Maharashtra SSC Mathematics (MSBSHSE) key terms

An equation of the form ax² + bx + c = 0 where a is not zero. Its solutions are found by factorising, completing the square, or using the quadratic formula.

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

The values of the variable that make the equation true. For a quadratic equation there are at most two roots, and their sum equals −b/a and product equals c/a.

The rule x = (−b ± √(b² − 4ac)) / 2a giving the roots of ax² + bx + c = 0. It works for every quadratic equation, whether or not it factorises neatly.

A list of numbers in which each term is obtained by adding a fixed constant to the previous term. The constant is called the common difference.

The fixed number added each time in an arithmetic progression. It is found by subtracting any term from the term that follows it, and may be negative or zero.

The term at position n in an arithmetic progression, given by tₙ = a + (n − 1)d, where a is the first term and d is the common difference.

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

A list of numbers in which each term after the first is obtained by multiplying the previous term by a fixed non-zero number called the common ratio.

The fixed multiplier between consecutive terms of a geometric progression. It is found by dividing any term by the term just before it.

The term at position n in a geometric progression, given by tₙ = a·r^(n−1), where a is the first term and r is the common ratio.

An equation such as ax + by + c = 0 in which each variable appears only to the first power. Its graph is a straight line, and a pair of them may be solved simultaneously.

Two or more equations that must be satisfied by the same values of the variables at once. They are solved by elimination, substitution, or by determinants.

A determinant method for solving simultaneous linear equations. Each variable is found as a ratio of two determinants, with the denominator being the coefficient determinant.

A single number calculated from a square array of numbers. For a 2×2 array with rows (a, b) and (c, d) its value is ad − bc.

A number between 0 and 1 measuring how likely an event is. For equally likely outcomes it equals the number of favourable outcomes divided by the total number of outcomes.

The set of all possible outcomes of a random experiment. For a single die throw it is {1, 2, 3, 4, 5, 6}, and its size is written n(S).

Any subset of the sample space, that is, a collection of outcomes we are interested in. It is written n(A) for the number of outcomes it contains.

The event that a given event A does not happen, written A′ or Ā. Its probability equals 1 minus the probability of A, so P(A) + P(A′) = 1.

Outcomes of an experiment that all have the same chance of occurring, such as the six faces of an unbiased die. The simple probability formula applies only in this situation.

The average of a set of observations, found by adding all the values and dividing by how many there are. For grouped data the class midpoints are used with their frequencies.

The middle value of observations arranged in order. For an even number of observations it is the mean of the two middle values; for grouped data a formula using cumulative frequency locates it.

The value that occurs most often in a set of data. For grouped data it lies in the modal class and is estimated with a formula using class boundaries and frequencies.

A running total of frequencies, obtained by adding each frequency to the sum of all frequencies before it. It is used to locate the median and to draw an ogive.

A shortcut for finding the mean of grouped data in which a convenient value is taken as the assumed mean and deviations from it are averaged, then added back.

The midpoint of a class interval, found by averaging its lower and upper class limits. It represents all observations falling in that class when computing grouped statistics.

A range such as 20–30 into which continuous data is grouped. Its width is the difference between the upper and lower class limits.

A graph of cumulative frequency plotted against the upper class boundaries. It rises steadily and is used to read off the median graphically.

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 test whether an angle is a right angle.

The side of a right-angled triangle opposite the right angle. It is the longest side, and its square equals the sum of the squares of the other two sides.

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

If a line is drawn parallel to one side of a triangle and cuts the other two sides, it divides them in the same ratio. Its converse is also true.

Triangles that are identical in shape and size, so all corresponding sides and angles are equal. Standard tests include SSS, SAS, ASA, AAS and the right-angle hypotenuse-side test.

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.

A line that cuts a circle at two distinct points. The part of the secant inside the circle is a chord of the circle.

The two tangent segments drawn from an external point to a circle are equal in length. This follows from the perpendicularity of each tangent to its radius.

A line segment whose endpoints both lie on a circle. A diameter is the longest chord, and a perpendicular from the centre to a chord bisects it.

A quadrilateral whose four vertices lie on a single circle. Its opposite angles are supplementary, adding to 180 degrees.

An angle formed inside a circle by two chords meeting at a point on the circumference. It equals half the measure of the arc it intercepts.

The six ratios sine, cosine, tangent, cosecant, secant and cotangent, defined from the sides of a right-angled triangle relative to a chosen acute angle.

In a right-angled triangle, the ratio of the side opposite a given acute angle to the hypotenuse. It is written sin θ and always lies between 0 and 1 for acute angles.

In a right-angled triangle, the ratio of the side adjacent to a given acute angle to the hypotenuse. It is written cos θ and equals sin(90° − θ).

In a right-angled triangle, the ratio of the side opposite a given acute angle to the side adjacent to it. It equals sin θ divided by cos θ.

An equation involving trigonometric ratios that is true for every value of the angle. The basic one is sin²θ + cos²θ = 1, from which two others follow.

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 in height-and-distance problems.

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 total area covering the outside of a sphere, given by 4πr² where r is the radius. A hemisphere's curved surface area is half of this.

The space enclosed by a sphere, given by (4/3)πr³ where r is the radius. The volume of a hemisphere is half of this value.

A solid with a circular base and a single curved surface meeting at an apex. Its slant height l, radius r and height h satisfy l² = r² + h².

Interest calculated on the principal plus previously earned interest, so the amount grows each period. The amount after n years is P(1 + R/100)^n.

Interest calculated only on the original principal for the whole period, using I = PRN/100 where P is principal, R the yearly rate and N the time in years.

Goods and Services Tax, an indirect tax added to the price of goods or services. It is split into CGST and SGST for sales within a state and IGST for sales between states.

The GST a trader has already paid on purchases, which can be set off against the GST collected on sales. Only the remaining balance is paid to the government.

The amount on which tax is charged, usually the selling price before tax is added. GST is calculated as a percentage of this value.