Mahesh

Students can go through AP Board 9th Class Maths Notes Chapter 2 Polynomials and Factorisation to understand and remember the concepts easily.

AP State Board Syllabus 9th Class Maths Notes Chapter 2 Polynomials and Factorisation

→ An algebraic expression ¡n which the variables involved have only non-negative integral powers is called a polynomial.
E.g.: 5x³ – 2x + 8

→ Polynomials can be classified as monomials. binomials, trinomials and polynomials based on the number of terms it contains.

→ A polynomial with a single term is a monomial.
E.g.: 2x, -5x², \(\frac{6}{7}\)x³ etc.

→ A polynomial with two terms is a binomial.
E.g.: 2x + 5y; -3x² + 5x etc.

→ A polynomial with three terms is a trinomial.
E.g.: 3x² + 5x – 8; 3x + 2y – 5z etc.

→ In general a polynomial may contain two or more than two terms.
E.g.: 5x + 8x² – 3x³ + 7

→ Degree of a polynomial ¡s the heighest degree of its variable terms.
E.g.: Degree of 3x² + 4xy³ + y² is 4.

→ Degree of a constant term is considered as zero.
E.g.: Degree of 4 is zero.

→ The general form of a polynomial is a₀xⁿ + a₁x^n-1 + a₂x^n-2 …….. + a_n-1x + a_n where a₀, a₁, a₂,…… a_n-1, a_n are constants and a₀ ≠ 0. Its degree is ‘n’.

→ Polynomials are again classified based on their degrees.

→ The zero of a polynomial p(x) is the value of x at which p(x) becomes zero (i.e.) p(x) = 0.
E.g.: Zero of 3x – 5 is x = \(\frac{5}{3}\)

→ To find the zero of a polynomial we equate the polynomial to zero and solve for the value of the variable.
E.g.: To find zero of 7x + 8.
7x + 8 = 0
7x = – 8
x = \(\frac{-8}{7}\)

→ Let p(x) be any polynomial of degree greater than or equal to one and let ‘a’ be any real number. If p(x) is divided by the linear polynomial (x – a), then the remainder is p(a). This is called the Remainder theorem.
E.g.: If p(x) = 4x³ + 3x + 8 then the remainder when it is divided by x – 1 is p(1).
i.e., p(1) = 4 + 3 + 8 = 15.

→ If p(x) is a polynomial of degree n ≥ 1 and ‘a’ is any real number, then
(i) (x – a) is a factor of p(x), if p(a) = 0
(ii) and its converse “if (x – a) is a factor of a polynomial p(x) then p(a) = 0. This is called Factor theorem”.
E.g.: Let p(x) = x² – 5x + 6 and p(2) = 2² – 5(2)+ 6 = 0 and hence (x – 2) is a factor of p(x) conversely; p(x) = x² + 7x + 10 and (x + 2) is a factor, then p(-2) = 0.

→ Algebraic identities

• (x + y)² = x² + 2xy + y²
• (x – y)² = x² – 2xy + y²
• (x + y) (x-y) = x² – y²
• (x + a) (x + b) = x² + (a + b) x + ab
• (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
• (x + y)³ = x³ + y³ + 3xy (x + y)
• (x – y)³ = x³ – y³ – 3xy (x – y)
• (x + y + z) (x² + y² + z² – xy – yz – zx) = x³ + y³ + z³ – 3xyz

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 Board 8th Class Maths Notes Chapter 15 Playing with Numbers to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 15 Playing with Numbers

→ Place value of the digits:
12, 34, 56, 789

→ Expanded form of the number:
7843 = 7 × 1000 + 8 × 100 + 4 × 10 + 3 × 1
= 7 × 10³ + 8 × 10² + 4 × 10¹ + 3 × 10⁰

→ Prime numbers: Numbers which are having the factors 1 and itself are called primes.
Ex: 2, 3, 5, 7, 11, 13, …. etc.

→ Divisibility Rules:
(i) Divisibility by 10: If the units digit of a number is ‘0’ then it is divisible by 10.
Ex: 10, 50, 100, 150, 1000 etc.

(ii) Divisibility by 5: If the units digit of a number is either ‘0 ‘or ‘5’ then it is divisible by 5.
Ex: 15, 20, 50, 90 …. etc.

(iii) Divisibility by 2: If the units digit of a number is 0, 2, 4, 6, 8 then it is divisible by 2.
Ex: 0, 16, 32, 48, 22 …. etc.

(iv) Divisibility by 3: If the sum of the digits of a number is divisible by 3 then the number is also divisible by 3.
Ex: 126 → 1 + 2 + 6 → \(\frac{9}{3}\) (R = 0) then 126 is divisible by 3.

(v) Divisibility by 6: If a number is divisible 2 and 3 then it is divisible by 6.

(vi) Divisibility by 4: If the last two digits of a number is divisible by 4, then the entire number is divisible by 4.

∴ 496 is divisible by 4.

(vii) Divisibility by 8: If the last 3 digits of a number is divisible by 8 then it is divisible by 8.

∴ 80952 is divisible by 8.

(viii) Divisibility by 7: If a number is divisible by 7 (2a + 3b + c) must be divisible by 7.
Where a = digit at the hundreds place, b = digit at the tens place and c = digit at ones place.

(ix) Divisibility by ’11’: If the difference of the sum of digits in odd places and sum of digits in even places of a number is divisible by 11. Then the number is divisible by 11.
Ex: 1234321

⇒ (1 + 3 + 3 + 1) – (2 + 4 + 2) = 8 – 8 = 0 → \(\frac{0}{11}\) (R = 0)
∴ 1234321 is divisible by 11.

Students can go through AP Board 8th Class Maths Notes Chapter 14 Surface Areas and Volume to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 14 Surface Areas and Volume

→ If l, b, h are-the dimensions of cuboid, then:

(i) its lateral surface area is 2h (l+ b)
(ii) its total surface area is 2 (lb + bh + hl)
(iii) its volume is l × b × h

→ Lateral surface area of a cube is 4a²

→ Total surface area of a cube is 6a²

→ Volume of a cube is side × side × side = a³

→ 1 cm³ = 1 ml

→ 1 l = 1000 cm³

→ 1 m³ = 1000000 cm³ = 1000 l = 1 kl (kilolitre)

Students can go through AP Board 8th Class Maths Notes Chapter 13 Visualizing 3-D in 2-D to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 13 Visualizing 3-D in 2-D

→ Polyhedron: Solid objects having flat surfaces.

→ Prism: The polyhedra have top and base as same polygon and other faces are rectangular (parallelogram).

→ Pyramid: Polyhedron which have a polygon as base and a vertex, rest of the faces are triangles. 4 3-D objects could be make by using 2-D nets.

→ Euler’s formula for polyhedra : E + 2 = F + V.

→ 3-D Objects made with cubes:
Observe the following solid shapes. Both are formed by arranging four unit cubes. If we observe them from different positions, it seems to be different. But the object is same.

→ Faces, Edges and Vertices of 3D-Objects:
Observe the walls, windows, doors, floor, top, corners etc of our living room and tables, boxes etc. Their faces are flat faces. The flat faces meet at its edges. Two or more edges meet at corners. Each of the corner is called vertex. Take a cube and observe it where the faces meet?

→ There are only five regular polyhedra, all of them are complex, often referred as Platonic solids as a tribute to Plato.

Note: Cube is the only polyhedron to completely fill the space.

→ Net diagrams of Platonic Solids

→ No. of Edges, Faces and Vertices of Polyhedrons:

By observing the last two columns of the above table, we can conclude that F + V = E + 2 for all polyhedra.

Students can go through AP Board 8th Class Maths Notes Chapter 12 Factorisation to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 12 Factorisation

→ Factorisation is a process of writing the given expression as a product of its factors.

→ A factor which cannot be further expressed as product of factors is an irreducible factor.

→ Expressions which can be transformed into the form:
a² + 2ab + b²;
a² – 2ab + b²;
a² – b² and x² + (a + b)x + ab can be factorised by using identities.

→ If the given expression is of the form x² + (a + b) x + ab, then its factorisation is (x + a)(x + b).

→ Division is the inverse of multiplication. This concept is also applicable to the division of algebraic expressions.

Gold Bach Conjecture:

→ Gold Bach found from observation that every odd number seems to be either a prime or the sum of a prime and twice a square.
Thus 21 = 19 + 2 or 13 + 8 or 3 + 18.

→ It is stated that up to 9000, the only exceptions to his statement are
5777 = 53 × 109 and 5993 = 13 × 641,
which are neither prime nor the sum of a primes and twice a square.

Students can go through AP Board 8th Class Maths Notes Chapter 11 Algebraic Expressions to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 11 Algebraic Expressions

→ There are number of situations in which we need to multiply algebraic expressions.

→ A monomial multiplied by a monomial always gives a monomial.

→ While multiplying a polynomial by a monomial, we multiply every term in the polynomial by the monomial.

→ In carrying out the multiplication of an algebraic expression with another algebraic expression (monomial/ binomial/ trinomial etc.) we multiply term by term i.e. every term of the expression is multiplied by every term in the another expression.

→ An identity is an equation, which is true for-all values of the variables in the equation. On the other hand, an equation is true only for certain values of its variables. An equation is not an identity.

→ The following are identities:
I. (a + b)² = a² + 2ab + b²
II. (a – b)² = a² – 2ab + b²
III. (a + b) (a -b) = a² – b²
IV. (x + a) (x + b) = x² + (a + b)x + ab

→ The above four identities are useful in carrying out squares and products of algebraic expressions. They also allow easy alternative methods to calculate products of numbers and so on.
Note:
We know that
(+) × (+) = +
(+) × (-) = –
(-) × (+) = –
(-) × (-) = +

Students can go through AP Board 8th Class Maths Notes Chapter 10 Direct and Inverse Proportions to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 10 Direct and Inverse Proportions

→ If x and y are in direct proportion, the two quantities vary in the same ratio.
i.e. if \(\frac{x}{y}\) = k or x = ky. We can write \(\frac{x_{1}}{y_{1}}\) = \(\frac{x_{2}}{y_{2}}\) [y₁, y₂ are values of y corresponding to the values x₁, x₂ of x respectively]

→ Two quantities x and y are said to vary in inverse proportion, if there exists a relation of the type xy = k between them, k being a constant. If y₁, y₂ are the values of y corresponding to the values x₁ and x₂ of x respectively, then x₁y₁ = x₂y₂ (= k), or = \(\frac{x_{1}}{x_{2}}\) = \(\frac{y_{2}}{y_{1}}\)

→ If one quantity increases (decreases) as the other quantity decreases (increases) in same proportion, then we say it varies in the inverse ratio of the other quantity. The ratio of the first quantity (x₁ : x₂) is equal to the inverse ratio of the second quantity (y₁ : y₂). As both the ratios are the same, we can express this inverse variation as proportion and it is called inverse proportion.

→ Sometimes change in one quantity depends upon the change in two or more other quantities in same proportion. Then we equate the ratio of the first quantity to the compound ratio of the other two quantities.

Students can go through AP Board 8th Class Maths Notes Chapter 9 Area of Plane Figures to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 9 Area of Plane Figures

→ Area of a triangle = \(\frac{1}{2}\) × base × height = \(\frac{1}{2}\) bh

→ Area of a quadrilateral = \(\frac{1}{2}\) × length of a diagonal × Sum of the lengths of the perpendiculars drawn from the remaining two vertices on the diagonal
= \(\frac{1}{2}\) d(h₁ + h₂)

→ Area of a trapezium = \(\frac{1}{2}\) × sum of the lengths of parallel sides × distance between them
= \(\frac{1}{2}\) h(a + b)

→ Area of a rhombus = Half of the product of diagonals = \(\frac{1}{2}\) d₁d₂

→ Angle at the centre of a circle = 360°

→ Area of a circle = πr²
Where ‘r’ is the radius of the circle, π = \(\frac{22}{7}\) or 3.14 nearly

→ Circumference of a circle = 2πr

→ Area of a circular path (or) Area of a Ring = π(R² – r²) or π(R + r) (R- r)
When R, r are radii of outer circle and inner circle respectively.

→ Width of the path w = R – r

→ Area of a sector A = \(\frac{x^{\circ}}{360^{\circ}}\) × πr² where x° is the angle subtended by the arc of the sector at the center of the circle and r is radius of the circle. (OR) A = \(\frac{lr}{2}\) where Tis thdength of the arc.
Length of the arc of a sector = \(\frac{x^{\circ}}{360^{\circ}}\) × 2πr

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

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