Bihar BSEB Matric 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 underlies the Euclidean algorithm for finding HCF.

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 factors to compute HCF and LCM.

The highest common factor of two or more numbers is the largest number that divides each of them exactly. Using prime factors, take the smallest power of every common prime.

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

A real number that cannot be written as a fraction of two integers. Its decimal expansion is non-terminating and non-repeating, for example the square root of 2 or pi.

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

A decimal expansion that stops after finitely many digits. A rational number in lowest terms has one exactly when its denominator has no prime factors other than 2 and 5.

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

A value of the variable at which the polynomial evaluates to zero. Geometrically these are the points where the graph cuts or touches the x-axis.

A polynomial of degree two, written ax² + bx + c with a not zero. Its graph is a parabola and it has at most two real zeroes.

For ax² + bx + c with a not zero, the zeroes α and β satisfy α + β = −b/a and αβ = c/a. These relations let you build a polynomial from its zeroes.

For polynomials p(x) and g(x) with g(x) not zero, there exist unique polynomials q(x) and r(x) with p(x) = g(x)q(x) + r(x), where the degree of r(x) is less than that of g(x).

A pair of linear equations that has at least one solution. The two lines either intersect at exactly one point or coincide, giving one or infinitely many solutions.

A pair of linear equations with no common solution, so the two lines are parallel and never meet. Algebraically the ratios a₁/a₂ and b₁/b₂ are equal while c₁/c₂ is not.

For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the pair has exactly one solution when a₁/a₂ ≠ b₁/b₂. The lines then intersect at a single point.

Solving a pair of linear equations by drawing both lines on the same axes and reading off the intersection point, or noting that the lines are parallel or coincident.

A technique for solving simultaneous equations in which one variable is removed by multiplying the equations by suitable numbers and then adding or subtracting them.

A technique in which one equation is rearranged to express a variable in terms of the other, and this expression is substituted into the second equation to get one equation in one variable.

For ax² + bx + c = 0 with a not zero, the quantity b² − 4ac. Its sign decides whether the roots are real and distinct, real and equal, or not real.

The classification of the solutions of a quadratic equation using the discriminant: positive gives two distinct real roots, zero gives two equal real roots, negative gives no real roots.

The roots of ax² + bx + c = 0 with a not zero are x = (−b ± √(b² − 4ac)) / 2a. It solves every quadratic equation once the discriminant is known to be non-negative.

Rewriting a quadratic expression as a perfect square plus a constant, for example x² + 6x + 1 = (x + 3)² − 8. It derives the quadratic formula and locates the vertex.

In an arithmetic progression, the fixed number obtained by subtracting any term from the term that follows it. It may be positive, negative or zero.

For an arithmetic progression with first term a and common difference d, the nth term is aₙ = a + (n − 1)d. It lets you find any term without listing the earlier ones.

The total of the first n terms of an arithmetic progression, given by Sₙ = (n/2)[2a + (n − 1)d] or Sₙ = (n/2)(a + l) where l is the last term.

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

If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those two sides in the same ratio. Its converse is also used in proofs.

Tests that prove two triangles are similar: AA (or AAA), SSS with proportional sides, and SAS with two proportional sides and the included angle equal. AA is the most frequently used.

In a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides. Its converse identifies a right angle from side lengths.

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

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

In a right triangle, ratios of pairs of sides: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent, with cosecant, secant and cotangent as their reciprocals.

Equations true for every angle in their domain, such as sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ. They are used to simplify and prove expressions.

Two angles adding to 90°. For such angles, sin(90° − θ) = cos θ, tan(90° − θ) = cot θ and sec(90° − θ) = cosec θ, which converts expressions into a single ratio.

The angle between the horizontal line of sight and the line of sight when an observer looks upward at an object. It is used with heights and distances problems.

The angle between the horizontal line of sight and the line of sight when an observer looks downward at an object. It equals the angle of elevation from the object to the observer.

A line that touches a 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 a circle. Tangents drawn from the same external point are equal in length.

A line that cuts a circle at two distinct points. It differs from a tangent, which meets the circle at only one point.

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

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

The region enclosed by a circle, equal to πr² where r is the radius. The circumference, the distance around the circle, is 2πr.

The portion of a cone or pyramid left when its top is cut off by 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 faces. For a cylinder it is 2πrh and for a cone it is πrl, where l is the slant height.

The sum of the curved surface area and the areas of all flat faces of a solid. For a closed cylinder it is 2πr(r + h).

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

The average of observations grouped into classes, computed by the direct method, assumed mean method, or step-deviation method using class marks and frequencies. It estimates the central value of the data.

The middle value of an ordered data set, found for grouped data by locating the median class and applying the formula involving cumulative frequency, class limits and class size.

The value or class that occurs most frequently. For grouped data it is computed from the modal class using the frequencies of the modal class and its neighbours.

The ratio of the number of favourable outcomes to the total number of equally likely outcomes. It always lies between 0 and 1, with 0 for an impossible event.

The event that a given event does not occur. Its probability equals 1 minus the probability of the original event, so the two probabilities add to 1.

The set of all possible outcomes of a random experiment. For a single die it is {1, 2, 3, 4, 5, 6}, and its size gives the denominator in classical probability.