“All Students Take Calculus”
Which ratios are positive in each quadrant
Quadrant I all positive, II sine, III tangent, IV cosine. Going anticlockwise from the first quadrant.
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Mathematics, Class 11 and 12 formula book
Every chapter on one page, to read straight through or print. 12 chapters, 208 formulas with every symbol defined, 43 memory hooks, and the mistakes that cost the most marks.
1Trigonometry
Trigonometry is the chapter the rest of mathematics keeps borrowing from. Calculus, vectors and three-dimensional geometry all assume you know these identities cold, so time spent here is repaid three times over.
Standard values
| What | Formula | Symbols and notes |
|---|---|---|
| sin | 0, ½, 1/√2, √3/2, 1 | at 0°, 30°, 45°, 60°, 90° |
| cos | 1, √3/2, 1/√2, ½, 0 | the sine row read backwards |
| tan | 0, 1/√3, 1, √3, not defined | tan = sin/cos, and cos 90° = 0 |
| Radians | 180° = π radian | 1 radian is about 57.3° |
Identities
| What | Formula | Symbols and notes |
|---|---|---|
| Pythagorean | sin²θ + cos²θ = 1 | the identity everything else is built from |
| Secant form | 1 + tan²θ = sec²θ | divide the first by cos²θ |
| Cosecant form | 1 + cot²θ = cosec²θ | divide the first by sin²θ |
| Sum of angles | sin(A + B) = sinA cosB + cosA sinB | cos(A + B) = cosA cosB − sinA sinB |
| Difference of angles | sin(A − B) = sinA cosB − cosA sinB | cos(A − B) = cosA cosB + sinA sinB |
| Tangent sum | tan(A + B) = (tanA + tanB)/(1 − tanA tanB) | undefined when the denominator is zero |
Multiple and half angles
| What | Formula | Symbols and notes |
|---|---|---|
| Double angle, sine | sin2A = 2 sinA cosA | also 2tanA/(1 + tan²A) |
| Double angle, cosine | cos2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A | three forms, choose what the question gives |
| Double angle, tangent | tan2A = 2tanA/(1 − tan²A) | undefined at 45° |
| Triple angle | sin3A = 3sinA − 4sin³A, cos3A = 4cos³A − 3cosA | the 3-4 and 4-3 pattern |
| Product to sum | 2 sinA cosB = sin(A + B) + sin(A − B) | used constantly in integration |
| Sum to product | sinC + sinD = 2 sin((C + D)/2) cos((C − D)/2) | and cosC + cosD = 2 cos((C+D)/2) cos((C−D)/2) |
Triangles and general solutions
| What | Formula | Symbols and notes |
|---|---|---|
| Sine rule | a/sinA = b/sinB = c/sinC = 2R | R is the circumradius |
| Cosine rule | cosA = (b² + c² − a²)/2bc | use when two sides and the included angle are known |
| Area of a triangle | Δ = ½ab sinC = √(s(s−a)(s−b)(s−c)) | the second is Heron's formula |
| General solution, sine | θ = nπ + (−1)nα | when sinθ = sinα |
| General solution, cosine | θ = 2nπ ± α | when cosθ = cosα |
| General solution, tangent | θ = nπ + α | when tanθ = tanα |
How to remember this chapter
“SOH CAH TOA”
The three basic ratios
Sine is Opposite over Hypotenuse, Cosine Adjacent over Hypotenuse, Tangent Opposite over Adjacent.
“Write the roots, then simplify”
The standard values
Write √0/2, √1/2, √2/2, √3/2, √4/2 for sine at 0, 30, 45, 60, 90 and simplify. Cosine is the same list backwards, so nothing needs memorising.
“Sine is odd, cosine is even”
Signs of negative angles
sin(−θ) = −sinθ but cos(−θ) = cosθ. Tangent, like sine, is odd.
The mistakes that cost the most
- Writing sin(A + B) as sinA + sinB. None of the ratios distribute over addition.
- Forgetting the (−1)n in the general solution for sine.
- Using the cosine rule with the wrong angle. The angle must be the one opposite the side that stands alone.
- Cancelling sinθ from both sides of an equation, which silently loses the roots where sinθ = 0.
- Working in degrees inside a calculus question. Differentiation formulas assume radians.
2Complex Numbers and Quadratic Equations
Complex numbers turn every quadratic into a solvable one. The modulus-argument form is the part that carries into physics and engineering, and De Moivre's theorem is the one result worth learning by heart.
Basics
| What | Formula | Symbols and notes |
|---|---|---|
| Imaginary unit | i = √(−1), i² = −1 | i³ = −i, i⁴ = 1, then it repeats |
| Standard form | z = a + ib | a is the real part, b the imaginary part |
| Conjugate | z̄ = a − ib | reflect across the real axis |
| Modulus | |z| = √(a² + b²) | the distance from the origin |
| Argument | θ = tan−1(b/a) | adjusted for the quadrant of the point |
| Polar form | z = r(cosθ + i sinθ) | r = |z| |
Operations
| What | Formula | Symbols and notes |
|---|---|---|
| Multiplication of moduli | |z₁z₂| = |z₁||z₂| | arguments add |
| Division | |z₁/z₂| = |z₁|/|z₂| | arguments subtract |
| Conjugate product | z·z̄ = |z|² | always real and non-negative |
| De Moivre's theorem | (cosθ + i sinθ)n = cos nθ + i sin nθ | for integer n |
| Triangle inequality | |z₁ + z₂| ≤ |z₁| + |z₂| | equality when the arguments are equal |
| Cube roots of unity | 1, ω, ω², with 1 + ω + ω² = 0 and ω³ = 1 | ω = (−1 + i√3)/2 |
Quadratic equations
| What | Formula | Symbols and notes |
|---|---|---|
| Roots | x = (−b ± √(b² − 4ac))/2a | for ax² + bx + c = 0 |
| Discriminant | D = b² − 4ac | D > 0 real distinct, D = 0 real equal, D < 0 complex |
| Sum of roots | α + β = −b/a | faster than solving |
| Product of roots | αβ = c/a | together these rebuild the equation |
| Forming an equation | x² − (sum)x + product = 0 | when the roots are known |
| Complex roots | occur in conjugate pairs | when the coefficients are real |
How to remember this chapter
“i cycles in fours”
Powers of i
i, −1, −i, 1, then it repeats. Divide the power by 4 and use the remainder.
“Multiply lengths, add angles”
Multiplying complex numbers
Moduli multiply and arguments add. Division does the reverse. Thinking of it as a rotation plus a stretch makes the algebra unnecessary.
“Check D first”
Any quadratic question
The discriminant tells you the nature of the roots before you solve anything, and that is often all the question wanted.
“Omega sums to zero”
Cube roots of unity
1 + ω + ω² = 0 and ω³ = 1. Nearly every ω question collapses using one of those two.
The mistakes that cost the most
- Writing √(−4) × √(−9) as √36. The rule √a × √b = √(ab) fails for negative numbers; the answer is −6.
- Reading the argument straight off tan−1(b/a) without checking the quadrant.
- Forgetting that complex roots come in conjugate pairs when the coefficients are real.
- Using D to find the roots when only their nature was asked, and losing time.
- Treating |z| as z. The modulus is a real, non-negative number.
Open Complex Numbers and Quadratic Equations on its own page
3Permutations, Combinations and Binomial Theorem
These two chapters share one skill: deciding whether order matters. Once that is settled the formulas are short, and the binomial theorem then rests on the same combination symbol.
Counting
| What | Formula | Symbols and notes |
|---|---|---|
| Fundamental principle | if one task has m ways and the next has n, together they have mn | multiply along, add between alternatives |
| Permutations | nPr = n!/(n − r)! | order matters, as in arranging |
| Combinations | nCr = n!/(r!(n − r)!) | order does not matter, as in selecting |
| Relation | nPr = nCr × r! | arrange after selecting |
| Symmetry | nCr = nC(n−r) | choosing r to keep is choosing n − r to leave |
| Pascal's rule | nCr + nC(r−1) = (n+1)Cr | the rule that builds Pascal's triangle |
Standard arrangements
| What | Formula | Symbols and notes |
|---|---|---|
| Circular permutations | (n − 1)! | one position is fixed to remove rotations |
| Necklace or garland | (n − 1)!/2 | a flip gives the same arrangement |
| With repeated items | n!/(p! q! r!) | p, q, r are the counts of each repeated item |
| Repetition allowed | nr | r places, n choices each time |
| At least one | total − none | almost always faster than counting cases |
Binomial theorem
| What | Formula | Symbols and notes |
|---|---|---|
| Expansion | (a + b)n = Σ nCr an−r br | n + 1 terms in all |
| General term | Tr+1 = nCr an−r br | note the r + 1, which trips most students |
| Middle term, n even | the (n/2 + 1)th term | one middle term |
| Middle term, n odd | the (n+1)/2 th and (n+3)/2 th terms | two middle terms |
| Sum of coefficients | put a = b = 1, giving 2n | a one-line trick |
| Greatest coefficient | the middle term's | for a symmetric expansion |
How to remember this chapter
“Order matters, P; order does not, C”
Choosing between the two
A password is a permutation because order changes it. A committee is a combination because it does not. Ask that one question first, every time.
“Circle loses one”
Circular arrangements
Around a table there is no first seat, so fix one person and arrange the rest: (n − 1)!. A garland halves it again because it can be flipped over.
“r plus one”
The general term of a binomial expansion
Tr+1 uses r, so the fourth term needs r = 3. Half the mistakes in this chapter are an off-by-one here.
“At least one means all minus none”
Counting problems
Counting the unwanted case and subtracting is nearly always shorter than counting the wanted ones.
The mistakes that cost the most
- Using permutations for a selection where order does not matter, which inflates the answer by r!.
- Forgetting to divide by the factorials of repeated letters when arranging a word.
- Using n! for a circular arrangement instead of (n − 1)!.
- Taking r as the term number in the binomial general term. The (r + 1)th term uses r.
- Missing that an odd power gives two middle terms, not one.
Open Permutations, Combinations and Binomial Theorem on its own page
4Sequences and Series
Arithmetic and geometric progressions are the most formula-driven chapter in Class 11, and also the most mechanical. Identify which progression you have, write the nth term, and nearly every question answers itself.
Arithmetic progression
| What | Formula | Symbols and notes |
|---|---|---|
| nth term | an = a + (n − 1)d | d is the common difference |
| Sum of n terms | Sn = (n/2)[2a + (n − 1)d] | also (n/2)(a + l), where l is the last term |
| Arithmetic mean | AM = (a + b)/2 | the middle term of three in AP |
| n terms in AP | take them as a − d, a, a + d | symmetric choice keeps the algebra short |
| Sum of first n natural numbers | n(n + 1)/2 | a special case with a = d = 1 |
Geometric progression
| What | Formula | Symbols and notes |
|---|---|---|
| nth term | an = arn−1 | r is the common ratio |
| Sum of n terms | Sn = a(rn − 1)/(r − 1) | for r ≠ 1 |
| Sum to infinity | S∞ = a/(1 − r) | only when |r| < 1 |
| Geometric mean | GM = √(ab) | for two positive numbers |
| Three terms in GP | take them as a/r, a, ar | again for symmetry |
Means and standard sums
| What | Formula | Symbols and notes |
|---|---|---|
| AM and GM inequality | AM ≥ GM | equal only when the numbers are equal |
| Harmonic mean | HM = 2ab/(a + b) | for two numbers |
| Relation | GM² = AM × HM | and AM ≥ GM ≥ HM |
| Sum of squares | Σn² = n(n + 1)(2n + 1)/6 | for the first n natural numbers |
| Sum of cubes | Σn³ = [n(n + 1)/2]² | the square of the sum of the first n numbers |
How to remember this chapter
“Difference or ratio”
Spotting the progression
Subtract consecutive terms: constant means AP. Divide them: constant means GP. Do both on the first three terms before anything else.
“Infinity needs a fraction”
Sum to infinity of a GP
S∞ = a/(1 − r) exists only when |r| < 1. If the ratio is 2, the sum simply grows without limit.
“AM beats GM beats HM”
The order of the three means
For positive unequal numbers, AM > GM > HM, and all three are equal only when the numbers are the same.
The mistakes that cost the most
- Using n as the number of terms when counting from a term other than the first.
- Applying the infinite GP sum when |r| ≥ 1, where no finite sum exists.
- Using the AP sum formula with the last term unknown. Use the 2a + (n − 1)d version instead.
- Forgetting that the GP sum formula divides by r − 1, so r = 1 needs the separate case S = na.
- Taking the geometric mean of two numbers of opposite sign, which has no real value.
5Straight Lines and Conic Sections
Coordinate geometry converts pictures into equations. The straight line forms are asked directly; the conic sections are asked as standard results, so the table of parabola, ellipse and hyperbola parameters is worth having in memory as one block.
Straight lines
| What | Formula | Symbols and notes |
|---|---|---|
| Distance between two points | d = √((x₂ − x₁)² + (y₂ − y₁)²) | from Pythagoras |
| Section formula | ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)) | internal division in the ratio m:n |
| Midpoint | ((x₁ + x₂)/2, (y₁ + y₂)/2) | the section formula with m = n |
| Slope | m = (y₂ − y₁)/(x₂ − x₁) = tanθ | θ is the angle with the positive x axis |
| Slope-intercept form | y = mx + c | c is the y intercept |
| Point-slope form | y − y₁ = m(x − x₁) | the most used form in practice |
| Two-point form | (y − y₁)/(y₂ − y₁) = (x − x₁)/(x₂ − x₁) | when two points are given |
| Intercept form | x/a + y/b = 1 | a and b are the intercepts |
| Parallel and perpendicular | m₁ = m₂ and m₁m₂ = −1 | the two conditions asked most |
| Angle between lines | tanθ = |(m₁ − m₂)/(1 + m₁m₂)| | zero when parallel |
| Distance from a point to a line | d = |ax₁ + by₁ + c|/√(a² + b²) | the line as ax + by + c = 0 |
Circle
| What | Formula | Symbols and notes |
|---|---|---|
| Standard equation | (x − h)² + (y − k)² = r² | centre (h, k), radius r |
| General equation | x² + y² + 2gx + 2fy + c = 0 | centre (−g, −f), radius √(g² + f² − c) |
| Tangent condition | the distance from the centre to the line equals the radius | the standard way to test tangency |
Conic sections
| What | Formula | Symbols and notes |
|---|---|---|
| Parabola | y² = 4ax | vertex at origin, focus (a, 0), directrix x = −a, e = 1 |
| Parabola, latus rectum | length 4a | the chord through the focus perpendicular to the axis |
| Ellipse | x²/a² + y²/b² = 1, a > b | foci (±ae, 0), e = √(1 − b²/a²), e < 1 |
| Ellipse, latus rectum | 2b²/a | and the sum of focal distances is 2a |
| Hyperbola | x²/a² − y²/b² = 1 | foci (±ae, 0), e = √(1 + b²/a²), e > 1 |
| Hyperbola, asymptotes | y = ±(b/a)x | and the difference of focal distances is 2a |
How to remember this chapter
“Minus one means perpendicular”
Testing two lines
Parallel lines have equal slopes; perpendicular lines have slopes multiplying to −1. A vertical line is the exception, having no slope at all.
“e tells the shape”
Identifying a conic
Eccentricity 0 is a circle, less than 1 an ellipse, exactly 1 a parabola, more than 1 a hyperbola. One number names the curve.
“Ellipse minus, hyperbola plus”
The two eccentricity formulas
Ellipse e = √(1 − b²/a²), hyperbola e = √(1 + b²/a²). The sign matches the sign in the curve's own equation.
The mistakes that cost the most
- Using the distance formula for a point and a line. That needs the perpendicular distance formula instead.
- Forgetting that a vertical line has an undefined slope, so the perpendicular test fails there.
- Mixing up 4a in the parabola with 2a in the ellipse when finding the latus rectum.
- Taking a as the larger denominator in a hyperbola. For a hyperbola a belongs to the positive term, whichever is larger.
- Reading the centre of a circle as (g, f) instead of (−g, −f).
6Matrices and Determinants
Matrices are mechanical marks: the rules are short, the arithmetic is the only risk, and solving a system of equations by matrices appears in almost every paper. Determinants supply the tests that decide whether a solution exists at all.
Matrix basics
| What | Formula | Symbols and notes |
|---|---|---|
| Order | a matrix with m rows and n columns is m × n | rows first, always |
| Addition | possible only for the same order | add corresponding elements |
| Multiplication | possible when columns of the first equal rows of the second | the result is m × p |
| Not commutative | AB ≠ BA in general | even when both products exist |
| Transpose | (AB)T = BTAT | the order reverses |
| Symmetric matrix | AT = A | skew-symmetric when AT = −A, with zero diagonal |
Determinants
| What | Formula | Symbols and notes |
|---|---|---|
| Two by two | |A| = ad − bc | for rows (a b) and (c d) |
| Property, row swap | swapping two rows changes the sign | two identical rows give zero |
| Property, scalar | multiplying one row by k multiplies |A| by k | so |kA| = kn|A| for an n × n matrix |
| Product rule | |AB| = |A||B| | determinants multiply |
| Singular matrix | |A| = 0 | no inverse exists |
| Area of a triangle | ½|determinant of the coordinate matrix| | three collinear points give zero |
Inverse and systems
| What | Formula | Symbols and notes |
|---|---|---|
| Adjoint | adj A is the transpose of the cofactor matrix | cofactor Cij = (−1)i+jMij |
| Inverse | A−1 = adj A/|A| | exists only when |A| ≠ 0 |
| Key identity | A(adj A) = |A|I | the fastest route in many proofs |
| Determinant of adjoint | |adj A| = |A|n−1 | for an n × n matrix |
| Matrix method | X = A−1B | for the system AX = B |
| Consistency | |A| ≠ 0 gives a unique solution | |A| = 0 gives either no solution or infinitely many |
| Cramer's rule | x = D₁/D, y = D₂/D, z = D₃/D | D is the determinant of the coefficients |
How to remember this chapter
“Rows then columns, always”
Reading the order of a matrix
A 2 × 3 matrix has 2 rows and 3 columns. The same order applies to the element aij, where i is the row.
“Inner numbers must match”
When multiplication is possible
For an m × n times an n × p, the two inner numbers agree and the answer takes the outer ones: m × p.
“Zero determinant, no inverse”
Singular matrices
A−1 divides by |A|, so a zero determinant means no inverse and no unique solution to the system.
“Checkerboard signs”
Cofactors
Signs alternate plus, minus, plus across the matrix starting from the top-left. The sign is (−1)i+j.
The mistakes that cost the most
- Assuming AB equals BA. Matrix multiplication is not commutative.
- Taking the transpose of a product without reversing the order.
- Dividing by a matrix. There is no division; multiply by the inverse instead.
- Forgetting that |kA| = kn|A|, not k|A|.
- Concluding no solution whenever |A| = 0. It may also mean infinitely many, and the two must be separated.
7Continuity and Differentiability
This chapter supplies the rules used everywhere else in calculus. Learn the standard derivatives as a block, then the four rules, and the rest of the year's differentiation is routine.
Continuity
| What | Formula | Symbols and notes |
|---|---|---|
| Continuous at a point | left limit = right limit = f(a) | all three must exist and agree |
| Differentiable implies continuous | but not the reverse | |x| is continuous at 0 and not differentiable there |
| Standard check | find the limits from both sides, then compare with the value | the method for every piecewise function |
Standard derivatives
| What | Formula | Symbols and notes |
|---|---|---|
| Power | d/dx (xn) = n xn−1 | the one used most |
| Trigonometric | d/dx (sin x) = cos x, d/dx (cos x) = −sin x | note the minus |
| More trigonometric | d/dx (tan x) = sec²x, d/dx (sec x) = sec x tan x | and cot gives −cosec²x |
| Exponential and log | d/dx (ex) = ex, d/dx (ln x) = 1/x | ex is its own derivative |
| General exponential | d/dx (ax) = ax ln a | reduces to ex when a = e |
| Inverse trigonometric | d/dx (sin−1x) = 1/√(1 − x²) | and d/dx (tan−1x) = 1/(1 + x²) |
Rules
| What | Formula | Symbols and notes |
|---|---|---|
| Product rule | (uv)' = u'v + uv' | differentiate one at a time |
| Quotient rule | (u/v)' = (u'v − uv')/v² | the numerator order matters |
| Chain rule | dy/dx = (dy/du)(du/dx) | for a function of a function |
| Implicit differentiation | differentiate both sides, then collect dy/dx | used when y is not isolated |
| Parametric form | dy/dx = (dy/dt)/(dx/dt) | when x and y are both given in terms of t |
| Logarithmic differentiation | take ln of both sides first | for forms like xx |
| Second derivative | d²y/dx² = d/dx(dy/dx) | not the square of the first derivative |
How to remember this chapter
“Differentiable is stronger”
Continuity and differentiability
Every differentiable function is continuous, but not the reverse. A sharp corner is continuous and has no single tangent.
“u dash v plus u v dash”
The product rule
Differentiate the first times the second, plus the first times the derivative of the second. Say it aloud once and the order sticks.
“Low d-high minus high d-low, over low squared”
The quotient rule
The numerator order matters, and reversing it flips the sign of the whole answer.
“Take log for a power of x”
Differentiating xx
Neither the power rule nor the exponential rule applies when the base and index both vary. Take logarithms first.
The mistakes that cost the most
- Using the power rule on ax. The variable is the index, so the exponential rule applies.
- Writing the second derivative as the square of the first.
- Forgetting the inner derivative in the chain rule, which is the most common single error in calculus.
- Reversing the numerator of the quotient rule.
- Assuming a continuous function must be differentiable everywhere.
8Applications of Derivatives
Derivatives stop being algebra here and start answering questions: how fast, how steep, and what is the best. Maxima and minima carry the most marks, and the second derivative test settles them quickly.
Rate of change
| What | Formula | Symbols and notes |
|---|---|---|
| Rate of change | dy/dx is the rate of y with respect to x | related rates chain through a common variable |
| Marginal cost | MC = dC/dx | the cost of producing one more unit |
| Approximation | f(x + Δx) ≈ f(x) + f'(x)Δx | used for small changes and errors |
| Percentage error | (Δy/y) × 100 ≈ (f'(x)Δx/y) × 100 | how an input error propagates |
Tangents and normals
| What | Formula | Symbols and notes |
|---|---|---|
| Slope of the tangent | m = dy/dx at the point | the derivative evaluated there |
| Equation of the tangent | y − y₁ = m(x − x₁) | the point-slope form |
| Slope of the normal | −1/m | perpendicular to the tangent |
| Horizontal tangent | dy/dx = 0 | and the tangent is vertical when dy/dx is undefined |
Increasing, decreasing and extrema
| What | Formula | Symbols and notes |
|---|---|---|
| Increasing | f'(x) > 0 on the interval | the curve rises left to right |
| Decreasing | f'(x) < 0 on the interval | the curve falls |
| Critical point | f'(x) = 0 or undefined | the only candidates for a maximum or minimum |
| First derivative test | sign of f' changes plus to minus at a maximum | minus to plus at a minimum |
| Second derivative test | f''(x) < 0 maximum, f''(x) > 0 minimum | inconclusive when f''(x) = 0 |
| Point of inflexion | f''(x) = 0 and the concavity changes | the curve switches from cup to cap |
| Absolute extrema | check the critical points and both endpoints | on a closed interval |
How to remember this chapter
“Sad face, happy face”
The second derivative test
Negative second derivative means concave down, a sad face, so a maximum. Positive means concave up, a happy face, so a minimum.
“Critical first, test second”
Finding extrema
Set the first derivative to zero to find candidates, then use the second derivative or a sign change to decide which is which.
“Do not forget the ends”
Absolute maximum on a closed interval
The largest value may sit at an endpoint where the derivative is not zero at all. Always compare the endpoints too.
The mistakes that cost the most
- Stopping at f'(x) = 0 and calling it a maximum without testing which it is.
- Ignoring the endpoints when asked for the absolute maximum or minimum on a closed interval.
- Using the second derivative test when f''(x) = 0, where it gives no answer.
- Taking the normal's slope as m rather than −1/m.
- Confusing a point of inflexion with an extremum. At an inflexion the curve keeps going, it only changes its bend.
9Integrals and Their Applications
Integration is differentiation read backwards, and the marks come from recognising the shape of the integrand. The standard forms and the two methods, substitution and by parts, cover nearly every question set.
Standard integrals
| What | Formula | Symbols and notes |
|---|---|---|
| Power rule | ∫xn dx = xn+1/(n + 1) + c | for n ≠ −1 |
| Reciprocal | ∫(1/x) dx = ln|x| + c | the exception to the power rule |
| Exponential | ∫ex dx = ex + c | and ∫ax dx = ax/ln a + c |
| Trigonometric | ∫sin x dx = −cos x + c, ∫cos x dx = sin x + c | the minus sits with sine |
| Secant squared | ∫sec²x dx = tan x + c | and ∫cosec²x dx = −cot x + c |
| Inverse forms | ∫dx/(1 + x²) = tan−1x + c | and ∫dx/√(1 − x²) = sin−1x + c |
| Difference of squares | ∫dx/(x² − a²) = (1/2a)ln|(x − a)/(x + a)| + c | partial fractions in disguise |
Methods
| What | Formula | Symbols and notes |
|---|---|---|
| Substitution | put u = g(x), then du = g'(x)dx | look for a function and its derivative together |
| Integration by parts | ∫u dv = uv − ∫v du | choose u by ILATE |
| Special by-parts result | ∫ex(f(x) + f'(x))dx = exf(x) + c | spot the pattern and skip the work |
| Partial fractions | split a rational function into simpler ones | then integrate term by term |
| Trigonometric powers | use identities to lower the power first | for example cos²x = (1 + cos2x)/2 |
Definite integrals and area
| What | Formula | Symbols and notes |
|---|---|---|
| Fundamental theorem | ∫ from a to b of f(x)dx = F(b) − F(a) | no constant of integration |
| Reversal | ∫ from a to b = −∫ from b to a | swapping the limits changes the sign |
| King's property | ∫ from 0 to a of f(x)dx = ∫ from 0 to a of f(a − x)dx | the most used property in exams |
| Even function | ∫ from −a to a of f(x)dx = 2∫ from 0 to a | when f(−x) = f(x) |
| Odd function | ∫ from −a to a of f(x)dx = 0 | when f(−x) = −f(x) |
| Area under a curve | A = ∫ y dx between the limits | take the modulus where the curve dips below the axis |
| Area between two curves | A = ∫(upper − lower)dx | find the intersection points first |
How to remember this chapter
“ILATE”
Choosing u in integration by parts
Inverse, Logarithmic, Algebraic, Trigonometric, Exponential. Whichever comes first in that list becomes u.
“Odd over a symmetric range is zero”
Definite integrals
If f(−x) = −f(x) and the limits are −a to a, the answer is zero without any work. Check the symmetry before integrating.
“Substitution wants a pair”
Spotting when to substitute
Look for a function and its own derivative in the same integrand. That pairing is the signal to substitute.
“Definite needs no c”
Constants of integration
The constant cancels between the two limits, so a definite integral never carries + c.
The mistakes that cost the most
- Using the power rule for 1/x, which needs the logarithm instead.
- Forgetting + c on an indefinite integral, a mark lost for nothing.
- Changing the variable in a substitution without changing the limits in a definite integral.
- Taking area as the plain integral when the curve crosses the x axis; the parts below count as positive area.
- Choosing u by habit rather than by ILATE, which turns a short problem into a long one.
10Differential Equations
A short, highly patterned chapter. Identify the type, apply its method, and the marks follow. Variable separable and linear equations between them cover the great majority of what is asked.
Basics
| What | Formula | Symbols and notes |
|---|---|---|
| Order | the highest derivative present | y'' gives order 2 |
| Degree | the power of the highest derivative, once free of radicals | not defined if it cannot be made polynomial |
| General solution | contains as many arbitrary constants as the order | a first order equation has one |
| Particular solution | the constants fixed by given conditions | substitute the initial values |
Methods
| What | Formula | Symbols and notes |
|---|---|---|
| Variable separable | write as f(y)dy = g(x)dx, then integrate both sides | the first thing to try |
| Homogeneous | substitute y = vx, so dy/dx = v + x dv/dx | when every term has the same total degree |
| Linear in y | dy/dx + Py = Q, with P and Q functions of x | the standard linear form |
| Integrating factor | IF = e∫P dx | multiply through by it |
| Linear solution | y × IF = ∫(Q × IF)dx + c | the result worth memorising |
| Linear in x | dx/dy + Px = Q | the same method with the roles swapped |
Forming equations
| What | Formula | Symbols and notes |
|---|---|---|
| From a family of curves | differentiate as many times as there are constants | then eliminate them |
| Growth and decay | dy/dt = ky gives y = y₀ekt | k positive for growth, negative for decay |
| Newton's law of cooling | dT/dt = −k(T − T₀) | the same shape of equation |
How to remember this chapter
“Separate first, always”
Which method to try
If the equation can be written with all the y terms on one side and all the x terms on the other, the rest is just two integrations.
“e to the integral of P”
The integrating factor
Get the equation into dy/dx + Py = Q form first, with the coefficient of dy/dx equal to one, then IF = e∫P dx.
“One constant per order”
How many constants to expect
A first order equation has one arbitrary constant in its general solution, a second order two. Use it as a check.
The mistakes that cost the most
- Finding the integrating factor before making the coefficient of dy/dx equal to one.
- Confusing order with degree. Order counts derivatives; degree is a power.
- Forgetting the constant of integration, which makes a general solution particular by accident.
- Substituting y = vx for an equation that is not homogeneous.
- Leaving the answer in terms of v after a homogeneous substitution instead of returning to y.
11Vectors and Three Dimensional Geometry
Vectors and 3D geometry are one chapter twice: every line and plane result is a vector identity written out. Learn the two products properly and the geometry becomes substitution.
Vector algebra
| What | Formula | Symbols and notes |
|---|---|---|
| Magnitude | |a| = √(a₁² + a₂² + a₃²) | for a = a₁î + a₂ĵ + a₃k̂ |
| Unit vector | â = a/|a| | magnitude one, same direction |
| Dot product | a·b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃ | a scalar |
| Perpendicular test | a·b = 0 | for non-zero vectors |
| Cross product | |a × b| = |a||b| sinθ | a vector perpendicular to both |
| Parallel test | a × b = 0 | the sine of zero is zero |
| Area of a triangle | ½|a × b| | and a parallelogram is |a × b| |
| Scalar triple product | [a b c] = a·(b × c) | volume of the parallelepiped; zero if coplanar |
| Projection | projection of a on b = (a·b)/|b| | a scalar length |
Lines in space
| What | Formula | Symbols and notes |
|---|---|---|
| Vector form | r = a + λb | a is a point on the line, b its direction |
| Cartesian form | (x − x₁)/a = (y − y₁)/b = (z − z₁)/c | a, b, c are direction ratios |
| Direction cosines | l = a/√(a² + b² + c²), and l² + m² + n² = 1 | cosines of the angles with the axes |
| Angle between two lines | cosθ = |b₁·b₂|/(|b₁||b₂|) | from the direction vectors |
| Shortest distance, skew lines | |(a₂ − a₁)·(b₁ × b₂)|/|b₁ × b₂| | zero when the lines intersect |
Planes
| What | Formula | Symbols and notes |
|---|---|---|
| Vector form | r·n̂ = d | n̂ is the unit normal, d the distance from the origin |
| Cartesian form | ax + by + cz = d | (a, b, c) are the direction ratios of the normal |
| Through three points | use the scalar triple product to eliminate | or take the cross product of two edge vectors |
| Angle between two planes | cosθ = |n₁·n₂|/(|n₁||n₂|) | the angle between their normals |
| Angle between a line and a plane | sinθ = |b·n|/(|b||n|) | note the sine, not the cosine |
| Distance from a point | |ax₁ + by₁ + cz₁ − d|/√(a² + b² + c²) | the three-dimensional version of the line formula |
How to remember this chapter
“Dot gives a number, cross gives a vector”
Telling the products apart
The dot product answers how much two vectors agree and is a scalar. The cross product answers what is perpendicular to both and is a vector.
“Zero dot perpendicular, zero cross parallel”
The two tests
A zero dot product means at right angles; a zero cross product means in line. cos 90° = 0 and sin 0° = 0 explain both.
“Line and plane uses sine”
The one angle formula that differs
Line to line and plane to plane use cosine. Line to plane uses sine, because the angle is measured to the surface while the normal sticks out of it.
The mistakes that cost the most
- Using cosine for the angle between a line and a plane. That one takes the sine.
- Treating the cross product as commutative. a × b = −(b × a).
- Reading the plane's coefficients as a point on it. They are the direction ratios of the normal.
- Forgetting that the scalar triple product being zero means the three vectors are coplanar.
- Mixing direction ratios with direction cosines. The cosines are the ratios divided by the magnitude.
12Probability
Probability rewards reading the question carefully more than any other chapter. Conditional probability, Bayes' theorem and the binomial distribution are the three results asked, and each has one standard shape of problem.
Basic rules
| What | Formula | Symbols and notes |
|---|---|---|
| Probability | P(A) = favourable outcomes / total outcomes | between 0 and 1 |
| Complement | P(A') = 1 − P(A) | the at least one route |
| Addition rule | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) | subtract the overlap once |
| Mutually exclusive | P(A ∩ B) = 0 | so the addition rule loses its last term |
| Independent events | P(A ∩ B) = P(A)P(B) | independence is about influence, not overlap |
Conditional probability
| What | Formula | Symbols and notes |
|---|---|---|
| Conditional probability | P(A|B) = P(A ∩ B)/P(B) | given that B has happened |
| Multiplication rule | P(A ∩ B) = P(B)P(A|B) | rearranged from the line above |
| Total probability | P(A) = ΣP(Ei)P(A|Ei) | over a partition of the sample space |
| Bayes' theorem | P(Ei|A) = P(Ei)P(A|Ei)/ΣP(Ej)P(A|Ej) | working backwards from an effect to a cause |
Distributions
| What | Formula | Symbols and notes |
|---|---|---|
| Random variable | a numerical outcome of an experiment | its distribution lists the probabilities |
| Mean | E(X) = Σxipi | also called the expectation |
| Variance | Var(X) = Σxi²pi − [E(X)]² | and the standard deviation is its square root |
| Binomial distribution | P(X = r) = nCr pr qn−r | n independent trials, q = 1 − p |
| Binomial mean and variance | mean = np, variance = npq | so the variance is never larger than the mean |
How to remember this chapter
“Given means divide”
Conditional probability
P(A|B) divides by P(B), because B has already happened and the sample space has shrunk to B alone.
“Bayes runs backwards”
When to use Bayes' theorem
If the question tells you the outcome and asks which box or machine it probably came from, it is Bayes. Forwards is total probability; backwards is Bayes.
“At least one means one minus none”
The fastest route in most questions
Counting every successful case is slow. Find the probability of none and subtract from 1.
“np and npq”
The binomial results
Mean np, variance npq. Since q is at most 1, the variance can never exceed the mean, which is a quick sanity check.
The mistakes that cost the most
- Treating mutually exclusive and independent as the same thing. Mutually exclusive events with non-zero probability are never independent.
- Forgetting to subtract P(A ∩ B) in the addition rule, which double counts the overlap.
- Using Bayes' theorem in the forward direction, where total probability is what is needed.
- Writing the variance as Σx²p without subtracting the square of the mean.
- Using the binomial distribution where the trials are not independent, such as drawing cards without replacement.
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