Math 2 (Grade 11)
Functions, matrices, polynomials and complex numbers, then inverse, radical and rational functions, sequences, probability and trigonometry.
Functions, matrices, polynomials and complex numbers, then inverse, radical and rational functions, sequences, probability and trigonometry.
Units of the book
| Unit | What to know |
|---|---|
| Functions and inequalities | functions, special and piecewise functions, linear programming |
| Matrices | order, operations, multiplication, determinants, Cramer’s rule, inverses |
| Polynomials | complex numbers, the discriminant, division, remainder and factor theorems, end behaviour |
| Inverse and radical functions | composition, inverses, nth roots |
| Rational functions | asymptotes, direct and inverse variation |
| Sequences and series | arithmetic, geometric, infinite series, the binomial theorem |
| Probability | permutations, combinations, independent and exclusive events, geometric probability |
| Trigonometry | trig ratios, radians, laws of sines and cosines, period and amplitude |
Formulas to remember
- det = ad − bc: 2×2 determinant
- i² = −1, i⁴ = 1: powers of i
- Δ = b² − 4ac: discriminant
- x₁ + x₂ = −b/a, x₁x₂ = c/a: sum and product of roots
- aₙ = a₁ + (n−1)d: arithmetic sequence
- aₙ = a₁ rⁿ⁻¹: geometric sequence
- S∞ = a₁/(1 − r): infinite series
- P(n,r) = n!/(n−r)!: permutations
- C(n,r) = n!/(r!(n−r)!): combinations
- 180° = π rad: radians
- a/sin A = b/sin B: law of sines
- c² = a² + b² − 2ab cos C: law of cosines
🧠 Memory kit: functions, matrices and polynomials
| Formula | In plain words | How to remember it |
|---|---|---|
| (m × n)(n × p) = m × p | You can multiply only when the inner numbers match; the outer numbers give the size. | “The inner two shake hands, the outer two stay.” |
| det [a b; c d] = ad − bc | 2×2 determinant: main diagonal minus the other diagonal. | Draw an X: “down from top-left minus down from top-right”. |
| A⁻¹ = (1/det) [d −b; −c a] | Inverse: swap a and d, change the signs of b and c, divide by the determinant. | “Swap, flip, divide.” If the determinant is 0 there is no inverse. |
| x = Dₓ / D, y = D y / D | Cramer’s rule: replace the unknown’s column with the constants. | The bottom is always the coefficient determinant D. |
| i¹ = i, i² = −1, i³ = −i, i⁴ = 1 | Powers of i repeat every 4. | Divide the power by 4 and use the remainder: i²⁰²⁶ → remainder 2 → −1. |
| (a + bi)(a − bi) = a² + b² | A number times its conjugate is real. | Use it to divide complex numbers: multiply top and bottom by the conjugate of the bottom. |
| x = (−b ± √(b² − 4ac)) / 2a | The quadratic formula. | Say it as one line: “negative b, plus or minus root of the discriminant, over two a”. |
| Δ = b² − 4ac | Positive → two real roots, zero → one repeated root, negative → two complex conjugate roots. | It is what sits under the root in the formula; a negative under the root gives i. |
| x₁ + x₂ = −b/a, x₁ · x₂ = c/a | Sum and product of the roots. | The sum “with the sign flipped”, the product “as it is”. |
| f(a) = الباقي | Remainder theorem: dividing f(x) by (x − a) leaves f(a); if that is 0, (x − a) is a factor. | Substitute the number that makes the divisor zero. |
| سلوك الطرفين | Even degree: both ends the same way; odd degree: opposite ways; a positive leading coefficient sends the right end up. | Picture x² (both arms up) and x³ (one arm down, one up). |
| (f ∘ g)(x) = f(g(x)) | Composition: do g first, then f. | Read from the right: the one nearest x works first. |
| f⁻¹: بدّل x و y | To find an inverse, swap x and y and solve for y. Its graph is the mirror image in y = x. | A function has an inverse only if it is one-to-one (horizontal line test). |
| ⁿ√(aᵐ) = a^(m/n) | An nth root is a fractional power. | “The root goes downstairs”: the index is the denominator. |
| f(−x) = f(x) | f(−x) = −f(x) | Even: symmetric in the y-axis (x², cos x). Odd: symmetric about the origin (x³, sin x). | The names come from the powers: x² is even, x³ is odd. |
| |x| | Absolute value less than → between; greater than → outside. | “Less = inside (and), greater = outside (or).” |
| × أو ÷ عدد سالب ⇒ اقلب المتباينة | Multiply or divide an inequality by a negative and the sign flips. | Test it: 2 −3. |
🧠 Memory kit: sequences, probability and trigonometry
| Formula | In plain words | How to remember it |
|---|---|---|
| aₙ = a₁ + (n − 1)d | nth term of an arithmetic sequence (add d each time). | From term 1 to term n you make only (n − 1) jumps. |
| Sₙ = n(a₁ + aₙ) / 2 | Arithmetic sum = number of terms × average of first and last. | Gauss’s trick: 1 + 2 + … + 100 = 50 pairs of 101. |
| aₙ = a₁ · rⁿ⁻¹ | nth term of a geometric sequence (multiply by r each time). | Same idea as arithmetic: adding becomes multiplying, multiplying becomes a power. |
| Sₙ = a₁(1 − rⁿ) / (1 − r) | Sum of n terms of a geometric series. | When rⁿ shrinks to 0 (|r| < 1) it becomes S∞ = a₁/(1 − r). |
| S∞ = a₁ / (1 − r), |r| | Sum to infinity; exists only when |r| < 1. | If |r| ≥ 1 the terms never shrink, so the sum never settles. |
| P(n, r) = n! / (n − r)! | Permutations: when order matters (places, codes, words). | P = Position: first and second are different. |
| C(n, r) = n! / (r!(n − r)!) | Combinations: when order does not matter (a committee, a team). | C = Committee: Ahmed then Ali = Ali then Ahmed. |
| (n − 1)! | n! / (a! b!) | Arrangements in a circle, and arrangements with repeated letters. | In a circle, fix one person and arrange the rest. |
| P(A ∪ B) = P(A) + P(B) − P(A ∩ B) | Probability of “A or B”. | Subtract the overlap because it was counted twice. |
| P(A ∩ B) = P(A) · P(B) | “A and B” for independent events. | “And” = multiply, “or” = add. |
| P(B | A) = P(A ∩ B) / P(A) | Conditional probability: B given that A happened. | The condition becomes the new “whole”, so it goes in the bottom. |
| sin = م/و, cos = ج/و, tan = م/ج | Trig ratios in a right triangle (opposite, adjacent, hypotenuse). | SOH-CAH-TOA. |
| sin 0°, 30°, 45°, 60°, 90° = √0/2, √1/2, √2/2, √3/2, √4/2 | Sines of the special angles. | Count 0, 1, 2, 3, 4 under the root and halve. Cosine is the same list backwards. |
| tan 30° = 1/√3, tan 45° = 1, tan 60° = √3 | Tangents of the special angles (tan = sin ÷ cos). | They climb 1/√3 → 1 → √3; tan 90° is undefined. |
| ASTC | Signs by quadrant: I all positive, II sin, III tan, IV cos. | “All Students Take Calculus”, anticlockwise from quadrant I. |
| rad = deg × π/180 | Degrees to radians. | 180° = π always; to radians × π/180, to degrees × 180/π. |
| s = rθ | Arc length when the angle is in radians. | Radians were made so this formula has no π. |
| a/sin A = b/sin B = c/sin C | Law of sines: use it when you know an angle and its opposite side. | Every side travels with its opposite angle as a pair. |
| c² = a² + b² − 2ab cos C | Law of cosines: two sides and the angle between, or all three sides. | Pythagoras with a correction; when C = 90° the correction is 0. |
| Area = ½ ab sin C | Triangle area from two sides and the included angle. | It is ½ × base × height, with height = b sin C. |
| y = a sin(bx): |a|, 2π/b | Amplitude |a|, period 2π/b (π/b for tan). | a stretches up and down; b squeezes sideways. |
- Permutation or combination? Ask: if two people swap, does the result change? Yes → permutation, no → combination.
- Size of a sample space: n coins → 2ⁿ, n dice → 6ⁿ. Counting principle: multiply the choices at each stage.
- Linear programming: the maximum or minimum is always at a vertex of the feasible region; test the vertices only.
A way of memorizing that sticks
- Understand before you memorize: read “in plain words” for each formula and say out loud what it measures and why each quantity is multiplied or divided. A formula you understand takes a minute to learn.
- Active recall: cover the formula column and write it from memory, then check. Writing from memory beats rereading by far.
- Spaced repetition: review the same sheet after 1 day, 3 days, a week and two weeks. Four short reviews lock it in until test day.
- Your own formula page: write every formula of a book on one page by hand. If it does not fit, you are writing things you can derive, so cut them.
- Teach it: explain the formula to a friend, or to yourself in a mirror, in two sentences. If you cannot explain it, you do not know it yet.
- Use it at once: after each group, press “Try one” in the patterns below. Memory sets when you use it.
Question patterns
Piecewise functions
Find which interval x lies in, then use only that rule.
Matrices
Order is rows × columns. Determinant ad − bc. You can multiply when the inner numbers match.
Complex numbers
Divide the power of i by 4 and use the remainder. When multiplying, expand and replace i² = −1.
Quadratics
Discriminant > 0: two real roots; = 0: one repeated; < 0: two complex.
Remainder theorem, end behaviour
The remainder on dividing by x − a is f(a). End behaviour comes from the degree and leading coefficient.
Composition and inverses
For f(g(x)) work out g first. For an inverse: swap x and y, then solve.
Radicals and rational exponents
a^(m/n) = (ⁿ√a)^m. What is under a square root must be ≥ 0.
Asymptotes and variation
Vertical: zeros of the denominator. Horizontal: compare degrees. Direct y = kx, inverse y = k/x.
Sequences, series and the binomial theorem
Constant difference → arithmetic; constant ratio → geometric. The coefficient of xᵏ in (x + 1)ⁿ is C(n, k).
Probability
Order matters → permutation; not → combination. Independent: multiply; exclusive: add. Geometric: ratio of areas.
Trigonometry
Know the values at 30°, 45°, 60°. Period 2π/|b|, amplitude |a|.
- In (a + bi)² remember i² = −1.
- The sum of the roots is −b/a, with a minus sign.
- An infinite geometric series has a sum only if |r|.
More patterns (figures and calculations)
Triangle area with sine
A = ½ab sin C
Law of cosines
c² = a² + b² − 2ab cos C
Systems of inequalities
Substitute the point into every inequality; it must satisfy all of them.
Sample spaces
Multiply the outcomes of each part.
Series sums
1 + 2 + … + n = n(n + 1)/2
Inverse matrices
A⁻¹ = (1/det) [d −b; −c a]
Even and odd functions
Only even powers → even; only odd powers → odd; a mix → neither.
Polynomial roots
Factor: a difference of squares, then a sum of squares gives complex roots.