RBSE Praveshika Mathematics key terms

For any positive integers a and b, there exist unique whole numbers q and r with a = bq + r and 0 ≤ r < b. It underlies the algorithm for finding highest common factors.

Every integer greater than 1 can be written as a product of primes in exactly one way apart from the order of the factors. This uniqueness justifies prime factorisation methods.

The highest common factor of two or more numbers is the largest number dividing each of them exactly. It equals the product of the lowest powers of common primes.

The lowest common multiple of two or more numbers is the smallest number divisible by each of them. For two numbers, HCF × LCM equals their product.

Writing a number as a product of prime numbers only, for example 84 = 2² × 3 × 7. It is the standard tool for computing HCF, LCM and testing perfect squares.

A real number that cannot be expressed as a ratio of two integers. Its decimal expansion is non-terminating and non-recurring, as with √2, √3 and π.

A number that can be written as p/q where p and q are integers and q is not zero. Its decimal expansion either terminates or recurs in a repeating block.

Any number representable by a point on the number line, including all rational and irrational numbers. The set is closed under addition and multiplication but not under square roots of negatives.

The highest power of the variable appearing with a non-zero coefficient in a polynomial. It fixes the maximum number of roots and the shape of the graph.

A value of the variable at which the polynomial evaluates to zero. For a quadratic ax² + bx + c the zeros have sum −b/a and product c/a.

A polynomial p(x) has (x − a) as a factor exactly when p(a) = 0. It converts factor questions into simple substitution checks.

When a polynomial p(x) is divided by (x − a), the remainder equals p(a). This avoids carrying out long division to find a remainder.

An equation true for all values of its variables, such as (a + b)² = a² + 2ab + b². Identities are used to expand, factorise and simplify expressions quickly.

A method for quadratics ax² + bx + c in which b is split into two numbers whose product is ac and whose sum is b, allowing grouping into two brackets.

An equation of the form ax + by + c = 0 with a and b not both zero. Its graph is a straight line and its solutions are ordered pairs (x, y).

A pair of linear equations that has at least one common solution. The lines either intersect at one point or coincide completely.

A technique for solving simultaneous equations in which one variable is expressed from one equation and inserted into the other, leaving a single-variable equation.

A technique for solving simultaneous equations in which one equation is multiplied so that a variable has equal coefficients, then the equations are added or subtracted to remove it.

The plane described by two perpendicular number lines, the x-axis and y-axis, meeting at the origin. Every point is located by an ordered pair of coordinates.

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

The point exactly halfway between (x₁, y₁) and (x₂, y₂) has coordinates ((x₁ + x₂)/2, (y₁ + y₂)/2). It is the average of the two sets of coordinates.

The point dividing the join of (x₁, y₁) and (x₂, y₂) internally in ratio m : n has coordinates ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

Triangles with exactly equal corresponding sides and angles. Standard tests are SSS, SAS, ASA, AAS and RHS; congruence preserves all lengths and angle measures.

Triangles whose corresponding angles are equal and whose corresponding sides are in the same ratio. Their areas are in the ratio of the squares of corresponding sides.

A line drawn parallel to one side of a triangle divides the other two sides in the same ratio. Its converse is used to prove that two lines are parallel.

In a right-angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides. Its converse tests whether a triangle is right-angled.

A line meeting a circle at exactly one point. The tangent is perpendicular to the radius drawn to the point of contact.

Any angle subtended at the circumference by a diameter is a right angle. This follows from the angle at the centre being twice the angle at the circumference.

A quadrilateral whose four vertices lie on a single circle. Each pair of opposite angles adds to 180 degrees.

The region enclosed by two radii and an arc. Its area is (θ/360) × πr² when θ is the central angle in degrees.

The portion of the circumference cut off by a central angle θ, measured as (θ/360) × 2πr. It is the curved boundary of the corresponding sector.

The space enclosed by a circular cylinder, equal to πr²h where r is the base radius and h the height. Volume is measured in cubic units.

One third of the volume of a cylinder with the same base and height, so V = (1/3)πr²h. The slant height relates to r and h by l² = r² + h².

The total outer area of a sphere, equal to 4πr². The curved surface of a hemisphere is 2πr² and its total surface area is 3πr².

A ratio of two sides of a right-angled triangle, such as sin, cos or tan of an acute angle. Values depend only on the angle, not on the triangle's size.

An equation involving trigonometric ratios that holds for every angle where it is defined, for example sin²θ + cos²θ = 1. Identities are used to simplify and prove expressions.

The angle measured upward from the horizontal to the line of sight when an observer looks at an object above eye level. It is used to find heights and distances.

The angle measured downward from the horizontal to the line of sight when an observer looks at an object below eye level. It equals the angle of elevation from the object.

The average of a grouped frequency distribution, computed as Σfᵢxᵢ / Σfᵢ using class marks, or by the assumed-mean or step-deviation shortcut.

The middle value of a distribution, found from the cumulative frequency table using l + [(N/2 − cf)/f] × h. It divides the data into two equal halves.

The midpoint of a class interval, calculated as the average of its lower and upper limits. It represents the whole class in mean calculations.

The ratio of the number of favourable equally likely outcomes to the total number of outcomes. It always lies between 0 and 1 inclusive.

The event consisting of all outcomes in which a given event does not occur. Its probability equals 1 minus the probability of the original event.

An irrational root of a rational number, written with a radical sign such as √5 or 2 + √3. Surds can often be simplified or rationalised.

Removing a surd from the bottom of a fraction by multiplying numerator and denominator by a suitable conjugate or surd. For 1/√a the multiplier is √a.

A rule for combining powers, such as aᵐ × aⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ. These laws allow expressions to be simplified and equations solved.

Interest calculated on the principal plus previously earned interest, giving A = P(1 + r/100)ⁿ. It grows faster than simple interest over the same period.

Interest computed only on the original principal, given by SI = P × r × t / 100. It does not grow on interest already earned.

Goods and Services Tax, an indirect tax charged as a percentage on the supply of goods or services. It is usually added to the marked price at the point of sale.

A reduction given on the marked price of an article, usually stated as a percentage. The selling price equals the marked price minus the discount.