Haryana BSEH Secondary Mathematics - Standard 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 computing HCF by repeated division.

A step-by-step method of finding the HCF of two numbers by repeatedly applying the division lemma: divide, then replace the pair by divisor and remainder, until the remainder becomes zero.

The highest common factor of two or more numbers is the largest number that divides each of them exactly. It equals the product of the smallest powers of common prime factors.

The least common multiple of two or more numbers is the smallest number divisible by each of them. It equals the product of the greatest powers of all prime factors involved.

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

Writing a number as a product of its prime factors, for example 360 = 2³ × 3² × 5. It is used to compute HCF, LCM and to test whether a fraction terminates.

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

A number expressible as p/q where p and q are integers and q ≠ 0. Its decimal expansion either terminates or repeats in a fixed block of digits.

A technique that assumes the opposite of the claim, derives an impossible consequence, and concludes the original claim is true. It is standard for proving √2, √3 and √5 irrational.

Two integers whose only positive common divisor is 1, that is, their HCF equals 1. For example, 8 and 15 are coprime even though neither is prime.

A decimal expansion that ends after finitely many digits. A rational number p/q in lowest terms has one exactly when q has no prime factor other than 2 and 5.

A value of the variable at which the polynomial evaluates to zero. Graphically it is the x-coordinate of a point where the curve meets the x-axis.

For a quadratic ax² + bx + c, the sum of zeros is −b/a and the product is c/a. For a cubic, the sum is −b/a, the pairwise sum c/a and the product −d/a.

For polynomials p(x) and g(x) with g(x) ≠ 0, there exist unique polynomials q(x) and r(x) such that p(x) = g(x)q(x) + r(x), where the degree of r(x) is less than that of g(x).

When a polynomial p(x) is divided by the linear polynomial x − a, the remainder equals p(a). Consequently, x − a is a factor exactly when p(a) = 0.

The number of real zeros of a polynomial equals the number of points where its graph cuts the x-axis. A quadratic may meet the axis twice, once, or not at all.

A pair of linear equations having at least one common solution. Graphically the lines either intersect at one point or coincide, giving a unique solution or infinitely many.

A pair of linear equations with no common solution, so their graphs are parallel lines. For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 this happens when a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

A pair of linear equations has exactly one solution when a₁/a₂ ≠ b₁/b₂, meaning the lines intersect at a single point. The solution is found by substitution, elimination or cross-multiplication.

A technique for solving simultaneous equations in which one variable is removed by adding or subtracting suitably multiplied versions of the equations, leaving a single-variable equation.

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.

For the quadratic ax² + bx + c = 0, the quantity D = b² − 4ac. It decides the nature of the roots: two distinct real roots if D > 0, equal roots if D = 0, no real roots if D < 0.

The classification of a quadratic's solutions as two distinct real, two equal real, or no real roots, determined solely by the sign of the discriminant b² − 4ac.

The roots of ax² + bx + c = 0 with a ≠ 0 are x = (−b ± √(b² − 4ac)) / 2a. It gives both roots directly whenever the discriminant is non-negative.

If α and β are the roots of ax² + bx + c = 0, then α + β = −b/a and αβ = c/a. These relations build a quadratic from given roots: x² − (α + β)x + αβ = 0.

For an arithmetic progression with first term a and common difference d, the nth term is aₙ = a + (n − 1)d. It grows linearly with n.

The constant amount added to each term of an arithmetic progression to obtain the next term. It equals any term minus the preceding term, and may be positive, negative or zero.

A sequence in which consecutive terms differ by a fixed constant. Examples include 3, 7, 11, 15 and any list whose successive differences are equal.

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

If a line parallel to one side of a triangle intersects the other two sides at distinct points, it divides those sides in the same ratio. Its converse is also true.

Triangles with equal corresponding angles and proportional corresponding sides. Their areas are in the ratio of the squares of any pair of corresponding sides.

Conditions sufficient to declare two triangles similar: AA (or AAA), SSS with proportional sides, and SAS with two proportional sides enclosing equal angles. Each avoids checking every angle and side.

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

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

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

For vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), the area is ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. A zero result shows the three points are collinear.

Three or more points that lie on a single straight line. In coordinate geometry they are detected when the area of the triangle they form equals zero.

A ratio of two sides of a right triangle defined for an acute angle: sine is opposite over hypotenuse, cosine adjacent over hypotenuse, tangent opposite over adjacent, with cosecant, secant and cotangent as reciprocals.

An equation in trigonometric ratios true for every angle where it is defined. Key examples are sin²θ + cos²θ = 1 and 1 + tan²θ = sec²θ.

The angle measured upward from the horizontal line of sight to an object above the observer. It is used with tangent ratios to find heights of towers and buildings.

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

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

The two tangent segments drawn to a circle from an external point are equal in length, and they subtend equal angles at the centre.

From an external point at distance d from the centre of a circle of radius r, the tangent length is √(d² − r²), obtained from the right triangle formed with the radius.

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

The distance along the curved part of a circle between two points. For central angle θ in degrees it equals (θ/360) × 2πr.

The region between a chord and its arc, found by subtracting the area of the triangle formed by the chord and 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 uses the radii of both circular ends and the height.

The total exposed outer area when two or more solids are joined. Hidden joined faces are excluded, so areas are added selectively rather than simply summed.

The total space occupied by joined solids, found by adding the volumes of the individual solids because volume is conserved when they are combined.

An average of grouped observations computed from class marks and frequencies. It can be found by the direct method, assumed mean method, or step-deviation method.

The middle value of ordered data, found for grouped data by locating the median class and applying median = l + [(n/2 − cf)/f] × h.

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

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

A smooth curve obtained by plotting cumulative frequency against class boundaries, either less-than or more-than type. The two ogives of a distribution intersect at the median.

For an experiment with equally likely outcomes, the probability of an event is the number of favourable outcomes divided by the total number of outcomes. It always lies between 0 and 1.

The event consisting of all outcomes not in a given event E. Its probability is P(not E) = 1 − P(E), so the two probabilities always sum to 1.

The set of all possible outcomes of a random experiment, such as {1, 2, 3, 4, 5, 6} for a die throw. Its size is the denominator in classical probability.