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Class 12 Maths — 15 skills

Everything a Class 12 child (ages 17–18) practises in BeadBop School Maths, aligned to NCERT. Each skill below has a one-line explanation and sample questions with answers — practise them free in the app with step-by-step help.

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Algebra

Relations and functions

A relation is reflexive if every (a, a) is in it, symmetric if (a, b) brings (b, a), transitive if (a, b) and (b, c) bring (a, c). To invert y = ax + b, solve for x: x = (y − b)/a.

Builds on: Relations and functions

Try it:

  1. f(x) = x − 1 and g(x) = x² + 1. Find (f∘g)(3). → 9
  2. On A = {1, 2, 3}, R = {(1, 1), (2, 1), (2, 3), (3, 1)}. Which describes R? → reflexive no, symmetric no, transitive yes

Matrices

In AB, the entry in row i, column j = (row i of A) · (column j of B): multiply pairs and add. An m × n matrix times an n × p matrix gives an m × p matrix.

Builds on: Relations and functions

Try it:

  1. A = [−2 1 ; −3 4], B = [5 −3 ; 3 2] (rows separated by ;). Find the entry in row 1, column 1 of AB. → −7
  2. A = [3 3 ; 4 0], B = [−4 0 ; −4 3] (rows separated by ;). Find the entry in row 2, column 2 of AB. → 0

Determinants

For a 2 × 2 matrix [a b ; c d], det = ad − bc. A 3 × 3 determinant expands along row 1 with signs + − +. A matrix is singular when its determinant is 0, and A⁻¹ = (1/det) × [d −b ; −c a].

Builds on: Matrices

Try it:

  1. A = [k 3 ; 4 −2] (rows separated by ;). For what value of k is A singular? → −6
  2. A = [k −3 ; −4 4] (rows separated by ;). For what value of k is A singular? → 3

Linear programming

The optimal value of Z = px + qy over a bounded feasible region occurs at a corner point: work out Z at every corner and pick the largest (or smallest).

Builds on: Linear inequalities

Try it:

  1. The corner points of a bounded feasible region are (0, 6), (1, 2), (6, 0), (6, 6). Find the minimum of Z = 6x + 9y. → 24
  2. The corner points of a bounded feasible region are (0, 0), (11, 0), (8, 4), (0, 5). Find the maximum of Z = 5x + y. → 55

Trigonometry

Inverse trigonometric functions

Principal values: sin⁻¹ gives −π/2 to π/2, cos⁻¹ gives 0 to π, tan⁻¹ gives between −π/2 and π/2. So sin⁻¹(−1/2) = −π/6 but cos⁻¹(−1/2) = 2π/3.

Builds on: Trigonometric functions

Try it:

  1. Find the principal value of sin⁻¹(−1/2). → −π/6
  2. Find the principal value of cos⁻¹(−1/2). → 2π/3

Calculus

Continuity

f is continuous at x = a when the left limit, the right limit and f(a) are all equal. Make the two pieces give the same value at the joining point.

Builds on: Limits

Try it:

  1. f(x) = (x² − 1)/(x − 1) for x ≠ 1, and f(1) = k. Find k so that f is continuous at x = 1. → 2
  2. f(x) = (x² − 25)/(x − 5) for x ≠ 5, and f(5) = k. Find k so that f is continuous at x = 5. → 10

Differentiation rules

Chain rule: d/dx (ax + b)ⁿ = n·a·(ax + b)ⁿ⁻¹. Product rule: (uv)′ = u′v + uv′. Quotient rule: (u/v)′ = (u′v − uv′)/v².

Builds on: Derivatives

Try it:

  1. y = (x − 3)(x² + 5). Find dy/dx at x = 2. → 5
  2. y = (x + 2)/(x − 5). Find dy/dx at x = 3 (as a fraction if needed). → −7/4

Applications of derivatives

Set f′(x) = 0 to find turning points; f″ > 0 means a minimum, f″ < 0 a maximum. f is increasing where f′(x) > 0. The slope of the tangent at x = a is f′(a).

Builds on: Differentiation rules

Try it:

  1. Find the minimum value of f(x) = 3x² + 6x − 6. → −9
  2. Find the minimum value of f(x) = x² + 8x + 12. → −4

Indefinite integrals

Power rule: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C. Standard forms: ∫cos kx dx = (1/k) sin kx + C, ∫eᵏˣ dx = (1/k) eᵏˣ + C. Check by differentiating your answer.

Builds on: Differentiation rules

Try it:

  1. Find ∫−12 sin 6x dx. → 2 cos 6x + C
  2. Find ∫12 cos 3x dx. → 4 sin 3x + C

Definite integrals

Find an antiderivative F, then ∫ from a to b of f(x) dx = F(b) − F(a). Example: ∫ from 0 to 2 of 3x² dx = 2³ − 0³ = 8.

Builds on: Indefinite integrals

Try it:

  1. Evaluate ∫ 0 1 (4x² − 4x + 3) dx. → 7/3
  2. Evaluate ∫ −1 4 (−2x² − x + 4) dx. → −185/6

Area under a curve

Area under y = f(x) from x = a to x = b is ∫ f(x) dx from a to b. Area between two curves = ∫ (upper − lower) dx between their meeting points.

Builds on: Definite integrals

Try it:

  1. Find the area (in square units) enclosed between y = x² and the line y = 16. → 256/3
  2. Find the area (in square units) enclosed between y = x² and the line y = 25. → 500/3

Differential equations

Order = the highest derivative; degree = the power of that highest derivative. To solve dy/dx = f(x), integrate both sides and use the given condition to find C.

Builds on: Indefinite integrals

Try it:

  1. Find the order and degree of (y‴)³ + (y″)² + 4y = 0. → order 3, degree 3
  2. Solve dy/dx = x/y (y > 0) with y = 9 when x = 0. Find y when x = 12. → 15

Shapes & patterns

Vectors

For a = a₁î + a₂ĵ + a₃k̂: |a| = √(a₁² + a₂² + a₃²) and a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ. Projection of a on b = a·b / |b|.

Builds on: Introduction to 3D geometry

Try it:

  1. Find the angle between a = −î − k̂ and b = −2î + 2ĵ. → 60°
  2. Find |a × b| for a = −3î + 2ĵ − 3k̂ and b = 2ĵ + 2k̂. → 2√43

Three-dimensional geometry

Direction cosines of a, b, c are a/r, b/r, c/r with r = √(a² + b² + c²). The distance from (x₁, y₁, z₁) to ax + by + cz = d is |ax₁ + by₁ + cz₁ − d| / √(a² + b² + c²).

Builds on: Vectors

Try it:

  1. Find the distance of the point (−2, 1, −3) from the plane 2x − 2y − z = −17. → 14/3
  2. Find the direction cosines of the vector −2î − 10ĵ + 11k̂. → −2/15, −2/3, 11/15

Data handling

Probability

P(A|B) = P(A ∩ B)/P(B). A and B are independent when P(A ∩ B) = P(A) P(B). Bayes: P(bag I | red) = P(I) P(red | I) / P(red). A binomial distribution has mean np and variance npq.

Builds on: Probability of events

Try it:

  1. X follows a binomial distribution with n = 12 and p = 1/3. Find the variance of X. → 8/3
  2. X follows a binomial distribution with n = 16 and p = 1/2. Find the variance of X. → 4

Questions parents ask

What maths does a Class 12 child learn?

In Class 12 (ages 17–18) BeadBop covers 15 skills: Relations and functions, Inverse trigonometric functions, Matrices, Determinants, Continuity, Differentiation rules, Applications of derivatives, Indefinite integrals, Definite integrals, Area under a curve, Differential equations, Vectors, Three-dimensional geometry, Linear programming, Probability.

How can my child practise Class 12 maths at home?

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