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Summary Algebra

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Course
Maths

Institution
12

 Quadratic Equation – Theory 1 – 10  Complex Number – Theory 11 – 25  Sets, Relations and Functions – Theory 26 – 46  Quadratic Equation – Exercise – 1 A1 – A5  Complex Number – Exercise – 1 A6 – A10  Sets, Relations and Functions – Exercise – 1 A1...

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Uploaded on July 26, 2021
Number of pages 110
Written in 2020/2021
Type Summary

quadratic equations
complex numbers
sets
relations
functions

Institution
12
Education
12
Course
Maths
School year 4

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CONTENTS
ALGEBRA-1 Pages

 Quadratic Equation – Theory 1 – 10
 Complex Number – Theory 11 – 25
 Sets, Relations and Functions – Theory 26 – 46
 Quadratic Equation – Exercise – 1 A1 – A5
 Complex Number – Exercise – 1 A6 – A10
 Sets, Relations and Functions – Exercise – 1 A11 – A16
 Quadratic Equation – Exercise – 2 B1 – B3
 Complex Number – Exercise – 2 B4 – B6
 Sets, Relations and Functions – Exercise – 2 B7 – B8
 Quadratic Equation – Exercise – 3 C1 – C8
 Complex Number – Exercise – 3 C9 – C16
 Sets, Relations and Functions – Exercise – 3 C17 – C23
 Quadratic Equation – Exercise – 4 D1 – D6
 Complex Number – Exercise – 4 D7 – D14
 Sets, Relations and Functions – Exercise – 4 D15 – D16

, QUADRATIC EQUATION

1. QUADRATIC EXPRESSION 3. NATURE OF ROOTS

The general form of a quadratic expression in x is,
(a) Consider the quadratic equation ax 2 + bx + c = 0
f (x) = ax + bx + c, where a, b, c R & a z 0.
2
where a, b, c R & a z 0 then;
and general form of a quadratic equation in x is,
(i) D > 0 roots are real & distinct (unequal).
ax + bx + c = 0, where a, b, c R & a z 0.
2

(ii) D = 0 roots are real & coincident (equal).
2. ROOTS OF QUADRATIC EQUATION
(iii) D < 0 roots are imaginary..
(a) The solution of the quadratic equation,
(iv) If p + i q is one root of a quadratic equation,
then the other must be the conjugate p – i q &
2
b ± b2 4ac
ax + bx + c = 0 is given by x =
2a vice versa. p, q R & i = -1 .
The expression D = b2 – 4ac is called the discriminant
of the quadratic equation. (b) Consider the quadratic equation ax 2 + bx + c = 0
where a, b, c Q & a z 0 then;
(b) If D & E are the roots of the quadratic equation
(i) If D > 0 & is a perfect square, then roots are
ax2 + bx + c = 0, then ;
rational & unequal.
(i) D + E = – b/a (ii) α β = c/a
(ii) If α = p + q is one root in this case, (where p
D
(iii) | α β |= .
|a|
is rational & q is a surd) then the other root
(c) A quadratic equation whose roots are D & E is
(x – D) (x – E ) = 0 i.e. must be the conjugate of it i.e. β = p - q & vice

x 2 – ( D + E ) x + DE = 0 i.e. versa.

x 2 – (sum of roots) x + product of roots = 0.

Remember that a quadratic equation cannot have
y (ax 2 bx c) { a(x D) (x E) three different roots & if it has, it becomes an

2
identity.
§ b · D
a ¨x ¸
© 2a ¹ 4a

1

,4. GRAPH OF QUADRATIC EXPRESSION 6. MAX. & MIN. VALUE OF QUADRATIC EXPRESSION

Consider the quadratic expression, y = ax 2 + bx + c, Maximum & Minimum Value of y = ax2 + bx + c occurs
at x = –(b/2a) according as :
a z 0 & a, b, c R then ;
For a > 0, we have :
(i) The graph between x, y is always a parabola.
If a > 0 then the shape of the parabola is
concave upwards & if a < 0 then the shape of
the parabola is concave downwards.

(ii) y > 0 x R, only if a>0&D<0
ª 4ac - b 2 ·
(iii) y < 0 x R, only if a<0&D<0 y« , f ¸¸
¬ 4a ¹

5. SOLUTION OF QUADRATIC INEQUALITIES

ax2 + bx + c > 0 a z 0 .

(i) If D > 0, then the equation ax2 + bx + c = 0 has D b
ymin at x , and y max o f
4a 2a
two different roots (x1 < x 2).

Then a > 0 x (–f, x1) (x2 , f)
For a < 0, we have :
a<0 x (x1, x2)

§ 4ac b 2 º
y ¨¨ f, »
© 4a ¼

P x
(ii) Inequalities of the form 0 can be
Q x
D b
ymax at x , and ymin o f
4a 2a
quickly solved using the method of intervals
(wavy curve).

2

, 7. THEORY OF EQUATIONS

If D1, D2, D3, ....., Dn are the roots of the nth degree
polynomial equation : Remainder Theorem : If f (x) is a polynomial, then
f (h) is the remainder when f (x) is divided by x – h.
f (x) = a0xn + a 1xn–1 + a2 xn–2 + ...... + an–1x + an = 0
Factor theorem : If x = h is a root of equation
f (x) = 0, then x–h is a factor of f (x) and conversely.
where a 0, a 1, ....... a n are all real & a0 z 0,

Then,

a 9. MAX. & MIN. VALUES OF RATIONAL EXPRESSION
¦ α1 = – a 1 ;
0
Here we shall find the values attained by a rational
a2
¦ α1 α 2 = a0
;
a1x 2 b1x c1
expresion of the form for real values
a 2 x 2 b2 x c2

a
¦ α1 α 2 α3 = – a 3 ; of x.
0
Example No. 4 will make the method clear.
............
10. COMMON ROOTS
an
α1 α 2 α 3.....α n = (–1) n
a0 (a) Only One Common Root

Let D be the common root of ax 2 + bx + c = 0 &
8. LOCATION OF ROOTS
a’x 2 + b’x + c’ = 0, such that a, a’ z 0 and a b’ z a’b.
Let f (x) = ax2 + bx + c, where a > 0 & a, b, c R.
Then, the condition for one common root is :
(i) Conditions for both the roots of f(x) = 0 to be
greater than a specified number ‘k’ are : (cac cca) 2 = (abc a cb) (bcc bcc).
Dt 0 & f (k) > 0 & (–b/2a) > k.
(ii) Conditions for both roots of f (x) = 0 to lie on (b) Two Common Roots
either side of the number ‘k’ (in other words Let D , E be the two common roots of
the numb er ‘k’ lies between the r oo ts o f
f (x) = 0 is: ax 2 + bx + c = 0 & a’x 2 + b’x + c’ = 0,
a f (k) < 0. such that a, a’ z 0.
(iii) Conditions for exactly one root of f (x) = 0 to lie
Then, the condition for two common roots is :
in the interval (k1, k2) i.e. k1 < x < k2 are :
D>0 & f (k1) . f (k2) < 0. a b c
a' b' c'
(iv) Conditions that both roots of f(x) = 0 to be
confined between the numbers k 1 & k 2 are
(k1 < k2) :
D t 0 & f (k1) > 0 & f (k2) > 0 & k1 < (–b/2a) < k2.

3

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