AP SSC Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Euclid's division lemma

For positive integers a and b, there exist unique whole numbers q and r with a = bq + r and 0 ≤ r < b; it is the basis for finding the HCF.

Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of primes, and this factorisation is unique except for the order in which the prime factors are written.

HCF–LCM product rule

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b; the rule does not extend directly to three or more numbers.

Terminating decimal test

A rational number p/q in lowest terms has a terminating decimal expansion only when q = 2^m 5^n for some non-negative integers m and n.

Union of two sets

A ∪ B is the set of all elements belonging to A or to B or to both; its cardinality is n(A) + n(B) − n(A ∩ B).

De Morgan's laws

For subsets A and B of a universal set, the complement of A ∪ B equals A′ ∩ B′, and the complement of A ∩ B equals A′ ∪ B′.

Number of subsets

A set with n distinct elements has 2^n subsets in total, of which 2^n − 1 are proper subsets and exactly one is the set itself.

Remainder theorem

If a polynomial p(x) is divided by x − a, the remainder equals p(a); a remainder of zero means that x − a is a factor of p(x).

Sum and product of zeros

For the quadratic polynomial ax² + bx + c with a ≠ 0, the zeros α and β satisfy α + β = −b/a and αβ = c/a.

Division algorithm for polynomials

For polynomials p(x) and g(x) with g(x) ≠ 0, we can write p(x) = g(x)·q(x) + r(x), where r(x) = 0 or degree of r(x) is less than degree of g(x).

nth term of an AP

In an arithmetic progression, the nth term is aₙ = a + (n − 1)d, where a is the first term and d is the constant common difference.

Sum of n terms of AP

The sum of the first n terms of an AP is Sₙ = (n/2)[2a + (n − 1)d], or Sₙ = (n/2)(a + l) when the last term l is known.

Sum of infinite GP

For a geometric progression with first term a and common ratio r where |r| < 1, the infinite sum is S∞ = a/(1 − r); the sum does not exist when |r| ≥ 1.

Distance formula

The distance between the points (x₁, y₁) and (x₂, y₂) in the Cartesian plane is √[(x₂ − x₁)² + (y₂ − y₁)²].

Section formula

The point dividing the segment joining (x₁, y₁) and (x₂, y₂) internally in the ratio m:n is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

Collinearity by area

Three points are collinear exactly when the area of the triangle formed by them is zero, that is ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)| = 0.

Unique solution condition

The pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 has exactly one solution when a₁/a₂ ≠ b₁/b₂; the two lines then intersect at a single point.

Inconsistent pair condition

A pair of linear equations has no solution when a₁/a₂ = b₁/b₂ ≠ c₁/c₂; the graphs are parallel lines that never meet, so the system is called inconsistent.

Cross-multiplication method

The cross-multiplication formula x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁) gives the solution of the pair when a₁b₂ − a₂b₁ ≠ 0.

Basic proportionality theorem

A line drawn parallel to one side of a triangle divides the other two sides in the same ratio; conversely, if it divides them proportionally it is parallel to the third side.

Areas of similar triangles

The ratio of the areas of two similar triangles equals the square of the ratio of any pair of corresponding sides, such as their corresponding medians or altitudes.

Tangent–radius theorem

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

Angle in a semicircle

The angle subtended by a diameter at any point on the remaining part of the circle is a right angle, that is, exactly 90 degrees.

Volume of a cone

For a cone of base radius r and height h, volume = ⅓πr²h; the slant height is l = √(r² + h²), curved surface area πrl and total surface area πr(l + r).

Sphere surface and volume

For a sphere of radius r, surface area = 4πr² and volume = (4/3)πr³; a solid hemisphere has total surface area 3πr².

Heron's formula

The area of a triangle with sides a, b and c is √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2 is the semi-perimeter.

Pythagorean identity

For any acute angle θ, sin²θ + cos²θ = 1; dividing this identity by cos²θ and by sin²θ gives 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.

Complementary angle ratios

For an angle θ measured in degrees, sin(90° − θ) = cos θ, tan(90° − θ) = cot θ and sec(90° − θ) = cosec θ.

Angle of elevation

The angle of elevation is the angle between the horizontal line through the observer's eye and the line of sight to an object lying above that horizontal level.

Empirical relationship

For moderately skewed frequency distributions, 3 × Median = Mode + 2 × Mean; this relation estimates one measure of central tendency when the other two are known.

Probability of an event

P(E) equals the number of favourable outcomes divided by the total number of equally likely outcomes; 0 ≤ P(E) ≤ 1 and P(E) + P(not E) = 1.

Mode of grouped data

Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, where l is the lower limit of the modal class, h the class size, f₁ the modal frequency, f₀ and f₂ the adjacent frequencies.