Revision reference
Every formula in the CBSE Class 10 syllabus, unit by unit — and next to each one, the short reason it is true. Formulas you understand are the ones you still remember in the exam hall.
Every number breaks into primes in exactly one way, and that single fact powers HCF and LCM.
HCF(a, b) × LCM(a, b) = a × b
Works for two numbers only, never for three.
a = b·q + r, 0 ≤ r < b
Euclid's division lemma — the basis of the HCF algorithm.
The coefficients of a polynomial already contain the sum and product of its roots — you rarely need to solve it.
α + β = −b/a
Sum of zeroes of ax² + bx + c.
αβ = c/a
Product of zeroes of ax² + bx + c.
x² − (α + β)x + αβ = 0
Rebuild a quadratic from its zeroes.
α + β + γ = −b/a, αβγ = −d/a
For a cubic ax³ + bx² + cx + d.
Comparing the ratios of coefficients tells you whether the two lines meet, run parallel, or are the same line.
a₁/a₂ ≠ b₁/b₂ → unique solution
Lines intersect once (consistent).
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → no solution
Parallel lines (inconsistent).
a₁/a₂ = b₁/b₂ = c₁/c₂ → infinite solutions
Same line drawn twice.
The discriminant is the part under the root sign — its sign decides how many real answers exist before you calculate anything.
x = (−b ± √(b² − 4ac)) / 2a
Quadratic formula for ax² + bx + c = 0.
D = b² − 4ac
D > 0: two distinct roots. D = 0: equal roots. D < 0: no real roots.
An AP climbs by the same step each time, so the n-th term is just the start plus that step taken (n − 1) times.
aₙ = a + (n − 1)d
n-th term; a is the first term, d the common difference.
Sₙ = n/2 · [2a + (n − 1)d]
Sum of the first n terms.
Sₙ = n/2 · (a + l)
Use when the last term l is known.
Distance is Pythagoras in disguise: the horizontal and vertical gaps are the two legs of a right triangle.
d = √((x₂ − x₁)² + (y₂ − y₁)²)
Distance between two points.
P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n))
Section formula for ratio m : n.
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Midpoint — the section formula with m = n.
Every ratio is a comparison of two sides of a right triangle, so the identities are just Pythagoras rewritten.
sin θ = opposite / hypotenuse
cosec θ is its reciprocal.
cos θ = adjacent / hypotenuse
sec θ is its reciprocal.
tan θ = sin θ / cos θ
cot θ is its reciprocal.
sin²θ + cos²θ = 1
Pythagoras applied to a triangle with hypotenuse 1.
1 + tan²θ = sec²θ
Divide the identity above by cos²θ.
1 + cot²θ = cosec²θ
Divide the same identity by sin²θ.
sin(90° − θ) = cos θ
Swapping the angle swaps opposite and adjacent.
These six columns cover almost every board question — learn the pattern 0, ½, 1/√2, √3/2, 1 for sine and read cosine backwards.
sin: 0, ½, 1/√2, √3/2, 1
At 0°, 30°, 45°, 60°, 90°.
cos: 1, √3/2, 1/√2, ½, 0
The same list reversed.
tan: 0, 1/√3, 1, √3, ∞
sin ÷ cos at each angle.
A tangent touches at one point only, and at that point it makes a perfect right angle with the radius.
Tangent ⟂ radius at the point of contact
Creates the right triangle most circle sums rely on.
Two tangents from an external point are equal
PA = PB.
Curved surface area is the side alone; total surface area adds the flat faces you can actually see.
Cylinder: CSA = 2πrh, TSA = 2πr(r + h), V = πr²h
Two circular ends plus a rolled rectangle.
Cone: CSA = πrl, TSA = πr(l + r), V = ⅓πr²h
l = √(r² + h²) is the slant height.
Sphere: SA = 4πr², V = 4/3·πr³
Hemisphere TSA = 3πr², V = ⅔πr³.
Cuboid: TSA = 2(lb + bh + hl), V = lbh
Cube: TSA = 6a², V = a³.
The mean shifts with every value, the median ignores extremes, and the mode just counts popularity.
Mean = Σfᵢxᵢ / Σfᵢ
Direct method for grouped data.
Mean = a + (Σfᵢdᵢ / Σfᵢ)
Assumed mean method; dᵢ = xᵢ − a.
Median = l + [(n/2 − cf)/f] × h
l = lower limit, cf = cumulative frequency before the class, h = class size.
Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h
f₁ is the modal class frequency.
3 Median = Mode + 2 Mean
Empirical relationship.
Probability is simply the share of outcomes you want out of all equally likely outcomes.
P(E) = favourable outcomes / total outcomes
Always between 0 and 1.
P(E) + P(not E) = 1
Useful when the opposite event is easier to count.
Each Class 10 chapter has an animated lesson, a worked example and a quiz that explains every answer.