A rational number \(\frac{p}{q}\) (in lowest form) results in:

• Terminating decimal: If prime factorization of q has only 2 or 5 or both as factors
• Non-terminating recurring decimal: Otherwise

Properties:

• Sum of a rational and an irrational number is irrational
• Product of a non-zero rational and an irrational number is irrational
• \(\sqrt{n}\) is irrational if n is not a perfect square

A polynomial in variable x is an expression of the form:

where \(a_n, a_{n-1}, …, a_1, a_0\) are constants and \(a_n \neq 0\)

Degree of the polynomial = \(n\)

If \(P(a) = 0\), then a is a zero of the polynomial P(x)

For a polynomial P(x) of degree n:

• P(x) can have at most n zeroes
• If a is a zero, then (x – a) is a factor of P(x)

For a quadratic polynomial \(ax^2 + bx + c\) with zeroes α and β:

For a cubic polynomial \(ax^3 + bx^2 + cx + d\) with zeroes α, β, and γ:

If P(x) and G(x) are two polynomials with G(x) ≠ 0, then there exist polynomials Q(x) and R(x) such that:

where R(x) = 0 or degree of R(x) < degree of G(x)

If a polynomial P(x) is divided by (x – a), then the remainder is P(a)

(x – a) is a factor of polynomial P(x) if and only if P(a) = 0

A pair of linear equations in two variables:

Consider the ratio \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)

• If \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\): Unique solution (Consistent and Independent)
• If \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\): No solution (Inconsistent)
• If \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\): Infinitely many solutions (Consistent and Dependent)

Methods to solve a pair of linear equations:

1. Substitution Method
2. Elimination Method
3. Cross-Multiplication Method
4. Graphical Method

For equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\):

For a pair of linear equations:

• Unique solution: Lines intersect at exactly one point
• No solution: Lines are parallel
• Infinitely many solutions: Lines coincide

A quadratic equation in variable x is an equation of the form:

The solutions of \(ax^2 + bx + c = 0\) are given by:

The discriminant is \(D = b^2 – 4ac\)

• If \(D > 0\): Two distinct real roots
• If \(D = 0\): Two equal real roots (repeated root)
• If \(D < 0\): No real roots (two complex roots)

If α and β are the roots of \(ax^2 + bx + c = 0\), then:

If α and β are the roots, the quadratic equation is:

For the equation \(ax^2 + bx + c = 0\):

• For real and equal roots: \(b^2 – 4ac = 0\)
• For real and unequal roots: \(b^2 – 4ac > 0\)
• For imaginary roots: \(b^2 – 4ac < 0\)

An Arithmetic Progression (AP) is a sequence in which:

• Each term differs from the preceding term by a constant (called common difference)
• If a is the first term and d is the common difference, the AP is: a, a+d, a+2d, a+3d, …

The general (nth) term of an AP is:

where a is the first term and d is the common difference

The sum of the first n terms of an AP is:

where l is the last (nth) term = a + (n-1)d

Two triangles are similar if:

• Their corresponding angles are equal
• Their corresponding sides are in the same ratio

If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

If DE || BC in triangle ABC, then:

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

If \(\frac{AD}{DB} = \frac{AE}{EC}\) in triangle ABC, then DE || BC.

1. AAA Criterion: If two angles of one triangle are respectively equal to two angles of another triangle, then the triangles are similar.
2. SSS Similarity: If the corresponding sides of two triangles are proportional, then the triangles are similar.
3. SAS Similarity: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the triangles are similar.

If two triangles are similar with the scale factor k, then:

In a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.

where c is the hypotenuse and a, b are the legs of the right triangle

Distance between points (x₁, y₁) and (x₂, y₂):

Coordinates of the point P(x, y) which divides the line segment joining the points A(x₁, y₁) and B(x₂, y₂) internally in the ratio m:n are:

Coordinates of the midpoint of the line segment joining the points (x₁, y₁) and (x₂, y₂):

Area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃):

Or using the determinant:

Coordinates of the centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃):

Three points (x₁, y₁), (x₂, y₂), and (x₃, y₃) are collinear if:

Or:

For an angle θ in a right-angled triangle:

For angles A and B:

Angle of Elevation: Angle made by the line of sight with the horizontal when the object is above the level of the eye.

Angle of Depression: Angle made by the line of sight with the horizontal when the object is below the level of the eye.

If h is the height of an object, d is the distance from the observer to the object, and θ is the angle of elevation:

If h is the height of the observer, H is the height of the object, d is the distance between them, and θ is the angle of elevation:

If h is the height of an object and θ is the angle of elevation from a point at distance d:

Properties of a tangent to a circle:

• A tangent to a circle is perpendicular to the radius at the point of contact.
• The lengths of the two tangents drawn from an external point to a circle are equal.

• From a point outside a circle: Two tangents can be drawn
• From a point on the circle: Only one tangent can be drawn
• From a point inside the circle: No tangent can be drawn

If from an external point P, a tangent PT and a secant PAB are drawn to a circle with center O, then:

where T is the point of contact of the tangent with the circle, and A and B are the points where the secant intersects the circle.

The length of the tangent from an external point P to a circle with center O and radius r is:

where PT is the length of the tangent and OP is the distance from the center to the external point

To divide a line segment in a given ratio m:n:

1. Draw a ray AX making an acute angle with AB
2. Locate m+n points on AX
3. Join the last point to B
4. Draw lines parallel to this line through the other points

To construct a tangent to a circle from an external point P:

1. Join OP, where O is the center of the circle
2. Draw the perpendicular bisector of OP
3. From P, draw a line perpendicular to the radius at the point of intersection with the circle

To construct a triangle similar to a given triangle with a given scale factor:

1. Draw a ray BX from vertex B of triangle ABC
2. Locate points B₁, B₂, …, Bₙ on BX such that BB₁ = B₁B₂ = … = Bₙ₋₁Bₙ
3. Join Bₖ to C (for required scale factor)
4. Draw a line through A parallel to BₖC
5. The line intersects BX at A’
6. Draw a line through A’ parallel to AC
7. This line intersects BC at C’
8. Triangle A’BC’ is similar to triangle ABC with the required scale factor

For a circle with radius r:

For a sector with angle θ (in radians) and radius r:

For a segment with angle θ (in radians) and radius r:

For a shaded region formed by combinations of geometric figures, use:

For a cube with side length a:

For a cuboid with dimensions l, b, and h:

For a cylinder with radius r and height h:

For a cone with radius r, height h, and slant height l \((l = \sqrt{r^2 + h^2})\):

For a sphere with radius r:

For a hemisphere with radius r:

For a frustum of a cone with radii r₁ and r₂ (r₁ > r₂), height h, and slant height l:

For combined solids:

For n observations x₁, x₂, …, xₙ:

For grouped data with midpoints x₁, x₂, …, xₙ and frequencies f₁, f₂, …, fₙ:

Using assumed mean a:

where d₁ = x₁ – a, d₂ = x₂ – a, etc.

Using assumed mean a and common factor h:

where u₁ = (x₁ – a)/h, u₂ = (x₂ – a)/h, etc.

For n observations arranged in ascending order:

For grouped data:

where:

• l = lower limit of median class
• n = sum of frequencies
• cf = cumulative frequency of class preceding the median class
• f = frequency of median class
• h = class width

Mode is the value that occurs most frequently in the data set.

For grouped data:

where:

• l = lower limit of modal class
• f₁ = frequency of modal class
• f₀ = frequency of class preceding the modal class
• f₂ = frequency of class succeeding the modal class
• h = class width

or

The median from an ogive (cumulative frequency curve) can be found by:

1. Locate the point on the y-axis corresponding to N/2 (where N is the total frequency)
2. Draw a horizontal line from this point to the ogive curve
3. From the point of intersection, draw a vertical line to the x-axis
4. The point where this vertical line meets the x-axis gives the median

Probability of an event E:

where S is the sample space (set of all possible outcomes)

• 0 ≤ P(E) ≤ 1 for any event E
• P(S) = 1 where S is the sample space
• P(∅) = 0 where ∅ is the empty set
• P(E’) = 1 – P(E) where E’ is the complement of E

For an event E, the probability that E does not occur is:

For events A and B:

If A and B are mutually exclusive (A ∩ B = ∅), then:

For a standard deck of 52 cards:

• P(drawing a spade) = 13/52 = 1/4
• P(drawing a face card) = 12/52 = 3/13
• P(drawing a red card) = 26/52 = 1/2
• P(drawing an ace) = 4/52 = 1/13

For a single fair die:

• P(rolling a specific number) = 1/6
• P(rolling an even number) = 3/6 = 1/2
• P(rolling a number greater than 4) = 2/6 = 1/3

For two dice:

• Total number of possible outcomes = 36
• P(sum equals 7) = 6/36 = 1/6
• P(sum equals 2) = 1/36