Kerala SSLC Mathematics flashcards

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

nth term of an arithmetic sequence

xₙ = f + (n − 1)d, where f is the first term and d the common difference; each term is obtained by adding d to the previous term.

Sum of first n terms

Sₙ = (n/2)[2f + (n − 1)d], which equals the number of terms multiplied by the average of the first and last terms of the sequence.

Sum of first n odd numbers

1 + 3 + 5 + … + (2n − 1) = n². The odd numbers form an arithmetic sequence with first term 1 and common difference 2.

Perpendicular from centre to chord

A perpendicular drawn from the centre of a circle to a chord bisects that chord; conversely, the line from the centre to the midpoint of a chord is perpendicular to it.

Angle in a semicircle

Any angle subtended by a diameter at a point on the circle measures 90°; conversely, a chord that subtends a right angle at the circumference is a diameter.

Cyclic quadrilateral property

In a quadrilateral whose vertices all lie on one circle, each pair of opposite angles sums to 180°, and every exterior angle equals the interior opposite angle.

Probability of an event

The number of favourable outcomes divided by the total number of equally likely outcomes; the value always lies between 0 and 1 inclusive.

Complement rule in probability

P(not E) = 1 − P(E). It is used when counting the outcomes that avoid a result is easier than counting the outcomes that produce it.

Sample space of two dice

Throwing two dice gives 36 equally likely ordered outcomes; the total 7 appears in 6 of them, making it the most probable sum.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = [−b ± √(b² − 4ac)]/(2a), obtained by completing the square on the left side.

Discriminant of a quadratic

D = b² − 4ac. If D > 0 there are two distinct real roots, if D = 0 the roots are equal, and if D < 0 there are no real roots.

Sum and product of roots

For ax² + bx + c = 0, the roots α and β satisfy α + β = −b/a and αβ = c/a, which lets you build or check the equation.

Tangent ratio

In a right triangle, tan θ = opposite side ÷ adjacent side = sin θ / cos θ; it is used when both legs of the triangle are involved.

Pythagorean identity in trigonometry

sin²θ + cos²θ = 1 for every angle θ; dividing throughout by cos²θ gives 1 + tan²θ = sec²θ.

Angle of elevation

The angle measured upward from the horizontal line of sight to an object above the observer; combined with tan it gives heights of towers and buildings.

Distance formula

The distance between (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²], which follows directly from the Pythagorean theorem.

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 they form is zero; compute that area from the coordinates and set it equal to zero.

Tangent–radius perpendicularity

A tangent to a circle is perpendicular to the radius drawn to the point of contact, so the radius, the tangent and the line to an external point form a right triangle.

Equal tangents from external point

The two tangent segments drawn from an external point to the same circle have equal lengths, and the line to the centre bisects the angle between them.

Alternate segment theorem

The angle between a tangent and a chord drawn from the point of contact equals the angle that the chord subtends in the alternate segment of the circle.

Volume of a cone

V = ⅓πr²h, one third of the volume of the cylinder with the same base radius and height; its slant height is √(r² + h²).

Surface area of a sphere

A sphere of radius r has surface area 4πr² and volume 4/3 πr³; a hemisphere has total surface area 3πr² including its flat circular face.

Volume conservation on recasting

When a solid is melted and recast into a different shape, the total volume stays the same, so equate the two volume formulas and solve for the unknown dimension.

Remainder theorem

The remainder when a polynomial p(x) is divided by (x − a) equals p(a); if p(a) = 0 then (x − a) is an exact factor.

Zeros and coefficients of quadratic

For ax² + bx + c, the sum of the zeros is −b/a and the product is c/a; these give the polynomial x² − (sum)x + product.

Factor theorem

A polynomial p(x) has (x − a) as a factor if and only if p(a) = 0, which gives a quick test for candidate factors.

Basic proportionality theorem

A line drawn parallel to one side of a triangle divides the other two sides in the same ratio; the converse of this result identifies parallel lines.

Ratio of areas of similar triangles

The areas of two similar triangles are in the ratio of the squares of any pair of corresponding sides, for example 9 : 16 when the sides are 3 : 4.

AA similarity criterion

If two angles of one triangle equal two angles of another, the triangles are similar and all three pairs of corresponding sides are proportional.

Mean of grouped data

Mean = Σfx / Σf, where x is the class mark, the midpoint of each class, and f is its frequency; the assumed-mean method simplifies the arithmetic.

Median of grouped data

Median = l + [(n/2 − cf)/f] × h, where the median class is the first class whose cumulative frequency reaches or passes n/2.