BSEB Matric Adv Math (Opt) 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 any integer a and positive integer b, there exist unique integers q and r such that a = bq + r with 0 ≤ r < b. This underlies the HCF algorithm and divisibility arguments.

Terminating decimal condition

A rational p/q in lowest terms has a terminating decimal expansion exactly when q's prime factors are only 2 and 5, i.e. q = 2^m · 5^n. Otherwise the decimal repeats.

HCF–LCM product rule

For two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. The identity fails for three or more numbers unless each pair is coprime.

Irrationality of square roots

If n is a positive integer that is not a perfect square, then √n is irrational. In particular √2, √3, √5 and √7 cannot be written as p/q.

Remainder Theorem

When a polynomial p(x) is divided by (x − a), the remainder equals p(a). Consequently (x − a) is a factor of p(x) exactly when p(a) = 0.

Sum and product of roots

For ax² + bx + c = 0 with roots α and β, α + β = −b/a and αβ = c/a. These let you build the equation x² − (α + β)x + αβ = 0.

Discriminant and nature of roots

For ax² + bx + c = 0, D = b² − 4ac. If D > 0 there are two distinct real roots; D = 0 gives equal roots; D < 0 gives no real roots.

Quadratic formula

The roots of ax² + bx + c = 0 with a ≠ 0 are x = [−b ± √(b² − 4ac)] / 2a. Substitute carefully and simplify the surd for the final answer.

nth term of an AP

For an arithmetic progression with first term a and common difference d, the nth term is aₙ = a + (n − 1)d. Use it to find the number of terms or an unknown term.

Sum of n terms of AP

Sₙ = (n/2)[2a + (n − 1)d], or Sₙ = (n/2)(a + l) when the last term l is known. The formula covers both finite and reverse-order sums.

Three terms in AP

Three numbers in arithmetic progression are conveniently written as a − d, a, a + d, whose sum is 3a. This trick simplifies problems giving the sum and product of three terms.

Sum of first n odd numbers

The sum 1 + 3 + 5 + … + (2n − 1) equals n². It is an AP with a = 1 and d = 2, so Sₙ = (n/2)(2n) = n².

Distance formula

The distance between points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]. Setting it equal to a given length produces an equation for an unknown coordinate.

Section formula

The point dividing the segment joining (x₁, y₁) and (x₂, y₂) internally in ratio m : n is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). The midpoint uses m = n = 1.

Perpendicular and parallel slopes

Two non-vertical lines are parallel when their slopes are equal, and perpendicular when the product of their slopes is −1. A horizontal line has slope 0.

Collinearity by area

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

Pythagorean trigonometric identity

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

Angle of elevation

The angle of elevation is the angle measured upward from the horizontal observer's line to the line of sight to an object above. It is used with tan to find heights.

Height from shadow length

If a vertical object casts a shadow of length s when the sun's elevation is θ, its height is h = s · tan θ. Draw a right triangle with the shadow as base.

Tangent perpendicular to radius

A tangent to a circle at point P is perpendicular to the radius drawn to P. This right angle links circle problems to Pythagoras and to the two-tangent theorem.

Equal tangents from external point

Tangents drawn from an external point to a circle are equal in length, and they make equal angles with the line joining the point to the centre.

Alternate segment theorem

The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment subtended by that chord.

Area of a sector

A sector of angle θ in a circle of radius r has area (θ/360°) × πr² and arc length (θ/360°) × 2πr. Subtract the triangle to get the segment area.

Basic Proportionality Theorem

If a line is drawn parallel to one side of a triangle and intersects the other two sides, it divides them in the same ratio. Its converse establishes parallelism.

Area ratio of similar triangles

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

Pythagoras theorem

In a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides. Its converse proves a triangle is right-angled.

Volume of a cone

A cone of base radius r and height h has volume (1/3)πr²h and slant height l = √(r² + h²). Curved surface area is πrl.

Surface area of a sphere

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

Volume conservation on recasting

When a solid is melted and recast into another shape, its volume is unchanged. Equate the original volume to the new shape's volume formula to find the unknown dimension.

Median of grouped data

Median = l + [(N/2 − cf)/f] × h, where l is the lower limit of the median class, cf the cumulative frequency before it, f its frequency and h the class width.

Empirical relation of averages

For moderately skewed distributions, Mode = 3 Median − 2 Mean. Use it to find the missing average when two of the three measures are given.

Complementary probability rule

For any event E, P(E) + P(not E) = 1, so P(not E) = 1 − P(E). Probabilities always lie between 0 and 1 inclusive.