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Mind Map Overal Idea Content Speed Notes Quick Coverage Content : (Scroll down till end of the page) Study Tools Audio, Visual & Digital Content Content … Key Terms Topic Terminology Term Important Tables Table: . Assessments Test Your Learning readmore
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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
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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)
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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.
(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.
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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
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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)
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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.
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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
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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
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An angle is positive if its rotation is in the anticlockwise and negative if its rotation is in the clockwise direction.
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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
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 =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
(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
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