RBSE Secondary Mathematics key terms

For any positive integers a and b, there exist unique whole numbers q and r such that a = bq + r, where 0 ≤ r < b. It is the basis for finding HCF and for the Fundamental Theorem of Arithmetic.

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 test irrationality and to compute HCF and LCM.

The largest positive integer that divides each of two or more given numbers without leaving a remainder. It can be found by prime factorisation or by repeated use of Euclid's division lemma.

The smallest positive integer that is a multiple of each of two or more given numbers. For two positive integers a and b, the product of their HCF and LCM equals a × b.

A real number that cannot be written as a ratio of two integers. Its decimal expansion is non-terminating and non-recurring, for example √2 or π. Sums and products of a rational and an irrational number are usually irrational.

A number expressible in the form p/q where p and q are integers and q ≠ 0. Its decimal expansion either terminates or recurs in a repeating block.

A decimal expansion that stops after a finite number of digits. A rational number in lowest terms has a terminating decimal exactly when its denominator has no prime factors other than 2 and 5.

A decimal expansion that continues without end but repeats a fixed block of digits, such as 0.333… or 0.142857142857…. Every such decimal represents a rational number and can be converted back into p/q form.

The highest power of the variable that appears with a non-zero coefficient in the polynomial. It fixes the maximum number of zeroes the polynomial can have and the shape of its graph.

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

A polynomial of degree two, written ax² + bx + c with a ≠ 0. Its graph is a parabola, and it has at most two zeroes, which may be real and distinct, real and equal, or not real.

For a quadratic polynomial ax² + bx + c, the sum of the zeroes equals −b/a and their product equals c/a. These relations let you build a polynomial from given zeroes or check a factorisation.

If p(x) and g(x) are polynomials with g(x) ≠ 0, then p(x) = g(x)·q(x) + r(x), where the remainder r(x) is either zero or has degree less than that of g(x).

A pair of linear equations that has at least one common solution, so their graphs meet in one point or coincide. Algebraically this happens when the ratios of coefficients satisfy a₁/a₂ ≠ b₁/b₂ or a₁/a₂ = b₁/b₂ = c₁/c₂.

A pair of linear equations with no common solution, whose graphs are parallel lines. The condition is a₁/a₂ = b₁/b₂ ≠ c₁/c₂, and the equations cannot be satisfied by any single pair of values.

Exactly one pair of values that satisfies both equations of a pair simultaneously. Graphically the two lines intersect at a single point, and the condition is a₁/a₂ ≠ b₁/b₂.

Solving a pair of linear equations by drawing both lines on the same axes and reading off the point of intersection. Parallel lines mean no solution; a single intersection means a unique solution; overlapping lines mean infinitely many solutions.

A technique for solving simultaneous equations in which one equation is rearranged to express one variable in terms of the other, and this expression is substituted into the second equation to obtain a single-variable equation.

A technique for solving simultaneous equations in which one equation, or a multiple of it, is added to or subtracted from the other so that one variable cancels, leaving an equation in one variable.

The quantity D = b² − 4ac for the quadratic equation ax² + bx + c = 0. Its sign tells whether the roots are real and distinct, real and equal, or not real.

The rule x = (−b ± √(b² − 4ac)) / 2a giving the roots of ax² + bx + c = 0 when a ≠ 0. It works for every quadratic equation, including those that do not factorise easily.

The classification of the solutions of a quadratic equation according to the discriminant: two distinct real roots when D > 0, two equal real roots when D = 0, and no real roots when D < 0.

The constant amount added to each term to obtain the next term in an arithmetic progression, usually denoted d. It may be positive, negative or zero and equals the difference of any two consecutive terms.

The general term of 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, given by Sₙ = (n/2)[2a + (n − 1)d], or by Sₙ = (n/2)(a + l) when the last term l is known.

Triangles whose corresponding angles are equal and whose corresponding sides are in the same ratio. The ratio of corresponding sides is the scale factor relating the two figures.

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 used to prove that a drawn line is parallel to a side.

Rules that establish similarity without checking every angle and side, namely AA (or AAA), SSS and SAS. Each criterion gives a sufficient condition for two triangles to be similar.

The distance between points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]. It comes from the Pythagoras theorem applied to the horizontal and vertical separations of the points.

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

The midpoint of the segment joining (x₁, y₁) and (x₂, y₂) has coordinates ((x₁ + x₂)/2, (y₁ + y₂)/2). It is the special case of the section formula with ratio 1 : 1.

A ratio of two sides of a right-angled triangle defined with respect to one acute angle, namely sine, cosine, tangent, cosecant, secant or cotangent. Each ratio depends only on the size of the angle, not on the triangle.

An equation involving trigonometric ratios that holds for every value of the angle for which the ratios are defined. The most used example is sin²θ + cos²θ = 1, with its secant and cosecant forms.

The angle between the horizontal line through the observer's eye and the line of sight to an object that lies above that horizontal level. It is used with right-triangle ratios to find heights and distances.

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. It equals the angle of elevation measured from the object to the observer.

A line that touches the circle at exactly one point, called the point of contact. It is perpendicular to the radius drawn to that point, and from an external point two equal tangents can be drawn.

The distance from an external point to the point of contact of a tangent drawn from it to a circle. It equals √(d² − r²), where d is the distance from the point to the centre and r is the radius.

The region enclosed by two radii and the arc between their endpoints. Its area is (θ/360) × πr², and the length of the bounding arc is (θ/360) × 2πr, where θ is the central angle in degrees.

The region cut off by a chord and the arc it subtends. 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 solid left when a cone is cut by a plane parallel to its base and the top portion is removed. Its volume is (1/3)πh(r₁² + r₁r₂ + r₂²), and its curved surface area is π(r₁ + r₂)l.

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

A measure of central tendency for data given in classes, computed by the direct method as Σfᵢxᵢ / Σfᵢ, or by the assumed-mean and step-deviation methods when the values are large.

The middle value of a grouped frequency distribution, found from the cumulative frequency table using the formula l + [(n/2 − cf)/f] × h, where l is the lower limit of the median class.

The value that occurs most often, estimated for grouped data by l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, where f₁ is the frequency of the modal class and h is its class width.

The ratio of the number of outcomes favourable to the event to the total number of equally likely outcomes of the experiment. Its value always lies between 0 and 1 inclusive.

The event that does not occur when a given event occurs, consisting of all outcomes not favourable to it. The probabilities of an event and its complement add up to 1.