SSLC Math Formulas in English

Here is the full list of SSLC Math Formulas in English

SETS AND FUNCTIONS

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}.
Venn_Diagram

{ } – Set: a collection of elements

n\left( A \right) – Cardinal number of set A

n0\left( A \right) – Cardinality: the number of elements of set A

\overset { - }{ A }  = { A }^{ c } – Complement of set A or elements not in A

U – Universal Set: set of all possible values

A\cup B – Union: in A or B (or both)

A\cap B – Intersection: in both A and B

A \subset B – Proper Subset: A has some elements of B

A \subseteq B – Subset: A has some (or all) elements of B

A\supset B – Proper Superset: A has B’s elements and more

A\subseteq B Superset: A has same elements as B, or more

\phi – Empty or Null set = {}

a\in A – Element of: a is in A

b\notin A – Not element of: b is not in A

Demorgan’s law

1. Commutative property:

A\cup B=B\cup A
A\cap B=B\cap A

2. Associative property:

A\cup \left( B\cup C \right) =(A\cup B)C
A\cap (B\cap C)=(A\cap B)\cap C

3. Distributive property:

A\cup (B\cap C)=(A\cup B)\cap (A\cup C)

A\cap (B\cup C)=(A\cap B)\cup (A\cap C)

4. De Morgan’s laws:

\overline { A\cup B } =\overline { A } \cap \overline { B }

\overline { A\cap B } =\overline { A } \cup \overline { B }

A-(B\cup C)=(A-B)\cap (A-C)

A-(B\cap C)=(A-B)\cup (A-C)

5. Cardinality of sets:

n(A\cup B)=n(A)+n(B)-n(A\cap B)

n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B\cap C)

6. Representation of functions:

A set of ordered pairs, a table , an arrow diagram, a graph

Types of functions

7.1. One-One function:
Every element in A has an image in B.

7.2 Onto function:
Every element in B has a pre-image in A.

7.3. One-One and onto function:
Both a one-one and an onto function.

7.4. Constant function:
Every element of A has the same image in B.

7.5. Identity function:
An identity function maps each element of A into itself.

SEQUENCES AND SERIES OF REAL NUMBERS

The sum of the terms of a sequence is called a series. While some sequences are simply random values, other sequences have a definite pattern that is used to arrive at the sequence’s terms. Two such sequences are the arithmetic and geometric sequences.
series

Arithmetic Sequence or Arithmetic Progression (A.P.)

1. General form a, a+d, a+2d, a+3d, . . . .

2. Three consecutive terms a‐d, a, a+d

3. The number of terms n=\frac { l-a }{ d } +1 

4. General term tn = a + (n ‐ 1 )d

5. The sum of the first n terms (if the common difference d is given)  Sn = [ 2a + (n ‐ 1)d ]

6. The sum of the first n terms (if the last term l is given) Sn = [ a + l]

Geometric Sequence or Geometric Progression (G.P.)

7. General form:
a, ar, ar2, ar3,  . . .  , arn‐1, arn, . . . .

8. General term:
tn = arn‐1

9. Three consecutive terms:
a/r, a, ar

10. The sum of the first n terms

Special Series
11. The sum of the first n natural numbers:
1 + 2 + 3 + . . . . + n  = \frac { n(n+1) }{ 2 }     

12. The sum of the first n odd natural numbers:
1 + 3 + 5 + . . . . + (2k‐1) = { n }^{ 2 }  

13. The sum of first n odd natural numbers (when the last term l is given):
1 + 3 + 5 + . . . . +   l={ \left( \frac { l+1 }{ 2 }  \right)  }^{ 2 }    

14. The sum of squares of first n natural numbers:
{ 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 }+{ K }^{ 2 } = \frac { n(n+1)(2n+1) }{ 6 }

15. The sum of cubes of the first n natural numbers:
{ 1 }^{ 3 }+{ 2 }^{ 3 }+{ 3 }^{ 3 }+{ K }^{ 3 } = { \left( \frac { n(n+1) }{ 2 }  \right)  }^{ 2 }

ALGEBRA

Algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics.

1.\quad { (a+b) }^{ 2 }={ a }^{ 2 }+2ab+{ b }^{ 2 }

2.\quad { (a-b) }^{ 2 }={ a }^{ 2 }-2ab+{ b }^{ 2 }

3.\quad { a }^{ 2 }-{ b }^{ 2 }=(a+b)(a-b)

4.\quad { a }^{ 2 }+{ b }^{ 2 }={ (a+b) }^{ 2 }-2ab

{ 5.\quad a }^{ 2 }+{ b }^{ 2 }={ (a-b) }^{ 2 }+2ab\\ \\ { 6.\quad a }^{ 3 }+{ b }^{ 3 }=(a+b)({ a }^{ 2 }+ab+{ b }^{ 2 })\\ \\ 7.\quad { a }^{ 3 }-{ b }^{ 3 }=(a-b)({ a }^{ 2 }+ab+{ b }^{ 2 })\\ \\ 8.\quad { a }^{ 3 }+{ b }^{ 3 }={ (a+b) }^{ 3 }-3ab(a+b)\\ \\ { 9.\quad a }^{ 3 }-{ b }^{ 3 }={ (a-b) }^{ 3 }+3ab(a-b)\\ \\ { 10.\quad a }^{ 4 }+{ b }^{ 4 }={ ({ a }^{ 2 }+{ b }^{ 2 }) }^{ 2 }-2{ a }^{ 2 }{ b }^{ 2 }\\ \\ { 11.\quad a }^{ 4 }-{ b }^{ 4 }=(a+b)(a-b)({ a }^{ 2 }+{ b }^{ 2 })\\ \\ 12.\quad { (a+b+c) }^{ 2 }={ a }^{ 2 }+{ b }^{ 2 }+c^{ 2 }+2(ab+bc+ca)\\ \\ 13.\quad (x+a)(x+b)={ x }^{ 2 }+(a+b)x+ab\\ \\ 14.\quad (x+a)(x+b)(x+c)={ x }^{ 3 }+(a+b+c){ x }^{ 2 }+(ab+bc+ca)x+abc

15.\quad Quadratic\quad Polynomials:\quad { ax }^{ 2 }+bx+c=0\\ \\ 16.\quad Sum\quad of\quad Zeros(\alpha +\beta )\quad =\quad -\quad \frac { coefficient\quad of\quad x }{ coeffienct\quad of\quad { x }^{ 2 } } \quad =\quad \left( \frac { -a }{ b }  \right) \\ \\ 17.\quad Product\quad of\quad Zeros(\alpha \beta )\quad =\quad \frac { constant\quad term }{ coeffienct\quad of\quad { x }^{ 2 } } \quad =\quad \left( \frac { c }{ a }  \right) \\ \\ 18.\quad Quadratic\quad Polynomials\quad with\quad Zeros\quad \alpha \quad and\quad \beta \quad :\quad { x }^{ 2 }-(\alpha +\beta )x+(\alpha \beta )

19.\quad Quadratic\quad equation\quad :\quad x=\frac { -b\frac { + }{ - } \sqrt { { b }^{ 2 }-4ac }  }{ 2a }

MATRICES

In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The dimensions of matrix (1) are 2 × 3, because there are two rows and three columns. The individual items in a matrix are called its elements or entries.

1. Row Matrix: A row vector or row matrix is a 1 × m matrix, that is, a matrix consisting of a single row of m elements

x = \left\lceil x1\quad x2\quad x3\quad ...\quad xm \right\rceil

2. Column Matrix: A column vector or column matrix is an m × 1 matrix, that is, a matrix consisting of a single column of m elements

x\quad =\quad \left( \begin{matrix} x1 \\ x2 \\ x3 \\ . \\ . \\ . \\ xm \end{matrix} \right)

3. Square Matrix: A matrix in which the number of rows and the number of columns are equal

I3\quad =\quad \left[ \begin{matrix} 1 & 3 & 5 \\ 7 & 9 & 11 \\ 13 & 15 & 17 \end{matrix} \right]

4. Diagonal Matrix : A square matrix in which all the elements above and below the leading diagonal are equal to zero

a\quad =\quad \left[ \begin{matrix} a11 & 0 & 0 \\ 0 & a22 & 0 \\ 0 & 0 & a33 \end{matrix} \right]

5. Scalar Matrix : A diagonal matrix in which all the elements along the leading diagonal are equal to a non-zero constant

\quad \left[ \begin{matrix} \lambda  & 0 & 0 \\ 0 & \lambda  & 0 \\ 0 & 0 & \lambda  \end{matrix} \right] =\lambda I3

6. Unit Matrix : A diagonal matrix in which all the leading diagonal entries are 1

{ I }_{ 1 }\quad =\quad \left[ 1 \right]

{ I }_{ 2 }\quad =\quad \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

{ I }_{ 3 }\quad =\quad \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right]

7. Null Matrix or Zero Matrix : A matrices has each of its elements is zero.

{ 0 }_{ 1,1 }\quad =\quad \left[ 0 \right]

{ 0 }_{ 2,2 }\quad =\quad \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

{ 0 }_{ 2,3 }\quad =\quad \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \right]

8. Transpose of a Matrix :A matrices has interchanging rows and columns of the matrix. The transpose of a matrix is a new matrix whose rows are the columns of the original. (This makes the columns of the new matrix the rows of the original). Here is a matrix and its transpose:

\\ \\ { \left[ \begin{matrix} 5 & 4 & 3 \\ 4 & 0 & 4 \\ 7 & 10 & 3 \end{matrix} \right]  }^{ T }\quad =\quad { \left[ \begin{matrix} 5 & 4 & 7 \\ 4 & 0 & 10 \\ 3 & 4 & 3 \end{matrix} \right]  }\\ \\

The superscript “T” means “transpose”.

9. Negative of a Matrix :The negation of a matrix is formed by negating each element of the matrix.

The negative of a matrix A = -1A:

-1\begin{pmatrix} 1 & 2 \\ 5 & 8 \end{pmatrix}\quad =\quad \begin{pmatrix} -1 & -2 \\ -5 & -8 \end{pmatrix}

It will not surprise you that:
A + (-A) = 0

10. Equality of Matrices : For two matrices to be equal, they must be of the same size and have all the same entries in the same places. For instance, suppose you have the following two matrices. For example:

A\quad =\quad \begin{pmatrix} 1 & 3 \\ -2 & 0 \end{pmatrix}\quad B\quad =\quad \begin{pmatrix} 1 & 3\quad 0 \\ -2 & 0\quad 0 \end{pmatrix}

Following matrices cannot be the same, since they are not the same size. Even if A and B are the following two matrices:

11. Two matrices of the same order, then the addition of A and B is a matrix C

12. If A is a matrix of order m x n and B is a matrix of order n x p, then the product matrix AB is m x p.

13. Properties of Matrix Addition:

Commutative: A +B = B + A

Associative: A + (B + C) = (A + B) +C

Existence of Additive Identity: A + O = O + A =A

Existence of Additive Inverse: A + (‐A) = (‐A) + A = O

14. Properties of Matrix Multiplication:

Not commutative in general: A B =  BA

Associative: A(BC) = (AB)C

Distributive over addition:
A(B + C) = AB + AC

(A + B)C = AC + BC

Existence of multiplicative identity A I = I A  = A

Existence of multiplicative inverse AB = BA = I

COORDINATE GEOMETRY

(coming soon)

GEOMETRY

Angle & Triangle

1. Basic Proportionality theorem or Thales Theorem: If a straight line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.

Basic Proportionality Theorem

For example in the above triangle ABC, a line drawn parallel to BC cuts AB and AC at P and Q respectively.
AD/DB=AF/FC

2. Converse of Basic Proportionality Theorem ( Converse of Thales Theorem):
If a straight line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.
Converse of Basic Proportionality Theorem

3. Angle Bisector Theorem : The internal (external) bisector of an angle of a triangle divides the opposite side internally (externally) in the ratio of the corresponding sides containing the angle.
Angle_Bisector_Theorem

4. Converse of Angle Bisector Theorem : If a straight line through one vertex of a triangle divides the opposite side internally (externally) in the ratio of the other two sides, then the line bisects the angle internally (externally) at the vertex.

4.1 Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

4.2 Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment.

5. Similar Triangles : Corresponding angles are equal (or) corresponding sides have lengths in the same ratio
5.1. Angle-Angle (AA) similarity criterion: If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

SImilar_Traingles_AA

If\measuredangle A\quad \cong \quad \measuredangle D\quad and\quad \measuredangle B\quad \cong \quad \measuredangle E

\\ Then\quad \triangle ABC\sim \triangle DEF

5.2. Side-Side-Side (SSS) similarity criterion for Two Triangles: In two triangles, if the sides of one triangle are proportional (in the same ratio) to the sides of the other triangle, then their corresponding angles are equal

Similar_Traingles_SSS

If\frac { AB }{ DE } =\frac { AC }{ DF } =\frac { BC }{ EF }

\\ Then\quad \triangle ABC\sim \triangle DEF

5.3. Side-Angle-Side (SAS) similarity criterion for Two Triangles: If one angle of a triangle is equal to one angle of the other triangle and if the corresponding sides including these angles are proportional, then the two triangles are similar.

Similar_Traingles_SAS

If\quad \measuredangle A\quad \cong \quad \measuredangle \quad D\quad and\quad if\quad \frac { AB }{ DE } =\frac { AC }{ DF\\  }

Then\quad \triangle ABC\quad \sim \quad \triangle DEF

6. Pythagoras theorem : In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the “Pythagoras or Pythagorean Equation”:

Pythagoras_Theorem

{ a }^{ 2 }+{ b }^{ 2 }={ c }^{ 2 } , where c represents the length of the hypotenuse and a and b the lengths of the triangle’s other two sides.

\\ c=\sqrt { { a }^{ 2 }+{ b }^{ 2 } }

7. Converse of Pythagorus Theorem: In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the first side is a right angle.

8. Tangent-Chord Theorem: If from the point of contact of tangent (of a circle), a chord is drawn, then the angles which the chord makes with the tangent line are equal respectively to the angles formed by the chord in the corresponding alternate segments.

9. Converse of Theorem: If in a circle, through one end of a chord, a straight line is drawn making an angle equal to the angle in the alternate segment, then the straight line is a tangent to the circle.

10. If two chords of a circle intersect either inside or out side the circle, the area of the rectangle contained by the segments of the chord is equal to the area of the rectangle contained by the segments of the other

Circles and Tangents

11. A tangent at any point on a circle is perpendicular to the radius through the point of contact.

Circle_Tangent-Perpendicular

Consider a circle with center O and let B be a point on the circle. Suppose A is a tangent line to the circle at B, that is A meets the circle only at B. Show that \bar { OB } \\ is perpendicular to A.

12. Only one tangent can be drawn at any point on a circle. However, from an exterior point of a circle two tangents can be drawn to the circle. It is observed that if two segments from the same exterior point are tangent to the circle, then they are congruent. If AP and BP are tangent to a Circle O then AP = BP.

Tangents_To_Circle

13. The lengths of the two tangents drawn from an exterior point to a circle are equal.

14. If two circles touch each other, then the point of contact of the circles lies on the line joining the centres.

15. If two circles touch externally, the distance between their centres is equal to the sum of their radii.

16. If two circles touch internally, the distance between their centres is equal to the difference of their radii.

TRIGNOMETRY

Trigonometry (from Greek trigōnon, “triangle” and metron, “measure”) is a branch of mathematics that studies relationships involving lengths and angles of triangles.
Traingle and Trignometry

Trigonometric Table

Trignometry-SSLC
* ND=Not Defined or Infinity

1. sin\theta \quad cosec\theta =1 ; sin\theta =\frac { 1 }{ cosec\theta  } ; cosec\theta =\frac { 1 }{ sin\theta  }

2. cos\theta \quad sec\theta =1 ; cos\theta =\frac { 1 }{ sec\theta  } ; sec\theta =\frac { 1 }{ cos\theta  }

3. tan\theta \quad cot\theta =1 ; tan\theta =\frac { 1 }{ cot\theta  } ; cot\theta =\frac { 1 }{ tan\theta  }

4. tan\theta =\frac { sin\theta  }{ cos\theta  }

5. cot\theta =\frac { cos\theta  }{ sin\theta  }

6. { sin }^{ 2 }\theta +{ cos }^{ 2 }\theta =1

7. { sec }^{ 2 }\theta -{ tan }^{ 2 }\theta =1

8. co{ sec }^{ 2 }\theta -{ cot }^{ 2 }\theta =1  

9. sin(90-\theta )=cos\theta \quad \quad cosec(90-\theta )=sec\theta

10. cos(90-\theta )=sin\theta \quad \quad sec(90-\theta )=cosec\theta

11. tan(90-\theta )=cot\theta \quad \quad cot(90-\theta )=tan\theta

MENSURATION

The measuring of geometric magnitudes, lengths, areas, and volumes. Here is the table of Mensuration Formula of various shapes and objects.

Table of mensuration formula

STATISTICS

(Coming Soon)

PROBABILITY

(Coming Soon)

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