Odisha BSE HSC Mathematics key terms

For any two positive integers a and b, there are unique whole numbers q and r with a = bq + r and 0 ≤ r < b. It gives the Euclidean algorithm for finding the HCF of two numbers.

Every composite number can be written as a product of primes, and this prime factorisation is unique apart from the order of the factors. It is the basis for finding HCF and LCM by factorisation.

The greatest positive integer that divides each of two or more given numbers exactly, leaving remainder zero. For example, the HCF of 12 and 18 is 6.

The least positive integer that is a multiple of each of two or more given numbers. For two numbers, the product of the numbers equals HCF × LCM.

A real number that cannot be expressed as a fraction of two integers, so its decimal expansion is neither terminating nor repeating. Examples include √2, √5 and π.

A number that can be written in the form p/q, where p and q are integers and q ≠ 0. Its decimal expansion either terminates or begins to repeat a block of digits.

For a polynomial in one variable, the highest power of the variable whose coefficient is not zero. The degree tells the maximum number of zeroes the polynomial can have.

A value of the variable that makes the polynomial equal to zero. On a graph it is the x-coordinate of a point where the curve cuts the x-axis.

For the quadratic equation ax² + bx + c = 0 with a ≠ 0, the value b² − 4ac. Its sign shows whether the roots are real and distinct, real and equal, or imaginary.

The formula x = (−b ± √(b² − 4ac)) / 2a giving the roots of ax² + bx + c = 0. It applies whenever a ≠ 0 and the discriminant is not negative.

A sequence in which each term is obtained by adding a fixed number to the term just before it. For example, 3, 7, 11, 15, ... is an arithmetic progression.

The fixed number added to each term of an arithmetic progression to produce the next term. It is found by subtracting any term from the term that follows it.

The expression aₙ = a + (n − 1)d, where a is the first term and d is the common difference. It gives any term of an arithmetic progression directly.

For an arithmetic progression, Sₙ = (n/2)[2a + (n − 1)d]. When the last term l is known, it can also be written as Sₙ = (n/2)(a + l).

An irrational number written with a radical sign, such as √7 or 3 + √2, that cannot be reduced to a rational number. Surds are simplified by taking perfect-square factors out of the root.

The expression by which a surd expression is multiplied so that the result has no surd in the denominator, for example √5 for 1/√5 or √a − √b for 1/(√a + √b).

Triangles whose corresponding angles are equal and whose corresponding sides are proportional. They have the same shape, though their sizes may differ.

If a line is drawn parallel to one side of a triangle and cuts the other two sides in distinct points, then it divides those two sides in the same ratio.

Triangles that are identical in both shape and size, so one can be placed exactly on the other. They are tested by the criteria SSS, SAS, ASA, AAS and RHS.

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Its converse is used to check whether a triangle has a right angle.

A line in the plane of a circle that meets the circle at exactly one point, the point of contact. The tangent is perpendicular to the radius at that point.

A quadrilateral all of whose vertices lie on one circle. A key property is that each pair of opposite angles adds up to 180°.

The angle subtended at a point on a circle by a diameter of that circle. It is always a right angle, that is, 90°.

A ratio of two sides of a right-angled triangle taken with respect to one acute angle, namely sine, cosine, tangent, cosecant, secant or cotangent.

An equation involving trigonometric ratios that holds for every angle for which the ratios are defined, such as sin²θ + cos²θ = 1.

The angle between the horizontal line through the observer's eye and the line of sight to an object that lies above the horizontal level.

The angle between the horizontal line through the observer's eye and the line of sight to an object that lies below that horizontal level.

The straight line along which an observer looks at an object. In height and distance problems it forms the hypotenuse of the right triangle used for calculation.

The solid left when the top of a cone or pyramid is cut off by a plane parallel to the base. A bucket is a familiar example of a conical frustum.

The distance measured along the sloping surface of a cone or frustum from the rim of a base to the apex, or between the two circular rims in a frustum.

The region enclosed by two radii of a circle and the arc between their ends. Its area is (θ/360) × πr² when the central angle θ is in degrees.

The region of a circle cut off by a chord, bounded by the chord and the arc it subtends. A minor segment is smaller than a semicircle, a major segment larger.

The average of a frequency distribution, found as Σfᵢxᵢ / Σfᵢ, where xᵢ are class marks. Assumed mean or step deviation methods give the same value with easier arithmetic.

The value that divides a grouped distribution into two equal parts. It is calculated as l + [(n/2 − cf)/f] × h, using the median class.

The value or class with the highest frequency in a distribution. For grouped data it is estimated by l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h from the modal class.

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

For equally likely outcomes, the number of outcomes favourable to the event divided by the total number of outcomes. It lies between 0 and 1.

The event made up of all outcomes of the experiment that are not in a given event E. Its probability equals 1 minus the probability of E.