Assessment Tools
Assign | Assess | Analyse
Quick Quiz
Objective Assessment
Question Bank
List Of Questions With Key, Aswers & Solutions
Back To Learn
Re – Learn
Go Back To Learn Again
Level: Board
Board Level
Algebraic Expressions and Identities | Study
Mind Map Overal Idea Content Speed Notes Quick Coverage Expressions are formed from variables and constants. Constant: A symbol having a fixed numerical value. Example: 2,, 2.1, etc. (Scroll down till end of the page) Study Tools Audio, Visual & Digital Content Variable: A symbol which takes various numerical values. Example: x, y, z, etc.… readmore
Mind Map
Overal Idea
Content
Speed Notes
Quick Coverage
Expressions are formed from variables and constants.
Constant: A symbol having a fixed numerical value.
Example: 2,, 2.1, etc. (Scroll down till end of the page)
Study Tools
Audio, Visual & Digital Content
Variable: A symbol which takes various numerical values. Example: x, y, z, etc.
Algebric Expression: A combination of constants and variables connected by the sign
+, -, and is called algebraic expression.
Terms are added to form expressions.
Terms themselves are formed as product of factors.
Expressions that contain exactly one, two and three terms are called monomials, binomials and trinomials respectively.
In general, any expression containing one or more terms with non-zero coefficients (and with variables having non- negative exponents) is called a polynomial.
Like terms are formed from the same variables and the powers of these variables are the same, too.
Coefficients of like terms need not be the same.
While adding (or subtracting) polynomials, first look for like terms and add (or subtract) them; then handle the unlike terms.
There are number of situations in which we need to multiply algebraic expressions: for example, in finding area of a rectangle, the sides of which are given as expressions.
Monomial: An expression containing only one term. Example: -3, 4x, 3xy, etc.
Binomial: An expression containing two terms. Example: 2x-3, 4x+3y, xy-4, etc.,
Polynomial: In general, any expression containing one or more terms with non-zero coefficients (and with variables having non-negative exponents).
A polynomial may contain any number of terms, one or more than one.
A monomial multiplied by a monomial always gives a monomial.
Multiplication of a Polynomial and a monomial:
While multiplying a polynomial by a monomial, we multiply every term in the polynomial by the mononomial.
Trinomial: An expression containing three terms.
Example:
3x+2y+5z, etc.
In carrying out the multiplication of a polynomial by a binomial (or trinomial), we multiply term by term, i.e., every term of the polynomial is multiplied by every term in the binomial (or trinomial).
Note that in such multiplication, we may get terms in the product which are like and have to be combined.
An identity is an equality, which is true for all values of the variables in the equality.
On the other hand, an equation is true only for certain values of its variables.
An equation is not an identity.
The following are the standard identities:
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab +b2
(a + b)(a – b) = a2 – b2
(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.
Coefficients: In the term of an expression any of the factors with the sign of the term is called the coefficient of the product of the other factors.
Terms: Various parts of an algebraic expression which are separated by + and – signs. Example: The expression 4x + 5 has two terms 4x and 5.
- Constant Term: A term of expression having no lateral factor.
- Like term: The term having the same literal factors. Example 2xy and -4xy are like terms.
(iii) Unlike term: The terms having different literal factors.
Example:
are unlike terms.
and 3xy
Factors: Each term in an algebraic expression is a product of one or more number (s) and/or literals. These number (s) and/or literal (s) are known as the factor of that term. A constant factor is called numerical factor, while a variable factor is known as
a literal factor. The term 4x is the product of its factors 4 and x.
Hindi Version Key Terms
Topic Terminology
Term
Important Tables
Table:
.
Assessments
Test Your Learning
Advanced Tools For Study Assess Interact FORCE AND PRESSURE | Assess
Assessment Tools
Assign | Assess | Analyse
Quick Quiz
Objective Assessment
Question Bank
List Of Questions With Key, Aswers & Solutions
Back To Learn
Re – Learn
Go Back To Learn Again
FORCE AND PRESSURE | Study
Mind Map Overal Idea Content Speed Notes Quick Coverage Force: A push or a pull, that changes or tends to change the state of rest or of uniform motion of an object or changes its direction or shape. A force arises due to the interaction between two objects. Force has magnitude as well as direction.Therefore… readmore
Mind Map
Overal Idea
Content
Speed Notes
Quick Coverage
Force: A push or a pull, that changes or tends to change the state of rest or of uniform motion of an object or changes its direction or shape.
A force arises due to the interaction between two objects.
Force has magnitude as well as direction.Therefore force is a vector quantity.
The SI unit of force is newton (Scroll down till end of the page)
Study Tools
Audio, Visual & Digital Content
A change in the speed of an object or the direction of its motion or both implies a change in its state of motion.
Force acting on an object may cause a change in its state of motion or a change in its shape.
Contact Non Contact Forces:
A force can act on an object with or without being in contact with it. Based on Contact the forces are classclassified as Contact Forces and Non Contact Forces.
Contact Forces: The forces act on a body when the source of force touches the body directly.
The point where the force is applied on an object is called the point of application of force (or point of contact).
Examples of Contact Forces:
(i) Muscular Force: The force exerted by the muscles of the body.
We use force acted by muscles of animals like Humans, bullocks, horses and camels to get our activities done.
(ii) Mechanical Force: The force acted by a machine.
Non-Contact Forces:
Non-Contact Forces: Forces which do not involve physical contact between two bodies on which they act.
Examples of Non-Contact Forces:
(i) Magnetic Force: A magnet exerts a non-contact force on objects made of iron, steel, cobalt or nickel.
(ii) Electrostatic Force: The force which result due to repulsion of similar charges or attraction of opposite charges.
(iii) Gravitational Forces: The force that exists between any two bodies by virtue of
Pressure
Pressure: Thrust acting per unit surface area is called pressure.
Thrust
Thrust is the force acting on an object perpendicular to its surface.
In SI system, pressure is measured in newton per square metre which is equal to 1 pascal (Pa).
Like solids, fluids (liquids and gases) also exert pressure.
A solid exerts pressure only in the downward direction due to its weight, whereas liquids and gases exert pressure inall directions.
Hence liquids and gases exert pressure on the walls of their container.
Atmospheric Pressure
Ttmosphere: The thick blanket of air that covers the earth is termed atmosphere.
The pressure exerted by the atmosphere is called atmospheric Pressure.
The tremendous atmospheric pressure surrounding us is not felt by us because the fluid pressure inside our bodies counter-balances the atmospheric pressure around us.
Hindi Version Key Terms
Topic Terminology
Term
Important Tables
Table:
.
Assessments
Test Your Learning
Advanced Tools For Study Assess Interact Heredity | Assess
Assessment Tools
Assign | Assess | Analyse
Quick Quiz
Objective Assessment
Question Bank
List Of Questions With Key, Aswers & Solutions
Back To Learn
Re – Learn
Go Back To Learn Again
How do Organisms Reproduce | Study
Mind Map Overal Idea Content Speed Notes Quick Coverage Content Study Tools Content … Key Terms Topic Terminology Term: Important Tables Topic Terminology Term: Assessments Test Your Learning readmore
Mind Map
Overal Idea
Content
Speed Notes
Quick Coverage
Content
Study Tools
Content …
Key Terms
Topic Terminology
Term:
Important Tables
Topic Terminology
Term:
Assessments
Test Your Learning
Advanced Tools For Study Assess Interact Introduction to Trigonometry | Assess
Assessment Tools
Assign | Assess | Analyse
Quick Quiz
Objective Assessment
Question Bank
List Of Questions With Key, Aswers & Solutions
Back To Learn
Re – Learn
Go Back To Learn Again
Introduction to Trigonometry | Study
Mind Map Overal Idea Content Speed Notes Quick Coverage Content : (Scroll down till end of the page) Study Tools Audio, Visual & Digital Content Speed Notes Quick Coverage Introduction To Trigonometry | Speed Notes Notes For Quick Recap An angle is positive if its rotation is in the anticlockwise and negative if its rotation… readmore
Mind Map
Overal Idea
Content
Speed Notes
Quick Coverage
Content : (Scroll down till end of the page)
Study Tools
Audio, Visual & Digital Content
Speed Notes
Quick Coverage
Introduction To Trigonometry | Speed Notes
Notes For Quick Recap
An angle is positive if its rotation is in the anticlockwise and negative if its rotation is in the clockwise direction.
(Scroll down to continue …)
Study Tools
Audio, Visual & Digital Content
Trigonometric Ratios
If one of the trigonometric ratios of an acute angle is known, the remaining trigonometric ration of the angle can be determined.
Two angles are said to be complementary, if their sum is 900 and each one of them is called the complement of the other.
sin (900 – θ) = Cos θ
Cos (900– θ)= Sin θ
tan (900– θ) = Cot θ
Cot(900– θ) = tan θ
sec (900– θ)= cosec θ
cosec (900– θ) = sec θ
Trigonometric Identities
An equation with trigonometric ratios of an angle θ, which is true for all values of ‘ θ ‘, for which the given trigonometric ratios are defined, is called an identity.
The three fundamental trigonometric identities are
- sin2 θ +cos2 θ = 1
⇒ sin2 θ =1-cos2 θ
⇒ sin2 θ =(1-cos θ)(1+cos θ)
⇒ (1- cos θ) = (sin2 θ) /(1+ cos θ)
⇒ (1+ cos θ) = (sin2 θ) /(1- cos θ)
⇒ cos2 θ + sin2 θ = 1
cos2 θ =1- sin2 θ
⇒ cos2 θ =(1- sin θ)(1+ sin θ)
⇒ (1+ sin θ) = (cos2 θ) /(1- sin θ)
⇒ (1- sin θ) = cos2 θ /(1+sin θ)
(b) sec2 θ = 1 + tan2 θ
⇒ sec2 θ – tan2 θ =1
⇒ (sec θ – tan θ)(sec θ + tan θ) = 1
⇒ (sec θ – tan θ) = 1/ (sec θ + tan θ)
⇒ (sec θ + tan θ) = 1/ (sec θ – tan θ)
⇒ sec2 θ – 1 = tan2 θ
⇒ (sec θ – 1)( sec θ – 1) = tan2 θ
(c) cosec2 θ = 1+cot2θ
⇒ cosec2 θ – cot2 θ = 1
⇒ (cosec θ – cot θ)(cosec θ + cot θ)=1
(cosec θ+ cot θ) =1cosec θ – cot θ
(cosec θ- cot θ) = 1cosec θ + cot θ
⇒ Cosec2 θ – 1 = cot2 θ
⇒ (Cosec θ – 1)( Cosec θ – 1) = Cot2 θ
Supportive Formulae:
(a+b)2=+a2+b2+2ab
(a-b)2 = a2+b2-2ab
(a+b)2+(a-b)2= 2 (a2+b2)
(a+b)2– (a-b)2= 4ab
(a-b)2– (a+b)2= – 4ab
(a+b)2 = (a-b)2+ 4ab
(a-b)2 = (a+b)2– 4ab
(a2-b2)=(a+b)(a-b)
a+b=(a2-b2) /(a-b)
a-b=(a2-b2) /(a-b)
(a+b)2= (a-b)2+ 4ab
Hindi Version Hindi Version Key Terms
Topic Terminology
Term
Important Tables
Table:
.
Assessments
Test Your Learning
Advanced Tools For Study Assess Interact LINES AND ANGLES | Study
Mind Map Overal Idea Content Speed Notes Quick Coverage Basic terms and Definitions 1. Point – A Point is that which has no component. It is represented by a dot. 2. Line – When we join two distinct points then we get a line. A line has no endpoints; it can be extended on both… readmore
Mind Map
Overal Idea
Content
Speed Notes
Quick Coverage
Basic terms and Definitions
1. Point – A Point is that which has no component. It is represented by a dot.
2. Line – When we join two distinct points then we get a line. A line has no endpoints; it can be extended on both sides infinitely.
3. Line Segment Line – Segment is the part of the line which has two endpoints.
4. Ray – Ray is also a part of the line that has only one endpoint and has no end on the other side.
5. Collinear points: Points lying on the same line are called Collinear Points.
6. Non-collinear points: Points which do not lie on the same line are called Non-Collinear Points. (Scroll to continue …)
(Scroll down till end of the page)
Study Tools
Audio, Visual & Digital Content
Angles
When two rays begin from the same endpoint then they form an Angle. The two rays are the arms of the angle and the endpoint is the vertex of the angle.
Types of Angles
Angle Notation Image Acute An angle which is between 0° and 90°. Right An angle which is exactly equal to 90°. Obtuse An angle which is between 90° and 180°. Reflex An angle which is between 180° and 360° Straight An angle which is exactly equal to 180°. Complete An angle which is exactly equal to 360°. Types of Angles Complementary and Supplementary Angles
Complementary Angles are the different angles whose sum is 90°.
Complementary Angles are the different angles whose sum is 180°.
Intersecting Lines and Non-intersecting Lines
There are two ways to draw two lines-
1. The lines which cross each other from a particular point are called Intersecting Lines.
2. The lines which never cross each other at any point are called Non-intersecting Lines. These lines are called Parallel Lines and the common length between two lines is the distance between parallel lines.
Pairs of Angles Axioms
1. If a ray stands on a line, then the sum of two adjacent angles formed by that ray is 180°.
This shows that the common arm of the two angles is the ray which is standing on a line and the two adjacent angles are the linear pair of the angles. As the sum of two angles is 180° so these are supplementary angles too.
2. If the sum of two adjacent angles is 180°, then the arms which are not common of the angles form a line.
This is the reverse of the first axiom which says that the opposite is also true.
Vertically opposite Angles Theorem
When two lines intersect each other, then the vertically opposite angles so formed will be equal.
AC and BD are intersecting each other so ∠AOD = ∠BOC and ∠AOB = DOC.
Parallel Lines and a Transversal
If a line passes through two distinct lines and intersects them at distant points then this line is called Transversal Line.
Here line “l” is transversal of line m and n.
Exterior Angles – ∠1, ∠2, ∠7 and ∠8
Interior Angles – ∠3, ∠4, ∠5 and ∠6
Pairs of angles formed when a transversal intersects two lines-
1. Corresponding Angles:
- ∠ 1 and ∠ 5
- ∠ 2 and ∠ 6
- ∠ 4 and ∠ 8
- ∠ 3 and ∠ 7
2. Alternate Interior Angles:
- ∠ 4 and ∠ 6
- ∠ 3 and ∠ 5
3. Alternate Exterior Angles:
- ∠ 1 and ∠ 7
- ∠ 2 and ∠ 8
4. Interior Angles on the same side of the transversal:
- ∠ 4 and ∠ 5
- ∠ 3 and ∠ 6
Transversal Axioms
1. If a transversal intersects two parallel lines, then
- Each pair of corresponding angles will be equal.
- Each pair of alternate interior angles will be equal.
- Each pair of interior angles on the same side of the transversal will be supplementary.
2. If a transversal intersects two lines in such a way that
- Corresponding angles are equal then these two lines will be parallel to each other.
- Alternate interior angles are equal then the two lines will be parallel.
- Interior angles on the same side of the transversal are supplementary then the two lines will be parallel.
Lines Parallel to the Same Line
If two lines are parallel with a common line then these two lines will also be parallel to each other.
As in the above figure if AB ∥ CD and EF ∥ CD then AB ∥ EF.
Angle Sum Property of a Triangle
1. The sum of the angles of a triangle is 180º.
∠A + ∠B + ∠C = 180°
2. If we produce any side of a triangle, then the exterior angle formed is equal to the sum of the two interior opposite angles.
∠BCD = ∠BAC + ∠ABC.
Hindi Version Key Terms
Topic Terminology
Term
Important Tables
Table:
.
Assessments
Test Your Learning
Advanced Tools For Study Assess Interact
