IMAT Mathematics Master Handbook & Compendium
An exhaustive, deep-dive compendium for the IMAT Mathematics section, containing detailed theoretical frameworks, operational derivations, step-by-step worked numerical examples, and logic warning systems.
🧭 Quick Navigator - 6 Core Mathematics Modules
1. Number Sets, Algebra & Equations
1.1 Hierarchy of Real Number Sets
A robust command of number sets and their nested relationships is fundamental. Real Numbers () consist of Rational Numbers () and Irrational Numbers ().
- Natural Numbers (): Positive counting integers: (excluding 0 for most standard set representations).
- Integers (): Whole numbers and their negatives: .
- Rational Numbers (): Numbers expressible as where and . Includes all terminating and repeating decimals.
- Irrational Numbers (): Decimals that are non-repeating and non-terminating (e.g. , , ).
📊 Diagram Visual: Venn Diagram showcasing the hierarchical embedding of number systems.
Repeating Decimal to Fraction Conversion Algorithm
📌 Concept & Application: The denominator has as many 9s as there are repeating digits in the period, followed by as many 0s as there are non-repeating decimal digits (antiperiod).
Express the repeating decimal as an irreducible rational fraction.
1. Identify parts: All digits = 123, Non-repeating digits = 1.
2. Period has 2 digits (2, 3) $\implies$ denominator starts with two 9s (99). Antiperiod has 1 digit (1) $\implies$ denominator ends with one 0 (990).
3. Apply the formula:
Final Answer: .
1.2 Radicals, Rationalization & Unnesting double radicals
Operations with roots require converting denominators to rational forms and unnesting complex double radical terms.
Rationalization of Denominators
📌 Concept & Application: Multiply both the numerator and denominator by the conjugate to eliminate the irrational root in the denominator.
Double Radical Unnesting Formula
📌 Concept & Application: This method simplifies nested roots. Always verify that a > 0 and a² - 4b is a perfect square.
Simplify the expression: .
1. Find two numbers $x$ and $y$ such that their sum $x+y=8$ and product $xy=15$.
2. The numbers are $5$ and $3$.
3. Apply the unnesting formula:
Final Answer: .
1.3 Polynomial expansions & Vieta's relations
Factoring polynomials and analyzing equations are critical algebraic skills.
Binomial Expansion Theorem (Pascal's Triangle)
📌 Concept & Application: Coefficients correspond directly to the n-th row of Pascal's Triangle.
Quadratic Equation Formulas & Vieta's Root Relations
📌 Concept & Application: Discriminant D = b² - 4ac dictates roots nature: D > 0 (2 real), D = 0 (1 real double), D < 0 (no real roots). Vieta's formulas find root sums and products directly from coefficients.
If $x_1$ and $x_2$ are roots of $2x^2 - 5x + 2 = 0$, evaluate the expression: .
1. By Vieta's formulas, find root sum and product:
2. Rewrite target expression in terms of sum and product:
3. Substitute the values:
Final Answer: .
2. Functions & Logarithms
2.1 Domain, Range, and Special Functions
A function maps elements from domain X to codomain Y. Finding the domain is a core task in coordinate graphing.
- Rational Fractions (): Denominator cannot be zero: .
- Even Roots (): Radicand must be non-negative: .
- Logarithms (): Argument must be strictly positive: .
Quadratic Function Vertex coordinates
📌 Concept & Application: The vertex (x_v, y_v) is the turning point of a parabola. If a > 0, it is a minimum; if a < 0, it is a maximum.
2.2 Logarithmic Rules & Inequalities
The logarithm is the inverse operation of exponentiation: (where base and argument ).
Solve the inequality: .
1. Domain check: Argument must be strictly positive:
2. Convert the inequality:
3. Solve the quadratic inequality:
4. Intersect with domain $D$:
Final Answer: .
3. Trigonometry
3.1 Radians, Right Triangles & Identities
Angles are measured in degrees or radians. The conversion relation is .
Fundamental Trigonometric Identities
📌 Concept & Application: These relations are derived from the Unit Circle equation x² + y² = 1 where coordinates are (cos θ, sin θ).
📊 Diagram Visual: The Unit Circle showing trigonometry coordinate properties.
| Trigonometric values for Key Angles | |||
|---|---|---|---|
| Angle | sin θ | cos θ | tan θ |
| 0 / 0° | 0 | 1 | 0 |
| π/6 / 30° | 1/2 | √3/2 | 1/√3 |
| π/4 / 45° | √2/2 | √2/2 | 1 |
| π/3 / 60° | √3/2 | 1/2 | √3 |
| π/2 / 90° | 1 | 0 | undefined |
3.2 Compound Angle & Laws of Triangles
Addition and Double-Angle Identities
📌 Concept & Application: Crucial for solving trigonometric equations and simplifying expressions.
Law of Sines and Law of Cosines
📌 Concept & Application: Used to solve general triangles. R represents the radius of the circumscribed circle.
In a triangle, side $b = 5$, side $c = 8$, and the enclosed angle $A = 60^\circ$. Calculate the length of side $a$.
1. State the Law of Cosines:
2. Substitute known values ($b=5, c=8, \\cos(60^\circ) = 0.5$):
Final Answer: Side $a = 7$.
4. Sets, Propositions & Logic
4.1 Set Operations & Logical Equivalence
Set theory and propositional logic form the core of IMAT Logical Reasoning questions.
De Morgan's Laws (Sets & Logic)
📌 Concept & Application: The complement of the union equals the intersection of the complements.
Logical Implication & Contrapositive Equivalence
📌 Concept & Application: An implication is logically equivalent to its contrapositive. Note that the converse (Q ⇒ P) and inverse (¬P ⇒ ¬Q) are NOT logically equivalent to the original statement!
- P is a Sufficient condition for Q (if P happens, Q must happen).
- Q is a Necessary condition for P (P cannot happen without Q).
5. Probability & Combinatorics
5.1 Permutations vs Combinations
Combinatorics calculates the number of ways events can occur. The choice of formula depends on whether the selection order matters.
Permutations (Order Matters) vs Combinations (Order Doesn't Matter)
📌 Concept & Application: Permutations are used for ordered arrangements (e.g. race finishes). Combinations are used for unordered selections (e.g. committee selections).
A group of 7 doctors contains 4 surgeons and 3 pediatricians. In how many ways can a committee of 3 doctors be formed if it must contain exactly 2 surgeons?
1. To meet the condition, select 2 surgeons from the 4 available, and 1 pediatrician from the 3 available.
2. Calculate combinations for surgeons:
3. Calculate combinations for pediatricians:
4. Apply the multiplication rule for independent selections:
Final Answer: There are 18 possible ways to form the committee.
5.2 Probability Laws & Conditional Probability
Addition Rule & Conditional Probability
📌 Concept & Application: If A and B are mutually exclusive, P(A ∩ B) = 0. P(A|B) is the probability of A occurring given that B has already occurred.
Independent Events vs Dependent Events
📌 Concept & Application: Independent trials occur when sampling WITH replacement. Dependent trials occur when sampling WITHOUT replacement.
An urn contains 5 red balls and 3 blue balls. If two balls are drawn sequentially without replacement, what is the probability that both balls are red?
1. Probability that the 1st ball is red: .
2. After drawing one red ball, 7 balls remain (4 red, 3 blue).
3. Conditional probability that the 2nd ball is red: .
4. Calculate combined intersection probability:
Final Answer: Probability is .
6. Plane & Solid Geometry
6.1 Euclidean Plane Geometry Theorems
Geometry questions test properties of triangles, polygons, and circles.
Polygons Interior Angles & Area ratios
📌 Concept & Application: If two geometric figures are similar with linear scale factor k, their areas scale as k² and volumes scale as k³.
6.2 Solid Geometry: Volume & Polyhedra
Euler's Polyhedral Formula & Space Diagonal
📌 Concept & Application: Euler's theorem connects Vertices (V), Edges (E), and Faces (F) for any simple convex polyhedron. Space diagonal d calculates distance between opposite corners of a rectangular box.
| Surface Area & Volume of Key 3D Shapes | ||
|---|---|---|
| Shape | Total Surface Area (A) | Volume (V) |
| Sphere | ||
| Cylinder | ||
| Cone | ||
A convex polyhedron has 12 vertices and 30 edges. Calculate the number of faces it possesses.
1. State Euler's Polyhedral Formula:
2. Substitute known values ($V = 12, E = 30$):
Final Answer: The polyhedron has 20 faces (it is an icosahedron).
7. Practice & Interactive Math Drills
Solidify your math skills with original practice problems and targeted exams.