
01. Real Numbers
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Lecture1.2

Lecture1.3

Lecture1.4

Lecture1.5

Lecture1.6

Lecture1.7

Lecture1.8

Lecture1.9

Lecture1.10


02. Polynomials
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Lecture2.1

Lecture2.2

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Lecture2.4

Lecture2.5

Lecture2.6

Lecture2.7

Lecture2.8

Lecture2.9

Lecture2.10

Lecture2.11

Lecture2.12


03. Linear Equation
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Lecture3.1

Lecture3.2

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Lecture3.6

Lecture3.7

Lecture3.8

Lecture3.9

Lecture3.10

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Lecture3.12


04. Quadratic Equation
10
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Lecture4.4

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Lecture4.6

Lecture4.7

Lecture4.8

Lecture4.9

Lecture4.10


05. Arithmetic Progressions
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Lecture5.2

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Lecture5.4

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Lecture5.6

Lecture5.7

Lecture5.8

Lecture5.9

Lecture5.10

Lecture5.11


06. Some Applications of Trigonometry
7
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Lecture6.4

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Lecture6.7


07. Coordinate Geometry
17
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Lecture7.4

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Lecture7.6

Lecture7.7

Lecture7.8

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08. Triangles
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Lecture8.18


09. Circles
9
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Lecture9.7

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Lecture9.9


10. Areas Related to Circles
12
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11. Introduction to Trigonometry
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Lecture11.2

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Lecture11.9


12. Surface Areas and Volumes
9
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Lecture12.2

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Lecture12.4

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Lecture12.6

Lecture12.7

Lecture12.8

Lecture12.9


13. Statistics
14
Lecture13.1

Lecture13.2

Lecture13.3

Lecture13.4

Lecture13.5

Lecture13.6

Lecture13.7

Lecture13.8

Lecture13.9

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Lecture13.12

Lecture13.13

Lecture13.14


14. Probability
9
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Lecture14.2

Lecture14.3

Lecture14.4

Lecture14.5

Lecture14.6

Lecture14.7

Lecture14.8

Lecture14.9


15. Construction
7
Lecture15.1

Lecture15.2

Lecture15.3

Lecture15.4

Lecture15.5

Lecture15.6

Lecture15.7

Chapter Notes – Quadratic Equation
(1) A polynomial of degree 2 is called a quadratic polynomial. The general form of a quadratic polynomial is ax2+bx+c, where a, b, c are real number such that a ≠0 and x is a real variable.
For Example: x2+5x+3, where a=1,b=5,c=3 are real number. So given equation is quadratic polynomial.
(2) If p(x)=ax2+bx+c,a≠0 is a quadratic polynomial and α is a real number, then p(α)=aα2+bα+c is known as the value of the quadratic polynomial p(α)
For Example: p(α)=α2+5α+3 in that equation if α=3 then p(α)=27. So 27 is a value of quadratic polynomial
(3) A real number α is said to be a zero of quadratic polynomial p(x)=ax2+bx+c, if p(α)=0.
For Example: p(x)=x2+6x+5
If x=(−5) then p(x)=0, So −5 is the zero of polynomial.
(4) If p(x)=ax2+bx+c is a quadratic polynomial, then p(x)=0 i.e., ax2+bx+c=0, a≠0 is called a quadratic equation.
For Example: p(x)=x2−8x+16 is a quadratic polynomial, then p(x)=0 i.e., x2−8x+16=0, a≠0 is called a quadratic equation.
(5) A real number α is said to be a root of the quadratic equation ax2+bx+c=0.
In other words, α is a root of ax2+bx+c=0 if and only if α is a zero of the polynomial p(x)=ax2+bx+c.
For Example: Suppose quadratic equation is 2x2−x−6=0.
If we put x=2 then p(x)=0, So 2 is a root of that given equation so here α=2.
(6) If ax2+bx+c=0, a≠0 is factorizable into a product of two linear factors, then the roots of the quadratic equation ax2+bx+c=0 can be found by equating each factor to zero.
For Example: The Given equation is
Now, Solving the above equation using factorization method.
(3x + 1) (3x – 2) =0
(3x + 1) = 0 or (3x – 2) = 0
3x = 1 or 3x = 2
or
Hence, and are the two roots of the given equation
For Example:The Given equation is –
Dividing through out by 2
Shifting the constant term to the right hand side.
Adding square of the half of coefficient of x on the both side.
Taking square root of both sides
Hence x= 3, and x=1/2 are the two root of the given equation
(8) The roots of the quadratic equation ax2+bx+c=0, a≠0 can be found by using the quadratic formula −b±b2−4ac√2a , provided that b2−4ac−−−−−−−√≥0.
For Example: the given equation in the form of ,
Where a= √3, b=10 c= 8√3
Therefore, the discriminant
D= (10)^{2} – 4 x √3 x (8√3)
D= 100 + 96
D= 196
Since, D > 0
Therefore, the roots of the given equation are real and distinct.
The real roots α and β are given by,
;
For,
Hence and are the two root of the given equation.
(9) Nature of the roots of quadratic equation ax2+bx+c=0, a≠0 depends upon the value of D=b2−4ac, which is known as the discriminate of the quadratic equation.
For Example: Value of D can be (i) D>0 , (ii) D=0 (iii) D<0.
(10) The quadratic equation ax2+bx+c=0 , a≠0 has:
(i) Two distinct real roots, if D ba4ac 0 two equal roots i.e. coincident real roots if D=b2−4ac>0
For Example: 16x^{2} = 24x + 1
16x^{2} – 24x – 1 = 0
The given equation is of the form of ax^{2} + bx + c = 0, where a = 16, b = 24, c = 1
Therefore, the discriminant D = b^{2} – 4ac
D= (24)^{2} – 4 x 16 x (1)
D= 576 + 64
D= 640
Since, D > 0
Therefore, the roots of the given equation are real and distinct.
The real roots α and β are given by,
For,
Hence and are the two root of the given equation.
(ii) Two equal roots i.e. coincident real roots, if D=b2−4ac=0.
For Example: 2x^{2} – 2√6x + 3 = 0
The given equation is of the form of ax^{2} + bx + c = 0, where a = 2, b = – 2√6, c = 3
Therefore, the discriminant D = b^{2} – 4ac
= ( 2√6)^{2} – 4 x 2 x 3
= 24 – 24
= 0
Since, D = 0
Therefore, the roots of the given equation are real.
The real and equal roots are given by and .
⇒
(iii) No real roots, if D=b2−4ac<0.
For Example: The given equation is
x^{2} + x + 2 = 0
The given equation is of the form of ax^{2} + bx + c = 0, where a = 1, b = 1, c = 2
Therefore, the discriminant
D = b^{2} – 4ac
D= (1)^{2} – 4 x 1 x 2
D= 1 – 8
D= 7
Since, D < 0
Therefore, the given equation has not real roots.
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