WBBSE Madhyamik Mathematics flashcards

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

Simple interest formula

Interest on a fixed principal only: I = PRT/100, with P principal, R annual rate percent, T time in years. Amount A = P + I.

Compound interest formula

For annual compounding, A = P(1 + R/100)^n and CI = A − P. Interest is added to principal each period, so growth is geometric rather than linear.

Successive discount rule

Two successive discounts of a% and b% are not equal to (a+b)%. They equal a single discount of a + b − ab/100 percent.

Mean proportional

If a:b = b:c then b is the mean proportional between a and c, and b² = ac. Example: between 4 and 9 the mean proportional is 6.

Profit percent base

Profit percent = (SP − CP)/CP × 100, computed on cost price unless the question states otherwise. Discount, by contrast, is computed on marked price.

Combined work time

If A alone finishes in x days and B in y days, working together they take xy/(x+y) days, because their one-day rates 1/x and 1/y add.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the roots are x = [−b ± √(b² − 4ac)]/(2a). The expression b² − 4ac under the root is the discriminant.

Discriminant and nature of roots

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

Sum and product of roots

For ax² + bx + c = 0, sum of roots = −b/a and product of roots = c/a. These let you build quadratics from given roots without solving.

Zero-sum cube identity

If a + b + c = 0 then a³ + b³ + c³ = 3abc. It follows from the factorisation of a³ + b³ + c³ − 3abc.

Unique solution condition

The pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 has a unique solution when a₁/a₂ ≠ b₁/b₂; equal ratios with equal c ratios give infinitely many solutions.

Rationalising a binomial surd

Multiply by the conjugate: 1/(√a + √b) = (√a − √b)/(a − b), since (√a + √b)(√a − √b) = a − b. Use a ≠ b.

Exterior angle of a triangle

An exterior angle of a triangle equals the sum of the two interior opposite angles. It is therefore greater than either of those two interior angles.

Thales basic proportionality theorem

A line drawn parallel to one side of a triangle intersects the other two sides so that the segments cut off are proportional, and it creates a triangle similar to the original.

Angle in a semicircle

An angle subtended by a diameter at any point on the remaining circle is a right angle, that is 90°. This is a special case of the angle-at-centre theorem.

Cyclic quadrilateral angle property

In a cyclic quadrilateral, each pair of opposite angles sums to 180°. Also, an exterior angle equals the interior opposite angle of the quadrilateral.

Triangle congruence criteria

Triangles are congruent by SSS, SAS, ASA, AAS, or RHS (right angle, hypotenuse, side). Equal angles alone (AAA) give similarity, not congruence.

Tangent-radius perpendicularity

A tangent to a circle is perpendicular to the radius at the point of contact. Also, tangents drawn from one external point are equal in length.

Sector arc length and area

For a sector of angle θ in a circle of radius r: arc length = (θ/360) × 2πr and area = (θ/360) × πr².

Heron's formula

Area of a triangle with sides a, b, c is √[s(s−a)(s−b)(s−c)], where s is the semi-perimeter (a+b+c)/2. Useful when no height is given.

Cone slant height relation

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

Sphere surface and volume

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

Pythagorean identity

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

Standard trigonometric values

sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2. Cosines are these values in reverse order, and tan 45° = 1.

Angle of elevation

The angle of elevation is measured upward from the horizontal observer line to the line of sight of an object above. Depression is measured downward instead.

Grouped data mean

Mean = Σfᵢxᵢ / Σfᵢ, where xᵢ are class midpoints and fᵢ their frequencies. The sum of frequencies Σfᵢ equals N, the total number of observations.

Grouped data median formula

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

Grouped data mode formula

Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, with f₁ the modal class frequency, f₀ the preceding and f₂ the following class frequency.

Empirical mean-median-mode relation

For moderately skewed frequency distributions, mode ≈ 3 median − 2 mean. It lets you estimate the missing one of the three averages.

Distance formula

Distance between (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]. Distance of a point (x, y) from the origin is √(x² + y²).

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)). Setting m = n gives the midpoint.

Collinearity by area

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