5 1

䉶 ñFŠªð‡ Mù£‚èœä‰¶ ñFŠªð‡ Mù£‚èœä‰¶ ñFŠªð‡ Mù£‚èœä‰¶ ñFŠªð‡ Mù£‚èœä‰¶ ñFŠªð‡ Mù£‚èœ1. èíƒèÀ‹ ꣘¹èíƒèÀ‹ ꣘¹èíƒèÀ‹ ꣘¹èíƒèÀ‹ ꣘¹èíƒèÀ‹ ꣘¹èèèèèÀ‹À‹À‹À‹À‹

1. ªõ¡ðìƒè¬÷Š ðò¡ð´ˆF H¡õ¼ùõŸ¬ø„ êKò£ âù «ê£Fˆ¶Š 𣘂è. A (B C) = (A B) (A C).

b˜¾ :

LHS: RHS

B C A B

A A C

A (B C) (A B) (A C)

------ (I) -------- (II)

I = II LHS = RHS

A (B C) = (A B) (A C).

2. A (B C) = (A B) (A C).

b˜¾:LHS RHS

B C A B

BA

C

BA

C

BA

C

BA

C

BA

C

BA

C

BA

C

A

C

B

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5 2

A A C

A (B C) (A B) (A C)

------ (I) -------- (II)

I = II LHS = RHS

A (B C) = (A B) (A C).

3. A\(B C) = (A\B) (A\C).b˜¾:

LHS RHS

B C A\B

A A\C

A\(B C) (A\B) (A\C)

------ (I) -------- (II)

I = IILHS = RHS

A\ (B C) = (A\B) (A\ C)

BA

C

BA

C

BA

C

BA

C

BA

C

BA

C

BA

C

A

C

B

A

C

B A

C

B

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5 3

4. A\(B C) = (A\B) (A\C)

b˜¾:

LHS RHS

(B C) (A\B)

A (A\C)

A\(B C) (A\B) (A\C)

---- (I) ------ (II)

I = IILHS = RHS

A\(B C) = (A\B) (A\C)

5. BA)BA( ′′=′ b˜¾:

LHS RHS

BA A′ = U - A

)BA( ′ = U - BA B′ = U - B

--- I

A

C

B BA

C

BA

C

A

C

B

BA

C

BA

C

A B U

A B U A B U

A B U

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5 4

BA ′′

-------- (II)

I = IILHS = RHS

BA)BA( ′′=′ 6. BA)BA( ′′=′

b˜¾:LHS RHS

BA A′ = U - A

)BA( ′ = U - BA B′ = U - B

---- (I)

BA ′′

-------- (II)

I = IILHS = RHS

BA)BA( ′′=′ 7. U = {-2, -1, 0, 1, 2, 3, .... 10}, A = {-2, 2, 3, 4, 5} B = {1, 3, 5, 8, 9} ̄ ñ£˜è¡ èí GóŠH MFè¬÷„ êKð£˜.

b˜¾ ¯ñ£˜è¡ èí GóŠ¹ i) BA)BA( ′′=′ ii) BA)BA( ′′=′

i) L.H.S. = )BA( ′A B = {-2, 2, 3, 4, 5} {1, 3, 5, 8, 9}

= {-2, 1, 2, 3, 4, 5, 8, 9}

)BA( ′ = U \ (A B)

= {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} \ {-2, 1, 2, 3, 5, 8, 9}= {-1, 0, 6, 7, 10} ------- (I)

A B U

B UA A B U

A B

A B

A B U

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5 5

R.H.S. = BA ′′

A′ = U\A

= {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} \ {-2,2,3, 4, 5}

= {-1, 0, 1, 6, 7, 8, 9, 10}

B′ = U \ B

= {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} \ {1, 3, 5, 8, 9}

= {-2, -1, 0, 2, 4, 6, 7, 10}

BA ′′ = {-1, 0, 1, 6, 7, 8, 9, 10} {-2, -1, 0, 2, 4, 6, 7, 10}

= {-1, 0, 6, 7, 10} ----- (II) I = II

)BA( ′ BA ′′

ii) BA)BA( ′′=′

L.H.S. = )BA( ′

)BA( = {-2, 2, 3, 4, 5} {1, 3, 5, 8, 9}

= {3, 5}

)BA( ′ = U \ (A B)

= {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} \ {3, 5}= {-2, -1, 0, 1, 2, 4, 6, 7, 8, 9, 10} ----- (I)

R.H.S. = BA ′′ A′ = U \ A

A′ = {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} \ {-2, 2, 3, 4, 5}

= {-1, 0, 1, 6, 7, 8, 9, 10}

B′ = U \ B

B′ = {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} \ {1, 3, 5, 8, 9}

= {-2, -1, 0, 2, 4, 6, 7, 10}

BA ′′ = {-1, 0, 1, 6, 7, 8, 9, 10} {-2, -1, 0, 2, 4, 6, 7, 10}

= {-2, -1, 0, 1, 2, 4, 6, 7, 8, 9, 10} ----- (II) I = II

BA)BA( ′′=′

8. A = {a, b, c, d, e, f, g, x, y, z}, B = {1, 2, c, d, e} , C = {d, e, f, g, z, y} A\(BC) = (A\B) (A\C)

b˜¾:BC = {1, 2, c, d, e} {d, e, f, g, z, y}

= {1, 2, c, d, e, f, g, y}

A\BC = {a, b, c, d, e, f, g, x, y, z} \ {1, 2, c, d, e, f, g, y}= {a, b, x, z} ---- (I)

A\B = {a, b, c, d, e, f, g, x, y, z}\ {1, 2, c, d, e}= {a, b, f, g, x, y, z}

A\C = {a, b, c, d, e, f, g, x, y, z} \ {d, e, f, g, z, y}= {a, b, c, x, z}

(A\B) (A\C) = {a, b, f, g, x, y, z} {a, b, c, x, z}

= {a, b, x, z} ---- (I) I = II

A\(BC) = (A\B) (A\C)

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5 6

9. å¼ ïèóˆF™ 85% ñ‚èœ îI¿‹, 40% «ð˜ ݃Aôº‹ ñŸÁ‹ 20% «ð˜ Þ‰F»‹ «ð²Aø£˜. 32% «ð˜ÝƒAôº‹ îI¿‹, 13% îI¿‹, Þ‰F»‹, 10% ݃Aôº‹ Þ‰F»‹ «ð²Aø£˜èœ. Í¡Á ªñ£N»‹«ðꈪîK‰îõ˜ êîiî‹ è£‡è.

îI› - T

݃Aô‹ - E

Þ‰F - H

{îI› ªñ£N «ð²ðõ˜} n(T) = 85%

{݃Aô ªñ£N «ð²ðõ˜} n(E) = 40%

{Þ‰F ªñ£N «ð²ðõ˜}n(H) = 20%

{îI¿‹, ݃Aôº‹ «ð²ðõ˜} n(T E) = 32%

{îI¿‹, Þ‰F»‹ «ð²ðõ˜} n(TH) = 13%

{݃Aôº‹, Þ‰F»‹ «ð²ðõ˜} n(E H) = 10%

{Í¡Á ªñ£N»‹ «ð²ðõ˜} n (T E H) = x

Í¡Á ªñ£N»‹ «ðêˆ ªîK‰îõ˜ êîiî‹40 + x + 32 - x + 13 - x + x - 2 + x - 3 + x + 10 - x = 100

95 - 5 + x = 10090+x = 100 x = 100 - 90 x = 10%

Í¡Á ªñ£N»‹ «ðêˆ ªîK‰îõ˜ êîiî‹ = 10%10. 170 õ£®‚¬èò£÷˜èO™ 115 «ð˜ ªî£¬ô‚裆C, 110 «ð˜ õ£ªù£L, 130 «ð˜ ðˆFK‚¬èè¬÷»‹

ðò¡ð´ˆ¶Aø£˜èœ â¡ð¬î å¼ M÷‹ðó GÁõù‹ è‡ìP‰î¶. 85 «ð˜ ªî£¬ô‚裆C ñŸÁ‹ðˆFK‚¬è, 75 «ð˜ ªî£¬ô‚裆C ñŸÁ‹ õ£ªù£L, 95 «ð˜ õ£ªù£L ñŸÁ‹ ðˆFK¬‚¬ò»‹ 70 «ð˜Í¡P¬ù»‹ ðò¡ð´ˆ¶Aø£˜èœ.(i) õ£ªù£L ñ†´‹ (ii) ªî£¬ô‚裆C ñ†´‹(iii) ªî£¬ô‚裆C ñŸÁ‹ ðˆFK‚¬è¬ò ðò¡ð´ˆF õ£ªù£L ðò¡ð´ˆî£îõ˜èœ 裇è.

ªî£¬ô‚裆C = Tõ£ªù£L = RðˆFK‚¬è = M

n(T) = 115n(R) = 110n(M) = 130n(TM) = 85n (T R) = 75n(RM) = 95n(T RM) = 70

i) õ£ªù£L ñ†´‹ = 10ii) ªî£¬ô‚裆C ñ†´‹ = 25iii) ªî£¬ô‚裆C ñŸÁ‹ ðˆFK‚¬èŠ ðò¡ð´ˆF õ£ªù£L ðò¡ð´ˆî£îõ˜ = 15

85 -

(32

-x+x

+13-

x)

85-4

5+x

40+x

32-x

40 - (32-x+x+10-x)

40-42+x

-2+x

x

10-x13-x

20-(13-x+x+10-x)20-23+x

-3+x

T îI›E݃Aô‹

H Þ‰F

115-

(70+

5+15

)11

5-90

25

75-705

110 - (5+70+25)

110 - 100

10

7095-70

2585-70

15

130-(15+70+25)130-110

20

TR

M

www.mathstimes.com

5 7

11. f(x) =

<<−≤≤−<≤−−

74:32

42:23

23:14 2

xx

xx

xx

H¡õ¼õùõŸ¬ø‚ 裇è, (i) f(5) + f(6)

ii) f(1) - f(-3) (iii) f(-2) - f(4) (iv) )1(f)6(f2)1(f)3(f

−−+

b˜¾:

f(x) = )6,5(

)4,3,2(

)1,0,1,2,3(

74:32

42:23

23:14 2 −−−

<<−≤≤−<≤−−

xx

xx

xx

i) f(5) + f(6) =?

f(x) = 2x - 3

f(5) = 2 x 5 - 3

= 10 - 3

f(5) = 7

f(6) = 2× 6 - 3

= 12 - 3

f(6) = 9

f(5) + f(6) = 7 + 9

f(5) + f(6) = 16

ii) f(1) - f(-3) = ?

f(x) = 4x2 - 1

f(1) = 4 x 12 - 1

= 4 - 1

f(1) = 3

f(-3) = 4 x (- 3)2 - 1

= 4 x 9 - 1

= 36 - 1

f(-3) = 35

f(1) - f(-3) = 3 - 35

f(1) - f(-3) = -32

iii) f(-2) - f(4)

f(x) = 4x2 - 1

f(-2) = 4x (- 2)2 - 1

= 4 x 4 - 1

= 16 - 1

f(-2) = 15

f(x) = 3x - 2

f(4) = 3 x 4 - 2

= 12 - 2

f(4) = 10

f(-2) - f(4) = 15 - 10

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5 8

f(-2) - f(4) = 5

iv) )1(f)6(f2)1(f)3(f

−−+

= ?

f(3) + f(-1)

f(x) = 3x - 2

f(3) = 3 x 3 - 2

= 9 - 2

f(3) = 7

f(x) = 4x (-1)2 - 1

= 4 - 1

f(-1) = 3

f(3) + f(-1) = 7 + 3

f(3) + f(-1) = 10

2f(6) - f(-1) = 10

2f(6) - f(1) = ?

2(x) = 2x - 3

f(6) = 2 x 6 - 3

= 12 - 3

f(6) = 9

2f(6) = 18

f(1) = 4 x 12 - 1

= 4 - 1

f(1) = 3

2f(6) - f(1) = 18 - 3

= 15

)1(f)6(f2)1(f)3(f

−−+

= 1510

Ans: 32

12. A= { 0, 1, 2, 3 } , B = { 1, 3, 5, 7, 9 } Þ¼ èíƒèœ f : A → B â¡Â‹ ꣘¹ f(x)=2x+1 âù‚

ªè£´‚èŠð†´œ÷¶. ބ꣘H¬ù (i) õK¬ê «ê£®èO¡ èí‹ (ii) ܆ìõ¬í (iii) Ü‹¹‚°PŠðì‹

(iv) õ¬óðì‹ ÝAòõŸø£™ °PŠH´è,

b˜¾

f(x) = 2x + 1

f (0) = 2 x 0 + 1 = 0+1 = 1

f (1) = 2 x 1 +1 = 2 + 1 = 3

f (2) = 2 x 2 + 1 = 4 + 1 = 5

f (3) = 2 x 3 + 1 = 6 + 1 = 7

(i) õK¬ê «ê£®èO¡ èí‹

{ (0, 1), (1, 3), (2, 5), (3, 7) }

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5 9

(ii) ܆ìõ¬í

x 0 1 2 3

f (x) 1 3 5 7

(iii) Ü‹¹‚°P ðì‹

A B

0 1

1 3

2 5

3 7

(iv) õ¬óðì‹

13. f(x) =

<≤−

<≤−<≤+

64103

4212

211

2 xx

xx

xx

(1, 6) = {x∈R; 1 ≤ 6)

(i) f(5) (ii) f (3) (iii) f (1) (iv) f(2)- f(4) (v) 2 f(5) - 3f (1) ñFŠ¹è¬÷‚ 裇è.

b˜¾:

f(x) = )5,4(

)3,2(

)1(

64103

4212

211

2

<≤−

<≤−<≤+

xx

xx

xx

i) f(x) = 3x2 - 10f(5) = 3 x 52 - 10

= 3 x 25 - 10= 75 - 10

f(5) = 65

ii) f(x) = 2x - 1

f(3) = 2 x 3 - 1

= 6 - 1

f(3) = 5

iii) f(x) = 1 + x

f(1) = 1 + 1

f(1) = 2

iv) f(2) - f(4)

f

www.mathstimes.com

6 0

f(x) 2x - 1

f(2) = 2 x 2 - 1

= 4 - 1

f(2) = 3

f(x) 3x2 - 10

f(4) = 3 x 42 - 10

= 3 x 16 - 10

= 48 - 10

f(4) = 38

f(2) - f(4) = 3 - 38

f(2) - f(4) = -35

v) 2f(5) - 3f(1)

2f(5) = 2 x 65

= 130

3f(1) = 3 x 2

= 6

2f(5) - 3f(1) = 130 - 6

2f(5) - 3f(1) = 124

14. A = {4, 6, 8, 10} B = {3, 4, 5, 6, 7} â¡è. f : A → B f(x) = 21

x + 1 õ¬óòÁ‚èŠð†´œ÷¶. (i) Ü‹¹‚°PŠðì‹

(ii) õK¬ê„ «ê£®èO¡ èí‹ (iii) ܆ìõ¬í (iv) õ¬óðì‹ ÝAòõŸP¡ Íô‹ °P‚è.

b˜¾:

f(x) = 21

x + 1

f(4) = 21

x 4 + 1 = 2 + 1 = 3

f(6) = 21

x 6 + 1 = 3 + 1 = 4

f(8) = 21

x 8 + 1 = 4 + 1 = 5

f(10) = 21

x 10 + 1 = 5 + 1 = 6

i) Ü‹¹‚°PŠðì‹

A B

4 3

6 4

8 5

10 6

ii) õK¬ê «ê£®èO¡ èí‹

f = {(4, 3) (6, 4) (8, 5) (10, 6)}

iii) ܆ìõ¬í

x 4 6 8 10f(x) 3 4 5 6

f

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6 1

iv) õ¬óðì‹

15. f(x) =

<<≤≤−

−<≤−

−+

++

62

25

57

1

5

122

x

x

x

x

x

xx

H¡õ¼õùõŸ¬ø‚ 裇è.

(i) 2f(-4) + 3f(2) ii) f(-7) -f(-3) iii) )1(f3)6(f)4(f2)3(f4

−−+−

f(x) = )5,4,3(

)2,1,0,1,2,3,4,5(

)6,7(

62

25

57

1

5

122

−−−−−−−

<<≤≤−

−<≤−

−+

++

x

x

x

x

x

xx

i) 2f(-4) + 3f(2)

f(x) = x + 5

f(-4) = - 4 + 5 = 1

2xf(-4)= 1 x 2

2f(-4) = 2

f(2) = 2 + 5 = 7

3 x f(2) = 7 x 3

3f(2) = 21

2f(-4) + 3f(2) = 2 + 21

2f(-4) + 3f(2) = 23

ii) f(-7) - f(-3)

f(x) = x2 + 2x + 1

f(-7) = (-7)2 + 2x (- 7) + 1

= 49 - 14 + 1

= 50 - 14

f(-7) = 36

f(x) x + 5

f(-3) = - 3 + 5

= 2

f(-3) = 2

f(-7) - f(-3) = 36 - 2

f(-7) - f (-3) = 34

654

32

1

1 2 3 4 5 6 7 8 9 10

(4,3)

(6,4)

(8,5)

(10,6)

x

y

O

www.mathstimes.com

6 2

iii) )1(f3)6(f)4(f2)3(f4

−−+−

f(x) = x + 5

f(-3) = - 3 + 5 = 2

4f(-3) = 2 x 4

4f(-3) = 8

f(x) = x - 1

f(4) = 4 - 1 = 3

2f(4) = 3 x 2

2f(4) = 6

4f(-3) + 2f(4) = 8 + 6

4f (-3) + 2f(4) = 14

f(x) = x2 + 2x + 1

f(-6) = (-6)2 + 2 x (- 6) + 1

= 36 - 12 + 1

= 37 - 12

f(-6) = 25

f(x) = x + 5

3(1) = 1 + 5 = 6

3f(1) = 6 x 3

3f(1) = 18

f(-6) - 3f(1) = 25 - 18

f(-6) - 3f(1) = 7

)1(f3)6(f)4(f2)3(f4

−−+−

= 7

14

M“: 2

16. A = { 6, 9, 15, 18, 21 }; B = { 1, 2, 4, 5, 6 } f : A → B â¡ð¶ f(x) =3

3−x âù õ¬óòÁ‚èŠð†´œ÷¶.

(i) Ü‹¹‚°Pðì‹ (ii) õK¬ê «ê£®èO¡ èí‹ (iii) ܆ìõ¬í (iv) õ¬óðì‹ °P‚辋.b˜¾:

f(x) = 3

3−x

f(6) = 3

36 − =

33

= 1

f(9) = 3

39 − =

36

= 2

f(15) = 3

315− =

312

= 4

f(18) = 3

318− =

315

= 5

f(15) = 3

321− =

318

= 6 www.mathstimes.com

6 3

i) Ü‹¹‚°PŠðì‹

A B

6 1

9 2

15 4

18 5

21 6

ii) õK¬ê «ê£®èO¡ èí‹

f = {(6, 1), (9, 2), (15, 4), (18, 5), (21, 6)}

iii) ܆ìõ¬í

x 6 9 15 18 21

f(x) 1 2 4 5 6

iv) õ¬óðì‹

17. A = {5, 6, 7, 8}, B = {-11, - 4, 7, -1, -7, -9, -13} â¡è. f = {(x, y); y = 3-2x, x∈A, y∈B}

i) f ¡ àÁŠ¹ ii) Üî¡ ¶¬í ñFŠðè‹ iii) i„êè‹

iv) âšõ¬è ꣘¹

b˜¾:

y = 3-2x

x = 5, y = 3-2 x 5 = 3 - 10 = -7

x = 6, y = 3 - 2 x 6 = 3 - 12 = -9

x = 7, y = 3 - 2 x 7 = 3 - 14 = -11

x = 8, y = 3 - 2 x 8 = 3 - 16 = -13

i) f ¡ àÁŠ¹èœ

f = {(5, -7), (6, -9), (7, -11), (8, -13)}

ii) ¶¬í ñFŠðè‹ = {-11, 4, 7, -10, -7, -9, -13}

iii) i„êè‹ = {-7, -9, -11, -13}

iv) ꣘H¡ õ¬è

å¡Á‚° å¡ø£ù ꣘¹

f

7654

32

1

3 6 9 12 15 18 21 24

(6,1)(9,2)

(15,4)

(18,5)

x

y

(21,6)

O

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6 4

2. ªñŒªò‡èO¡ ªî£ì˜õK¬èèÀ‹ ªî£ì˜èÀ‹ªñŒªò‡èO¡ ªî£ì˜õK¬èèÀ‹ ªî£ì˜èÀ‹ªñŒªò‡èO¡ ªî£ì˜õK¬èèÀ‹ ªî£ì˜èÀ‹ªñŒªò‡èO¡ ªî£ì˜õK¬èèÀ‹ ªî£ì˜èÀ‹ªñŒªò‡èO¡ ªî£ì˜õK¬èèÀ‹ ªî£ì˜èÀ‹

1. å¼ Ã†´ˆ ªî£ì˜ õK¬êJ™ 10 ñŸÁ‹ 18õ¶ àÁŠ¹èœ º¬ø«ò 41 ñŸÁ‹ 73 âQ™ 27õ¶ àÁŠ¬ð‚裇è.b˜¾:

ªè£´‚èŠð†ì¶ t10 = 41 a + 9d = 41 ---- (1) t18 = 73 a + 17d = 73 ---- (2)(1) - (2) - 8d = 32

d = 832

−−

d = 4 ä (1) ™ HóFJì a + 9d = 41a + 9 x 4 = 41

a + 36 = 41 a = 5

a = 41 - 36a = 5

t27 = a + 26 d a = 5 , d = 4 âQ™ t27 = 5 + 26 (4)

= 5 + 104 t27= 109

2. a, b, c ÝAòù Æ´ˆ ªî£ì˜ õK¬êJ™ Þ¼ŠH¡ bc1

, ca1

, ab1

ÝAòù å¼ Ã†´ˆ ªî£ì˜õK¬êJ™

Þ¼‚°‹ âù GÁ¾è.b˜¾:

a, b, c å¼ A.P.÷ abc.

abca

, abcb

, abcc

»‹ å¼ A.P.

bc1

, ca1

, ab1

»‹ å¼ A.P.

3. å¼ ªð¼‚°ˆ ªî£ì˜ õK¬êJ¡ 裋 àÁŠ¹ 32

ñŸÁ‹ Üî¡ ãö£õ¶ àÁŠ¹8116

âQ™ ÜŠªð¼‚°ˆ

ªî£ì˜ õK¬ê¬ò 裇è.

t4 = 32

ar3 = 32

--- (1)

t7 = 8116

ar6 = 8116

--- (2)

(2) ÷ (1) 3

6

ar

ar = 32

8116

r6-3 = 8116

x 23

r3 = 278

r3 =

3

32

www.mathstimes.com

6 5

r = 32 ä (1) ™ HóFJì

ar3 = 32

a x

3

32

= 2/3

a = 32

x 23

x 23

x 23

a = 49

ªð¼‚°ˆ ªî£ì˜ õK¬ê = a, ar, ar2 .....

49

,

49

32

,

49 2

32

.....

4. å¼ ªð¼‚°ˆ ªî£ì˜õK¬êJ™ ºî™ ñŸÁ‹ Ýø£õ¶ àÁŠ¹èœ º¬ø«ò 31

, 729

1 âQ™ ÜŠªð¼‚°ˆ

ªî£ì˜õK¬ê¬ò‚ 裇è.

a = 31

t6 = 729

1 ar5 =

7291

31

r5 = 729

1

r5 = 729

1 x 3

= 243

1

r5 = 53

1

r5 =

5

31

r = 31

ªð¼‚°ˆ ªî£ì˜ õK¬ê,= a, ar, ar2 .....

= 31

,

31

31

, 31 2

31

......

= 31

, 91

, 271

.....

www.mathstimes.com

6 6

5. å¼ ªð¼‚°ˆ ªî£ì˜ õK¬êJ™ 4õ¶ ñŸÁ‹ 7õ¶ àÁŠ¹èœ º¬ø«ò 54 ñŸÁ‹ 1458 âQ™,܈ªî£ì˜õK¬ê¬ò‚ 裇è.

t4 = 54 ar3 = 54 ---- (1)t7 = 1458 ar6 = 1458 ---- (2)

(2) ÷ (1) 3

6

ar

ar =

541458

r3 = 27r3 = (3)3

r = 3 ä (1) ™ HóFJìar3 = 54a(3)3 = 54

a = 33354

××a = 2

ªð¼‚°ˆ ªî£ì˜ õK¬ê a, ar, ar2

= 2, (2) (3), (2) (3)2 ....= 2, 6, 18 .....

6. 8 Ý™ õ°ð´‹ ܬùˆ¶ Í¡Pô‚è Þò™â‡èO¡ Ã´î™ è£‡è.Í¡Pô‚è Þò™ â‡èœ 100, 101, ..... 999.a = 104, d = 8, l = 992

ð®1 :

n =

−

da

+ 1

=

−

8104992

+1

=

8888

+ 1

= 111 + 1n = 112

ð® 2:

Sn = 2n

[a + l]

S112 = 2

112[104 + 992]

= 56 x 1096S112 = 61376

7. 9Ý™ õ°ð´‹ ܬùˆ¶ Í¡Pô‚è Þò™ â‡èO¡ Ã´î™ è£‡.Í¡Pô‚è â‡èœ 100, 101, ... 999.a = 108, d = 9, l = 999

ð® 1 :

n =

−

da

+ 1

=

−

9108999

+1

12

8 100 + 4 = 104

8

20

16

4+4

125

8 999 -7 = 992

8

19

16

39

32

7

11

8 100 + 8

9

10

9

1+8

111

8 999 - 0

9

9

9

9

9

0www.mathstimes.com

6 7

=

9891

+ 1

= 99 + 1n = 100

ð® 2:

Sn = 2n

[a + l ]

S100 = 2

100[108 + 999]

= 56 x 1107

S100 = 55350

8. 300‚°‹ 500‚°‹ Þ¬ì«ò»œ÷ 11 Ý™ õ°ð´‹ ܬùˆ¶ Þò™ â‡èO¡ ÆìŸðô¬ù‚ 裇è.

a = 308, d = 11, l = 495

ð® 1 :

n =

−

da

+ 1

=

−

11308495

+1

=

11187

+ 1

= 17 + 1n = 18

ð® 2:

Sn = 2n

[a + l ]

S18 = 2

18[308 + 495]

= 9 x 803S18 = 7227

9. 100‚°‹ 200‚°‹ Þ¬ì«ò»œ÷ 5Ý™ õ°ðì£î ܬùˆ¶ Þò™ â‡èO¡ ÆìŸðô¬ù‚ 裇è.5Ý™ õ°ð´‹ Þò™ â‡èO¡ Ã´î™ 105 + 110 + ..... +195, a = 105, d = 5, l = 195ð® 1 :

n =

−

da

+ 1

=

−

5105195

+1

=

590

+ 1

= 18 + 1n = 19

Sn = 2n

[a + l ]

S19 = 2

19[105 + 195]

27

11 300 + 8 =308

22

80

77

3+8

45

11 499 - 4 = 495

44

59

55

4

www.mathstimes.com

6 8

= 19 x 150S19 = 2850

ð® 2 :Þò™ â‡èO¡ Ã´î™ 101 + 102 + ... 199

2)1n(n

n+=

101 + 102 + .... + 199= (1 + 2 + .... + 199) - (1 + 2 + .... + 100)

= 2

200199× -

2101100×

= 19900 - 5050= 14850

ð® 3:5Ý™ õ°ðì£î â‡èO¡ Ã´î™ = 14850 - 2850

= 1200010. 6 + 66+ 666 +... â‹ ªî£ìK™ ºî™ n àÁŠ¹èO¡ Ã´î™ è£‡.

Sn = 6 + 66 + 666 + .... n àÁŠ¹èœ õ¬ó= 6 (1 + 11 + 111 + ....n àÁŠ¹èœ õ¬ó)

= 96

(9 + 99 + 999 + ..... n àÁŠ¹èœ õ¬ó)

= 32

[(10 - 1) + (100 - 1) + (1000 - 1) ..... n àÁŠ¹èœ õ¬ó]

= 32

[(10 + 100 + 1000 + .... n àÁŠ¹èœ õ¬ó) - n] a = 10, r = 10>1

= 32

−

−−

n110

)110(10 n

Sn = 1r

)1r(a n

−−

= 110

)110(10 n

−−

Sn = 32

−−

n9

)110(10 n

11. 7 + 77 + 777 +... â‹ ªî£ìK™ ºî™ n àÁŠ¹èO¡ Ã´î™ è£‡. Sn = 7 + 77 + 777 + ....n àÁŠ¹èœ õ¬ó

= 7 (1 + 11 + 111 + .... n àÁŠ¹èœ õ¬ó)

= 97

(9 + 99 + 999 + ..... n àÁŠ¹èœ õ¬ó)

= 97

[(10 - 1) + (100 - 1) + (1000 - 1) ..... n àÁŠ¹èœ õ¬ó]

= 97

[(10 + 100 + 1000 + .... n àÁŠ¹èœ õ¬ó) - n] a = 10, r = 10>1

= 97

−

−−

n110

)110(10 n

Sn = 1r

)1r(a n

−−

= 110

)110(10 n

−−

Sn = 97

−−

n9

)110(10 n

12. 1 + 11 + 111 +... 20 àÁŠ¹èœ õ¬ó â‹ ªî£ìK¡ Ã´î™ è£‡. Sn = 1 + 11 + 111 + .... n àÁŠ¹èœ õ¬ó

= 91

(9 + 99 + 999 + ..... n àÁŠ¹èœ õ¬ó)

= 91

[(10 - 1) + (100 - 1) + (1000 - 1) ..... n àÁŠ¹èœ õ¬ó]www.mathstimes.com

6 9

= 91 [(10 + 100 + 1000 + .... n àÁŠ¹èœ õ¬ó) - n]

Sn =91

−

−−

n110

)110(10 n

a = 10, r = 10>1

S20 = 91

−

−−

20110

)110(10 20

Sn = 1r

)1r(a n

−−

= 91

−−

209

)110(10 20

S20 =

−−

920

)110(8110 20

13. Ã´î™ è£‡ 162 + 172 + 182 .... + 252

6)1n2()1n(n

n2 ++=

162 + 172 + 182 .... + 252 = (12 + 22 + ... + 252 ) - (12 + 22 .... + 152)

= 6

3116156

512625 ××−××

= 25 x 13 x 17 - 5 x 8 x 31= 5525 - 1240= 4285

14. Ã´î™ è£‡ 162 + 172 + ... 352

6)1n2()1n(n

n2 ++=

162 + 172 + .... + 352 = (12 + 22 + ... + 352 ) - (12 + 22 .... + 152)

= 6

3116156

713635 ××−××

= 35 x 6 x 17 - 5 x 8 x 31= 14910 - 1240= 13670

15. 11 ªê.e, 12 ªê.e, 13 ªê.e .... 24 ªê.e ÝAòõŸ¬ø º¬ø«ò ð‚è Ü÷¾è÷£è‚ ªè£‡ì 14ê¶óƒèO¡ ªñ£ˆîŠðóŠ¹ è£‡è.

ðóŠ¹ = 112 +122 + 132 +.... + 242

6)1n2()1n(n

n2 ++=

112 + 122 +132 .... + 242 = (12 + 22 + ... + 242 ) - (12 + 22 .... + 102)

= 6

2111106

492524 ××−××

= 4 x 25 x 49 - 5 x 11 x 7= 4900 - 385= 4515

ðóŠ¹ = 4515 ªê.e2

16. 12 ªê.e, 13 ªê.e...... 23 ªê.e ÝAòùõŸ¬ø º¬ø«ò ð‚è Ü÷¾è÷£è‚ ªè£‡ì 12 ê¶óƒèO¡ªñ£ˆîŠðóŠð÷¾‚ 裇è.b˜¾ :

ðóŠ¹ = 122 + 132 + 142 + .... + 232

6)1n2()1n(n

n2 ++= www.mathstimes.com

7 0

122 + 132 + .... + 232 = (12 + 22 + .... +232) - (12 + 22 + .... + 112)

= 6

2312116

472423 ××−××

= 23 x 4 x 47 - 22 x 23= 4324 - 506= 3818

ðóŠ¹ = 3818 ªê.e2.17. 16 ªê.e, 17 ªê.e, 18 ªê.e..... 30 ªê.e ÝAòùõŸ¬ø º¬ø«ò ð‚è Ü÷¾è÷£è‚ ªè£‡ì 15

èù„ê¶óƒèO¡ èù Ü÷¾èO¡ Ã´î™ è£‡è.b˜¾ :

èùÜ÷¾ = 163 + 173 + 183 + .... + 303

23

2)1n(n

n

+=

163 + 173 + 183 + .... + 303 = (13 + 23 + .... +303) - (13 + 23 + .... + 153)

= 22

21615

23130

×−

×

= (15 x 31)2 - (15 x 8)2

= (465)2 − (120)2

= (465 + 120) (465 - 120)

= 585 x 345

èù Ü÷¾ = 201825 ªê.e3

18. å¼ Ã†´ˆ ªî£ì˜ õK¬êJ™ Ü´ˆî´ˆî Í¡Á àÁŠ¹èO¡ Ã´î™ 6 ñŸÁ‹ ÜõŸP¡ ªð¼‚°ˆ

ªî£¬è &120 âQ™ Ü‹Í¡Á ⇂¬÷‚ 裇è.

a - d, a, a+d â¡ðù Æ´ˆªî£ì˜ õK¬êJ¡ Í¡Á àÁŠ¹èœ â¡è.

Ã´î™ = 6

a - d + a + a + d = 6

3a = 6

a = 6/3

a = 2

ªð¼‚°ˆ ªî£¬è= - 120

(a-d) (a) (a+d) = -120

(a2 - d2) a = -120

a = 2 ä HóFJì

(22 - d2)2 = -120

4 - d2 = 2120−

-d2 = -60 -4d2 = 64

d = 88×

d = ± 8a = 2 ñŸÁ‹ d = 8 âQ™Í¡Á â‡èœ = 2- 8, 2, 2+8 (Ü™ô¶) = -6, 2 10

a = 2 ñŸÁ‹ d = -8 âQ™

Í¡Á â‡èœ = 2 - (-8), 2, 2 - 8 = 10, 2, -6 www.mathstimes.com

7 1

19. 5 + 11 + 17 + .... + 95 â¡ø ªî£ìK¡ Ã´î™ è£‡.

a = 5, d = 11-5 = 6; l = 95

ð® 1:

n =

−

da

+ 1

=

−6

595+1

=

6

90 + 1

= 15 + 1n = 16

ð® 2:

Sn = 2n

[a + l ]

S16 = 2

16[5+95]

= 7 x 100S16 = 800

3. ÞòŸèEî‹ÞòŸèEî‹ÞòŸèEî‹ÞòŸèEî‹ÞòŸèEî‹1. c‚è™ º¬ø¬ò ðò¡ð´ˆFˆ b˜‚è 101x + 99y = 499, 99x + 101y = 501

101x + 99y = 499 --- (1)99x + 101y = 501 --- (2)

(1)+(2) 200x + 200y = 1000

÷ 200x + y = 5 --- (3)

(1)-(2) 2x - 2y = -2

÷ 2x - y = -1 ---- (4)

(3)+(4) 2x = 4x = 4/2 = 2

(1) ™ HóFJì2 + y = 5

y = 5 - 2y = 3

x = 2y = 3

2. è£óEŠð´ˆ¶è : x3 - 2x2 - 5x + 61 1 -2 -5 6

0 1 -1 -63 1 -1 -6 0 (x-1) è£óE

0 3 61 2 0 (x-3) è£óE

(x +2) è£óE(x - 1), (x - 3), (x+2) è£óEèœ Ý°‹. www.mathstimes.com

7 2

3. è£óEŠð´ˆ¶è 4x3 - 7x + 3

1 4 0 -7 3

0 4 4 -3

4 4 -3 0 (x - 1) è£óE

4x2 + 4x - 3 = (2x + 3) (2x -1) è£óEèœ

∴ (x-1), (2x-1), (2x+3) è£óEèœ.

4. è£óEŠð´ˆ¶è: x3 - 7x + 6

1 1 0 -7 6

0 1 1 -6

2 1 1 -6 0 (x-1) è£óE

0 2 6

1 3 0 (x-2) è£óE

(x+3) is a factor

∴ (x-1), (x-2), (x+3) are factors.

5. è£óEŠð´ˆ¶è: x3 - 3x2 - 10x + 24

2 1 -3 -10 24

0 2 -2 -24

-3 1 -1 -12 0 (x - 2) è£óE

0 -3 12

1 -4 0 (x+3) è£óE

(x-4) è£óE

(x-2), (x + 3), (x - 4) è£óEèœ(°PŠ¹ : å¼ è£óE è‡ìH¡, ã¬ùòõŸ¬ø è£óEŠð´ˆî ÞòôM™¬ôªòQ™ ÜŠð®«ò â¿î¾‹

ðJŸC 3.5™ àœ÷ IV, VIII ñŸÁ‹ XI Þ‰î õ¬è¬ò„ ꣼‹)

6. P = y+xx

, Q = yy+x âQ™ 22 QP

Q2QP

1

−−

− 裇è.

22 QP

Q2QP

1

−−

− = QP

1− − )QP()QP(

Q2−+

= )QP()QP(Q2QP−+

−+

= )QP()QP(QP

−+−

= QP

1+

=

yy

y

1

++

+ xxx

= yy

1

++

xx

= 1 www.mathstimes.com

7 3

7. õ˜‚è Íô‹ 裇è (x2 - 25) (x2 + 8x + 15) (x2 -2x -15)= (x + 5) (x - 5) (x + 3) (x + 5) (x - 5) (x + 3)= (x+5)2 (x - 5)2 (x + 3)2

õ˜‚è Íô‹ = |(x + 5) (x - 5) (x + 3)|8. õ˜‚è Íô‹ 裇è 9x4 + 12x3 + 10x2 + 4x + 1

3 2 13 9 12 10 4 1

96 2 12 10

12 46 4 1 6 4 1

6 4 10

õ˜‚è Íô‹ = |3x2 + 2x + 1|

9. õ˜‚è Íô‹ 裇è x4 - 10x3 + 37x2 - 60x + 36

1 -5 6

1 1 -10 37 -60 36

1

2 -5 -10 37

-10 25

2 -10 6 12 -60 36

12 -60 36

0

õ˜‚è Íô‹ = |x2 - 5x + 6|

10. õ˜‚è Íô‹ 裇è 4+25x2−12x−24x3+16x4

F†ì ܬñŠH™ â¿î

16x4 - 24x3 + 25x2 - 12x + 4

4 − 3 2

4 16 − 24 25 − 12 4

16

8 −3 −24 25

−24 9

8 −6 2 16 − 12 4

16 − 12 40

õ˜‚è Íô‹ = |4x2 − 3x + 2|

11. m - nx + 28x2 + 12x3 + 9x4 Ýù¶ å¼ º¿ õ˜‚è‹ âQ™ m , n 裇è.

Þøƒ° õK¬êJ™ â¿î

9x4 + 12x3 + 28x2 - nx + m

3 2 43 9 12 28 -n m

96 2 12 28

12 46 4 4 24 -n m

24 16 160

º¿ õ˜‚è‹ Ýîô£™, m = 16, n = -16 www.mathstimes.com

7 4

12. ax4 - bx3 + 40x2 + 24x + 36 å¼ º¿ õ˜‚è‹ âQ™ a ñŸÁ‹ b 裇è.

ãÁõK¬êJ™ â¿î

36 + 24x + 40x2 - bx3 + ax4

6 2 3

6 36 24 40 -b a

36

12 2 24 40

24 4

12 4 3 36 -b a

36 12 9

0

º¿ õ˜‚è‹ Ýîô£™ a = 9

b = -12

13. å¼ â‡ Üî¡ î¬ôWN ÝAòõŸP¡ Ã´î™ 551âQ™ ܉î â‡¬í‚ è£‡è.

b˜¾ :܉î ⇠= xî¬ôWN = 1/x

Ã´î™ = 5 51

x

x1+ =

526

x

x 12 +=

526

5(x2 +1) = 26x

5x2 + 5 - 26x = 0

5x2 - 26x + 5 = 0

(5x - 1) (x - 5)= 0

5x - 1 = 0 or x = 5

x = 1/5 or x = 5

x = {1/5, 5}

14. êñ¡ð£´ (1+m2) x2 + 2mcx+c2 - a2 = 0 ¡ Íôƒèœ êñ‹ âQ™ c2 = a2 (1+m2) âù GÁ¾è.

êñ¡ð£´ (1+m2) x2 + 2mcx+c2 - a2 = 0

a = 1 + m2, b = 2mc, c = c2 - a2

Íôƒèœ êñ‹ = b2 - 4AC = 0

(2mc)2 - 4 (1+m2) (c2 - a2) = 0

4m2c2 - 4 (c2 - a2 + m2c2 - m2a2) = 0

4m2 c2 - 4c2 + 4a2 - 4m2c2 + 4m2 a2 = 0

-4c2 = -4a2 - 4m2a2 = 0

÷ −−−−− 4

c2 = a2 + m2 a2

c2 = a2 (1+m2)

∴ c2 = a2 (1+m2) GÏH‚èŠð†ì¶

°PŠ¹: ðJŸC 3.5 ñŸÁ‹ 3.13 º¿õ¶‹ 𣘂辋. www.mathstimes.com

7 5

4. ÜEèœÜEèœÜEèœÜEèœÜEèœ

1.

21

53 ñŸÁ‹

−

−31

52 ÜEŠªð¼‚è¬ôŠ ªð£¼ˆ¶ å¡Á‚ªè£¡Á «ï˜ñ£Á ÜE âù GÁ¾è.

b˜¾:

21

53

−

−31

52=

+−−+−−

6522

151556

=

10

01 = I

−

−31

52

21

53=

+−+−

−−6533

101056

=

10

01 = I

«ï˜ñ£Á ÜEò£°‹.

2. A =

37

25 B =

−

−57

23 å¡Á‚ªè£¡Á ªð¼‚è™ «ï˜ñ£Á ÜE âù GÁ¾è.

b˜¾ :

AB =

37

25

−

−57

23

=

+−−+−−

15142121

10101415 =

10

01= I

BA =

−

−57

23

37

25

=

+−+−

−−15143535

661415=

10

01= I

«ï˜ñ£Á ÜEò£°‹

3. A =

04

23 ñŸÁ‹ B =

23

03 âQ™ AB , BA 裇. ܬõ êññ£è Þ¼‚°ñ£?

b˜¾:

AB =

04

23

23

03

=

++++

00012

4069 =

012

415

BA =

23

03

04

23

=

++++

0689

0609=

BA =

618

69

AB ≠ BA

www.mathstimes.com

7 6

4. A =

−

5

4

2

ñŸÁ‹ B = (1 3 -6) âQ™ (AB)T = BTAT GÁ¾è.

b˜¾:

AB =

−

5

4

2

(1 3 -6)

=

−−

−−

30155

24124

1262

(AB)T =

−−−−

302412

15126

542

----- (1)

BT =

− 6

3

1

AT = ( -2 4 5)

BT AT =

− 6

3

1

( -2 4 5)

=

−−−−

302412

15126

542

--- (2)

(AB)T = BT AT êK𣘂èŠð†ì¶.

5. A =

37

25 B =

−

−11

12 âQ™ (AB)T = BT AT êKð£˜.

b˜¾:

AB =

37

25

−

−11

12

=

+−−+−−

37314

25210

AB =

−−

411

38

(AB)T =

−− 43

118------ (1)

BT =

−

−11

12

www.mathstimes.com

7 7

AT =

32

75

BTAT =

−

−11

12

32

75

=

+−+−−−

3725

314210

BTAT =

−− 43

118----- (2)

(1) = (2)

(AB)T = BT AT êK𣘂èŠð†ì¶.

6. A =

−32

11âQ™ A2 - 4A + 5I2 = 0 GÁ¾è.

b˜¾

A2 = A x A =

−32

11

−32

11

=

+−+−−−

9262

3121

=

−−78

41

4A = 4

−32

11

=

−128

44

5I2 = 5

10

01

=

50

05

A2 - 4A = 5I2 =

−−78

41 -

−128

44 +

50

05

=

+−+−++−+−−

5127088

044541

=

00

00

A2 - 4A + 5I2 = 0 GÁõŠð†ì¶.

7. A =

− 41

23, B =

−76

52 C =

− 35

11 âQ™ A(B+C) = AB + AC â¡ð¬î êKð£˜.

b˜¾:

B + C =

−76

52 +

− 35

11 =

−101

61

www.mathstimes.com

7 8

A(B+C) =

− 41

23

−101

61

=

+−+++−

40641

201823

=

−345

381----- (1)

AB =

− 41

23

−76

52

=

+−+++−

285242

1415126

=

2326

296

AC =

− 41

23

− 35

11

=

+−−−+−

121201

63103

=

−−

1121

97

AB + AC =

2326

296 +

−−

1121

97

=

−345

381----- (2)

A (B+C) = AB + AC êK𣘂èŠð†ì¶

5. Ýòˆªî£¬ôõ®Mò™Ýòˆªî£¬ôõ®Mò™Ýòˆªî£¬ôõ®Mò™Ýòˆªî£¬ôõ®Mò™Ýòˆªî£¬ôõ®Mò™

1. A(2,-2) ñŸÁ‹ B(-7, 4) â¡ø ¹÷Oè¬÷ ެ킰‹ «è£†´ˆ¶‡¬ì Í¡Á êñð£èƒè÷£èŠ HK‚°‹¹œOè¬÷‚ 裇è.

P , Q â¡ðù «è£†´ˆ¶‡´ AB ä Í¡Á êñð£èƒè÷£è AP = PQ = QB â¡øõ£Á HK‚°‹¹œOèœ â¡è. Ýè«õ P â¡ð¶ AB ä 1 : 2 â¡ø MAîˆF½‹ Q â¡ð¶ AB¬ò 2 : 1 â¡øMAîˆF½‹ à†¹øñ£è HK‚A¡øù, HK¾ ňFóˆF¡ ð®

P =

+

−+×+

×+−×21

)2(2)41(,

21)22()7(1

=

−+−3

44,

347

= (-1, 0)

Ýè«õ ¹œO P â¡ð¶ (-1, 0) Ý°‹.

QPA B

www.mathstimes.com

7 9

«ñ½‹ Q =

+

−×+×+

×+−×12

)2(1)42(,

12)21()7(2

= (-4, 2)

Ýè«õ ¹œO Q â¡ð¶ (-4, 2) Ý°‹.

2. A (-4, 0) ñŸÁ‹ (0,6) â¡ø ¹œOè¬÷ ެ킰‹ «è£†´ˆ¶‡¬ì ° êñð£èƒè÷£èŠ HK‚°‹

¹œOè¬÷‚ 裇è.

P,Q,R â¡ðù AB â¡ø «è£†´ˆ¶‡¬ì ° êñð£èƒè÷£è HK‚°‹ ¹œOèœ â¡è.

Q â¡ð¶ AB ¡ ¬ñòŠ¹œO

âù«õ Q =

++−2

60,

204

= (-2, 3)

Pâ¡ð¶ AQ ¡ ¬ñòŠ¹œO

P =

+−−2

30,

224

P =

−23

,26

=

−23

,3

R â¡ð¶ QB ¡ ¬ñòŠ¹œO

R =

++−2

63,

202

=

−29

,1

Ýè«õ «î¬õò£ù ¹œOèœ P =

−23

,3 Q = (-2, 3) R =

−29

,1 .

3. (-5, 1) ñŸÁ‹ (2, 3) â¡ø ¹œOè¬÷ ެ킰‹ «è£†´ˆ¶‡®¬ù y Ü„² HK‚°‹ MA

ñŸÁ‹ HK‚°‹ ¹œO¬ò»‹ 裇è.

A (-5, 1) ñŸÁ‹ B(2, 3) â¡ðù ªè£´‚èŠð†ì ¹œOèœ

P (0, y) â¡ð¶ AB¬ò l : m â¡ø MAîˆF™ à†¹øñ£è HK‚Aø¶ â¡è.

P(0, y) = P

+

×+×+

−×+×m

)1m)3(,

m)5(m()2(

---- (1)

P(0, y) = P

++

+−

mm3

,mm52

---- (1)

x Ü„²ˆªî£¬ô¾è¬÷ êñŠð´ˆî

mm52

+−

= 0 2l - 5m = 0 m

= 25

âù«õ «î¬õò£ù MAî‹ l:m = 5 : 2 «ñ½‹ I L¼‰¶  ªðÁõ¶

P(0,y) = P

+

×+×25

)12()35(,0

QPA B

R

(2,3)(-5,1)

x

y

P(0,y)

www.mathstimes.com

8 0

= P

7

17,0

Ýè«õ, y Ü„² HK‚°‹ ¹œO

7

17,0 Ý°‹,

l : m = 5 : 2 ñŸÁ‹ P(0, y) = P (0, 17/4)4. å¼ º‚«è£íˆF¡ º¬ùèœ (1, -1) , (0, 4) ñŸÁ‹ (-5, 3) âQ™ Ü‹º‚«è£íˆF¡ ï´‚«è£´èO¡

c÷ƒè¬÷‚ èí‚A쾋.b˜¾:

A (1, -1), B (0, 4) C (-5, 3) â¡ðù º‚«è£íˆF¡ à„Cèœ D, E, F â¡ðù º¬ø«ò BC, AC,AB¡ ¬ñòŠ¹œOèœ â¡è.

âù«õ BC¡ ¬ñòŠ¹œO D =

+−2

34,

250

= D

−27

,25

AC¡ ¬ñòŠ¹œO E =

+−−2

31,

251

= E (-2, 1)

AB¡ ¬ñòŠ¹œO F =

+−+2

41,

201

= F

23

,21

Ýè«õ ï´‚«è£´ AD¡ c÷‹ AD = 22

27

125

1

−−+

+

= 22

29

27

−+

=

481

449 + =

4130

ï´‚«è£´ BE¡ c÷‹ = 22 )41()02( −+− = 94 + = = 13

ï´‚«è£´ CF¡ c÷‹ = 22

323

521

−+

+ = 22

23

211

−+

=

49

4121+ =

4130

Ýè«õ ΔABC¡ ï´‚«è£´èO¡ c÷ƒèœ 2

130, 13 ,

2130

Ý°‹.

5. (6, 9), (7, 4), (4,2) ñŸÁ‹ (3,7) ÝAòõŸ¬ø º¬ùè÷£è‚ ªè£‡ì èóƒèO¡ ðóŠð÷¾è¬÷‚裇è.

ªè£´‚èŠð†ì ¹œOè¬÷ è®è£ó º¡«ù£†ì F¬ê‚° âF˜F¬êJ™ ܬñ»ñ£Á õK¬êò£è ðìˆF™°P‚辋A(4,2), B(7,4), C (6, 9), D(3,7).èó‹ ABCD¡ ðóŠ¹

=

27942

43674

21

= ( ) ( )[ ]28272414642631621 +++−+++

= [ ]9312721 −

= 3421×

= 17Ýè«õ èó‹ ABCD ¡ ðóŠ¹ 17ê.Üô°èœ

y

x

C(6,9)

B(7,4)

D(3,7)

A(4,2)

www.mathstimes.com

8 1

6. (-4, 5) (0,7) (5,-5) ñŸÁ‹ (-4,-2) º¬ùè÷£è‚ ªè£‡ì èóƒèO¡ ðóŠð÷¾ 裇.

ªè£´‚èŠð†ì ¹œOè¬÷ è®è£ó ºœ«÷£†ì F¬ê‚° âF˜F¬êJ™ ܬñ»ñ£Á õK¬êò£è ðìˆF™

°P‚辋.

A(-4, 5) B(0,7) C(5,-5) D(-4,-2) â¡è. èó‹ ABCD ¡ ðóŠ¹.

=

−−−−−−

25752

44054

21

= 21

[(20+35+0+8) - (-10 + 0 - 28 - 20)]

= 21

[63 + 58]

= 21

x 121

= 60.5Ýè«õ, èó‹ ABCD¡ ðóŠ¹ 60.5 ê.Üô°èœ.

7. (2, -5) (3, -4) ñŸÁ‹ (9, k) å¼ «è£ì¬ñ âQ™ k ¡ ñFŠ¬ð‚ 裇.

A(2, -5) B(3, -4) C (9, k) â¡ðù ªè£´‚èŠð†ì ¹œOèœ â¡è.

Í¡Á ¹œOèÀ‹ å«ó «ï˜‚«è£†®™ ܬñõ ΔABC ¡ ðóŠ¹ Ì„Cò‹ Ý°‹.

Δ = 21

−−− 5k45

2932 = 0

Δ = 21

[(-8 + 3k - 45) - (-15 - 36 + 2k)] = 0

Δ = 21

[-53 + 3k + 51 - 2k] = 0

Δ = -2 + k = 0k - 2 = 0

k = 2

Ýè«õ k ¡ ñFŠ¹ 2.

8. å¼ º‚«è£íˆF¡ º¬ùèœ (0,- 1), (2,1) ñŸø‹ (0,3) âQ™ Þî¡ ð‚èƒèO¡ ï´Š¹œOè¬÷ެ툶 à¼õ£‚°‹ º‚«è£íˆF¡ ðóŠ¹è£‡. «ñ½‹ Þ„CPò º‚«è£íˆF¡ ðóŠð÷MŸ°‹ªè£´‚èŠð†ì º‚«è£íˆF¡ ðóŠð÷MŸ° àœ÷ MAîˆ¬î‚ è£‡.

A (0,- 1) B(2,1) ñŸÁ‹ C(0,3) â¡ðù º‚«è£íˆF¡ à„Cèœ «ñ½‹ D, E, F â¡ðù º¬ø«ò BC,

CA ñŸÁ‹ AB ¡ ï´Š¹œOèœ â¡è.

BC ¡ ï´Š¹œO D =

++2

31,

202

= D (1, 2)

AC ¡ ï´Š¹œO E =

−+2

13,

200

= E (0, 1)

AB ¡ ï´Š¹œO F =

+−+2

11,

220

= F (1, 0)

ΔDEF¡ ðóŠ¹ =

2012

1101

21

= ( ) ( )[ ]01020121 ++−++ = 1 ê.Üô°èœ

D(-4,5) D(0,7)

D(5,-5)A(-4,-2)

O

y

x

www.mathstimes.com

8 2

ΔABC¡ ðóŠ¹ =

− 1131

2002

21

= ( ) ( )[ ]20000621 −+−++

= 4 ê.Üô°èœÝè«õ ΔDEF ñŸÁ‹ ΔABC¡ ðóŠ¹èO¡ MAî‹ 1 : 4 Ý°‹.

9. ΔABC ¡ º¬ùèœ A(1, 8) B (-2, 4), C (8, -5). «ñ½‹ M , N â¡ðù º¬ø«ò AB , AC ÞõŸP¡ï´Š¹œOèœ âQ™ MN ¡ ꣌¬õ‚ 裇. ެ裇´ MN ñŸÁ‹ BC ÝAò «ï˜‚«è£´èœÞ¬í âù‚ 裆´è.

A(1, 8) B (-2, 4), C (8, -5) â¡ðù º‚«è£íˆF¡ º¬ùèœ

AB ¡ ï´Š¹œO M =

+−2

48,

221

=

−6,

21

AC¡ ï´Š¹œO N =

+−+2

85,

281

N =

23

,29

MN ¡ ꣌¾ M1 =

21

29

623

+

−

=

2102123 −

= 10

9− ----- I

BC ¡ ꣌¾ M2

= 2845

+−−

= 10

9−--- II

I ñŸÁ‹ II L¼‰¶ M1 = M

2

Ýè«õ, «ï˜‚«è£´èœ BC ñŸÁ‹ MN ÝAò¬õ Þ¬íò£°‹.

10. (6 , 7), (2 , -9) ñŸÁ‹ (-4 , 1) ÝAòù å¼ º‚«è£íˆF¡ º¬ùèœ âQ™ º‚«è£íˆF¡ ï´‚«è£´èO¡

꣌¾è¬÷‚ 裇è.

A(6 , 7), B(2 , -9) ñŸÁ‹ C(-4 , 1) ÝAò¬õ º‚«è£íˆF¡ º¬ùèœ

«ñ½‹ D, E,F â¡ðù º¬ø«ò BC, CA ñŸÁ‹ AB ¡ ï´Š¹œOèœ â¡è. Ýè«õ AD, BE ñŸÁ‹ CF

ÝAò¬õ ΔABC ï´‚«è£´è÷£°‹.

BC ¡ ï´Š¹œO D =

+−−2

19,

242

= (-1, -4)

CA ¡ ï´Š¹œO E

++−2

71,

264

E = (1, 4)

AB ¡ ï´Š¹œO F

−+2

97,

226

F = (4, -1)

âù«õ AD¡ ꣌¾ = 6174

−−−−

= 711

−−

= 711

A(1,8)

N

C (8,-5)B (-2,4)

M

C(-4,1)

E

A(6,7) C(2,-9)

D

D

www.mathstimes.com

8 3

BE¡ ꣌¾ = 2194

−+

= 1

13−

= -13

CF¡ ꣌¾ = 4411

+−−

= 82−

= 41−

Ýè«õ, ï´‚«è£´èO¡ ꣌¾èœ 711

, -13, 41−

.

11. A (-2, 3), B (a, 5) ÝAò ¹œOè¬÷ ެ킰‹ «ï˜‚«è£´ ñŸÁ‹ C (0, 5) D (-2, 1) ÝAò ¹œOè¬÷ެ킰‹ «ï˜‚«è£´ ÝAòù Þ¬í«è£´èœ âQ™ ‘a’ ¡ ñFŠ¹ 裇.

«ï˜‚«è£´èœ AB ñŸÁ‹ CD Þ¬í â¡ð Üî¡ ê£Œ¾èœ êñ‹.Ýè«õ AB ¡ ꣌¾ = CD ¡ ꣌¾

AB¡ ꣌¾ = 2a

22a35

+=

+−

CD ¡ ꣌¾ = 24

0251

−−=

−−−

= 2

AB¡ ꣌¾ = CD ¡ ꣌¾

2a2+ = 2

a + 2 = 1a = 1 - 2a = -1

Ýè«õ, a ¡ ñFŠ¹ -112. (2, 2) â¡ø ¹œO õN„ ªê™õ¶‹, ªõ†´ˆ¶‡´èO¡ Ã´î™ 9 Ý辋 ªè£‡ì «ï˜‚«è£´èO¡

êñ¡ð£´è¬÷‚ 裇.b˜¾:a, b â¡ðù º¬ø«ò «ï˜‚«è£†®¡ x ñŸÁ‹ y ¡ ªõ†´ˆ¶‡´èœ â¡è.Ýè«õ a + b = 9 Ü™ô¶ b = 9 - a

ªõ†´ˆ¶‡´ ܬñŠHô£ù «ï˜‚«è£†®¡ êñ¡ð£†®¡ ð® by

a+x

= 1 ---- I

Þ‚«è£´ (2, 2) â¡ø ¹œO õN„ªê™õ  ªðÁð¶ a9

2a2

−+ = 1

a2 - 9a + 18 = 0

(a - 6) (a - 3) = 0 a = 6 Ü™ô¶ a = 3

a = 3 â‹ «ð£¶ 6y

3+x

= 1 2x + y - 6 = 0

a = 6 â‹ «ð£¶ 3y

6+x

=1 x + 2y - 6 = 0

13. A(-2, 6) B (3, -4) ÝAò ¹œOè¬÷ ެ킰‹ «ï˜‚«è£†´ˆ¶‡¬ì P â¡ø ¹œO à†¹øñ£è 2:3

â¡ø MAîˆF™ HK‚A¡ø¶. ¹œO P õNò£è„ ªê™½‹ ꣌¾ 23 à¬ìò «ï˜‚«è£†®¡ êñ¡ð£†¬ì‚

裇.ABä à†¹øñ£è 2 : 3 â¡ø MAîˆF™ HK‚°‹ ¹œO P â¡è.

Ýè«õ ¹œO P

++−

+−+

32)6(3)4(2

,32

)2(3)3(2

= (2, 0) www.mathstimes.com

8 4

꣌¾ 23

Ý辋 (0,2) â¡ø ¹œO õNò£è¾‹ ªê™½‹ «ï˜‚«è£†®¡ êñ¡ð£´

y - 2 = 23

(x - 0)

2y - 4 = 3x3x - 2y + 4 = 0

14. 3x - y + 9 = 0, x + 2y = 4 ÝAò «ï˜‚«è£´èœ ªõ†´‹ ¹œO»ì¡, 2x + y - 4 = 0 , x - 2y + 3 = 0 ÝAò«ï˜‚«è£´èœ ªõ†´‹ ¹œO¬ò ެ킰‹ «ï˜‚«è£†®¡ êñ¡ð£†¬ì‚ 裇.

3x - y = -9 ----- (I) x + 2y = 4 ---- (II)

2x + y = 4 ---- (III) x - 2y = -3 --- (IV)

(I x 2) + II3x - y = -9x + 2y = 4

(I x 2) + II6x - 2y = -18 Put x = -2 we get x - 2y = -4 -2 +2y = 4 7x = -14 2y = 4 + 2

x = -2 y = 6/2 y = 3ªõ†´‹ ¹œO (-2, 3)III, IVä b˜‚è 2x + y = 4 ---- (III)

x - 2y = -3 ---- (IV)

(III x 2) + IV 4x + 2y = 8 Put x = 1

x - 2y = -3 1 - 2y = -35x = 5 x = 1 -2y = -4

y = 2ªõ†´‹ ¹œO (1, 2)(-2, 3) ñŸÁ‹ (1, 2) ä ެ킰‹ «ï˜‚«è£†®¡ êñ¡ð£´

212

323y

++=

−− x

32

13y +=

−− x

x + 3y - 7 = 0

M =

+−−2

82,

253

= (-1, 3)15. ΔABC ¡ º¬ùèœ A(2, -4), B(3, 3) ñŸÁ‹ C(-1, 5) âQ™ BL¼‰¶ õ¬óòŠð´‹ °ˆ¶‚«è£†´ õN„

ªê™½‹ «ï˜‚«è£†®¡ êñ¡ð£†¬ì‚ 裇.

3x-y

+9 =

0

x+2y

= 4

2x+y

-4 =

0

x-2y

+3 =

0

A(2,-4)

B(3,3) C(-1,5)

D

www.mathstimes.com

8 5

A(2, -4), B(3, 3) ñŸÁ‹ C(-1, 5) â¡ðù º‚«è£íˆF¡ à„Cèœ BD â¡ð¶ º‚«è£íˆF¡ à„CBJL¼‰¶ õ¬óòŠð´‹ °ˆ¶‚«è£´ â¡è.

AC ¡ ꣌¾ 2145

−−+

= 3

9− = -3

Ýè«õ, °ˆ¶‚«è£´ BD ¡ ꣌¾ 31

(AC ⊥ BD)

꣌¾ 31Ý辋 (3, 3) â¡ø ¹œO õN„ªê™õ¶ñ£ù «ï˜‚«è£†®¡ êñ¡ð£´.

y - 3 = 31

(x - 3)

3y - 9 = x - 3 x - 3y + 6 = 0

16. ΔABC ¡ º¬ùèœ A (-4, 4) B(8,4) C (8,10) âQ™ A JL¼‰¶ õ¬óòŠð´‹ ï´‚«è£†´ õN„ ªê™½‹«ï˜‚«è£†®¡ êñ¡ð£†¬ì‚ 裇.

º¬ù AJL¼‰¶ õ¬óòŠð´‹ ï´‚«è£´ AD â¡è.

BC ¡ ¬ñòŠ¹œO D

++2104

,2

88

D = (8, 7)

A (-4, 4) D (8,7) ÝAò ¹œOè¬÷ ެ킰‹ ï´‚«è£´ AD ¡ êñ¡ð£´

484

474y

++=

−− x

4y - 16 = x + 4

x - 4y + 20 = 0

6. õ®Mò™õ®Mò™õ®Mò™õ®Mò™õ®Mò™

(50 ñFŠªð‡èÀ‚° «ñ™ ªðø M¼‹¹ðõ˜èÀ‚°50 ñFŠªð‡èÀ‚° «ñ™ ªðø M¼‹¹ðõ˜èÀ‚°50 ñFŠªð‡èÀ‚° «ñ™ ªðø M¼‹¹ðõ˜èÀ‚°50 ñFŠªð‡èÀ‚° «ñ™ ªðø M¼‹¹ðõ˜èÀ‚°50 ñFŠªð‡èÀ‚° «ñ™ ªðø M¼‹¹ðõ˜èÀ‚°)1. å¼ õ†ìˆF¡ ¹œO A ™ õ¬óòŠð´‹ ªî£´«è£´ PQ â¡è. AB â¡ð¶ õ†ìˆF¡  â¡è.

«ñ½‹ ∠ BAC = 54o ñŸÁ‹ ∠ BAQ = 62o â¡Á ܬñ»ñ£Á õ†ìˆF¡ «ñ™ àœ÷ ¹œO C âQ™

∠ ABC 裇è,

PQ å¼ ªî£´«è£´ AB .

∠ BAQ = ∠ ACB = 62o («îŸø‹)

∠ BAC = ∠ ABC + ∠ ACB = 180o

(º‚«è£íˆF¡ Í¡Á «è£íƒèœ)

54 + ∠ ABC+62o = 180o

∠ ABC +- 116 = 180

∠ ABC = 180 - 116 = 64o

∠ ABC = 64o

2. ðìˆF™ TP å¼ ªî£´«è£´, A , Bõ†ìˆF¡ e¶œ÷ ¹œOèœ ∠ BTP = 72o ñŸÁ‹ ∠ ATB = 43o âQ™

∠ ABT 裇è. (Ap. 13)

TP å¼ ªî£´«è£´

TB 

∠ BTP = ∠ BAT = 72o («îŸø‹)

∠ BTP + ∠ ABT + ∠ BAT = 180o

(º‚«è£íˆF¡ Í¡Á «è£íƒèœ)

43o + ∠ ABT + 72o = 180o

C(8,10)

A(-4,4) B(-4,4)

D

T P

72o43o

?

AB

www.mathstimes.com

8 6

∠ ABT + 115 = 180

∠ ABT = 180 - 115

= 65o

∠ ABT = 65o

3. E ñŸÁ‹ F â¡ø ¹œOèœ º¬ø«ò ΔPQR ¡ ð‚èƒèœ PQ ñŸÁ‹ PR ÝAòõŸP¡ e¶ ܬñ‰¶œ÷ù.EF||QR â¡ð¬î„ êKð£˜. PE = 3.9 ªê.e, EQ = 3 ªê.e, PF = 3.6 ªê.e ñŸÁ‹ FR = 2.4 ªê.e

FRPF

EQPE =

4.26.3

39.3 =

3.9 x 2.4 = 3 x 3.6

9.36 ≠ 10.8

âù«õ EF || QR (Þ¬í Ü™ô)

4. E ñŸÁ‹ F â¡ø ¹œOèœ º¬ø«ò ΔPQR ¡ ð‚èƒèœ PQ ñŸÁ‹ PR ÝAòõŸP¡ e¶ ܬñ‰¶œ÷ù.EF || QR â¡ð¬î„ êKð£˜. PE = 4 ªê.e, QE = 4.5 ªê.e, PF = 8 ªê.e ‹Á‹ RF = 9 ªê.e

(ºòŸC ªêŒè)

5. AD â¡ð¶ ΔABC J™ ∠ A ¡ «è£í Þ¼êñ ªõ†® Ý°ñ£ âù„ «ê£F‚è, AB = 4 ªê.e, AC = 6 ªê.e

BD = 1.6 ªê.e ñŸÁ‹ CD = 2.4 ªê.e

ACAB

DCBD =

64

4.26.1 =

1.6 x 6 = 2.4 x 4

9.6 = 9.6

âù«õ AD â¡ð¶ ∠ A ¡ «è£í Þ¼êñªõ†®

6. AD â¡ð¶ ΔABC J™ ∠ A ¡ «è£í Þ¼êñ ªõ†® Ý°ñ£ âù„ «ê£F‚è. AB = 6 ªê.e, AC = 8 ªê.e,

BD = 1.5 ªê.e ñŸÁ‹ CD = 3 ªê.e (ºòŸC ªêŒè)

1. ð®‚è «õ‡®ò «îŸøƒèœ

1. Ü®Šð¬ì MAî îêñ «îŸø‹ (Ü™ô¶) «î™v «îŸø‹ (Oct.14, Ap. 14, Ju. 13)

2. «è£í Þ¼êñªõ†® «îŸø‹ (Oct. 13, Ap. 12)

3. Hî£èóv «îŸø‹. (Ap. 13, Ju. 12)

2. å¼ CÁõ¡ ¬õóˆF¡ °Á‚° ªõ†´ «îŸøõ®M™ ðìˆF™ 裆®òõ£Á å¼ ð†ì‹ ªêŒî£¡. Þƒ°AE = 16 ªê.e, EC = 81ªê.e. Üõ¡ BD â¡Ÿ °Á‚°‚ °„CJ¬ùŠ ðò¡ð´ˆî M¼‹¹Aø£¡. °„CJ¡

c÷‹ âšõ÷õ£è Þ¼‚è «õ‡´‹. (Govt. Model Question)

ΔEAD ~ ΔEDC

âù«õ EDEA

= ECED

ED2 = EA x EC ED2 = 16 x 81

ED = 8116×

= 4 x 9 ED = 36 ED = BE = 36 cm

P

F

RQ

E

3

3.9 3.6

2.4

C

B

A

D

E

81

16

A

6

CB

4

D1.6 2.4

www.mathstimes.com

8 7

âù«õ BD = 36 + 36 = 72

BD = 72 ªê.e

°„CJ¡ c÷‹ = 72 ªê.e

3. å¼ î£ñ¬óŠ Ìõ£ù¶ î‡a˜ ñ†ìˆFŸ° «ñ™ 20 ªê.e àòóˆF™ àœ÷¶. ¡ eFŠð°F î‡a˜ñ†ìˆFŸ° W«ö àœ÷¶. 裟Á i²‹ «ð£¶  îœ÷Šð†´ î£ñ¬óŠ Ìõ£ù¶ ¡ Ýó‹ðG¬ôJL¼‰¶ 40 ªê.e ÉóˆF™ î‡a¬óˆ ªî£´Aø¶. Ýó‹ð G¬ôJ™ î‡a˜ ñ†ìˆFŸ° W«öàœ÷ ¡ c÷‹ 裇è.

b˜¾:

BD = î‡aK¡ W«ö ¡ c÷‹ = x ªê.e â¡è.

ΔBCD ™ Hî£èóv «îŸøˆF¡ð®

BC2 = AB2 + AC2

(x + 20)2 = x2 + 402

x2 + 40x + 400 = x2 + 1600

x2 + 40x + 400 = x2 + 1600

40x = 1600 - 400

40 = 1200

x = 40

1200 = 30

x = 30 cm

î‡a˜ ñ†ìˆFŸ° W«ö àœ÷ ¡ c÷‹ = 30 ªê.e

4. ªêšõè‹ ABCD¡ à†¹ø ¹œO O ML¼‰¶ ªêšõèˆF¡ º¬ùèœ A, B, C, D Þ¬í‚èŠð†´œ÷ù

âQ™ OA2 + OC2 + OB2 + OD2 âù GÁ¾è. (Ju. 14)

5. å¼ Þ¬íèóˆF¡ â™ô£Š ð‚èƒèÀ‹ å¼ õ†ìˆF¬ù ªî£´ñ£ù£™ ÜšM¬íèó‹ å¼ ê£Œê¶óñ£°‹

âù GÁ¾è. (March 15)

6. ABCD â¡ø èó‹, Üî¡ â™ô£ ð‚èƒèÀ‹ å¼ õ†ìˆ¬î ªî£´ñ£Á ܬñ‰¶œ÷¶. AB = 6 ªê.e,

BC = 6.5 ªê.e ñŸÁ‹ CD = 7ªê.e âQ™ AD¡ c÷ˆ¬î‚ 裇è.

7. å¼ GöŸðì‚ è¼MJ½œ÷ ðì„ ²¼O™ å¼ ñóˆF¡ H‹ðˆF¡ c÷‹ 35 I.e, ªô¡²‚°‹ ð섲¼À‚°‹Þ¬ìŠð†ì Éó‹ 42 I.e. «ñ½‹, ªô¡RL¼‰¶ ñ󈶂° àœ÷ Éó‹ 6 e âQ™, GöŸðì‹â´‚èŠð´‹ ñóˆF¡ ð°FJ¡ c÷‹ 裇è. (Score Model Question)

7. º‚«è£íMò™º‚«è£íMò™º‚«è£íMò™º‚«è£íMò™º‚«è£íMò™

1. «ï˜‚°ˆî£ù å¼ ñóˆF¡ «ñ™ð£è‹ 裟Pù£™ ºPŸ¶, Ü‹ ºP‰î ð°F W«ö M¿‰¶ Mì£ñ™ñóˆF¡ à„C î¬ó»ì¡ 30o «è£íˆ¬î ãŸð´ˆ¶Aø¶. ñóˆF¡ à„C Üî¡ Ü®JL¼‰¶ 30 eªî£¬ôM™ î¬ó¬òˆ ªî£´Aø¶ âQ™ ñóˆF¡ º¿ àòóˆ¬î‚ 裇è.

ñóˆF¡ àòó‹ x e + ye â¡è (Oct-12, July-14)âF˜ð‚è‹

ΔABC ™, tanθ = ---------------------Ü´ˆ¶œ÷ ð‚è‹

tan 30o = 30x

3

1=

30x

3 x = 1 x 30

x = 3

30

A

40cm C

B

x cm

A

20cm

x+20

30m

A30o

C

B30m

x

y

www.mathstimes.com

8 8

= 3.3

330 =

3330

= 10 3 e

x = 10 3 e

Ü´ˆ¶œ÷ ð‚è‹ Cosθ = ----------------------------

è˜í‹

cos30o = y30

23

= y30

3 y= 2 x 30

y = 3.3

3302 ×× =

33302 ××

= 20 3

y = 20 3 e

ñóˆF¡ àòó‹ = x + y

= 10 3 + 20 3 e

ñóˆF¡ àòó‹ = 30 3 e

2. æ˜ ÜF«õèŠ «ð£˜ Mñ£ù‹ î¬ó ñ†ìˆFL¼‰¶ 3000 e àòóˆF™ ñŸªø£¼ ÜF«õèŠ «ð£˜Mñ£ùˆ¬î «ï˜ «ñô£è‚ èì‚Aø¶. Üšõ£Á èì‚°‹ «ð£¶ î¬óñ†ìˆF™ å¼ °PŠH†ì ¹œOJL¼‰¶Üî¡ ãŸø‚«è£íƒèœ º¬ø«ò 60o ñŸÁ‹ 45o âQ™ Ü‰î «ïóˆF™ 2õ¶ «ð£˜ Mñ£ù‹ ñŸÁ‹ ºî™

«ð£˜ Mñ£ù‹, ÝAòõŸPŸ° Þ¬ìŠð†ì Éó‹ èí‚A´è. ( 3 =1.732) (Ju. 13)

Þ¬ìŠð†ì Éó‹ = h e â¡èΔOAC ™

âF˜ð‚è‹tanθ = ------------------------

Ü´ˆ¶œ÷ ð‚è‹

tan 60o = OC

3000

3 = OC

3000

3 OC = 3000

OC = 3

3000

= 33

33000

×

× =

333000

= 1000 3

OC = 1000 3 e

âF˜ ð‚è‹

ΔOCB ™ tanθ = ------------------------Ü´ˆ¶œ÷ ð‚è‹

tan45o = OC

h3000 −

30

00 m

www.mathstimes.com

8 9

1 = 31000

3000 h−

1 x 1000 3 = 3000 - h h = 3000 - 1000 3

= 3000 - 1000 x 1.732

= 3000 - 1732

h = 1268 e

Þ¬ìŠð†ì Éó‹ = 1268e

3. 500 e àòóˆF™ ðø‰¶ ªè£‡®¼‚°‹ ªýL裊ìK™ å¼õ˜ æ˜ ÝŸP™ Þ¼ è¬óèO™ «ïªóFó£è

àœ÷ Þ¼ ªð£¼†è¬÷ 30o , 45o Þø‚è‚ «è£íƒèO™ 裇Aø£˜ âQ™ ÝŸP¡ Üèô‹ 裇è.

( 3 = 1.732 ) (Ap. 14)

ÝŸP¡ Üèô‹ = x e + y e

ΔABC J™

âF˜ð£‚è‹

tanθ = ------------------------- Ü´ˆ¶œ÷ ð‚è‹

tan 30o = x

500

3

1 =

x500

x = 500 3

x = 500 x 1.732= 866.00

x = 866 e

âF˜ð£‚è‹

In ΔACD, tanθ = ------------------------- Ü´ˆ¶œ÷ ð‚è‹

tan45o = y500

1 = y500

1 x y = 500

y = 500 e

ÝŸP¡ Üèô‹ = x e + y e

= 866 + 500

= 1366 e4. 700 e àòóˆF™ ðø‰¶ ªè£‡®¼‚°‹ ªýL裊ìK™ å¼õ˜ æ˜ ÝŸP™ Þ¼ è¬óèO™ «ïªóFó£è

àœ÷ Þ¼ ªð£¼†è¬÷ 30o , 45o Þø‚è‚ «è£íƒèO™ 裇Aø£˜ âQ™ ÝŸP¡ Üèô‹ 裇è.

( 3 = 1.732 )

Mù£ ⇠3äŠ «ð£ô«õ b˜¾ è£íô£‹

60 ñFŠªð‡èÀ‚° «ñ™ ªðø M¼‹¹‹ ñ£íõ˜èœ ðJŸC ªêŒò «õ‡®ò Cô èí‚°èœ1. ð‚è‹ 210, â.è£. 7.6 (Ap.13)2. ð‚è‹ 213, â.è£. 7.12 (Oct. 13, Ap. 15)3. ð‚è‹ 214, ðJŸC 7.1™ 5 (Score model question III)

B DC

45o30o

A

30o 45o

500m

xm ym

www.mathstimes.com

9 0

4. ð‚è‹ 220, â.è£. 7.20 (Oct. 14)5. ð‚è‹ 221, â.è£. 7.22 (Score model question V)6. ð‚è‹ 226 ðJŸC 7.2™ 10 (10) (Ap. 13)7. ð‚è‹ 226 ðJŸC 7.2™ 12 (Ap. 12)8. ð‚è‹ 227, ðJŸC 7.2™ 17 (Ju. 12)9. ð‚è‹ 227, ðJŸC 7.2™ 16 (Score model question IV)

8. Ü÷Mò™Ü÷Mò™Ü÷Mò™Ü÷Mò™Ü÷Mò™

1. 120 ªê.e c÷º‹, 84 ªê.e M†ìº‹ ªè£‡ì å¼ ê£¬ô¬ò êñŠð´ˆ¶‹ ༬÷¬ò‚ ªè£‡´ å¼M¬÷ò£†´ˆ Fì™ êñŠð´ˆîŠð´Aø¶. M¬÷ò£†´ˆ Fì¬ô êñŠð´ˆî Þš¾¼¬÷ 500 º¿„²ŸÁ‚蜲öô «õ‡´‹. M¬÷ò£†´ˆFì¬ô êñŠð´ˆî å¼ ê.e†ì¼‚° 75 ¬ðê£ iî‹, Fì¬ô„ êñŠð´ˆî

Ý°‹ ªêô¬õ‚ 裇è. ( π = 722

)

b˜¾:Þƒ° 2r = 84 ªê.e r = 42 ªê.e, h = 120 ªê.e༬÷J¡ å¼ º¿„²ŸPù£™ êñŠð´ˆîŠð´‹ FìL¡ ðóŠ¹ = 2πrh

= 2 x722

x 42 x 120

= 31680 ªê.e2

500 º¿„²ŸÁèO™ êñŠð´ˆîŠð´‹ FìL¡ ðóŠ¹ = 31680 x 500= 15840000 ªê.e2

= 10000

15840000 e2

= 1584 e2

1 ê.e†ì¼‚° êñŠð´ˆî Ý°‹ ªêô¾ = 75 ¬ðê£âù«õ M¬÷ò£†´ˆ Fìì¬ô êñŠð´ˆî Ý°‹ ªñ£ˆî ªêô¾ = 1584 x 0.75

= Ï. 11882. å¼ F‡ñ «ï˜õ†ì ༬÷J¡ ªñ£ˆîŠ¹øŠðóŠ¹ 660ê.ªê.e Üî¡ M†ì‹ 14 ªê.e âQ™, Üš¾¼¬J¡

àòóˆ¬î»‹, õ¬÷ðóŠ¬ð»‹ 裇è.b˜¾:

Þƒ° 2r = 14 ªê.e r = 7 ªê.e

ªñ£ˆîŠ¹øŠðóŠ¹ = 660 ê.ªê.e

2πr (h+r) = 660

2 x 722

x 7 (h + 7) = 660

h = 222

660×

- 7

= 15 - 7

= 8 ªê.e

༬÷J¡ õ¬÷ðóŠ¹ = 2πrh = 2 x 722

x 7 x 8 = 352 ªê.e2

3. å¼ F‡ñ‚ ËH¡ Ýó‹ ñŸÁ‹ ꣻòó‹ º¬ø«ò 20 ªê.e ñŸÁ‹ 29 ªê.e âQ™ ܈F‡ñ‚ ËH¡èù Ü÷¬õ‚ 裇è.b˜¾:

Þƒ° r = 20 ªê.e , l = 29 ªê.e

h = 22 r− www.mathstimes.com

9 1

= 22 2029 −

= 400841−

= 441 h = 21 ªê.e

ËH¡ èùÜ÷¾ = 31

πr2h

= 31

x 722

x 20 x 20 x 21

= 8800 ªê.e3.4. å¼ «ï˜õ†ì‚ ËH¡ Þ¬ì‚è‡ìˆF¡ Þ¼¹øº‹ ܬñ‰î õ†ì MO‹¹èO¡ ²Ÿø÷¾èœ º¬ø«ò 44

ªê.e ñŸÁ‹ 8.4π ªê.e â¡è. Üî¡ àòó‹ 14 ªê.e âQ™ ÜšM¬ì‚è‡ìˆF¡ èù Ü÷¬õ‚ 裇è.b˜¾:

Þƒ° 2πR = 44 ªê.e ñŸÁ‹ 2πr = 8.4 π ªê.e

2 x 722

x R = 44 2r = 8.4

R = 222744

××

r = 4.2 ªê.e

R = 7 ªê.e

Þ¬ì‚è‡ìˆF¡ èùÜ÷¾ = 31

πh (R2 + r2 + Rr)

= 14722

31 ×× (72 + 4.22 + 7 x 4.2)

= 344

(49 + 29.4 +17.64)

= 344

x 96.04

= 1408.58 ªê.e3

5. å¼ F‡ñ ñóŠªð£‹¬ñò£ù¶ ܬó‚«è£÷ˆF¡ «ñ™ ˹ Þ¬í‰î õ®M™ àœ÷¶. ܬó‚«è£÷‹ñŸÁ‹ ˹ ÝAòõŸP¡ Ýó‹ 3.5 ªê.e «ñ½‹ ªð£‹¬ñJ¡ ªñ£ˆî àòó‹ 17.5 ªê.e âQ™

ÜŠªð£‹¬ñ îò£K‚èŠðò¡ð´ˆîŠð†ì ñóˆF¡ èùÜ÷¬õ 裇è. ( π = 722

)

b˜¾:Þƒ°Ü¬ó‚«è£÷‹ ˹

r = 3.5 ªê.e h = 3. 5ªê.eh = 17.5 - 3.5

èù Ü÷¾ = 32

πr3 = 14 ªê.e

= 32

π x 3.5 x 3.5 x 3.5 èù Ü÷¾ = 31

πr2h

= 31

π x 3.5 x 3.5 x 14

ñóˆF¡ èùÜ÷¾ = ܬó‚«è£÷ˆF¡ èùÜ÷¾ + ËH¡ èùÜ÷¾

= 32

π x 3.5 x 3.5 x 3.5 + 31

π x 3.5 x 3.5 x 14www.mathstimes.com

9 2

= 31

π x 3.5 x 3.5 [2 x 3.5 + 14]

= 31

x 722

x 3.5 x 3.5 [7+14]

= 31

x 722

x 3.5 x 3.5 x 21

= 22 x 3.5 x 3.5= 296.5 è.ªê.e

6. å¼ Ãì£óñ£ù¶ ༬÷J¡ e¶ ˹ Þ¬íˆî õ®M™ àœ÷¶. Ãì£óˆF¡ ªñ£ˆî àòó‹ 13.5 eñŸÁ‹ M†ì‹ 28e. «ñ½‹ ༬÷Š ð£èˆF¡ àòó‹ 3 e âQ™ Ãì£óˆF¡ ªñ£ˆî ¹øŠðóŠ¬ð‚裇è.

b˜¾:

༬÷ ˹

h = 3e h = 14 m

2r = 28e h = 13.5 - 3

r = 14e = 10.5 m

õ¬÷ðóŠ¹ = 2πrh l = 22 hr +

= 2π x 14 x 3 = 22 5.1014 +

= 84π = 25.110196 +

= 25.306

= 17.5 mõ¬÷ðóŠ¹

= πrl= π x 14 x 17.5 m= 245π

Ãì£óˆF¡ ªñ£ˆî ¹øŠðóŠ¹ = ༬÷J¡ õ¬÷ðóŠ¹ + ËH¡ õ¬÷ðóŠ¹= 84 π + 245 π= 329 π

= 329 x 722

= 1034 ê.ªê.e7. èOñ‡¬íŠ ðò¡ð´ˆF å¼ ñ£íõ¡ 48 ªê.e àòóº‹ 12 ªê.e Ýóº‹ ªè£‡ì «ï˜õ†ì F‡ñ‚

Ë¬ð„ ªêŒî£˜. ܂ˬð ñŸªø£¼ ñ£íõ˜ å¼ F‡ñ‚ «è£÷ñ£è ñ£ŸPù£˜. Üšõ£Á ñ£ŸøŠð†ì¹Fò «è£÷ˆF¡ Ýóˆ¬î‚ 裇è.

b˜¾:

˹ «è£÷‹ r1 = 12 ªê.e r = ?h = 48 ªê.e«è£÷ˆF¡ èùÜ÷¾ = ËH¡ èùÜ÷¾

3 m

28 m

13.5

m

r

12 cm

48 c

m

www.mathstimes.com

9 3

ie 34

πr3 = 31

πr12 h

4r3 = r12 h

4r3 = 12 x 12 x 48

r3 = 12

481212 ××

r3 = 12 x 12 x 12 r = 12 ªê.e

8. 8 ªê.e M†ìº‹ 12 ªê.e àòóº‹ ªè£‡ì å¼ «ï˜õ†ì F‡ñ Þ¼‹¹‚ Ëð£ù¶ ༂èŠð†´ 4 I.eÝóºœ÷ F‡ñ‚ «è£÷ õ®õ °‡´è÷£è õ£˜‚èŠð†ì£™ A¬ì‚°‹ «è£÷ õ®õ °‡´èO¡â‡E‚¬è¬ò‚ 裇è.b˜¾:

r , h â¡ð¶ ËH¡ Ýó‹, àòó‹ Ý°‹. r1 â¡ð¶ «è£÷ˆF¡ Ýó‹ Ý°‹.˹ «è£÷‹ h = 12 cm = 120 I.e r1 = 4I.e

2r = 8 ªê.e r = 4ªê.e

= 40 I.e ËH¡ èùÜ÷¾

A¬ì‚°‹ «è£÷ õ®õ °‡´èO¡ â‡E‚¬è = ------------------------------«è£÷ˆF¡ èùÜ÷¾

= 31

2

r34

hr31

π

π

= 31

2

r4

hr

= 4444

1204040×××

××

= 7509. ñíô£™ GóŠðŠð†ì å¼ à¼¬÷ õ®õ õ£OJ¡ àòó‹ 32 ªê.e ñŸÁ‹ Ýó‹ 18 ªê.e Ü‹ñí™

º¿õ¶‹ î¬óJ™ å¼ «ï˜õ†ì‚ ˹ õ®M™ ªè£†ìŠð´Aø¶. Üšõ£Á ªè£†ìŠð†ì ñíŸ Ã‹H¡

àòó‹ 24 ªê.e âQ™ ܂ËH¡ Ýó‹ ñŸÁ‹ ꣻòóˆ¬î‚ 裇è.

b˜¾:

༬÷ (õ£O) ˹ (ñí™)

h = 32 ªê.e h = 24 ªê.e

r = 18 ªê.e r = ?

༬÷J¡ è÷Ü÷¾ = πr2h ËH¡ èùÜ÷¾ = 31

πr2h

= π x 18 x 18 x 32 = 31

π x r2 x 24

ËH¡ èù Ü÷¾ = ༬÷J¡ èùÜ÷¾

31

π x r2 x 24 = π x 18 x 18 x 32

r2 x 8 = 18 x 18 x 32

r2 = 8

321818 ××

r2 = 18 x 18 x 4r = 18 x 2

= 36 ªê.e

www.mathstimes.com

9 4

l = 22 rh +

= 22 3624 +

= 1296576 +

= 1872

= 1312 ªê.e

10. 14 e M†ìº‹ 20 e Ýöºœ÷ å¼ AíÁ ༬÷ õ®M™ ªõ†ìŠð´Aø¶. Üšõ£Á ªõ†´‹«ð£¶«î£‡®ªò´‚èŠð†ì ñ‡ Yó£è ðóŠðŠð†´ 20 e x 14 e Ü÷¾èO™ Ü®Šð‚èñ£è‚ ªè£‡ì 弫ñ¬ìò£è ܬñ‚èŠð†ì£™, Ü‹«ñ¬ìJ¡ àòó‹ 裇è.b˜¾:

༬÷ (AíÁ) èù„ªêšõè‹ («ñ¬ì)

h = 20 e l = 20 e 2r = 14 e b = 14 e r = 7 e h1 = ?

èùÜ÷¾ = πr2 h èùÜ÷¾ = l bh

= 722

x 7 x 7 x 20 = 20 x 14 x h

= 22 x 7 x 20èù„ªêšõèˆF¡ èùÜ÷¾ = ༬÷J¡ èùÜ÷¾

20 x 14 x h = 22 x 7 x 20

h = 142020722

×××

= 11 e

11. ¹œOJò™¹œOJò™¹œOJò™¹œOJò™¹œOJò™

1. 18, 20, 15, 12, 25 ñ£Á𣆴‚ªè¿ 裇è.n = 5

x = 5

2512152018 ++++=

590

= 18

x d = x - x d2

18 0 020 2 415 -3 912 -6 3625 7 49

98

σ = n

2d = 5

98= 6.19 ~ 4.428

C.V.= x

σ x 100% =

18428.4

x 100

C.V. = 24.6%

www.mathstimes.com

9 5

2. 20, 18, 32, 24, 26 ñ£Á𣆴‚ ªè¿ 裇è. n = 5

x = 5

2624321820 ++++

= 5

120= 24

x d = x - x d2

20 -4 1618 -6 3632 8 6424 0 026 2 4

120

σ = n

2d = 5

120= 24 ~ 4.9

C.V = x

σ x 100% =

249.4

x 100

C.V. = 20.4%3. 20, 14, 16, 30, 21, 25 F†ìMô‚è‹ è£‡è.

x = 6

252130161420 +++++=

6126

= 21

x d = x - x d2

20 -1 114 -7 4916 -5 2530 9 8121 0 025 4 16

172

σ = n

2d = 6

172= 6.28

σ ~ 5.34. 62, 58, 53, 50, 63, 52, 55 F†ìMô‚è‹ è£‡è.

n = 7

x =7

55526350535862 ++++++

= 7

393 = 56

x d = x - x d2

62 6 3658 2 453 -3 950 -6 3663 7 4952 -4 1655 -1 1

151

www.mathstimes.com

9 6

σ = n

2d = 7

151= 5.21

σ ~ 4.95. 10, 20, 15, 8, 3, 4 F†ìMô‚è‹ èí‚A´è.

n = 6

x = 6

438152010 +++++ =

660

= 10

x d = x - 10 d2

10 0 020 10 10015 5 258 -2 43 -7 494 -6 36

214

σ = n

2d=

6214

= 6.35

σ ~ 5.96. 38, 40, 34, 31, 28, 26, 34 F†ìMô‚è‹ èí‚A´è.

n = 7

x = 7

34262831344038 ++++++=

7231

= 33

x d = x - x d2

38 5 2540 7 4934 1 131 -2 428 -5 2526 -7 4934 1 1

154

σ = n

2d=

7154

= 22

σ ~ 4.69

12. Gè›îè¾Gè›îè¾Gè›îè¾Gè›îè¾Gè›îè¾

1. Í¡Á ï£íòƒèœ ²‡´‹ªð£¿¶ êKò£è Þ¼ Ì‚èœ Ü™ô¶ °¬ø‰îð†ê‹ å¼ î¬ô A¬ì‚èGè›îè¾ ò£¶?

S = {(HHH, HHT, HTH, THH, TTT, TTH, THT, HTT}

n(S) = 8

êKò£ù Þ¼ Ì‚èœ: A = {HTT, TTH, THT}, n (A) = 3

P(A) = 83

°¬ø‰î ð†ê‹ å¼ î¬ô: B = {HTT, THT, TTH, HHT, HTH, THH, HHH}

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9 7

n(B) = 7, P(B) = 87

AB = {HTT, TTH, THT}, n(AB) = 3

P(AB) = 83

P(AB) = P(A) + P(B) - P(AB) 83

+87

- 83

= 87

2. å¼ ðè¬ì Þ¼º¬ø ༆ìŠð†ì¶. °¬ø‰î¶ å¼ à¼†ìLô£õ¶ ⇠5 A¬ì‚è Gè›îè¾ è£‡è.S = {(1,1) ..... (6, 6)} n(S) = 36ºî™ ༆ìL™ 5: A = {(5, 1) (5, 2) (5, 3) (5,4) (5,5) (5,6)}

n(A) = 6 , P(A) = 366

Þó‡ì£‹ ༆ìL™ 5: B = {(1, 5) (2, 5) (3, 5) (4, 5) (5, 5) (6, 5)}

n(B) = 6, P(B) = 366

AB = {(5,5)} n (AB) = 1 P(AB) = 361

P(AB) = P(A) + P(B) − P(AB) 366

+366

-361

= 3611

3. ENTERTAINMENT â¡ø ªê£™L™ ݃Aô àJªó¿ˆ¶ Ü™ô¶ T Ýè Þ¼‚è Gè›îè¾?S = {E, N, T, E, R, T, A I, N, M, E, N, T} N(S) = 13݃Aô àJªó¿ˆ¶ : A = {E, E, A, I, E}, n (A) = 5

P(A) = 135

T ⿈¶ : B = {T, T, T}, n(B) = 3

P(B) = 133

n(AB) = 0 , P (AB) = 0

P(AB) = P(A) + P(B) 135

+ 133

= 138

4. 52 Y†´èœ å¼ Þó£ê£ Ü™ô¶ å¼ ý£˜† Y†´ â´‚è Gè›îè¾?n(S) = 52

ý£˜† : A n(A) = 13 , P(A) = 5213

Þó£ê£ : B n(B) = 4, P(B) = 524

n(AB) = 1, P(AB) = 521

P(AB) = P(A) + P(B) - P(AB)

5213

+ 524

- 521

= 5216

= 134

5. å¼ ¬ðJ™ 10 ªõœ¬÷, 5 輊¹, 3 ð„¬ê, 2 CõŠ¹ ð‰¶. å¼ ð‰¶ ªõœ¬÷ Ü™ô¶ 輊¹ Ü™ô¶ð„¬ê Ýè Þ¼‚è Gè›îè¾

n(S) = 10 + 6 + 10 = 26

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9 8

ªõœ¬÷: A n(A) = 10, P(A) = 2610

輊¹ : B n(B) = 6 , P(B) = 266

P(AB) = P(A) + P(B) =2610

+266

= 2616

= 138

6. å¼ ðè¬ì Þ¼º¬ø ༆ìŠð´‹. ºîô£õ¶ ༆ìL™ Þó†¬ìŠð¬ì ⇠ܙô¶ ºè Ã´î™ 8.S = {(1, 1) ...... (1,6) } ; n (S) = 36

Þó†¬ìŠ ð¬ì â‡èœ A¬ì‚è : A = { (2, 1) (2, 2) (2, 3) (2,4) (2,5) (2,6)(4, 1) (4, 2) (4, 3) (4,4) (4,5)(4,6)(6, 1) (6, 2) (6, 3) (6,4) (6,5) (6,6)}

n(S) = 18, P (A) = 3618

Ã´î™ 8 A¬ì‚è : B = {(2, 6) (3,5) (4,4) (5,3) (6,2)}

n(B) = 5, P(B) = 365

AB = {(2, 6) (4, 4) (6, 2)}n(AB) = 3 , P(AB) = 363

P(AB) = P(A) + P(B) −−−−− P(AB)

= 3618

+ 365

−−−−− 363

= 3620

= 95

7. å¼ ¹Fò ñA›‰¶ õ®õ¬ñŠHŸ° M¼¶ Gè›îè¾ 0.25, âKªð£¼œ ðò¡ð£†®¡ Gè›îè¾ 0.35, Þ¼M¼¶èÀ‹ 0.15, i) °¬ø‰î¶ ãî£õ¶ å¼ M¼¶ ii) å«ó å¼ M¼¶ ñ†´‹

P(A) = 0.25, P(B) = 0.35, P(AB) = 0.15

i) P(AB) = P(A) + P(B) −−−−− P (AB) = 0.25 + 0.35 −−−−− 0.15

= 0.45

ii) )BA(P)BA(P + = P(A) −−−−− P(AB) + P(B) −−−−− P(AB)

= 0.25 −−−−− 0.15 + 0.35 −−−−− 0.15

= 0.10 + 0.20

= 0.3

8. å¼ ñ£íM ñ¼ˆ¶õ è™ÖK Gè›îè¾ 0.16, ªð£PJò™ è™ÖK Gè›îè¾ 0.24, Þ¼ è™ÖKèÀ‹ 0.11i) ãî£õ¶ å¼ è™ÖK ii) å«ó å¼ è™ÖK ñ†´‹

P(A) = 0.16, P(B) = 0.24, P(AB) = 0.11

i) P(AB) = P(A) + P(B) −−−−− P (AB) = 0.16 + 0.24 −−−−− 0.11

= 0.40 - 0.11 = 0.29

ii) )BA(P)BA(P + = P(A) −−−−− P(AB) + P(B) −−−−− P(AB)

= 0.16 −−−−− 0.11 + 0.24 −−−−− 0.11

= 0.05 + 0.13 = 0.18

9. 52 Y†´èœ, ܉î Y†´ v«ð´ Ü™ô¶ Þó£ê£ õ£è Ü™ô¶ CõŠ¹ Gø„ Y†ì£è Þ¼Šð Gè›îè¾ ò£¶?

n(S) = 52Þó£ê£ v«ð´ CõŠ¹ Gø„ Y†´

n(A) = 4, n(B) = 13, n(C) = 26,

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9 9

P(A) = 524

P(B) = 5213

P(C) =5226

n(AB) = 1 n(BC) = 13 n(AC) = 2 n(ABC) = 1

P(AB) = 521

P(BC)5213

P(AC) = 522

P(ABC) = 521

P(ABC) = P(A) + P(B) + P(C) −−−−− P(AB) −−−−− P(BC) −−−−− P(AC) + P(ABC)

= 524

+ 5213

+ 5226

−521

−−−−−5213

− − − − −522

+ 521

= 5228

= 137

10. å¼ ¬ðJ™ 10 ªõœ¬÷, 5 èÁŠ¹, 2 CõŠ¹ àœ÷¶. å¼ ð‰¶ ªõœ¬÷ Ü™ô¶ èÁŠ¹ Ü™ô¶ ð„¬êGø‹

n(S) = 10 + 5 + 3 + 2 = 20ªõœ¬÷: A 輊¹: B ð„¬ê: Cn(A) = 10 n(B) = 5 n(C) = 3

P(A) = 2010

P(B) = 205

P(C) = 203

P(ABC) = P(A) + P(B) + P(C)

= 2010

+ 205

+203

= 2018

=109

11. P(A) =54

, P(B) = 32

, P(C) 73

, P(AB)= 158

, P(BC) = 72

, P(AC) =3512

, P(ABC) = 358

,

P(ABC) = ?

P(ABC) = 54

+ 32

+ 73

−−−−−158

−−−−− 72

−−−−− 3512

+ 358

= 105

24363056457084 +−−−++ =

105101

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100

9. ªêŒº¬ø õ®Mò™ªêŒº¬ø õ®Mò™ªêŒº¬ø õ®Mò™ªêŒº¬ø õ®Mò™ªêŒº¬ø õ®Mò™

ð®ð®ð®ð®ð®1: àîMŠðì‹ õ¬óè. Ýó‹ OT , Éó‹ OP ä °P‚è.

ð®ð®ð®ð®ð® 2 : O ¬õ ¬ñòñ£è‚ ªè£‡´ ªè£´‚èŠð†ì ÝóˆF¡ Ü÷M™ õ†ì‹

õ¬óè.

ð®ð®ð®ð®ð® 3: OML¼‰¶ OP™ P â¡ø ¹œO¬ò‚ °P‚è.

ð®ð®ð®ð®ð® 4: OP ¡ ï´‚°ˆ¶‚«è£´ õ¬óè.

ð®ð®ð®ð®ð® 5: Mä ¬ñòñ£è¾‹ OM ä Ýóñ£è¾‹ à¬ìò Þó‡ì£õ¶ õ†ì‹ õ¬óè.

ð®ð®ð®ð®ð® 6 : Þó‡´ õ†ìƒèÀ‹ T, T’ â¡ø Þ¼ ¹œOèO™ ªõ†´A¡øù.

PT , PT’ ä Þ¬í. ÜõŸP¡ c÷ƒè¬÷ Ü÷‰¶ ⿶è.

°PŠ¹: i) M¬ì¬ò êK𣘂è â¡Á «è†ì£™ ñ†´‹ êK𣘂è.

ii) õ†ìˆ¶‚° ªõO«ò àœ÷ ¹œOJL¼‰¶ õ†ìˆ¶‚° Þ¼

ªî£´«è£´èœ õ¬óòô£‹.

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101

1. Ýó‹ 3 ªê.e Ü÷M™ å¼ õ†ì‹ õ¬óè. ¬ñòˆFL¼‰¶ 7 ªê.e ªî£¬ôM½œ÷ å¼ ¹œOJL¼‰¶

õ†ìˆ¶‚° ªî£´«è£´èœ õ¬óè. Üî¡ c÷ƒè¬÷ Ü÷‰¶ ⿶è.

Fair diagram

2. Ýó‹ 4.2 ªê.e Ü÷M™ å¼ õ†ì‹ õ¬óè. õ†ìˆF¡ e¶ ãî£õ¶ å¼ ¹œO¬ò‚ °Pˆ¶, ¬ñòŠ¹œO¬òŠ

ðò¡ð´ˆF ÜŠ¹œOJ™ å¼ ªî£´«è£´ õ¬óè.

Fair diagram

T1

T2

P7 cm

3cm

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O

3 cm

7 cm

6.5 cm

T1

T2

PM

6.5 cm

O

4.2 cmO P

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T1

M

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102

3. M†ì‹ 10 ªê.e Ü÷M™ å¼ õ†ì‹ õ¬óè. ¬ñòˆFL¼‰¶ 13 ªê.e ªî£¬ôML¼‰¶ õ†ìˆFŸ°PA , PB â¡ø Þ¼ ªî£´«è£´èœ õ¬óè. ÜõŸP¡ c÷ƒè¬÷ Ü÷‰¶ ⿶è.

Fair Diagram

4. Ýó‹ 6 ªê.e Ü÷¾œ÷ õ†ìˆF¡ ¬ñòˆFL¼‰¶ 10 ªê.e ªî£¬ôM™ àœ÷ å¼ ¹œOJL¼‰¶õ†ìˆFŸ° Þ¼ ªî£´«è£´èœ õ¬óè. ÜõŸP¡ c÷ƒè¬÷ Ü÷‰¶ ⿶è.

Fair Diagram

A

B

P10 cm

6cm

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O

A

B

P13 cm

5 cm

àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹

O

13 cm

A

B

MO

12 cm

12 cm

5 cm

P

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103

10 cm P

6cm

B

A

O M

8 cm

8 cm

5. Ýó‹ 3 ªê.e Ü÷¾œ÷ õ†ìˆF¡ ¬ñòˆFL¼‰¶ 9 ªê.e ªî£¬ôM™ àœ÷ å¼ ¹œOJL¼‰¶õ†ìˆFŸ° Þó‡´ ªî£´«è£´èœ õ¬óè.

Fair Diagram

A

B

P9 cm

3cm

àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹

O

3 cm

9 cm

8.5 cm

A

B

PM

8.5 cm

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104

º‚«è£íƒèœ õ¬ó‚«è£íƒèœ õ¬ó‚«è£íƒèœ õ¬ó‚«è£íƒèœ õ¬ó‚«è£íƒèœ õ¬óî™1. AB = 6 ªê.e ∠ C = 40o ñŸÁ‹ à„C CJL¼‰¶ AB‚° õ¬óòŠð†ì °ˆ¶‚«è£†®¡ c÷‹ 4.2 ªê.e

ªè£‡ì ΔABC õ¬óè.

Fair Diagram

2. ΔABC ™ BC = 5 ªê.e∠ A = 45o ñŸÁ‹ à„C AL¼‰¶ BC‚° õ¬óòŠð†ì ï´‚«è£†®¡ c÷‹ 4 ªê.eâù Þ¼‚°‹ð® ΔABC õ¬óè.

Fair Diagram

A B

C

40o

6cm

4.2c

m

àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹

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C C’

O40o

M

Y

X

H

K

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cm

>

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50o

B C

A

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5cm

4cm

O

X

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105

3. Ü®Šð‚è‹ BC = 5.5ªê.e ∠ A = 60o ñŸÁ‹ à„C AJL¼‰¶ õ¬óòŠð†ì ï´‚«è£´ AM¡ c÷‹ 4.5 ªê.eªè£‡ì ΔABC õ¬óè.

Fair diagram

4. PQ = 4ªê.e ∠ R = 25o ñŸÁ‹ à„C R L¼‰¶ PQ‚° õ¬óòŠð†ì °ˆ¶‚«è£†®¡ c÷‹ 4.5ªê.e â¡øÜ÷¾èœ ªè£‡ì ΔPQR õ¬óè.

Fair Diagram

B C

A

60o

5.5cm

4.5c

m

àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹àîMŠðì‹

M

B

A

C5.5cm

4.5c

m

60o

X

Y

M

K

A’

O

>

>

P Q

R

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4cm

4c

m

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P Q

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R’

X

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M

H

O

K

>

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106

5. ΔPQR ™ Ü®Šð‚è‹ PQ = 6ªê.e ∠ R = 60o ñŸÁ‹ à„C R L¼‰¶ PQ‚° õ¬óòŠð†ì °ˆ¶‚«è£†®¡c÷‹ 4 ªê.e âù Þ¼‚°ñ£Á ΔPQR õ¬óè.

Fair Diagram

6. BC = 5ªê.e ∠ BAC = 40o ñŸÁ‹ à„C A JL¼‰¶ BC‚° õ¬óòŠð†ì ï´‚«è£†®¡ c÷‹ 6 ªê.e â¡øÜ÷¾èœ ªè£‡ì ΔABC õ¬óè. à„C AJL¼‰¶ õ¬óòŠð†ì °ˆ¶‚«è£†®¡ c÷‹ 裇è.

Fair Diagram

P Q

R

60o

6cm

4c

m

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o

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R’

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107

10. õ¬óðìƒèœõ¬óðìƒèœõ¬óðìƒèœõ¬óðìƒèœõ¬óðìƒèœ

1. W›‚裵‹ ܆ìõ¬í‚°ˆ î°‰î õ¬óðì‹ õ¬ó‰¶ ñ£PèO¡ ñ£Á𣆴ˆ ñ¬ò‚ 裇è.x 2 3 5 8 10y 8 12 20 32 40

«ñ½‹, x=4 âQ™ y¡ ñFŠ¬ð‚ 裇è.b˜¾ : x = 4 âQ™ y = 16 Ý°‹

^

^

80

75

70

65

60

55

50

45

40

35

30

25

20

15

10

5

1 2 3 4 5 6 7 8 9 10

y

x′ x^

^

(2,8)

(3,12)

(5,20)

(8,32)

(10,40)

Scalex-axis 1 cm = 1 unity-axis 1 cm = 5 units

y = 4x

^

^

16

O>

>

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108

2. å¼ IFõ‡® 憴ðõ˜ A â¡ø ÞìˆFL¼‰¶ B â¡ø ÞìˆFŸ° å¼ Yó£ù «õèˆF™ å«ó õNJ™ªõš«õÁ èO™ ðòí‹ ªêŒAø£ .̃ Üõ˜ ðòí‹ ªêŒî «õè‹ ÜˆÉóˆF¬ù‚ èì‚è â´ˆ¶‚ªè£‡ì«ïó‹ ÝAòùõŸ¬øŠ ðŸPò Mõóƒèœ («õè&è£ô) H¡õ¼‹ ܆ìõ¬íJ™ ªè£´‚èŠð†´œ÷ù.

«õè‹ (A.e/ñE) 2 4 6 10 12«ïó‹ (ñEJ™) 60 30 20 12 10«õè&è£ô õ¬óðì‹ õ¬ó‰¶ ÜFL¼‰¶ i) Üõ˜ ñE‚° 5 A.e «õèˆF™ ªê¡ø£™ Éóˆ¬î‚ èì‚èÝ°‹ ðòí «ïó‹ ii) Üõ˜ Þ‚°PŠH†ì Éóˆ¬î 40 ñE «ïóˆF™ èì‚è â‰î «õèˆF™ ðòE‚è«õ‡´‹. ÝAòùõŸ¬ø‚ 裇è,b˜¾: i) x = 5 âQ™ y = 24, ii) y = 40 âQ™ x = 3

^x′ x1 2 3 4 5 6 7 8 9 10

^

100

90

80

70

60

50

40

30

20

10

y Scalex-axis 1cm = 1 unity-axis 1cm = 10 units

-10

-20

-30

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(ñ£P

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(ñ£P

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«õè‹ (A.e)«õè‹ (A.e)«õè‹ (A.e)«õè‹ (A.e)«õè‹ (A.e)

xy = 120

^

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(3, 40)

(4,30)

(5, 24)(6, 20)

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109

3. õ£ƒèŠð†ì «ï£†´Š ¹ˆîèƒèO¡ â‡E‚¬è ñŸÁ‹ ÜîŸè£ù M¬ô Mõó‹ H¡õ¼‹ ܆ìõ¬íJ™

îóŠð†´œ÷¶.

«ï£†´Š¹ˆîèƒèO¡ â‡E‚¬è (x) 2 4 6 8 10 12

M¬ô Ï. (y) 30 60 90 120 150 180

ÞîŸè£ù õ¬óðì‹ õ¬ó‰¶ Üî¡ Íô‹ i) ã¿ «ï£†´Š¹ˆîèƒèO¡ M¬ô¬ò‚ 裇è.

ii) Ï.165‚° õ£ƒèŠð†ì «ï£†´Š¹ˆîèƒèO¡ â‡E‚¬è¬ò‚ 裇è.

b˜¾: x = 7 âQ™ y = Ï.105 y = Ï.165 âQ™ x = 13

^

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210

195

180

165

150

135

120

105

90

75

60

45

30

15

2 4 6 8 10 12 14 16 18 20

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(2,30)

(4,60)

(6,90)

(8,120)

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110

4. x 1 3 5 7 8

y 2 6 10 14 16

«ñŸè‡ì ܆ìõ¬íJ™ àœ÷ MõóˆFŸ° õ¬óðì‹ õ¬ó‰¶ Üî¡ Íô‹

i) x = 4 âQ™ y ¡ ñFŠ¬ð»‹ ii) y = 12 âQ™ x¡ ñFŠ¬ð»‹ 裇è.

b˜¾ : i) x = 4 âQ™ y = 8

ii) y = 12 âQ™ x = 6

^

^

20

18

16

14

12

10

8

6

4

2

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111

5. «õ¬ôò£†èO¡ â‡E‚¬è (x) 3 4 6 8 9 16

èO¡ â‡E‚¬è (y) 96 72 48 36 32 18

܆ìõ¬íJ™ ªè£´‚èŠð†´œ÷ MõóˆFŸè£ù õ¬óðì‹ õ¬óè. Üî¡Íô‹ 12 «õ¬ôò£†èœÜš«õ¬ô¬ò º¿õ¶ñ£è ªêŒ¶ º®‚è Ý°‹ èO¡ â‡E‚¬è¬ò‚ 裇è.

b˜¾ : x = 12 âQ™ y = 24

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96

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84

78

72

66

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42

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112

1. å¼ õƒA ͈®ñèQ¡ ¬õŠ¹ˆªî£¬è‚° 10% îQõ†® î¼Aø¶. ¬õŠ¹ˆªî£¬è‚°‹ ÜîŸ°æ˜ Ý‡´‚° A¬ì‚°‹ õ†®‚°‹ Þ¬ì«òò£ù ªî£ì˜H¬ù‚ 裆ì å¼ õ¬óðì‹ õ¬óè. Üî¡Íô‹ i) Ï. 650 ¬õŠ¹ˆªî£¬è‚° A¬ì‚°‹ õ†® ñŸÁ‹ii) Ï.45 õ†®ò£è‚ A¬ì‚è õƒAJ™ ªê½ˆîŠðì «õ‡®ò ¬õŠ¹ˆªî£¬è ÝAòùõŸ¬ø‚ 裇è.

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îQõ†® (y) 10 20 30 40 50 60 70 80

õ¬óðìˆFL¼‰¶ x = Ï. 650 âQ™ y = Ï. 65 y = Ï. 45 âQ™ x = Ï. 450

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113

2. å¼ «ð¼‰¶ ñE‚° 40 A.e «õèˆF™ ªê™Aø¶ ÞKò Éó è£ô ªî£ì˜H¬ù õ¬óðì‹ õ¬óè.Þ¬îŠðò¡ð´ˆF 3 ñE «ïóˆF™ ÞŠ«ð¼‰¶ ðòEˆî Éóˆ¬î‚ 致H®. (June 13, June 14)b˜¾:

è£ô‹ (ñEJ™) 1 2 3 4 5 6 7

¶ó‹ y (A.e) 40 80 120 160 200 240 280

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114

3. å¼ L†ì˜ ð£L¡ M¬ô Ï.15 â¡è. ð£L¡ Ü÷¾‚°‹ M¬ô‚°‹ àœ÷ˆ ªî£ì˜H¬ù‚ 裆´‹õ¬óðì‹ õ¬óè. Üî¬ùŠ ðò¡ð´ˆF

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M¬ô Ï. y 15 30 45 60 75 90 105

i) MAî êñ ñ£PL 15 ii) 3 L†ì˜ ð£L¡ M¬ô Ï.45

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115

4. xy = 20, x, y > 0 â¡ø õ¬óðì‹ õ¬óè. õ¬óðìˆFL¼‰¶ x = 5 âQ™ y ¡ ñFŠ¬ð»‹ y = 10 âQ™x¡ ñFŠ¬ð»‹ 裇è.

b˜¾:xy = 20‚è£ù ܆ìõ¬í

x 1 2 4 5 10 20

y 20 10 5 4 2 1

x = 5 âQ™ y = 4, y = 10 âQ™ x = 2

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