Maths (UM)

MA THEMA TICS CLASS IX

2 3

2.5

2.1 2.2 2.3 2.4 2.6 2.7 2.82.9

2.7 2.8

2.75

2.71 2.72 2.73 2.74 2.762.772.782.79

2.77 2.78

2.775

2.771 2.772 2.773 2.774 2.7762.7772.7782.779

Class IX

MATHEMATICS

NA TIO L

A R

C E

U S

D E

E A

F RC

O H

LI &

CN TR

U A

O IC NIE NT GA TS

S

T

R E

C

FREE

SCERT TELANGANA

SCERT TELANGANA

SCERT

TELANGANA

SCERT

TELANGANA

SCERT

TELANGANA

SCERT

TELANGANA

SCERT

TELANGANA

SCERT

TELANGANA

SCERT

TELANGANA

SCERT TELANGANA

SCERT

TELANGANA

SCERT

TELANGANA

1.1

Q Z W N

Z I Z

Z = {... -3, -2, -1, 0, 1, 2, 3, …}

p q Q

q 0 q p

q 0

Real Numbers 1

7, 100

0,7

N 100

W ^0,7,-3, 100

0,7, 100 Z -3,

Q

SCERT TELANGANA

q = 1 p = -15 15

=...

q 0 q p

q p q 0

q p (Universal factor)

'0'

1, 2, 3, 4, ...

(Pictorially)

SCERT TELANGANA

p/q 0,7,10,-4 .2

.i .ii .iii .i .ii

b

q p .iii

x '0'

SCERT TELANGANA

b a I

b a b = 4 a = 3

4 3

:II

2 + 1 = 3 4 3

4 3 3 =

4 3 5 4 3

5 + 1 = 6

4 3

5 3 2 .i

10 8/11 3/11 .ii

SCERT TELANGANA

(ii) (i)

q p 3.28 5

3.28 =

=

=

2/3 = 0.666 = 0.6

SCERT TELANGANA

q p

x = 1.626262... (1) 100 100x = 162.6262... (2)

(1) (2)

100x = 162.6262...

x = 1.6262...

_______________

99x = 161 x =

(1)

...

SCERT TELANGANA

1.1 (a) .1 (b)

.2 .i .ii .iii .iv .v 2 1 .3

.4 .5 .6

(iv (iii (ii (i .I

(iv (iii (ii (i .II

q p .7

3.25 (iv 10.25 (iii 15.4 (ii 0.36 (i

.8

(iv (iii (ii (i

.9

(iv (iii (ii (i

SCERT TELANGANA 1.2

x² = 2 (iii) 3x = 4 (ii) x² = 4 (i)

2 2 2 2 x (i)

3 3x = 4 (ii)

x² = 2 (iii)

2 :I :II III :IV

SCERT TELANGANA

= 1.4142135623731...

; q p

= 1.7320508075689

= 2.2360679774998...

Q' S

(1) 2.1356217528... (2) ,

Theodorus (800BC) Sulba Sutra

n

SCERT TELANGANA

(d) (c) (d) (c)

d c

3.142857 3.140845

3.140845 < < 3.142857

3.1416 (476-500AD)

1.24

=3.14159265358979323846264338327950 …..

= 3.14159 .... 3.14 14

1829 14

SCERT TELANGANA

1.3

OB= 1 OABC O

(i)

O O OB O OB =

K

(i)

(ii)

OD (ii) OB 1 BD

D

SCERT

TELANGANA

OD =

L O C OD O

n

[5² = (2)² + (1)² ]

(Real Numbers) 1.3

q p

p

q

R Real Numbers

R

, 7, 8, 2, 7, 100, , ,

, 9, 999...

SCERT TELANGANA

= 0.20

0.201201120111..., 0.24114111411114…, 0.25231617181912..., 0.267812147512 … 4

ab b a

b a

= 4 3

= =

(ii) (i)

(iv) (iii)

= (i)

= 6

SCERT

TELANGANA

= = 9 3 = 6

(iii)

= (iv)

1.2

.1

30.232342345… (iii) (ii) (i)

0.3030030003... (vi) 11.2132435465 (v) 7.484848… (iv) .2 .3 0.77 0.7 .4

3 .5

.6 .7

2 3 2 .8

.9 (i) (ii) (iii)

n (iv)

n (v)

SCERT TELANGANA

OP O :1

OP PQ :2

OP = PQ = 1

OQ = Q Q O :3

OQ QR :4

OR = R O :5

OR RS 6

:7 ....

...

1.4

3 2 2.776

SCERT TELANGANA

(i) 10 3 2

(Magnifying 2.1, 2.2, 2.3

(i) 3 2 Glass)

2.8 2.7 2.8 2.7 2.776

2.72 2.71 10

(iii)

2.78 2.77 2.776

10 (ii)

(iii)

2.776 2.772 2.771

(ii)

SCERT

TELANGANA

4 11 3.5888

1.3

2.874 .1

3 5.28 .2

1.5

:I

:II

:III

:IV

3.5888

SCERT TELANGANA

O O

2

13 (iv) (iii) (ii) (i)

(i) = 5(1.414…) = 7.070…

(ii) = = 3.535… i

(iii) = 21+1.732… = 22.732…

(iv) = 3.1415… + 3 = 6.1415…

14

=

=

=

s q

qs q - s q + s s 0)

SCERT TELANGANA

15

b a (i)

(ii) b 0

(iii) (iv) (v) (vi) (vii)

16

(i) (ii)

(iii) (iv)

(i) =

(ii)

(iii) =

(iv) =

SCERT TELANGANA

1.5.1

= 1.4142135...

=

= 2

R.F

R.F

(iii) (ii) (i)

SCERT TELANGANA

18

=

19

20

=

SCERT TELANGANA

1.5.2

(i) (ii) (iii)

(iv) (v) (vi)

n m b a b 0 a 0 m b a

(i) (ii)

(iii) (iv)

(i) (ii) (iii) (iv)

n n^th

3 9 3² = 9

5² = 25

2 8 2³ = 8

SCERT

TELANGANA

16 2⁴ = 16

2 32 2⁵ = 32

2 64 2⁶ = 64

a n b aⁿ = b

n a > 0

n a b bⁿ = a

q p a > 0

(ii) (i)

(v) (iv) (iii)

20

(iv) (iii) (ii) (i)

(i) (ii)

= (iii)

= (iv)

(16)^1/2 (i) (128)^1/7 (ii) (343)^1/5 (iii)

SCERT TELANGANA

(Surd)

n a n

a n a n a^1/n

n

7

8

.1

.2

.2

iv) v) vi) vii)

1.4

SCERT TELANGANA

x, y, z .3

(i) (ii) (iii)

(iv) (v) (vi) t⁴ = 256

.4 .5

(i) (ii) (iii) (iv)

.6

(i) (ii) (iii) (iv)

.7

.8 (i) (ii) (iii)

(iv) (v) (vi)

.9

b a b a .10

(i) (ii)

SCERT

TELANGANA

q 0 q p .1

q 0 q p .2

.3 .4 .5 .6

qs q- s q + s s q .7

n .8

b a .9

(i) (ii)

(iii) (iv)

(v) (vi)

b a .10

q p b > 0 a > 0 .11

(i) (ii) (iii)

(iv)

n a n .12

SCERT

TELANGANA

2.1

x x

x²

q p 10

`10p r

` 10r `10q

s²

lbh lb

3xy, x²+2x, x³ x² + 4x + 3, r², ax + b

-1

Polynomials and Factorisation 2

SCERT TELANGANA

2.2

x, y, z ...

x ...

P = 4s

s 4

SCERT TELANGANA

ax, by, cz .... a, b, c, ...

( a r b i t r a r y c o n s t a n t ) a,b,c, . . . .

x y 2x² + 3xy + 5y²

2.3

(i) 3x² + 7x + 5 (ii) 3x²y² + 4xy + 7

5 7x 3x² 3x² + 7x + 5 7x 3 x² 3x² + 7x + 5

x⁰ 5 7

2 3x² + 7x + 5 3x² +7x+5

xy 3 x²y³ 3x²y³ + 4xy + 7

2 + 3 = 5 3x²y³ 7 x⁰y⁰ 4

5 x²y³+ 4xy + 7 x⁰

5x⁰ 5

n 3 2 1

SCERT TELANGANA

x n

(0)

.1 (i) 7x³ + 5x² + 2x 6 (ii) 7 x + 3x²

(iii) 5p - (iv) 2 (v) 5xy²

x² .2

(i) 15 3x + 2x² (ii) 1 x² (iii) x² 3x + 5 (iv)

(i)

n^th n

SCERT TELANGANA

(ii)

2x 5 3x

p(x), q(x) r(x) x

p(x) = 3x² + 2x + 1 .1

q(x) = x³ 5x² + x 7

r(y) = y⁴ 1 15 p .2

t(z) = z² + 5z + 3

x, y x² y² x² + 2xy + y² x + y

x³ + y³ + z³ x² + y² + z² 3

SCERT TELANGANA

2.1

.1 (i) x⁵ x⁴ + 3 (ii) x² + x 5 (iii) 5

(iv) 3x⁶ + 6y³ 7 (v) 4 y² (vi) 5t

.2 (i) 3x² 2x + 5 (ii) x² + (iii) p² 3p + q (iv)

(v) (vi) x¹⁰⁰ + y¹⁰⁰

x³ .3

(i) x³ + x + 1 (ii) 2 x³ + x² (iii) (iv) 2x³ + 5

(v) (vi) (vii) (viii) 4

.4

(i) 5x² + x 7 (ii) x x³ (iii) x² + x + 4 (iv) x 1

(v) 3p (vi) r²

.5

2 (i)

(ii)

3 (iii)

(iv) 2 x² + 2x + y² (v) r² (vi)

10 .6

2.4

p(x) = x² + 5x + 4 -

x = 1 p(x)

1 x p(x)

SCERT TELANGANA

10 p(x) x = 1 p(x)

x = 1 x = 0 p(x)

p( 4)

s( 1)

(i) p(x) = 4x² 3x + 7 , x = 1 (ii) q(y) = 2y³ 4y + , y = 1 (iii) r(t) = 4t⁴ + 3t³ t² + 6 , t = p , (iv) s(z) = z³ - 1 , z = 1

(v) p(x) = 3x² + 5x 7, x = 1 (vi) q(z) = 5z³ 4z + , z = 2

r(t) = t 1 r(1) , r(1) = 1 1 = 0

r(1) = 0 p(x) p(x) = 0

SCERT TELANGANA

p(x)

f(x) = x + 1

x + 1 = 0 (0) x + 1

x f (x) = 0 x f (x) x = 1

x + 1 = 0 x + 1 1 f (x) 1

3 x 3 = 3x⁰

3x⁰

x + 2 p( 2) p(2) p(1) p(x) = x + 2 1 -

p(x) = x + 2

1 x

p(1) = 1 + 2 = 3

2 x

p(2) = 2 + 2 = 4

1 x

p( 1) = + 2 = 1

2 x

p( 2) = + 2 = 0

2 x + 2 1, 2, 1

p(x) = 3x + 1 2 - p(x) p(x) = 0

3x + 1 = 0 3x = 1 x =

x² 5x + 6 .2 2x 3 .1 x + 5 .3

SCERT TELANGANA

3x + 1

2x 1 3 -

2x 1 = 0 p(x) = 0 p(x)

x =

P

p(x) p(x) = ax + b

p(x) = 0 p(x)

ax + b = 0 ax = b x =

x =

p(x) = ax + b

SCERT TELANGANA

x² 3x + 2 1 2 4 - p(x) = x² 3x + 2

2 x p(2) = (2)² 3(2) + 2

= 4 6 + 2 = 0 1 x p(1) = (1)² 3(1) + 2

= 1 3 + 2 x² 3x + 2 1 2

x² 3x + 2

a x² + 2x a 3 5 -

p(x) = x² + 2x a

p(3) = 0 3

x² + 2x a = 0

x = 3 (3)² + 2 (3) a = 0

9 + 6 a = 0

15 a = 0 a = 15

a = 15

x² + 1 .1

n .2

2.2

4x² 5x + 3 (i) x = 0 (ii) x = 1 (iii) x = 2 (iv) x =

SCERT TELANGANA

p(2) p(1) p(0) .2

.3

.4

p,q

a p(x) = 2x² 3x + 7a 2 .5 b a f (x) = 2x³ - 3x² + ax + b 1 0 .6

2.5

8 3 25 3 25 (i)

25 = (8 x 3) + 1

20 = (4 x 5) + 0 5 20

5 20 20 5

SCERT TELANGANA

x 3x³ + x² + x (ii) (3x³ + x² + x) x =

= 3x² + x + 1

3x³ + x² + x x

3x³ + x² + x = x(3x² + x + 1)

3x³ + x² + x

(2x² + x + 1) x (iii)

(2x² + x + 1) x =

= 2x + 1 +

= x ¹

2x + 1 +

(2x² + x + 1) = [x x (2x + 1)] + 1

'1'

(1) x (2x + 1)

2x² + x + 1 x

'y' 3y³ + 2y² + y .1 2p 4p² + 2p + 2 .2

x + 1 3x² + x - 1 6 - (x) = x + 1 p(x) = 3x² 3x 1

q(x) p(x)

SCERT TELANGANA

(x + 1)3x :2 (x + 1) 3x = 3x² + 3x

(-2x) 3x² + x 3x² + 3x :3 2x 2 = (x + 1)( 2) :4

1 2x 1

'1' :5

(+1) (3x 2)

3x² + x 1 = (x + 1) (3x 2) + 1

= 1 x p(x) p(x) = 3x² + x - 1

p(x) = 3x² + x - 1

= 3(+1) + ( 1) 1 = 1 (x + 1) p(x) = 3x² + x 1 i.e: x = 1 (x + 1) 1 p( 1)

(x + 1) 2x⁴ 4x³ 3x 7 7 - f (x) = 2x⁴ 4x³ 3x 1

x2x⁴

2x⁴ 2x³ 2x³ (x 1) 2x³

p( 1)

1

SCERT

TELANGANA

2x³ 2x² 2x 5 - 6 1 (x 1)

f (x) x = 1 f (x) = 2x⁴ 4x³ 3x 1

f (1) = 2(1)4 4(1)³ 3(1) 1

= 2(1) 4(1) 3(1) 1

= 2 4 3 1

= 6

f(x) (x - 1) f (x)

a p(x)

p(a) (x a) p(x)

p(x)

r(x) q(x) g(x) = x a p(x)

g(x) 0 g (x) p(x) g(x) p(x)

g (x) r(x) r(x) = 0 r(x) q(x)

p(x) = g(x) . q(x) + r(x)

p(x) = (x a) . q(x) + r(x) g(x) = (x a) (x a) r(x) (x a)

K r(x) r(x) = 0

r(x) = K x

p(x) = (x a) q(x) + K

p(a) = (a a) q(a) + K x = a

= 0 + K

SCERT

TELANGANA

(x + 1) x³ + 1 8 - p(x) = x³ + 1

[x + 1 = 0, x = 1] 1 x + 1

1 x

p( 1) = ( 1)³ + 1

= 1 + 1

= 0 x + 1 x³ + 1

x + 1 x³ + 1 (x³ + 1) (x + 1)

x³ 2x² 5x + 4 (x 2) 9 - p(x) = x³ 2x² 5x + 4

'0' x (x 2) (x 2)

x 2 = 0 x = 2 p(2) = (2)³ 2(2)² 5(2) + 4

= 8 2(4) 10 + 4

= 8 8 10 + 4

= 6.

x³ 2x² 5x + 4 (x 2)

2y + 1 p(y) = 4y³ = 4y² y 1 10 -

(2y + 1) p(y) p(y) (2y + 1)

2y + 1

SCERT

TELANGANA

y p(y)

= 0

(2y + 1) p(y) p(y) (2y + 1)

(x - 2) 2x³ + 5x + a ax³ 3x² - 13 11 - a q(x) = 2x³ - 5x + a p(x) = ax³ + 2x² - 3x² - 13

(x - 2) q(x) p(x) p(2) = q(2)

a(2)³ + 3(2)² 13 = 2(2)³ 5(2) + a 8a + 12 13 = 16 10 + a

8a 1 = a + 6 8a a = 6 + 1 7a = 7

a = 1

2.3

x³ + 3x² + 3x + 1 .1

(i) x + 1 (ii) x 1/2 (iii) x (iv) x + p (v) 5 + 2x

x - p x³ px² + 6x p .2

SCERT TELANGANA

2x 3 2x² 3x + 5 .3

x 2/3 9x³ 3x² + x 5 .4 x 2 x³ + x² 4x + a 2x³ + ax² + 3x 5 .5

a

a (x + 2) x³ 2x² a x³ ax² + 5 .6

g(x) = x 2 f (x) = x⁴ 3x² + 4 .7

g(x) = 1 2x p(x) = x³ 6x² + 14x 3 .8

2 x + 2 2 (x 2) 2x³ +3x² + ax + b .9

b a

2.6

q(x) p(x) q(x)

p(x) g(x) = 2x + 1 p(x) = 4x³ + 4x² x 1

4x³ + 4x² x 1 = q(x) (2x + 1) + 0 p(x) = q(x) (2x + 1)

p(x) g(x) = 2x + 1

(x a) (i) a n > 1 p(x)

(ii) p(a) = a p(x)

p(a) = 0 p(x) (x a)

p(x) = (x a) q(x) + p(a)

p(x) = 0 (i)

SCERT

TELANGANA

p(x) (iii) (ii)

= 0 p(a) = 0

12 -

2

K 13 -

P(x)

SCERT TELANGANA

27 81 + 6 + 4K = 0 48 + 4K = 0

4K = 48

K = 12

x¹⁰ 1 14 -

= 1 1 = 1 1

= 0 = 0

ax + bx + c

=

= prx² + (ps + qr) x + qs a = pr

b = ps + qr c = qs

qs ps b (ps) (qr) = (pr)(qs)

= ac

b ax² = bx + c

(x 1)

xⁿ 1

SCERT TELANGANA

15 -

pq = 3 x 6 = 18 p + q = 1 q p

18 p + q = 11 9 2 (1, 18), (2, 9), (3, 6)

1. 6x² + 19x + 5 2. 10m² 31m 132 3. 12x² + 11x + 2

x² 3x + 2 2x⁴ - 6x³ + 3x² + 3x 2 16 -

x² 3x + 2 = x² 2x x + 2

= x(x 2) 1(x 2)

= (x 2) (x 1).

(x 2) 2x⁴ 6x³ 3x² + 3x 2 x² 3x + 2

2x⁴ 6x³ 3x² + 3x 2 (x 1) p(x) = 2x⁴ 6x³ 3x² + 3x 2

p(2) = 2(2)⁴ 6(2)³ + 3(2)² + 3(2) 2

= 2(16) 6(8) + 3(4) + 6 2

= 32 48 + 12 + 6 2

= 50 50

SCERT TELANGANA

p(x) (x 2) p(2) = 0 p(1) = 2(1)⁴ 6(1)³ + 3(1)² + 3(1) 2

= 2(1) 6(1) + 3(1) + 3 2

= 2 6 + 3 + 3 2

= 8 8

= 0

p(x) (x - 1) p(1) = 0

x² 3x + 2 p(x) (x 1) (x 2)

p(x) = 2x⁴ 6x³ + 3x² + 3x x³ 23x² + 142x 120 17 - p(x) = x³ 23x² + 142x 120

p(1) = 0

p(x) (x 1) x² - 22x + 120 (x 1) p(x)

x³ 23x² + 142x 120 = x³ x² 22x² + 22x + 120x 120 = x²(x 1) 22x (x 1) + 120 (x 1) = (x 1) (x² 22x + 120)

x² 22x + 120 x² - 22x + 120 = x² 12x 10x + 120

= x(x 12) 10 (x 12) = (x 12) (x 10)

x³ 23x² + 142x 120 = (x 1) (x 10) (x 12).

b a b a a | b

b a b a a b

n n

SCERT TELANGANA

2.4

(x + 1) .1

(i) x³ x² x + 1 (ii) x⁴ x³ + x² x + 1 (iii) x⁴ + 2x³ + 2x² + x + 1 (iv) x³ x² (3 ) x +

f (x) g(x) .2

(i) f (x) = 5x³ + x² 5x - 1, g(x) = x + 1 (ii) f (x) = x³ + 3x² + 3x + 1, g(x) = x + 1 (iii) f (x) = x³ - 4x² + x + 6, g(x) = x 2 (iv) f (x) = 3x³ + x² 20x + 12, g(x) = 3x 2 (v) f (x)= 4x³ + 20x² + 33x + 18, g(x) = 2x + 3

x³ 3x² 10x + 24 (x 4) (x 2) (x + 3) .3 x³ 6x² 19x + 84 (x 7) (x 4) (x 3) .4

p = r px² - 5x + r (x 2) .5

a + c +e = b + d = 0 ax⁴ + bx³ + cx² + dx + e (x² 1) .6 .7 (i) x³ 2x² x + 2 (ii) x³ 3x² 9x 5

(iii) x³ + 13x² + 32x + 20 (iv) y³ + y² y 1

a = b c = 0 x + 1 bx² + ax + c ax² + bx + c .8 a (x a) x² + 3x 18 x² x 6 .9 y³ 2y² 9y 18 (y 3) .10

2.6

(x + y)² x² + 2xy + y² I (x y)² x² 2xy + y² II

(x + y) (x - y) x² y² III -

SCERT TELANGANA

(x a) (x + b) x² (a + b) x + ab IV -

(x y)² x : I x y : II (x y)² : III

= x² - [(x y) y + (x y) y + y²]

= x² - xy + y² xy + y² y²

= x² - 2xy + y²

(i) (x + y)² x² + 2xy + y² (ii)(x + y)(x y) = x² y² (iii)(x + a)(x + b) x² + (a + b) x + ab (iv) (x+a)(x+2)(x+c)=x (a+b+c)x²+(ab+bc+ca)x+abc

(i) (x + 5) (x + 5) (ii) (P - 3) ( P + 3) (iii) (4 - 1) (4 - 1) (iv) (t + 2) (t + 4) (v) 102 98 (vi) (x+1) (x+2) (x+3)

18 - (i) x² + 5x + 4 (ii) 9x² 25

(iii) 25a² + 40ab + 16b² (iv) 49x² 112xy + 64y²

x² + 5x + 4 = x² + (4 + 1)x + (4) (1) (i) (x + a) (x + b) x² + (a + b)x + ab

(x + 4) (x + 1)

9x² 25 = (3x)² (5)² (ii) x² y² (x + y) (x y) III

9x² 25 = (3x + 5) (3x 5)

SCERT TELANGANA

(iii) 25a² + 40ab + 16b² = (5a)² + 2(5a) (4b) + (4b)²

x² + 2xy + y²

y = 4b x = 5a (x + y)² x² + 2xy + y² I

25a² + 40ab + 16b² (5a + 4b)

= (5a + 4b) (5a + 4b).

49x² 112xy + 64y² (iv) 49x² = (7x)², 64y² = (8y)²

112 xy = 2(7x) (8y) II

(x y)² x² 2xy + y²

49x² 112xy + 64y² = (7x)² 2(7x) (8y) + (8y)² = (7x 8y)²

= (7x 8y) (7x 8y)

(i) 49a² + 70ab + 25b² (ii)

(iii) t² 2t + 1 (iv) x² + 3x + 2

x + y + z I

(x + y + z)² (x + y + z)² = (t + z)² x + y = t

= t² + 2tz + z² I

= (x + y)² + 2(x + y) z + z² 't'

= x² + 2xy + y² + 2xz + 2yz + z² x² + y² + z² + 2xy + 2yz + 2zx

SCERT TELANGANA

(x + y + z)²

[(x + y) + z]² = (x + y)² + 2(x + y) (z) + (z)² [ I ] = x² + 2xy + y² + 2xz + 2yz + z²

= x² + y² + z² + 2xy + 2yz + 2xz

V : (x + y + z)² x² + y² + z² + 2xy + 2yz + 2zx

(2a + 3b + 5)² 19 -

(x + y + z)² z = 5 y = 3b x = 2a

V (2a + 3b + 5)² = (2a)² + (3b)² + (5)² + 2(2a)(3b) + 2(3b) (5) + 2(5) (2a)

= 4a² + 9b² + 25 + 12ab + 30b + 20a

(5x y + z) (5x y + z) 20 - (5x y + z) (5x y + z) = (5x y + z)²

= [5x + ( y) + z]² V (5x + ( y) + z)² = (5x)² + ( y)² + (z)² + 2(5x) ( y) + 2( y) (z) + 2(z) (5x)

= 25x² + y² + z² 10xy 2yz + 10zx

4x² + 9y² + 25z² 12xy 30yz + 20zx 21 -

4x² + 9y² + 25z² 12xy 30yz + 20zx

= [(2x)² + ( 3y)² + (5z)² + 2(2x)( 3y) + 2( 3y)(5z) + 2(5z)(2x)]

SCERT

TELANGANA

V (x + y + z)² x² + y² + z² + 2xy + 2yz + 2zx

= (2x 3y + 5z)²

= (2x 3y + 5z) (2x 3y + 5z)

(p + 2q 3z)² (i) (4x 2y 3z)² (ii) 4a² + b² + c² 4ab + 2bc 4ca (iii)

(x + y)² (x + y)³ I

(x + y)³ = (x + y)² (x + y) = (x² + 2xy + y²) (x + y)

= x(x² + 2xy + y²) + y(x² + 2xy + y²) = x³ + 2x²y + xy² + x²y + 2xy² + y³ = x³ + 3x²y + 3xy² + y³

= x³ + 3xy (x + y) + y³ = x³ + y³ + 3xy(x + y).

(x + y)³ x³ + y³ + 3xy (x + y) VI

(x y)³

(x y)³ x³ y³ 3xy (x y) VII x³ 3x²y 3xy² y³

SCERT TELANGANA

22 - (i) (2a + 3b)³ (ii) (2p 5)³

x = 2a, y = 3b (x + y)³ (i)

IV (2a + 3b)³ = (2a)³ + (3b)³ + 3(2a)(3b) (2a + 3b)

= 8a³ + 27b³ + 18ab (2a + 3b) = 8a³ + 27b³ + 36a²b +54 ab² = 8a³ + 36a²b + 54ab² + 27b³.

x = 2p, y = 5 (x + y)³ (ii)

VII (2p 5)³ = (2p)³ (5)³ 3(2p)(5) (2p 5)

= 8p³ 125 30p (2p 5)

= 8p³ 125 60p² + 150p = 8p³ 60p² + 150p 125.

23 -

(i) (103)³ (ii) (99)³

(i) (103)³ = (100 + 3)³ (x + y)³ x³ + y³ + 3xy(x + y)

= (100)³ + (3)³ + 3(100) (3) (100 + 3)

= 1000000 + 27 + 900(103)

= 1000000 + 27 + 92700

= 1092727.

(99)³ = (100 1)³ (ii) (x y)³ x³ y³ 3xy(x y)

= (100)³ (1)³ 3(100)(1) (100 1)

= 1000000 1 300 (99)

SCERT

TELANGANA

= 1000000 1 29700

= 970299.

8x³ + 36x²y + 54xy² + 27y³ 24 -

8x³ + 36x²y + 54xy² + 27y³ = (2x)³ + 3(2x)² (3y) + 3(2x) (3y)² + (3y)³ (x + y)³ x³ + 3x²y + 3xy² + y³ VI

= (2x + 3y)³

= (2x + 3y) (2x + 3y) (2x + 3y).

(x + 1)³ (1) (3m 2m)³ (2) a³ 3a²b + 3ab² b³ (3) (x + y + z) (x² + y² + z² xy yz zx)

= x(x² + y² + z² xy yz zx) + y(x² + y² + z² xy yz zx)

+ z(x² + y² + z² xy yz zx)

= x³ + y³ + z³ 3xyz

(x + y + z) (x² + y² + z² xy yz xz) x³ + y³ + z³ 3xyz VIII

(2a + b + c) (4a² + b² + c² 2ab bc 2ca) 25 -

SCERT TELANGANA

= (2a + b + c) [(2a)² + b² + c² (2a)(b) (b)(c) (c) (2a)]

VIII (x + y + z) (x² + y² + z² xy yz zx) x³ + y³ + z³ 3xyz

= (2a)³ + (b)³ + (c)³ 3(2a) (b) (c)

= 8a³ + b³ + c³ 6abc

a³ 8b³ 64c³ 24abc 26 -

a³ 8b³ 64c³ 24abc = (a)³ + ( 2b)³ + ( 4c)³ 3(a)( 2b)( 4c) VIII x³ + y³ + z³ 3xyz (x + y + z) (x² + y² + z² xy yz zx)

= (a 2b 4c) [(a)² + ( 2b)² + ( 4c)² (a) ( 2b) 2b) ( 4c) ( 4c) (a)]

= (a 2b 4c) (a² + 4b² + 16c² + 2ab 8bc + 4ca).

(a b c) (a² + b² + c² ab + bc ca) (1) 27a³ + b³ + 8c³ 18abc (2)

2x² + 9x 5 27 - b l

= 2x² + 9x 5 lb = 2x² + 9x 5

= 2x² + 10x x 5

= 2x(x + 5) 1(x + 5)

= (x + 5) (2x 1)

SCERT TELANGANA

(x + 5) = (2x 1) = x = 1, l = 6, b = 1 x = 2, l = 7, b = 3 x = 3, l = 8, b = 5 ...

...

2.5

.1

.2

.3

.4

SCERT TELANGANA

a² + b² + c² ab + bc + ca = 26 a + b + c = 9 .6 .7 (i) (99)³ (ii) (102)³ (iii) (998)³ (iv) (1001)³

.8 (i) 8a³ + b³ + 12a²b + 6ab² (ii) 8a³ b³ 12a²b + 6ab²

(iii) 1 64a³ - 12a + 48a² (iv)

(i) x³ + y³ = (x + y) (x² xy + y²) (ii) x³ y³ = (x y) (x²+ xy + y²) .9

(i) 27a³ + 64b³ (ii) 343y³ 1000 .10

27x³ + y³ + z³ 9xyz .11

x³ + y³ + z³ 3xyz = .12

x³ + y³ + z³ = 3xyz x + y + z = 0 .13 .14 (i) ( 10)³ + (7)³ + (3)³ (ii) (28)³ + ( 15)³ + ( 13)³

(iii) (iv) (0.2)³ (0.3)³ + (0.1)³

.15 (i) 4a² + 4a 3 (ii) 25a² 35a + 12

.16 (i) 3x³ 12x (ii) 12y² + 8y - 20.

a = b 2(a²+b²) = (a+b)² .17

SCERT

TELANGANA

x p(x) .1 a₀, a₁, a₂, .... a_n p(x) = a_nxⁿ + a_{n 1}x^{n 1} + ... + a₂x² + a₁x + a₀,

n a_n 0 x⁰, x¹, x², ... xⁿ p(x) a_{n 1}x^{n 1}... a₀

.2 .3

'a' p(x) 'a' p(a) = 0 .4

p(x) = 0

.5

(x a) p(x) p(x) .6

P(a)

(x a) p(a) = 0 p(a) = 0 p(x) x a .7

.8 (i) (x + y + z)² x² + y² + z² + 2xy + 2yz + 2zx

(ii) (x + y)³ x³ + y³ + 3xy(x + y) (iii) (x y)³ x³ y³ 3xy(x y)

(iv) x³ + y³ + z³ 3xyz (x + y + z) (x² + y² + z² xy yz zx) (v) x³ + y³ (x + y) (x² xy + y²)

(vi) x³ y³ (x y) (x² + xy + y²) (vii) x⁴ + y⁴ ((x+y)² + y²)((x - y)²+y²)

SCERT

TELANGANA

The Elements of Geometry 03

3.1

3.2

metrein (geo)

SCERT

TELANGANA

(8,15,17) (5,12,13) (3,4,5)

13 "The Elements" (325-265 BC)

(Euclid's Elements of Geometry) 3.3

i

ii

iii

SCERT

TELANGANA

23 "The Elements"

2 300

SCERT TELANGANA

(AXIOMS & POSTULATES) 3.3.1

C>P R P C P

C = P + R R

(Whole)

B B A

SCERT TELANGANA

B A .1

B A B A

1

PQ .2

PQ PQ

2

S R Q P 5 4.5 4 3 4 .3

SCERT TELANGANA

12

3 .4

4

5

PQ 2 1 CD AB PQ

PQ CD AB 180^O

9 2000

John Play Fair

SCERT TELANGANA

3.3.2

John Play fair - 1748-1819

l P P l

(Legendre)

1+ 2+ 3=180o

(Posidominus)

(Proclus)

(Proclus)

SCERT TELANGANA

Gold Bach

(theorem)

465

AC-AB=BC C A B A,B,C 1

AB+BC AC

AB+BC=AC

AC-AB=BC AC

AB + BC - AB = BC

1 PQ

SCERT TELANGANA

P

QP Q PQ

PQR Q P R R

Q PQ=QR P PQ=PR

PQR PQ=QR=RP

Q P

3 m l

B A

D A

XY AB D C AC=XD 4

AB=XY AB C AB=2AC

XY D XY = 2XD AC = XD

AB = XY

SCERT TELANGANA

3.1

.1 (i) (ii) (iii) (iv) (v)

.2 (a) (b) (c) (d)

AB>AC (e)

AH>AB+BC+CD .3

PQ= 1/2 PR PQ=QR R P Q .4

5.2 .5

.6

PQ Q P Q P .7

m l n .8

m l 180 2 1

3= 4 2= 4 1= 3 .9

1= 2

AB=BC BY=1/2 BC BX = 1/2 AB .10

BX=BY

SCERT TELANGANA

Bolyai Lobachevsky Carl Fedrick Gauss V

BN AN AB BN AN (ii)

90^O AB N

A+ B=180^O 180^O NAB + 90^O = 180^O

"The Elements"

SCERT TELANGANA

A + B + C + D + E + F .1

"a" .2

SCERT TELANGANA

4.1

(ii) (i)

UPQ CD BC AB ST RS PQ

ABC EAB PQR

(Optics)

Lines and Angles

4

SCERT TELANGANA

4.2

AB AB A B

AB

n m l

SCERT TELANGANA

'O'

360⁰

(Vertex)

4.2.1

SCERT

TELANGANA

4.2.2

4.1

.1 (i) (ii) (iii) (iv) (v)

.2

.3 (i)

BA AB (ii)

BA AB (iii)

(iv)

SCERT TELANGANA

(vi) (vii) (viii)

.4

7:00 (iii) 6 (ii) 9 (i)

4.3

90⁰

(Complementry angle)

(90⁰ x⁰) x⁰ x⁰

62⁰ (1)

28⁰ = 90⁰ 62⁰ 62⁰ 90⁰

180⁰

x⁰ (Supplimentry Angle) x⁰

(180⁰ x⁰)

SCERT

TELANGANA

4:5 (2) 5x 4x

4x 5x 90⁰

50⁰ 40⁰ 310⁰ 50⁰ 180⁰ 180⁰ 260⁰100⁰ 240⁰ 120⁰

(Conjugtive Angle) 360⁰

x⁰ 270⁰

.1

.2 54⁰

0⁰

SCERT TELANGANA

'O' (i)

(ii)

(iv) (iii) (ii) (i) (i)

(ii)

(Javelin)

180⁰

180⁰

SCERT TELANGANA

4.3.1

180⁰ 180⁰

SCERT TELANGANA

(iv)

AOC+ COB=125⁰+55⁰=180⁰

180⁰

360⁰

4.3.2

4 3 2 1

SCERT TELANGANA

4.1

O CD AB

(i) (ii)

(1) (2)

(2) (1)

SCERT TELANGANA

'O' .2

x (3)

180⁰ O

O RS PQ (4)

SCERT TELANGANA

(5) AOB

AOB

OR PQ OS (6)

OT

PQ OS

SCERT

TELANGANA

(7)

OP

(1)

(2)

(2) (3)

(3) (1) (4)

SCERT

TELANGANA

4.2

O EF CD AB .1

z y x x:y:z 2:3:5

x .2

O CD AB .3

BOD = 40o AOC + BOE = 70o

COE BOE

O MN XY .4

c a:b 2:3 POY = 90o

SCERT TELANGANA

PQR = PRQ .5 PQS = PRT

x y w z .6

AOB

PQ OR PQ .7

OR OP OS ROS =1 ( QOS POS)2

ZYP YQ P XY XYZ = 64° .8

QYP XYQ

4.4

n m l

l

Q P n m l

n m l

SCERT TELANGANA

(a)

(ii) (i)

4 (iv) (iii)

(b)

(ii) (i)

(c)

(ii) (i)

(d)

(ii) (i)

(e)

(ii) (i)

m l

(i)

SCERT

TELANGANA

R Q

(1)

(2)

(2) (1)

4.2

4.3

n m l .1

SCERT TELANGANA

x .2

SCERT TELANGANA

C B

BCS ABQ

SC QB AD

RS PQ

GH EF GH EF RS PQ

(1)

120⁰ 120⁰

R Q

(1) (2)

(2) (1)

SCERT TELANGANA

4.4 4.4.1

n m l

t n m l

n m

4.5

(i) BCS (ii)

SCERT TELANGANA

x (8)

CE E

c b a z y x (9)

z 70⁰ y 110⁰ x 70⁰ c 65⁰ b 65⁰ a 115⁰

x CD AB GH EF (10)

SCERT

TELANGANA

(11)

M PQ AB

XM

(1)

(2)

(2) (1)

(12)

C B

(i) (ii) (iii)

SCERT

TELANGANA

BE

(1)

CF (2)

CF BE (3)

(3) (2) (1)

y x (13)

z

G BE

125⁰

35⁰

.1 .2 .3 .4

SCERT

TELANGANA

4.3

8 1 l || m .1

_________ l || m (i)

_________ 1 5 (ii)

_________ 5 + 8 = 180° (iii) _________ 1 + 8 = 180° (iv)

_________ 8 1 (v)

y : z = 3 : 7 CD || EF AB || CD .2 x

E F + C D AB | | C D .3

GEF AGE GED = 126°

FGE

PQR = 110° PQ || ST .4 QRS RST = 130°

R ST

n m B A m || n .5

n m C

ACB

SCERT TELANGANA

r || s p || q b a .6

c || d a || b .7

2 (ii) 1 (i)

.8 y x

.9 y x

y x .10

y x .11

SCERT TELANGANA

APQ = 50° AB|| CD .13 y x PRD = 127°

RS PQ .14

B PQ AB

C RS BC

AB || CD CD

CD AB EF AB || CD .15

y x H G

F E t AB || CD .16

2 : 1 = 5 : 4

SCERT TELANGANA

z y x AB || CD .17

z y x AB || CD .18

x AB || CD .19

4.5

180⁰

.1

SCERT TELANGANA

180⁰ 4.6 ABC

BA CE C D BC

(1)

CE AB (2)

(3)

QR

S

(1)

(2) (2) (1)

SCERT

TELANGANA

x (4x 14)⁰ (3x 5)⁰ (2x)⁰ (14) 180⁰

70⁰ 68⁰ 42⁰

(15)

(iii) (ii) (i)

(i) Q PA

PQ (ii)

180⁰

PR (iii)

SCERT TELANGANA

x (16) ABCD

E AC

(17)

y x (18)

D BC

SCERT

TELANGANA

35o + 100o + yo = 180o

y x (19)

D BC

A CE

y x (20)

E AC AB (21)

CO BO D

O BO

SCERT

TELANGANA

CO

(2) (3)

(2) (1) (3)

(4)

4.4

z y x .1

SCERT TELANGANA

4 = 5 AS || BT .2

1 AST

y x BC || DE AB || CD .3

CD DA BE DA .4

m =~ 3

BC AD y x .5

y x .6

y x .7

T S RQ QP .8

PQT = 110° SPR = 135°

PRQ

ZO YO XYZ = 54° X = 62° .9

OZY XZY XYZ

SCERT TELANGANA

BAC = 35° AB || DE .10

DCE CDE = 53°

T RS PQ .11

TSQ = 75° RPT = 95° PRT = 40°

SQT

B = 50° ABC .12

AC AB C = 70°

z

SQR = 28° PQ || SR PQ PS .13

y x QRT = 65°

D AC ABC .14

A A B = 2 : 3 BCD = 125°

B

BAC = 35° BC || DE .15

BCE = 102°

(i) BCA (ii) ADE (iii) CED

SCERT TELANGANA

BAC = 36° AB AC .16 AEC = 40° ADB = 45°

AB (iii) ACB (ii) ABC (i) EAC (iv)

y x .17

180⁰

180⁰

SCERT TELANGANA

180⁰

(Golden Traingle)

36⁰ 72⁰

= |AB| / |BC| = 1.618 ...

SCERT TELANGANA

Co-Ordinate Geometry 05

5.1

7^OC -6^OC

H F E D C B A

A H CAF H A

(ii) (i)

SCERT TELANGANA

6

D C B A

6

6

8

(6,8) 7

SCERT TELANGANA

(1590-1650)

Clip

5.1

.1

3 (i)

2 (ii)

K (iii) (iv) (v)

SCERT TELANGANA

5.2

O Origin

(iii) x XX'

y YY'

SCERT TELANGANA

YY' XX' O X

OY' OX' Y

Y X

Q³ Q² Q¹

Q⁴ XY

5.2.1

B A

B A

Q³ B Q¹ A

X B A

BF BE AD Y AC

X Y A (i)

A AD=OC=5

x

y A AC=OD=3 y Y A (ii)

(5,3) A

SCERT TELANGANA

-4 X OE=BF=4 X Y B (iii) x B

-3 Y OF=EB=3 Y X B (iv)

B (-4,3) y B

X x (a)

abscissa x

2 x P

-3 x Q

Y y (b)

cordinate y

-2 y P

4 y Q

(-3,4) Q (2,-2) p

5.2.2

Y X .1

y x 1

(i) P (8, 8) (ii) Q(6, -8)

P(8, 8) (i)

8 y 8 x

X 8 y p

8 X Y 8 y

Q(6,-8) II

y = -8 x = 6

Y X 6 Y,Q

8 X

SCERT

TELANGANA

2

P 4 P 4 X X P 1

Y Y

3 P 3

( 4 , 3 ) P

5 -4 Q 2

(-4,5) Q

R 3

(-2,4) R 4 -2

-5 4 S 4

(4,-5) S

3

X 3 Y A

3 A

(3,0) A O Y

(2,0) B (i)

(-1,0) C (ii) (-2.5,0) D(iii) (-4,0) E (iv)

Y X X X

Y=0 X

SCERT TELANGANA

X .1

4

Y +5 X P (i)

5 Y 0 X P

(0,5) P

(0,3.5) Q (ii)

(0,1) R (iii)

(0,-2) S (iv)

(0,-5) T (v)

Y Y

x 0 Y X Y

5.2.3

X X Y Y 0

Y X

(0,0) ' O '

(0,-5) (0,y) (0,x) .1

SCERT TELANGANA

5

SCERT TELANGANA

5.2 Quadrant .1

.2

.3

.4

Y L (i)

Y Q (ii)

(-2,2) (iii)

SCERT TELANGANA

(5,-4) (iv)

x N (v)

X M (vi)

.5

Y (i)

Y (ii)

(iii) ( ) (2,-3) (iv) (-5,-8) (v)

( ) y< 0 x<0 (-x,-y) (vi)

SCERT TELANGANA

5.3

(4,6) P

Y 4 X

P(4,6) X

' O ' Y

X

A 4 X

Y 6 A

(4,6) P y x

SCERT TELANGANA

7 (i) M (-2 , 4) (ii) A (-5 , -3) (iii) N (1 , -6)

Y X

M (i)

2 X 'O' :M(-2 , 4)

4 Y

A(-5 , -3) (ii)

X 5 O

3 Y

N(1 , -6) (iii)

1 X

6 Y

SCERT TELANGANA