RBSE Praveshika Mathematics cheat sheet

Euclid's division lemma: for integers a and b with b > 0 there exist unique q, r with a = bq + r and 0 ≤ r < b.

Euclid's algorithm finds HCF by repeated division: HCF(a, b) = HCF(b, r) where r is the remainder of a ÷ b.

Fundamental Theorem of Arithmetic: every integer greater than 1 factors uniquely into primes up to order.

HCF × LCM = product of the two numbers, for any two positive integers a and b.

Rational numbers have terminating decimals only when the denominator in lowest terms is of the form 2^m · 5^n.

A number is irrational if its decimal is non-terminating and non-repeating; √p is irrational for every prime p.

To compare surds √a and √b, compare a and b directly since √ is increasing on non-negative reals.

Every real number has exactly one position on the number line; between any two reals lies another real.

Remainder Theorem: the remainder when p(x) is divided by (x − a) is p(a).

Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0.

Quadratic formula: roots of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac)) / 2a.

For ax² + bx + c: sum of roots = −b/a and product of roots = c/a.

Discriminant D = b² − 4ac; D > 0 gives two distinct real roots, D = 0 equal roots, D < 0 no real roots.

Algebraic identities: (a + b)² = a² + 2ab + b²; a² − b² = (a + b)(a − b); a³ ± b³ = (a ± b)(a² ∓ ab + b²).

Splitting the middle term: for ax² + bx + c find two numbers with product ac and sum b.

A polynomial of degree n has at most n real roots and exactly n roots counted with multiplicity over the complex numbers.

Standard form: ax + by + c = 0 with a, b not both zero; its graph is a straight line.

A pair a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 is consistent with a unique solution when a₁/a₂ ≠ b₁/b₂.

Infinitely many solutions when a₁/a₂ = b₁/b₂ = c₁/c₂ (coincident lines).

No solution when a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel distinct lines).

Substitution method: solve one equation for one variable and substitute into the other.

Elimination method: multiply equations to match a coefficient, then add or subtract to cancel that variable.

Cross-multiplication: x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁).

A single linear equation in two variables has infinitely many solutions, forming a line.

Distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²).

Section formula for point dividing AB in ratio m:n internally: ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

Midpoint of A(x₁, y₁) and B(x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Area of triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃) = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.

Collinear points give zero area by the triangle-area formula.

Slope of line through two points: m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes, perpendicular lines have m₁m₂ = −1.

Equation of a line with slope m through (x₁, y₁): y − y₁ = m(x − x₁).

A point on the x-axis has y = 0; a point on the y-axis has x = 0.

Angle sum of a triangle is 180°; each exterior angle equals the sum of the two remote interior angles.

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

Pythagoras' theorem: in a right triangle, the square on the hypotenuse equals the sum of squares on the other two sides.

Two triangles are similar if their corresponding angles are equal (AA), sides are proportional (SSS), or two sides are proportional with included angle equal (SAS).

In similar triangles, ratio of areas equals the square of the ratio of corresponding sides.

Tangent–radius theorem: a tangent to a circle is perpendicular to the radius at the point of contact.

Tangents drawn from an external point to a circle are equal in length.

Angle in a semicircle is a right angle; the angle between a tangent and a chord equals the angle in the alternate segment.

Area of a triangle = ½ × base × height; Heron's formula: √(s(s − a)(s − b)(s − c)) with s = (a + b + c)/2.

Area of trapezium = ½(a + b)h, where a and b are parallel sides and h the perpendicular height.

Area of circle = πr²; circumference = 2πr; area of sector = (θ/360)πr²; arc length = (θ/360)2πr.

Volume of cuboid = lbh; surface area = 2(lb + bh + hl); volume of cube = a³.

Volume of cylinder = πr²h; curved surface = 2πrh; total surface = 2πr(r + h).

Volume of cone = (1/3)πr²h; slant height l = √(r² + h²); curved surface = πrl.

Volume of sphere = (4/3)πr³; surface area = 4πr²; hemisphere volume = (2/3)πr³.

When dimensions scale by factor k, lengths scale by k, areas by k², and volumes by k³.

sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent = sin θ/cos θ.

Reciprocals: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.

Identity: sin²θ + cos²θ = 1; dividing by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives 1 + cot²θ = cosec²θ.

Standard values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2; sin 60° = √3/2, tan 60° = √3.

sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ.

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

In height-and-distance problems, draw the right triangle and apply tan = height/distance for the given angle.

Mean of ungrouped data = (sum of observations)/(number of observations).

Median of n ordered values: the middle value if n is odd, the average of the two middle values if n is even.

Mode is the value with the highest frequency; a data set may have more than one mode.

For grouped data, mean = Σfᵢxᵢ / Σfᵢ where xᵢ are class marks.

Median of grouped data = l + [(n/2 − cf)/f] × h, with l the lower limit of the median class, cf the cumulative frequency before it, f its frequency, h the class width.

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

Empirical relation for moderately skewed data: Mode = 3 Median − 2 Mean.

Probability of event E = (favourable outcomes)/(total equally likely outcomes), with 0 ≤ P(E) ≤ 1 and P(not E) = 1 − P(E).

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

Fractional index: a^(m/n) = the nth root of aᵐ; also a^(m/n) = (a^(1/n))ᵐ.

Rationalising factor of √a is √a; of a + √b is a − √b; of √a + √b is √a − √b.

√a · √b = √(ab) and √a/√b = √(a/b) for non-negative a and b.

Simple interest = PRT/100; amount = P + SI.

Compound amount = P(1 + R/100)ⁿ; compound interest = amount − principal.

Profit% or loss% is always calculated on the cost price: % = (profit or loss)/CP × 100.

Discount is calculated on the marked price; selling price = marked price − discount.

Euclid's division lemma: for integers a and b with b > 0 there exist unique q, r with a = bq + r and 0 ≤ r < b.

Euclid's algorithm finds HCF by repeated division: HCF(a, b) = HCF(b, r) where r is the remainder of a ÷ b.

Fundamental Theorem of Arithmetic: every integer greater than 1 factors uniquely into primes up to order.

HCF × LCM = product of the two numbers, for any two positive integers a and b.

Rational numbers have terminating decimals only when the denominator in lowest terms is of the form 2^m · 5^n.

A number is irrational if its decimal is non-terminating and non-repeating; √p is irrational for every prime p.

To compare surds √a and √b, compare a and b directly since √ is increasing on non-negative reals.

Every real number has exactly one position on the number line; between any two reals lies another real.

Remainder Theorem: the remainder when p(x) is divided by (x − a) is p(a).

Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0.

Quadratic formula: roots of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac)) / 2a.

For ax² + bx + c: sum of roots = −b/a and product of roots = c/a.

Discriminant D = b² − 4ac; D > 0 gives two distinct real roots, D = 0 equal roots, D < 0 no real roots.

Algebraic identities: (a + b)² = a² + 2ab + b²; a² − b² = (a + b)(a − b); a³ ± b³ = (a ± b)(a² ∓ ab + b²).

Splitting the middle term: for ax² + bx + c find two numbers with product ac and sum b.

A polynomial of degree n has at most n real roots and exactly n roots counted with multiplicity over the complex numbers.

Standard form: ax + by + c = 0 with a, b not both zero; its graph is a straight line.

A pair a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 is consistent with a unique solution when a₁/a₂ ≠ b₁/b₂.

Infinitely many solutions when a₁/a₂ = b₁/b₂ = c₁/c₂ (coincident lines).

No solution when a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel distinct lines).

Substitution method: solve one equation for one variable and substitute into the other.

Elimination method: multiply equations to match a coefficient, then add or subtract to cancel that variable.

Cross-multiplication: x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁).

A single linear equation in two variables has infinitely many solutions, forming a line.

Distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²).

Section formula for point dividing AB in ratio m:n internally: ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

Midpoint of A(x₁, y₁) and B(x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Area of triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃) = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.

Collinear points give zero area by the triangle-area formula.

Slope of line through two points: m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes, perpendicular lines have m₁m₂ = −1.

Equation of a line with slope m through (x₁, y₁): y − y₁ = m(x − x₁).

A point on the x-axis has y = 0; a point on the y-axis has x = 0.

Angle sum of a triangle is 180°; each exterior angle equals the sum of the two remote interior angles.

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

Pythagoras' theorem: in a right triangle, the square on the hypotenuse equals the sum of squares on the other two sides.

Two triangles are similar if their corresponding angles are equal (AA), sides are proportional (SSS), or two sides are proportional with included angle equal (SAS).

In similar triangles, ratio of areas equals the square of the ratio of corresponding sides.

Tangent–radius theorem: a tangent to a circle is perpendicular to the radius at the point of contact.

Tangents drawn from an external point to a circle are equal in length.

Angle in a semicircle is a right angle; the angle between a tangent and a chord equals the angle in the alternate segment.

Area of a triangle = ½ × base × height; Heron's formula: √(s(s − a)(s − b)(s − c)) with s = (a + b + c)/2.

Area of trapezium = ½(a + b)h, where a and b are parallel sides and h the perpendicular height.

Area of circle = πr²; circumference = 2πr; area of sector = (θ/360)πr²; arc length = (θ/360)2πr.

Volume of cuboid = lbh; surface area = 2(lb + bh + hl); volume of cube = a³.

Volume of cylinder = πr²h; curved surface = 2πrh; total surface = 2πr(r + h).

Volume of cone = (1/3)πr²h; slant height l = √(r² + h²); curved surface = πrl.

Volume of sphere = (4/3)πr³; surface area = 4πr²; hemisphere volume = (2/3)πr³.

When dimensions scale by factor k, lengths scale by k, areas by k², and volumes by k³.

sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent = sin θ/cos θ.

Reciprocals: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.

Identity: sin²θ + cos²θ = 1; dividing by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives 1 + cot²θ = cosec²θ.

Standard values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2; sin 60° = √3/2, tan 60° = √3.

sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ.

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

In height-and-distance problems, draw the right triangle and apply tan = height/distance for the given angle.

Mean of ungrouped data = (sum of observations)/(number of observations).

Median of n ordered values: the middle value if n is odd, the average of the two middle values if n is even.

Mode is the value with the highest frequency; a data set may have more than one mode.

For grouped data, mean = Σfᵢxᵢ / Σfᵢ where xᵢ are class marks.

Median of grouped data = l + [(n/2 − cf)/f] × h, with l the lower limit of the median class, cf the cumulative frequency before it, f its frequency, h the class width.

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

Empirical relation for moderately skewed data: Mode = 3 Median − 2 Mean.

Probability of event E = (favourable outcomes)/(total equally likely outcomes), with 0 ≤ P(E) ≤ 1 and P(not E) = 1 − P(E).

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

Fractional index: a^(m/n) = the nth root of aᵐ; also a^(m/n) = (a^(1/n))ᵐ.

Rationalising factor of √a is √a; of a + √b is a − √b; of √a + √b is √a − √b.

√a · √b = √(ab) and √a/√b = √(a/b) for non-negative a and b.

Simple interest = PRT/100; amount = P + SI.

Compound amount = P(1 + R/100)ⁿ; compound interest = amount − principal.

Profit% or loss% is always calculated on the cost price: % = (profit or loss)/CP × 100.

Discount is calculated on the marked price; selling price = marked price − discount.