Assam HSLC Class 10 Elective Advanced 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 integers a and b with b positive, there exist unique integers q and r with a = bq + r and 0 ≤ r < b. The remainder r is zero exactly when b divides a.

HCF-LCM product rule

For two positive integers a and b, HCF(a, b) multiplied by LCM(a, b) equals a times b. The rule does not extend to three or more numbers.

Terminating decimal criterion

A rational number in lowest terms has a terminating decimal expansion exactly when its denominator has no prime factor other than 2 and 5. Otherwise the expansion repeats forever.

Remainder theorem

The remainder obtained on dividing a polynomial p(x) by (x − a) equals p(a). Consequently p(x) is divisible by (x − a) precisely when p(a) is zero.

Sum and product of roots

For the quadratic ax² + bx + c = 0 with roots α and β, the sum α + β equals −b/a and the product αβ equals c/a, linking coefficients to roots.

Rational root theorem

Any rational zero of a polynomial with integer coefficients must be of the form p/q, where p divides the constant term and q divides the leading coefficient.

Discriminant test

For ax² + bx + c = 0 the discriminant D = b² − 4ac decides the nature of roots: positive gives two distinct real roots, zero gives equal real roots, negative gives no real roots.

Consistency of linear pairs

Two linear equations have one solution when coefficient ratios of x and y differ, infinitely many when all three ratios agree, and none when only the constant ratio differs.

Completing the square

Writing ax² + bx + c as a(x + b/2a)² + (c − b²/4a) converts a quadratic to vertex form and directly gives the turning point of its parabola.

Sum of an arithmetic progression

For first term a, common difference d and n terms, the sum equals (n/2)[2a + (n − 1)d], or equivalently (n/2)(first term + last term).

Sum of infinite geometric progression

A geometric progression with first term a and common ratio r has a finite sum to infinity equal to a/(1 − r), defined only when the absolute value of r is less than one.

Geometric mean condition

Three positive numbers a, b, c are in geometric progression exactly when b² = ac. Then b, the geometric mean of a and c, equals the square root of ac.

Sum of first n squares

The sum 1² + 2² + ... + n² equals n(n + 1)(2n + 1)/6. It is proved by induction and is used to sum series whose general term is quadratic.

Telescoping series

A series whose general term can be written as f(r) − f(r + 1). Its partial sum collapses to f(1) minus the last remaining term, giving the total easily.

Section formula

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

Perpendicular line slopes

Two non-vertical lines are perpendicular exactly when the product of their slopes equals −1, so the slope of a perpendicular is the negative reciprocal of the original slope.

Distance from point to line

The perpendicular distance from (x₁, y₁) to the line Ax + By + C = 0 equals |Ax₁ + By₁ + C| divided by the square root of A² + B².

Pythagorean trigonometric identity

For every angle θ, sin²θ + cos²θ = 1. Dividing by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = cosec²θ.

Angle of depression

The angle measured downward from the horizontal line of sight to an object below the observer. In problems it equals the angle of elevation from the object to the observer.

Angle in a semicircle

Any angle subtended at a point on a circle by a diameter is a right angle. This is a special case of the angle at the centre being twice the angle at the circumference.

Basic proportionality theorem

A line drawn parallel to one side of a triangle intersects the other two sides so that the segments are proportional. Its converse establishes parallelism from proportional segments.

Equal tangents from external point

The two tangent segments drawn from an external point to a circle are equal in length, and each is perpendicular to the radius at its point of contact.

Volume of a cone

A cone of base radius r and perpendicular height h has volume one third of pi r squared h. Its slant height is the square root of r² + h².

Surface area of a sphere

A sphere of radius r has surface area 4πr² and volume (4/3)πr³. A hemisphere has curved surface 2πr² and total surface 3πr².

Volume conservation on recasting

When a solid is melted and reshaped into another solid, the volume stays the same. Equating the two volume formulas determines the unknown dimension.

Coefficient of variation

The ratio of standard deviation to mean, expressed as a percentage. Because it is unit-free, it compares dispersion of data sets whose means differ greatly.

Standard deviation from raw data

The standard deviation equals the square root of Σ(x − x̄)²/n, equivalently the square root of (Σx²/n) − x̄², measuring spread about the mean.

Empirical mean-median-mode relation

For moderately skewed frequency distributions, mode is approximately 3 times median minus 2 times mean. The relation is approximate and fails for heavily skewed data.

Complement rule in probability

The probability that an event does not occur equals 1 minus the probability that it occurs. This is useful when counting failures is simpler than counting successes.

Independent events multiplication rule

Events A and B are independent when the probability of both occurring equals the product of their separate probabilities. Drawing without replacement usually destroys independence.

Inclusion-exclusion for two sets

For finite sets A and B, the number of elements in their union equals n(A) + n(B) − n(A ∩ B), correcting for elements counted twice in the sum.

Bijective function

A function that is both one-one and onto its codomain. Each element of the codomain is the image of exactly one element of the domain, so the function is invertible.