Karnataka SSLC Mathematics (KSEAB) key terms

For any positive integers a and b, there exist unique whole numbers q and r such that a = bq + r, where r satisfies 0 ≤ r < b. It underpins the Euclidean algorithm for finding highest common factors.

Every composite number can be expressed as a product of primes, and this prime factorisation is unique apart from the order of the factors. It justifies using prime factorisation to compute HCF and LCM.

The largest positive integer that divides each of two or more given numbers without leaving a remainder. For two numbers a and b, it equals the product of the smallest powers of common primes.

The smallest positive integer that is a multiple of each of two or more given numbers. For two positive integers a and b, the relation HCF(a, b) × LCM(a, b) = a × b always holds.

A real number that cannot be written as a ratio of two integers. Its decimal expansion is non-terminating and non-recurring; familiar examples include √2, √3, π and 2 + √5.

A number expressible in the form p/q where p and q are integers and q is not zero. Its decimal expansion either terminates or repeats in a fixed block of digits.

A decimal expansion that stops after a finite number of digits. A rational number p/q in lowest terms has a terminating decimal exactly when the prime factors of q are only 2 and 5.

The fixed amount obtained by subtracting any term of an arithmetic progression from the term that follows it. Denoted d, it may be positive, negative or zero.

A list of numbers in which each term after the first is obtained by adding a constant value to the previous term. Examples include 5, 8, 11, 14 and 20, 15, 10, 5.

The term occupying position n in an arithmetic progression, given by aₙ = a + (n − 1)d, where a is the first term and d is the common difference. It allows any term to be found without listing all earlier terms.

The total of the first n terms of an arithmetic progression, calculated as Sₙ = (n/2)[2a + (n − 1)d] or equivalently Sₙ = (n/2)(a + l) when the last term l is known.

A pair of linear equations that has at least one solution. The graphs either intersect at exactly one point or coincide, corresponding to a unique solution or infinitely many solutions respectively.

A pair of linear equations with no common solution, so their graphs are parallel lines that never meet. For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 this happens when a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

A single ordered pair (x, y) that satisfies both equations of a pair of linear equations simultaneously. It occurs when the ratio of x-coefficients differs from the ratio of y-coefficients, giving intersecting lines.

The situation where every solution of one linear equation is also a solution of the other, so the two equations represent the same line. The condition is a₁/a₂ = b₁/b₂ = c₁/c₂.

An algebraic technique for solving simultaneous equations in which one variable is expressed in terms of the other from one equation and that expression is substituted into the second equation.

A method of solving simultaneous equations in which one equation or both are multiplied by suitable numbers so that the coefficients of one variable become equal, then the equations are added or subtracted to remove that variable.

The highest power of the variable appearing with a non-zero coefficient in a polynomial. It determines the polynomial's name, such as linear, quadratic or cubic, and the maximum number of its zeroes.

A value of the variable at which the polynomial evaluates to zero. Geometrically, the zeroes of p(x) are the x-coordinates of the points where the graph of y = p(x) meets the x-axis.

For a quadratic polynomial ax² + bx + c, the sum of zeroes equals −b/a and the product of zeroes equals c/a. These identities allow a polynomial to be formed from its zeroes.

For a quadratic equation ax² + bx + c = 0, the quantity b² − 4ac, which reveals the nature of the roots: positive gives two distinct real roots, zero gives equal real roots, and negative gives no real roots.

The rule x = [−b ± √(b² − 4ac)] / 2a giving the roots of ax² + bx + c = 0 when a is not zero. It works for every quadratic equation with real roots.

The values of x that make ax² + bx + c equal to zero. A quadratic equation has at most two roots, which may be real and distinct, real and equal, or non-real depending on the discriminant.

Triangles whose corresponding angles are equal and whose corresponding sides are in the same ratio. Similarity is usually established by the AA, SSS or SAS criteria rather than by checking every angle and side.

Also called Thales' theorem: if a line is drawn parallel to one side of a triangle and intersects the other two sides, it divides those two sides in the same ratio. Its converse is also used to prove lines parallel.

If two angles of one triangle are equal to two corresponding angles of another triangle, then the triangles are similar. Since the angles of a triangle sum to 180°, equality of two angles forces the third to match.

The constant ratio between corresponding side lengths of two similar figures. It equals the ratio of perimeters, while the ratio of areas equals the square of the scale factor.

The rule giving the coordinates of the point that divides the segment joining (x₁, y₁) and (x₂, y₂) internally in the ratio m : n, namely ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

The distance between points (x₁, y₁) and (x₂, y₂) in the coordinate plane, given by √[(x₂ − x₁)² + (y₂ − y₁)²]. It follows from the Pythagorean theorem applied to the horizontal and vertical separations.

The coordinates of the point exactly halfway between (x₁, y₁) and (x₂, y₂), namely ((x₁ + x₂)/2, (y₁ + y₂)/2). It is the section formula with the ratio 1 : 1.

A ratio of two sides of a right triangle defined with respect to one acute angle, such as sine, cosine, tangent, cosecant, secant and cotangent. Each ratio depends only on the size of the angle, not on the triangle's scale.

An equation involving trigonometric ratios that is true for every value of the angle for which the ratios are defined. Standard examples include sin²θ + cos²θ = 1 and 1 + tan²θ = sec²θ.

The angle measured upward from the horizontal line of sight of an observer to an object lying above that horizontal level. It is used together with right-triangle ratios to find heights and distances.

The angle measured downward from the horizontal line of sight of an observer to an object lying below that horizontal level. It equals the angle of elevation of the observer as seen from the object.

A line that touches the circle at exactly one point, called the point of contact. At that point the tangent is perpendicular to the radius drawn to it.

The distance from an external point to the point of contact on the circle. The two tangent segments drawn from the same external point to a circle are always equal in length.

The single point at which a tangent line meets a circle. The radius drawn to this point is perpendicular to the tangent, a fact used in most circle problems.

The region of a circle bounded by two radii and the arc between them. Its area equals (θ/360) × πr², where θ is the central angle in degrees.

The portion of a circle's circumference cut off by a given central angle. It equals (θ/360) × 2πr, so the full circumference corresponds to an angle of 360°.

The region between a chord and the arc it cuts off. Its area is found by subtracting the area of the triangle formed by the chord and the two radii from the area of the corresponding sector.

The part of a cone or pyramid left after cutting off the top with a plane parallel to the base. Its volume is (1/3)πh(r₁² + r₁r₂ + r₂²) for a conical frustum.

The area of only the curved or lateral part of a solid, excluding flat bases. For a cylinder it is 2πrh and for a cone it is πrl, where l is the slant height.

The amount of three-dimensional space enclosed by a solid, measured in cubic units. Standard results include πr²h for a cylinder and (4/3)πr³ for a sphere.

The average of a grouped frequency distribution, computed by the direct method as Σfᵢxᵢ / Σfᵢ, or by the assumed mean or step-deviation methods when the values are large.

The middle value of a grouped distribution, found using the formula 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 size.

The class interval with the highest frequency in a grouped distribution. The mode is then calculated from this class using the formula l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.

The midpoint of a class interval, obtained by averaging its upper and lower limits. It represents the whole class in the direct-method calculation of the mean.

The running total of frequencies up to and including a given class. It is used to locate the median class and to draw an ogive.

A graph of a cumulative frequency distribution, plotted as cumulative frequency against class boundaries. It may be drawn as a less-than or more-than curve, and the median is read at cumulative frequency n/2.

A number between 0 and 1 giving the likelihood of an event, calculated for equally likely outcomes as the number of favourable outcomes divided by the total number of outcomes.

The event that a given event does not occur, written as E′ or 'not E'. Its probability equals 1 minus the probability of the event, so P(E) + P(E′) = 1.

An event that is certain to happen whenever the experiment is performed, such as getting a number less than 7 when a fair die is rolled. Its probability is 1.

An event that cannot occur in a given experiment, such as drawing a red card from a pack containing only black cards. Its probability is 0.