UPMSP High School Mathematics flashcards

33 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 positive b, there exist unique integers q and r such that a = bq + r with 0 ≤ r < b. It underlies the Euclidean algorithm for HCF.

Terminating decimal condition

A rational p/q in lowest terms has a terminating decimal expansion exactly when the prime factorisation of q contains only 2s and 5s, that is q = 2^m·5^n.

HCF and LCM relation

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. This lets you find one value when the other three are known.

Remainder theorem

When a polynomial p(x) is divided by (x − a), the remainder equals p(a). This gives the remainder without performing long division.

Factor theorem

A polynomial p(x) has (x − a) as a factor if and only if p(a) = 0. It converts factor questions into substitution and solving.

Sum and product of roots

For ax² + bx + c = 0 with roots α and β, the sum α + β = −b/a and the product αβ = c/a. These hold whether or not roots are real.

Cubic root relations

For ax³ + bx² + cx + d = 0 with roots α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a, and αβγ = −d/a.

Unique solution condition

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

Parallel lines condition

The two linear equations have no solution when a₁/a₂ = b₁/b₂ but c₁/c₂ differs, since the lines are parallel and never meet.

Coincident lines condition

The pair has infinitely many solutions when a₁/a₂ = b₁/b₂ = c₁/c₂. Both equations represent the same line, so every point on it satisfies both.

Cross-multiplication formula

For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁), provided the last denominator is nonzero.

Quadratic formula

The roots of ax² + bx + c = 0, a ≠ 0, are x = (−b ± √(b² − 4ac))/(2a). Real roots exist only when the discriminant b² − 4ac is non-negative.

Discriminant and nature of roots

For ax² + bx + c = 0, D = b² − 4ac. If D > 0 roots are real and distinct, if D = 0 they are equal, and if D < 0 no real roots exist.

Completing the square form

ax² + bx + c can be written as a(x + b/2a)² + (c − b²/4a). This locates the vertex and shows the minimum or maximum value of the quadratic.

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 when the position of a term is asked.

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. Each formula needs different given data.

AP mean condition

Three numbers a, b, c are in arithmetic progression exactly when 2b = a + c, that is b is the arithmetic mean of a and c.

nth term from Sₙ

If the sum of the first n terms is Sₙ, then the nth term aₙ = Sₙ − Sₙ₋₁ for n ≥ 2, and the first term a₁ = S₁.

Distance formula

The distance between points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]. It follows from the Pythagoras theorem applied to coordinate differences.

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)).

Area of a triangle by coordinates

Area = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. If the result is zero, the three points are collinear.

Basic Proportionality Theorem

A line drawn parallel to one side of a triangle intersects the other two sides in the same ratio. Its converse establishes parallelism from equal ratios.

Similar triangle area ratio

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-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Its converse identifies a right angle.

Equal tangents from external point

Two tangents drawn to a circle from an external point are equal in length, and the line joining the point to the centre bisects the angle between them.

Tangent perpendicular to radius

At the point of contact, a tangent to a circle is perpendicular to the radius drawn to that point. This is the basis of most tangent problems.

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.

Tangent construction from external point

Draw the circle with the segment joining the external point to the centre as diameter. Its intersections with the given circle are the points of contact.

Dividing a segment in ratio

Draw a ray from one endpoint, mark m + n equal divisions, join the last to the other endpoint, and draw a parallel through the mth mark.

Pythagorean 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 between the horizontal and the line of sight when the observer looks upward at an object. It is used with tan in height problems.

Cone slant height

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

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, h the class width.