AP SSC 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: HCF(a, b) = HCF(b, r), repeated until the remainder is 0; the last non-zero divisor is the HCF.

Fundamental Theorem of Arithmetic: every composite number has a prime factorisation that is unique apart from the order of the factors.

HCF(a, b) × LCM(a, b) = a × b for two positive integers; this identity fails in general for three numbers.

A rational number p/q in lowest terms has a terminating decimal iff q = 2^m 5^n; otherwise its decimal is non-terminating and recurring.

√p is irrational for every prime p; the standard proof is by contradiction, writing √p = a/b with HCF(a, b) = 1.

Rational + irrational = irrational; non-zero rational × irrational = irrational; but irrational ± irrational may be rational.

Logarithms: log_a(mn) = log_a m + log_a n, log_a(m/n) = log_a m − log_a n, log_a m^k = k log_a m, log_a a = 1, log_a 1 = 0.

Roster form lists the elements; set-builder form states a property, e.g. {x : x ∈ N, x < 10}.

A ∪ B = {x : x ∈ A or x ∈ B}; A ∩ B = {x : x ∈ A and x ∈ B}; disjoint sets satisfy A ∩ B = ∅.

n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(C∩A) + n(A∩B∩C).

A − B = {x : x ∈ A, x ∉ B}; n(A − B) = n(A) − n(A ∩ B).

Complement A′ = μ − A; De Morgan: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

A set with n elements has 2^n subsets in all, 2^n − 1 proper subsets and 2^n − 2 non-empty proper subsets.

A ⊆ B iff every element of A lies in B; equal sets have identical elements, whereas equivalent sets only have equal cardinality.

A zero of p(x) is a value k with p(k) = 0; a polynomial of degree n has at most n zeros.

Remainder theorem: dividing p(x) by x − a leaves the remainder p(a).

Factor theorem: x − a is a factor of p(x) iff p(a) = 0.

Division algorithm: p(x) = g(x)q(x) + r(x), where r(x) = 0 or deg r < deg g.

For ax² + bx + c (a ≠ 0): α + β = −b/a and αβ = c/a.

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

Polynomial from given zeros: x² − (α + β)x + αβ; cubic: x³ − (α+β+γ)x² + (αβ+βγ+γα)x − αβγ.

The graph of a quadratic meets the x-axis twice, once or not at all according as D = b² − 4ac is > 0, = 0 or < 0.

AP: aₙ = a + (n − 1)d, with common difference d = aₙ − aₙ₋₁ constant.

Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(a + l), and aₙ = Sₙ − Sₙ₋₁.

Three terms in AP: a − d, a, a + d; four terms: a − 3d, a − d, a + d, a + 3d.

In an AP the sums of equidistant terms are equal: a₁ + aₙ = a₂ + aₙ₋₁ = …

Arithmetic mean of a and b is (a + b)/2; a, b, c are in AP iff 2b = a + c.

GP: aₙ = ar^(n−1); Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1, and Sₙ = na when r = 1.

S∞ = a/(1 − r) exists only when |r| < 1.

Sums: Σn = n(n+1)/2; Σ(2n − 1) = n²; Σn² = n(n+1)(2n+1)/6; Σn³ = [n(n+1)/2]².

Distance formula: √[(x₂ − x₁)² + (y₂ − y₁)²]; distance of (x, y) from the origin is √(x² + y²).

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

Section formula (internal division in m:n): ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)); the midpoint is the case m = n.

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

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

Slope m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes, and perpendicular lines satisfy m₁m₂ = −1.

A point on the x-axis has the form (x, 0); a point on the y-axis has the form (0, y).

To find the ratio in which a point divides a segment, substitute its coordinates in the section formula and solve for m:n.

Standard form ax + by + c = 0, where a and b are not both zero; the graph is a straight line.

Intersecting lines give exactly one solution (consistent); parallel lines give none (inconsistent); coincident lines give infinitely many (dependent).

Unique solution iff a₁/a₂ ≠ b₁/b₂.

No solution iff a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

Infinitely many solutions iff a₁/a₂ = b₁/b₂ = c₁/c₂.

Cross-multiplication: x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁), valid when a₁b₂ − a₂b₁ ≠ 0.

A two-digit number with tens digit a and units digit b equals 10a + b; on reversing the digits it becomes 10b + a.

Basic Proportionality Theorem: DE ∥ BC in △ABC gives AD/DB = AE/EC; the converse states that equal ratios imply DE ∥ BC.

Similarity criteria: AA (or AAA), SSS (all three pairs of sides proportional) and SAS (two sides proportional with equal included angle).

Ratio of areas of similar triangles = (ratio of corresponding sides)²; ratio of perimeters = ratio of corresponding sides.

Triangles on the same base and between the same parallels are equal in area; a diagonal bisects a parallelogram.

Midpoint theorem: the segment joining the midpoints of two sides is parallel to the third side and half its length.

Pythagoras theorem: in a right triangle, hypotenuse² = sum of the squares of the other two sides; the converse also holds.

Sides of a 45°–45°–90° triangle are in the ratio 1:1:√2; of a 30°–60°–90° triangle in the ratio 1:√3:2.

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

Angle at the centre = 2 × angle at the remaining circumference; angle in a semicircle = 90°; opposite angles of a cyclic quadrilateral are supplementary.

Cuboid: TSA = 2(lb + bh + hl), lateral surface = 2h(l + b), V = lbh; cube: TSA = 6a², V = a³.

Cylinder: CSA = 2πrh, TSA = 2πr(r + h), V = πr²h.

Cone: l = √(r² + h²), CSA = πrl, TSA = πr(l + r), V = ⅓πr²h.

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

Hemisphere: CSA = 2πr², TSA = 3πr², volume = (2/3)πr³.

Frustum: l = √[h² + (r₁ − r₂)²], CSA = π(r₁ + r₂)l, V = ⅓πh(r₁² + r₂² + r₁r₂).

Heron's formula: area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2.

1 m³ = 1000 litres, 1 litre = 1000 cm³, 1 hectare = 10,000 m².

Volume of material in a hollow cylinder = πh(R² − r²), where R and r are the outer and inner radii.

sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent; cosec, sec and cot are the respective reciprocals.

sin 0° = 0, sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1; the cos values run in the reverse order.

tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3; tan 90° is not defined.

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

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

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

In height-and-distance problems tan(angle) = height / horizontal distance; equal angles of elevation from two points imply equal distances from the foot.

For 0° ≤ θ ≤ 90°, 0 ≤ sin θ ≤ 1 and 0 ≤ cos θ ≤ 1, while sec θ ≥ 1 and cosec θ ≥ 1.

Mean = Σx/n for ungrouped data and Σfx/Σf for grouped data; assumed-mean method: x̄ = a + Σfd/Σf.

Median of ungrouped data is the middle value after sorting; for even n it is the average of the two middle values.

Median of grouped data = l + [(n/2 − cf)/f] × h, where cf is the cumulative frequency of the class preceding the median class.

Mode of grouped data = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, with f₁ the frequency of the modal class.

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

Class mark = (lower limit + upper limit)/2; class size h = upper limit − lower limit.

P(E) = number of favourable outcomes / total number of equally likely outcomes; 0 ≤ P(E) ≤ 1 and P(E) + P(E′) = 1.

The sum of the probabilities of all elementary events of a random experiment is 1; a sure event has probability 1 and an impossible event 0.