Assam HSLC Class 10 Elective Advanced Mathematics cheat sheet

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

Fundamental Theorem of Arithmetic: every integer greater than 1 factors uniquely into primes apart from the order of the factors.

HCF(a, b) × LCM(a, b) = a × b for positive integers a, b; this fails for three or more numbers.

Irrationality test: if a prime p divides n² then p divides n. Hence √p is irrational for every prime p, and √n is irrational unless n is a perfect square.

Rational numbers have terminating decimal expansions exactly when, in lowest terms, the denominator has no prime factor other than 2 and 5.

For positive reals a, b: AM ≥ GM ≥ HM, with equality only when a = b; AM = (a+b)/2, GM = √(ab), HM = 2ab/(a+b).

A number is rational iff its decimal expansion is terminating or eventually repeating; non-terminating non-repeating decimals are irrational.

Between any two distinct real numbers there are infinitely many rationals and infinitely many irrationals (density of Q and of irrationals in R).

Remainder Theorem: the remainder when p(x) is divided by (x − a) is p(a); p(x) is divisible by (x − a) exactly when p(a) = 0.

Factor Theorem: (x − a) is a factor of p(x) iff p(a) = 0; repeated use plus the Rational Root Theorem locates rational zeros.

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

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

A polynomial of degree n has at most n real zeros, and a real polynomial of odd degree has at least one real zero.

Division algorithm for polynomials: p(x) = g(x)q(x) + r(x) with deg r < deg g, valid whenever deg g ≥ 1.

Symmetric expressions in roots are computed from sum and product: α² + β² = (α + β)² − 2αβ; α³ + β³ = (α + β)³ − 3αβ(α + β).

A pair a₁x + b₁y = c₁, a₂x + b₂y = c₂ has a unique solution when a₁/a₂ ≠ b₁/b₂, infinitely many when a₁/a₂ = b₁/b₂ = c₁/c₂, and none when a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

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

Discriminant D = b² − 4ac: D > 0 gives two distinct real roots, D = 0 gives equal real roots, D < 0 gives no real roots (a conjugate complex pair).

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

Completing the square: ax² + bx + c = a(x + b/2a)² + (c − b²/4a), which yields the vertex (−b/2a, c − b²/4a) of the parabola.

For equations reducible to quadratic form, substitute a single variable for the repeated expression, solve, then reject values excluded by the original denominators.

Word problems: fix one unknown, express the second from the given relation, form one equation, then check that the solution satisfies every stated condition (lengths positive, ages consistent).

AP: nth term aₙ = a + (n − 1)d; sum Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(a + l) where l is the last term.

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

If a, b, c are in AP then 2b = a + c; the arithmetic mean of two numbers is their average.

GP: nth term aₙ = arⁿ⁻¹; sum Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1, and Sₙ = na when r = 1.

Sum to infinity of a GP exists only when |r| < 1 and equals a/(1 − r).

If a, b, c are in GP then b² = ac; the geometric mean of positive a and c is √(ac).

Choosing terms: for three terms in AP use a − d, a, a + d; for three in GP use a/r, a, ar, which simplifies the conditions given.

Sum of the first n natural numbers: Σn = n(n + 1)/2.

Sum of squares: Σn² = n(n + 1)(2n + 1)/6; sum of cubes: Σn³ = [n(n + 1)/2]².

These standard results are proved by induction: verify for n = 1 and show the statement for n implies it for n + 1.

Sum of the first n odd numbers is n²; sum of the first n even numbers is n(n + 1).

A series whose nth term is a polynomial in n of degree k has a partial-sum formula of degree k + 1 in n.

Telescoping sums: write each term as a difference f(r) − f(r + 1); the partial sum collapses to first term minus last term.

Σ notation is linear: Σ(auₙ + bvₙ) = aΣuₙ + bΣvₙ over the same range of indices.

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

Section formula: the point dividing the join of A and B internally in ratio m : n is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

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

Forms of a line: slope-intercept y = mx + c; point-slope y − y₁ = m(x − x₁); two-point and intercept form x/a + y/b = 1.

General form Ax + By + C = 0 has slope −A/B; distance from (x₁, y₁) to it is |Ax₁ + By₁ + C|/√(A² + B²).

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

Angle θ between lines of slopes m₁, m₂ satisfies tan θ = |(m₁ − m₂)/(1 + m₁m₂)|.

Fundamental identity: sin²θ + cos²θ = 1, hence 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.

Standard values: sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2; cos runs the same values in reverse; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

Complementary angles: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, sec(90° − θ) = cosec θ.

Signs by quadrant follow ASTC: All positive in I, sin only in II, tan only in III, cos only in IV.

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

Height-and-distance problems: draw the vertical height h, mark the horizontal distance d, and use tan θ = h/d; combine two right triangles when the observer moves.

In a right triangle, tan θ = opposite/adjacent, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse.

Basic Proportionality (Thales) Theorem: a line parallel to one side of a triangle divides the other two sides in the same ratio, and the converse also holds.

Similar triangles: corresponding angles equal and corresponding sides proportional; areas are in the ratio of the squares of corresponding sides.

Pythagoras theorem: in a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides; the converse identifies right angles.

Angle in a semicircle is a right angle; angles in the same segment of a circle are equal; the angle at the centre is twice the angle at the circumference on the same arc.

Tangent–radius: a tangent is perpendicular to the radius at the point of contact; tangents drawn from an external point are equal in length.

Tangent–secant (alternate segment): the angle between a tangent and a chord equals the angle in the alternate segment.

Constructions must retain arcs and construction lines: bisect an angle, draw a perpendicular bisector, divide a segment in a given ratio, and draw tangents from an external point.

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 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); curved surface π(r₁ + r₂)l with l = √[h² + (r₁ − r₂)²].

Cube and cuboid: cuboid surface 2(lb + bh + hl), volume lbh; cube surface 6a², volume a³, diagonal a√3.

When a solid is melted and recast, volume is conserved: equate initial volume to final volume to find the unknown dimension.

For a composite solid, add volumes of the parts but count only exposed surfaces when computing surface area.

Mean of ungrouped data: x̄ = Σx/n; for a frequency distribution x̄ = Σfx/Σf.

Median of n ordered observations is the middle value when n is odd and the average of the two middle values when n is even.

Mode is the most frequent value; an empirical relation for moderately skewed data is mode ≈ 3 median − 2 mean.

Range = maximum − minimum; it uses only two observations and ignores the rest of the data.

Variance of ungrouped data: σ² = Σ(x − x̄)²/n = Σx²/n − x̄²; standard deviation σ is its positive square root.

Coefficient of variation = (σ/x̄) × 100, allowing comparison of dispersion between data sets with different means.

For grouped data with classes, use class marks as x, and locate the median class by cumulative frequency before applying the median formula.

Classical probability: P(E) = number of favourable outcomes / total number of equally likely outcomes.

For any event E, 0 ≤ P(E) ≤ 1; P(impossible event) = 0 and P(sure event) = 1.

Complement rule: P(not E) = 1 − P(E).

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the last term is zero.

A standard deck has 52 cards: 26 red, 26 black, 4 suits of 13, and 4 aces; face cards are J, Q, K (12 in all).

Two dice give 36 equally likely ordered outcomes; sums 2 and 12 each occur once, while sum 7 occurs six times.

Independent events satisfy P(A ∩ B) = P(A) × P(B); drawing without replacement makes successive draws dependent.

For finite sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B); for three sets subtract the pairwise intersections and add back the triple intersection.

Number of subsets of a set with n elements is 2ⁿ; the number of proper subsets is 2ⁿ − 1.

A × B = {(a, b) : a ∈ A, b ∈ B} with n(A × B) = n(A) · n(B); A × B = B × A only when A = B.

A relation from A to B is a subset of A × B; its domain is the set of first components and its range the set of second components.

A function f : A → B assigns exactly one element of B to each element of A; equal images for different inputs are allowed, but one input cannot have two images.

A function is one-one (injective) if f(x₁) = f(x₂) forces x₁ = x₂, and onto (surjective) if every element of B is an image; both make it bijective.

Composition (g ∘ f)(x) = g(f(x)) is defined only when the range of f lies within the domain of g.