Mahesh

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.

Students can go through AP SSC 10th Class Maths Notes Chapter 13 Probability to understand and remember the concepts easily.

AP State Syllabus SSC 10th Class Maths Notes Chapter 13 Probability

→ Theory of probability has its origin date back to 16th century.

→ J. Cardan, an Italian physician and mathematician wrote the first book on probability named “The Book of Games of Chance”.

→ James Bernoulli (1654 – 1705), A.De Moivre (1667 – 1754) and Pierre Simon Laplace (1749 – 1827) made significant contribution to the theory of probability.

→ Experimental or empirical probability : The probability estimated on the basis of results of an actual experiment is called experimental probability of empirical probability.
Eg : An unbiased coin is tossed for 1000 times, head turned up for 455 times and tail turned up 545 times, then the probability or likelyhood of getting a head is = \(\frac{455}{1000}\) = 0.455.

Thus experimental probability = \(\frac{\text { No. of trials in which the event happened }}{\text { Total no. of trials }}\)

→ Classical or Theoretical probability: Classical probability of an event (E) is defined Number as

P(E) = \(\frac{\text { Number of outcomes favourable to E }}{\text { No. of all possible outcomes of the experiment }}\)

This definition was given by ‘Pierre Simon Laplace’.
Eg: The probability of getting a head when a coin is tossed is given by Number of outcomes favourable to this event getting a head = 1 Number of all possible outcomes of this experiment = 2 (Head, Tail)

∴ P(E) = \(\frac{\text { No. of favourable outcomes }}{\text { Total events }}\) = \(\frac{1}{2}\)

Note: If an experiment is conducted for many number of times, then the experimental probability may become closer and closer to theoretical probability.

→ The probability of a sure event is 1.

→ The probability of an impossible event is zero.

→ The probability of an event E is a number P(E) such that 0 ≤ P(E) ≤ 1.

→ An event having only one outcome is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is 1.

→ For any event E, P(E) + P(\(\overline{\mathrm{E}}\)) = 1, where E and \(\overline{\mathrm{E}}\) are complementary events.

→ Playing cards and their probability : A deck of playing cards consists of 52 cards which are divided into four suits of 13 cards each.
They are:

→ The cards in each suit are:

Eg : When a card is drawn at random from a deck of cards then

• Getting a black or red card – equally likely exhaustive events.
• Getting an ace or king – mutually exclusive.
  
• Getting an ace or a hearts – not mutually exclusive since the hearts contain an ace.
  

→ When a coin is tossed, the outcomes are H, T (Head, Tail).

→ When a dice is thrown the outcomes are 1, 2, 3, 4, 5 and 6.

→ When two dice are thrown, the outcomes are

→ If a coin is tossed n-times or n – coins are tossed simultaneously, then the number of total outcomes = 2ⁿ.

→ If a dice is thrown for n – times or n – dice are thrown simultaneously then the number of total outcomes = 6ⁿ.

Students can go through AP SSC 10th Class Maths Notes Chapter 12 Applications of Trigonometry to understand and remember the concepts easily.

AP State Syllabus SSC 10th Class Maths Notes Chapter 12 Applications of Trigonometry

→ If a person is looking at an object then the imaginary line joining the object and the eye of the observer is called the line of sight or ray of view.

→ An imaginary line parallel to earth surface and passing through the point of observation is called the horizontal.

→ If the line of sight is above the horizontal then the angle between them is called “angle of elevation”.

→ If the line of sight is below the horizontal then the angle between them is called the angle of depression.

→ Useful hints to solve the problems:

1. Draw a neat diagram of a right triangle or a combination of right triangles if necessary.
2. Represent the data given on the triangle.
3. Find the relation between known values and unknown values.
4. Choose appropriate trigonometric ratio and solve for the unknown.

→ The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios.

→ To use this application of trigonometry, we should know the following terms.

→ The terms are Horizontal line, Line of Sight, Angle of Elevation and Angle of Depression.

→ Horizontal line: A line which is parallel to earth from observation point to object is called “horizontal line”.

→ Line of Sight (or) Ray of Vision: The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer.

→ Angle of Elevation: The line of sight is above the horizontal line then angle between the line of sight and the horizontal line is called “angle of elevation”.

Note:

1. If the angle of observer moves towards the perpendicular line (pole/tree/ building), then angle of elevation increases and if the observer moves away from the perpendicular line (pole/tree/building), then angle of elevation decreases.
2. If height of tower is doubled and the distance between the observer and foot of the tower is also doubled, then the angle of elevation remains same.
3. If the angle of elevation of sun above a tower decreases, then the length of shadow of a tower increases.

→ Angle of Depression: The line of sight is below the horizontal line then angle between the line of sight and the horizontal line is called angle of depression.

Note:

1. The angle of elevation and depression are always acute angles.
2. The angle of elevation of a point P as seen from a point ‘O’ is always equal to the angle of depression of ‘O’ as seen from P.

→ Points to be kept in mind:
I. Trigonometric ratios in a right triangle:

II. Trigonometric ratios of some specific angles:

→ Solving Procedure:
When we want to solve the problems of height and distances, we should consider the following :

1. All the objects such as tower, trees, buildings, ships, mountains, etc. shall be considered as linear for mathematical convenience.
2. The angle of elevation or angle of depression is considered with reference to the horizontal line.
3. The height of the observer is neglected, if it is not given in the problem.
4. To find heights and distances, we need to draw figures and with the help of these figures we can solve the problems.

Students can go through AP SSC 10th Class Maths Notes Chapter 11 Trigonometry to understand and remember the concepts easily.

AP State Syllabus SSC 10th Class Maths Notes Chapter 11 Trigonometry

→ In our daily life, we can measure the heights, distances and slopes by using some mathematical techniques.

→ The mathematical techniques which come under a branch of mathematics is called ‘trigonometry’.

→ “Trigonometry” is the study of relationships between the sides and angles of a triangle.

→ Early astronomers used to find out the distances of the stars and planets from the Earth. Even today, most of the technologically advanced methods used in engineering and physical sciences are based on trigonometrical concepts.

→ Naming the sides in a right triangle:
Let’s take a right triangle ABC as shown in the figure.

Consider ∠CAB as A where angle ‘A’ is acute.
Since AC is the longest side, it is called “Hypotenuse”.

→ Now observe the position of side BC with respect to angle A. It is opposite to angle ‘A’ and we can call it as “Opposite side of angle A”.

→ And the remaining side AB can be called as “Adjacent side of angle A”.

→ Trigonometric Ratios:
The ratios of the sides of a right angled triangle with respect to its acute angles, are called Trigonometric ratios.

→ Consider a right angled triangle ABC having right angle at B as shown in the given figure.

Then, trigonometric ratios of the angle A in right angled triangle ABC are defined as follows:

→ There are three more ratios defined in trigonometry which are considered as multiplicative inverse of the above three ratios.

→ Multiplicative inverse of “sine A” is “cosecant A”.
i.e., cosec A = \(\frac{1}{\sin A}\) = \(\frac{\text { Hypotenuse }}{\text { Opposite side of the angle } A}\)

→ Multiplicative inverse of “cosine A” is “secant A”.
i.e., sec A = \(\frac{1}{\cos A}\) = \(\frac{\text { Hypotenuse }}{\text { Adjacent side of the angle } A}\)

→ Multiplicative inverse of “tangent A” is “cot A”.
i.e., cot A = \(\frac{1}{\tan A}\) = \(\frac{\text { Adjacent side of the angle } A}{\text { Opposite side of the angle } A}\)

Note:
i) The values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangles, if the angle remains the same.
ii) Each trigonometric ratio is a real number and has no unit.
iii) “sin θ” is one symbol and sin, cos, tan etc., cannot be separated from θ.
iv) If one of the trigonometric ratios of an acute angle is known, the remaining trigonometric ratios of angle can be easily determined.
v)

→ Trigonometric ratios of some specific angles:
The values of various trigonometric ratios of 0°, 30°, 45°, 60° and 90°.

Note:
i) The values of “sin θ” and “cos θ” always lie between ‘0’ and ‘1’.
ii) In the case of tan θ, the values increase from 0 to ∞ (not determinate).
iii) In the case of cot θ, the values decrease from ∞ to 0.
iv) In the case of cosec θ, the values decrease from ∞ to 1.
v) In the case of sec θ, the values increase from 1 to ∞.

→ Trigonometric ratios of complementary angles:
Two angles are said to be complementary, if their sum is equal to 90°.
In a right angled triangle, if ∠B = 90°, then ∠A + ∠C = 90° i.e., ∠A and ∠C form a pair of complementary angles.
If ‘θ’ is an acute angle, then we can prove that
sin (90 – θ) = cos θ
cos (90 – θ) = sin θ
tan (90 – θ) = cot θ
cot (90 – θ) = tan θ
sec (90 – θ) = cosec θ
cosec (90 – θ) = sec θ

→ Trigonometric Identity: An identity equation having trigonometric ratios of an angle is called trigonometric identity. It is true for all the values of the angles involved in it.
We have three major trigonometric identities. They are
i) sin² A + cos² A = 1
ii) sec² A – tan² A = 1
iii) cosec² A – cot² A = 1
Note: sin² θ = (sin θ)² but sin θ² ≠ (sin θ)²

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

AP State Syllabus SSC 10th Class Maths Notes Chapter 10 Mensuration

→ A solid is a geometrical shape with three dimensions namely length, breadth and height.
Eg:

→ A solid has two types of area namely,
a) Lateral Surface Area (L.S.A.)
b) Total Surface Area (T.S.A,)

→ In general, L.S.A. of a solid is the product of its base perimeter and height.
Eg : L.S.A. of a cuboid = 2h(l + b)
L.S.A. of a cylinder = 2πrh

→ The T.S.A. of a solid is the sum of L.S.A. and the areas of its top and base.
Eg : T.S.A. of a cylinder = 2πrh + 2πr²
= 2πr(r + h)

→ In general, the volume of a solid is the product of its base area and height.
V = A. h
Eg: Volume of a cube = a² . a = a³
Volume of a cylinder = πr² . h = πr²h

→ The volume of solid formed by joining two basic solids is the sum of volumes of the constituents.

→ Surface area of the combination of solids: In calculating the surface area of the solid which is a combination of two or more solids, we can’t add the surface areas of all its constituents, because some part of the surface area disappears in the process of joining them.

→ Surface areas and volume of different solid shapes:

→ Some solid figures and their combination shapes:

Students can go through AP SSC 10th Class Maths Notes Chapter 3 Polynomials to understand and remember the concepts easily.

AP State Syllabus SSC 10th Class Maths Notes Chapter 3 Polynomials

→ Polynomial: An algebraic expression in which the variables involved have only non-negative integer power is called a polynomial.
Ex: 2x + 5, 3x² + 5x + 6, -5y, x³, etc.

→ Polynomials are constructed using constants and variables.

→ Coefficients operate on variables, which can be raised to various powers of non negative integer exponents.
, etc. are not polynomials.

→ General form of a polynomial having n^th degree is p(x) = a₀xⁿ + a₁x^n-1 + a₂x^n-2 …….. + a_n-1x + a_n where a₀, a₁, a₂,…… a_n-1, a_n are real coefficients and a₀ ≠ 0.

→ Degree of a polynomial:
The exponent of the highest degree term in a polynomial is known as its degree.
In other words, the highest power of x in a polynomial f(x) is called the degree of a polynomial f(x).
Example:
i) f(x) = 5x + \(\frac{1}{3}\) is a polynomial in the variable x of degree 1.
ii) g(y) = 3y² – \(\frac{5}{2}\)y + 7 is a polynomial in the variable y of degree 2.

→ Zero polynomial: A polynomial of degree zero is called zero polynomial that are having only constants.
Ex: f(x) = 8, f(x) = –\(\frac{5}{2}\)

→ Linear polynomial: A polynomial of degree one is called linear polynomial.
Ex: f(x) = 3x + 5, g(y) = 7y – 1, p(z) = 5z – 3.
More generally, any linear polynomial in variable x with real coefficients is of the form f(x) = ax + b, where a and b are real numbers and a ≠ 0.
Note: A linear polynomial may be a monomial or a binomial because linear polynomial f(x) = \(\frac{7}{5}\)x – \(\frac{5}{2}\) is a binomial, whereas the linear polynomial g(x) = \(\frac{2}{5}\) x is a monomial.

→ Quadratic polynomial: A polynomial of degree two is called quadratic polynomial.
Ex: f(x) = 5x², f(x) = 7x² – 5x, f(x) = 8x² + 6x + 5.
More generally, any quadratic polynomial in variable x with real coefficients is of the form f(x) = ax² + bx + c, where a, b and c are real numbers and a ≠ 0.
Note: A quadratic polynomial may be a monomial or a binomial or a trinomial.
Ex: f(x) = \(\frac{1}{5}\)x² is a monomial, g(x) = 3x² – 5 is a binomial and
h(x) = 3x² – 2x + 5 is a trinomial.

→ Cubic polynomial: A polynomial of degree three is called cubic polynomial.
Ex: f(x) = 8x³, f(x) = 9x³ + 5x²
f(y) = 11y³ – 9y² + 7y,
f(z) = 13z³ – 12z² + 11z + 5.

→ Polynomial of degree ‘n’ in standard form: A polynomial in one variable x of degree n is an expression of the form f(x) = a_nxⁿ + a_n-1x^n-1 + …….. + a₁x + a₀ where a₀, a₁, a₂,…… a_n, a_n are constants and a_n ≠ 0.
In particular, if a₀ = a₁ = a₂ = …… = a_n = 0 (all the constants are zero; we get the constants zero), we get the zero polynomial which is not defined.

→ Value of a polynomial at a given point: If p(x) is a polynomial in x and α is a real number. Then the value obtained by putting x = a in p(x) is called the value of p(x) at x = α.
Ex : Let p(x) = 5x² – 4x + 2, then its value at x = 2 is given by
p(2) = 5(2)² – 4(2) + 2 = 5(4) – 8 + 2 = 20 – 8 + 2 = 14 Thus, the value of p(x) at x = 2 is 14.

→ Graph of a polynomial: In algebraic or in set theory language, the graph of a polynomial f(x) is the collection (or set) of all points (x, y) where y = f(x).
i) Graph of a linear polynomial ax + b is a straight line.
ii) The graph of a quadratic polynomial (ax² + bx + c) is U – shaped, called parabola.

→ If a > 0 in ax² + bx + c, the shape of parabola is opening upwards ‘∪’.

→ If a < 0 in ax² + bx + c, the shape of parabola is opening downwards ‘∩’

→ Relationship between the zeroes and the coefficient of a polynomial:

Note: Formation of a cubic polynomial : Let α, β, and γ be the zeroes of the polynomial.
∴ Required cubic polynomial = (x – α) (x – β) (x – γ).

→ How to make a quadratic polynomial with the given zeroes : Let the zeroes of a quadratic polynomial be α and β.
∴ x = α, x = β
Then, obviously the quadratic polynomial is (x – α) (x – β) i.e., x² – (α + β)x 4- ap.
i.e., x² – (sum of the zeroes) x + product of the zeroes.

→ Division Algorithm : If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that, p(x) = g(x) × q(x) + r(x)
i.e., Dividend = Divisor × Quotient + Remainder
where, r(x) = 0 or degree of r(x) < degree of g(x). This result is known as the division algorithm for polynomials.

→ Some useful relations:
α² + β² = (α + β)² – 2αp
(α – β)² = (α + β)² – 4αβ
α² – β² = (α + β) (α – β) = (α + β)\(\sqrt{(\alpha+\beta)^{2}-4 \alpha \beta}\)
α³ + β³ = (α + β)³ – 3αβ(α + β)
α³ – β³ = (α – β)³ + 3αβ(α – β)

Students can go through AP SSC 10th Class Maths Notes Chapter 2 Sets to understand and remember the concepts easily.

AP State Syllabus SSC 10th Class Maths Notes Chapter 2 Sets

→ Set theory is comparatively a new concept in mathematics.

→ This theory was developed by George Cantor.

→ A well defined collection of objects or ideas is known as “set”.

→ Well defined means that:
i) All the objects in the set should have a common feature or property.
ii) It should be possible to decide whether any given object belongs to the set or not.
Example or comparision of well defined and not well defined collections:

Not well defined collections	Well defined collections
i) A family of rich persons	i) A family of persons having more than one crore rupees
ii) A group of tall students	ii) A group of students, with height 160 cm or more
iii) A group of numbers	iii) A group of even natural numbers less than 15

→ Some more examples of well defined collections:
i) Vowels of English alphabets, namely a, e, i, o, u.
ii) Odd natural numbers less than 11, namely 1, 3, 5, 7, 9.
iii) The roots of the equation x² – 3x + 2 = 0, i.e., 1 and 2.

→ Objects, elements and members of a set are synonymous words.

→ Sets are usually denoted by the capital letters like A, B, C, X, Y, Z, etc.

→ An object belonging to a set is known as a member/element/individual of the set.

→ The elements of a set are represented by small case letters,
i.e., a, b, c, , x, y, z, etc.

→ If ‘b’ is an element of a set A, then we say that ‘b’ belongs to A.

→ The word ‘belongs to’ is denoted by the Greek symbol ‘∈’.

→ Thus, in a notation form, ‘b’ belongs to A is written as b ∈ A and ‘c’ does not belong to ‘A’ is written as c ∉ A.

→ Representation of sets: Sets are generally represented by the following two methods.
i) Roster (or) Tabular form
ii) Rule method (or) Set builder form.

→ Roster (or) Tabular form: In this form, all elements of the set are written, separated by commas, within curly brackets.
Example:
i) The set of all natural numbers less than 5 is represented as N = {l,2,3,4}
ii) The set of all letters in the word “JANUARY” is represented as B = {A, J, N, R, U, Y}
Note:
a) In a set notation, order is not important.
b) The elements of a set are generally not repeated in a particular set.

→ Set builder form (or) Rule method: In this method, a set is described by using a representative and stating the property (or) properties which the elements of the set satisfy, through the representative.
Example:
i) Set of all natural numbers less than 5.
A = {x : x ∈ N, x < 5}
ii) Set of vowels of the English alphabet.
V = {x : x is a vowel in the English alphabet)
Note: It may be observed that we describe the set by using a symbol (x or y or z etc.) for elements of the set.

→ Types of set:

→ Empty set (or) Null set (or) Void set: A set, which does not contain any element is called an empty set (or) a null set (or) a void set.

→ Empty set is denoted by ∅ (or) { }

Example :
A = (x : x is a natural number smaller than 1}
B = {x : x² – 2 = 0 and x is a rational number}
C = (x : x is a man living on the moon}
Note: ∅ and { 0 } are two different sets. {0} is a set containing the single element ‘0’ while { } is a null set.

→ Singleton set: A set consisting of a single element is called a singleton set.
Examples:
{ 0 }, {- 7} are singleton sets.

→ Finite set: A set which is possible to count the number of elements of that set is called finite set.
Example – 1 : The set {3, 4, 5, 6} is a finite set, because it contains a definite number of elements i.e., only 4 elements.
Example – 2 : The set of days in a week is a finite set.

Example – 3 : An empty set, which does not contain any element (no element) is also a finite set.

→ Infinite set: A set whose elements cannot be listed, that type of set is called infinite set.

Example : i) B = {x : x is an even number}
ii) J = {x : x is a multiple of 7}
iii) The set of all points in a plane. s|s A set is infinite if it is not finite.

→ Equal sets: Two sets are said to be equal, if they have exactly the same elements.
For example: The set A and B are having same elements i.e., watch, ring, flower are said to be equal sets.

→ Cardinal number: The number of elements in a set is called the cardinal number of the set.
Example: Consider the finite set A = {1, 2, 4}
Number of elements in set ‘A’ is 3.
It is represented by n(A) = 3

→ Universal set: A set which consists of all the sets under consideration (or) discussion is called the universal set. (or) A set containing all objects or elements and of which all other sets or subsets.
It is usually denoted by ∪ (or) μ.

The universal set is usually represented by rectangles.
Example:
i) The set of real numbers is universal set for number theory.
Here ‘R’ is a universal set.
ii) If we want to study various groups of people of our state, universal set is the set of all people in Andhra Pradesh.

→ Subset: If every element of first set (A) is also an element of second set (B), then first set (A) is said to be a subset of second set (B).

→ It is represented as A ⊂ B.
Example :
Set A = {2, 4, 6, 8} is a subset of .
Set B = {1,2, 3, 4,5, 6, 7, 8}

→ Empty set is a subset of every set.

→ Every set is a subset of itself.

→ Consider ‘A’ and ‘B’ are two sets, if A ⊂ B and B ⊂ A ⇔ A = B.

→ A set doesn’t change if one or more elements of the set are repeated.

→ If A ⊂ B, B ⊂ C ⇒ A ⊂ C.

→ Venn Diagrams: Venn-Euler diagram or simply Venn diagram is a way of representing the relationships between sets.

→ These diagrams consist of rectangles and closed curves usually circles.
Example: Consider that U = {1, 2, 3, ……, 10} is
the universal set of which, A = {2, 4, 6, 8, 10} is a subset.
Then the Venn diagram is as:

→ Basic operations on sets: In sets, we define the operations of union, intersection and difference of sets.

→ Union of sets: The union of two or more sets is the set of all those elements which are either individual (or) both in common.

→ In symbolic form, union of two sets A and B is written as A ∪ B and usually read as “A union B”.

→ Set builder form of A ∪ B is A ∪ B = (x : x ∈ A or x ∈ B}

→ The union of the sets can be represented by a Venn diagram as shown (shaded portion).

→ It is evident from the definition that A ⊆ A ∪ B; B ⊆ A ∪ B.

→ Roster form of union of sets : Let A = {a, e, i, o, u} and B = (a, i, u} then A ∪ B = {a, e, i, o, u} ∪ { a, i, u} = {a, e, i, o, u}

→ Intersection of sets: The intersection of two sets A and B is the set of all those elements which belong to both A and B.

→ We denote intersection by A ∩ B.

→ We read A ∩ B as “A intersection B”.

→ Symbolically, we write A ∩ B = (x : x ∈ A and x ∈ B}

→ The intersection of A and B can be illustrated in the Venn diagram as shown in the shaded portion in the adjacent figure.

→ The intersection of A and B can be illustrated in the Roster form:
If A = {5, 6, 7, 8} and B = {7, 8, 9, 10} then A ∩ B = {7, 8}

→ Disjoint set: Consider A and B are two finite sets and if there are no common element in both A and B. Such set is known as disjoint set (or A ∩ B = ∅).
(or)
Two sets (finite) are said to be disjoint sets if they have no common elements. That is if the intersection of two sets is a null set they are disjoint sets.

→ The disjoint sets can be represented by means of the Venn diagrams as shown in the adjacent figure.

→ Difference of sets: The difference of sets A and B is the set of elements which belong to ‘A’ but do not belong to ‘B’.

→ We denote the difference of A and B by A – B or simply “A minus B”.

→ Set builder form of A – B is (x : x ∈ A and ∉ B}

→ Venn-diagram of A – B is

→ Venn-diagram of B – A is

→ A – B ≠ B – A

→ Fundamental theorem on sets:
If A and B are any two sets then n (A ∪ B) = n (A) + n (B) – n (A ∩ B) where
n (A ∪ B) = number of elements in the set (A ∪ B), also called cardinal number of A ∪ B
n (A) = number of elements in the set A also called cardinal number of A
n (B) = number of elements in the set B also called cardinal number of B