Sec 4 IP Finals Math Summary (2023 batch)

Update: we got A2 for math CONTENTS Surds Inequalities Simultaneous inequalities Indices and Exponential Functions Solving Graphs Logarithms Solving Proving not defined Graphs Discriminant and Roots Prove there are no real roots. For y to be always increasing/decreasing. To find values where (expressions) have no intersection Find range of values of k for which f(x) is always positive/negative Probability Polynomials and partial fractions Factor and remainder theorem Splitting into partial fractions Improper fractions Special methods of substitution Circular Measure Circle properties Circle angle properties Proving circles (do not) intersect Trigonometric Functions and Graphs Drawing parabola Finding asinbx+c Transformation Translation Scalling // Reflection (Further) Trigo identities and equations

Proving equations General steps General methods Finding cosb given cosa Diagrams Solving Bearings Solving Modulus Solving (Further) Differentiation and Applications Solving Finding minimum/maximum area/volume Finding change over time (Further) Integration and Applications Solving Proving an integral is not defined (i.e has an asymptote) Special methods Tangents and Normals Find equation of tangent at x=k Kinematics Linear Law Binomial Theorem The General Term Finding unknowns given first few terms Vectors Surds Always rationalise denominator. CHECK VALIDITY FOR ALL EQUATIONS especially when there is x 2 or y 2 . Inequalities Simultaneous inequalities 1) Solve separately 2) Both solutions are related by ‘and’ not ‘or’

3) Use number line to find common values. 4) Check Validity (LHS=RHS) Indices and Exponential Functions Solving 1) e |x| < 2 a) |x| < ln2 b) x < ln2 or x > -ln2 c) -ln2 < x < ln2 2) Replacing ln(f(x)), log(f(x)), x y with k to solve quadratically. Graphs Logarithms Solving 1) Use properties of log/ln to simplify. a) log (ab) = loga + logb b) log (a/b) = loga - logb c) log a b = lgb/lga d) alogb = logb a 2) Equate both sides to a single log/ln 3) Equate one side to log/ln.

Proving not defined 1) Since (expression) is not defined when (denominator) = 0, 2) (expression) is not defined when (solve denominator). 3) Since (equation) is not continuous at (solved value in denominator), 4) It is not possible to evaluate … Graphs Discriminant and Roots Prove there are no real roots. (Often coupled with asking for proving no other turning points (dy/dx = 0)) 1) Prove D < 0 For y to be always increasing/decreasing. 1) Find dy/dx 2) For y to be increasing, dy/dx > 0. For y to be decreasing, dy/dx < 0 a) (f(x)) 2 > 0 b) e x > 0 c) For dy/dx > 0 or dy/dx < 0, D < 0 To find values where (expressions) have no intersection 1) Equate both into one equation

2) No intersection = no real roots = D < 0 Find range of values of k for which f(x) is always positive/negative 1) Convert to ax 2 +bx+c 2) Use D < 0, a > x or a < x a) To get a k 2 +b k +c 3) Solve for range of k 4) Probability If P(A) × P(B) = P(A ☹ B), A and B are independent events. If not, they are not independent events. If P(A ☹ B) = 0, A and B are mutually exclusive events. Polynomials and partial fractions Factor and remainder theorem 1) f(x) leaves a remainder of k when divided by x → f(0) = k 2) (x-1) is a factor of f(x) → f(1) = 0 3) Fully factorise whenever possible Splitting into partial fractions Improper fractions Long division 🙂

Special methods of substitution 1) Convert to original form either by × or ÷ Circular Measure Circle properties

Circle angle properties

Proving circles (do not) intersect 1) Add radius of both circles 2) Find distance between centre of both circles a) (1) < (2), do not intersect b) (1) > (2), intersect at 2 points c) (1) = (2), intersect at 1 point Trigonometric Functions and Graphs Drawing parabola 1) Sub x=0 to find y-intercepts 2) Determine direction of graph a) y 2 > 0, f(x) > 0, x > 0 or x < 0 3) Sub y=0 to find x-intercept Finding asinbx+c 1) y=asinbx+c or acosbx+c a) Period = / b= 2π 𝑏 2π 𝑝𝑒𝑟𝑖𝑜𝑑 Transformation Translation Translation by a units in the positive/negative x/y direction. 1) y=f(x-a), increase x values by a 2) y=f(x+a), decrease x values by a 3) y-a=f(x), increase y values by a 4) y+a=f(x), decrease y values by a Scalling // Scaling parallel to x/y-axis by a factor of 1/a 1) y=f(a), multiply x values by 1/a 2) ay=f(x), multiply y values by 1/a Reflection Reflection in the x/y-axis. 1) y=f(-x), reverse sign for all x

2) -y=f(x), reverse sign for all y (Further) Trigo identities and equations Proving equations General steps 1) Get rid of ‘1’s using general methods 2) Combine 2 fractions into 1 3) Expand complicated expression (ax 2 +bx+c) a) Usually complicated expressions will cancel out b) E.g (sin 3 x, cot 4 x, cos 3 xsinxcot 2 x) 4) Simplify fraction General methods 1) Addition formula 2) Double angle formula 3) Trigo identities a) sin 4 x + cos 4 x ≠ 1 4) Multiply both numerator and denominator with same expression (introduce new expression) 5) Add and subtract same expression on numerator to match denominator (dividing into 2 fractions) Finding cosb given cosa 1) Use triangle to find sin/cos/tan values 2) Use addition formula (or memorise) a) Take note of positive and negative Diagrams Solving 1) Pythagoras Theorem 2) Addition formula (given in formula sheet)