H2 Mathematics Checklist

Graphing techniques

  1. Sketch the shapes of the graphs such as logarithm, exponential, trigonometric, rational functions and conics (Cartesian or parametric form).

  2. Identify the features of each graph such as asymptotes, intercepts, turning points, centre (conics), length of radius or major/minor axis (conics).

  3. Finding intersection of graphs either using GC or algebraic method.

  1. Sketch the shapes of the graphs such as logarithm, exponential, trigonometric, rational functions and conics (Cartesian or parametric form).

  2. Identify the features of each graph such as asymptotes, intercepts, turning points, centre (conics), length of radius or major/minor axis (conics).

  3. Finding intersection of graphs either using GC or algebraic method.

Graphing Transformation

  1. Manipulate the equation of a graph that undergoes linear transformation such as translation, scaling and reflection.

  2. Sketch the resultant graph that has undergone transformations such as  

Inequalities & SLE

  1. Solve an inequality using number line method (for rational functions).

  2. Solve an inequality using graphical method.

  3. Solve basic inequalities involving exponential, logarithm, surds, etc using either graphical or algebraic approach.

  4. Read a context and formulate simultaneous equations to solve for unknown.

Functions

Functions

  1. Use vertical line test to verify if an equation represents a function.

  2. Sketch the graph of a function under a given domain and find its range.

  3. Use horizontal line test to check if the inverse function exists.

  4. Find the equation, domain and range of an inverse function and sketch its graph in relation to the graph of the original function.

  5. Check if a composite of two functions exists.

  6. Find the equation, domain and range of a composite function.

  7. Familiar with piecewise and periodic functions and perform the above points.

AP & GP

  1. Recognise or prove that a sequence or series is an AP or GP.

  2. Find the first term, common difference (AP) or common ratio (GP) and the position or the number of terms in an AP of GP.

  3. Use the various formula to evaluate the value of a specific term or the sum of an AP or GP.

  4. Determine whether a GP converges or not and hence find its sum to infinity (if it exists).

  5. Apply concepts of AP and GP in contextual based problem (e.g. money interest)

Summation (sigma notation)

  1. Evaluate a series using special formulas such as sum of AP or GP, or using Method of Difference.

  2. Apply special properties of summation such as

 

 

 

   3. Perform substitution to relate a new series to another series with known answer.                                          E.g.

Replace r with r + 1

Differentiation

  1. Determine whether a graph is increasing or decreasing using  f '(x).

  2. Determine the concavity of a graph using f "(x).

  3. Use differentiation to find the maximum or minimum of a quantity.

  4. Find tangent and normal of a curve (through Cartesian or parametric form).

  5. Use differentiation to calculate connected rate of change.

MaclAUrin Series

  1. Perform binomial expansion of the form             , where n is a rational number.

  2. Find the general term in a binomial expansion and the range for which the expansion is valid.

  3. Obtain the Maclaurin series of a function through repeated differentiation.

  4. Obtained the Maclaurin series of a function using standard series in MF26.

  5. Perform small angle approximation for trigonometric functions.

Integration

  1. Solve integral using standard formulas.

  2. Solve integral using substitution and integration by parts.

  3. Use integration to evaluate area involving single or two curves with respect to x and y axes.

  4. Use area of rectangles to estimate the area between a curve and an axis.

  5. Use integration to evaluate volume involving single or two curves with respect to x and y axes.

Differential equations

  1. Construct a differential equation based on the context.

  2. Know the difference between a general solution and particular solution.

  3. Solve a differential equation using direct integration, variable separation and substitution methods.

  4. Sketch the graphs of the family of solution curves.

  5. Evaluate the limiting properties of the solution (if necessary).

Vectors

  1. Know what is a position vector and calculate the vector between two points.

  2. Check whether two vectors are parallel and if three points are collinear.

  3. Use ratio theorem to evaluate the position vector of a point given a ratio.

  4. Evaluate the scalar (dot) and vector (cross) product of two vectors,                                     .

  5. Apply scalar product to find the angle between two vectors, check if two vectors are perpendicular, length of projection of one vector onto another, and the respective vector projection.

  6. Apply vector product to find the perpendicular distance from a point to a vector, area of triangle and parallelogram and a third vector that is perpendicular to two known vectors.

  7. Find the vector equation and Cartesian equation of a vector line.

  8. Check whether two lines are parallel, intersecting or skew.

  9. Find the angle, intersection point and distance (if parallel) between two lines.

  10. Find the foot of perpendicular and the perpendicular distance from a point to a line.

  11. Find the equation of a plane in vector, scalar product or Cartesian form.

  12. Find the angle, the intersection and distance (if parallel) between two planes.

  13. Check whether a line is parallel, intersecting or lies on a plane.

  14. Find the angle, intersection point and distance (if parallel) between a line and a plane.

  15. Find the perpendicular distance from a point to the plane, the foot of perpendicular from a point to the plane and the reflection of the point about the plane.

Complex numbers

  1. Perform addition, subtraction, multiplication and division of complex numbers in Cartesian form.

  2. Finding roots of a polynomial.

  3. Convert a complex number from Cartesian form to trigo or exponential form and vice versa.

  4. Apply the properties of modulus and argument to evaluate the product, division and powers of a complex number.

  5. Describe the geometrical effect of multiplying, dividing and raising powers of complex numbers.

Permutation and combination

  1. Able to apply multiplication (sequential steps) and addition rule (cases) correctly.

  2. Able to identify if a question is about permutation or combination.

  3. Know when and how to apply the formula

       correctly.

   4. Apply special techniques such as "insertion method", "complement method" and "grouping method"

       for special permutation situation.

Probability

  1. Identify the correct type of events such as "complement", "union", "intersection", "conditional", "independent" and "mutually exclusive" events and their respective formulas.

  2. Identify the correct strategy to solve the probability problem as as "P&C", "Tree diagram", "Venn Diagram" and "Frequency Table".

Discrete Random Variables

  1. Apply probability concepts and present the probability distribution in a table format.

  2. Calculate the expectation, E(X) and variance Var(X) of a discrete probability distribution.

  3. Apply properties of E(X) and Var(X) such as 

Binomial Distribution

  1. Determine if a random variable has a binomial distribution using the conditions for a binomial distribution.

  2. Solve probability problems involving binomial distribution using GC (binompdf, binomcdf) or the binomial pdf formula

   3. Calculate the expectation E(X), variance Var(X) and the mode of a binomial distribution.

Normal Distribution

  1. Calculate probability problems involving normal distribution using GC (normalcdf, invnorm).

  2. Know when to convert a non-standard normal random variable to the standard normal random variable, Z using the formula

  3. Evaluate the distribution, mean and variance for a linear combination of normal random variables.

Distribution of sample mean

  1. Know the conditions when is the distribution of the sample mean, 

        (a) normal​, (b) approximately normal (CLT) or (c) neither.

   2. Know how to calculate the mean and variance of       and calculate probabilities involving 

       and normal distribution.

Hypothesis Tests

  1. Write the appropriate null and alternative hypothesis statements.

  2. Know the definitions of "level of significance" and "p value".

  3. Know how to calculate the value of test statistic and critical value.

  4. Carry out the hypothesis test (Z test) using p value method or critical value method.

  5. Write appropriate conclusion statements for the hypothesis test.

Linear Regression and Correlation

  1. Use GC and plot scatter diagram and comment on the relationship between two sets of data.

  2. Compute the correlation coefficient and comment on the linear relationship between two sets of data.

  3. Explain, in context, if a linear model is suitable and hence obtain the equation of the least squared regression line.

  4. Explain, in context, the meaning of "a" and "b" in the regression line equation y = a + bx.

  5. Carry out estimation using the equation of the regression line and comment whether the estimate is reliable or not.

  6. Convert a non-linear equation for the relationship between two variables into a linear one.

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