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Quadratic Equation MC Sir1.. Theory of Equations : Relation between Roots and Coefficients of Cubic and Higher 1 and Coefficients of Cubic and Higher Polynomials 4.. Maximum and Minimum

Quadratic Equation MC Sir

1 Introduction, Graphs

2 Inequality

3 Theory of Equations : Relation between Roots

and Coefficients of Cubic and Higher

1

and Coefficients of Cubic and Higher

Polynomials

4 Identity

5 Infinite Roots, Common Roots

6 Maximum and Minimum Values of Quadratic and

Rational Function

7 General 20 in x and y

factorized in two linears

9 Location of Roots

10.Modulus Inequality

11.Logarithm Inequality

Quadratic Equation MC Sir

No of Questions

3

2008 2009 2010 2011 2012

Cubic Polynomial

a = leading coefficient

d = absolute term

Roots of Quadratic Equation

7

ax 2 + bx + c = 0

Sum of roots = – b/a

Product of roots = c/a

Different Graphs of

Quadratic Expression

9

Parabola y

In general graph of y = ax 2 + bx + c ;

11

In general graph of y = ax 2 + bx + c ;

y = 0 for only one

value of x (root)

13

In General

Q.

cuts the x axis at 2 distinct point)

15

D = 0 Q.

2 1

y

In General

a < 0 mouth facing downward

Q.

a < 0 mouth facing downward

D = 0 (one real root) parabola touch the x axis

19

Leading Coefficient < 0

Y

X

In General

Q.

the x-axis at two distinct points.

21

Co-ordinate of vertex

x =

y =

Nature of Roots

23

Nature of Roots

If D is a perfect square, then roots are rational.

p

-25

If p + iq is one root of a quadratic equation,

then the other root must be the conjugate

(a) are real and negative

(b) have negative real parts

(c) have positive real parts

27

Both the roots of the equation

(x – b) (x – c) + (x – a) (x – c) + (x – a) (x – b) = 0 are always

Q.

are always

Let , be the roots of the equation

The number of points of intersection of two

then the interval in which a lies is

Assignment 1

If a, b R, a 0 and the quadratic equation

ax 2 – bx + 1 = 0 has imaginary roots then

a + b + 1 is :

Q.2

a + b + 1 is :

[Multiple Objective Type]

The graph of the quadratic polynomial;

If a, b, c R such that a + b + c = 0 and a c, then prove that the roots of

Find the value of a for which the roots of the

equal.

Q.5

41

For what values of m does the equation

Q.6

For what values of m does the equation

Q.7

43

Relation between root and Coefficient of Quadratic Equation

Formation of Quadratic Equation

45

Form a Quadratic Equation with rational coefficients whose one root is tan75°

Q.

Form a Quadratic Equation with rational coefficients whose one root is tanπ/8

Q.

(x 2 – 5x + 6) (x 2 – 6x + 5) 0

Type – 3

61

2 – x – x 2 0

Type – 3

3x 2 – 7x + 4 0

Type – 3

63

Rules :

Type – 4

Repeated Linear Factor

Rules :

Get rid of even power

odd power treat as linear

Type – 4

Repeated Linear Factor

x (x + 6) (x + 2) (x – 3) > 0

Type – 5

Rational Inequality

Type – 5

Rational Inequality

69

Type – 5

Rational Inequality

Type – 5

Rational Inequality

71

Type – 5

Rational Inequality

Type – 5

Rational Inequality

73

Type – 5

Rational Inequality

Q.

75

Q.

Q.

77

Let y = Find all the real values of x for which y takes Q.

Find the set of all x for which

[IIT-JEE 1987] Q.

79

Solve |x 2 + 4x + 3| + 2x + 5 = 0 [IIT-JEE 1988] Q.

Example

Let a and b be the roots of the equation

Fill in the blank :

Fill in the blank :

If the products of the roots of the equation

Q.

then the roots are real for k = … .

[IIT-JEE 1984]

If x, y and z are real and different and

is always

Q.

is always

[IIT-JEE 1979]

85

If one root is square of the other root of the

Assignment 2

87

The sum of all the value of m for which the

If and are the roots of the equation

The set of values of 'a' for which the inequality, (x - 3a) (x - a - 3) < 0 is satisfied

If and are the roots of a(x 2 – 1) + 2bx = 0 then, which one of the following are the roots

of the same equation?

Solve the following Inequality Q.5

Solve the following Inequality Q.5

93

Solve the following Inequality

Q.5

Solve the following Inequality Q.5

95

Double Inequality

Solve the following Inequality

(ii) Q.

(ii)

Solve the following Inequality

(iv) Q.

(iv)

real roots then show that c (a + b + c) > 0

Q Find a,

109

x R

111

Example

Q If x = 3 +

113

equal root.

Show that :

equation (l – m) x 2 – 5 (l + m) x – 2(l – m) = 0 are

(c) real and unequal (d) none of these

[IIT-JEE 1979]

117

equations 3x + my = m, 2x – 5y = 20 has solution satisfying the conditions x > 0, y > 0

[IIT-JEE 1980]

2

of p, q, r and s.

[IIT-JEE 1979]

121

Assignment 3

Solve the following inequalities

123

Solve the following inequalities

Solve the following inequalities

125

Solve the following inequalities

Solve the following inequalities

127

Solve the following inequalities

Q.7 For what values of c does the equation

Q.8 For what values of a does the equation

possess equal roots ? Solve the following inequalities

possess equal roots ?

Q.9 Find the value of k for which the curve

Solve the following inequalities

131

Q.10 Find the least integral value of k for which

two different real roots.

Solve the following inequalities

two different real roots.

Q.11 If the equation 4x 2 – 4(5x + 1) + p 2 = 0 has

one root equals to two more than the other, then the value of p is equal to

Solve the following inequalities

133

then the value of p is equal to

Q.12 Possible values of x simultaneously satisfying

the system of inequalities Solve the following inequalities

Note

Prove that above is an identity

Q.

Example

Prove that above is an identity

Quadratic With One Root Zero

Product of root = = 0

c = 0

139

Quadratic With Both Root Zero

Sum of root = Product of root = 0

b = 0, c = 0

Quadratic With One Root Infinite

a = 0

141

Quadratic With Both Root ∞

a = 0, b = 0, c 0

roots infinite Find p & q

143

Symmetric Function

Then f( , ) is called symmetric function of ,

(ii) f( , ) = cos ( - ) (iii) f ( , ) = sin ( - ) (iv) f ( , ) = ( 2 - )

145

Condition of Common Root

Condition for both Roots Common

a

1 x 2 + b

1 x 2 + c

1 = 0 a

2 x 2 + b

2 x 2 + c

2 = 0

147

Condition for One Root Common

149

Example

b R

151

that there uncommon roots are roots of

153

155

have only one root common then show that quadratic equation containing their other common roots is

157

Fill in the blank :

Assignment 4

Q.1 Find value of k for which the equation

have a common root

161

Q.2 If x be the real number such that x 3 + 4x + 8.

Q.3 If every solution of the equation

value of (a + b) is equal to

163

Q.4 If x 2 + 3x + 5 = 0 & ax 2 + bx + c = 0 have

minimum value of a + b + c

Q.5 Determine the values of m for which the

may have a common root.

165

Q.6 Q.7 For what value of a is the difference

(a – 4) x – 2 = 0 equal to 3 ?

Q.7 Find all values of a for which the sum of the

equal to the sum of the squares of its roots.

167

Q.8 For what values of a do the equations

have a root in common ?

Maximum & Minimum

Value of Quadratic Equation

minimum at point where x =

according as a < 0 or a > 0.

Maximum and Minimum value can be

obtained by making a perfect square.

169

Q.

171

Q.

Range of Linear

y = ax + b ;a 0

y R

Q y = f(x) = x + 1

Example

175

Range of

y =

Example

177

Example

Example

179

Example

, Find range of y

Range of

Assume y

Check for common roots in numerator & denominator

Form Quadratic Equation

to zero

Find range of following

Q.

183

Find range of following

Q.

Find range of following

Q.

185

Find range of following

Q.

Find range of following

Q.

187

Find range of following

Q.

Assignment 5

189

Q.1 Find the range of the function f(x) = x 2 – 2x – 4

Q.2 Find the least value of

191

Q.3 Find Range

Q.4 Find the domain and Range of

193

General 2° in x & y

Condition of General 2° in x & y

to be Resolved into two linear

Factors

195

Step 1 : factorize purely 2°

Step 2 : Add constant to both the linear

Step 3 : Compare coefficient of x & coefficient of y &

satisfied by real values of x & y then show

199

Theory of Equation

Sum & Product of Root

taken 1 at a time

= -d/a

201

Sum of root taken 2 at a time

Bi Quadratic

203

Sum of root taken 2 at a time

Sum of root taken 3 at a time

205

Note

207

Find

209

211

Example

Location of Roots

Type -1

Both roots of a quadratic equation are greater

than a specified number

( , ) > d

215

greater than 3

217

the quantity ‘a’.

a > 0

f(d) < 0

d

Q Find k for which 1 root of the equation is

greater than 2 and other is less than 2

Example

x – (k + 1) x + k + k – 8 = 0

221

Q Find the set of value of ‘a’ for which zeroes

of the quadratic polynomial

Example

(a + a + 1) x + (a – 1)x + a are located on either side of 3.

Q Find a for which one root is positive, one is

Example

223

Q Find a for which both root lie on either side

of -1 of quadratic

Example

(a – 5a + 6) x – (a – 3) x + 7 = 0

Type - 3

Both roots lies between two fixed number

d < < < e

225

Type - 4

Both roots lies on either side of two fixed number

< d < e <

one root of the equation

Type - 6

If f (p) f(q) < 0

lies between (p, q)

Q Find p for which the expression

one real x

all positive x

Q Show that for any real value of a

one negative x.

239

least one negative x, find all values of a

has at least one positive root.

241

Q Find p for which the least value of

243

Modulas Inequality

Q.

249

| x | < x (- , )

| x | > x (- , - ) ( , )

Q (| x – 1 | – 3) (| x + 2 | – 5) < 0

251

Q | x – 5| > | x 2 – 5x + 9 |

Q.

253