Mahesh

Students can go through AP Board 8th Class Maths Notes Chapter 8 Exploring Geometrical Figures to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 8 Exploring Geometrical Figures

→ Shapes are said to be congruent if they have same shape and size.

→ Shapes are said to be similar if they have same shapes but in different size.

→ If we flip, slide or turn the congruent/similar shapes their congruence/similarity remain the same.

→ Some figures may have more than one line of symmetry.

→ Symmetry is of three types namely line symmetry, rotational symmetry and point symmetry.

→ With rotational symmetry, the figure is rotated around a central point so that it appears two or more times same as original.

→ The number of times for which it appears the same is called the order.

→ The method of drawing enlarged or reduced similar figures is called Dialation.

→ The patterns formed by repeating figures to fill a plane without gaps or overlaps are called tessellations.

→ Flip: Flip is a transformation in which a plane figure is reflected across a line, creating a mirror image of the original figure.

→ After a figure is flipped or reflected, the distance between the line of reflection and each point on the original figure is the same as the distance between the line of reflection and the corresponding point on the mirror image.

→ Rotation: “Rotation “means turning around a center.
The distance from the center to any point on the shape stays the same. Every point makes a circle around the center.

There is a central point that stays fixed and everything else moves around that point in a circle.
A “Full Rotation” is 360°.

→ Now observe the following geometrical figures.

In all the cases if the first figure in the row is moved, rotated and flipped do you find any change in size and shape? No, the figures in every row are congruent they represent the same figure but oriented differently.

→ If two shapes are congruent, still they remain congruent if they are moved or rotated. The shapes would also remain congruent if we reflect the shapes by producing their mirror images.

→ We use the symbol ≅ to represent congruency.

Students can go through AP Board 8th Class Maths Notes Chapter 7 Frequency Distribution Tables and Graphs to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 7 Frequency Distribution Tables and Graphs

→ The Central Tendencies are 3 types. They are

1. Arithmetic Mean
2. Median
3. Mode

→ Information, available in the numerical form or verbal form or graphical form that helps in taking decisions or drawing conclusions is called Data.

→ Arithmetic mean of the ungrouped data = (short representation) where ∑x_i represents the sum of all x_is, where ‘i’ takes the values from 1 to n.

→ Arithmetic mean = Estimated mean + Average of deviations

→ Mean is used in the analysis of numerical data represented by unique value.

→ Median represents the middle value of the distribution arranged in order.

→ The median is used to analyse the numerical data, particularly useful when there are a few observations that are unlike mean, it is not affected by extreme values.

→ Mode is used to analyse both numerical and verbal data.

→ Mode is the most frequent observation of the given data. There may be more than one mode for the given data.

→ Representation of classified distinct observations of the data with frequencies is called ‘Frequency Distribution’ or ‘Distribution Table’.

→ Difference between upper and lower boundaries of a class is called length of the class denoted by ‘C’.

→ In a class the initial value and end value of each class is called the lower limit and upper limit respectively of that class.

→ The average of upper limit of a class and lower limit of successive class is called upper boundary of that class.

→ The average of the lower limit of a class and upper limit of preceding class is called the lower boundary of the class.
The progressive total of frequencies from the last class of the table to the lower boundary of particular class is called Greater than Cumulative Frequency (G.C.F).

→ The progressive total of frequencies from first class to the upper boundary of particular class is called Less than Cumulative Frequency (L.C.F.).

→ Histogram is a graphical representation of frequency distribution of exclusive class intervals. When the class intervals in a grouped frequency distribution are varying we need to construct rectangles in histogram on the basis of frequency density.
Frequency density = \(\frac{\text { Frequency of class }}{\text { Length of that class }}\) × Least class length in the data

→ Frequency polygon is a graphical representation of a frequency distribution (discrete/ continuous).

→ Infrequency polygon or frequency curve, class marks or mid values of the classes are taken on X-axis and the corresponding frequencies on the Y-axis.

→ Area of frequency polygon and histogram drawn for the same data are equal.

→ A graph representing the cumulative frequencies of a grouped frequency distribution against the corresponding lower/upper boundaries of respective class intervals is called Cumulative Frequency Curve or “Ogive Curve”.

Students can go through AP Board 8th Class Maths Notes Chapter 6 Square Roots and Cube Roots to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 6 Square Roots and Cube Roots

→ The product of two same numbers is called its square.
Ex: 1) x × x = x²
2) 6 × 6 = 6² = 36

→ The digits in the units place of a square number are 0, 1, 4, 5, 6, 9.

→ If the digits 2,3, 7 or 8 are in the units place of an umber then it is not a perfect square.

→ If there are ‘n’ digits in a number then the no.of digits in its square = 2n or (2n -1).

→ Sum of ‘n ‘ consecutive odd numbers = n²

→ The square of any odd number say ‘n’ can be expressed as the sum of two consecutive numbers as

→ If a, b, c are any three positive integers and a² + b² = c² then a, b, c are called Pythagorean triplets. Ex: (3, 4, 5) (5, 12, 13).

→ If a square number is expressed, as the product of two equal factors, then one of the factors is called the square root of that square number. Thus, the square root of 169 is 13. It can be expressed as √169 = 13 (symbol used for square root is √). Thus it is the inverse operation of squaring.

→ If the same number is multiplied itself by 3 times then it is called a cube of a number. Ex: cube of x = x × x × x = x³

→ If a cube number is expressed, as the product of 3 equal factors, then one of he factors is called the cube root of that number.
Symbol for cube root is \(\sqrt[3]{ }\)
Ex: \(\sqrt[3]{64}=\left(4^{3}\right)^{1 / 3}\) = 4

→

Students can go through AP Board 9th Class Maths Notes Chapter 1 Real Numbers to understand and remember the concepts easily.

AP State Board Syllabus 9th Class Maths Notes Chapter 1 Real Numbers

→ Numbers of the form \(\frac{p}{q}\) where p and q are integers and q ≠ 0 are called rational numbers, represented by ‘Q’.

→ There are infinitely many rational numbers between any two integers.
E.g.: 3 < \(\frac{19}{6}\), \(\frac{20}{6}\), \(\frac{21}{6}\), \(\frac{22}{6}\), \(\frac{23}{6}\), ……. < 4

→ There are infinitely many rational numbers between any two rational numbers.
E.g.: \(\frac{3}{4}\) < \(\frac{29}{8}\) < \(\frac{71}{16}\) < \(\frac{81}{14}\) ……. < \(\frac{13}{2}\)

→ To find the decimal representation of a rational number we divide the numerator of a rational number by its denominator.
E.g.: The decimal representation of \(\frac{5}{6}\) is

∴ \(\frac{5}{6}\) = 0.833 …. = 0.8 \(\overline{3}\)

→ Every rational number can be expressed as a terminating decimal or as a non-terminating repeating decimal. Conversely every terminating decimal or non¬terminating recurring decimal can be expressed as a rational number.
E.g.: 1.6 \(\overline{2}\) = \(\frac{161}{99}\)

→ A rational number whose denominator consists of only 2’s or 5’s or a combination of 2’s and 5’s can be expressed as a terminating decimal.
E.g. : \(\frac{13}{32}\) can be expressed as a terminating decimal (∵ 32 = 2 × 2 × 2 × 2 × 2)
\(\frac{7}{125}\) can be expressed as a terminating decimal (∵ 125 = 5 × 5 × 5)
\(\frac{24}{40}\) can be expressed as a terminating decimal (∵ 40 = 2 × 2 × 2 × 5)

→ Numbers which can’t written in the form \(\frac{p}{q}\) where p and q are integers and q ≠ 0, are called irrational numbers.
E.g.: √2, √3, √5,….. etc.
The decimal form of an irrational number is neither terminating nor recurring decimal.

→ Irrational numbers can be represented on a number line using Pythagoras theorem.
E.g.: Represent √2 on a number line.

→ If ‘n’ is a natural number which is not a perfect square, then √n is always an irrational number.
E.g.: 2, 3, 5, 7, 8, …… etc., are not perfect squares.
∴ √2, √3, √5, √7 and √8 are irrational numbers.

→ We often write π as \(\frac{22}{7}\) there by π seems to be a rational number; but π is not a rational number.

→ The collection of rational numbers together with irrational numbers is called set of Real numbers.
R = Q ∪ S

→ If a and b are two positive rational numbers such that ab is not a perfect square, then , √ab is an irrational number between ‘a’ and b’.
E.g.: Consider any two rational numbers 7 and 4.
7 × 4 = 28 is not a perfect square; then √28 lies between 4 and 7.
i.e., 4 < √28 < 1

→ If ‘a’ is a rational number and ‘b’ is arty irrational number then a + b, a – b, a.b or \(\frac{a}{b}\) is an irrational number.
E.g.: Consider 7 and √5 then 7 + √5, 7 – √5, 7√5 and \(\frac{7}{\sqrt{5}}\)= are all irrational numbers.

→ If the product of any two irrational numbers is a rational number, then they are said to be the rationalising factor of each other.
E.g.: Consider any two irrational number 7√3 and 5√3.
7√3 × 5√3 = 7 × 5 × 3 = 105 a rational number.
Also 7√3 × √3 = 21 – a rational number.
5√3 × √3 = 15 – a rational number.
So the rationalising factor of an irrational number is not unique.

→ The general form of rationalising factor (R.F.) of (a ± √b} is (a ∓ √b). They are called conjugates to each other.

→ Laws of exponents:

i) a^m × aⁿ = a^m+n
e.g.: 5⁴ . 5^-3 = 5^4+(-3) = 5¹ = 5

ii) (a^m)ⁿ = a^mn
e.g.: (4³)² = 4^3×2 = 4⁶

iii)
 = a^m-n if (m > n)
= 1 if m = n
= \(\frac{1}{a^{n-m}}\) if (m < n)

iv) a^m . b^m = (ab)^m
e.g.:(-5)³ . (2)³ = (-5 × 2)³ =(-10)³

v) \(\frac{1}{a^{n}}\) = a^-n
e.g.: (6)^-3 = \(\frac{1}{6^{3}}\) = \(\frac{1}{216}\)

vi) a⁰ = 1
e.g.: \(\left(\frac{-3}{4}\right)^{0}\) = 1
Where a, b are rationals and m, n are integers.

→ Let a, b be any two rational numbers such that a = bⁿ then b = \(\sqrt[n]{a}\) = \((\mathrm{a})^{1 / \mathrm{n}}\)
Here ‘b’ is called n^th root of a.
e.g.: 4² = 16 then \(16^{1 / 2}\) or \(\sqrt[2]{16}\)
3⁴ = 81 then 3 = \(\sqrt[4]{81}\) or \((81)^{1 / 4}\)

→ Let ‘a’ be a positive number and n > 1 then \(\sqrt[n]{a}\) i.e., n^th root of a is called a surd.

Students can go through AP SSC 10th Class Maths Notes Chapter 9 Tangents and Secants to a Circle to understand and remember the concepts easily.

AP State Syllabus SSC 10th Class Maths Notes Chapter 9 Tangents and Secants to a Circle

→ The locus of points which are joined by a curve and are equidistant from a fixed point is called a circle. The fixed point here is called the centre of the circle.
(or)
A simple closed curve consisting of all points in a plane which are equidistant from a fixed point is called a circle. The fixed point is its centre and the fixed distance is its radius.

→ The path followed by a circular object is a straight line.

→ The line segment joining any two points on a circle is called a ‘chord’. The longest of all chords of a circle passes through the centre and is called a diameter.

\(\overline{\mathrm{AB}}\) is a chord and \(\overline{\mathrm{PQ}}\) is a diameter.
\(\overline{\mathrm{OP}}\) is the radius of the circle,
diameter = 2 × radius d = 2r
r = \(\frac{d}{2}\)

→ There are three different possibilities for a given line and a circle.

Case (i): The line PQ and the circle have no point in common (or) they do not touch each other.
Case (ii): The line PQ and the circle have two common points (or)
The line PQ intersects the circle at two distinct points A and B. Here the line PQ is a “secant” of the circle.
Case (iii): The line PQ touches the circle at an unique point A (or) there is one and only one point common to both the line and circle.
Here \(\stackrel{\leftrightarrow}{\mathrm{PQ}}\) is called a tangent to the circle at ‘A’.

→ The word tangent is derived from the Latin word “TANGERE” which means “to touch” and was introduced by Danish mathematician“Thomas Fineke” in 1583.

→ There is only one tangent to the circle at one point.

→ The tangent at any point of a circle is perpendicular to the radius through the point of contact.

The radius OP is perpendicular to \(\stackrel{\leftrightarrow}{\mathrm{AB}}\) at P.
i.e, OP ⊥ AB.

→ Construction of a tangent to a circle:
Draw a circle with centre ‘O’.
Take a point ‘P’ on it. Join OP.
Draw a perpendicular line to OP through ‘P’.

Let it be \(\stackrel{\leftrightarrow}{\mathrm{XY}}\)
XY is the required tangent to the given circle passing through P.

→ Let ‘O’ be the centre of the given circle and \(\overline{\mathrm{AP}}\) is a tangent through A where OA is the radius, then the length of the tangent AP = \(\sqrt{\mathrm{OP}^{2}-\mathrm{OA}^{2}}\).

→ Two tangents can be drawn to a circle from an external point.

→ Let ‘O’ be the centre of the circle and P is an exterior point. There are exactly two tangents to the circle through P.
\(\overline{\mathrm{PA}}\) and \(\overline{\mathrm{PB}}\) are the tangents.

Here the lengths of the two tangents drawn from the external points are equal.
\(\overline{\mathrm{PA}}\) = \(\overline{\mathrm{PB}}\)

→ Construction of tangents to a circle from an external point:
Step – 1: Draw a circle with centre ‘O’ and with given radius.
Step – 2: Mark a point ‘P’ in the exterior of the circle and join ‘OP’.
Step – 3: Draw the perpendicular bisector \(\stackrel{\leftrightarrow}{\mathrm{XY}}\) to \(\overline{\mathrm{OP}}\), intersecting at M.
Step – 4: Taking M as centre, MP or OM as radius, draw a circle which intersects the given circle at A and B.

Step – 5: Join PA and PB. PA and PB are the required tangents.

→ Consider a circle with centre ‘O’. PA and PB are the tangents from an exterior point ‘P’. Then, the centre of the circle lies on the bisector of the angle between two tangents drawn from the exterior point P.
∠OPA = ∠OPB

→ Consider two concentric circles with centre ‘O’. Let the chord \(\overline{\mathrm{AB}}\) of the larger/ bigger circle just touches the smaller circle, then it is bisected at the point of contact with the smaller circle.
In the figure, \(\overline{\mathrm{AB}}\) is the chord of bigger circle touching the smaller circle at P then AP = PB.

→ If AP and AQ are two tangents to a circle with centre ‘O’, then ∠PAQ = 2∠OPQ = 2∠OQP.

→ If a circle touches the sides of a quadrilateral ABCD at points P, Q, R and S then AB + CD = BC + DA.
i.e., sum of the opposite sides are equal.

→ The region enclosed by a secant/chord and an arc is called a ‘segment of the circle’.

Case (i): If the arc is a minor arc then the segment is a minor segment.
Case (ii): If the arc is a semi arc then the segment is a semi circle.
Case (iii): If the arc is a major arc then the segment is a major segment.

→ Area of a segment between the chord AB and whose arc makes an angle ‘x’ at the centre = \(\frac{x}{360}\) × πr²

i.e., Area of the segment APB = (Area of the corresponding sector OAPB) – (Area of the corresponding triangle OAB).

Students can go through AP Board 8th Class Maths Notes Chapter 4 Exponents and Powers to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 4 Exponents and Powers

Laws of Exponents:

→ a × a × a …… m times = a^m

In a^m, a is called base; m is called exponent/ power.

→ a^m × aⁿ = a^m+n

→ \(\frac{\mathrm{a}^{\mathrm{m}}}{\mathrm{a}^{\mathrm{n}}}\) = a^m-n (m > n)
⇒ \(\frac{1}{a^{n-m}}\) (m < n)

→ (ab)^m = a^m . b^m

→ a⁰ = 1

→ a^-n = \(\frac{1}{a^{n}}\)

→ aⁿ = \(\frac{1}{a^{-n}}\)

→ \(\left(\frac{a}{b}\right)^{m}\) = \(\frac{a^{m}}{b^{m}}\)

→ \(\left(a^{m}\right)^{n}\) = a^mn

→

→ \(\sqrt[n]{a}\) = \((\mathrm{a})^{1 / \mathrm{n}}\)

Students can go through AP SSC 10th Class Maths Notes Chapter 8 Similar Triangles to understand and remember the concepts easily.

AP State Syllabus SSC 10th Class Maths Notes Chapter 8 Similar Triangles

→ The geometrical figures which have the same shape but are not necessarily of the same size are called similar figures.

→ The heights and distances of distant objects can be found using the principles of similar figures.

→ Two polygons with same number of sides are said to be similar if their corresponding angles are equal and their corresponding sides are in proportion.

→ A polygon in which all sides and all its angles are equal is called a regular polygon. Eg.:

→ The ratio of the corresponding sides is referred to as scale factor or representative factor.

→ All squares are similar.

→ All circles are similar.

→ All equilateral triangles are similar.

→ Two congruent figures are similar but two similar figures need not be congruent.

→ A square ABCD and a rectangle PQRS are of equal corresponding angles, but their corre¬sponding sides are not in proportion.

∴ The square ABCD and the rectangle PQRS are not similar.

→ The corresponding sides of a square ABCD and a rhombus PQRS are equal but their corresponding angles are not equal. So they are not similar.

→ If a line is drawn parallel to one side of a triangle intersecting the other two sides at two distinct points then the other two sides are divided in the same ratio.

In △ABC; DE // BC then \(\frac{AD}{DB}\) = \(\frac{AE}{EC}\).
This is called Basic proportionality theorem (or) Thale’s theorem.

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

In △ABC, a line ‘l’ intersecting AB in D and AC in E
such that \(\frac{AD}{DB}\) = \(\frac{AE}{EC}\) then l // BC.
This is converse of Thale’s theorem.

→ Two triangles are similar, if
i) their corresponding angles are equal.
ii) their corresponding sides are in the same ratio.

→ If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio or proportional and hence the two triangles are similar.

In △ABC, △DEF
∠A = ∠D
∠B = ∠E
∠C = ∠F
⇒ \(\frac{AB}{DE}\) = \(\frac{BC}{EF}\) = \(\frac{AE}{DF}\)
∴ △ABC ~ △DEF (A.A.A)

→ If in two triangles, sides of one triangle are proportional to the sides of other triangle, then their corresponding angles are equal and hence the two triangles are similar.

In △ABC, △DEF
if \(\frac{AB}{DE}\) = \(\frac{BC}{EF}\) = \(\frac{AE}{DF}\)
⇒ ∠A = ∠D
∠B = ∠E
∠C = ∠F
Hence, △ABC ~ △DEF (S.S.S)

→ If two angles of a triangle are equal to two corresponding angles of another triangle then the two triangles are similar.

In △ABC, △DEF
if ∠A = ∠D
∠B = ∠E
⇒ ∠C = ∠F (By Angle Sum property)
∴ △ABC ~ △DEF (A.A)

→ If one angle of a triangle is equal to one angle of other triangle and the sides including these angles are proportional, then the two triangles are similar.

In △ABC, △DEF if ∠A = ∠D, and
\(\frac{AB}{DE}\) = \(\frac{AC}{DF}\)
⇒ △ABC ~ △DEF (S.A.S)

→ The ratio of areas of two similar triangles is equal to the ratio corresponding sides.

→ If a perpendicular is drawn from the vertex, containing the right angle of a right angled – triangle to the hypotenuse, then the triangles on each side of perpendicular are similar to one another and to the original triangle. Also the square of the perpendicular is equal to the product of the lengths of the two parts of the hypotenuse.
In △ABC, ∠B = 90°
BD ⊥ AC

Then △ADB ~ △BDC ~ △ABC and
BD² = AD . DC

→ Pythagoras theorem: In a right angled triangle, the square of hypotenuse is equal to the sum of the squares of other two sides.

In △ABC; ∠A = 90°
AB² + AC² = BC²

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

In △ABC, if
AC² = AB² + BC² then ∠B = 90°
This is converse of Pythagoras theorem.

→ Baudhayan Theorem (about 800 BC):
The diagonal of a rectangle produces itself the same area as produced by its both sides (i.e., length and breadth).
In rectangle ABCD,

area produced by the diagonal AC = AC • AC
= AC²
area produced by the length = AB • BA = AB
area produced by the breadth = BC • CB = BC²
Hence, AC² = AB² + BC².

→ A sentence which is either true or false but not both is called a simple statement.

→ A statement formed by combining two or more simple statements is called a compound statement.

→ A compound statement of the form “If …… then ……” is called a Conditional or Implication.

→ A statement obtained by modifying the given statement by ‘NOT’ is called its negation.

Students can go through AP Board 8th Class Maths Notes Chapter 5 Comparing Quantities Using Proportion to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 5 Comparing Quantities Using Proportion

→ Two simple ratios are expressed like a single ratio as the ratio of product of antecedents to product of consequents and we call it compound ratio of the given two simple ratios.
a : b and c : d are any two ratios, then their compound ratio is \(\frac{a}{b}\) × \(\frac{c}{d}\) = \(\frac{ac}{bd}\) i.e. ac : bd.

→ A percentage(%) compares a number to 100. The word percent means “per every hundred” or “out of every hundred”. 100% = \(\frac{100}{100}\) it is also a fraction with denominator 100.

→ Discount is a decrease percent of marked price. Price reduction is called rebate or discount. It is calculated on marked price or list price.

→ Profit or loss is always calculated on cost price. Profit is an example of increase percent of cost price and loss is an example of decrease percent of cost price.

→ VAT will be charged on the selling price of an item and will be included in the bill.
VAT is an increase percent on selling price.

→ Simple interest is an increase percent on the principal.

→ Simple interest (I) = \(\frac{P \times T \times R}{100}\)
where P = Principal, T = Time inyears, R = Rate of interest.

→ Amount = Principal + Interest = P + \(\frac{P \times T \times R}{100}\) = P\(\left(1+\frac{T \times R}{100}\right)\)

→ Compound interest allows you to earn interest on interest.

→ Amount at the end of ‘n’ years using compound interest is A = P \(\left(1+\frac{R}{100}\right)^{n}\)

→ The time period after which interest is added to principal is called conversion period.
When interest is compounded halfyearly, there are two conversion periods in a year, each after 6 months. In such a case, ha If year rate will be half of the annual rate.

→ Note: 1.615 : 1 is called as golden ratio.
In ancient Greece, artists and architects believed there was a particular rectangular shape that looked very pleasing to the eye. For rectangles of this shape, the ratio of long side to the short side is roughly 1.615 : 1. This ratio is very close to what is known as golden ratio.

Students can go through AP Board 8th Class Maths Notes Chapter 3 Construction of Quadrilaterals to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 3 Construction of Quadrilaterals

→ A closed four sided polygon is called a quadrilateral.

→ A quadrilateral has 4 sides, 4 vertices, 4 angles and 2 diagonals.

→ The sum of the 4 angles of a quadrilateral is 360°.

Type of a quadrilateral	No. of individual measurements
1. Quadrilateral	5
2. Trapezium	4
3. Parallelogram	3
4. Rectangle	3
5. Rhombus	2
6. Square	1

→ Quadrilateral and their types:

→ Five independent measurements are required to draw a unique quadrilateral.

→ A quadrilateral can be constructed uniquely, if
a) The lengths of four sides and one angle are given
b) The lengths of four sides and one diagonal are given
c) The lengths of three sides and two diagonals are given
d) The lengths of two adjacent sides and three angles are given
e) The lengths of three sides and two included angles are given

→ The two special quadrilaterals, namely rhombus and square can be constructed when two diagonals are given.

Students can go through AP Board 8th Class Maths Notes Chapter 2 Linear Equations in One Variable to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 2 Linear Equations in One Variable

→ An algebraic equation is equality of algebraic expressions involving variables and constants.

→ If the degree of an equation is one then it is called a linear equation.

→ If a linear equation has only one variable then it is called a linear equation in one variable or simple equation. ‘

→ The value which when substituted for the variable in the given equation makes L.H.S. = R.H.S. is called a solution or root of the given equation.

→ Just as numbers, variables can also be transposed from one side of the equation to the other side.
Note: When we transpose terms
‘+’ quantity becomes ’-‘ quantity,
‘-‘ quantity becomes ‘+’ quantity.
‘×’ quantity becomes ‘÷’ quantity.
‘÷’ quantity becomes ‘×’ quantity.
Also
Also,
(+) × (+) = +
(+) × (-) = –
(-) × (+) = –
(-) × (-) = +

Students can go through AP Board 8th Class Maths Notes Chapter 1 Rational Numbers to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 1 Rational Numbers

→ The numbers which are expressed in the form of \(\frac{p}{q}\) where p and q are integers and q ≠ 0, are called “Rational Numbers” which are denoted by the letter ‘Q’.

→ Rational numbers are closed under the operations of addition, subtraction and multiplication.

→ Rational numbers are not closed on division.

→ Whole numbers:

→ Whole numbers:

→ The additive inverse of a is – a. (∵ a + (-a) = 0)

→ The multiplicative inverse of a is \(\frac{1}{a}\). (∵ a × \(\frac{1}{a}\) = 1)

→ The operations addition and multiplications are

1. Commutative for rational numbers.
2. Associative for rational numbers.

→ ‘0’ is the additive identity for rational number.

→ ‘1’ is the multiplicative identity for rational number.

→ A rational number and its additive inverse are opposite in their sign.

→ The multiplicative inverse of a rational number is its reciprocal.

→ Distributivity of rational numbers a, b and c is a(b + c) = ab + ac and a(b – c) = ab – ac.

→ Rational numbers can be represented on a number line.

→ There are infinite number of rational numbers between any two given rational numbers.

→ The concept of mean help us to find rational numbers between any two rational numbers.

→ The decimal representation of rational numbers is either in the form of terminating decimal or non-terminating recurring decimals.

Students can go through AP SSC 10th Class Maths Notes Chapter 14 Statistics to understand and remember the concepts easily.

AP State Syllabus SSC 10th Class Maths Notes Chapter 14 Statistics

→ Statistics is a branch of mathematics which deals with collection, organisation, presentation, analysis and interpretation of numerical data.

→ Data is a collection of actual information which is used to make logical inferences.

→ Arithmetic Mean of raw data:
The Arithmetic Mean (A.M.) of a raw data viz. x₁, x₂, x₃, ……., x_n is the sum of values of all observations divided by the number of observations.
Arithmetic Mean (A.M.) =
Eg.: Sita secured 23, 24, 24, 22 and 20 marks in a test. Her mean marks are
A.M. = \(\frac{23+24+24+22+20}{5}\) = \(\frac{113}{5}\) = 22.6

→ A.M. by direct method:
Let x₁, x₂, x₃, ……., x_n be observations with respective frequencies f₁, f₂, ……, f_n
i.e., x₁ occurs for f₁ times, x₂ occurs for f₂ times, ….., x_n occurs for f_n times.

→ For a grouped data, it is assumed that the frequency of each class interval is centered around its mid-point and the A.M. is given by A.M. = \(\frac{\Sigma \mathrm{f}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}}{\Sigma \mathrm{f}_{\mathrm{i}}}\)

→ A.M. by deviation method, \(\overline{\mathbf{x}}=\mathbf{a}+\frac{\Sigma \mathbf{f}_{\mathbf{i}} \mathbf{d}_{\mathbf{i}}}{\Sigma \mathbf{f}_{\mathbf{i}}}\)
where, a – assumed mean
d_i – deviation = x_i – a.
Step – 1: Choose ‘a’ from the central values.
Step – 2: Obtain d_i by subtracting a from x_i.
Step – 3: Multiply f_i and d_i.
Step – 4: Find ∑f_id_i and ∑f_i .
Step – 5: Find \(\overline{\mathbf{x}}=\mathbf{a}+\frac{\Sigma \mathbf{f}_{\mathbf{i}} \mathbf{d}_{\mathbf{i}}}{\Sigma \mathbf{f}_{\mathbf{i}}}\)

→ A.M. by step-deviation method:

Step – 1: Choose ‘a’ from mid values.
Step – 2: Obtain u_i = \(\frac{x_{i}-a}{h}\).
Step – 3: Multiply f_i and u_i.
Step – 4: Find Ef_iu_i and Sf_i.
Step – 5: Find \(\overline{\mathrm{x}}=\mathrm{a}+\left(\frac{\Sigma \mathrm{f}_{\mathrm{i}} \mathrm{u}_{\mathrm{i}}}{\Sigma \mathrm{f}_{\mathrm{i}}}\right) \times \mathrm{h}\)

→ Mode : Mode is the size of variable which occurs most frequently.

→ Mode of a grouped data:

Where, l – lower boundary of the modal class,
h – size of the modal class interval,
f₁ – frequency of modal class.
f₀ – frequency of the class preceding the modal class.
f₂ – frequency of the class succeeding the modal class.

→ Median: Median is defined as the measure of the central items when they are in descending or ascending order of magnitude.

→ Median for a grouped data:

where,
l – lower boundary of median class,
n – number of observations.
cf – cumulative frequency of class preceding the median class.
f – frequency of median class.
h – size of the median class.

→ Cumulative frequency curve or an ogive:
First we prepare the cumulative frequency table, then the cumulative frequencies are plotted against the upper or lower limits of the corresponding class intervals. By joining the points the curve so obtained is called a cumulative frequency or ogive.
Ogives are of two types.

1. Less than ogive: Plot the points with the upper limits of the classes as abscissa and the corresponding less than cumulative frequencies as ordinates. The points are joined by free hand smooth curve to give less than cumulative frequency curve or the less than ogive. It is a rising curve.
2. Greater than ogive: Plot the points with the lower limits of the classes as abscissa and the corresponding greater than cumulative frequencies as ordinates. Join the points by a free hand smooth curve to get the greater than ogive. It is a falling curve.

When the points are joined by straight lines, the figure obtained is called cumulative frequency polygon.

→ Median can be obtained from cumulative frequency curve: From \(\frac{n}{2}\) frequency draw a line parallel to X-axis cutting the curve at a point. From this point draw a perpendicular to the axis. The point at which the perpendicular meets the X – axis determines the median.

Less than type and greater than type curves intersects at a point. From this point of intersection if we draw a perpendicular on the X-axis then this cuts X-axis at some point. This point gives the median.