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ISC Class 12 Maths Formula Sheet 2027: All Chapters, with Common Traps
This ISC Class 12 Maths formula sheet covers every chapter in the official CISCE ISC 2027 Mathematics (860) syllabus, organised by its seven units and their marks. Each chapter has a compact table and a short "how to use / common traps" note. The 2027 syllabus no longer has the old Section B / Section C choice: all seven units, including Vectors, 3D Geometry and Linear Programming, are for every candidate. Topics that have left the syllabus, such as regression and commerce applications of calculus, are flagged at the end.
A formula sheet is for revision, not for learning a chapter the first time. If a chapter still feels shaky, one-to-one help from our ISC home tutors in Gurgaon for Class 11 and 12 can rebuild it, and this page then becomes your daily recall tool. For the full topic list and marks, see our ISC Class 12 Maths syllabus 2027 guide. The page is built to print cleanly: use your browser's Print option.
How to use this ISC Class 12 Maths formula sheet
Use the sheet for short, daily recall sessions. The units and marks below come from the official CISCE ISC Year 2027 Mathematics (860) syllabus. In Class XII, Paper I (Theory) is 3 hours and 80 marks, and Paper II is Project Work for 20 marks (two projects of 10 marks each).
| Unit (ISC 2027 Class XII) | Official weightage | Chapters on this page |
|---|---|---|
| 1. Relations and Functions | 10 marks | Types of relations and functions; inverse trigonometric functions |
| 2. Algebra | 10 marks | Matrices; determinants |
| 3. Calculus | 35 marks | Continuity and differentiation; applications of derivatives; integrals; area under curves; differential equations |
| 4. Vector Algebra | 5 marks | Vectors, dot and cross products |
| 5. Three-Dimensional Geometry | 6 marks | Lines and planes |
| 6. Linear Programming | 5 marks | Graphical method in two variables |
| 7. Probability | 9 marks | Conditional probability, Bayes' theorem, random variables |
| Total | 80 marks |
The official ISC 2027 Mathematics specimen question paper has 20 compulsory questions in four sections by question length: A (20 one-mark parts, 20 marks), B (2-mark questions, 14 marks), C (3-mark questions, 21 marks) and D (5-mark questions, 25 marks), with internal choice in three questions each in B, C and D. Mathematical tables and graph paper are provided.
- Weight your time by marks. Calculus is 35 of 80 marks, so it gets two sections on this page and should get close to half of your formula revision.
- Cover and recall. Cover the right-hand column, write each result from memory, and add anything you miss to an error list.
- Learn the conditions. Inverse trigonometric identities hold only on stated intervals; the determinant tests for consistency have three cases; a function must be one-one and onto to be invertible. The trap notes point these out.
1. Relations and functions (10 marks)
Relations and functions
| Idea | Definition or result |
|---|---|
| Reflexive | (a, a) ∈ R for every a ∈ A |
| Symmetric | (a, b) ∈ R ⇒ (b, a) ∈ R |
| Transitive | (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R |
| Equivalence relation | Reflexive, symmetric and transitive |
| Identity, empty, universal relations | {(a, a) : a ∈ A}; ∅; A × A |
| One-one (injective) | f(x1) = f(x2) ⇒ x1 = x2 |
| Onto (surjective) | Range = codomain; every y has some x with f(x) = y |
| Invertible | f is invertible ⇔ f is one-one and onto; then f(f−1(y)) = y and f−1(f(x)) = x |
| Composite function | (g ∘ f)(x) = g(f(x)); in general g ∘ f ≠ f ∘ g; (g ∘ f)−1 = f−1 ∘ g−1 |
| Graph of the inverse | Mirror image of the graph of f in the line y = x |
Inverse trigonometric functions
| Function | Domain | Principal value range |
|---|---|---|
| sin−1x | [−1, 1] | [−π/2, π/2] |
| cos−1x | [−1, 1] | [0, π] |
| tan−1x | R | (−π/2, π/2) |
| cot−1x | R | (0, π) |
| sec−1x | |x| ≥ 1 | [0, π] except π/2 |
| cosec−1x | |x| ≥ 1 | [−π/2, π/2] except 0 |
| Property | Formula |
|---|---|
| Complementary pairs | sin−1x + cos−1x = π/2; tan−1x + cot−1x = π/2; sec−1x + cosec−1x = π/2 |
| Reciprocals | sin−1x = cosec−1(1/x); cos−1x = sec−1(1/x); tan−1x = cot−1(1/x) for x > 0 |
| Negatives | sin−1(−x) = −sin−1x; tan−1(−x) = −tan−1x; cos−1(−x) = π − cos−1x; cot−1(−x) = π − cot−1x |
| Conversion (0 ≤ x < 1) | sin−1x = cos−1√(1 − x2) = tan−1[x/√(1 − x2)] |
| Sum and difference of tan−1 | tan−1x + tan−1y = tan−1[(x + y)/(1 − xy)], xy < 1; tan−1x − tan−1y = tan−1[(x − y)/(1 + xy)], xy > −1 |
| Sum of sin−1 and cos−1 (standard conditions) | sin−1x ± sin−1y = sin−1[x√(1 − y2) ± y√(1 − x2)]; cos−1x ± cos−1y = cos−1[xy ∓ √(1 − x2)√(1 − y2)] |
| 2tan−1x | = sin−1[2x/(1 + x2)] for |x| ≤ 1; = cos−1[(1 − x2)/(1 + x2)] for x ≥ 0; = tan−1[2x/(1 − x2)] for |x| < 1 |
| 3tan−1x | = tan−1[(3x − x3)/(1 − 3x2)] for |x| < 1/√3 |
| 2sin−1x and 2cos−1x | 2sin−1x = sin−1[2x√(1 − x2)] for |x| ≤ 1/√2; 2cos−1x = cos−1(2x2 − 1) for 0 ≤ x ≤ 1 |
| 3sin−1x and 3cos−1x | 3sin−1x = sin−1(3x − 4x3) for |x| ≤ ½; 3cos−1x = cos−1(4x3 − 3x) for ½ ≤ x ≤ 1 |
How to use / common traps: the answer to any inverse trigonometric question must lie in the principal range. sin−1(sin 2π/3) is π/3, not 2π/3, and tan−1(tan 3π/4) is −π/4. cos−1(−½) is 2π/3, not −π/3. The sum formulas have conditions; check xy < 1 before using the tan−1 sum. Example: tan−1(½) + tan−1(⅓) = tan−1[(5/6)/(5/6)] = tan−11 = π/4. In functions, "f ∘ g" means apply g first: with f(x) = x2 and g(x) = x + 1, f(g(2)) = 9 but g(f(2)) = 5. To prove a function is not onto, name one value in the codomain that is never reached; to prove it is not one-one, give two inputs with the same output.
2. Algebra: matrices and determinants (10 marks)
Matrices
| Result | Formula |
|---|---|
| Product is defined | A (m × n) × B (n × p) gives an m × p matrix; AB ≠ BA in general |
| Zero product | AB = O does not imply A = O or B = O (the syllabus asks for examples) |
| Transpose | (A′)′ = A; (A + B)′ = A′ + B′; (kA)′ = kA′; (AB)′ = B′A′ |
| Symmetric and skew symmetric | A′ = A; A′ = −A (diagonal entries 0) |
| Any square matrix | A = ½(A + A′) + ½(A − A′), symmetric part plus skew symmetric part |
| Inverse | AB = BA = I ⇒ B = A−1; the inverse is unique if it exists (prove); (AB)−1 = B−1A−1; (A′)−1 = (A−1)′ |
| Singular and non-singular | |A| = 0: singular, no inverse; |A| ≠ 0: non-singular, invertible |
Determinants
| Result | Formula (A of order n; syllabus goes up to 3 × 3) |
|---|---|
| Minor and cofactor | Mij = determinant after deleting row i and column j; Cij = (−1)i + jMij |
| Expansion | |A| = a11C11 + a12C12 + a13C13 (along any row or column) |
| Key properties | |A′| = |A|; swapping two rows changes the sign; two identical rows give 0; Ri → Ri + kRj leaves |A| unchanged; |AB| = |A||B| |
| Scalar multiple | |kA| = kn|A| |
| Adjoint | adj A = transpose of the cofactor matrix; A(adj A) = (adj A)A = |A|I |
| Adjoint properties | |adj A| = |A|n − 1; adj(AB) = (adj B)(adj A); |A−1| = 1/|A| |
| Inverse | A−1 = (adj A)/|A|, |A| ≠ 0 |
| Area of a triangle | ½ |det[[x1, y1, 1], [x2, y2, 1], [x3, y3, 1]]|; collinear points give 0 |
| Solving AX = B | X = A−1B when |A| ≠ 0 (unique solution) |
| Consistency | |A| ≠ 0: consistent, unique solution. |A| = 0 and (adj A)B ≠ O: inconsistent. |A| = 0 and (adj A)B = O: infinitely many solutions or none (check further) |
How to use / common traps: |kA| = kn|A|, not k|A|. For a 3 × 3 matrix with |A| = 7, |adj A| = 72 = 49 and |2A| = 8 × 7 = 56. When solving equations by the matrix method, write the system as AX = B with the unknowns in the same order in every row, then find A−1 and multiply in the right order (A−1B, not BA−1). Example: x + y + z = 6, x − y + z = 2, 2x + y − z = 1 has |A| = 6 and gives x = 1, y = 2, z = 3; substitute back into all three equations as a check, which takes 30 seconds and catches most cofactor sign slips. The area formula needs the modulus; a negative determinant is not a negative area.
3a. Calculus: continuity, differentiability and applications of derivatives (part of 35 marks)
Continuity and differentiability
| Result | Statement |
|---|---|
| Continuity at x = a | limx→a− f(x) = limx→a+ f(x) = f(a) |
| Removable discontinuity | The limit exists but is not equal to f(a), or f(a) is not defined; redefine f(a) to remove it |
| Differentiability at x = a | Left-hand derivative = right-hand derivative |
| Link | Differentiable ⇒ continuous, but not conversely: |x| is continuous at 0 but not differentiable there |
| Greatest integer function [x] | Discontinuous (so not differentiable) at every integer |
Standard derivatives and rules
| f(x) | f′(x) | f(x) | f′(x) |
|---|---|---|---|
| xn | nxn − 1 | sin−1x | 1/√(1 − x2) |
| sin x | cos x | cos−1x | −1/√(1 − x2) |
| cos x | −sin x | tan−1x | 1/(1 + x2) |
| tan x | sec2x | cot−1x | −1/(1 + x2) |
| cot x | −cosec2x | sec−1x | 1/[|x|√(x2 − 1)] |
| sec x | sec x tan x | cosec−1x | −1/[|x|√(x2 − 1)] |
| cosec x | −cosec x cot x | ex | ex |
| logex | 1/x | ax | ax logea |
| Rule | Formula |
|---|---|
| Product and quotient | (uv)′ = u′v + uv′; (u/v)′ = (u′v − uv′)/v2 |
| Chain rule | dy/dx = (dy/du)(du/dx) |
| Implicit functions | Differentiate both sides with respect to x, treating y as a function of x, then collect dy/dx |
| Parametric form | dy/dx = (dy/dt)/(dx/dt) |
| Second derivative, parametric | d2y/dx2 = [d/dt(dy/dx)] ÷ (dx/dt) |
| One function with respect to another | d(u)/d(v) = (du/dx)/(dv/dx), for example sin x3 with respect to x3 gives cos x3 |
| Logarithmic differentiation | For y = uv: log y = v log u, then differentiate; y = xx gives y′ = xx(1 + log x) |
Applications of derivatives
| Result | Formula or test |
|---|---|
| Rate of change | dy/dt = (dy/dx)(dx/dt); for a circle dA/dt = 2πr dr/dt; for a sphere dV/dt = 4πr2 dr/dt |
| Increasing and decreasing | f′(x) > 0 on an interval ⇒ strictly increasing; f′(x) < 0 ⇒ strictly decreasing |
| Tangent at (x1, y1) | y − y1 = m(x − x1), m = dy/dx at the point |
| Normal | y − y1 = (−1/m)(x − x1); if m = 0 the normal is x = x1; if the tangent is vertical the normal is y = y1 |
| Angle between two curves | tan θ = |(m1 − m2)/(1 + m1m2)| at the point of intersection; orthogonal if m1m2 = −1 |
| Critical points | f′(c) = 0 or f′(c) does not exist |
| First derivative test | f′ changes + to −: local maximum; − to +: local minimum; no change: neither (point of inflexion possible) |
| Second derivative test | f′(c) = 0 and f″(c) < 0: local maximum; f″(c) > 0: local minimum; f″(c) = 0: test fails, use the first derivative test |
| Absolute extrema on [a, b] | Compare f at all critical points in (a, b) and at a and b |
How to use / common traps: in parametric second derivatives, students often differentiate dy/dx with respect to t and stop. With x = t2 and y = t3, dy/dx = 3t/2, but d2y/dx2 = (3/2) ÷ 2t = 3/(4t), not 3/2. In rate problems, substitute the particular value only after differentiating: if r grows at 0.5 cm/s, dA/dt at r = 10 cm is 2π × 10 × 0.5 = 10π cm2/s. For absolute extrema, always check the end points. Example: f(x) = x3 − 6x2 + 9x + 1 has f′(x) = 3(x − 1)(x − 3), a local maximum f(1) = 5 and a local minimum f(3) = 1; on [0, 4] the end values f(0) = 1 and f(4) = 5 tie with them, so the absolute maximum is 5 and the absolute minimum is 1, each reached twice. The angle between y = x2 and y = x at (1, 1) uses slopes 2 and 1: tan θ = 1/3, so θ ≈ 18.4°. Our guide to application of derivatives for ISC Class 12 works through the word-problem types in detail.
From our tutors: in maxima and minima word problems, the calculus is usually fine and the marks go on the set-up. The function is written in two variables and never reduced to one, or the domain is missed, so an end point or an impossible value slips through. We ask students to write three lines before differentiating: what is being maximised, the one variable it depends on, and the allowed range of that variable. In our experience this habit fixes more errors than any extra practice of the derivative itself.
3b. Calculus: integrals, area and differential equations (part of 35 marks)
Standard indefinite integrals (add + C)
| Integral | Result |
|---|---|
| ∫xn dx (n ≠ −1); ∫(ax + b)n dx | xn + 1/(n + 1); (ax + b)n + 1/[a(n + 1)] |
| ∫(1/x) dx; ∫ex dx; ∫ax dx | log|x|; ex; ax/log a |
| ∫sin x dx; ∫cos x dx | −cos x; sin x |
| ∫sec2x dx; ∫cosec2x dx | tan x; −cot x |
| ∫sec x tan x dx; ∫cosec x cot x dx | sec x; −cosec x |
| ∫tan x dx; ∫cot x dx | log|sec x|; log|sin x| |
| ∫sec x dx; ∫cosec x dx | log|sec x + tan x|; log|cosec x − cot x| |
| ∫dx/(x2 + a2) | (1/a) tan−1(x/a) |
| ∫dx/(x2 − a2) | (1/2a) log|(x − a)/(x + a)| |
| ∫dx/(a2 − x2) | (1/2a) log|(a + x)/(a − x)| |
| ∫dx/√(a2 − x2) | sin−1(x/a) |
| ∫dx/√(x2 ± a2) | log|x + √(x2 ± a2)| |
| ∫√(a2 − x2) dx | (x/2)√(a2 − x2) + (a2/2) sin−1(x/a) |
| ∫√(x2 + a2) dx | (x/2)√(x2 + a2) + (a2/2) log|x + √(x2 + a2)| |
| ∫√(x2 − a2) dx | (x/2)√(x2 − a2) − (a2/2) log|x + √(x2 − a2)| |
| ∫f′(x)[f(x)]n dx; ∫f′(x)/f(x) dx | [f(x)]n + 1/(n + 1); log|f(x)| |
| Integration by parts | ∫u v dx = u∫v dx − ∫[u′∫v dx] dx; choose u in the order inverse trig, log, algebraic, trig, exponential |
| A useful by-parts result | ∫ex[f(x) + f′(x)] dx = exf(x) |
Methods for the special forms named in the syllabus
| Form | Method |
|---|---|
| sin2x, cos2x, sin3x, cos3x, sin4x, cos4x | Reduce the power: sin2x = (1 − cos 2x)/2; cos2x = (1 + cos 2x)/2; sin3x = (3 sin x − sin 3x)/4; cos3x = (3 cos x + cos 3x)/4; square the double-angle forms for fourth powers |
| 1/(ax2 + bx + c) and 1/√(ax2 + bx + c); √(ax2 + bx + c) | Complete the square, then use a standard form above |
| (px + q)/(ax2 + bx + c), (px + q)/√(ax2 + bx + c), (px + q)√(ax2 + bx + c) | Write px + q = A·d/dx(ax2 + bx + c) + B, find A and B, split |
| Partial fractions (degree of f < degree of g) | Distinct linear: A/(x − a) + B/(x − b); repeated: A/(x − a) + B/(x − a)2 + …; quadratic factor: (Ax + B)/(x2 + k) + … |
| Partial fractions (degree of f ≥ degree of g) | Divide first, then split the proper remainder |
| 1/(a + b cos x), 1/(a + b sin x), 1/(a cos x + b sin x + c) | Substitute t = tan(x/2): cos x = (1 − t2)/(1 + t2), sin x = 2t/(1 + t2), dx = 2 dt/(1 + t2) |
| 1/(a cos x + b sin x) | Write a cos x + b sin x = r sin(x + α), with r = √(a2 + b2), then integrate cosec |
| (a cos x + b sin x)/(c cos x + d sin x) | Numerator = A(denominator) + B(derivative of denominator) |
| 1/(a cos2x + b sin2x + c) | Divide top and bottom by cos2x, put t = tan x |
| (x2 ± 1)/(x4 + 1), 1/(x4 + 1), √tan x, √cot x | Divide by x2 and put t = x ∓ 1/x; for √tan x put tan x = t2 to reach the same form |
Definite integrals (properties listed in the syllabus)
| Property | Statement |
|---|---|
| Fundamental theorem (without proof) | ∫ab f(x) dx = F(b) − F(a), where F′ = f |
| Dummy variable; reversing limits | ∫ab f(x) dx = ∫ab f(t) dt; ∫ab f = −∫ba f |
| Splitting | ∫ab f = ∫ac f + ∫cb f (use at the corners of |x| and [x]) |
| King's property | ∫ab f(x) dx = ∫ab f(a + b − x) dx; special case ∫0a f(x) dx = ∫0a f(a − x) dx |
| Doubling | ∫02a f = 2∫0a f if f(2a − x) = f(x); 0 if f(2a − x) = −f(x) |
| Even and odd | ∫−aa f = 2∫0a f if f is even; 0 if f is odd |
Area under curves
| Result | Formula |
|---|---|
| Area with the x-axis | ∫ab |y| dx; split where the curve crosses the axis |
| Area with the y-axis | ∫cd |x| dy |
| Between two curves | ∫ab [f(x) − g(x)] dx, with f the upper curve; a and b from the points of intersection |
| Checks you can do in your head | Circle x2 + y2 = a2: πa2; ellipse x2/a2 + y2/b2 = 1: πab; y2 = 4ax cut by its latus rectum x = a: 8a2/3; between y2 = 4ax and x2 = 4ay: 16a2/3 |
The syllabus lists the curves as lines, circles, parabolas, ellipses, polynomial functions, the modulus function, and exponential and logarithmic functions.
Differential equations
| Type | Method |
|---|---|
| Order and degree | Order = highest derivative; degree = power of that derivative once the equation is a polynomial in derivatives (not defined otherwise) |
| Formation | Differentiate as many times as there are arbitrary constants, then eliminate them; order = number of constants |
| Variable separable (and reducible to it) | Write as g(y) dy = f(x) dx and integrate both sides |
| Homogeneous (first order, first degree) | dy/dx = F(y/x): put y = vx, dy/dx = v + x dv/dx, then separate |
| Linear in y | dy/dx + Py = Q (P, Q functions of x): IF = e∫P dx; y × IF = ∫Q × IF dx + C |
| Linear in x | dx/dy + Px = Q (P, Q functions of y): IF = e∫P dy; x × IF = ∫Q × IF dy + C |
How to use / common traps: in integrals, the most frequent loss is a dropped "+ C" or a missing modulus in log|x|. Complete the square carefully: ∫dx/(x2 + 4x + 13) = ∫dx/[(x + 2)2 + 9] = (1/3) tan−1[(x + 2)/3] + C. For partial fractions, (x + 2)/[(x − 3)(x + 1)] = (5/4)/(x − 3) − (1/4)/(x + 1); check by putting x = 0 on both sides. King's property turns ∫0π/2 sin x/(sin x + cos x) dx into half of ∫0π/2 1 dx, which is π/4, and ∫−11 x3 cos x dx is 0 at once because the integrand is odd. For areas, sketch first and shade: the area between y = x and y = x2 is ∫01 (x − x2) dx = 1/6, and swapping the curves gives −1/6, which is a sign you have the upper curve wrong. In linear differential equations, make the coefficient of dy/dx equal to 1 before reading off P: for dy/dx + y/x = x2, IF = x and xy = x4/4 + C. Second-order differential equations are explicitly excluded.
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| Result | Formula |
|---|---|
| Magnitude and unit vector | a = a1î + a2ĵ + a3k̂: |a| = √(a12 + a22 + a32); â = a/|a| |
| Direction cosines | l = a1/|a|, m = a2/|a|, n = a3/|a|; l2 + m2 + n2 = 1 |
| Position vector of AB | AB = (position vector of B) − (position vector of A) |
| Section formula (m : n) | Internal: (mb + na)/(m + n); midpoint (a + b)/2 |
| Dot product | a·b = |a||b| cos θ = a1b1 + a2b2 + a3b3; a ⊥ b ⇔ a·b = 0 |
| Projections of a on b | Scalar projection a·b/|b|; vector projection (a·b/|b|2) b |
| Cross product | a × b = |a||b| sin θ n̂ = det[[î, ĵ, k̂], [a1, a2, a3], [b1, b2, b3]]; b × a = −(a × b) |
| Areas | Parallelogram with sides a, b: |a × b|; with diagonals d1, d2: ½|d1 × d2|; triangle: ½|a × b| |
| Collinear vectors | a × b = 0, or a = λb |
How to use / common traps: the projection of a on b divides by |b|, not |a|. Example: a = î + 2ĵ + 2k̂ and b = 3î + 4k̂ give a·b = 11, so the projection of a on b is 11/5, and a × b = 8î + 2ĵ − 6k̂, so the triangle on a and b has area ½√104 = √26. In the cross-product determinant, the middle (ĵ) term takes a minus sign. The syllabus excludes proofs of geometrical theorems by vector methods.
5. Three-dimensional geometry (6 marks)
| Result | Formula |
|---|---|
| Direction ratios of PQ | (x2 − x1, y2 − y1, z2 − z1); divide by PQ for the direction cosines |
| Axes and coordinate planes | x-axis: y = 0, z = 0; xy-plane: z = 0 (and similarly for the others) |
| Line through a point, parallel to b | Vector: r = a + λb; Cartesian: (x − x1)/b1 = (y − y1)/b2 = (z − z1)/b3 |
| Line through two points | r = a + λ(b − a); (x − x1)/(x2 − x1) = (y − y1)/(y2 − y1) = (z − z1)/(z2 − z1) |
| Angle between two lines | cos θ = |b1·b2|/(|b1||b2|); perpendicular: a1a2 + b1b2 + c1c2 = 0; parallel: direction ratios proportional |
| Shortest distance, skew lines | d = |(a2 − a1)·(b1 × b2)|/|b1 × b2| |
| Shortest distance, parallel lines | d = |b × (a2 − a1)|/|b| |
| Coplanar (intersecting) lines | (a2 − a1)·(b1 × b2) = 0 |
| Distance of a point P from a line r = a + λb | |(p − a) × b|/|b|; or take the foot of the perpendicular as a general point on the line |
| Plane, normal form | r·n̂ = d; lx + my + nz = d (d ≥ 0 is the distance from the origin) |
| Plane, one point form | (r − a)·n = 0; A(x − x1) + B(y − y1) + C(z − z1) = 0 |
| Plane, intercept form | x/a + y/b + z/c = 1 |
| Normal to Ax + By + Cz = D | Direction ratios A, B, C |
| Distance of (x1, y1, z1) from Ax + By + Cz = D | |Ax1 + By1 + Cz1 − D|/√(A2 + B2 + C2) |
| Angle between two planes | cos θ = |n1·n2|/(|n1||n2|) |
| Angle between a line and a plane | sin φ = |b·n|/(|b||n|) |
| Line meets plane | Substitute the general point (x1 + b1λ, y1 + b2λ, z1 + b3λ) into the plane and solve for λ |
How to use / common traps: the angle between a line and a plane uses sin, because n is perpendicular to the plane; using cos gives the complement. Example: the distance of (2, 3, −5) from x + 2y − 2z = 9 is |2 + 6 + 10 − 9|/3 = 3. For the lines r = (î + ĵ) + λ(2î − ĵ + k̂) and r = (2î + ĵ − k̂) + μ(3î − 5ĵ + 2k̂), b1 × b2 = 3î − ĵ − 7k̂ and (a2 − a1)·(b1 × b2) = 10, so the shortest distance is 10/√59. If that numerator had come out as 0, the lines would intersect. Before using the skew-lines formula, check the lines are not parallel; parallel lines make b1 × b2 = 0 and need the other formula.
6. Linear programming (5 marks)
| Term or result | Meaning |
|---|---|
| Objective function | Z = ax + by, to be maximised or minimised |
| Constraints | Linear inequalities, plus x ≥ 0, y ≥ 0; the syllabus limits problems to three non-trivial constraints |
| Feasible region | The common region satisfying all constraints; bounded or unbounded; if empty, the problem is infeasible |
| Corner point method | If an optimum exists, it occurs at a corner point of the feasible region: evaluate Z at every corner |
| Bounded region | Both a maximum and a minimum exist |
| Unbounded region | If M is the largest corner value, it is the maximum only if ax + by > M has no point in common with the region (similarly for a minimum with <); otherwise no optimum exists |
How to use / common traps: find every corner, including those on the axes and the intersection of each pair of constraint lines. Example: maximise Z = 3x + 2y subject to x + y ≤ 4, x + 3y ≤ 6, x, y ≥ 0. The corners are (0, 0), (4, 0), (3, 1) and (0, 2), with Z = 0, 12, 11 and 4, so the maximum is 12 at (4, 0). Write the table of corner values in your answer; it is where the method marks sit. In word problems, define x and y in words and include the non-negativity constraints.
7. Probability (9 marks)
| Result | Formula |
|---|---|
| Addition theorem | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) |
| Conditional probability | P(A | B) = P(A ∩ B)/P(B), P(B) ≠ 0 |
| Multiplication theorem | P(A ∩ B) = P(A) P(B | A) = P(B) P(A | B) |
| Independent events | P(A ∩ B) = P(A) P(B); then A′ and B′ are also independent |
| Neither event | P(A′ ∩ B′) = 1 − P(A ∪ B) |
| Total probability (E1, …, En a partition) | P(A) = Σ P(Ei) P(A | Ei) |
| Bayes' theorem | P(Ei | A) = P(Ei) P(A | Ei) / Σ P(Ej) P(A | Ej) |
| Probability distribution | Each pi ≥ 0 and Σ pi = 1 |
| Mean of a random variable | E(X) = Σ xipi |
How to use / common traps: "independent" and "mutually exclusive" are different: mutually exclusive events with non-zero probabilities can never be independent. With independent A and B, P(A) = 0.3 and P(B) = 0.4, P(A ∪ B) = 0.3 + 0.4 − 0.12 = 0.58. For Bayes' theorem, draw a two-level tree first. Example: bag 1 has 4 red and 6 black balls, bag 2 has 7 red and 3 black, a bag is chosen at random and a red ball is drawn. P(red) = ½(0.4) + ½(0.7) = 0.55, and P(bag 2 | red) = 0.35/0.55 = 7/11. For a random variable, check that the probabilities add to 1 before finding the mean; if a question gives them in terms of k, that check is how you find k. The number of heads when two fair coins are tossed has mean 0 × ¼ + 1 × ½ + 2 × ¼ = 1.
From our tutors: the three units that were once optional (Vectors, 3D Geometry and Linear Programming) are now compulsory and together worth 16 marks. Students who studied from older material sometimes treat them as a quick extra, and they are among the easiest marks on the paper if the formulas are secure. We give them a fixed slot every week from the start of Class 12, rather than leaving them to the last month.
Formulas beyond the ISC 2027 Class 12 syllabus
An earlier CISCE Mathematics syllabus split the Class XII paper into Section A (compulsory) and a choice of Section B or Section C, with Section C covering applications of calculus in commerce and linear regression. The ISC 2027 syllabus has seven units for all candidates and does not name the topics below. Many formula sheets, JEE books and older guides still include them. For the ISC 2027 board paper, give them low priority, and check the current syllabus on cisce.org before relying on older material.
| Formula or topic | Where it used to appear | ISC 2027 Class XII syllabus |
|---|---|---|
| Cost, average cost, marginal cost, revenue, marginal revenue, profit and break-even point | Earlier Section C (Application of Calculus) | Not named |
| Lines of regression, regression coefficients byx and bxy, bxy × byx = r2, method of least squares | Earlier Section C (Linear Regression) | Not named |
| Variance and standard deviation of a random variable | Probability (the earlier syllabus named "mean and variance") | Mean only is named |
| Binomial distribution P(X = r) = nCrprqn − r, mean np, variance npq | Probability in many books | Not named in the theory syllabus; appears only as a suggested project |
| Rolle's theorem and Lagrange's mean value theorem | Continuity and differentiability in many books | Not named |
| Approximations using differentials, dy ≈ f′(x) dx | Applications of derivatives in many books | Not named |
| Definite integral as the limit of a sum | Integrals in many books | Not named |
| Cramer's rule | Determinants | Not named; the matrix method X = A−1B is |
| Scalar triple product [a b c] as a topic; volume of a parallelepiped | Vectors | Not named as a topic; the same expression still appears inside the shortest-distance and coplanarity formulas |
| Second-order differential equations | Differential equations | Explicitly excluded |
| Proofs of geometrical theorems using vectors | Vectors | Explicitly excluded |
"Not named" means the topic is absent from the syllabus text; it does not promise that no question will ever touch the idea, and CISCE can revise the syllabus. If you are preparing for JEE alongside the board exam, keep these on your JEE list: our JEE Maths formula sheet covers the wider set. Our guide to ISC specimen papers 2027 explains how to use older papers without wasting time on topics that have left the syllabus.
Printing and daily revision
- Use your browser's Print option. The table of contents and booking boxes are hidden when you print, so the sheet prints as formulas and notes only.
- To keep a copy on your phone or laptop, choose "Save as PDF" as the printer. We do not offer a separate download.
- Revise for 10 to 15 minutes a day, covering the formula column. Give calculus about half the rotation, since it carries 35 of 80 marks.
- After each formula session, solve two short questions from the specimen paper on anything you missed.
- Choosing a book to practise from? See our guide to the best books for ISC Class 12 Maths, and check that any book you use is written for the 2027 syllabus.
Doing Physics too? Our ISC Class 12 Physics formula sheet follows the same format for all nine Physics units.
Frequently asked questions
Does this ISC Class 12 Maths formula sheet cover the whole 2027 syllabus?
It covers the key formulas and results for every chapter in the seven units of the official ISC Year 2027 Mathematics (860) Class XII syllabus. The paper also tests proofs, set-up of word problems and graphs, so pair the sheet with the official specimen paper.
Can I download this as a PDF?
We do not offer a separate download. The page is built to print: use your browser's Print option and choose "Save as PDF" if you want a copy on your device.
Is there still a choice between Section B and Section C in ISC Maths 2027?
No. The ISC 2027 syllabus lists seven units for all candidates, and the 2027 specimen paper has 20 compulsory questions in Sections A to D, which are grouped by marks per question, not by topic. Internal choice is given in three questions each in Sections B, C and D.
Which unit carries the most marks in ISC Class 12 Maths?
Calculus, with 35 of the 80 theory marks. Relations and Functions and Algebra carry 10 each, Probability 9, Three-Dimensional Geometry 6, and Vector Algebra and Linear Programming 5 each.
Is linear regression in the ISC Class 12 Maths syllabus for 2027?
No. Linear regression and the commerce applications of calculus were in the earlier Section C. The 2027 syllabus does not name them, so they are in the flagged table above.
Are mathematical tables provided in the ISC Maths exam?
The official 2027 specimen paper says mathematical tables and graph paper are provided. Check the instructions on your actual board paper, as they are the final word.
How much is the ISC Maths project worth?
Paper II, Project Work, is worth 20 marks: two projects of 10 marks each, marked for format, content, findings and a viva by a visiting examiner.
How should my child revise Maths formulas each day?
Ten to fifteen minutes a day, with calculus in most sessions, works better than one long session a week. Cover the formula column, write from memory, then solve two short questions on anything missed. A home tutor can make this routine stick and check the working, not just the final answer.
Want a tutor to turn this ISC Class 12 Maths formula sheet into marks? Book a free ISC Maths demo with Ajay Vatsyayan Classes, Saraswati kunj II, Wazirabad, Sector 52, Gurugram, Haryana 122003. Male and female tutors are available, at home across Gurgaon or online. You can also see our Class 12 tuition in Gurgaon options.
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