JAMB MathematicsStudy notes & practice

What you should be able to do

• Convert whole numbers and fractions between bases
• Work with remainders in modular arithmetic

Understand it

Place value depends on the base. In base b, places to the left of the point represent powers b⁰, b¹, b² and so on; places to the right represent b⁻¹, b⁻² and so on. Each digit must be smaller than the base.

To convert from base ten, divide the whole-number part repeatedly by the new base and read the remainders upwards. Convert a fractional part by repeated multiplication, reading the whole-number parts in order.

Working modulo n means comparing remainders after division by n. For example, 17 and 5 are congruent modulo 12. Perform ordinary addition or multiplication and then reduce the result to its remainder.

Watch out: A base-two number cannot contain the digit 2.

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What you should be able to do

• Calculate fractions, ratios and percentage changes
• Apply stated interest, share and tax rates

Understand it

Use a common denominator to add or subtract fractions. Multiplication works directly across numerators and denominators; division by a fraction means multiplication by its reciprocal. A ratio divides a total into a stated number of equal parts.

Percentage change is change divided by the original value, multiplied by 100. Profit percentage normally uses cost price; discount percentage uses the marked price. Simple interest is I = Prt/100 when r is an annual percentage and t is in years.

For an examination tax or share question, use the rate and price supplied. Percentage error is absolute error divided by the accepted value, multiplied by 100. Keep extra digits during working and round the final answer to the requested accuracy.

Watch out: A 20% rise followed by a 20% fall does not restore the starting value.

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What you should be able to do

• Use index and logarithm laws
• Simplify surds and standard form

Understand it

For the same positive base, multiply powers by adding indices and divide by subtracting indices. Raising a power to another power multiplies the indices. A negative index indicates a reciprocal; a fractional index indicates a root.

logₐx = y means aʸ = x, where a is positive and not 1 and x is positive. Products become sums of logarithms, quotients become differences, and powers become multipliers. Change base using logₐx = logᵦx/logᵦa.

Simplify a surd by removing square factors. Rationalise a denominator using a suitable surd or conjugate. For a nonzero number, standard form is A×10ⁿ with 1 ≤ |A| < 10 and integer n. For positive numbers this becomes 1 ≤ A < 10; it is useful for very large and very small quantities.

Watch out: log(a+b) is not log a + log b.

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What you should be able to do

• Interpret set notation and complements
• Count members of overlapping sets

Understand it

A set is a collection of distinct objects. A subset contains only members of another set. The universal set fixes the objects under discussion; the complement of A contains its members that are not in A.

A ∪ B includes members of A or B or both. A ∩ B includes members common to both. For two finite sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

In a Venn diagram, fill the most restricted overlaps first. For three sets, begin with the central overlap, then the pair-only regions, then the single-only regions. Count the outside region only when the question includes neither set.

Watch out: Students in both groups must not be counted twice in the union.

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What you should be able to do

• Factorise polynomials and use the remainder theorem
• Solve linked equations and rearrange formulae

Understand it

A polynomial is built from constant multiples of non-negative whole-number powers of a variable. Collect like terms before multiplying, factorising or evaluating it. To rearrange a formula, perform the same inverse operation on both sides.

The remainder when f(x) is divided by x−a is f(a). Therefore x−a is a factor exactly when f(a)=0. Factor a cubic by finding one root, dividing by its linear factor and then factorising the resulting quadratic.

To solve one linear and one quadratic equation together, express one variable from the linear equation and substitute into the quadratic. A graph crosses or touches the x-axis at real roots; the y-intercept is f(0).

Watch out: A factor x+2 corresponds to the root −2, not +2.

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What you should be able to do

• Build direct, inverse and joint variation equations
• Determine constants from given data

Understand it

If y varies directly as x, write y=kx. If y varies inversely as x, write y=k/x. The constant k is found from a known pair of values before predicting another pair.

Joint variation combines variables: y varies directly as x and z means y=kxz. If y varies directly as x and inversely as z², write y=kx/z². Read the words before choosing the formula.

Partial variation includes a fixed amount and a varying part, such as a delivery charge y=a+bx. Two data pairs are usually needed to determine both a and b. A proportional relationship passes through the origin; partial variation need not.

Watch out: Inverse variation uses a constant product, not a constant ratio.

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What you should be able to do

• Solve inequalities and show solution intervals
• Interpret inequalities on number lines and graphs

Understand it

Solve a linear inequality much like an equation, but reverse its sign when multiplying or dividing by a negative number. Strict signs < and > exclude an endpoint; ≤ and ≥ include it.

For a quadratic inequality, factorise and mark the roots on a number line. Test the sign in each interval. A positive-leading quadratic is positive outside two distinct real roots and negative between them.

A two-variable linear inequality describes a half-plane. Draw the boundary line, choose a test point such as (0,0) when it is not on the line, and shade the side that satisfies the inequality.

Watch out: Do not reverse an inequality merely because a term is moved across the sign.

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What you should be able to do

• Find terms and sums of APs and GPs
• Recognise constant differences and ratios

Understand it

An arithmetic progression has a constant difference d. Its nth term is a+(n−1)d and the sum of n terms is n[2a+(n−1)d]/2. Check that the number of terms is a whole number.

A geometric progression has a constant ratio r. Its nth term is arⁿ⁻¹. For r≠1, its finite sum is a(1−rⁿ)/(1−r); when r=1 the sum is na.

An infinite GP has a finite sum a/(1−r) only when |r|<1. A negative ratio produces alternating signs. Do not assume every sequence is an AP or GP: test consecutive differences or ratios first.

Watch out: The nth term contains n−1 steps after the first term.

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What you should be able to do

• Evaluate an operation defined by a rule
• Check identity, inverse and algebraic properties

Understand it

A binary operation combines two members of a set using a stated rule. For a*b=a+b+ab, the symbol * does not mean ordinary multiplication. Substitute both values into the whole rule.

Closure means the result remains in the set. Commutativity means a*b=b*a, while associativity means (a*b)*c=a*(b*c). Test the actual rule; an operation can have one property without the other.

An identity e satisfies a*e=e*a=a for every member. An inverse of a combines with a to give that identity. Whether an inverse exists depends on the operation and the set.

Watch out: The identity is not always 1; it depends on the rule.

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What you should be able to do

• Perform matrix operations and evaluate determinants
• Find a 2 by 2 inverse when it exists

Understand it

A matrix arranges numbers in rows and columns. Addition and subtraction require equal dimensions. To multiply A by B, the number of columns of A must equal the number of rows of B; each entry is a row-by-column sum.

For a 2 by 2 matrix [[a,b],[c,d]], the determinant is ad−bc. Its inverse is (1/(ad−bc))[[d,−b],[−c,a]], provided the determinant is not zero. A zero determinant means the inverse does not exist.

For a 3 by 3 determinant, expand along a convenient row or column using alternating signs. Matrix multiplication is generally not commutative. A square identity matrix has 1s on the main diagonal and 0s elsewhere.

Watch out: Subtract bc from ad; do not add the two diagonal products.

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What you should be able to do

• Use angle facts in lines, polygons and circles
• Link constructions to geometric properties

Understand it

Angles on a straight line total 180° and angles around a point total 360°. Parallel lines produce equal corresponding and alternate angles; co-interior angles total 180°. A triangle has angle sum 180°.

The interior-angle sum of an n-sided polygon is (n−2)180°. A regular polygon has equal sides and angles, and each exterior angle is 360°/n. Construction methods use equal radii, angle bisectors and perpendicular bisectors.

In a circle, the angle at the centre is twice the angle at the circumference standing on the same arc. Opposite angles in a cyclic quadrilateral total 180°. A tangent is perpendicular to the radius at contact; intersecting chords satisfy the product-of-segments rule.

Watch out: Equal angles in the same segment must stand on the same chord.

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What you should be able to do

• Calculate areas, volumes and surface areas
• Use circle angles and spherical distances

Understand it

Keep length, area and volume units separate. A circle has area πr² and circumference 2πr. A sector of angle θ degrees has area θπr²/360 and arc length θ2πr/360. For a minor segment, subtract the triangle formed by the two radii and chord from the minor sector. The corresponding major segment is the whole circle minus that minor segment.

A prism has volume equal to cross-sectional area times length. A cylinder has volume πr²h; a cone has πr²h/3; a sphere has 4πr³/3. For a composite solid, add or subtract the volumes and count only exposed surfaces.

On a spherical Earth of radius R, a central angle θ radians subtends distance Rθ along a great circle. A parallel at latitude φ has radius R cosφ. Longitude distances along that parallel therefore use R cosφ rather than R.

Watch out: A diameter must be halved before it is used as a radius.

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What you should be able to do

• Identify common two-dimensional loci
• Combine geometric conditions

Understand it

A locus is the set of points satisfying a condition. Points a fixed distance from a fixed point form a circle. Points equally distant from two fixed points lie on the perpendicular bisector of the segment joining them.

Points a fixed distance from a straight line form two parallel lines, one on either side. Points equally distant from two intersecting lines lie on their angle bisectors. A locus may contain more than one branch.

When two conditions apply together, construct both loci and identify their intersections. Compass arcs preserve equal distances, while a straightedge joins known points. Keep construction arcs visible when a practical construction is requested.

Watch out: A locus is the entire set of allowed positions, not one example point.

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What you should be able to do

• Find distances, midpoints and gradients
• Write equations of straight lines

Understand it

For points (x₁,y₁) and (x₂,y₂), the midpoint is ((x₁+x₂)/2,(y₁+y₂)/2). Their distance is √[(x₂−x₁)²+(y₂−y₁)²], obtained from Pythagoras.

The gradient is (y₂−y₁)/(x₂−x₁) for a non-vertical line. Parallel non-vertical lines have equal gradients. Perpendicular lines with defined nonzero gradients have product −1.

A line of gradient m through (x₁,y₁) has equation y−y₁=m(x−x₁). In y=mx+c, c is the y-intercept. A vertical line instead has equation x=k and an undefined gradient.

Watch out: A horizontal line has gradient 0; a vertical line has undefined gradient.

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What you should be able to do

• Choose trigonometric rules for triangles
• Interpret bearings and elevation angles

Understand it

In a right triangle, sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse and tanθ=opposite/adjacent. The side names depend on the angle chosen. Pythagoras applies only to right triangles.

For any triangle, a/sinA=b/sinB=c/sinC. The cosine rule is a²=b²+c²−2bc cosA, and area=½bc sinA. Use a sketch to match a side with its opposite angle; sine-rule problems may have two possible triangles.

Bearings are measured clockwise from north and normally written with three digits. Elevation and depression angles are measured from a horizontal line. Sine and cosine graphs repeat every 360° or 2π radians; convert units before calculating.

Watch out: A bearing of 060° is measured from north, not from east.

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What you should be able to do

• Interpret a derivative as a limiting rate
• Differentiate powers and basic trigonometric functions

Understand it

A limit describes the value a function approaches as its input approaches a point. A removable factor may allow a limit to exist even when the original expression is undefined at that point.

The derivative gives instantaneous rate of change and the gradient of a tangent. For y=xⁿ, dy/dx=nxⁿ⁻¹. A constant differentiates to zero; differentiate a sum term by term.

With angles in radians, d(sinx)/dx=cosx, d(cosx)/dx=−sinx and d(tanx)/dx=sec²x. For a composite expression, the chain rule also multiplies by the derivative of its inner expression.

Watch out: Differentiating xⁿ decreases the power by one; integrating increases it.

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What you should be able to do

• Apply derivatives to motion and optimisation
• Classify stationary points

Understand it

If displacement is s(t), velocity is ds/dt and acceleration is d²s/dt². The variables and units matter: a derivative with respect to time is a rate per unit time.

A stationary point has first derivative zero. A positive second derivative indicates a local minimum and a negative second derivative indicates a local maximum. If the second derivative is zero, another test is needed.

For an optimisation question, write the quantity to maximise or minimise in one variable, state the allowed domain and test stationary points and relevant endpoints. An algebraic stationary point outside the physical domain is not a valid answer.

Watch out: Zero first derivative alone does not prove that a point is a maximum.

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What you should be able to do

• Find antiderivatives
• Use definite integrals to calculate signed area

Understand it

Integration reverses differentiation. For n≠−1, ∫xⁿdx=xⁿ⁺¹/(n+1)+C. An indefinite integral needs the constant C because many functions have the same derivative.

For angles in radians, ∫cosx dx=sinx+C and ∫sinx dx=−cosx+C. Check an antiderivative by differentiating it. Keep coefficients outside the integral when convenient.

A definite integral from a to b equals F(b)−F(a). It gives signed area: regions below the x-axis contribute negatively. For total geometric area, split at x-axis crossings and add positive area magnitudes.

Watch out: The power rule above does not apply to x⁻¹.

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What you should be able to do

• Construct and interpret frequency tables and charts
• Distinguish categorical and continuous displays

Understand it

A frequency table counts how often values or classes occur. Categories are separate labels, while continuous data can take values across an interval. Choose a display that fits the type of data.

A bar chart has separate bars for categories. A histogram uses touching intervals; when class widths differ, bar height must be frequency density, equal to frequency divided by class width. Histogram area then represents frequency.

In a pie chart, sector angle is category frequency divided by total frequency, multiplied by 360°. Read scales carefully and check whether a graph begins at zero before comparing visual differences.

Watch out: With unequal histogram widths, height alone does not show frequency.

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What you should be able to do

• Calculate location measures
• Estimate medians from grouped data

Understand it

The arithmetic mean is the total of all values divided by their number. For a frequency table it is Σfx/Σf. For grouped data, using class midpoints gives an estimate rather than the exact mean.

The median is the middle value after sorting. For an even number of observations, average the two middle values. The mode is the most frequent value or category; a dataset can have more than one mode.

Cumulative frequency adds frequencies successively and helps locate medians and quartiles. On a grouped cumulative-frequency curve, use the appropriate fraction of the total on the frequency axis and read the estimated data value.

Watch out: Sort observations before locating the median.

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What you should be able to do

• Measure the spread of data
• Interpret changes in location and scale

Understand it

The range is largest minus smallest value. It is easy to calculate but uses only two observations. Mean absolute deviation averages the absolute distances of observations from their mean.

Population variance is Σ(x−mean)²/n, or Σf(x−mean)²/Σf for a frequency table. Standard deviation is its non-negative square root and has the same units as the data. Use a sample formula only when the question specifies that convention.

Adding the same constant to every value shifts the mean but leaves variance unchanged. Multiplying every value by k multiplies standard deviation by |k| and variance by k². Spread compares consistency, not which dataset has a higher average.

If every observation is the same, all deviations from the mean are zero. Both variance and standard deviation are then zero.

Watch out: Variance has squared units; standard deviation returns to the original units.

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What you should be able to do

• Count selections and arrangements
• Handle repeated objects and circular arrangements

Understand it

Use multiplication when a task has successive independent choices. Factorial n! counts arrangements of n distinct objects. A permutation selects and orders r objects: nPr=n!/(n−r)!.

A combination selects without order: nCr=n!/[r!(n−r)!]. Choose the method by asking whether swapping selected objects creates a new outcome. A committee is normally a combination; assigning different offices is a permutation.

For arrangements with repeated indistinguishable objects, divide by the factorials of the repeated counts. Distinct people around an unmarked circular table have (n−1)! arrangements when rotations count as the same; extra symmetry rules must be stated.

Watch out: Do not count AB and BA twice when only selecting a pair.

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What you should be able to do

• Calculate probabilities from equally likely outcomes
• Distinguish independent and mutually exclusive events

Understand it

For equally likely outcomes, probability equals favourable outcomes divided by total outcomes. Probabilities lie between 0 and 1, and the complement rule is P(not A)=1−P(A). Experimental probability uses observed relative frequency.

For any two events, P(A or B)=P(A)+P(B)−P(A and B). Mutually exclusive events cannot happen together, so their overlap is zero. Independence instead means that knowing one occurred does not change the probability of the other.

For independent events, multiply probabilities to find the probability of both. Without replacement, probabilities often change after the first selection; update both the number of favourable objects and the total.

Watch out: Independent and mutually exclusive do not mean the same thing.

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