WBBSE Madhyamik Mathematics cheat sheet

Simple interest: I = PRT/100, where P is principal, R the annual rate in percent, T the time in years; amount A = P + I.

Compound interest compounded annually: A = P(1 + R/100)^n; CI = A − P. For half-yearly compounding use rate R/2 with 2n periods.

Percentage change: increase of x% then decrease of x% gives net change of −x²/100 percent, always a net loss for x > 0.

Profit and loss are computed on cost price unless stated otherwise: profit% = (SP − CP)/CP × 100; discount is computed on marked price.

Successive discounts of a% and b% equal a single discount of a + b − ab/100 percent.

Ratio and proportion: if a:b = c:d then ad = bc; for continued proportion a:b = b:c, b² = ac and b is the mean proportional.

Partnership: profit shares are proportional to capital × time invested, not capital alone.

Time and work: if A alone takes x days and B alone y days, together they take xy/(x+y) days; rates add, times do not.

Speed conversion: 1 km/h = 5/18 m/s; for a train of length L crossing a platform of length P, time = (L + P)/speed.

Quadratic formula: for ax² + bx + c = 0 (a ≠ 0), x = [−b ± √(b² − 4ac)]/(2a); the discriminant D = b² − 4ac decides two real, one real, or no real roots.

Root relations: sum of roots = −b/a and product = c/a; a quadratic with given roots α, β is x² − (α+β)x + αβ = 0.

Identities: (a+b)² = a² + 2ab + b²; a² − b² = (a+b)(a−b); a³ ± b³ = (a ± b)(a² ∓ ab + b²); a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca).

If a + b + c = 0 then a³ + b³ + c³ = 3abc.

Linear equations in two variables: solve by elimination or substitution; the pair a₁x+b₁y+c₁=0, a₂x+b₂y+c₂=0 has a unique solution when a₁/a₂ ≠ b₁/b₂.

Ratio problems: if x varies directly as y then x = ky; if inversely, xy = k.

Surds: rationalise by multiplying by the conjugate, e.g. 1/(√a + √b) = (√a − √b)/(a − b).

Laws of indices: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 for a ≠ 0.

Angle sum of a triangle is 180°; of a convex polygon with n sides it is (n − 2) × 180°, and the exterior angles sum to 360°.

An exterior angle of a triangle equals the sum of the two interior opposite angles.

Congruence criteria: SSS, SAS, ASA (with AAS), and RHS for right triangles; AAA gives similarity, not congruence.

Basic proportionality (Thales) theorem: a line parallel to one side of a triangle divides the other two sides proportionally.

In a right triangle the midpoint of the hypotenuse is equidistant from all three vertices.

Angle in a semicircle is a right angle; the angle at the centre is twice the angle at the circumference on the same arc.

Cyclic quadrilateral: opposite angles are supplementary, and the exterior angle equals the interior opposite angle.

Tangents from an external point are equal in length, and the tangent is perpendicular to the radius at the point of contact.

Circle: circumference 2πr, area πr²; sector of angle θ has arc length (θ/360)×2πr and area (θ/360)×πr².

Triangle area = ½ × base × height; Heron's formula with semi-perimeter s: area = √[s(s−a)(s−b)(s−c)].

Cylinder: curved surface 2πrh, total surface 2πr(r+h), volume πr²h.

Cone: slant height l = √(r²+h²), curved surface πrl, total surface πr(l+r), volume ⅓πr²h.

Sphere: surface area 4πr², volume (4/3)πr³; hemisphere total surface 3πr².

Trigonometric ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = sin θ/cos θ.

Standard values: sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2; cos is these in reverse order; tan 45° = 1.

Identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.

Angle of elevation is measured upward from the horizontal to the line of sight; angle of depression is measured downward from the horizontal.

Mean of grouped data: x̄ = Σfᵢxᵢ / Σfᵢ, using class midpoints xᵢ.

Median of grouped data: median = l + [(N/2 − cf)/f] × h, where l is lower boundary, cf cumulative frequency below the median class, f its frequency, h class width.

Mode of grouped data: mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.

Empirical relation for moderately skewed data: mode ≈ 3 median − 2 mean.

Ogive (cumulative frequency curve) reads off the median at cumulative frequency N/2.

Distance between points (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)² + (y₂−y₁)²]; distance from origin is √(x²+y²).

Section formula: point dividing the join of (x₁,y₁) and (x₂,y₂) in ratio m:n internally is ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)); midpoint is the m = n case.

Slope of the line through two points is (y₂−y₁)/(x₂−x₁); parallel lines have equal slopes and perpendicular lines have slopes whose product is −1.

Area of a triangle with given vertices = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|; zero area means the points are collinear.

Simple interest: I = PRT/100, where P is principal, R the annual rate in percent, T the time in years; amount A = P + I.

Compound interest compounded annually: A = P(1 + R/100)^n; CI = A − P. For half-yearly compounding use rate R/2 with 2n periods.

Percentage change: increase of x% then decrease of x% gives net change of −x²/100 percent, always a net loss for x > 0.

Profit and loss are computed on cost price unless stated otherwise: profit% = (SP − CP)/CP × 100; discount is computed on marked price.

Successive discounts of a% and b% equal a single discount of a + b − ab/100 percent.

Ratio and proportion: if a:b = c:d then ad = bc; for continued proportion a:b = b:c, b² = ac and b is the mean proportional.

Partnership: profit shares are proportional to capital × time invested, not capital alone.

Time and work: if A alone takes x days and B alone y days, together they take xy/(x+y) days; rates add, times do not.

Speed conversion: 1 km/h = 5/18 m/s; for a train of length L crossing a platform of length P, time = (L + P)/speed.

Quadratic formula: for ax² + bx + c = 0 (a ≠ 0), x = [−b ± √(b² − 4ac)]/(2a); the discriminant D = b² − 4ac decides two real, one real, or no real roots.

Root relations: sum of roots = −b/a and product = c/a; a quadratic with given roots α, β is x² − (α+β)x + αβ = 0.

Identities: (a+b)² = a² + 2ab + b²; a² − b² = (a+b)(a−b); a³ ± b³ = (a ± b)(a² ∓ ab + b²); a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca).

If a + b + c = 0 then a³ + b³ + c³ = 3abc.

Linear equations in two variables: solve by elimination or substitution; the pair a₁x+b₁y+c₁=0, a₂x+b₂y+c₂=0 has a unique solution when a₁/a₂ ≠ b₁/b₂.

Ratio problems: if x varies directly as y then x = ky; if inversely, xy = k.

Surds: rationalise by multiplying by the conjugate, e.g. 1/(√a + √b) = (√a − √b)/(a − b).

Laws of indices: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 for a ≠ 0.

Angle sum of a triangle is 180°; of a convex polygon with n sides it is (n − 2) × 180°, and the exterior angles sum to 360°.

An exterior angle of a triangle equals the sum of the two interior opposite angles.

Congruence criteria: SSS, SAS, ASA (with AAS), and RHS for right triangles; AAA gives similarity, not congruence.

Basic proportionality (Thales) theorem: a line parallel to one side of a triangle divides the other two sides proportionally.

In a right triangle the midpoint of the hypotenuse is equidistant from all three vertices.

Angle in a semicircle is a right angle; the angle at the centre is twice the angle at the circumference on the same arc.

Cyclic quadrilateral: opposite angles are supplementary, and the exterior angle equals the interior opposite angle.

Tangents from an external point are equal in length, and the tangent is perpendicular to the radius at the point of contact.

Circle: circumference 2πr, area πr²; sector of angle θ has arc length (θ/360)×2πr and area (θ/360)×πr².

Triangle area = ½ × base × height; Heron's formula with semi-perimeter s: area = √[s(s−a)(s−b)(s−c)].

Cylinder: curved surface 2πrh, total surface 2πr(r+h), volume πr²h.

Cone: slant height l = √(r²+h²), curved surface πrl, total surface πr(l+r), volume ⅓πr²h.

Sphere: surface area 4πr², volume (4/3)πr³; hemisphere total surface 3πr².

Trigonometric ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = sin θ/cos θ.

Standard values: sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2; cos is these in reverse order; tan 45° = 1.

Identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.

Angle of elevation is measured upward from the horizontal to the line of sight; angle of depression is measured downward from the horizontal.

Mean of grouped data: x̄ = Σfᵢxᵢ / Σfᵢ, using class midpoints xᵢ.

Median of grouped data: median = l + [(N/2 − cf)/f] × h, where l is lower boundary, cf cumulative frequency below the median class, f its frequency, h class width.

Mode of grouped data: mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.

Empirical relation for moderately skewed data: mode ≈ 3 median − 2 mean.

Ogive (cumulative frequency curve) reads off the median at cumulative frequency N/2.

Distance between points (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)² + (y₂−y₁)²]; distance from origin is √(x²+y²).

Section formula: point dividing the join of (x₁,y₁) and (x₂,y₂) in ratio m:n internally is ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)); midpoint is the m = n case.

Slope of the line through two points is (y₂−y₁)/(x₂−x₁); parallel lines have equal slopes and perpendicular lines have slopes whose product is −1.

Area of a triangle with given vertices = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|; zero area means the points are collinear.

Simple interest: I = PRT/100, where P is principal, R the annual rate in percent, T the time in years; amount A = P + I.

Compound interest compounded annually: A = P(1 + R/100)^n; CI = A − P. For half-yearly compounding use rate R/2 with 2n periods.

Percentage change: increase of x% then decrease of x% gives net change of −x²/100 percent, always a net loss for x > 0.

Profit and loss are computed on cost price unless stated otherwise: profit% = (SP − CP)/CP × 100; discount is computed on marked price.

Successive discounts of a% and b% equal a single discount of a + b − ab/100 percent.

Ratio and proportion: if a:b = c:d then ad = bc; for continued proportion a:b = b:c, b² = ac and b is the mean proportional.

Partnership: profit shares are proportional to capital × time invested, not capital alone.

Time and work: if A alone takes x days and B alone y days, together they take xy/(x+y) days; rates add, times do not.

Speed conversion: 1 km/h = 5/18 m/s; for a train of length L crossing a platform of length P, time = (L + P)/speed.

Quadratic formula: for ax² + bx + c = 0 (a ≠ 0), x = [−b ± √(b² − 4ac)]/(2a); the discriminant D = b² − 4ac decides two real, one real, or no real roots.

Root relations: sum of roots = −b/a and product = c/a; a quadratic with given roots α, β is x² − (α+β)x + αβ = 0.

Identities: (a+b)² = a² + 2ab + b²; a² − b² = (a+b)(a−b); a³ ± b³ = (a ± b)(a² ∓ ab + b²); a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc-ca).

If a + b + c = 0 then a³ + b³ + c³ = 3abc.

Linear equations in two variables: solve by elimination or substitution; the pair a₁x+b₁y+c₁=0, a₂x+b₂y+c₂=0 has a unique solution when a₁/a₂ ≠ b₁/b₂.

Ratio problems: if x varies directly as y then x = ky; if inversely, xy = k.

Surds: rationalise by multiplying by the conjugate, e.g. 1/(√a + √b) = (√a − √b)/(a − b).

Laws of indices: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 for a ≠ 0.

Angle sum of a triangle is 180°; of a convex polygon with n sides it is (n − 2) × 180°, and the exterior angles sum to 360°.

An exterior angle of a triangle equals the sum of the two interior opposite angles.

Congruence criteria: SSS, SAS, ASA (with AAS), and RHS for right triangles; AAA gives similarity, not congruence.

Basic proportionality (Thales) theorem: a line parallel to one side of a triangle divides the other two sides proportionally.

In a right triangle the midpoint of the hypotenuse is equidistant from all three vertices.

Angle in a semicircle is a right angle; the angle at the centre is twice the angle at the circumference on the same arc.

Cyclic quadrilateral: opposite angles are supplementary, and the exterior angle equals the interior opposite angle.

Tangents from an external point are equal in length, and the tangent is perpendicular to the radius at the point of contact.

Circle: circumference 2πr, area πr²; sector of angle θ has arc length (θ/360)×2πr and area (θ/360)×πr².

Triangle area = ½ × base × height; Heron's formula with semi-perimeter s: area = √[s(s−a)(s−b)(s−c)].

Cylinder: curved surface 2πrh, total surface 2πr(r+h), volume πr²h.

Cone: slant height l = √(r²+h²), curved surface πrl, total surface πr(l+r), volume ⅓πr²h.

Sphere: surface area 4πr², volume (4/3)πr³; hemisphere total surface 3πr².

Trigonometric ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = sin θ/cos θ.

Standard values: sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2; cos is these in reverse order; tan 45° = 1.

Identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.

Angle of elevation is measured upward from the horizontal to the line of sight; angle of depression is measured downward from the horizontal.

Mean of grouped data: x̄ = Σfᵢxᵢ / Σfᵢ, using class midpoints xᵢ.

Median of grouped data: median = l + [(N/2 − cf)/f] × h, where l is lower boundary, cf cumulative frequency below the median class, f its frequency, h class width.

Mode of grouped data: mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.

Empirical relation for moderately skewed data: mode ≈ 3 median − 2 mean.

Ogive (cumulative frequency curve) reads off the median at cumulative frequency N/2.

Distance between points (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)² + (y₂−y₁)²]; distance from origin is √(x²+y²).

Section formula: point dividing the join of (x₁,y₁) and (x₂,y₂) in ratio m:n internally is ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)); midpoint is the m = n case.

Slope of the line through two points is (y₂−y₁)/(x₂−x₁); parallel lines have equal slopes and perpendicular lines have slopes whose product is −1.

Area of a triangle with given vertices = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|; zero area means the points are collinear.

Simple interest: I = PRT/100, where P is principal, R the annual rate in percent, T the time in years; amount A = P + I.

Compound interest compounded annually: A = P(1 + R/100)^n; CI = A − P. For half-yearly compounding use rate R/2 with 2n periods.

Percentage change: increase of x% then decrease of x% gives net change of −x²/100 percent, always a net loss for x > 0.

Profit and loss are computed on cost price unless stated otherwise: profit% = (SP − CP)/CP × 100; discount is computed on marked price.

Successive discounts of a% and b% equal a single discount of a + b − ab/100 percent.

Ratio and proportion: if a:b = c:d then ad = bc; for continued proportion a:b = b:c, b² = ac and b is the mean proportional.

Partnership: profit shares are proportional to capital × time invested, not capital alone.

Time and work: if A alone takes x days and B alone y days, together they take xy/(x+y) days; rates add, times do not.

Speed conversion: 1 km/h = 5/18 m/s; for a train of length L crossing a platform of length P, time = (L + P)/speed.

Quadratic formula: for ax² + bx + c = 0 (a ≠ 0), x = [−b ± √(b² − 4ac)]/(2a); the discriminant D = b² − 4ac decides two real, one real, or no real roots.

Root relations: sum of roots = −b/a and product = c/a; a quadratic with given roots α, β is x² − (α+β)x + αβ = 0.

Identities: (a+b)² = a² + 2ab + b²; a² − b² = (a+b)(a−b); a³ ± b³ = (a ± b)(a² ∓ ab + b²); a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca).

If a + b + c = 0 then a³ + b³ + c³ = 3abc.

Linear equations in two variables: solve by elimination or substitution; the pair a₁x+b₁y+c₁=0, a₂x+b₂y+c₂=0 has a unique solution when a₁/a₂ ≠ b₁/b₂.

Ratio problems: if x varies directly as y then x = ky; if inversely, xy = k.

Surds: rationalise by multiplying by the conjugate, e.g. 1/(√a + √b) = (√a − √b)/(a − b).

Laws of indices: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 for a ≠ 0.

Angle sum of a triangle is 180°; of a convex polygon with n sides it is (n − 2) × 180°, and the exterior angles sum to 360°.

An exterior angle of a triangle equals the sum of the two interior opposite angles.

Congruence criteria: SSS, SAS, ASA (with AAS), and RHS for right triangles; AAA gives similarity, not congruence.

Basic proportionality (Thales) theorem: a line parallel to one side of a triangle divides the other two sides proportionally.

In a right triangle the midpoint of the hypotenuse is equidistant from all three vertices.

Angle in a semicircle is a right angle; the angle at the centre is twice the angle at the circumference on the same arc.

Cyclic quadrilateral: opposite angles are supplementary, and the exterior angle equals the interior opposite angle.

Tangents from an external point are equal in length, and the tangent is perpendicular to the radius at the point of contact.

Circle: circumference 2πr, area πr²; sector of angle θ has arc length (θ/360)×2πr and area (θ/360)×πr².

Triangle area = ½ × base × height; Heron's formula with semi-perimeter s: area = √[s(s−a)(s−b)(s−c)].

Cylinder: curved surface 2πrh, total surface 2πr(r+h), volume πr²h.

Cone: slant height l = √(r²+h²), curved surface πrl, total surface πr(l+r), volume ⅓πr²h.

Sphere: surface area 4πr², volume (4/3)πr³; hemisphere total surface 3πr².

Trigonometric ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = sin θ/cos θ.

Standard values: sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2; cos is these in reverse order; tan 45° = 1.

Identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.

Angle of elevation is measured upward from the horizontal to the line of sight; angle of depression is measured downward from the horizontal.

Mean of grouped data: x̄ = Σfᵢxᵢ / Σfᵢ, using class midpoints xᵢ.

Median of grouped data: median = l + [(N/2 − cf)/f] × h, where l is lower boundary, cf cumulative frequency below the median class, f its frequency, h class width.

Mode of grouped data: mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.

Empirical relation for moderately skewed data: mode ≈ 3 median − 2 mean.

Ogive (cumulative frequency curve) reads off the median at cumulative frequency N/2.

Distance between points (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)² + (y₂−y₁)²]; distance from origin is √(x²+y²).

Section formula: point dividing the join of (x₁,y₁) and (x₂,y₂) in ratio m:n internally is ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)); midpoint is the m = n case.

Slope of the line through two points is (y₂−y₁)/(x₂−x₁); parallel lines have equal slopes and perpendicular lines have slopes whose product is −1.

Area of a triangle with given vertices = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|; zero area means the points are collinear.

Simple interest: I = PRT/100, where P is principal, R the annual rate in percent, T the time in years; amount A = P + I.

Compound interest compounded annually: A = P(1 + R/100)^n; CI = A − P. For half-yearly compounding use rate R/2 with 2n periods.

Percentage change: increase of x% then decrease of x% gives net change of −x²/100 percent, always a net loss for x > 0.

Profit and loss are computed on cost price unless stated otherwise: profit% = (SP − CP)/CP × 100; discount is computed on marked price.

Successive discounts of a% and b% equal a single discount of a + b − ab/100 percent.

Ratio and proportion: if a:b = c:d then ad = bc; for continued proportion a:b = b:c, b² = ac and b is the mean proportional.

Partnership: profit shares are proportional to capital × time invested, not capital alone.

Time and work: if A alone takes x days and B alone y days, together they take xy/(x+y) days; rates add, times do not.

Speed conversion: 1 km/h = 5/18 m/s; for a train of length L crossing a platform of length P, time = (L + P)/speed.

Quadratic formula: for ax² + bx + c = 0 (a ≠ 0), x = [−b ± √(b² − 4ac)]/(2a); the discriminant D = b² − 4ac decides two real, one real, or no real roots.

Root relations: sum of roots = −b/a and product = c/a; a quadratic with given roots α, β is x² − (α+β)x + αβ = 0.

Identities: (a+b)² = a² + 2ab + b²; a² − b² = (a+b)(a−b); a³ ± b³ = (a ± b)(a² ∓ ab + b²); a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca).

If a + b + c = 0 then a³ + b³ + c³ = 3abc.

Linear equations in two variables: solve by elimination or substitution; the pair a₁x+b₁y+c₁=0, a₂x+b₂y+c₂=0 has a unique solution when a₁/a₂ ≠ b₁/b₂.

Ratio problems: if x varies directly as y then x = ky; if inversely, xy = k.

Surds: rationalise by multiplying by the conjugate, e.g. 1/(√a + √b) = (√a − √b)/(a − b).

Laws of indices: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 for a ≠ 0.

Angle sum of a triangle is 180°; of a convex polygon with n sides it is (n − 2) × 180°, and the exterior angles sum to 360°.

An exterior angle of a triangle equals the sum of the two interior opposite angles.

Congruence criteria: SSS, SAS, ASA (with AAS), and RHS for right triangles; AAA gives similarity, not congruence.

Basic proportionality (Thales) theorem: a line parallel to one side of a triangle divides the other two sides proportionally.

In a right triangle the midpoint of the hypotenuse is equidistant from all three vertices.

Angle in a semicircle is a right angle; the angle at the centre is twice the angle at the circumference on the same arc.

Cyclic quadrilateral: opposite angles are supplementary, and the exterior angle equals the interior opposite angle.

Tangents from an external point are equal in length, and the tangent is perpendicular to the radius at the point of contact.

Circle: circumference 2πr, area πr²; sector of angle θ has arc length (θ/360)×2πr and area (θ/360)×πr².

Triangle area = ½ × base × height; Heron's formula with semi-perimeter s: area = √[s(s−a)(s−b)(s−c)].

Cylinder: curved surface 2πrh, total surface 2πr(r+h), volume πr²h.

Cone: slant height l = √(r²+h²), curved surface πrl, total surface πr(l+r), volume ⅓πr²h.

Sphere: surface area 4πr², volume (4/3)πr³; hemisphere total surface 3πr².

Trigonometric ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = sin θ/cos θ.

Standard values: sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2; cos is these in reverse order; tan 45° = 1.

Identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.

Angle of elevation is measured upward from the horizontal to the line of sight; angle of depression is measured downward from the horizontal.

Mean of grouped data: x̄ = Σfᵢxᵢ / Σfᵢ, using class midpoints xᵢ.

Median of grouped data: median = l + [(N/2 − cf)/f] × h, where l is lower boundary, cf cumulative frequency below the median class, f its frequency, h class width.

Mode of grouped data: mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.

Empirical relation for moderately skewed data: mode ≈ 3 median − 2 mean.

Ogive (cumulative frequency curve) reads off the median at cumulative frequency N/2.

Distance between points (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)² + (y₂−y₁)²]; distance from origin is √(x²+y²).

Section formula: point dividing the join of (x₁,y₁) and (x₂,y₂) in ratio m:n internally is ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)); midpoint is the m = n case.

Slope of the line through two points is (y₂−y₁)/(x₂−x₁); parallel lines have equal slopes and perpendicular lines have slopes whose product is −1.

Area of a triangle with given vertices = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|; zero area means the points are collinear.

Simple interest: I = PRT/100, where P is principal, R the annual rate in percent, T the time in years; amount A = P + I.

Compound interest compounded annually: A = P(1 + R/100)^n; CI = A − P. For half-yearly compounding use rate R/2 with 2n periods.

Percentage change: increase of x% then decrease of x% gives net change of −x²/100 percent, always a net loss for x > 0.

Profit and loss are computed on cost price unless stated otherwise: profit% = (SP − CP)/CP × 100; discount is computed on marked price.

Successive discounts of a% and b% equal a single discount of a + b − ab/100 percent.

Ratio and proportion: if a:b = c:d then ad = bc; for continued proportion a:b = b:c, b² = ac and b is the mean proportional.

Partnership: profit shares are proportional to capital × time invested, not capital alone.

Time and work: if A alone takes x days and B alone y days, together they take xy/(x+y) days; rates add, times do not.

Speed conversion: 1 km/h = 5/18 m/s; for a train of length L crossing a platform of length P, time = (L + P)/speed.

Quadratic formula: for ax² + bx + c = 0 (a ≠ 0), x = [−b ± √(b² − 4ac)]/(2a); the discriminant D = b² − 4ac decides two real, one real, or no real roots.

Root relations: sum of roots = −b/a and product = c/a; a quadratic with given roots α, β is x² − (α+β)x + αβ = 0.

Identities: (a+b)² = a² + 2ab + b²; a² − b² = (a+b)(a−b); a³ ± b³ = (a ± b)(a² ∓ ab + b²); a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca).

If a + b + c = 0 then a³ + b³ + c³ = 3abc.

Linear equations in two variables: solve by elimination or substitution; the pair a₁x+b₁y+c₁=0, a₂x+b₂y+c₂=0 has a unique solution when a₁/a₂ ≠ b₁/b₂.

Ratio problems: if x varies directly as y then x = ky; if inversely, xy = k.

Surds: rationalise by multiplying by the conjugate, e.g. 1/(√a + √b) = (√a − √b)/(a − b).

Laws of indices: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 for a ≠ 0.

Angle sum of a triangle is 180°; of a convex polygon with n sides it is (n − 2) × 180°, and the exterior angles sum to 360°.

An exterior angle of a triangle equals the sum of the two interior opposite angles.

Congruence criteria: SSS, SAS, ASA (with AAS), and RHS for right triangles; AAA gives similarity, not congruence.

Basic proportionality (Thales) theorem: a line parallel to one side of a triangle divides the other two sides proportionally.

In a right triangle the midpoint of the hypotenuse is equidistant from all three vertices.

Angle in a semicircle is a right angle; the angle at the centre is twice the angle at the circumference on the same arc.

Cyclic quadrilateral: opposite angles are supplementary, and the exterior angle equals the interior opposite angle.

Tangents from an external point are equal in length, and the tangent is perpendicular to the radius at the point of contact.

Circle: circumference 2πr, area πr²; sector of angle θ has arc length (θ/360)×2πr and area (θ/360)×πr².

Triangle area = ½ × base × height; Heron's formula with semi-perimeter s: area = √[s(s−a)(s−b)(s−c)].

Cylinder: curved surface 2πrh, total surface 2πr(r+h), volume πr²h.

Cone: slant height l = √(r²+h²), curved surface πrl, total surface πr(l+r), volume ⅓πr²h.

Sphere: surface area 4πr², volume (4/3)πr³; hemisphere total surface 3πr².

Trigonometric ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = sin θ/cos θ.

Standard values: sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2; cos is these in reverse order; tan 45° = 1.

Identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.

Angle of elevation is measured upward from the horizontal to the line of sight; angle of depression is measured downward from the horizontal.

Mean of grouped data: x̄ = Σfᵢxᵢ / Σfᵢ, using class midpoints xᵢ.

Median of grouped data: median = l + [(N/2 − cf)/f] × h, where l is lower boundary, cf cumulative frequency below the median class, f its frequency, h class width.

Mode of grouped data: mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.

Empirical relation for moderately skewed data: mode ≈ 3 median − 2 mean.

Ogive (cumulative frequency curve) reads off the median at cumulative frequency N/2.

Distance between points (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)² + (y₂−y₁)²]; distance from origin is √(x²+y²).

Section formula: point dividing the join of (x₁,y₁) and (x₂,y₂) in ratio m:n internally is ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)); midpoint is the m = n case.

Slope of the line through two points is (y₂−y₁)/(x₂−x₁); parallel lines have equal slopes and perpendicular lines have slopes whose product is −1.

Area of a triangle with given vertices = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|; zero area means the points are collinear.