 2 2 4. By a – b = (a + b)(a – b) Solving Linear Equations Solving Algebraic Fractions Changing the Subject of a Formula Solving of Simultaneous Equations 1. Substitution Method 2. Elimination Method 3. Graphical Method 4. Variation Direct Variation: y = kx, where k is a constant Inverse Variation: y = k/x, where k is a constant Joint Variation: y = kxz, where k is a constant Combined Variation: y = kx/z where k is a constant Special thanks to Ms Tay C.H for sharing this guide. 1 5. Mensuration Perimeter and Area of some Geometrical Shapes: Geometrical Figure Perimeter/ Circumference Area 2 Square 4 x (Length of each Side) (Length of each side) Rectangle 2 x (Length + Breadth) Length x Breadth Triangle ½ x Base x Height or ½ab sin C Sum of length of each side or ½bc sin A or ½ac sin B Parallelogram/ Rhombus Base x Height Trapezium ½ (Sum of // sides) x Height 2 Circle 2r or d r o o 2 Sector of Circle (/360 x 2r) + 2r /360 x r where (/360o x 2r) = Arc Length Surface Area and Volume of some Solids: Solid Surface Area Volume 2 3 Cube 6 x (Length) (Length) Cuboid 2 x ( Length x Breadth) x Length x Breadth x Height (Length x Height) x ( Breadth x Height) 2 2 Cylinder 2rh + 2r r h where Curved Surface Area = 2rh Prism (Perimeter of Cross-section) x Length Cross-section Area x Length + 2(Cross-section Area) Pyramid Sum of all Faces + Base Area 1/3 x Base Area x Height 2 2 Cone r(Slant Length) + r 1/3r h where Curved Surface Area = r(Slant Length) 2 3 Sphere 4r 4/3r 2 2 2 3 Hemisphere 2r + r = 3r 2/3r 6. Coordinate Distance between 2 points: Geometry AB = ( x2 x1 ) 2 ( y 2 y1 ) 2 Midpoint of 2 points: x1 x2 y1 y 2 M= , 2 2 Gradient of a Straight Line: y 2 y1 m= x 2 x1 Gradient of some Straight Lines: 1. m> 0: Upward Sloping Gradient 2. m< 0: Downward Sloping Gradient 3. m = Undefined: Vertical Straight line // to y-axis 4. m = 0: Horizontal Straight line // to x-axis Gradient of parallel lines: // lines have equal gradients Collinear Points: If 3 points A, B, C are collinear, Gradient of AB = Gradient of BC = Gradient of AC General Equation of Straight Line form: y = mx + c , [where m = gradient, c = y-intercept] Equations of Straight lines // to axes: 1. y = c is a horizontal straight line // to the x-axis where c is the y-intercept, 2. x = a is a vertical straight line // to the y-axis where a is the x-intercept. Equations of the axes: 1. y =0 is the equation of the x-axis 2. x = 0 is the equation of the y-axis Proofs for a given quadrilateral: [Refer to Topic 12 on Properties of Quadrilaterals.] Proof that a given triangle is a right-angled triangle: 2 2 2 Use Pythagoras Theorem [i.e. c = a + b ] 7. Solutions to General Quadratic Equation form: 2 Quadratic ax + bx + c = 0, [where a, b and c are real numbers with a 0] Equations Methods of Solving Quadratic Equations: 1. By Factorisation 2. Square Root Property 3. Completing the Square 4. By Quadratic Formula b b 2 4ac The Quadratic Formula: x 2a Solving Problems involving Quadratic Equations Special thanks to Ms Tay C.H for sharing this guide. 2 n 8. Graphs and Shapes of Graphs of the form y = ax , where n = -2, -1, 0, 1, 2, 3 Graphical Drawing of Straight Linear graphs requires only 3 points Solutions of Drawing of special lines, i.e. y = 0, x = 0, y = c, and x = a, y = x, etc. Equations 2 Drawing of Quadratic Graphs of the form y = ax + bx + c, where a 0, b & c are constants: 1. If a > 0, graph is u-shaped and has a minimum point, 2. If a < 0, graph is n-shaped and has a maximum point 3 2 Drawing of Cubic Graphs of the form y = ax + bx + cx + d, where a 0, b, c & d are constants a Drawing of Reciprocal Graphs of the form y = , where a & b are constants, (x + b)0 xb a Drawing of Graphs of the form y = , where a is a constant and x 0 x2 n Drawing of Exponential functions of the form y = a , where a is a positive number Graphical Solutions of Equations: 1. Locating of Intersection points of 2 graphs to obtain solutions. 2. Inserting of suitable straight line graphs to given curve to obtain solutions. Gradient of a curve at a given Point: 1. Construct a tangent to the given point. 2. Locate two points of the tangent. 3. Use gradient formula to derive gradient at that point Obtaining readings from graphs drawn 9. Kinematics Distance-Time Graphs: 1. Gradient: Instantaneous speed 2. Straight-line Gradient: Uniform speed 3. Horizontal or zero Gradient: object at rest or stationary 4. Curved gradient: Gradient of tangent to a point gives the speed at that point Application of Average Speed Formula Speed-Time Graphs: 1. Gradient: Instantaneous acceleration of a moving object 2. Positive Gradient: Acceleration 3. Negative Gradient: Deceleration or Retardation 4. Zero Gradient: Constant Speed 5. Curved Gradient: Gradient of tangent to a point gives the acceleration at that point 6. Area under Graph: Distance Traveled Conversion Graphs 10. Congruent and Congruent Triangles: Similar 2 Triangles are congruent if they have exactly the same shape and size Triangles Congruency symbol: Methods of Proving Triangles are Congruent: 1. All 3 corresponding sides are equal. (SSS) 2. Two sides and the including angle are equal. (SAS) 3. Two angles and a corresponding side are equal. (ASA/ AAS/ SAA) 4. Both s have a right angle, equal hypotenuse and another side, which is equal. (RHS) Definition of Similar Triangles: 2 triangles are similar if corresponding angles on the figures are equal and the ratio of the corresponding sides is the same for all lengths. [i.e. both have same shape but not necessarily same size.] Conditions for Similar Triangles: 1. All 3 corresponding angles are equal. (AAA) 2. All corresponding sides are in the same ratio. [i.e. if ABC is similar to PQR, then AB/PQ = BC/QR =AC/PR] 3. 2 pairs of corresponding sides are n the same ratio and the included angles between them are equal. [i.e. if ABC is similar to PQR, then AB/PQ = BC/QR & B = Q] 11. Area and Areas of Similar Figures: Volume of 2 A1 l1 Similar Figures A2 l 2 , [where A1 and A2 =Areas of the figure l1 and l2 are their respective lengths] Volumes of Similar Figures: 3 V1 l1 V2 l 2 , [where V1 and V2 =Volumes of the figure l1 and l2 are their respective lengths] Special thanks to Ms Tay C.H for sharing this guide. 3 12. Symmetry Line Symmetry and Rotational Symmetry Symmetries of Triangles: Type Properties No. Of Line/s Order of Rotational of Symmetry Symmetry Isosceles 2 equal sides 1 1 2 equal angles opposite the 2 equal sides Equilateral 3 equal sides 3 3 o All angles = 60 Symmetries of Quadrilaterals: Type Properties No. Of Line/s Order of Rotational of Symmetry Symmetry Square 4 equal sides 4 4 o All angles = 90 Opposite sides are // o Diagonals bisect at 90 o Rectangle All angles = 90 2 2 Opposite sides are // & equal Diagonals bisect each other Parallelogram Opposite sides are // & equal 0 2 Opposite angle are equal Diagonals bisect each other Rhombus 4 equal sides 2 2 Opposite sides are // o Diagonals bisect at 90 Kite 2 pairs of equal adjacent sides 1 1 1 pair of equal angles Diagonals bisect at right angles A diagonal bisect a pair of opposite angles Trapezium 1 pair of // sides 0 1 Symmetries of Circles: A circle has infinite number of lines of symmetry and infinite number of orders of Rotational symmetry. Symmetries of Regular Polygons: Type No. of sides No. of Line/s of Order of Rotational Symmetry Symmetry Pentagon 5 5 5 Hexagon 6 6 6 Heptagon 7 7 7 Octagon 8 8 8 Nonagon 9 9 9 Decagon 10 10 10 13. Trigonometry Application of Pythagoras’ Theorem for right-angled triangles: 2 2 2 c =a +b , [where c is the side opposite to the right angle(hypotenuse), a & b are the other 2 adjacent sides] o Trigonometric Ratios of an acute angle (< 90 ) for a right-angled triangle: opposite adjacent opposite sin = , cos = , tan = hypotenuse hypotenuse adjacent [where hypotenuse = side opposite right angle, Opposite = side opposite the angle , Adjacent = side next to the angle ] o o Trigonometric Ratios of Obtuse Angles (90 <<180 ): o o o sin = sin(180 - ), cos = - cos(180 - ), tan = -tan(180 - ) Special Trigonometric Angles: o o o o o o o o o 30 45 60 30 45 60 30 45 60 sin 1 1 cos 1 1 tan 1 1 3 3 3 2 2 2 2 2 2 3 3 Facts about any triangle: o 1. Sum of angles = 180 2. The shortest side is opposite the smallest angle and the longest side is opposite the largest angle. 3. The sum of the length of the two shorter sides is always the length of the longest side. Special thanks to Ms Tay C.H for sharing this guide. 4 The Sine Rule: a b c sin A sin B sin C or sin A sin B sin C a b c Apply Sine Rule when: 1. Any 2 angles and 1 side are known or 2. 2 sides and 1 angle opposite one of those angles are known The Cosine Rule: a 2 b 2 c 2 2bc cos A or b 2 a 2 c 2 2ac cos B or c 2 a 2 b 2 2ab cosC Apply Cosine Rule when: 1. 2 sides and the included angle are known or 2. 3 sides are known Area of Triangle: ½ab sin C or ½bc sin A or ½ac sin B Apply formula when 2 sides and the included angle are known 14. Bearings and Bearings are always: Three- 1. measured from the north Dimensional 2. in a clockwise direction o o Problems 3. is written as a 3-digit number [i.e. 000 to 360 ] Angle of Elevation: The angle formed between the line of sight and the horizontal line when an observer look up from the horizontal platform at an object. Angle of Depression: The angle formed between the line of sight and the horizontal line when an observer look Down from the horizontal platform at an object. Three-Dimensional Problems: To solve for 3-D problems, 1. Useful to extract relevant 2-D planes out from the 3-D solid given. 2. Look for right-angled triangles in the relevant planes. 3. Apply Pythagoras Theorem or TOA CAH SOH to find the unknown. 4. For non-right-angled triangles, apply Sine Rule or Cosine Rule. 15. Geometrical Types of Angles: Properties of Acute Angles Right Angles Obtuse Angles Reflex Angles o o o o o o Angles and < 90 = 90 > 90 , < 180 > 180 , < 360 Circles Geometrical Properties of Angles: o 1. Complementary Angles: 2 angles add up to 90 o 2. Supplementary Angles: 2 angles add up to 180 [i.e. adj. s on a str. Line] o 3. Angles at a point add up to 360 . [i.e. s at a point] 4. Vertically opposite angles are equal. [i.e. vert. Opp. s] 5. Angles formed by parallel lines cut by a transversal: (a) Corresponding angles are equal [i.e. corr. s] (b) Alternate angles are equal [i.e. alt. s] o (c) Interior angles add up to 180 . [i.e. int. s] Angle Properties of a Triangle: o 1. The sum of the 3 angles of a triangle adds up to 180 . [i.e s sum of ] 2. The exterior angle of a triangle is equal to the sum of the interior opposite angles. [i.e. ext. of ] 3. An isosceles triangle has 2 equal angles opposite the 2 equal sides. [i.e base s of isos. ] o 4. An equilateral triangle has 3 equal sides and 3 equal angles, each of 60 . [i.e. of equi. ] Symmetrical Properties of Circles: 1. The perpendicular bisector of a chord passes through the center of the circle. 2. Equal chords of a circle are equidistant from the center of the circle. Special thanks to Ms Tay C.H for sharing this guide. 5 Angle Properties of Circles: 1. An angle at center of a circle is twice the angle at the circumference subtended by the same arc . [i.e. at center = 2 at circumference] x x x 2x 2x 2x 2. The angle at the circumference in a semicircle is a right angle. [i.e. rt. in semicircle] 3. Angles in the same segment of a circle are equal. [ i.e. s in the same segment] x x Cyclic Quadrilaterals: o 1. The opposite angles of a cyclic quadrilateral add up to 180 .[i.e. opp.s of a cyclis quad.] o o d a + c = 180 and b + d = 180 a c b 2. The exterior angle o a cyclic quadrilateral is equal to the interior opposite angle. [i.e. ext. of a cyclic quad.) b a=b a Tangent Theorems: A tangent to a circle is perpendicular to the radius at the point of contact. [i.e. tan rad.] Tangents from an external point: 1. Tangents from an external point to a circle are equal. A i.e. PA = PB 2. The tangents subtend equal angles at the center. a b i.e. POA = POB O a P 3. The line joining the external point to the center of the circle b bisects the angle between the tangents. i.e. APO = BPO B Special thanks to Ms Tay C.H for sharing this guide. 6 The Alternate Segment Theorem: An angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. [i.e. in alt. Segment] A B BCY = BAC and ACX = ABC X Y C 16. Polygons Names of Polygons [Refer to Topic 12.] Interior Angles of a Polygon: 1. For a n-sided polygon, sum of interior angles = (n 2)180 o ( n 2)180 o 2. For a regular n-sided polygon, each interior angle = n Exterior Angles of a Polygon: o 1. Sum of all the exterior angles of any polygon = 360 . 360 o 2. For a regular n-sided polygon, each exterior angle = n 17. Linear Symbols of Inequalities Inequalities Properties of Inequalities Number Line Representation to illustrate solutions of Linear Inequalities Solutions to satisfy more than one set of given linear inequalities. [i.e. Intersection region/ range] 18. Simple Constructions: Construction 1. Line segments, angles, triangles, circles and quadrilaterals. and Loci 2. Perpendicular Bisector of a Line 3. Angle Bisector 4. Perpendicular Line from a point 5. A straight line parallel to another straight line, a given distance from it 6. Circumcircle of a triangle: [i.e. A circle that passes through the 3 vertices of a triangle.] Steps for constructing a circumcircle of a triangle ABC- i) Construct perpendicular bisectors of any 2 sides of the triangle ABC. ii) The 2 perpendiculars bisectors intersects at O. iii) With O as the center, draw a circle with radius OA (or OB or OC). 7. Incircle of a triangle: [i.e. A circle that is completely inside a triangle and whose circumference touched each of the 3 sides of the triangle. Steps for constructing a incircle PQR- i) Construct the angle bisectors of any 2 angles of triangle PQR. ii) The 2 angle bisectors intersect at O. iii) Construct perpendicular line from O to PQ (or QR or PR) to obtain point X (intersection point) . iv) With center O and radius OX, draw a circle. Construction/ Description of Loci: 1. The locus of a point that is of a constant distance, r cm from a fixed point, O is a circle with center at O and radius r. 2. The locus of a point that is equidistant from 2 fixed points, A and B, is the perpendicular bisector of AB. 3. The locus of a point that is equidistant from two fixed intersecting lines, AB and BC is the angle bisector of ABC. 4. The locus of a point that is fixed distance, x cm away from a given straight line AB is a pair of lines parallel to AB and x cm away from AB. 5. The locus of a point P, that moves such that the area of triangle ABP is a constant, is a pair of lines parallel to straight line AB and h cm away from AB. [where h is the height of triangle ABP] o 6. The locus of point P, which moves such that a straight line AB subtends an angle of 90 is the circumference of a circle with AB as the diameter. o 8. The locus of a point P, which moves such that a straight line AB subtends an angle of x , is a pair of identical arcs on either side of AB. 9. Shading regions to show locus of a point P. [* Refer to appendix attached ] 19. Statistics Stem-and-Leaves Diagram Dot Diagram Pie Chart: Displays the given data using sectors of a circle. The angles in each sector are proportional to the frequency represented. Bar Chart: Representation of data using vertical/horizontal bars, often with gaps, for each category. The height/length of each bar represents the frequency. Special thanks to Ms Tay C.H for sharing this guide. 7 Histogram: A vertical bar chart with no gaps in between. The area represents the frequency. Histogram with unequal class intervals: Area of rectangle bar represents the frequency. Frequency density is used for construction of Class Frequency the bars. i.e. Frequency density = Class Width Frequency Polygon: A line graph joining the midpoints of the top of each bar of a histogram. The midpoints at both ends are joined to the horizontal axis to complete the graph. Mean, Median and Mode: 1. The mean of a set of data is the average that can be found by dividing the total of all values by the number of values. Total of all Values Mean No. of Values 2. The median of a set of data is the middle value when the data is arranged in ascending order. If there are 2 middle numbers, the median is the average of the 2 numbers. 3. The mode of a set of data is the value which occurs most often. 4. Mean of Grouped Data: To calculate the estimate mean when data are grouped into intervals. Mean, x fx i.e. Estimated mean = (sum of fx)/(sum of f) f [where x = mid-value of the class interval, f = frequency of the interval] 20. Cumulative Cumulative Frequency Curve: Frequency Drawn by plotting the cumulative frequency against the upper class limits. Distribution Median, Quartiles and Percentiles: 1. When a set of data is divided into 100 equal parts, each part is 1% of the total frequency and is called a percentile. 2. The Quartiles [i.e. median, lower quartile and upper quartile] are formed when the set of data is divided into 4 equal parts. 3. Median = 50th Percentile = Q2 Lower Quartile = 25 th Percentile = Q1 th Upper Quartile = 75 Percentile = Q3 Interquartile range = Upper Quartile – Lower Quartile = Q3 – Q1 21. Vectors Vector Notation The Magnitude of a Vector i.e. AB Length of AB Equal Vectors: 2 vectors are equal if they have the same magnitude and direction. Negative Vectors: Negative vectors are vectors having same magnitude but opposite in direction. Addition of Vectors using Triangle Law and Parallelogram Law. Subtraction of Vectors Scalar Multiplication of a Vector Special thanks to Ms Tay C.H for sharing this guide. 8 Parallel Vectors: 1. If a vector a is parallel to vector b , then a k b . ~ ~ ~ ~ 2. If the vector a is not parallel to vector b and h a k b , then h 0 and k 0 . ~ ~ ~ ~ 3. If m a n b h a k b and vector a is not parallel to vector b , then m h and ~ ~ ~ ~ ~ ~ nk. 4. If the points A, B and C are collinear, then AB k BC . Column Vectors Laws of Column Vectors: p r For any 2 column vectors a and b , q s ~ ~ p r If a = b , then and p r and q s . 1. q s ~ ~ p r p r p r p r 2. q s q s and a b q s q s a b ~ ~ ~ ~ p mp 3. m a m , where m is a scalar. q mq ~ p r mp nr 4. q s mq ns , where m and n are scalars. m a n b m n ~ ~ Magnitude of a Column Vector: u The magnitude of a column vector AB is given by v AB u 2 v 2 Position Vectors: The position vector of any point A is the vector from the origin O to that given point A, denoted by OA . 22. Transformation Isometric and Non-isometric Transformation: Isometric Non-isometric [i.e. Original size and shape is preserved] [i.e. Original size and shape isnot preserved] 1. Reflection 1. Enlargement 2. Rotation 2. Stretching 3. Translation 3. Shearing Reflection: 1. Figure and image are symmetrical about the line of refection. 2. Line of reflection is the perpendicular bisector of the line joining any point to its image. 3. Points lie on the line of reflection are invariant points. [i.e. points that remain unchanged after undergoing transformation.] 4. To describe a reflection, state the equation of the line of reflection. Rotation: 1. Rotates all points of a figure about a fixed point called the „center of rotation‟ through a given angle, in a given direction [i.e. clockwise or anti-clockwise]. 2. To describe a rotation, state the angle of rotation, the direction and the center of rotation. Special thanks to Ms Tay C.H for sharing this guide. 9 Translation: 1. Moves all the points of an object exactly the same distance in the same direction. a 2. Written in terms of a column vector , where a is the number of units moved along b the x-axis and b is the number of units moved along the y-axis. 3. The vector equation representing a translation is x1 x a x a a x1 y y b y b , where b is the translation vector and is y 1 1 x the image of . y 4. To describe a translation, state the translation column vector. Enlargement: 1. Changes the size and shape of the original object without changing its shape. 2. To describe an enlargement, state the center of enlargement and the scale factor, k. length of corresponding side of image 3. Scale Factor, k = length of the side of object Area of image 4. k 2 Or Area of original figure Area of image k 2 ( Area of original figure ) 5. If k > 1, the image is enlarged. If 0 < k < 1, the image becomes smaller. If k > 0 (positive), the object lies on the same side of the center of enlargement. If k < 0 (negative), the object lies on the opposite side of the center of enlargement. Stretching: 1. A stretch is parallel to the x-axis will move a point parallel to the x-axis, over a distance from the given invariant line. 2. A stretch is parallel to the y-axis will move a point parallel to the y-axis, over a distance from the given invariant line. 3. To describe a stretch, state the stretch factor, k and the invariant line (or the direction of the stretch). 4. If A1 is the image of A under a stretch, then Dis tan ce of A1 from in var iant line Stretch Factor, k = Dis tan ce of A from in var iant line 5. Area of image = k X (Area of Object) For a double stretch parallel to the x-axis and y-axis with stretch factors k 1 and k2 respectively. Area of image = k ! k2 X (Area of Object) Shearing: 1. Under a shear, all points move parallel to the given invariant line. 2. Points on the opposite sides of the invariant line move in the opposite directions. 3. To describe a shear, state the shear factor, k and the invariant line (or the direction of the shear). 4. If A1 is the image of A under a shear, then Dis tan ce of A from A1 Shear Factor, k = Dis tan ce of A from in var iant line 5. Area of image is the same as the area of the object. [i.e. area remains unchanged] Combined Transformation: The order of transformation is important when combining transformation. 23. Probability Definition: If an event E can occur in m ways out of a total of n equally likely possible ways, then Number of ways the event E can occur (m) P( E ) Total number of all possible (n) Properties of Probability: 1. P(E) = 0 if E is an impossible event. 2. P(E) = 1 if E is a sure event. 3. 0 P(E ) 1 [i.e. any probability lies between 0 and 1; closer to 1implies more likely to happen.] 4. P(E) + P(E‟) = 1 or P(E) = 1 – P(E‟) Addition Law for Probability: For 2 mutually exclusive events A and B [i.e. events can happen at the same time], P(A or B) = P(A) + P(B) Special thanks to Ms Tay C.H for sharing this guide. 10 Multiplication Law for Probability: If A and B are independent events, [i.e. occurrence of one event does not influence the occurrence of the other.] P(A and B) = P(A) X P(B) 24. Number Number Sequence Patterns and Investigative problems Problem Logical Analysis Solving Special thanks to Ms Tay C.H for sharing this guide. 11