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Class 10 Maths — 15 skills

Everything a Class 10 child (ages 15–16) 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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Numbers

HCF and LCM by prime factors

Write each number as a product of primes. HCF = common primes with the smaller powers; LCM = all primes with the bigger powers. Also HCF × LCM = product of the two numbers.

Builds on: Prime factorisation

Try it:

  1. Find the HCF of 60 and 45. → 15
  2. Find the LCM of 98 and 28. → 196

Algebra

Zeros of a quadratic polynomial

For ax² + bx + c: sum of zeros = −b/a and product of zeros = c/a. A quadratic with sum S and product P is x² − Sx + P.

Builds on: Polynomials and remainder theorem

Try it:

  1. Find the sum of the zeros of x² + x − 6. → −1
  2. Find the product of the zeros of 2x² − 4x − 48. → −24

Pair of linear equations

Make the x (or y) numbers equal and subtract to eliminate one letter. For a₁x + b₁y = c₁ and a₂x + b₂y = c₂: a₁/a₂ ≠ b₁/b₂ → one solution; a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → no solution; all three equal → infinitely many.

Builds on: Linear equations in two variables, Two-step equations

Try it:

  1. Solve: 5x − 2y = 9 and 3x + 3y = −3 → x = 1, y = −2
  2. How many solutions does this pair have? 5x + y = −9 and 15x + 3y = −27 → infinitely many solutions

Quadratic equations

Factorise and set each bracket to 0, or use x = (−b ± √D) / 2a with D = b² − 4ac. D > 0: two different real roots; D = 0: two equal roots; D < 0: no real roots.

Builds on: Zeros of a quadratic polynomial, Factorising polynomials

Try it:

  1. Solve: x² + 4x − 12 = 0 → x = −6 or 2
  2. Find the discriminant of 4x² − 8x + 4 = 0 and the nature of its roots. → D = 0, two equal real roots

Arithmetic progressions

In an AP with first term a and common difference d: nth term aₙ = a + (n − 1)d, and the sum of n terms Sₙ = n/2 × [2a + (n − 1)d].

Builds on: Two-step equations

Try it:

  1. Find the 28th term of the AP: 5, 14, 23, 32, … → 248
  2. Find the 26th term of the AP: −10, −5, 0, 5, … → 115

Shapes & patterns

Similar triangles

If DE ∥ BC in triangle ABC, then AD/DB = AE/EC (basic proportionality theorem). For similar triangles, sides are in one ratio and areas are in the square of that ratio.

Builds on: Triangles and congruence, Proportion

Try it:

  1. In △ABC, DE ∥ BC with D on AB and E on AC. AD = 6 cm, DB = 4 cm and AE = 6 cm. Find EC (in cm). → 4
  2. △ABC ~ △PQR. AB = 5 cm and PQ = 7 cm. If the area of △ABC is 100 cm², find the area of △PQR (in cm²). → 196

Distance, section and mid-point

Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]. Mid-point = ((x₁ + x₂)/2, (y₁ + y₂)/2). The point dividing AB in m : n is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

Builds on: Coordinate geometry, Pythagoras theorem

Try it:

  1. Find the mid-point of A(−1, 3) and B(3, 11). → (1, 7)
  2. Find the mid-point of A(−6, −1) and B(2, 9). → (−2, 4)

Tangents to a circle

A tangent is perpendicular to the radius at the point of contact, so OP² = radius² + tangent². The two tangents from an outside point are equal in length.

Builds on: Angles in circles, Pythagoras theorem

Try it:

  1. The tangent from a point P to a circle of radius 7 cm is 24 cm long. How far is P from the centre (in cm)? → 25
  2. The tangent from a point P to a circle of radius 24 cm is 7 cm long. How far is P from the centre (in cm)? → 25

Trigonometry

Trigonometric ratios

sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Learn the table: sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, tan 45° = 1.

Builds on: Pythagoras theorem

Try it:

  1. What is the exact value of sin 90° ? → 1
  2. Evaluate: tan²60° + 4 cos²30° → 6

Trigonometric identities

sin²A + cos²A = 1, sec²A − tan²A = 1, cosec²A − cot²A = 1. Complementary angles: sin(90° − A) = cos A and tan(90° − A) = cot A.

Builds on: Trigonometric ratios

Try it:

  1. Evaluate: cot 19° ÷ tan 71° → 1
  2. If sin 3A = cos (A − 14°), where 3A is acute, find A (in degrees). → 26

Heights and distances

Draw the right triangle: tan(angle of elevation) = height ÷ distance. tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

Builds on: Trigonometric ratios

Try it:

  1. A tower is 12 m tall. From a point on the ground, the angle of elevation of its top is 30°. How far is the point from the foot of the tower? → 12√3 m
  2. From a point 4 m from its foot, the angle of elevation of the top of a building is 60°. Find the height of the building. → 4√3 m

Measurement

Arc length and sector area

For a sector of angle θ: arc length = θ/360 × 2πr and area = θ/360 × πr². With π = 22/7, choose r as a multiple of 7.

Builds on: Circumference and area of a circle

Try it:

  1. A sector of a circle of radius 42 cm has angle 180°. Find its arc length (in cm). (π = 22/7) → 132
  2. A sector of a circle of radius 35 cm has angle 45°. Find its area (in cm²). (π = 22/7) → 481.25

Combined solids and conversion

When a solid is melted and recast, the volume stays the same. Sphere = 4/3 πr³, cylinder = πr²h, cone = 1/3 πr²h, hemisphere = 2/3 πr³.

Builds on: Cone, sphere and hemisphere

Try it:

  1. A metal sphere of radius 12 cm is melted and recast into a cylinder of radius 6 cm. Find the height of the cylinder (in cm). → 64
  2. A toy is a cone of radius 21 cm and height 9 cm on a hemisphere of the same radius. Find its volume (in cm³). (π = 22/7) → 23562

Data handling

Mean, mode and median of grouped data

Mean = Σfx ÷ Σf using class marks x. Mode = l + (f₁ − f₀)/(2f₁ − f₀ − f₂) × h. Median = l + (n/2 − cf)/f × h.

Builds on: Mean, median and mode

Try it:

  1. Find the mode of this grouped data (class: frequency): 20–40: 4, 40–60: 9, 60–80: 6, 80–100: 13, 100–120: 4 → 88.75
  2. Find the mode of this grouped data (class: frequency): 0–10: 12, 10–20: 13, 20–30: 9, 30–40: 12, 40–50: 4 → 12

Probability with cards, dice and coins

P(event) = favourable outcomes ÷ total outcomes. A deck has 52 cards (4 suits of 13; 12 face cards), two dice give 36 outcomes, three coins give 8.

Builds on: Experimental probability

Try it:

  1. One card is drawn from a well-shuffled deck of 52 cards. What is the probability that it is a red card? → 1/2
  2. One card is drawn from a well-shuffled deck of 52 cards. What is the probability that it is a jack or a queen? → 2/13

Questions parents ask

What maths does a Class 10 child learn?

In Class 10 (ages 15–16) BeadBop covers 15 skills: HCF and LCM by prime factors, Zeros of a quadratic polynomial, Pair of linear equations, Quadratic equations, Arithmetic progressions, Similar triangles, Distance, section and mid-point, Trigonometric ratios, Trigonometric identities, Heights and distances, Tangents to a circle, Arc length and sector area, Combined solids and conversion, Mean, mode and median of grouped data, Probability with cards, dice and coins.

How can my child practise Class 10 maths at home?

Start with the free starting-point check in BeadBop. It recommends the right skill, explains it step by step and gives practice that adapts to your child.

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