CMA (Cost & Management Accountancy) Foundation: Fundamentals of Business Mathematics and Statistics Syllabus

How do you solve a linear equation in one variable such as $ax+b=0$?

Isolate the variable: $ax=-b$, so $x=-\dfrac{b}{a}$ (for $a\neq 0$). A linear equation has exactly one solution and its highest power of the variable is 1.

State the quadratic formula for $ax^{2}+bx+c=0$.

$x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$, where $a\neq 0$. The two roots come from the $\pm$ sign.

What does the discriminant of a quadratic equation tell you about its roots?

For $ax^{2}+bx+c=0$, the discriminant is $D=b^{2}-4ac$. If $D>0$ the roots are real and distinct; if $D=0$ real and equal; if $D<0$ the roots are complex (imaginary).

State the relationship between the roots and coefficients of $ax^{2}+bx+c=0$.

If $\alpha,\beta$ are the roots, then the sum $\alpha+\beta=-\dfrac{b}{a}$ and the product $\alpha\beta=\dfrac{c}{a}$.

Name two methods to solve a pair of simultaneous linear equations.

Common methods are: (1) Substitution, (2) Elimination (addition/subtraction), (3) Cross-multiplication, and (4) graphical method. They find values of variables satisfying both equations simultaneously.

By cross-multiplication, what are the solutions of $a_{1}x+b_{1}y+c_{1}=0$ and $a_{2}x+b_{2}y+c_{2}=0$?

$\dfrac{x}{b_{1}c_{2}-b_{2}c_{1}}=\dfrac{y}{c_{1}a_{2}-c_{2}a_{1}}=\dfrac{1}{a_{1}b_{2}-a_{2}b_{1}}$, provided $a_{1}b_{2}-a_{2}b_{1}\neq 0$.

Define the derivative of a function $f(x)$ from first principles.

$f'(x)=\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$, the instantaneous rate of change of $f$ with respect to $x$ (the slope of the tangent).

State the power rule, constant rule and constant-multiple rule for differentiation.

$\dfrac{d}{dx}x^{n}=nx^{n-1}$; the derivative of a constant is $0$; $\dfrac{d}{dx}\!\left[c\,f(x)\right]=c\,f'(x)$.

State the product rule and quotient rule of differentiation.

Product: $\dfrac{d}{dx}(uv)=u'v+uv'$. Quotient: $\dfrac{d}{dx}\!\left(\dfrac{u}{v}\right)=\dfrac{u'v-uv'}{v^{2}}$.

State the power rule for integration of $x^{n}$.

$\displaystyle\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C$, for $n\neq -1$. When $n=-1$, $\displaystyle\int \frac{1}{x}\,dx=\ln|x|+C$.

What is the relationship between differentiation and integration (Fundamental Theorem of Calculus)?

Integration is the reverse of differentiation. If $\frac{d}{dx}F(x)=f(x)$, then $\int f(x)\,dx=F(x)+C$, and $\int_{a}^{b}f(x)\,dx=F(b)-F(a)$.

State the first-derivative and second-derivative test conditions for maxima and minima.

At a stationary point $f'(x)=0$. If $f''(x)<0$ the point is a local maximum; if $f''(x)>0$ it is a local minimum; if $f''(x)=0$ the test is inconclusive (point of inflection possible).

Outline the process for solving an applied maxima/minima problem.

1) Express the quantity to optimize as a function of one variable. 2) Differentiate and set $f'(x)=0$ to find critical points. 3) Use the second-derivative test to classify. 4) Evaluate the function to get the maximum/minimum value.

Distinguish between primary data and secondary data in statistics.

Primary data are collected first-hand by the investigator for a specific purpose (surveys, experiments). Secondary data are already collected by someone else and reused (published reports, government records).

Define classification, tabulation and frequency distribution in data presentation.

Classification arranges data into groups/classes by common characteristics. Tabulation presents data systematically in rows and columns. A frequency distribution shows the number of observations (frequency) falling in each class.

Name the three principal measures of central tendency.

The arithmetic mean (average), the median (middle value), and the mode (most frequent value). They are typical/representative values around which data cluster.

State the formula for the arithmetic mean of ungrouped and grouped data.

Ungrouped: $\bar{x}=\dfrac{\sum x}{n}$. Grouped: $\bar{x}=\dfrac{\sum f x}{\sum f}$, where $f$ is the frequency and $x$ the class midpoint.

How is the median of grouped data computed?

$\text{Median}=L+\dfrac{\frac{N}{2}-C}{f}\times h$, where $L$ = lower boundary of median class, $N=\sum f$, $C$ = cumulative frequency before median class, $f$ = frequency of median class, $h$ = class width.

How is the mode of grouped data computed, and state the empirical relation among mean, median and mode.

$\text{Mode}=L+\dfrac{f_{1}-f_{0}}{2f_{1}-f_{0}-f_{2}}\times h$. Empirical relation: $\text{Mode}=3\,\text{Median}-2\,\text{Mean}$.

State the formula for the Geometric Mean and Harmonic Mean of $n$ values.

Geometric mean $GM=\left(x_{1}x_{2}\cdots x_{n}\right)^{1/n}=\sqrt[n]{\prod x_{i}}$. Harmonic mean $HM=\dfrac{n}{\sum \frac{1}{x_{i}}}$.

What are measures of dispersion and why are they used?

Measures of dispersion quantify the spread or variability of data about a central value. They include range, quartile deviation, mean deviation, standard deviation and variance, indicating consistency/reliability of the average.

Define range and coefficient of range.

Range $=L-S$ (largest value minus smallest value). Coefficient of range $=\dfrac{L-S}{L+S}$. It is the simplest measure of dispersion.

Define quartile deviation (semi-interquartile range) and its coefficient.

Quartile deviation $QD=\dfrac{Q_{3}-Q_{1}}{2}$, where $Q_{1}$ and $Q_{3}$ are the first and third quartiles. Coefficient of QD $=\dfrac{Q_{3}-Q_{1}}{Q_{3}+Q_{1}}$.

State the formula for standard deviation of ungrouped data.

$\sigma=\sqrt{\dfrac{\sum (x-\bar{x})^{2}}{n}}$, or equivalently $\sigma=\sqrt{\dfrac{\sum x^{2}}{n}-\left(\dfrac{\sum x}{n}\right)^{2}}$. Variance is $\sigma^{2}$.

Define the coefficient of variation and state its use.

$CV=\dfrac{\sigma}{\bar{x}}\times 100\%$. It is a relative (unit-free) measure of dispersion used to compare variability of two or more data sets; lower $CV$ means greater consistency.

State the classical (mathematical) definition of probability.

$P(E)=\dfrac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}=\dfrac{m}{n}$, where $0\leq P(E)\leq 1$. $P(E)=0$ means impossible, $P(E)=1$ means certain.

State the addition theorem of probability for two events.

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$. For mutually exclusive events ($A\cap B=\varnothing$): $P(A\cup B)=P(A)+P(B)$.

State the multiplication theorem of probability and the condition for independence.

$P(A\cap B)=P(A)\cdot P(B\mid A)=P(B)\cdot P(A\mid B)$. For independent events, $P(A\cap B)=P(A)\cdot P(B)$.

Define conditional probability of $A$ given $B$.

$P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}$, for $P(B)>0$. It is the probability of $A$ occurring given that $B$ has already occurred.

State Bayes' theorem for events.

$P(A_{i}\mid B)=\dfrac{P(A_{i})\,P(B\mid A_{i})}{\sum_{j}P(A_{j})\,P(B\mid A_{j})}$, where the $A_{j}$ form a partition of the sample space. It updates prior probabilities $P(A_{i})$ to posterior probabilities given evidence $B$.