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1
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(iii)(iii)

, Q C
. , Q C - . ( ) , .

.
, ,
.
, .
, .
.

.
.
.

Preliminary_XI_TM.indd 3 19/04/18 7:20 PM
(iv)

• , ,
• (S CC), . •
, (S S c C C c , P ).
• ,
C.A., I.C.W.A, C.S.

• , . • , • , • . . .C.C , .
M.Sc.

• . •

M.L.
–
• , ,
Ph.D.,
• , , . M.S.W
1.1 1
1.2 12
1.3 - 28
2 42-86
2.2 49
2.3 61
2.4 68
2.5 73
3 87-127
3.1 88
3.3 97
3.4 103
4 128-161
4.1 131
4.2 136
4.3 143
4.4 150
5 162-216
5.1 163
5.2 178
5.3 196
217-225
226-229

(vi)
– 1. (15 )
- - - .
- - - - . –
- . 2. (20 )
( ) - (n ) - , . - - - . .
3. (20 )
. -
- . - - - . , - - - - . .
4. (25 )
- - 0º 360º θ - .
- sine, cosine - . - . .
5. (25 )
- - . - - - . - - .

z z z z z z

(1683) . . (1841) . 1750 .
. . (x1, y1) (x2, y2) (x3, y3) ,
1
CH-1_TM.indd 1 19/04/18 6:38 PM
2
( ) ( ) ( )x y y x y y x y y 2 1
1 2 3 2 3 1 3 1 2- + - + -[ ]
. .
2 1
.
. . . () , , , .
. ..

1.1
.
, , C, .
, A 1 5
4 8

A A m × n .
, A = 1 5
4
8
2
6 = G A 2 × 3

m × n A .
CH-1_TM.indd 2 19/04/18 6:38 PM
a a a
R
T
SSSSSSSSSSSSSS
V
X
WWWWWWWWWWWWWW
, , ...... ; , , .. ..a i m j n1 2 1 2ij mxn = =6 @ . aij - .

. , ............ ..........A a a a a ai n n11 12 13 1 1 1×= 6 @ ; B 1 2 X1 2= 7 A

.
, . . .
=
G

.
,

, O = ,06 @ O = 0 0
0 0= G , O =

, .
, 8
4
2
2 A = = G 2
B = 3 2 2
CH-1_TM.indd 3 19/04/18 6:38 PM
4

.

.
, A= 1
0
0
2
2
0
3
4

.
, A= 5
0
0
0
2
0
0
0
1 > H 3 .

.
, A = 5
0
0
0
5
0
0
0
5 > H 3 .

“ ” .
, I 1
0
0
1 2 = = G 2 .
I 1 0 0
W W WW
3 .

A aij= 6 @ k kA kaij= 6 @ (i, j ).
A k A k .
, A= 1 3
2 2
4 8-
4
4
8

A a m nij= #6 @ . A A aij m n- = - #6 @ .
, A 2 0
5 5
7 6=
5 5
7 6- =
= G .

A, B (i)
(ii) .

A, B , . A + B A, B
.
A = [aij]m× n B = [bij]m × n , A + B = [aij+ bij]m × n
A − B = A + (−B) = [aij]m× n+ [− bij]m × n= [aij− bij]m × n .

A, B A , B . A B .
A aij= 6 @ m # p B bij= 6 @ p # n AB C cij= 6 @ m # n .

A aij= 6 @ m # n . A A . AT . AT n # m .
, 1
1
2
5
3
4
6
6

, . .
1.2
n A , A . | A | det (A) Δ .
A A .
A = a a
a a

1
4
4-
( )( ) ( ) 2
1
4
- = - -
= 8+4 =12
.
| A | = |[aij]| n .
aij aij . aij Mij .
. aij
Aij
( )A M1ij i j
a a a
a a a
a a a
11
21
31
12
22
32
13
23
33
11 12 13, ,a a a
.
(i) a11 M11 a
a
a
22
32
23
a11 A11 ( ) M a
a
a
11
22
32
23
33
(ii) a12 M12 a
a
a
21
31
23
33
a
a
21
31
23
33
(iii) a13 M13 = a
a
a
21
31
22
a13 A13 ( ) M a
a
a
21
31
22
32
1 4
2 3 -

.
1 = M11 = 3
2 = M12 = 4
4 = M21 = 2
3 = M22 = 1
1 = A11 = ( ) M1 31 1 11- =+
–2 = A12 = ( ) M1 41 2 12- =-+
4 = A21 = ( ) M1 22 1 21- =+
3 = A22 = ( ) M1 12 2 22- =+
3
2
4
1
2
1
2
5
0

.
CH-1_TM.indd 7 19/04/18 6:38 PM
3 M11 = 2 1
5
1 M12 = 2 4
5
2 M13 = 2 4
2
2 M21 = 1 1
2
2 M22 = 3 4
2
5 M23 = 3 4
1
4 M31 = 1 2
2
1 M32 = 3 2
2
2
1
2 6 2 4= - =
3 A11 = ( ) M M1 51 1 11 11- = =-+
1 A12 = ( ) M M1 201 2 12 12- =- =+
2 A13 = ( ) M M1 61 3 13 13- = =-+
2 A21 = ( ) M M1 22 1 21 21- =- =+
2 A22 = ( ) M M1 82 2 22 22- = =-+
5 A23 = ( ) M M1 12 3 23 23- =- =+
4 A31 = ( ) M M1 13 1 31 31- = =+
1 A32 = ( ) M M1 113 2 32 32- =- =-+
0 A33 = ( ) M M1 43 3 33 33- = =+
.
,
a a a
a a a
a a a
Δ = a11A11+ a12A12+ a13A13 ()
a11M11−a12M12+ a13M13 (R1 )
Δ = a11A11+ a21A21+ a31A31 ()
a11M11−a21M21+ a31M31 (C1 )

1
4
2
3
1
4
0
0
1
1
4
2
3
1
4
0
0
- = 1 (1 ) –2 (2 ) +
4(4 )
= 1 3 0 1 0 1 3
2 4 1 0 4 0 4 1
= 0 –0 –52 = –52.
1. , , . 2. ()
. 3. ()
. 4. ()
k k .
5. () .
6. () .
7. () () .
CH-1_TM.indd 9 19/04/18 6:38 PM
c
2 2 2 2 2 2+ + + = 0 .
3 4
2 5
1 7
6 8
4 11
3 4
2 5
1 7
6 8
4 11
8 13!+
= x
x
a
y
y
b
z
z
c
x
a
a
y
b
b
z
c
c
= 0 + 0 = 0
x
x
x
x1
1 -

.
x = 1, x = 2, x = 3

a
b
c
a
b
c
1
1
1
2
2
2
= (a–b) (b–c) (c–a) .
a
b
c
a
b
c
1
1
1
2
2
2
+^ ^h h
= 0 0a b b c b c a b- - - + + - +^ ^ ^h h h6 @" ,
= a b b c c a- - -^ ^ ^h h h.

1. .
(i) 5 0
20 1- (ii)
3. : x2
0 2=
CH-1_TM.indd 11 19/04/18 6:38 PM
7. a
8. a
= .

A , |A| = 0 .
A , |A| ≠ 0
.
.
A B
AB BA
.
A = 1 2
= 4 – 4 = 0
` A .

8
4
2
3 = G .
CH-1_TM.indd 12 19/04/18 6:38 PM
= 24 – 8 = 16 ! 0
` A
A
, A . adj A
adj A = Aij T6 @ . Aij6 @
.

(i) adjA A n 1= - n A .
(ii) kA k An= n A .
(iii) ( )adj kA k adjAn 1= - n A .
(iv) ( ) ( )A adj A adj A A A I= =
(v) ,AdjI I I= . (vi) adj (AB) = (adj B)(adj A)
(vii) AB A B=
A = 2 1
Aij = (–1)i+jMij
A22 = (–1)2+2M22 = 1
-> H
A n . AB=BA =I (I n
) B B A . A A 1- .
, A (adj A) = (adj A)A = A I
( A A adj A
1c mA = I
adjA 1
A A
adjA 11 =-

(i) B A A B
B A A B1 1&= =- -
(ii) AA A A I1 1= =- -
(iii) .
(iv) A–1 , A .
(v) I I1 =- , I .
(vi) ( )AB B A1 1 1=- - - (vii) A A 11 =-
A
A I A A2 1+= = -

A = 2
= 16 ≠ 0
A , A 1- .
adj A = 2
A = 2 3
- - = 0
A . A 1- .

A = 2
= ( ) ( ) ( )2 5 4 2 4 0- - - +
= 10 8 0- + + = 2 0!-
A , A 1- .
; ;A A A5 8 411 21 31= - = =-
; ;A A A2 2 012 22 32= =- =
; ;A A A0 2 213 23 33= =- =
Aij6 @ = 5
2 A = = G A kA I O2
2- + = k
. A 1- .
A = 2 1
7
4
12
7
2
1
3
2
1
0
0
1
0
0
0
7 2 1
1
1
2
-= G , ( )AB B A1 1 1=- - -
A = 1
` A 1-
- = + =
AB = 1
adj(AB) = 1
- = G … (2)
(1) (2) ( )AB B A1 1 1=- - - .

35 1
35 6
35 11
35 6
35 1
35 5
35 25
35 10
.
V
X
= 1 4 1
3 2 2
7 3 1
>
H
1 > H = I
AB = BA = I
A B .

1. 2
1
3
4 A = = G A .
2. 1
1
1
3
4
3
3
3
4
A = > H A(adj A) = |A| I , A 1- .
3. .
(i) 1 2
1
4
2 =
-
- = G , adj (AB) = (adj B)(adj A)
5. A
> H , (adj A)A = O .
6. If A 1
=
-
-
-
-
> H , A A .
7. A
CH-1_TM.indd 20 19/04/18 6:38 PM
5 4
5 1
5 1
5 2
5 3
5 2
5 1
5 1
5 4
V
X

.
9. 3
7
8
9 B = = G , ( )AB B A1 1 1=- - - .
10. 1
2
9
1
7
3
4
11
m> H m .
11. X
-
> H , Y X 1= - p,q .
, , ......x x xn1 2 n .
n .
....a x a x a x bn n11 1 12 2 1 1+ + + =
....a x a x a x bn n21 1 22 2 2 2+ + + = ……………………………… ………………………………
...a x a x a x bm m mn n m1 1 2 2+ + + =

a a a
11 12 1
21 22 2
R
T
R
T
V
X
V
X
V
X
W W W W W WW
. AX = B … (1) .
A =
a a a
V
X
; X = . .
V
X
B = . .
V
X
(1)
X = A–1B
2x + 5y = 1
3x + 2y = 7
AX = B
7 = G
adj A = 2
:
; ;x y z x y z x y z3 2 3 8 2 1 4 3 2 4- + = + - = - + =
x
X = A–1B
A = 3
8
1
[Aij] = 1

4 , 3 2 3 0, 2 , 4 , 6 560,
6 , 2 3 380 , .
, , x, y, z .
4x + 3y + 2z = 320
2x + 4y + 6z = 560
6x + 2y + 3z = 380.
AX = B
320
560
CH-1_TM.indd 24 19/04/18 6:38 PM
[Aij] = 0
= 20.
= 40
= 60
CH-1_TM.indd 25 19/04/18 6:38 PM

– 1 G G .. “11th BUSINESS MATHS” GeoGebra “Simultaneous Equations” .
- 2 . . () . .
CH-1_TM.indd 26 19/04/18 6:38 PM
27

1. : 2x + 3y – 5 = 0 ; x – 2y + 1 = 0
2. :
(i) 3x – y + 2z = 13 ; 2x + y – z = 3 ; x + 3y – 5z = –8
(ii) x – y + 2z = 3 ; 2x + z = 1 ; 3x + 2y + z = 4.
(iii) 2x – z = 0 ; 5x + y = 4 ; y + 3z = 5
3. A, B, C 2009 ,

A B C
9 10 2 800
15 5 4 900
6 10 3 850
A, B, C .
4. A, B, C x, y z . P 4 C 3 A 5 B . Q 3 B . 2 A 1 C . R 1 A . 4 B 6 C . P,Q,R 6,000, 5,000, 13,000 A,B C .
5. 20. 2 , , , 23 .
46 .
6. . . .
CH-1_TM.indd 27 19/04/18 6:38 PM

(`)A B C
– –
W. . . .
.
- . . . . , , . ( , , , ) .
A1 A2 . . , , . . . , , .
, .
CH-1_TM.indd 28 19/04/18 6:38 PM
29
aij Aj Ai . i, j = 1, 2
x1 x2 A1 A2 .
d1 d2 A1 A2 .

a11+ a12+ d1 = x1; a21+ a22+ d2 = x2 ... (1)
bij = x a
j
b11 = ; ; ; x a
b x a
b x a
b x a
2
22= = =
(1) .
b11x1+ b12x2+ d1 = x1 b21x1+ b22x2+ d2 = x2
.
b x b x1 11 1 12 2- -^ h = d1
1b x b x2221 1 2- + -^ h = d2

b
b
b
b
x
x
1
2
= G
B = , b
X=(I–B)-1D
B [Technology Matrix] .
CH-1_TM.indd 29 19/04/18 6:39 PM
30
-
.
B ,
(i) I – B .
(ii) I B- .

.
.
.
0 8
0 9
0 2
0 7 = G -
.
B = .
I B- .
- [Hawkins-simon] .. .

2016 , .
CH-1_TM.indd 30 19/04/18 6:39 PM

1 500 1,600 400 2,500
2 1,750 1,600 4,650 8,000
250 4,800 - -
, - .
a11 = 500 a12 = 1600 x1 = 2500
a21 = 1750 a22 = 1600 x2 = 8000
b11 = 0.20 x a
8000 1600
2500 1750
7 1
8000 1600
0 8
0 7
0 2
0 8-
-= G
, I B- = .
.
.
.
= 0.64 – 0.14
= 0.50 >0
I B- . |I–B| . - (Hawkins-simon) . .

P1 P2 . ( ) .
CH-1_TM.indd 31 19/04/18 6:39 PM

P1 10 25 15 50
P2 20 30 10 60
P1 35 P2 42 .
a11 = 10 a12 = 25 x1 = 50
a21 = 20 a22 = 30 x2 = 60
b11 = x a
5 2
12 5
2 1
V
X
W W WW , .
I B- = 5 4
= 0 30 7 2
(I B) . I B| . . .
adj (I–B) = 2 1
5 2
12 5
5 4
adj I B 1 -
-
( |I – B|≠0, (I – B)–1 )
= ( )adj I B1 30 7 -
= 7 30 2
X = (I–B)–1D, D = 35 42 = G
= 7 1 15
12 2 12
204 => = =H G G
P1 150 , P2 204 .

. . 0.4 0.7 . 0.1 0.6 . .
5 2. 100 50 .
0.4 0.1 50
0.7 0.6 100
5 2 -
B = .
.
.
.
34
= (0.6)(0.4)–(–0.7)(–0.1)
= 0.24 – 0.07 = 0.17
- . .
adj (I–B) = .
adj I B 1 -
100 = G
0 7
0 1
0 6

= 5( ) + 2( )
= 5(176.5) + 2(558.8)

.
0 30 0 33= G

.
CH-1_TM.indd 34 19/04/18 6:39 PM
35
. .
0 9 0 80= G

.
0 25 0 67= G

.
4. A B 0.4 A 0.7 B 1 A . 0.1 A 0.7 B 1 B .
. 68 A 10.2 B .
5. .
X Y
X 30 40 50 120
Y 20 10 30 60
, 80 40
6. .
1 2
1 4 3 13 20
2 5 4 3 12
(i)
(ii) 1 23 .
CH-1_TM.indd 35 19/04/18 6:39 PM
36
7. X Y .

X Y X 15 10 10 35 Y 20 30 15 65
X 12 Y 18 .

1. x x x
1 2 3
0 = 0 x
( a ) 0, –1 ( b ) 0, 1 ( c ) – 1, 1 ( d ) –1, –1
2. x y y z z x
x y z
y z x
2 2 2

(a) x y z (b) x + y + z (c) 2x + 2y + 2z (d) 0
3. 2 6 1
(a) –18 (b) 18 (c) –7 (d) 7
4. If 1 3 2
2 1 3
3 2 1
1 2 3
2 3 1

5. a
b c
0 0
(a) abc (b) 0 (c) a2 b2 c2 (d) –abc
6. A 3 |kA|
(a) k A (b) – k A (c) k A3 (d) – k A3
CH-1_TM.indd 36 19/04/18 6:39 PM
7. adj AB] g =
(a) adjA adjB (b) adjA adjBT T (c) adjB adjA (d) adjB adjAT T
8. 5 4
(a) 30 7 2
7 2 1
1
30 2 1
9. A a c
b d= e o ad bc 0!- , A 1-
(a) ad bc d c
b a
b a
b a
b a
-e o
10. -
(a) 1 (b) 3 (c) 4 (d) 2
11.
(a) . (b)
(c) . . (d)
12.
(a) 1 1
1 2-
-e o
1 2e o
(a) 2 5
5 3
5 2
(a) 4 1
2 1-
0 2e o (d)
2 0e o
15. A, B ,
(a) A I2 = & A A1 =- (b) I I1 =-
(c) I f AX B= , X B A1= -
(d) A 3 adj A A 2=
16. x x
(a) 5 (b) 4 (c) 0 (d) –3
17. 2 A , det A 1-^ h
(a) de t ( A ) (b) det A 1 ] g (c) 1 (d) 0
18. A 3 3 A 4= , A 1-
(a) 4 1 (b) 16
1 (c) 2 (d) 4
19. A 3 A 3= , adjA
(a) 81 (b) 27 (c) 3 (d) 9
20. x y x
1 1 1
21. A = cos sin
(a) cos4 2i (b) 4 (c) 2 (d) 1
22. a a a
11
21
31
12
22
32
13
23
33
D = Aij aij, D
(a) a A a A a A11 31 12 32 13 33+ + (b) a A a A a A11 11 12 21 13 31+ +
(c) a A a A a A21 11 22 12 23 13+ + (d) a A a A a A11 11 21 21 31 31+ +
CH-1_TM.indd 38 19/04/18 6:39 PM
(a) 6 5- (b) 6
5 (c) 5 16- (d) 5
16
20 15
(a) –5 (b) –125 (c) –25 (d) 0
25. ()
(a) 0 (b) 2 (c) 1 (d) 3

1. : x
10042 10046 10050
10043 10047 10051
3. 5 5 5
5 5 5
5 5 5
ab
0 0
0 22
5. A 1 3 1
1 4 1
1 7 1
R
T
SSSSSSSS
V
X
WWWWWWWW A adjA adjA A A I3= =^ ^ ]h h g .
6 . A = 3
7 . A = 1 2
1 3
-> H , A A I O4 52 2- + = A 1- .
8. : , ,x y z x y y z2 2 0 2 1- + = - = - = .
9. 2 1 `70. 1 `70. 3 , 2 1 `170. .
CH-1_TM.indd 39 19/04/18 6:39 PM
40
10. A B ( ) .

A B
A 50 75 75 200
B 100 50 50 200
A 300 B 600
z .
z A = 11 12
a a a a
A a a a a11 22 21 12= -
z A aij , A aij , . Mij .
z . aij Aij ( ) .A M1ij
i j ij= - +
Δ= a11A11+ a12A12+ a13A13 a11M11−a12M12+ a13M13
z adj A = Aij T7 A Aij7 A
.
z adjA A n 1= - , n A .
z A adjA adjA A A I= =^ ^ ]h h g .
z adjI = I, I .
z adj(AB) = (adj B)(adj A).
z A | A | = 0 .
z A | A | ! 0 .
CH-1_TM.indd 40 19/04/18 6:39 PM
41
z A n . AB = BA = I B B A . A A–1 .
z A A A adjA11 =- .
z - (Hawkkins - Simon) -
(i) I B- .
(ii) I B- .
Determinant
Scalar Transpose of a Matrix
Inverse Matrix Analysis
Singular Matrix Non-Singular Matrix
Triangular Matrix Digonal Matrix
Output

z z . z
. z
z
z
. z .

-
. . “Algebra” “al-Jabr” . - (Al-Khwarizm) . , .

b x b x b a x a x a
m m m
n n n
0 1 1
0 1 1
] ] g g
,[q(x)≠0] .
2
43
, , . , .
.
, q x p x ] ] g g
= f x q x r x
+] ] ]g g g
2 1 1
x 1 3 - x 2
2 -
.
i.e., x x1 3
- ] ]g g
- ] ]g g = x x1
2 - + -
.
p(x) q(x) x q x
p x ] ] g g
.

q x p x ] ] g g
q(x)
, ax b cx d+ +] ]g g ax b A
cx d B
+ + +
. A B .
x x x
x A
x B
1 2- - + = - + -] ] ] ]g g g g .

x B
1 1
1 12 - = - + +^ h A, B
x 1 1
44
x x1 1- +] ]g g
1 = A x B x1 1+ + -] ]g g ... (1)
x =1 , 1 = A 2] g
` A = 2 1
` B = – 2 1
7 1 2 - +
x x5 62 - + = x x2 3- -] ]g g
x x x 5 6
7 1 2 - +
- = x A 2-] g + x
B 3-] g …(1)
(x–2)(x–3),
x7 1- = A x B x3 2- + -] ]g g … (2)
x = 3 (2)
21 – 1 = B (1)
x = 2 (2)
14 1- = A 1-] g
& A = 13-
A B (1)
x x x 5 6
7 1 2 - +
- = x x2 13
7 1 2 - +
- = x x x
x x x
4 2 - +
+ ^ ]h g .

x x4 12 - +^ ]h g= x x x2 2 1- + +] ] ]g g g
` x x x 4 1
4 2 - +
C 2 2 1- + + + +] ] ]g g g ...(1)
(x–2)(x+2)(x+1)
x 4+ = A x x B x x C x x2 1 2 1 2 2+ + + - + + - +] ] ] ] ] ]g g g g g g ... (2)
x = –2 (2)
2 4- + = A B C0 4 1 0+ - - +] ] ] ]g g g g
` B = 2 1
x = 2 (2)
2 + 4 = A B C4 3 0 0+ +] ] ] ]g g g g
` A = 2 1
x = –1 (2)
1 4- + = A B C0 0 3 1+ + -] ] ] ]g g g g
` C = 1-
4 2 - +
1 2 2
1 1
1 - + + - +] g

( ) ( )
q x p x , ( )q x ax b+] g
n , ax b n+] g
Ch-2_TM.indd 45 19/04/18 6:56 PM
, x x x 4 1 9 7
2+ + +

x B
x C
1 1 1 2- + + + +] g A, B C .
x x x
x B
x C
1 1 1 2- + + + +] g x = A x B x x C x1 1 1 12+ + - + + -] ] ] ]g g g g ... (1)
x = 1 (1)
1 = A B C1 1 0 02+ + +] ] ]g g g
` A = 4 1
1- = A B C0 0 1 1+ + - -] ] ]g g g
C` = 2 1
(1)
A B C- - = 0
1 2- + +
Let x x x 2 3
1 2- + +
2 2 32- + - + +] ] ]g g g ... (1)
x x2 32- +] ]g g
x 1+ = A x x B x C x2 3 3 2 2- + + + + -] ] ] ]g g g g …(2)
Ch-2_TM.indd 46 19/04/18 6:56 PM
2+1 = A B C0 5 0+ +] ] ]g g g
` B = 5 3
x = –3 (2)
–3+1 = A B C0 0 3 2 2+ + - -] ] ]g g g
` C = 25 2-
x2 (2)
0 = A C+
` A = 25 2
,A B C (1) ,
x x x 2 3
1 2- + +
5 2 3
25 3 2
2- +] ]g g .
x x1 2 9
x B
x C
1 2 2 2- + + + +] ] ]g g g … (1)
x x1 2 2- +] ]g g
` 9 = A x B x x C x2 1 2 12+ + - + + -] ] ] ]g g g g … (2)
x = –2 (2)
9 = A B C0 0 2 1+ + - -] ] ]g g g
` C = –3
` A = 1
48
x2 (2)
0 = A B+
B = A-
` B = –1
,A B C (1)
x x1 2 9
2 1
2 3
2- - + - +] g

q x p x ] ] g g
, , q x] g ax bx c2 + +
, , q x p x ] ] g g

ax bx c Ax B 2 + +
+ .

2- + +
] ^g h .
x 12 +^ h .
x x x 1 1 2 1
2- + +
1 12- + + + . ... (1)
x x1 12- +] ^g h
2x+1 = A x Bx C x1 12 + + + -^ ] ]h g g ... (2)
x = 1 (2)
2+1 = A 1 1 0+ +] g ` A = 2
3
x = 0 (2)
0+1 = A C0 1 0 1+ + + -] ] ]g g g 1 = A – C
` C = 2 1
49
x2 (2) ,
A B+ = 0
` B = 2 3-
,A B C (1)
x x x 1 1 2 1
2- + +
1 2 3
1 2 3
1
:
1. x x x 3 2
3 7 2 - +
+ ] ]g g 3. x x1 2
1 2- +] ]g g
4. x 1 1
2 - 5. x x
2 6 2
2 2- +
+ ] ]g g
2 + +^ ]h g

n , n - , n . n! n .
n, n! = n n n1 2- -] ]g g … 3 2 1# #
,
5! = 5#4#3#2#1 =120
4! = 4#3#2#1=24
3! = 3#2#1=6
0! = 1 2! = 2#1=2
1! = 1
n! = n n n1 2- -] ]g g … 3 2 1# #
= !n n 1-] g
= n n n1 2- -] ]g g !
, 8! = 8×(7×6×5×4×3×2×1)
= 8×7!
(i) ! ! 6 7 (ii) !
(iii) ! ! !
# # # #
# # # #

7! 5! .
7! = 7×6!

! !9 1
10 1+ = !
n 11
! !9 1
P1
. 2.1
, . .
, P1, P2 P3
. S1 S2 .
, P1 P2 P3
, 3 . , S1 S2 , ( ) . 3 × 2 = 6
, , , .
2.1
m n , , m × n .
Ch-2_TM.indd 51 19/04/18 6:57 PM

‘NOTE’ , ,
‘NOTE’ ,
1 2 3 4 2.2
, , . , 3 , . 4 3 2 1 24# # # = .

, ,
, 4 , , . , = 4 4 4 4# # # =256

. 100 10
1 2 3 2.3
Ch-2_TM.indd 52 19/04/18 6:57 PM
53
. , 1 9 9 . . 10 , 0
9 . 8 .
, = 9#9#8 = 648 .

, . 10 , 8 . , . .
.
(i) 10 ()
(ii) 8
10 10 , 8 8 . 10+8 = 18 .
, .
, , m, n , (m+n) . .
2.2
, , m, n , (m+n) . .
Ch-2_TM.indd 53 19/04/18 6:57 PM

, 6 , 5 .
6 6 .
5 5 . 5+6=11 .

1. ! ! ! x
6 1
7 1
8+ = x-
2. n = 5 r = 2 ! ( ) ! !
r n r n - - .
3. If (n+2)! = 60[(n–1)!], n - .
4. 0 9 67 , 5 ?
5. 5, 6, 7, 8 9 . , 1000

, “, , ” “ ” (Fruit salad) . . .
, 395 (number lock) . , ., 395 .
Ch-2_TM.indd 54 19/04/18 6:57 PM
55
395 , 3 – 5 – 9 9-5-3 . 3 - 9 - 5 , .
, ( ) .
2.3
n r n r .
, 1, 2, 3 6 .
123, 132, 231, 213, 312, 321
0 < r < n, n r P(n, r ( ) nPr .
.
: npr = ( ) ! !
n r n -

5P3 P(8, 5)
5P3 = ! !
Ch-2_TM.indd 55 19/04/18 6:57 PM
! ! !
1 - = -
(iv) npn = ! !
! ! !n n
n n n0- = =] g (v) npr = n(n–1) (n–2) ... [n–(r–1)]

(i) 8P3 = 8 7 6# # = 336.
(ii) 5P4 = 5 4 3 2# # # = 120.

5 7
5 7 ,
= 7P5 = ! !

“LOGARITHMS” , ( ) 4
LOGARITHMS 10
n = 10
4 , r = 4
Ch-2_TM.indd 56 19/04/18 6:57 PM
= 10 × 9 × 8 × 7

nPr = 360, n, r
nPr = 360 = 36 × 10
n = 6 r = 4.

n r nr

1, 2, 3, …… 9, ,
9
n = 9 r = 3
= nr
= 93 = 729

n p , q . p + q = n n ! !
! p q n
58
! ! ... ! !

(i) MISSISSIPPI (ii) MATHEMATICS ,
(i) MISSISSIPPI = 11

M I 4 S 4 P 2 .
n = ! ! ! !
4 4 2 11
(ii) MATHEMATICS = 11 M 2
T 2 A 2
= ! ! !
! 2 2 2 11
, n
. . n! , . .
A, B, C, D . 4! . 4!
, ABCD, BCDA, CDAB,
Ch-2_TM.indd 58 19/04/18 6:57 PM
59
DABC .
. 2.4
D
B
AC
, ‘4’ ! !4 4 3=
, n , = (n–1)!

,
!n 2 1-] g .

8
(i) (ii) .
(i) 8 = 8P8 = 8!
(ii) , (8–1)! = 7 ( , )

10 ,
Ch-2_TM.indd 59 19/04/18 6:57 PM
60
.
, . ,
! ! !n 2 1
2 10 1

‘RANK’ .

, ,

.

,
s
‘RANK’ , 4!
A, K, N, R .
A- =3!=6
K- =3!=6
N- =3!=6
RAK =1! = 1
RANK =0! = 1
` RANK 6 + 6 + 6 + 1+1 = 20

1. nP4 = 12(nP2), , n - ,
2. , 5 3
Ch-2_TM.indd 60 19/04/18 6:57 PM
61
3. 0 9 10 , 35 , 6 , ,
4. “ASSASSINATION”
5. (a) 8 , (b) 8
6. ‘CHAT’

. . A, B, C D . A , B A . . 2 6 . AB, AC, AD, BC, BD, CD . . , 4C2 = 1 2
4 3 × × = 6

2.4
n r .
n r nCr . n ≠ 0 r = 0 .
nCr
r n r n -] g , 0 < r < n
5 3 5C3 .
5C3 = ! ! !
- = =] bg l = 10 .
Ch-2_TM.indd 61 19/04/18 6:57 PM

( )
( ))
8C2
8C2 = 2 1 8 7 × × = 28

2 1-] g
(iv) nCx = nCy , x = y x + y = n.
(v) nCr = nCn–r .
(vii) nCr = !r npr

nC4 = nC6 12Cn .
nCx = nCy , x + y = n
nC4 = nC6 ,
n = 4 + 6 = 10

nPr = 720 nCr = 120 r .
Ch-2_TM.indd 62 19/04/18 6:57 PM
120 = !r 720
( r = 3.

15C3r = 15Cr+3 r .
15C3r = 15Cr+3
nCx = nCy ( x + y = n,
3r + r + 3 = 15

32 , .
32 4
= 32C4
= ! ! !
4 28 32 ] g .

(), () . 10 . , 8 , () - 5 , , ,
Ch-2_TM.indd 63 19/04/18 6:57 PM
64
() 8 10C8 . ()- 10 5 10C5 .
10C8 × 10C5 = 10C2 × 10C5
= 2 1 10 9 × × × × ×
5 4 3 2 1 10 9 8 7 6
× × × × × × = 11340 .

4 , 2 (wicket keeper) 16 11 . 3 11
11 .
(i) 3 , 1 7
×C C C4 2 10×3 1 7
C C C4 2 10× ×3 1 7 = C C C4 2 10× ×1 1 3
= 960
(ii) 3 , 2 6
C C C4 2 10× ×1 2 6
C C C4 2 10× ×1 2 6 = C C C4 2 10× ×1 2 4
= 840
(iii) 4 , 1 6 C C C4 2 104 1 6# #
C C C4 2 104 1 4# # = 420
(iv) 4 , 2 5
C C C4 2 104 2 5# #
Ch-2_TM.indd 64 19/04/18 6:57 PM
65
C C C4 2 104 2 5# # = 252 .
, = 960+840+420+ 252 = 2472 .

nC4 2] g= n C2 3+] g , n .
( )nC4 2 = n C2 3+] g
( )n n4 1 2 1
# - = n n n
+ +] ] ]g g g
n n9 142 - + = 0

n C2 n+] g = 45 n .
n C2 n+] g = 45
n C2 n n2+ + -] g = 45
n C2 2+] g = 45
( )n n 2
n = –11, 8
n, .
` n = 8
1
2

– 1 G G .. “11th BUSINESS MATHS” GeoGebra “Combination Exercise” .
- 2
“Show solution” . “Short method” .
Ch-2_TM.indd 66 19/04/18 6:57 PM

1. nPr = 1680, nCr = 70 n r .
2. 8C4 + 8C3 = 9C4 .
3. 21
4.
5. 7 4 , 3 ,
6. 4 2
7. 18 . , 9 , 3 ,
8. 44
9. 15 11
(i) .
(ii) .
(iii) .
10. 6 4 5 ,
(i)
(ii)
.
Ch-2_TM.indd 67 19/04/18 6:57 PM
68
n
.
, , , .

n P(n) .
: n =1 P(1) . ( )
: n = k ( k , )
n = k + 1 .
, P(k) P(k+1) . P(n) .

1 + 2 + 3 +… + n = n n 2
1+] g , ( n N! ) .
P(n)
n N! P(n) = 1+2+3+……….n = n n 2
1+] g .
1: n =1
P(1) LHS = 1
LHS = RHS
` P(1) .
2: n = k .
Ch-2_TM.indd 68 19/04/18 6:57 PM
i.e., P(k) .
P(k) = 1 + 2 + 3+… + k = k k 2
1+] g .
3: P(k + 1)
LHS = P(k + 1) = 1+2+3+……….+ k + (k +1)
= P(k) + k +1
= k k 2
1 2+ +] ]g g = RHS
` P(k +1)
P(k) P(k +1) .
` P(n) n N! .
1+ 2 + 3 +… +n = ,n n n N2 1
! +] g .

1 + 3 + 5 …+ (2n–1) = n2 ( n N! ) .
P(n) 1 + 3+ 5 …+ (2n–1) = n2 .
n = 1
P(1) LHS = 1
P(1) RHS = 12
P(k) .
i.e., P(k) = 1 + 3 + 5 ... + (2k–1) = k2
P(k + 1) .
Ch-2_TM.indd 69 19/04/18 6:57 PM
70
P(k + 1) LHS= 1 + 3+ 5 …+(2k–1) + (2k+1) = P k k2 1+ +] ]g g
= k k2 12 + +
P(k) P(k + 1) .
` n P(n) .

12+22+32+……….+ n2 = n n n 6
1 2 1+ +] ]g g ( n N! ) .
P(n) : 12 + 22 + 32 +… +n2 = n n n 6
1 2 1+ +] ]g g
n = 1 P(1) LHS = 12
= 1
P(1) RHS = 6 1 1 1 2 1+ +] ]g g
= 1
P(k)
P(k) = 12 + 22 + 32 +… +k2 = k k k 6
1 2 1+ +] ]g g
P(k + 1) LHS = 12 + 22+ 32+… + k2 + (k+1)2 = p k k 1 2+ +] ]g g
= k k k k6 1 2 1 1 2+ +
+ +] ] ]g g g
1 2 1 6 1 2+ + + +] ] ]g g g
= k k k k
= k k k
= k k k 6
Ch-2_TM.indd 70 19/04/18 6:58 PM
` P(k) P(k + 1) .
` P(n) .

23n–1 7 ( n N! ) .
P(n) = 23n–1
` P(1) = 23 –1
i.e., P(1) .
2: n = k .
i.e., P( ) = 23k –1 7 .
m N! 23k –1 = 7m . ... (1)
3: P( + 1)
P(k +1) = 2 k3 1+] g –1
= 2 1k3 3 -+] g
= 2 2 1k3 3$ -
= 2 7 2 1k k3 3$ + -^ h
= ( )m m2 7 7 7 2k k3 3$ + = + & 7
, P (k +1) .
P(k) P (k +1) .
` n P (n) .
Ch-2_TM.indd 71 19/04/18 6:58 PM

n2 + n ( n N! ) .
P(n) “n2+ n .
n = 1
P(1) = 12 + 1 = 1 + 1 = 2 , .
P(k) .
i.e., P(k) = k2 + k
` P(k) = k2 + k = 2m, m N! …(1)
P(k + 1)
P(k + 1) = k k1 12+ + +] ]g g= k k k2 1 12 + + + +
= k k k2 22 + + +
= 2(m + k + 1) ( )
` (k + 1)2 + (k + 1)
` P(k) P(k + 1) .
` n P(n) .

n N!
1. 13 + 23 + 33+… +n3 = ( )n n 4
12 2+ .
2. 1.2 + 2.3 + 3.4 +… + n(n + 1) = n n n 3 1 2+ +] ]g g .
3. 4 + 8 + 12 + … + 4n = 2n(n+1).
4. 1 + 4 + 7 + … + (3n –2) = n n 2
3 1-] g .
5. 32n –1 8 .
6. a bn n- a – b .
7. 52n –1 24 .
8. n(n + 1) (n+2) 6 .
9. 2n > n.

(Binomial) .
, , , ,x y a b x y p p y5 2 1 5 4 7 1
2+ - + + +^ ] c c dh g m m n .
.
(x + y )n axbyc . b c b + c = n . a .
... 4 (x + a)2 - . ... 6 (x + a)3 - . 1544- ‘ ’ (Michael Stifle) ‘ ’ .
(Blaise Pascal) (19 1623 – 19 1662) , , . 1652 “ ” . .
(x + a)n - , .
Ch-2_TM.indd 73 19/04/18 6:58 PM
74
x a n+] g n N!
()
x, a , n N! x a n+] g x a n+] g = ...nC x a nC x a nC x a nC x an n n
r n r r
nC x a nC x a nC x an n
n n

n = 0 (x+a)0 = 1 n = 1 x a C x C a x a1 10 1+ = + = +] g n = 2 x a C x C x C a x xa a2 2 2 2a2
0 2
1 2 2 2 2+ = + + = + +] g
n = 3 x a C x C x a C xa C a3 3 3 33 0
3 1
2 2
2 3
3+ = + + +] g = x x a xa a3 33 2 2 3+ + + ...
x a n+] g - . .
n = 0 1
n = 3 1 3 3 1
n = 4 1 4 6 4 1
n = 5 1 5 10 10 5 1
. 2.5

(i) (x + a)n n+1 . (ii) (x + a)n , x a
n
Ch-2_TM.indd 74 19/04/18 6:58 PM
75
(iii) , , , , ... , ...nC nC nC nC nC nCr n0 1 2 3 . , ,C C C ,0 1 2 , ..., , ..., ,C C Cr n3 , .
(iv) ,nC nCr n r= - , r =0, 1, 2, …, n (x + a)n - .
(v) = 2n
(vi) (1+x)n, = 2n–1
(vii) (x+a)n tr+1 = nC x ar n r r-
x a n+] g (i) (x+a)n n (n+1 )
tn 2 1+ .
(ii) (x+a)n- n (n+1 ) . tn
2 1+ tn
2 3+ .
(x + a)n - . tr 1+ . tr 1+ r . x tr 1+ x r tr 1+ x .

x a n+] g = ... ...nC x a nC x a nC x a nC x a an n n r
n r r n 0
0 1
1 1 2
2 2+ + + + + +- - - ….(1)
(i) (x–a) , a –a
x a n-] g = ... ...nC x a nC x a nC x a nC x a an n n r
n r r n 0
0 1
1 2
2 2+ - + - + + - + + -- - -] ] ] ]g g g g
= ... ( ) ... ( )nC x a nC x a nC x a nC x a a1 1n n n r r
n r r n n 0
0 1
1 2
2 2- + + + - + + -- - -] ] ] ]g g g g ......(2)
. (ii) a = 1 (1)
... ...x nC x nC x nC x nC x1 1n r
r n
(iii) x = – x (3)
... ... (– )x nC x nC x nC x nC x1 1 1 1n r
r r n

(2x + 3y)5 .
x a n+] g = ...nC x a nC x a nC x a nC x an n n r
n r r 0
n n
-
x y2 3 5+^ h = C x C x y C x y C x y5 2 5 2 3 5 2 3 5 2 30 5
1 4
C x y C y5 2 3 5 34 4
5 5
+] ^ ^g h h
= x x y x y x y32 5 16 3 2 1 5 4 8 9 3 2 1
5 4 3 4 27× ×
× × × ×5 4 3 2 2 3+ + + +] ^ ^ _ _g h h i i
x y y5 2 81 2434 5+] g
= x x y x y x y xy y32 240 720 1080 810 2435 4 3 2 2 3 4 5+ + + + +

x x 12 2
4 +c m .
x x 12 2
4 +c m = x C x x C x x C x x C x4 1 4
1 4
1 4
+ + + +^ ^ ^ c ^ c ch h h m h m m
= x x x x x4 6 4 4 18 4 2 4 8+ + + + +

(101)5 .
(101)5 = (100 + 1)5
= (100)5 + C5 1 (100)4 + C5 2 (100)3 + C5 3 (100)2 + C5 4 (100) + C5 5
= 10000000000+ 5(100000000) + 10(1000000) + 10(10000) + 5(100) + 1
= 10000000000 + 500000000 + 10000000 + 100000 + 500 + 1
= 10510100501

x x 3 2
10 -c m 5 .
x a n+] g tr 1+ = nC x ar n r r- . ... (1)
x x 3 2
10 -c m 5 (1) n 10 r 4
t4 1+ = t C x x10 3
5 4 6
2 4
= -] cg m ( n = 10 , x = x , a = – x 3 2 )
= C x x10 3

x x 22 10
-b l .
x a n+] g tr 1+ = nC x ar n r r- . ... (1)

x x 22 10
-b l x a n+] g
n = 10, ,x x a x 22= = - r = 5 1
t t5 1 6=+ = C x x10 2 5
2 5 5-^ bh l
= C x x10 2 5
10 5

x y3 9 9
+b l - .
x y3 9 9
+b l (x + a)n n = 9 ( ) .
`
` ,t t 2 2 n n1 3+ +
,t t 2 2 9 1 9 3+ + .
Ch-2_TM.indd 77 19/04/18 6:58 PM
78
, t5 t6 .
(x + a)n
tr + 1 = nC x ar n r r- ... (1)
r = 4 (1) t t4 1 5=+ = C x y9 3 94 5 4 $b ^l h
= C x y9 3 94 5
5 4 4$
3 9× × × × × ×
= x y3402 5 4
r = 5 (1) t t5 1 6=+ = C x y9 3 95 4 5 $b ^l h
= C x y9 3 95 4
4 5 5$
= x y91854 4 5

x x2 32 11 -b l x10- .
(x + a)n tr 1+ = nC x ar n r r- ... (1)
x x2 32 11 -b l (x + a)n
tr 1+ = C x x11 2 3 r
r r2 11 --^ bh l
= C x x11 2 3 1 r
r r r r
11 2 11 -- - ] b] g lg
= C x11 2 1 3r r r r r11 22 3$-- -] g
x10 - 22–3r = 10 .
& r = 4
r = 4 (1) , t5 = C x11 2 34 11 4 4 10- = C x11 2 34
7 4 10$

x x2 2 1 2
9 +c m - x .
Ch-2_TM.indd 78 19/04/18 6:58 PM
79
(x + a)n - tr 1+ = nC x ar
n r r-
9 +c m (x + a)n .
` tr 1+ = C x x9 2 3 1
r r
r 9
r r r
r r r
- ... (1)
x - x ‘ 0 ’ 9 – 3r = 0
` r = 3
t3 1+ = C x9 3 2
3 5
3 5
6 = 1792

1. .
(i) a b2 3 4-] g (ii) x y 1 7
+c m (iii) x x 1 2
6 +c m
2. :
(i) (101)4 (ii) (999)5
3. (x–2y)13 5 .
4. .
(i) x x 1 11
+b l (ii) x x3 2 2 8
+b l (iii) x x2 32 3
10 -c m
5. x -
(i) x x3 22 9
-b l (ii) x x 2 2
15 -c m (iii) x x2 12 12
+b l
6. x x 1 n2
+b l x - ! ...,
n n1 3 5 2 1 2n$ $ -] g .
7. x1 n2+] g - ! . . …( )
n n x1 3 5 2 1 2n n- .
Ch-2_TM.indd 79 19/04/18 6:58 PM
1
2

– 1 G G .. “11th BUSINESS MATHS” GeoGebra “Pascal’s Triangle” .
- 2 “Pascal’s Triangle” , “View Pascal Triangle” “View Pascal’s Numbers” , “view Combination” C - . .
:
https://ggbm.at/qKj9gSTG (or) scan the QR Code
Ch-2_TM.indd 80 19/04/18 6:58 PM
1. nC nC3 2= nc4
(a) 2 (b) 3 (c) 4 (d) 5
2. np2 = 20 n -
(a) 3 (b) 6 (c) 5 (d) 4
3. 5 4
(a) 4! (b) 20 (c) 25 (d) 5
4. nPr = 720 nCr] g , r -
(a) 4 (b) 5 (c) 6 (d) 7
5. ,
(a) 25 (b) 52 (c) 10 (d) 2 5
6. n -
(a) nC2 (b) nC 22 - (c) nC n2 - (d) nC 12 -
7. n N! n n n n1 2 3+ + +] ] ]g g g
(a) 2 (b) 6 (c) 20 (d) 24
8. n x a n+] g
(a) n (b) n + 1 (c) n – 1 (d) 2n
9. n nC1+ nC2+ nC3+ ………+ nCn
(a) 2n (b) 2n –1 (c) n2 (d) n2–1
10. x y2 7-^ h , x3
(a) 3 (b) 4 (c) 5 (d) 6
Ch-2_TM.indd 81 19/04/18 6:58 PM
11. x x 1 10
+b l
(a) C x10 1 4 b l (b) C10 5 (c) C10 6 (d) C x10 7
4
12. x x 2 6
+b l
(a) 156 (b) 165 (c) 162 (d) 160
13. 3 2 8 +^ h
(a) 81 (b) 16 (c) 8 2 (d) 27 3
14. x x kx
2 1 1
+ - = + + -] ]g g k
(a) 9 (b) 11 (c) 5 (d) 7
15. NUMBER
(a) 206 (b) 133 (c) 216 (d) 300
16. ,
(a) 18 (b) 12 (c) 9 (d) 6
17. 10 .
(a) 240 (b) 120 (c) 1024 (d) 100
18. C C C C C C C C C C5 5 5 5 5 5 5 5 5 50 1 1 2 2 3 3 4 4 5+ + + + + + + + +] ] ] ] ]g g g g g
(a) 26 –2 (b) 25 –1 (c) 28 (d) 27
19. 9
(a) 10! (b) 9! (c) 9 × 9! (d) 10 × 10!
20. “ CHEESE ”
(a) 120 (b) 240 (c) 720 (d) 6
Ch-2_TM.indd 82 19/04/18 6:59 PM
83
21. 13 , (a) 715 (b) 78 (c) 286 (d) 13
22. “ EQUATION ” , , () (a) 7! (b) 3! (c) 8! (d) 5!
23. 256 , (a) 8 (b) 7 (c) 6 (d) 9
24. , n r (a) rn (b) nr (c) !
! n r

1. : x x x 1 3 5 7 - +
+ ] ]g g
2. : x x x 3 2 4
2 - + -
3. : x x x x
2 3 6 14 27
2
2
+ - - -
] ]g g 4. : x x x
x x 2 1
5 8 5 2
5.
(i) ! ! 6 7 (ii) !
! 5 8 (iii) ! !
! 6 3 9
6. A, B, C , D, E, F 6 5 a) b) c) E d) C A B .
Ch-2_TM.indd 83 19/04/18 6:59 PM
84
7. 20 , 4 . . , .
8. 9 , 8 7 2 , 2 2
9. 3 , 2 2 (signal) .
10. x x 2 2
17 +c m x11 .
z n , n - , n
. n ! n .
z n n ! = n ( n – 1) ( n – 2) … 3× 2× 1
z 0! = 1
z m n ( m + n ) .
z n r .
z ! !np n r
! ! !
! ! !np n n
n n n0n = - = =] g z ( )( )…[ ( )]np n n n n r1 2 1r = - - - -
z n r n r
Ch-2_TM.indd 84 19/04/18 6:59 PM
85
z ! ! !
p q n n p , q
. p + q = n n (clock wise) (anti- clockwise) , (n-1)!
z , !n
2 1-] g .
z n r .
z n ncr .
z ! ! ! ,nC r n r
n r n0r 1#= -^ h z nCn = 1 = nC0 .
z nC1 = n.
2 1-^ h
z nCx = nCy , x = y x + y = n.
z nCr = nCn r- .
z nC nC n C1r r r1+ = +- ^ h
z ( ) …x a nC x a nC x a nC x a nC x a nC x an o
n o n n r
n r r n
r
n
=
-/
z (x+a)n n+1 .
z x a n .
z , , , …… …nC nC nC nC nC nCr n0 1 2 3 , , , … …C C C C C Cr n0 1 2 3 . .
z r = 0, 1, 2,…n nC nCr n r= - , (x+a)n . , , ,nC nC nC nC nC nCn n n0 1 1 2 2= = =- -
Ch-2_TM.indd 85 19/04/18 6:59 PM
86
z 2n .
z = 2n-1
z x a n+^ h t nC x ar r n r r
1 =+ - .
Binomial
Principle of counting Multiplication Principle of counting
Linear factor Mathematical Induction Factorial Coecient Independent term
Combination Partial fractions Pascal’s Triangle Middle term Circular Permutation Permutations Rational Expression
Ch-2_TM.indd 86 19/04/18 6:59 PM

z z z z z z z
z , z z , ,

(1596-1650)
“Geometry” “geo” “metron’’ . “geo” , “metron’’ .
, , , , . , , .
, , (1596- 1650), , , “la Geometry’’ 1637 . , . .
3
Ch-3_TM.indd 87 19/04/18 7:04 PM
88
. .

3.1
.

a
3.1
, .
,
(i) C(h, k) , P(x, y) , . c .
(ii) A, B AB .
3.2
A B

.

y- .
Ch-3_TM.indd 88 19/04/18 7:04 PM
89
,P x y1 1^ h . ,P x y1 1^ h Y - A .
OP = 3 AP
OP2 = 9AP2
2
1 2- = 0
` ,P x y1 1^ h x y8 02 2- =

(2, –3) (3, –4) .
A(2, –3) B(3, –4)
,P x y1 1^ h .
PA = PB .
PA2 = PB2
2 1
2- + +^ ^h h x x y y4 4 6 91
2 1 1
2 1 1
i.e., x y 61 1- - = 0
,P x y1 1^ h x y 6- - = 0.

(–5, 1) (3, 2) .
A(–5, 1) B(3, 2) .
Ch-3_TM.indd 89 19/04/18 7:04 PM
90
,P x y1 1^ h .
APB+ = 90o .
TAPB
BA2 = PA2+PB2
(3+5)2 + (2–1)2 = x y x y5 1 3 21 2
1 2
1 2
1 2+ + - + - + -^ ^ ^ ^h h h h
65 = x x y y x x y y10 25 2 1 6 9 4 41 2
1 1 2
1 1 2
1 1 2
1+ + + - + + - + + - +
x y x y2 2 4 6 39 651 2
1 2
1 2
1 1+ + - - = 0
,P x y1 1^ h x y x y2 3 132 2+ + - - = 0

1. (1, 3) , x-, .
2. (3, –2) , 4 .
3. , (2,1) (1,2) 2:1 , .
4. (7, –6) (3,4) , x - .
5. A(–1, 1) B(2, 3) APB 8 . P .

, ,
. , , . , 10 . 11 , , .
Ch-3_TM.indd 90 19/04/18 7:04 PM
(i)
m y c y = mx+c
(ii)
,P x y1 1^ h m y y m x x1 1- = -^ h
(iii)
, , ,A x y B x y1 1 2 2^ ^h h y y
y y x x x x
2 1
= - - .
,A x y1 1^ h , ,B x y2 2^ h
x x x
y y y
1 1 1
= .
(iv) x- a y- b , , a
x b y
1+ = .
(v) ax by c 0+ + = . a, b c . a, b .
y
l1 l2
2i 1i
3.3
l1 l2 , P- :l y m x c1 1 1= + :l y m x c2 2 2= +
.
1i 2i l1 l2 x- .
Ch-3_TM.indd 91 19/04/18 7:05 PM
92
tanm1 1i= tanm2 2i= .
l1 l2 i , 1 2i i i= - ( 3.3)
` tani = tan 1 2i i-^ h
= tan tan tan tan 1 1 2
1 2
i i
i i
1 2 + -
1 1 2
1 2 + -
(>0) , l1 l2 i . m m
m m 1 1 2
1 2 + - -
(<0) , i
( i i ) . m1=m2 .
( i i i ) , , –1 , m1 m2=–1. ( m1 m2 )
x- y- ,
m m 11 2 = - .
x- m1=0, y- m2 .

x y2 3 0- + = x y 2 0+ + = .
m1 m2 x y2 3 0- + = x y 2 0+ + = .
m1 = 2, m2 = –1
Ch-3_TM.indd 92 19/04/18 7:05 PM
93
i
tan i = m m m m 1 1 2
1 2 + -
i = tan 31- ^ h

(i) P(l, m) ax by c 0+ + = - d a b
al bm c 2 2= +
+ +
(ii) (0, 0) ax by c 0+ + = - d a b
c 2 2= +

x y 5 0- + = P(2, 2) .
x y 5 0- + =
P(2, 2) = 1 1
2 2 5 2 2+
- +
= 2 5 =
2 5
(0, 0) = 1 1 5
2 5
2 2+ = =
2 5
(0, 0) (2, 2)

4 r , 3,
.
m1 m2 i
tan i = m m m m 1 1 2
1 2 + -
Ch-3_TM.indd 93 19/04/18 7:05 PM
& m2 = 2 1

l1 l2 ‘P’ , P . l1 l2
, .
, . , .

a x b y c1 1 1+ + = 0 " (1)
a x b y c2 2 2+ + = 0 " (2)
a x b y c 03 3 3+ + = " (3)
.
a a a
b b b
c c c

,x y x y3 4 13 0 8 11 33- - = - = x y2 3 7 0- - = . .
x y3 4 13- - = 0 ... (1)
x y8 11- = 33 ... (2)
Ch-3_TM.indd 94 19/04/18 7:05 PM
x y2 3 7- - = 0 ... (3)
a a a
b b b
c c c
= –66 + 40 + 26 = 0
& .

(1) #2 & 6x–8y =26
(3) #3 & 6x–9y =21 ___________ y = 5 ___________ y=5 (2) 8x = 88
x = 11

,x y x y3 5 11 0 5 3 7 0- - = + - = x ky 0+ = , k- .
.
a a a
b b b
c c c
& 34k = 68. ` k = 2.

, 2012 `20,000 - , . 2017 `25,000 . ,
(i) y- x- x, y - .
(ii) 2020 .
y , x- .
(x) (y)
2012(x1) 20,000(y1) 2017(x2) 25,000(y2)
2020 ?
x, y-
y y y y 2 1
1 - -
1 - -
, , ,y
- = x
y = 1000(2020)–19,92,000
y = `28000

1. 2 1 3 .
2. (4, 1) x y3 4 12 0- + = .
Ch-3_TM.indd 96 19/04/18 7:05 PM
97
3. ,x y x4 0 3 2 0+ - = + = x y3 3 16 0- + = .
4. ;x y x y3 4 13 2 7 1+ = - =- ax y 14 0- - = ‘a’ .
5. 80 , 2,20,000 . 125 2,87,500 . - , - . 95 .

l x m y n 01 1 1+ + = l x m y n 02 2 2+ + = .

l x m y n1 1 1+ +^ h l x m y n2 2 2+ +^ h = 0
l l x l m l m xy m m y l n l n x m n m n y n n 01 2 2
1 2 2 1 1 2 2
1 2 2 1 1 2 2 1 1 2+ + + + + + + + =^ ^ ^h h h
,
ax hxy by gx fy c2 2 2 02 2+ + + + + =
. ( , , , , ,a b c f g h ) .

x, y ax hxy by2 02 2+ + = ... (1) .

y m x1= y m x2= .

Ch-3_TM.indd 97 19/04/18 7:05 PM
y m x y m x1 2- -^ ^h h = 0
& m m x m m xy y1 2 2
1 2 2- + +^ h = 0 ... (2)
(1), (2) .
` m m a
m m h b2
11 2 1 2 =
- + =^ h m m b
a 1 2& = = b
a
m m b h2
1 2+ = - = b h2-

ax hxy by2 02 2+ + =
m1, m2 .
` m m b h2
1 2 = .
i .
tan m m m m 1 1 2
1 2i = + -
= ± a b
+ -
‘i ’
` i = tan a b h ab21 2
+ -- < F

(i) ax hxy by gx fy c2 2 2 02 2+ + + + + =
i , i = tan a b h ab21 2
+ -- < F
(ii) h ab2 = .
(iii) , a b 0+ =
x2 + y2 = 0
Ch-3_TM.indd 98 19/04/18 7:05 PM
99

a h g
h b f
g f c
.
ax hxy by gx fy c2 2 2 02 2+ + + + + =
– – –abc fgh af bg ch2 02 2 2+ = .

x y2 1 0+ - = , x y2 5 0+ - = .

x y2 1+ -^ h x y2 5 0+ - =^ h x xy x xy y y x y2 4 2 2 10 5 5 02 2+ - + + - - - + =
x xy y x y2 5 2 11 7 5 02 2+ + - - + =

x xy y x y2 5 3 6 7 4 02 2+ + + + + = . .

x xy y x y2 5 3 6 7 4 02 2+ + + + + =
ax hxy by gx fy c2 2 2 02 2+ + + + + =
, , , ,a b h g f2 3 2 5 3 2
7= = = = = c = 4

– – –abc fgh af bg ch2 02 2 2+ = .
– – –abc fgh af bg ch2 24 2 105
2 49 27 25 02 2 2+ = + - - - =
Ch-3_TM.indd 99 19/04/18 7:05 PM
100

‘θ’ .
i = tan a b h ab21 2
+ -- < F = tan

ax hxy by2 02 2+ + = h ab8 92 = .
m1 , m2
ax hxy by22 2+ + = 0 .
` m m1 2+ = b h2- m m b
a 1 2 =
` m m21 1+ = b h2- m m b
a21 1$ =
a2 1 2 =
-b l = b a
& b h 9 8

x xy y x y2 7 3 5 5 2 02 2+ + + + + = . .
x xy y x y2 7 3 5 5 2 02 2+ + + + + =
ax hxy by gx fy c2 2 2 02 2+ + + + + = ,
, , , , ,a b h g f c2 3 2 7
2 5
= 2 4 1
8 25- - + = 0

x xy y2 7 32 2+ + = x xy xy y2 6 32 2+ + +
= x x y y x y2 3 3+ + +^ ^h h = x y x y3 2+ +^ ^h h
x xy y x y2 7 3 5 5 22 2+ + + + + = x y l x y m3 2+ + + +^ ^h h .
x – , 2l + m = 5 (1)
y – , l + 3m = 5 (2)
(1) (2) m = 1 l = 2
,
x y3 2 0+ + = x y2 1 0+ + = .

x xy y x y4 12 9 18 27 8 02 2- + + - + = . .
x xy y x y4 12 9 18 27 8 02 2- + + - + =
a = 4, b = 9, h = –6
h ab 36 36 02 - = - =
.
x xy y4 12 92 2- + = x y2 3 2-^ h
x xy y x y4 12 9 18 27 82 2- + + - + = 0
& x y x y2 3 9 2 3 82- + - +^ ^h h = 0 .
2x–3y = z .
2x–3y+1 = 0 2x–3y+8 = 0
2x–3y+1 = 0 2x–3y+8 =0 .

x xy y4 02 2+ + = .
x xy y4 02 2+ + =
a = 1, b = 1 h = 2.
i
θ = tan a b h ab21 2
+ -- < F
= tan 31- ^ h
i = 3 r

k- x xy y x y k2 5 2 15 18 02 2+ + + + + = ?
a = 2, b = 2, h = 2 5 , g = 2
15 , f = 9, c = k.
,
abc fgh af bg ch2 2 2 2+ - - - = 0
i.e., k k4 2 675 162 2
225 4 25+ - - - = 0
& 9k = 252 ` k = 28

1. ax xy y x y c5 6 12 52 2+ - + + + = 0 a c - .
2. x xy y x y12 10 2 14 5 22 2- + + - + = 0 . .
3. x xy y x y4 12 9 6 9 22 2+ + - - + =0 . .
4. x xy y x y3 5 2 17 102 2- - + + + =0 .

3.2
. ,
.

y
xMLO
r(h, k) C R
P(x, y) C(h, k) , ‘r’ .
P(x, y) .
CP = r
CP2 = r2
(x–h)2 + (y–k)2 = r2 .
x2 + y2 = r2 .
Ch-3_TM.indd 103 19/04/18 7:06 PM

(3, –1) 4 .
x h y k2 2- + -^ ^h h = r2
(h, k) = (3, –1) r = 4
x y3 12 2- + +^ ^h h = 16
x x y y6 9 2 12 2- + + + + = 16
x y x y6 2 62 2+ - + - = 0

, 3 .
x2 + y2 = r2 r = 3
x2 + y2 = 9

3.5
A(x1, y1) B(x2, y2)
,A x y1 1^ h , ,B x y2 2^ h
P(x, y) .
APB = 90c
` (AP ) (BP ) = –1
x x y y
x x y y
- -d dn n = –1
y y y y1 2- -^ ^h h = x x x x1 2- - -^ ^h h
& x x x x y y y y1 2 1 2- - + - -^ ^ ^ ^h h h h = 0
.
Ch-3_TM.indd 104 19/04/18 7:06 PM
105

(2, 4), (3, –2) , .

x x x x y y y y1 2 1 2- - + - -^ ^ ^ ^h h h h = 0
,x y1 1^ h = (2, 4) ,x y2 2^ h = (3, –2)
` x x y y2 3 4 2- - + - +] ] _ _g g i i = 0
x y x y5 2 22 2+ - - - = 0
x y gx fy c2 22 2+ + + + = 0 ,
g, f , c .
x y gx fy2 22 2+ + + = –c
x gx g g y fy f f2 22 2 2 2 2 2+ + - + + + - = –c
x g g y f f2 2 2 2+ - + + -^ ^h h = –c
x g y f2 2+ + +^ ^h h = g f c2 2+ -
x g y f2 2 - - + - -^ ^h h7 7A A = g f c2 2 2
+ -7 A
x h y k r2 2 2- + - =^ ^h h
,g f- -^ h , g f c2 2+ - .

ax by hxy gx fy c2 2 2 02 2+ + + + + =
(i) a = b () x2 = y2 .
(ii) h = 0 () xy - =0 (xy ).
Ch-3_TM.indd 105 19/04/18 7:06 PM

x y x y8 6 24 02 2+ - + - = .
x y x y8 6 242 2+ - + - = 0
g = –4 , f = 3 , c = –24
= , ,C g f C 4 3- - = -^ ^h h
r = g f c2 2+ -
= 16 9 24+ + = 7 .

a, b a x by b xy x y2 2 4 4 12 2- + + - + + -] ]g g = 0 .
a x by b xy x y2 2 4 4 12 2- + + - + + -] ]g g = 0
,
(i) xy = 0 & b–2 = 0
` b = 2
(ii) x2 = y2 & a–2 = b
a – 2 = 2 & a = 4
x y x y2 2 4 4 1 02 2+ + + - =

x y ax by2 2+ + + = 0 (1, 2) (1, 1) ‘a’ ‘b’ - .
x y ax by2 2+ + + = 0 (1, 2) (1, 1)
Ch-3_TM.indd 106 19/04/18 7:06 PM
107
1 + 4 + a + 2b = 0 1 + 1+ a + b = 0
& a b2+ = –5 (1)
a + b = –2 (2)
(1) (2) - a = 1, b, = –3 .

x y x y2 6 12 2+ + - + = 0 ax y2 2+ + = 0 ‘a’- .
C(–1, 3)
ax y2 2+ + = 0 .
–a + 6 + 2 = 0
a = 8

x y x y6 42 2+ - + –12 = 0 (7, –5) . .
A(7, –5) .
x y x y6 4 122 2+ - + - = 0
(7, –5)
x y x y6 4 122 2+ - + - = 7 5 6 7 4 5 122 2+ - - + - -] ] ]g g g
= 49 + 25 – 42 – 20 – 12 = 0
` (7, –5)
g = –3 f = 2
` = C(3, –2)
B (x, y) .
Ch-3_TM.indd 107 19/04/18 7:06 PM
7 2 5 3 2+ -
= -c ^m h
x = –1 y = 1
` (–1 , 1) .

(0,0), (1, 2) (2, 0) .
x y gx fy c2 22 2+ + + + = 0 .
(0, 0)
c = 0 ... (1)
(1, 2)
( ) ( )g f c1 2 2 1 2 22 2+ + + + = 0
2g + 4f + c = – 5 ... (2)
(2, 0)
( ) ( )g f c2 0 2 2 2 02 + + + + = 0
4g + c = –4 ... (3)
4g = –4 & g = –1
(1), (2) (3) – ,
,g f1 4 3= - = - , c = 0
`
( )x y x y2 1 2 4 3 02 2+ + - +
- +b l = 0
, x y x y2 2 4 32 2+ - - = 0
Ch-3_TM.indd 108 19/04/18 7:06 PM

r . P(x,y) . x - OP i .
x - PM .
3.6
i
X
Y
, cos cosr x x r&i i= =
sin sinr x y r&i i= =
x = r cosi , y = r sini x y r2 2 2+ = . ‘i ’ 0 2# #i r .

x y2 2+ = 25 .
r2 = 25 & r = 5
x = r cosi , y = r sini
, ,cos sinx y5 5 0 2& # #i i i r= =

1. . (i) (3,5) 5
(ii) (0, 0) 2
2. (i) x y2 2+ = 16
(ii) x y x y22 4 25 02 2+ - - + =
(iii) x y x y5 5 4 8 16 02 2+ + - - =
(iv) x x y y2 5 2 1 0+ - + - - =] ] ^ ^g g h h
Ch-3_TM.indd 109 19/04/18 7:06 PM
110
3. 16r (–3, –2) .
4. (1,4) (2,3) .
5. (0,1),(4,3) (1, –1) .
6. (1, 0), (0, 1) x+y = 1 .
7. x+y =6 x+2y = 4 (2, 6) .
8. (4, 7) (–2, 5) .
9. x = 3 cosi , y =3 sini , 0 2# #i r , .

3.7
P(x1, y1)
,x y1 1^ h
x y gx fy c2 22 2+ + + + = 0
( ) ( )xx yy g x x f y y c1 1 1 1+ + + + + + = 0 .
:
x y a2 2 2+ = ,x y1 1^ h xx yy a1 1
2+ =

x y gx fy c2 22 2+ + + + = 0 x2 xx1 , y2 yy1 x x x 2
1+ y y y 2
1+ ( ,x y1 1 ) .

(–2, 5) x y x y3 8 172 2+ + - + = 0 .
Ch-3_TM.indd 110 19/04/18 7:06 PM
111
,x y1 1^ h
xx yy x x y y3 2 1 8 2
1 171 1 1 1# #+ + + - + +^ ^h h = 0
,x y1 1^ h = (–2, 5)
( ) ( )x y x y2 5 2 3 2 4 5 17- + + - - + + = 0
x y x y2 5 2 3 3 4 20 17- + + - - - + = 0
x y x y4 10 3 6 8 40 34- + + - - - + = 0
x y2 12- + = 0 .

P ,x y1 1^ h x y gx fy c2 22 2+ + + + = 0
PT = x y gx fy c2 21 2
1 2
1 1+ + + +
3.8
P(x1, y1)

(i) P PT2 = 0
(ii) P , PT2 > 0
(iii) PT2 < 0 P
Ch-3_TM.indd 111 19/04/18 7:06 PM

x y x y8 4 82 2+ + + + = 0 (2, 3) .
x y gx fy c2 22 2+ + + + = 0 ,x y1 1^ h
x y gx fy c2 21 2
1 2
1 1+ + + +
= x y x y8 4 81 2
1 2
1 1+ + + +
= ( ) ( )2 3 8 2 4 3 82 2+ + + + [ , ( , )x y 2 31 1 =^ h ]
= 49

P(0,1), Q(5,9), R(–2, 3) S(2, 2) x y x y4 4 82 2+ - + - 0 ,
x y x y4 4 82 2+ - + - = 0
PT2 = x y x y4 4 81 2
1 2
1 1+ - + -
P(0, 1) PT2 = 0+ 1 + 0 + 4–8 = –3 < 0
Q(5, 9) QT2 = 25 + 81 –20 + 36 –8 = 114 > 0
R(–2, 3) RT2 = 4 + 9 + 8 + 12 – 8 = 25 > 0
S(2, 2) ST2 = 4 + 4 – 8 + 8 – 8 = 0
P , Q R , S .
y = mx + c x y a2 2 2+ =
( )c a m12 2 2= + .
x y k3 4 0+ - = x y 642 2+ - = 0
.
Ch-3_TM.indd 112 19/04/18 7:06 PM
113
x y 64 02 2+ - = x y k3 4 0+ - =
( )c a m12 2 2= +
a2 = 64 , m = 4 3- , c = k
4
( )c a m k1 16 64 1 16 92 2 2
2 &= + = +b l
k2 = 64 × 25
k = ± 40

1. (–2, –2) x y x y4 4 82 2+ - + - = 0 .
2. P(1, 0), Q(2, 1) R(2, 3) x y x y4 6 92 2+ - - + = 0 ,
3. (1 , 2) x y x y2 4 92 2+ - + + = 0 .
4. x y P3 4+ - = 0 x y 162 2+ = P .

3.9

(

) 3.3
, , , , .
F . l . P , ,PM
FP e= P - . ‘e’ .
Ch-3_TM.indd 113 19/04/18 7:06 PM
114
,
a) e = 1, , b) e < 1, , .
c) e > 1, , .
ax hxy by gx fy c2 2 2 02 2+ + + + + = ,
(i) abc fgh af bg ch2 02 2 2+ - - - = .
(ii) a = b h = 0 .
, , ax hxy by gx fy c2 2 2 02 2+ + + + + =
(i) –h ab 02 = ,
(ii) –h ab 0<2 , .
(iii) –h ab 0>2 , .

3.4
, , .
y2 = 4 ax . .
y ax42 =
3.10
y
115
F . F(a,0) .
l . x = - a.
. y ax42 = x - . x - y = 0 y ax42 = .
, , . V(0,0) .
, , .
. .
, , . x = a.
4a
1. : y ax a4 02 2-= 6 @
x > 0, , y . x ≤ 0 .
3.11
116
2. : x ay a4 02 2= 6 @ y < 0 x . y ≥ 0 .
3.12
x
y
3. : x ay a4 02 2-= 6 @ y > 0 , x . y ≤ 0 .
3.13
x
y
y2 = 4ax y2 = –4ax x2 = 4ay x2 = –4ay
y = 0 y = 0 x = 0 x = 0
V(0, 0) V(0, 0) V(0, 0) V(0, 0)
F(a, 0) F(–a, 0) F(0, a) F(0, –a)
x = –a x = a y = –a y = a
4a 4a 4a 4a
x = a x = –a y = a y = –a
Ch-3_TM.indd 116 19/04/18 7:07 PM
117

xoy x – . (X - ) y- (Y- ). P(x,y) xoy . XO Yl P(X,Y) .
3.14
y Y
xoy - Ol - (h, k) .
xoy P- :
OL = OM + ML
y = Y + k
` XO Yl P - .
–X x h= –Y y k=

(h, k) , x- x–h y- y–k .
–
XOY V(0, 0) , xoy (h, k) .
Ch-3_TM.indd 117 19/04/18 7:07 PM
118
XOY . Y aX42 = .

X = x – h Y = y – k . xoy y k a x h42- = -^ ^h h .
.
y k a x h42- =- -^ ^h h ( )
x h a y k42- = -^ ^h h ( )
x h a y k42- =- -^ ^h h ( )

x y 2 0- + = (1, 3) .
3.15
P(x, y)
F(1, 3)
F (1, 3)
x y 2 0- + =
x y 2 0- + =
,P x y^ h .
PM FP = 1
= x x y y2 1 6 92 2- + + - +
= x y x y2 6 102 2+ - - +
PM = + x y
1 1 2
2 2
= + + - - +^ h
FP2 = PM2
x y x y2 2 4 12 202 2+ - - + = x y xy y x2 4 4 42 2+ - - + +

& x y xy x y2 8 8 162 2+ + - - + = 0

y x122 = - , , , ,
3.16
F(–3, 0) X
Y y ax42 =-
4a = 12, a = 3.
.
, ,F a F 3 0q- = -^ ^h h
, ,V h k V 0 0=^ ^h h
x a=
i.e., x = 3
x - y = 0.
= 4a =12

x p p10 20 2= - - . .
x = p p10 20 2- -
= p p20 10 5 52- - + + -
x 5- = p p10 252- + -
Ch-3_TM.indd 119 19/04/18 7:07 PM
2
: https://ggbm.at/qKj9gSTG (or) scan the QR Code

– 1 G G .. “11th BUSINESS MATHS” GeoGebra “Locus-Parabola” .
- 2 “Locus-Parabola” 4 , . “a” . play/pause .
1
X P2=- P X2 =-
.
` x = 5 P = 5 .

x x y6 4 21 02 + - + = , , , ,
4y = x x6 212 + +
,X x Y y3 3= + = -
x X 3= - y Y 3= +
X2 = 4Y
a = 1
(X, Y) (x, y) ,x X y Y3 3= - = +
y = 0 Y = 0 y = 3
V(0,0) V(0,0) V(–3,3)
F(0, a) F(0, 1) F(–3, 4)
(y=–a) Y = –1 y = 2
(4a) 4(1) = 4 4
Ch-3_TM.indd 121 19/04/18 7:07 PM

, x P5 15= - . .
x2 = 5p – 15
= 5(p – 3)
X aP42& = X x= , P p 3= -
` ( , )X P0 0= =
V(0, 3) .

1. F(-1,-2) , –x y4 3 2 0+ = .
2. y kx2 = (4,-2) .
3. – –y y x8 8 24 02 + = , , , .
4. , , , .
(a) y x202 = (b) x y82 = (c) x y162 =-
5. x x x5
1 6 1002 - + , . , .
6. x ` y () . y x x10 152= - + - , , .
Ch-3_TM.indd 122 19/04/18 7:07 PM

1. ,ax hxy by2 02 2+ + =
m1 , m2 , m m1 2+
(a) b h2 (b) b
h2- (c) a h2 (d) – a
h2
2. –x xy y7 4 02 2+ =
(a) tan 3 11- b l (b) tan 2
11- b l (c) tan 5 331- c m (d) tan 33
51- d n
3. x y2 3 5 0- - = x y3 4 7 0- - = , (a) ( –1, 1) (b) (1,1) (c) (1, –1) (d) (–1, –1)
4. x y3 2 1 0+ - = x - (a)3 (b) 2 (c) 3
1 (d) 2 1
5. x y7 5 8 0+ - =
(a) 5 7 (b) – 5
7 (c) 7 5 (d) – 7
5
6. P
(a) y x 1= (b) y x=- (c) y x= (d) y x
1= -
7. x y2 7 0+ + = , P (a) x y2 2 0+ + = (b) x y2 1 0- + = (c) x y2 2 0- + = (d) x y3 1 0+ + =
8. kx xy y3 2 02 2+ - = k =
(a) 2 1 (b) – 2
1 (c) 2 (d) –2
9. x y ax by 4 02 2+ + + - = (1, –2) (a) 3 (b) 2 (c) 4 (d) 1
10. (4, 5) x y 162 2+ =
(a) 4 (b) 5 (c) 16 (d) 25
Ch-3_TM.indd 123 19/04/18 7:07 PM
124
11. x y162 = (a) (4 ,0) (b) (–4 , 0) (c) (0, 4) (d) (0, –4)
12. y x252 =- .
(a) 25 (b) –5 (c) 5 (d) –25
13. x y x y2 2 9 02 2+ - + - = (a) (1 ,1) (b) (–1, –1) (c) (–1,1) (d) (1, –1)
14. , x - (a) x ax y2 02 2- + = (b) y ay x2 02 2- + =
(c) x y a2 2 2+ = (d) x ay y2 02 2- + =
15. (–a, –b) a b2 2-
(a) x y ax by b2 2 2 02 2 2+ + + + = (b) x y ax by b2 2 2 02 2 2+ + + - =
(c) x y ax by b2 2 2 02 2 2+ - - - = (d) x y ax by b2 2 2 02 2 2+ - - + =
16.
(a) x y 02 2- = (b) x y 02 2+ = (c) xy c= (d) xy 0=
17. ax xy y4 2 02 2+ + = ‘a’
(a) 2 (b) –2 (c) 4 (d) –4
18. x y 162 2+ = , y ()
(a) 4 (b) 16 (c) ±4 (d) ±16
19. 8π (2, 2)
(a) x y2 2 42 2- + - =^ ^h h (b) x y2 2 162 2- + - =^ ^h h (c) x y4 4 22 2- + - =^ ^h h (d) x y 42 2+ =
20. (3, –4) x
(a) x y3 4 42 2- + - =^ ^h h (b) x y3 4 162 2- + + =^ ^h h (c) x y3 4 162 2- + - =^ ^h h (d) x y 162 2+ =
Ch-3_TM.indd 124 19/04/18 7:07 PM
125
21. , x -, y - x = 6 ,
(a) 6 (b)3 (c)12 (d) 4
22.
(a) 3 (b)2 (c) 0 (d) 1
23.
(a) (b) (c) (d)
24. y ax42 =
(a) a (b)2a (c) 4a (d) 3a
25. y x2 =-
(a) x4 1 0+ = (b) x4 1 0- = (c) x 4 0- = (d) x 4 0+ =

1. P, (2,3) (1, 5) P
2. 500 1000 `6000 `9000 . x, y (y) (x) .
3. , –

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