RBSE Secondary Mathematics cheat sheet

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

Euclid's algorithm for HCF: repeatedly apply a = bq + r, replacing (a, b) by (b, r) until r = 0; the last non-zero divisor is the HCF.

Fundamental Theorem of Arithmetic: every integer greater than 1 is a unique product of primes up to order of factors.

HCF(a, b) × LCM(a, b) = a × b holds for two positive integers; for three numbers no such single product identity exists.

A rational number p/q in lowest terms has a terminating decimal expansion exactly when q = 2^m · 5^n with m, n non-negative integers.

If q has any prime factor other than 2 or 5, the decimal expansion of p/q is non-terminating and repeating.

√n is irrational for every prime n and, more generally, for any positive integer n that is not a perfect square.

Irrationality proofs use contradiction: assume √p = a/b in lowest terms, square, and show p divides both a and b.

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.

For quadratic ax² + bx + c with zeros α and β: α + β = −b/a and αβ = c/a.

For cubic ax³ + bx² + cx + d with zeros α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a.

A quadratic polynomial with given zeros α, β is k(x² − (α + β)x + αβ) for any non-zero constant k.

Geometrically, the zeros of p(x) are the x-coordinates of the points where y = p(x) cuts the x-axis.

A polynomial of degree n has at most n real zeros; the graph can cross the x-axis at most n times.

Division algorithm for polynomials: p(x) = g(x)·q(x) + r(x) with either r(x) = 0 or deg r < deg g.

For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, compare a₁/a₂, b₁/b₂, c₁/c₂ to decide consistency.

a₁/a₂ ≠ b₁/b₂ gives exactly one solution: the lines intersect at a single point.

a₁/a₂ = b₁/b₂ = c₁/c₂ gives infinitely many solutions: the lines are coincident.

a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives no solution: the lines are parallel.

Graphical method: draw both lines and read off the intersection point; parallel lines mean no solution.

Substitution: solve one equation for one variable and substitute into the other to obtain a single-variable equation.

Elimination: multiply equations to equalise a coefficient, then add or subtract to eliminate that variable.

Word problems: form two equations from two conditions, then solve and check the answer against the original wording.

Standard form ax² + bx + c = 0 with a ≠ 0; the discriminant is D = b² − 4ac.

D > 0 gives two distinct real roots; D = 0 gives two equal real roots; D < 0 gives no real roots.

Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a).

Factorisation method: split the middle term into two parts whose product is ac and whose sum is b.

Sum of roots α + β = −b/a and product αβ = c/a.

A quadratic with given roots α, β is x² − (α + β)x + αβ = 0.

Completing the square: write ax² + bx + c as a(x + b/2a)² + (c − b²/4a) to solve or to locate the vertex.

For real-world problems, discard roots that are negative when the unknown represents a length, age, or count.

AP: each term differs from the previous by a fixed common difference d; aₙ = a + (n − 1)d.

Sum of the first n terms: Sₙ = (n/2)[2a + (n − 1)d].

Equivalent sum formula: Sₙ = (n/2)(a + l) when the last term l is known.

The nth term from the end of a finite AP with last term l is l − (n − 1)d.

If a, b, c are in AP then 2b = a + c; the middle term is the arithmetic mean of the other two.

Sₙ is a quadratic in n with no constant term; conversely, a sequence with Sₙ of that form is an AP.

The first n natural numbers sum to n(n + 1)/2; the first n odd numbers sum to n².

To test whether a number is a term of an AP, set aₙ equal to it and check that the resulting n is a positive integer.

Basic Proportionality Theorem: a line parallel to one side of a triangle divides the other two sides in the same ratio.

Converse of BPT: if a line divides two sides of a triangle in the same ratio, it is parallel to the third side.

Two triangles are similar if their corresponding angles are equal (AA) or corresponding sides are proportional (SSS).

SAS similarity requires two pairs of proportional sides with the included angles equal.

Ratio of areas of similar triangles equals the square of the ratio of any pair of corresponding sides.

Ratio of corresponding medians, altitudes, and perimeters of similar triangles equals the similarity ratio.

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

Converse of Pythagoras: if a² + b² = c² in a triangle, the angle opposite side c is a right angle.

Distance between (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²].

Distance of a point (x, y) from the origin is √(x² + y²).

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

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

Centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3).

Collinearity test: three points are collinear if the sum of two side lengths equals the third, or if the triangle's area is zero.

Points on the x-axis have y = 0; points on the y-axis have x = 0.

A point on the perpendicular bisector of a segment is equidistant from the segment's endpoints.

Ratios in a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

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, sin 45° = 1/√2, sin 60° = √3/2; cos values run in reverse order.

tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3; tan 90° is undefined.

Complementary angles: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, sec(90° − θ) = cosec 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, mark the known angle, and use tan when both legs are involved.

A tangent to a circle is perpendicular to the radius at the point of contact.

From an external point, the two tangents to a circle are equal in length.

The lengths of tangents from an external point P satisfy PA = PB, and OP bisects the angle between them.

Number of tangents: none from an interior point, exactly one from a point on the circle, exactly two from an exterior point.

A parallelogram circumscribing a circle is a rhombus.

Construction of a tangent at a point on a circle: draw the radius to that point, then erect a perpendicular at the point.

Construction of tangents from an external point: draw the circle with OP as diameter; its intersections with the given circle are the points of contact.

To divide a segment in a given ratio, draw a ray at an acute angle, mark equal steps with a compass, and join corresponding points.

Circumference of a circle of radius r is 2πr; area is πr².

Area of a sector of angle θ is (θ/360) × πr².

Length of an arc of angle θ is (θ/360) × 2πr.

Area of a minor segment equals sector area minus triangle area, where the triangle has sides r, r and included angle θ.

Area of a major sector is πr² minus the area of the corresponding minor sector.

Perimeter of a sector is arc length plus 2r; perimeter of a segment is arc length plus chord length.

Distance covered by a wheel in one revolution equals its circumference; revolutions = total distance ÷ circumference.

Area swept by a clock hand of length r in time t is (θ/360)πr², where θ is the angle turned in that time.

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 (1/3)πr²h.

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

Hemisphere: curved surface 2πr², total surface 3πr², volume (2/3)πr³.

Frustum of a cone: volume (1/3)πh(r₁² + r₁r₂ + r₂²), slant height l = √[h² + (r₁ − r₂)²].

When a solid is melted and recast, volume is conserved; equate initial and final volumes.

When a cone, sphere, or cylinder is immersed in a cylinder of water, the rise in level times the base area equals the volume of the solid.

Composite solids: add the volumes of the parts and add only the surfaces that remain exposed.

Mean of raw data is Σxᵢ/n; for a frequency table it is Σfᵢxᵢ/Σfᵢ.

Median of n ordered values: the middle value when n is odd; the average of the two middle values when n is even.

Mode is the value with the highest frequency; a data set may have more than one mode or none.

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

Class mark of a class interval is (lower limit + upper limit)/2 and is used as xᵢ in grouped mean calculations.

Cumulative frequency is used to locate the median class; the median class is the first class whose cumulative frequency reaches n/2.

Probability of an event E is P(E) = number of favourable outcomes ÷ total number of equally likely outcomes.

0 ≤ P(E) ≤ 1, P(impossible event) = 0, P(sure event) = 1, and P(E) + P(not E) = 1.