11

Þó‡´ ñFŠªð‡ Mù£‚èœÞó‡´ ñFŠªð‡ Mù£‚èœÞó‡´ ñFŠªð‡ Mù£‚èœÞó‡´ ñFŠªð‡ Mù£‚èœÞó‡´ ñFŠªð‡ Mù£‚èœ

1. èíƒèÀ‹ ꣘¹èÀ‹1. èíƒèÀ‹ ꣘¹èÀ‹1. èíƒèÀ‹ ꣘¹èÀ‹1. èíƒèÀ‹ ꣘¹èÀ‹1. èíƒèÀ‹ ꣘¹èÀ‹

1. A = {4, 6, 7, 8, 9}, B = {2, 4, 6}, C = {1, 2, 3, 4, 5, 6} âQ™ A (B C) 裇è.

b˜¾:

A = {4, 6, 7, 8, 9}, B = {2, 4, 6}, C = {1, 2, 3, 4, 5, 6}

BC = {2, 4, 6} {1, 2, 3, 4, 5, 6}

= {2, 4, 6}

∴ A (B C) = {4, 6, 7, 8, 9} {2, 4, 6}

= {2, 4, 6, 7, 8, 9}

2. A = {10, 15, 20, 25, 30, 35, 40, 45, 50}, B = {1, 5, 10, 15, 20, 30}, C = {7,8, 15, 20, 35, 45, 48}

âQ™ A\(BC) 裇è.

b˜¾:

(BC) = {1, 5, 10, 15, 20, 30} {7,8, 15, 20, 35, 45, 48}= {15, 20}

A\(BC) = {10, 15, 20, 25, 30, 35, 40, 45, 50} \ {15, 20}

= {10, 25, 30, 35, 40, 45, 50}

3. P = {a, b, c}, Q = {g, h, x, y} and R = {a, e, f, s} âQ™ R\(PQ) = ?

b˜¾:

PQ = {a, b, c} {g, h, x, y} = { }

R\(PQ) = {a, e, f, s} \ { } = {a, e, f, s}

4. = {4, 8, 12, 16, 20, 24, 28}, A = {8, 16, 24} and B = {4, 16, 20, 28} âQ™

)BA( ′ ñŸÁ‹ )BA( ′ 裇è.

b˜¾:

AB = {8, 16, 24} {4, 16, 20, 28}= {4, 8, 16, 20, 24, 28}

)BA( ′ = \ )BA( = { 4, 8, 12, 16, 20, 24, 28} \ {4, 8, 16, 20, 24, 28}

= {12}

BA = {8, 16, 24} {4, 16, 20, 28}

= {16}

∴ )BA( ′ = \ )BA( = {4, 8, 12, 16, 20, 24, 28} \ {16}= {4, 8, 12, 20, 24, 28}

5. A = {-10, 0, 1, 9, 2, 4, 5}, B = {-1, -2, 5, 6, 2, 3, 4} â¡ø èíƒèÀ‚° ªõ†´ ðKñ£ŸÁ ð‡¹ à¬ìòî£â¡ð¬î êKð£˜.b˜¾:

AB = BA

AB = {-10, 0, 1, 9, 2, 4, 5} {-1, -2, 5, 6, 2, 3, 4}= {2, 4, 5} ---- (1)

BA = {-1, -2, 5, 6, 2, 3, 4} {-10, 0, 1, 9, 2, 4, 5}= {2, 4, 5} ---- (2)

(1) = (2)

AB = BA

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12

6. A = {4, 6, 7, 8, 9}, B = {2, 4, 6}, C = {1, 2, 3, 4, 5, 6} A (BC) 裇è.

b˜¾:

BC = {2, 4, 6} {1, 2, 3, 4, 5, 6}= {1, 2, 3, 4, 5, 6}

A (BC) = {4, 6, 7, 8, 9} {1, 2, 3, 4, 5, 6}= {4, 6}

7. A = {l, m, n, o, 2, 3, 4, 7} , B = {2, 5, 3, -2, m,n, o, p} âQ™ èíƒèO™ ªõ†´, ðKñ£ŸÁŠð‡¹ à¬ìò¶â¡ð¬î êKð£˜.

b˜¾:

AB = BA

A B = {l, m, n, o, 2, 3, 4, 7} {2, 5, 3, -2, m,n, o, p}.

= {m,n, o, 2, 3} ----- (1)

BA = {2, 5, 3, -2, m,n, o, p} {l, m, n, o, 2, 3, 4, 7}

= {m, n, o, 2, 3} ----- (2) (1) = (2)

A B = BA.8. A = {5, 10, 15, 20}, B = {6, 10, 12, 18, 24} , C = {7, 10, 12, 14, 21, 28} âQ™ A\(B\C) = (A\B)\C âù

êKð£˜.b˜¾:

B\C = {6, 10, 12, 18, 24} \ {7, 10, 12, 14, 21, 28}

= {6, 18, 24}A\(B\C) = {5, 10, 15, 20} \ {6, 18, 24}

= {5, 10, 15, 20} ----- (1)A\B = {5, 10, 15, 20} \ {6, 10, 12, 18, 24}

= {5, 15, 20}

(A\B)\C = {5, 15, 20} \ {7, 10, 12, 14, 21, 28}= {5, 15, 20} ----- (2)

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

9. A ⊂ B âQ™ ªõ¡ðìƒè¬÷Š ðò¡ð´F AB , A\B 裇è,

b˜¾:

A\B = φ AB = A if A ⊂ B

10. (BC)\A ¡ ªõ¡ðì‹ õ¬óè.b˜¾:

BC (BC) \ A

BB

123456123456123456123456123456A A

AB

C

AB

C

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13

11. A ⊂ B âQ™ AB=B âù‚ 裆´è.b˜¾:

AB = AA â¡ð¶ B ¡ à†èí‹

12. X = {1, 2, 3, 4}, g = {(3,1), (4, 2), (2,1)} â¡ø àø¾ X L¼‰¶ X‚° å¼ ê£˜ð£°ñ£ âù Ýó£Œè. M¬ì‚°

ãŸø è£óí‹ î¼è.

b˜¾:

X L¼‰¶ X‚° ꣘ð£è£¶

è£óí‹:

g = {(3,1), (4, 2), (2,1)} . 1 â‹ X¡ àÁŠHŸ° Gö™ ༠ޙ¬ô. âù«õ g ꣘ð™ô.

13. X = {1, 2, 3, 4}, Y = {1, 3, 5, 7, 7, 9} â¡ð¶ X L¼‰¶ Y ‚è£ù àø¾ {(1,1), (1, 3), (3, 5), (3,7), (5, 7)}âù õ¬óòÁ‚èŠð†´œ÷¶. ꣘ð£°ñ£? ꣘ð™ô âQ™ Üî¡ è£óí‹ î¼è.

b˜¾: X Y

X →Y â¡ð¶ ꣘ð™ôñFŠðèˆF™ 2, 4 â¡ø â‡EŸ° Gö™ ༠ޙ¬ô.1, 3‚° 2 Gö™ ༠àœ÷¶. âù«õ Þ¶ ꣘ð£è£¶.

14. f = {(12, 2), (13, 3), (15, 3), (14, 2) (17, 17)} â¡ø ꣘H™ 2, 3 ÝAòõŸP¡ º¡ ༂è¬÷‚ 裇è.b˜¾ :

2¡ º¡ ༠= 12 ñŸÁ‹ 143¡ º¡ ༠= 13 ñŸÁ‹ 15

15. A = {1, 4, 9, 16} L¼‰¶ B = {-1, 2, -3, -4, 5, 6}‚° f = {(1, -1) (4, 2), (9,-3), (16, -4)} â¡ø àø¾ å¼ê£˜ð£°ñ£? ꣘¹ âQ™ i„êè‹ è£‡è.b˜¾:

f = {(1, -1), (4, 2), (9, -3) (16, -4)}A¡ 嚪õ£¼ àÁŠ¹‹ B ¡ å«ó å¼ àÁŠ¹ì¡ ªî£ì˜¹Šð´ˆîŠð†´œ÷¶. âù«õ ꣘¹ Ý°‹.f ¡ i„êè‹ = {-1, 2, -3, -4}

16. W«ö ªè£´‚èŠð†´œ÷ Ü‹¹‚°PŠðì‹ å¼ ê£˜H¬ù‚ °P‚Aøî£ âù Ýó£Œè.

b˜¾:«ñŸè‡ì Ü‹¹‚°PŠðìˆF™ A ¡ 嚪õ£¼ àÁŠHŸ°‹ å«ó å¼ Gö™ ༠àœ÷¶. âù«õ Þ¶

å¼ ê£˜ð£°‹.17. A = {1, 2, 3, 4, 5}, B = N ñŸÁ‹ f : A → B Ýù¶ f(x) = x2 âù õ¬óòÁ‚èŠð†´œ÷¶. âQ™ f ¡ i„êè‹

裇è. ꣘H¡ õ¬è¬ò‚ 裇.

b˜¾:

12345

13579

abcd

x

y

z

B 123456789123456789123456789123456789123456789123456789123456789123456789123456789123456789

B

A A

A f B

f

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A = {1, 2, 3, 4, 5}

B = {1, 2, 3, 4, ....}

f(x) = x2

f(1) = 12 = 1

f(2) = 22 = 4

f(3) = 32 = 9

f(4) = 42 = 16

f(5) = 52 = 25

f ¡ i„ê‹ = {1, 4, 9, 16, 25}ªõš«õø£ù àÁŠ¹èœ ªõš«õÁ Gö™ ༂è«÷£´ ªî£ì˜¹ð´ˆîŠð†´œ÷¶. âù«õ Þ¶ å¡Á‚°å¡ø£ù ꣘¹ Ý°‹.

18. A = {1, 3, 9, 16} L¼‰¶ B = {-1, 2, -3, -4, 5, 6} ‚° f = {(1, 2), (4, 5), (9,-4), (16, 5)} â¡ø àø¾ ꣘ð£°ñ£âù Ýó£Œè. ꣘¹ âQ™ i„êè‹ è£‡è.b˜¾:

f = {(1, 2), (4, 5), (9,-4), (16, 5)}A ¡ 嚪õ£¼ àÁŠ¹‹ B ¡ å«ó å¼ àÁŠ¹ì¡ ªî£ì˜¹Šð´ˆîŠð†´œ÷¶. âù«õ f å¼

꣘𣰋.f ¡ i„êè‹ = {2, 5, -4}

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

1.21

,41 −

, 1, -2, ..... ªð¼‚°ˆ ªî£ì˜ õK¬êJ™ 10õ¶ àÁŠ¬ð»‹, ªð£¶ MA 裇è.

a = 41

; r = 1

2tt

= 4121−

= 21− x 4 = - 2

tn = arn-1

t10 = 41

(- 2)10-1

= 41

(-2)9 = (-2)7

t10 = (-2)7

2. å¼ ªî£ì˜ õK¬êJ™ nõ¶ àÁŠ¹ 2n2 - 3n + 1 âQ™ ªî£ìK¡ 7õ¶ àÁŠ¬ð‚ 裇è.an = 2n2 - 3n + 1a7 = 2(7)2 - 3(7) + 1

= 2 x 49 - 21 + 1= 98 - 20

a7 = 783. 125, 120, 115, 110..... â¡ø ªî£ìK¡ 15õ¶ àÁŠ¬ð‚ 裇è.

a = 125; d = 120 - 125 = -5tn = a + (n-1)dt15 = 125 = (15-1) (-5)

= 125 + 14 (-5)= 125 - 70

t15 = 554. 4, 9, 14 ...... â¡ø Æ´ˆªî£ì˜ õK¬êJ¡ 17õ¶ àÁŠ¹ 裇.

a = 4; d = 9 - 4 = 5tn = a + (n - 1)dt17 = 4 + (17 - 1) (5)

= 4 + 16 (5)= 4 + 80

t17 = 84

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15

5.6

17...

23

,67

,65

,21

â¡ø Æ´ˆ ªî£ì˜ õK¬êJ¡ ºî™ àÁŠ¬ð»‹ ªð£¶ MˆFò£êˆ¬î»‹ 裇è.

a = 21

; d = 65

- 21

= 6

35 −=

62

= 31

ºî™ àÁŠ¹ a = 21

ªð£¶ MˆFò£ê‹ d = 31

6. Í¡Á â‡èO¡ MAî‹ 2 : 5 : 7 â¡è. ºîô£‹ ⇠Þó‡ì£‹ â‡EL¼‰¶ 7ä èNˆ¶Š ªðøŠð´‹â‡ ñŸÁ‹ Í¡ø£‹ ⇠ÝAòù å¼ Ã†´ˆªî£ì˜ õK¬ê¬ò ãŸð´ˆFù£™ Üšªõ‡è¬÷‚裇è.

Í¡Á â‡èœ = 2x, 5x, 7x â¡è 2x, 5x - 7, 7x å¼ A.P. âQ™

t2 - t1 = t3 - t2 (5x - 7) - 2x = 7x - (5x - 7)

3x - 7 = 2x + 7 3x - 2x = 7 + 7

x = 14 Üšªõ‡èœ = 2x, 5x, 7x.

= 2 x 14, 5 x 14, 7 x 14= 28, 70, 98

7. å¼ ªð¼‚°ˆ ªî£ì˜ õK¬êJ¡ ºî™ àÁŠ¹ 3 ñŸÁ‹ ä‰î£õ¶ àÁŠ¹ 1875 âQ™ Üî¡ ªð£¶MAî‹ è£‡è.

a = 3; tn = arn-1

t5 = 1875 (3) (r4) = 1875

r4 = 3

1875

= 625 r4 = 54

r = 5 r = 5

8. 1, 2, 4, 8..... â¡ø ªð¼‚°ˆ ªî£ì˜õK¬êJ™ 1024 âˆî¬ùò£õ¶ àÁŠ¹?a = 1; r = 2/1 = 2 ; tn = arn-1

tn = 1024 (1) (2)n-1 = 1024 2n x 2-1 = 1024

2n x 21

= 210

2n = 210 x 21

2n = 211

n = 119. a, b, c å¼ Ã†´ˆ ªî£ì˜ õK¬êJ™ Þ¼ŠH¡ (a - c)2 = 4 (b2 - ac) âù GÁ¾è.

a, b, c å¼ A.P. âQ™t2 - t1 = t3 - t2b - a = c - bb + b = c + a2b = c + a

Þ¼¹øº‹ õ˜‚è‹ â´‚è4b2 = (c+a)2

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16

4b2 = a2 + 2ac + c2

Þ¼¹øº‹ (- 4ac) ä Æ´è a2 + 2ac + c2 - 4ac = 4b2 - 4ac

a2 - 2ac + c2 = 4 (b2 - ac) (a - c)2 = 4 (b2 - ac)

10. å¼ Ã†´ˆ ªî£ìK™ Sn = 1275 ñŸÁ‹ ºî™ àÁŠ¹ a = 3 ªð£¶MˆFò£ê‹ d = 4 âQ™ n¡ ñFŠ¹è£‡è.

Sn = 1275

2n

[2a + (n-1)d] = 1275

2n

[2(3) + (n-1)4] = 1275

2n

[6 + 4n - 4] = 1275

2n

[2 + 4n] = 1275

2n

x 2 [1+2n] = 1275

n [1+2n] = 1275 2n2 + n - 1275 = 0(n - 25) (2n + 51) = 0n - 25 = 0 (or) 2n + 51 = 0n = 25 2n = -51

n = -51/2 (°¬ø ⇠A¬ìò£¶)

∴ n = 25

11. ⇠13 Ý™ õ°ð´‹ ßK‚è I¬è º¿ â‡èO¡ â‡E‚¬è¬ò‚ 裇è.

ßKô‚è I¬è º¿ â‡èœ: 11, 12, 13 .... 99

13 Ý™ õ°ð´‹ â‡èœ = 13, 26, .... 91

a = 13; d = 26 - 13 = 13; l = 91

n =

−d

a + 1

=

−13

1391 + 1

=

1378

+ 1 = 6 + 1

n = 712. å¼ Ì‰«î£†ìˆF™ ºî™ õK¬êJ™ 23 «ó£ü£„ ªê®èœ, Þó‡ì£‹ õK¬êJ™ 21 «ó£ü£„ ªê®èœ

Í¡ø£‹ õK¬êJ™ 19 «ó£ü£„ ªê®èœ â¡ø º¬øJ™ «ó£ü£„ ªê®èœ å¼ ªî£ì˜ õK¬ê ܬñŠH™àœ÷ù. è¬ìC õK¬êJ™ 5 «ó£ü£„ ªê®èœ Þ¼ŠH¡, ܊̉«î£†ìˆF™ âˆî¬ù õK¬êèœ àœ÷ù?

23, 21, 19, ...., 5a = 23; d = 21 - 23 = -2; l = 5

n =

−d

a + 1

=

−−

2235

+ 1

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17

=

−

−2

18 + 1

= 9 + 1 = 10܊̉«î£†ìˆF™ 10 õK¬êèO™ «ó£ü£„ ªê®èœ àœ÷ù.

13. 2010™ å¼õ˜ ݇´ áFò‹ Ï. 30000 âùŠ ðEJ™ «ê¼Aø£˜. «ñ½‹ 嚪õ£¼ õ¼ìº‹ Ï.600ä݇´ áFò àò˜õ£èŠ ªðÁAø£˜. Üõ¼¬ìò ݇´ áFò‹ â‰î õ¼ìˆF™ Ï.39000 ÝèÞ¼‚°‹?

30000, 30600, ..... , 39000 å¼ A.P.÷ 100 300, 306, .... 390 å¼ A.P.

a = 300; l = 390, d = 306 - 300 = 6

n =

−d

a + 1

=

−6

300390 + 1

=

6

90+1

= 15 + 1n = 16

16õ¶ ݇®™ áFò‹ Ï.39000 Ý°‹.݇´ áFò‹ Ï.39000 ä 2025‹ ݇´ ªðÁõ£˜.

14. 25,23,2 ..... â¡ø Æ´ˆ ªî£ì˜ õK¬êJ¡ 12õ¶ àÁŠ¹ ò£¶?

a = 2 , d = 223 − = 22 , n = 12 tn = a + (n - 1) d

t12= 2 + (12-1) 22

= 2 + 224 - 22

t12= 223

15. ....12518

,256

,52

â¡ø ªð¼‚°ˆ ªî£ì˜ õK¬êJ¡ ªð£¶ MA ñŸÁ‹ Üî¡ ªð£¶ àÁŠ¬ð»‹

裇è.

a = 52

; r = 52256

= 256

x 25

= 53

ªð£¶ MAî‹ r = 53

ªð£¶ àÁŠ¹ tn = arn-1

=

52 1n

53

−

, n = 1, 2, 3 .....

16. 0.02, 0.006, 0.0018..... â¡ø ªð¼‚°ˆ ªî£ì˜ õK¬êJ¡ ªð£¶ MA ñŸø‹ ªð£¶ àÁŠ¬ð»‹è£‡è.

a = 0.02, r = 02.0

006.0 = 0.3 =

103

ªð£¶ MAî‹ r = 103

, ªð£¶ àÁŠ¹ tn = arn-1

= (0.02) 1n

103

−

, n = 1, 2,3....

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18

17. ºî™ 125 Þò™ â‡èO¡ ôˆ 裇,

n = 2

)1n(n +

1 + 2 + ..... + 125 = 2126125×

= 125 x 63= 7875

18. ºî™ 75 I¬è º¿‚èO¡ Ã´î™ è£‡

n = 2

)1n(n +

1 + 2 + ..... + 75 = 2

7675×

= 75 x 38= 2850

19. 1 + 3 + 5 + .... , 25 àÁŠ¹èœ õ¬ó Ã´î™ è£‡.

1n2 − = n2

1 + 3 + 5 + .... , 25 àÁŠ¹èœ õ¬ó = 252

= 62520. 31 + 33 + ..... + 53 â¡ø ªî£ìK¡ Ã´î™ è£‡.

1n2 − = 2

21

+

31 + 33 + .... + 53 = (1 + 3 + .... + 53) - (1 + 3 + .... + 29)

= 22

2129

2153

+−

+

= 22

230

254

−

= 272 - 152

= (27 + 15) (27 - 15)= 42 x 12= 504.

21. 13 + 23 + 33 + .... + 203 â¡ø ªî£ìK¡ Ã´î™ è£‡.

3n = 2

2)1n(n

+

13 + 23 + 33 + ..... + 203 = 2

22120

×

= [10 x 21]2

= (210)2

= 4410022. 13 + 23 + 33 + .... + n3 = 36100 âQ™ 1 + 2 + 3 + .... + n ¡ ñFŠ¬ð‚ 裇.

3n = [ ]2n13 + 23 + 33 + ..... + n3 = 36100

3n = 36100

[ ]2n = 36100 [ 3n = [ ]2n ]

n = 36100

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19

= 1919× = 19

1 + 2 + .... + n = 1923. 2 + 4+ 6 + ....+100 â¡ø ªî£ìK¡ Ã´î™ è£‡è.

2 + 4 + 6 + .... + 100= 2 (1 + 2 + 3 + ..... +50)

= 2

×2

5150

+=2

)1n(nn

= 50 x 51= 2550

24. 7 + 14 + 21 + .... + 490 â¡ø ªî£ìK¡ Ã´î™ è£‡.7 + 14 + 21 + .... + 490 = 7 [1 + 2 + 3 + .... + 70]

= 7

×2

7170

+=2

)1n(nn

= 7 x 35 x 7= 17395

25. å¼ «î£†ì‚è£ó˜ êKõè õ®M™ ²õ˜ å¡P¬ù ܬñ‚è F†ìI´Aø£˜. êKõèˆF¡ c‡ì ºî™õK¬ê‚° 97 ªêƒèŸèœ «î¬õŠð´Aø¶. H¡¹ 嚪õ£¼ õK¬êJ¡ Þ¼¹øº‹ Þó‡®ó‡´ ªêƒèŸèœ°¬øõ£è ¬õ‚è «õ‡´‹. Üšõ®õ¬ñŠH™ 25 õK¬êèO¼ŠH¡, Üõ˜ õ£ƒè «õ‡®ò ªêƒèŸèO¡â‡E‚¬è âˆî¬ù?

97 + 93 + 89 + .... 25 àÁŠ¹èœa = 97; d = -4; n = 25

Sn = 2n

[ 2a + (n-1)d]

S25 = 2

25[2 (97) + (24) (-4)]

= 2

25(194 - 96)

= 2

25 x 98

= 12251225 ªêƒèŸèœ «î¬õŠð´Aø¶

26. å¼ è®è£ó‹ å¼ ñE‚° «î¬õŠð´Aø¶. 强¬ø 2 ñE‚° Þ¼º¬ø, Í¡Á ñE‚° Í¡Áº¬ø â¡øõ£Á ªî£ì˜‰¶ êKò£è 嚪õ£¼ ñE‚°‹ åL â¿‹¹‹ âQ™, å¼ ï£O™ Ü‚è®è£ó‹âˆî¬ù º¬ø åL â¿Š¹‹?å¼ ï£¬÷‚° è®è£ó‹ ñE Ü®‚°‹ º¬ø = 2 (1+2 +.... 12)

= 2

×21312

= 156

3. ÞòŸèEî‹3. ÞòŸèEî‹3. ÞòŸèEî‹3. ÞòŸèEî‹3. ÞòŸèEî‹

1. å¼ Þ¼ð® ð™½ÁŠ¹‚ «è£¬õJ¡ Ì„CòƒèO¡ Ã´î™ &4 ñŸÁ‹ Üî¡ ªð¼‚èŸðô¡ 3 âQ™,Ü‚«è£¬õ¬ò‚ 裇è.

α + β = -4, αβ = 3

Ü‚«è£¬õ P(x) = x2 - (α+β) x + αβ= x2 - (-4) x + 3

= x2 + 4x + 3

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20

2. x3 + x2 - 7x - 3 â¡ð¬î x - 3 Ý™ õ°‚°‹ «ð£¶ A¬ì‚°‹ ß¾ ñŸÁ‹ eF 裇è.

3 1 1 -7 -3

0 3 12 15

1 4 5 12

ß¾ = x2 + 4x + 5

eF = 12

3. x3 - 6x2 + 11x - 6 â¡ø ð™½ÁŠ¹‚ «è£¬õ‚° x - 1 å¼ è£óE âù GÁ¾è.

P(x)= x3 - 6x2 + 11x - 6

P(x) = (1)3 - 6 (1)2 + 11(1) - 6

= 1 - 6(1) + 11 - 6

= 1 - 6 + 11 - 6

= 12 - 12

= 0

∴(x-1) è£óE Ý°‹.

4. e.ªð£.õ. 裇è. x2y, x3y, x2 y2

e.ªð£.õ. = x2y

5. e.ªð£.ñ. 裇è ) a2bc, b2ca, c2ab, ii) am+1, am+2, am+3

i) e.ªð£.ñ.= a2 b2 c2

ii) e.ªð£.ñ.. = am+3

6. âOò õ®MŸ° ²¼‚°è : 287205

++

xx

287205

++

xx

= )4(7)4(5

++

xx

= 75

7. õ˜‚è Íô‹ 裇è.

i) 1412

864

SW64

zy81xii) 121(x - a)4 (x - b)6 (x - c)12

i) õ˜‚è Íô‹ = 76

432

SW8

zy9 x

ii) õ˜‚è Íô‹ = |11 (x - a)2 (x - b)3 (x - c)6 |

8. ÍôƒèO¡ ñ¬ò Ýó£Œè. i) x2 - 11x - 10 = 0 ii) 9x2+12x+4=0

i) ñ 裆® Δ = b2 - 4aca = 1, b = -11, c = -10 = (-11)2 - 4 (1) (-10)

= 121 + 40= 161

Δ > 0. Íôƒèœ ªñŒ, êññ™ô.

ii) ñ 裆® Δ = b2 - 4aca = 9, b = 12, c = 4

= (12)2 - 4(9) (4)= 144 - 144 = 0

Δ = 0 Íôƒèœ ªñŒ, êññ£ù¬õ

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21

9. Íôƒèœ ªñŒ ñŸÁ‹ êñ‹ âQ™ k ä‚ è£‡è. 2x2 - 10x + k = 0

Íôƒèœ ªñŒ, êñ‹ â¡ð b2 - 4ac = 0a = 2, b = -10, c = k

(-10)2 - 4 (2) (k) = 0100 - 8k = 0

- 8k = -100

k = 8

100

k = 2

25

10. 37 + ñŸÁ‹ 37 − ÝAòõŸ¬ø Íôƒè÷£‚ ªè£‡ì ޼𮄠êñ¡ð£´ 裇è.

Íôƒèœ 37 + ñŸÁ‹ 37 −

ÍôƒèO¡ Ã´î™ = 37 + + 37 − = 14

ÍôƒèO¡ ªð¼‚è÷™ = ( 37 + ) ( 37 − ) = 49 - 3 = 46

êñ¡ð£´x2 - (ÍôƒèO¡ ôî™) x + ÍôƒèO¡ ªð¼‚è™ = 0

x2 - 14 x + 46 = 0

4. ÜEèœ4. ÜEèœ4. ÜEèœ4. ÜEèœ4. ÜEèœ

1. A =

−− 129

073

526

841

âQ™ (i) ÜEJ¡ õK¬ê

ii) a13 , a42 àÁŠ¹ iii) 2 â¡ø àÁŠ¹ ܬñ‰¶œ÷ G¬ô ÝAòõŸ¬ø‚ 裇è.

b˜¾ :i) ÜEõK¬ê = 4 x 3ii) a13 àÁŠ¹ = 8

a42 àÁŠ¹ = -2iii) 2 â¡ø àÁŠ¹ ܬñ‰¶œ÷ G¬ô a22 = 2

2. aij = |2i - 3j| â¡ø àÁŠ¹è¬÷‚ ªè£‡ì õK¬ê 2 x 3 àœ÷ ÜE A = [aij] J¬ù ܬñ‚辋.b˜¾:

A =

232221

131211

aaa

aaa

a11 = |2 (1) - 3 (1)| = |2 - 3| = |-1| = 1a12 = |2 (1) - 3 (2)| = |2 - 6| = |-4| = 4a13 = |2 (1) - 3 (3)| = |2 − 9| = |-7| = 7a21 = |2 (2) - 3 (1)| = |4 - 3| = |1| = 1a22 = |2 (2) - 3 (2)| = |4 - 6| = |-2| = 2a23 = |2 (2) - 3 (3)| = |4 − 9| = |-5| = 5

A =

521

741

3. H¡õ¼õùõŸ¬ø‚ ªè£‡´ 2 x 2 õK¬ê ÜE 裇è.

i) aij = ij

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22

a11 = 1 x 1 = 1 a12 = 1 x 2 = 2

=∴

42

21A

a21 = 2 x 1 = 2 a22 = 2 x 2 = 4ii) aij = 2i - j

a11 = 2(1) - 1 = 2 - 1 = 1a12 = 2(1) - 2 = 2 - 2 = 0a21 = 2(2) - 1 = 4 - 1 = 3a22 = 2(2) - 2 = 4 - 2 = 2

A =

23

01

iii) aij = jiji

+−

a11 = 1111

+−

= 20

= 0 a12 = 2121

+−

= 31−

a21 = 1212

+−

= 31

a22 = 2222

+−

= 40

= 0

A =

−031

310

4. A =

− 431

258 âQ™ AT ñŸÁ‹ (AT)T .

b˜¾:

A =

− 431

258

AT =

−

4

3

1

2

5

8

(AT)T =

− 431

258

5. A =

−−

8

4

2

906

745

311

âQ™ i) ÜEJ¡ õK¬ê‚ 裇. ii) a24 ñŸÁ‹ a32 àÁŠ¹è¬÷ ⿶è.

iii) 7 àÁŠ¹ ܬñ‰¶œ÷ G¬ó ñŸÁ‹ Gó¬ô‚ 裇.

b˜¾ :

i) ÜE õK¬ê 3 x 4

ii) a24 = 4 a32 = 0

iii) àÁŠ¹ 7 ܬñ‰¶œ÷ G¬ô = 2 x 3

6. H¡õ¼‹ ÜEèO¡ õK¬êè¬÷‚ 裇.

i)

−

−432

511 ¡ ÜE õK¬ê = 2 x 3

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23

ii)

9

8

7

¡ ÜE õK¬ê = 3 x 1

iii)

−−

542

116

623

¡ ÜE õK¬ê = 3 x 3

iv) (3 4 5) ¡ ÜE õK¬ê = 1 x 3

v)

−

4

7

3

2

6

9

2

1

¡ ÜE õK¬ê = 4 x 2

7. 30 àÁŠ¹èœ ªè£‡ì ÜE‚° âšõ¬è õK¬êèœ Þ¼‚è Þò½‹?

b˜¾:

1 x 3030 x 12 x 1515 x 2 Gó™ 1 2 3 53 x 10 G¬ó 30 15 10 610 x 35 x 66 x 5

8. A=

0

1

3

5

4

1

âQ™ A J¡ G¬ó Gó™ ñ£ŸÁ ÜE¬ò‚ 裇è.

b˜¾: AT =

013

541

9. A =

−−653

542

321

âQ™ (AT)T = A êKð£˜.

b˜¾ :

AT =

−−653

542

321

(AT)T =

−−653

542

321

= A

(AT)T = A êK𣘂èŠð†ì¶

10.

195

45x =

1y5

z53 âQ™ x, y, z ñFŠ¹ 裇.

b˜¾: X = 3, Y = 9, Z = 4

11. A =

−

−563

421 âQ™ 3A ä‚ è£‡è.

3A = 3

−

−563

421

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24

=

−

−)5(3)6(3)3(3

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

=

−

−15189

1263

12. A =

−2401

3265, B =

−3282

7413 A + B 裇è.

b˜¾:

A + B =

−2401

3265 +

−3282

7413

=

++++++−−+32248021

73421635

A + B =

5683

10258

13. A =

− 59

32 −

−17

51 âQ™ A ¡ Ã†ì™ «ï˜ñ£Á ÜE¬ò‚ 裇è.

b˜¾:

A =

− 59

32 −

−17

51

=

−−−−

−−1)(579

5312 =

−

−616

21

A ¡ Ã†ì™ «ï˜ñ£Á =

−

−616

21

14. A =

15

23 B =

−34

18 âQ™ C = 2A + B 裇.

b˜¾:

C = 2

15

23 +

−34

18

=

210

46+

−34

18 =

++−+

32410

1486

C =

514

314

15. A =

−−

95

24 ñŸÁ‹ B =

−− 31

28 âQ™ 6A - 3B 裇.

b˜¾:

6A - 3B = 6

−−

95

24 −−−−− 3

−− 31

28

=

−×−×

××−

−××−××

3313

2383

9656

2646

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25

=

−−

5430

1224 +

−− 93

624

=

+−+−−−

954330

6122424

=

−−

4533

180

16. A =

− 69

31 âQ™ AI = IA = A â¡ð¬î„ êK𣘂è. Þƒ° I â¡ð¶ õK¬ê 2 ªè£‡ì Üô° ÜE

b˜¾ :

AI =

− 69

31

10

01

=

−+++

6009

3001

−

−

1

0)69(

0

1)69(

1

0)31(

0

1)31(

=

− 69

31

AΙ = A

IA =

10

01

− 69

31

=

−+++

6090

0301

−

−

6

3)10(

9

1)10(

6

3)01(

9

1)01(

=

− 69

31

ΙA = A∴ AI = IA = A .

17. H¡õ¼õùõŸPŸ° ÜE ªð¼‚è™ è£‡.

i) ( )

−

4

512 = (10 - 4) = (6)

ii)

−15

23

72

14

++−−

75220

143412

( ) ( )

( ) ( )

−

−

7

115

2

415

7

123

2

423

−1222

118

www.mathstimes.com

26

iii)

−

−014

392

−−

1

7

2

2

6

4

=

+−++−++−0780616

36346548 ( ) ( )

−

−−−

1

7

2

392

2

6

4

392

=

−122

6440

iv) ( )723

6−

− =

−

−216

4212

18. A =

−−

3

4

7

0

2

8

B =

−−

−516

239 âQ™ AB ñŸÁ‹ BA 裇.

b˜¾:

AB =

−−

3

4

7

0

2

8

−−

−516

239

=

−−+−−−+−++−−

15030180

204462418

35167244272

=

−−−

−

15318

2426

511730

BA =

−−

−516

239

−−

3

4

7

0

2

8

=

−−−+++−−++154420248

612630672

BA =

−−

6150

6978

5. Ýòˆªî£¬ô õ®Mò™5. Ýòˆªî£¬ô õ®Mò™5. Ýòˆªî£¬ô õ®Mò™5. Ýòˆªî£¬ô õ®Mò™5. Ýòˆªî£¬ô õ®Mò™

1. (3, 0) (-1, 4) ÝAò ¹œOè¬÷ ެ킰‹ «è£†´ˆ ¶‡®¡ ï´Š¹œO¬ò‚ 裇è.(x1, y1) (x2, y2) â¡ø ¹œOè¬÷ ެ킰‹ «è£†´ˆ ¶‡®¡ ï´Š¹œO

M(x,y) =

++2

yy,

22121 xx

(3, 0) (-1, 4) â¡ø ¹œOè¬÷ ެ킰‹ «è£†´ˆ¶‡®¡ ï´Š¹œO

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27

M(x,y) = M

+−2

40,

213

= M (1, 2)

2. A (-3, 5) ñŸÁ‹ B (4, -9) ÝAò ¹œOè¬÷ ެ킰‹ «è£†´ˆ¶‡¬ì P(-2, 3) â¡ø ¹œO à†¹øñ£è

â‰î MAîˆF™ HK‚°‹?

ªè£´‚èŠð†ì ¹œOèœ A (-3, 5) and B (4, -9).

P (-2, 3) â¡ø ¹œO AB ä l : m â¡ø MAîˆF™ à†¹øñ£è HK‚A¡ø¶.

HK¾„ ňFóŠð®

P

++

++

mmyy

,mm 1212

xx

= P (-2, 3)

x1 = -3, y

1 = 5, x

2 = 4, y

2 = -9

++−

+−

mm59

,m

m34

= (-2, 3)

x Ü„² ªî£¬ô¬õ Þ¼¹øº‹ êñŠð´ˆîm

m34+−

= -2

6l = m

m

= 61

l : m = 1 : 6

âù«õ P â¡ø ¹œO AB ä à†¹øñ£è 1 : 6 â¡ø MAîˆF™ HK‚A¡ø¶.3. A (4, -6) B (3, -2) ñŸÁ‹ C (5, 2) ÝAòõŸ¬ø à„Cè÷£è‚ ªè£‡ì º‚«è£íˆF¡ ï´‚«è£†´ ¬ñò‹

裇.(x

1, y

1) (x

2, y

2) ñŸÁ‹ (x

3, y

3) ÝAò ¹œOè¬÷ à„Cè÷£è‚ ªè£‡ì º‚«è£íˆF¡ ï´‚«è£†´

¬ñò‹ G (x, y) â¡è.

G (x, y) = G

++++3

yyy,

3321321 xxx

(4, -6) (3, -2) ñŸÁ‹ (5, 2) ÝAòõŸ¬ø à„Cè÷£è à¬ìò º‚«è£íˆF¡ ï´‚«è£†´ ¬ñò‹

G (x, y) = G

+−−++3

2263

534 = G (4, -2)

4. å¼ õ†ìˆF¡ ¬ñò‹ (-6, 4) Üšõ†ìˆF¡ å¼ M†ìˆF¡ å¼ º¬ù ÝFŠ¹œO âQ™ ñŸªø£¼º¬ù¬ò‚ 裇è.M†ìˆF¡ å¼ º¬ù ÝFŠ¹œO (0, 0), ñŸªø£¼ º¬ù (x, y) õ†ìˆF¡ ¬ñò‹ M†ìˆF¡ ¬ñòŠ¹œOÝ°‹.b˜¾: ¬ñòŠ¹œO = (&6, 4)

Ýè«õ

++2

y0,

20 x

= (-6, 4)

x, y Ü„²ˆ ªî£¬ô¾è¬÷ Þ¼¹øº‹ êñŠð´ˆî,  ªðÁõ¶

2x

= -6 x = - 12

2y

= 4 y = 8

âù«õ M†ìˆF¡ ñŸªø£¼ º¬ù (-12, 8)5. ¹œO (1, 3) ä ï´‚«è£†´ ¬ñòñ£è‚ ªè£‡ì º‚«è£íˆF¡ Þ¼ º¬ùèœ (-7, 6) ñŸÁ‹ (8, 5)

âQ™ º‚«è£íˆF¡ Í¡ø£õ¶ º¬ù¬ò‚ 裇 (Apr. 12)b˜¾ :

º‚«è£íˆF¡ à„Cèœ (-7, 6), (8, 5) ñŸÁ‹ ï´‚«è£†´ ¬ñò‹ (1, 3) âù ªè£´‚èŠð†´œ÷¶Í¡ø£õ¶ à„C (x, y)

(0,0) (x, y)(-6,4)

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28

Ýè«õ

++++−3

y56,

387 x

= (1, 3)

x, y Ü„²ˆ ªî£¬ô¾è¬÷ Þ¼¹øº‹

31+x

= 1 ñŸÁ‹311y +

= 3

x = 2 y = -2âù«õ º‚«è£íˆF¡ Í¡ø£õ¶ à„C (2, -2).

6. (7,3) (6,1) (8,2) ñŸÁ‹ (p, 4) â¡ðù æ˜ Þ¬íèóˆF¡ õK¬êŠð® ܬñ‰î à„Cèœ âQ™ p ¡ ñFŠ¹è£‡.b˜¾:

Þ¬íèóˆF¡ à„Cèœ A(7,3) B(6,1) C(8,2) ñŸÁ‹ D (p,4) æ˜ Þ¬íèóˆF¡ ͬô M†ìƒèœå¡¬øªò£¡Á Þ¼êñ‚ÃP´‹.

âù«õ

++2

23,

287

=

++2

41,

2p6

+25

,2

p6=

25

,2

15

x Ü„²ˆ ªî£¬ô¾è¬÷ Þ¼¹øº‹ êñŠð´ˆî

2

p6 +=

215

p = 97. (3, 4) ñŸÁ‹ (–6, 2) ÝAò ¹œOè¬÷ ެ킰‹ «è£†´ˆ ¶‡®¬ù ªõOŠ¹øñ£è 3 : 2 â¡ø

MAîˆF™ HK‚°‹ ¹œOJ¡ Ü„²ˆ ªî£¬ô¾è¬÷‚ 裇.

b˜¾ :

A (3, 4) ñŸÁ‹ B (-6, 2) ÝAò ¹œOèœ ªè£´‚èŠð†ì¬õ. AB ä 3:2 â¡ø MAîˆF™ ªõOŠ¹øñ£è

HK‚°‹ ¹œO P(x, y)

HK¾„ ňFóˆF¡ð® l = 3 x1 = 3 x

2 = -6

m = 2 y1 = 4 y

2 = 2

−−

−−

mmyy

,mm 1212

xx

= (x, y)

(x, y) =

−−−1

86,

1618

(x,y) = (-24, -2)

8. (-3, 5) ñŸÁ‹ (4, -9) ÝAò ¹œOè¬÷ ެ킰‹ «è£†´ˆ ¶‡®¬ù à†¹øñ£è 1:6 â¡ø MAîˆF™HK‚°‹ ¹œOJ¡ Ü„²ˆ ªî£¬ô¬õ 裇.

b˜¾:

A (-3, 5) ñŸÁ‹ B (4, -9) ªè£´‚èŠð†ì ¹œOèœ P(x, y) â¡ð¶ AB ä 1 : 6 â¡ø MAîˆF™à†¹øñ£è HK‚°‹ ¹œO

++

++

mmyy

,mm 1212

xx

= (x, y) x1 = 3 x

2 = 4 l = 1, m = 6

(x, y)=

+

×+−+

−+×61

)56()9(1,

61)3(6)41(

y1 = 5 y

2 = -9

(x,y) =

−721

,714

= (-2, 3)

A (7,3) B (6,1)

C (8, 2)D (p, 4)

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29

9. A (6,7), B (-4, 1) ñŸÁ‹ C (a, -9) ÝAòõŸ¬ø º¬ùè÷£è‚ ªè£‡ì ΔABC ¡ ðóŠ¹ 68ê.Üô°èœâQ™ ‘a’¡ ñFŠ¬ð‚ 裇è.

ΔABC ¡ ðóŠ¹ =21

7917

6a46

−−

Δ = 21

[(6+36+7a) - (-28 + a - 54) ]= 68

(42+7a) - (a - 82) = 136

6a = 12

a = 2

10. A (2, 3), B(4,0) ñŸÁ‹ C(6, -3) ÝAò ¹œOèœ å«ó «ï˜‚«è£†®™ ܬñ‰¶œ÷ù âù GÏH.

ΔABC¡ ðóŠ¹ = 3303

2642

21

−

Δ = 21

[ (0 - 12 + 18) - (12 + 0 - 6)]

= 21

[6 - 6]

= 0

âù«õ, ªè£´‚èŠð†ì ¹œOèœ å«ó «ï˜‚«è£†®™ ܬñ‰¶œ÷ù.

11. «ï˜‚«è£†®¡ ꣌¾ 3

1âQ™, Ü‚«è£†®¡ ꣌¾‚«è£í‹ 裇.

θ â¡ð¶ «ï˜‚«è£†®¡ ꣌¾‚«è£íªñQ™ Þî¡ ê£Œ¾ m = tanθ.0o ≤ θ ≤ 180o θ ≠ 90o

tanθ = 3

1 θ = 30o

12. «ï˜‚«è£†®¡ ꣌¾ «è£í‹ 45o âQ™, Ü‚«è£†®¡ ꣌¬õ‚ 裇.

꣌¾‚«è£í‹ θ âQ™ «ï˜‚«è£†®¡ ꣌¾ m = tanθ âù«õ m = tan45o

m = 1

13. ê¶ó‹ ABCD¡ ð‚è‹ AB Ýù¶ x-Ü„²‚° Þ¬íò£è àœ÷¶ âQ™ (i) AB¡ ꣌¾ ii) BC¡ ꣌¾

iii) ͬôM†ì‹ AC¡ ꣌¾

i) ð‚è‹ ABÝù¶ x-Ü„²‚° Þ¬í â¡ð AB¡ ꣌¾ m = 0

ii) BC ⊥ AB â¡ð BC, x Ü„²ì¡ ãŸð´ˆ¶‹ «è£í‹ θ = 90o

ꣻòó‹ m = 90o õ¬óòÁ‚èŠðì£î¶

ii) ͬôM†ì‹ ACÝù¶ ∠ DABä Þ¼êñ‚ÃP´‹ âù«õ ∠ BAC = 45o

θ = 45o

Ýè«õ ͬôM†ì‹ AC¡ ꣌¾ m = tanθ = tan45o = 1

14. êñð‚è º‚«è£í‹ ABC¡ ð‚è‹ BCÝù¶ x-Ü„CŸ° Þ¬í âQ™ AB ñŸÁ‹ BC ÝAòõŸP¡ê£Œ¾è¬÷‚ 裇.

êñð‚è ΔABC ™ ð‚è‹ BC Ýù¶ x-Ü„²‚° Þ¬í.

«ñ½‹ ∠ ABC = 60o. âù«õ ð‚è‹ AB¡ ꣌¾ m = tan60o = 3

«ñ½‹ BC Ýù¶ x Ü„²‚° Þ¬í â¡ð

BC¡ ꣌¾ m tan60o = 3

y

x

A

C

B

D

PO

45o

y

x

C

A

B60o

60o60o

60o

O P

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30

15. (a, 1) (1, 2) ñŸÁ‹ (0, b+1) â¡ðù å«ó «ï˜‚«è£†®™ ܬñ»‹ ¹œOèœ âQ™ b1

a1 + = 1 âù GÁ¾è.

A (a, 1) B (1, 2) ñŸÁ‹ C (0, b+1) â¡ðù ªè£´‚èŠð†ì ¹œOèœ

AB ¡ ꣌¾ m1 =

a112

−−

= 10

21b−

−+

BC ¡ ꣌¾ m2 =

a11− =

11b

−−

Í¡Á ¹œOèÀ‹ å«ó «ï˜‚«è£†®™ ܬñ»‹ â¡ð m1 = m

2

(1-a) (b-1) = -1(a - 1) (b -1) = 1ab - a - b + 1 = 1ab - a - b = 0ab = a + b

abb

aba + = 1

Þ¼¹øº‹ ab™ õ°‚èb1

a1 + = 1

16. (3, -4) â¡ø ¹œO õN„ ªê™½‹ ñŸÁ‹ Ýò Ü„²èÀ‚° Þ¬íò£è ܬñ‰î «ï˜‚«è£´èO¡êñ¡ð£´è¬÷‚ 裇.

(3, -4) â¡ø ¹œO õN„ ªê¡Á x Ü„²‚° Þ¬íò£è¾‹ àœ÷

«ï˜‚«è£´èœ l ñŸÁ‹ l’

l ¡ â™ô£Š ¹œOèO¡ y Ü„²ˆ ªî£¬ô¾‹ -4 Ý°‹.

âù«õ, «ï˜‚«è£´ì¡ êñ¡ð£´ y = -4

l’ e¶œ÷ â™ô£Š¹œOèœ x Ü„²ˆªî£¬ô¾‹ 3 Ý°‹.

âù«õ l’ ¡ êñ¡ð£´ x = 3.

17. x Ü„CL¼‰¶ 5 Üô°èœ ªî£¬ôM™ àœ÷¶‹ x Ü„²‚° Þ¬íò£ù¶ñ£ù «ï˜‚«è£´èO¡

êñ¡ð£´è¬÷‚ 裇.

x Ü„²‚° Þ¬íò£ù «ï˜‚«è£†®¡ êñ¡ð£´ y = k

x Ü„²‚° Þ¬íò£è¾‹ x Ü„CL¼‰¶ 5 Üô°èœ ÉóˆF™

ܬñ‰¶œ÷¶ñ£ù «ï˜‚«è£†®¡ êñ¡ð£´èœ

y = 5 , y = -5

y - 5 = 0 , y + 5 = 0

18. (-5,-2) â¡ø ¹œO õN„ ªê™õ¶‹ Ýò Ý„²èÀ‚° Þ¬íò£ù¶ñ£ù «ï˜‚«è£´èO¡ êñ¡ð£´‚¬÷‚

裇.

x Ü„²‚° Þ¬íò£è¾‹ (-5, -2) â¡ø ¹œO õN„ ªê™õ¶ñ£ù

«ï˜‚«è£†®¡ êñ¡ð£´ y = -2

y Ü„²‚° Þ¬íò£è¾‹ (-5, -2) â¡ø ¹œO õN„ ªê™õ¶ñ£ù

«ï˜‚«è£†®¡ êñ¡ð£´ x = -5

l’

l

x

y

O

(3,-4)

y =-4

x = 3

x

y

y =-5

x = 5

x

(-5,-2)

y

O

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31

19. å¼ «ï˜‚«è£´ y Ü„¬ê ÝFŠ¹œO‚° «ñô£è 3 Üô°èœ ÉóˆF™ ªõ†´Aø¶ ñŸÁ‹ tanθ = 1/2âQ™ Ü‰î «ï˜‚«è£†®¡ êñ¡ð£†¬ì 裇.

꣌¾ m = tanθ =21

y ªõ†´ˆ¶‡´ c = 3

꣌¾ ªõ†´ˆ¶‡´ ܬñŠH™ «ï˜‚«è£†®¡ êñ¡ð£´ y = mx + c

Ýè«õ, «î¬õò£ù «ï˜‚«è£†®¡ êñ¡ð£´ y = 21

x + 3

2y = x + 6 x - 2y + 6 = 0

20. P (1, -3) Q (-2, 5) ñŸÁ‹ R (-3, 4) ÝAò º¬ùè¬÷‚ ªè£‡ì .ΔPQR ™ º¬ù R L¼‰¶ õ¬óòŠð´‹ï´‚«è£†®¡ êñ¡ð£†¬ì‚ 裇.

P (1, -3) Q (-2, 5) ñŸÁ‹ R (-3, 4) ÝAò¬õ ΔPQR ¡ º¬ùèœ M â¡ð¶ PQ ¡ ï´Š¹œO â¡è.

âù«õ M =

+−−2

53,

221

=

−1,

21

R (-3, 4) ñŸÁ‹

−1,

21

ÝAò ¹œOè¬÷ ެ킰‹

ï´‚«è£´ RM ¡ êñ¡ð£´

414y

−−

= 3213+−

−x 3

4y−−

= 5

)3(2 +x

6x + 5y - 2 = 021. (3, 4) â¡ø ¹œO õN„ªê™õ¶‹, ªõ†´ˆ¶‡´èO¡ MAî‹ 3 : 2 âù àœ÷¶ñ£ù «ï˜‚«è£†®¡

êñ¡ð£†¬ì‚ 裇.a, b â¡ðù º¬ø«ò «ï˜‚«è£†®¡ x ñŸÁ‹ y ¡ ªõ†´ˆ¶‡´èœ.Ýè«õ a : b = 3 : 2 «ñ½‹ a = 3k ñŸÁ‹ b = 2kªõ†´ˆ¶‡´ ܬñŠHô£ù «ï˜‚«è£†´ êñ¡ð£´

k3x

+ k2y

= 1 ---- (I)

Þ‚«è£´ (3, 4) â¡ø ¹œO õN„ªê™õ k24

k33 + = 1

k2

k1 + = 1 k = 3

k = 3 âù I ™ HóFJì

9x

+ 6y

= 2x + 3y -18 = 0

22. 3x + 2y - 12 = 0, 6x + 4y + 8 = 0 ÝAò «ï˜‚«è£´èœ Þ¬í âù GÁ¾è.

3x + 2y - 12 = 0 ¡ ꣌¾ m1 =

23−

Þšõ£Á 6x + 4y + 8 = 0 ¡ ꣌¾ m2 =

46−

= 23−

m1 = m

2 Ýè«õ, ÞšM¼ «ï˜‚«è£´èœ Þ¬íò£°‹.

23. x + 2y + 1 = 0 , 2x - y + 5 = 0 ÝAò «ï˜‚«è£´èœ å¡Á‚° å¡Á ªêƒ°ˆî£ù¬õ âù GÁ¾è.

x + 2y + 1 = 0 ¡ ꣌¾ m1 =

21−

2x - y + 5 = 0 ¡ ꣌¾ m2 =

12

−−

= 2

âù«õ ꣌¾èO¡ ªð¼‚èŸðô¡ m1 x m

2 =

21−

x 2 = -1

P (1,-3)

R (-3,4)Q (-2,5)

S

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32

Ýè«õ ÞšM¼ «ï˜‚«è£´èœ å¡Á‚ªè£¡Á ªêƒ°ˆî£ù¬õ24. ΔABC ¡ º¬ùèœ A (2, 1) B (6, -1) C (4, 11) â¡è. AJL¼‰¶ õ¬óòŠð´‹ °ˆ¶‚«è£†®¡ êñ¡ð£†¬ì‚

裇.

BC¡ ꣌¾ = 64111

−+

= 2

12−

= - 6

AD â¡ð¶ BC‚°„ ªêƒ°ˆ¶. âù«õ AD¡ ꣌¾ 61

AD ¡ êñ¡ð£´ y - y1 = m (x - x

1)

y - 1 = 61

(x - 2)

6y - 6 = x - 2

«î¬õò£ù «ï˜‚«è£†®¡ êñ¡ð£´ x - 6y + 4 = 0 Ý°‹.

25. 3x - y + 7 = 0 â¡ø «ï˜‚«è£†®Ÿ° Þ¬íò£ù¶ (1, -2) â¡ø ¹œO õN„ ªê™õ¶ñ£ù «ï˜‚«è£†®¡êñ¡ð£†¬ì‚ 裇..

3x - y + 7 = 0 â¡ø «ï˜‚«è£†®Ÿ° Þ¬íò£è ªê™½‹ «ï˜‚«è£†®¡ êñ¡ð£´ 3x - y + k = 0Þ‚«è£´ (1, -2) õN„ªê™õ 3(1) + 2 + k = 0 k = -5 Ýè«õ «î¬õò£ù «ï˜‚«è£†®¡ êñ¡ð£´3x - y - 5 = 0

26. ꣌¾ «è£í‹ 45o ñŸÁ‹ y ªõ†´ˆ¶‡´ 2/5 ÝAòõŸ¬ø‚ «ï˜‚«è£†®¡ êñ¡ð£†¬ì‚ ÃÁè.

«î¬õò£ù «ï˜‚«è£†®¡ ꣌¾ m = tan45o = 1

y ªõ†´ˆ¶‡´ C = 52

꣌¾ ªõ†´ˆ¶‡´ ܬñŠ¬ð‚ ªè£‡ì «ï˜‚«è£†®¡ êñ¡ð£´ y = mx+c

y = x +52 y =

52x5 +

âù«õ «î¬õò£ù «ï˜‚«è£†®¡ êñ¡ð£´ 5x - 5y + 2 = 0

27. ꣌¾‚«è£í‹ 30o ªè£‡ì (4, 2) , (3, 1) ÝAò ¹œOè¬÷ ެ킰‹ «ï˜‚«è£†´ˆ¶‡®¡ ï´Š¹œOõN„ ªê™½‹ «è£†®¡ êñ¡ð£†¬ì‚ 裇.

(4, 2) ñŸÁ‹ (3, 1) ÝAò ¹œOèO¡ ï´Š¹œO =

23

,27

꣌¾ m = tan30o = 3

1

꣌¾ ¹œO õ®õˆF¡ 𮠫裆®¡ êñ¡ð£´ y - y1 = m (x - x

1)

y - 23

= 3

1

−27

x (2y - 3) 3 (2x - 7)

23y2 −

= 3

1

−2

72x2 3 y - 3 3 = 2x - 7

(2y - 3) 3

1 = (2x - 7) âù«õ 2x - 2 3 y + (3 3 -7) = 0

A (2,1)

C(4,11)B(6,-1)D

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33

6. õ®Mò™õ®Mò™õ®Mò™õ®Mò™õ®Mò™

1. ΔABC™ DE||BC ñŸÁ‹ DBAD

= 32

. AE = 3.7 ªê.e âQ™ EC¬ò‚ 裇è. (June 12, June 14)

ΔABC , DE || BC

âù«õ DBAD

= ECAE

(«î™v «îŸø‹)

32

= EC

7.3

2 x EC = 3 x 3.7

EC = 2

37.3 × = 5.55 ªê.e

EC = 5.55 ªê.e

2. D ñŸÁ‹ E ÝAò ¹œOèœ º¬ø«ò ΔABC ¡ ð‚èƒèœ AB ñŸÁ‹ AC èO™ DE || BC â¡P¼‚°ñ£Á

ܬñ‰¶œ÷ù, AD = 6ªê.e, DB = 9 ªê.e ñŸÁ‹ AE = 8ªê.e âQ™ AC ä‚ è£‡è.

ΔABC ™ DE || BC.

âù«õ DBAD

= ECAE

96

= EC8

6 x EC = 8 x 9

EC = 6

98×

EC = 12 ªê.eAC = AE + EC

= 8 + 12 = 20

= 20 ªê.e

3. D, E ÝAò ¹œOèœ º¬ø«ò ΔABC ¡ ð‚èƒèœ AB ñŸÁ‹ AC èO™ DE || BC â¡P¼‚°ñ£Á

ܬñ‰¶œ÷ù. AD = 8 ªê.e, AB = 12 ªê.e ñŸÁ‹ AE = 12 ªê.e âQ™ CE ä 裇è.

ΔABC ™ DE || BC.

âù«õ DBAD

= ECAE

48

= EC

21

8 EC = 4 x 12

EC = 8124×

EC = 6ªê.e

4. ΔABC ™ ∠ A â¡ø «è£íˆF¡ à†¹ø Þ¼êñ ªõ†® AD Ýù¶ ð‚è‹ BC ¬ò D ™ ê‰F‚Aø¶.

BD = 2.5 ªê.e. AB = 5 ªê.e ñŸÁ‹ AC = 4.2 ªê.e âQ™ DC ä‚ è£‡è. (Ap. 12, Oct. 12, 13)

ΔABC ™ AD ò£ù¶ ∠ A ¡ à†¹øñ£è Þ¼êñªõ†®

A

D

B C

E3.7

ªê.e

ªê.e

ªê.e

ªê.e

ªê.e

?

A

D

B C

E

8

?9

6

A

D

B C

E

12

?4ªê.eªê.eªê.eªê.eªê.e

8ªê.eªê.eªê.eªê.eªê.e

12 ª

ê.e

ªê.e

ªê.e

ªê.e

ªê.e

www.mathstimes.com

34

DCBD

ACAB = («è£í Þ¼êñ ªõ†® «îŸø‹)

2.45

= DC

5.2

DC x 5 = 2.5 x 4.2

DC = 5

2.45.2 × = 2.1 ªê.e

DC = 2.1 ªê.e5. AD â¡ð¶ ΔABC ™ ∠ A ¡ à†¹Ÿ «è£í Þ¼êñ ªõ†®. ܶ BC ä D™ ê‰F‚Aø¶. BD = 2ªê.e

AB = 5 ªê.e, DC = 3 ªê.e âQ™ AC 裇è.

∠ A¡ à†¹ø Þ¼êñªõ†® AD

âù«õACAB

= DCBD

AC5

= 32

2 x AC = 3 x 5

AC = 2

5x3= 7.5 ªê.e

AC = 7.5 ªê.e

6. AD â¡ð¶ ΔABC ™ ∠ A ¡ à†¹ø Þ¼êñªõ†®. ܶ BC ¬ò D™ ê‰F‚Aø¶. AB = 5.6 ªê.e,

AC = 6 ªê.e ñŸÁ‹ DC = 3 ªê.e âQ™ BC 裇è.

∠ A ¡ à†¹øñ£è Þ¼êñªõ†® AD.

âù«õDCBD

=ACAB

3BD

= 66.5

6 x BD = 5.6 x 3

BD = 6

36.5 × = 2.8ªê.e

BD = 2.8 ªê.e

BC = BD + DC

BC = 2.8 + 3 ªê.e = 5.8 ªê.e

BC = 5.8 ªê.e

7. MP â¡ð¶ ΔMNO ™ ∠ M ¡ ªõOŠ¹ø Þ¼êñªõ†® «ñ½‹ Þ¶ NO ¡ c†CJ¬ù P ™ ê‰F‚Aø¶.

MN = 10 ªê.e, MO = 6ªê.e, NO = 12ªê.e âQ™ OP 裇è. (July 13, Oct. 14)∠ M ¡ ªõOŠ¹øñ£è Þ¼êñªõ†® MP Ý°‹.

âù«õ MOMN

= OPNP

OP = x â¡è.PN = PO + ON = x + 12

âù«õ x

x 12+=

610

6 (12+x) = 10 x x6x + 72 = 10x

A

B C

5ªê.e

ªê.e

ªê.e

ªê.e

ªê.e ?

3ªê.eªê.eªê.eªê.eªê.e2ªê.eªê.eªê.eªê.eªê.e D

A

B C

5ªê.e

ªê.e

ªê.e

ªê.e

ªê.e

?2.5ªê.eªê.eªê.eªê.eªê.e D

4.2 ªê.e

ªê.e

ªê.e

ªê.e

ªê.e

A

B C

5.6ª

ê.e

ªê.e

ªê.e

ªê.e

ªê.e 5

ªê.e

ªê.e

ªê.e

ªê.e

ªê.e

3ªê.eªê.eªê.eªê.eªê.e? D

Px ªê.eªê.eªê.eªê.eªê.eO12ªê.eªê.eªê.eªê.eªê.eN

10ªê.e

ªê.e

ªê.e

ªê.e

ªê.e 6

ªê.e

ªê.e

ªê.e

ªê.e

ªê.e

MQ

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35

72 = 10x - 6x72 = 4x4x = 72

x = 4

72 = 18

OP = 18 ªê.e8. W›‚裵‹ ðìˆF™ x ¡ ñFŠ¬ð‚ 裇è.

AB, CD õ†ì èœ

PA x PB = PC x PD

4 x x = 3 x 8

x = 4

83×

x = 6

x = 6

9. å¼ õ†ìˆF™ AB,CD â¡Â‹ Þ¼ ï£‡èœ å¡¬øªò£¡Á à†¹øñ£è P™ ªõ†®‚ªè£œA¡øù.

CP = 4ªê.e, AP = 8 ªê.e, PB = 2 ªê.e âQ™ PD 裇è. (Apr. 14)AB , CD õ†ì èœ

P™ à†¹øñ£è ªõ†®‚ ªè£œA¡øù.

AP x PB = CP x PD. 8 x 2 = 4 x PD

4 x PD = 8 x 2

PD = 4

28× = 4

PD = 4ªê.e

10. å¼ õ†ìˆF™ AB, CD â¡Â‹ Þ¼ ï£‡èœ å¡¬øªò£¡Á à†¹øñ£è PJ™ ªõ†®‚ ªè£œA¡øù.

AP = 12 ªê.e, AB = 15ªê.e, CP = PD âQ™ CD ä‚ è£‡è.

AP + PB = 15 ªê.e

12 + PB = 15 ªê.e

PB = 15 - 12

= 3

CP = PD

AB, CD ÝAò ï£‡èœ P ™ ªõ†®‚ªè£œA¡øù.PA x PB = PC x PD12 x 3 = PC x PC [PC = PD]

36 = PC2

PC2 = 36

PC = 36 = 6

CD = PC + PD = 6 + 6 = 12 ªê.eCD = 12 ªê.e

11. W›‚裵‹ ðìˆF™ x ¡ ñFŠ¹ 裇è.

P

A D

BC3

x4

8

3ªê.e

ªê.e

ªê.e

ªê.e

ªê.e

P

A D

BC

4 2

8?

P

A D

BC

12

15 ªê.eªê.eªê.eªê.eªê.e

A

Dx

B

C

P2

45

www.mathstimes.com

36

AB , CD ÝAò ï£‡èœ ªõO«ò ªõ†®‚ªè£œA¡øù,PA x PB = PC x PD9 x 4 = (2 + x) x 2

(2 + x) x 2 = 9 x 4

2 + x = 2

49×

2 + x = 18

x = 18 - 2 = 16

x = 16

12. AB , CD â¡ø Þ¼ ï£‡èœ õ†ìˆFŸ° ªõO«ò P â¡ø ¹œOJ™ ªõ†®‚ ªè£œA¡øù. AB = 4ªê.e., BP = 5 ªê.e ñŸÁ‹ PD = 3 ªê.e âQ™ CD ¬ò‚ 裇è.

CD = x ªê.e â¡è.

PA x PB = PC x PD

(4 + 5) x 5 = (x + 3) x 3

9 x 5 = 3x + 9

3x + 9 = 45

3x = 45 - 9

3x = 36

x = 3

36 = 12

CD = 12 ªê.e

13. AB , CD â¡ø Þ¼ ï£‡èœ õ†ìˆFŸ° ªõO«ò P â¡øŠ ¹œOJ™ ªõ†®‚ ªè£œA¡øù. BP = 3ªê.e, CP = 6 ªê.e ñŸÁ‹ CD = 2 ªê.e âQ™ AB 裇è.

Let AB = x cm

PA x PB = PC x PD

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

3x x 9= 6 x 4

3x + 9 = 24

3x = 24 - 9

3x = 15

x = 3

15 = 5

AB = 5 ªê.e

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

1. 200 e c÷ºœ÷ ËLù£™ å¼ è£Ÿø£® è†ìŠð†´ ðø‰¶ ªè£‡®¼‚Aø¶. ܉î Ë™ î¬ó ñ†ìˆ¶ì¡

30o «è£í‹ ãŸð´ˆFù£™ 裟ø£® î¬óñ†ìˆFL¼‰¶ âšõ÷¾ àòóˆF™ ðø‚Aø¶ âù‚ 裇è,BC = àòó‹ = x e â¡èAC = Ë™ = 200 eθ = 30o

ΔABC J™âF˜ð‚è‹

sinθ = ------------------ è˜í‹

A

D6

B

C

P4

3?

2

A

D?

B

C

P3

54

www.mathstimes.com

37

sin30o = 200

x

21

= 200

x

2 x x = 1 x 200

x = 2

200 = 100

x = 100 e

î¬óñ†ìˆFL¼‰¶ àòó‹ = 100e

2. ²õK™ ꣌ˆ¶ ¬õ‚èŠð†ì å¼ ãEò£ù¶ î¬ó»ì¡ 60o «è£íˆ¬î ãŸð´ˆ¶A¡ø¶. ãEJ¡ Ü®²õŸPL¼‰¶ 3.5 e ÉóˆF™ àœ÷¶ âQ™ ãEJ¡ c÷‹ 裇è. (Oct 12, Apr. 13, June 15)

AC = ãEJ¡ c÷‹ = x e â‡è,

AB = 3. 5 e

∠ BAC = 60o

Ü´. ð‚è‹cos60o = ----------------

è˜í‹

cos60o = x5.3

21

=x5.3

1x x = 2 x 3.5 x = 7

ãEJ¡ c÷‹ = 7e.

3. 30 e c÷ºœ÷ å¼ è‹ðˆF¡ GöL¡ c÷‹ 10 3 e âQ™ ÅKòQ¡ ãŸø‚«è£í Ü÷¬õ‚ 裇è.

(Mar. 12, Mar. 14)BC = è‹ðˆF¡ c÷‹ = 30 e

AB = GöL¡ c÷‹ = 10 3 m, θ = ?

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

Ü´ˆ¶œ÷ð‚è‹

tan θ = 310

30

= 3

3 =

3

3.3

tan θ = 3

tan60 = 3 âù«õ θ = 60o

ÅKòQ¡ ãŸø‚ «è£í‹ = 60o.

4. å¼ «è£¹óˆF¡ Ü®JL¼‰¶ 30 3 e ªî£¬ôM™ GŸ°‹ å¼ ð£˜¬õò£÷˜ Ü‚«è£¹óˆF¡ à„CJ¬ù

30o ãŸø‚«è£íˆF™ 裇Aø£˜. î¬óñ†ìˆFL¼‰¶ ÜõϬìò A¬ìG¬ôŠ 𣘬õ‚«è£†®Ÿ° àœ÷Éó‹ 1.5 e âQ™ «è£¹óˆF¡ àòó‹ 裇è.

AD = «è£¹óˆF¡ àò˜ = x + 1.5 e

BC = DE = 30 3 e

BA

C

Ë™ 2

00eeeee

30o

?

BA

C

ãE x

ªê.e

ªê.e

ªê.e

ªê.e

ªê.e

60o 3.5eeeee

î¬ó²õ˜

BA

C

10 3 eeeee

θθθθθ

30eeee e

www.mathstimes.com

38

ΔABC ™ ∠ ABC = 30o

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

Ü´ˆ¶œ÷ ð‚è‹

tan θ = 330

x

3

1 = 330

x

x 3 = 30 3

x = 3

330= 30

x = 30 e«è£¹óˆF¡ àòó‹ = x + 1.5 e

= 30 + 1.5 = 31. 5 e5. å¼ ²¬ñ á˜FJL¼‰¶ ²¬ñ¬ò Þø‚è ã¶õ£è 30o ãŸø‚ «è£íˆF™ å¼ ê£Œ¾ î÷‹ àœ÷¶.

꣌¾î÷ˆF¡ à„C î¬óJL¼‰¶ 0.9e àòóˆF™ àœ÷¶ âQ™ ꣌¾î÷ˆF¡ c÷‹ â¡ù? (Oct. 14,Mar. 15)

AC = ꣌¾î÷ˆF¡ c÷‹ = x e â¡è.BC = 0.9e∠ CAB = 30o

âF˜ð‚è‹sinθ = -------------------

è˜í‹

sin30o = x9.0

21

= x9.0

1 x x = 0.9 x 2x = 1.8 e

꣌¾ î÷ˆF¡ c÷‹ = 1.8 e.

6. àòó‹ 150 ªê.e àœ÷ å¼ CÁI å¼ M÷‚°‚ è‹ðˆF¡ º¡ G¡øõ£Á 150 3 ªê.e c÷ºœ÷ Gö¬ô

ãŸð´ˆ¶Aø£œ âQ™ M÷‚°‚ è‹ðˆF¡ à„CJ¡ ãŸø‚«è£í‹ 裇è. (June 12)BC = CÁIJ¡ àòó‹ = 150 ªê.e

AB = Gö™ = 150 3 ªê.e, θ = ?

âF˜ð‚è‹tanθ = -----------------------

Ü´ˆ¶œ÷ ð‚è‹

tanθ = 3150

150

tanθ = 3

1

tan30o = 3

1

tan30o = 3

1. âù«õ θ = 30o

M÷‚°‚ è‹ðˆF¡ à„CJ¡ ãŸø‚«è£í‹ = 30o.

DE

A

30 3 m

30o

1.5

m

CB 30 3 m

gir

l

1.5

mx

m

BA

C

30o

0.9m

꣌¾ˆî÷‹x

m

î¬ó

BA

C

θθθθθ

CÁI

Gö™

150 3 cm

150c

m

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39

7. θθ+

θθ

seccos

eccossin

= 1 â¡ø ºŸªø£¼¬ñ¬ò GÁ¾. (June 12)

LHS = θθ+

θθ

seccos

eccossin

=

θ

θ+

θ

θ

cos1

cos

sin1

sin

= 1cos.cos

1sin.sin θθ+θθ

= sin2 θ + cos2 θ= 1= RHS

8. GÁ¾è θ+θ−

sin1sin1

= secθ - tanθ. (Oct. 12, June 14)

LHS = θ+θ−

sin1sin1

= θ−θ−×

θ+θ−

sin1sin1

sin1sin1

(¶¬íJò â‡í£™ ªð¼‚è)

= θ−

θ−2

2

sin1

)sin1(

=θθ−

2

2

cos

)sin1(

= 2

cossin1

θ

θ−

= θθ−

cossin1

= θcos1

- θθ

cossin

= secθ - tanθ= RHS. âù GÁõŠð´Aø¶.

9. GÁ¾è. θ+θ−

cos1cos1

= cosecθ - cotθ

LHS = θ+θ−

cos1cos1

= )cos1()cos1()cos1()cos1(

θ−θ+θ−θ−

(¶¬íJò â‡í£™ ªð¼‚è)

= θ−

θ−22

2

cos1

)cos1(

= θθ−

2

2

sin

)cos1(

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40

= 2

sincos1

θ

θ−

= θθ−

sincos1

= θθ−

θ sincos

sin1

= cosecθ - cotθ= RHS âù GÁõŠð´Aø¶

10. GÁ¾è θ−θθtansec

cos = 1 + sinθ. (June 13)

LHS = θ−θθtansec

cos

=

θθ−

θ

θ

cossin

cos1

cos

=

θθ−

θ

cossin1

cos

= θ−θθ

sin1cos.cos

= θ−

θsin1

cos2

= θ−θ−

sin1sin1 2

= θ−

θ−sin1sin1 22

= )sin1()sin1()sin1(

θ−θ−θ+

= 1 + sinθ= RHS

11. GÁ¾è θ+θθ

coteccossin

= 1 - cosθ (Oct. 14)

LHS = θ+θθ

coteccossin

=

θθ+

θ

θ

sincos

sin1

sin

=

θθ+

θ

sincos1

sin

= θ+θθ

cos1sin.sin

= θ+

θcos1

sin2 =

θ+θ−

cos1cos1 2

= θ+

θ−cos1cos1 22

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41

= )cos1()cos1()cos1(

θ+θ−θ+

= 1 - cosθ= RHS âù GÁõŠð´Aø¶

12. GÁ¾è θ+θ 22 eccossec = tanθ + cotθ (Mar. 14, Mar. 15)

LHS = θ+θ 22 eccossec sec2θ = 1 + tan2θ

= )cot1()tan1( 22 θ++θ+ cosec2θ = 1 + cot2θ

= θ++θ 22 cot2tan tanθ .cotθ = tanθ x θtan1

= θ+θθ+θ 22 cotcot.tan2tan a2 + 2ab + b2 = (a + b)2

= 2)cot(tan θ+θ

= tanθ + cotθ= RHS

13. GÁ¾è θ−

θ=θ

θ+cos1

sinsec

sec1 2 (Oct. 13)

L.H.S. = θθ+

secsec1

=

θ

θ+

cos1cos

11

=

θ

θ+θ

cos1

cos1cos

= 1xcoscosx)1(cos

θθ+θ

= 1 + cosθ

= (1 + cosθ) )cos1()cos1(

θ−θ−

(¶¬íJò â‡í£™ ªð¼‚è)

= θ−θ−

cos1cos1 2

= θ−

θcos1

sin2

= R.H.S. âù GÁõŠð´Aø¶14. GÁ¾è (sin6θ + cos6θ) = 1- 3sin2θ cos2θ (Mar. 12)

LHS = sin6θ + cos6θ a3 + b3 = (a + b)3 - 3ab (a + b)= (sin2θ)3 + (cos2θ)3

= (sin2θ + cos2θ )3 - 3sin2θ cos2θ (sin2θ + cos2θ )= (1)3 - 3sin2 θ. cos2θ x 1= 1 - 3sin2 θ.cos2 θ= RHS âù GÁõŠð´Aø¶

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42

15. GÁ¾è. θ−θ

cos1sin

= cosecθ + cotθ.

L.H.S. = θ−θ

cos1sin

= )cos1()cos1()cos1(sin

θ+θ−θ+θ

= θ−

θ+θ2cos1

)cos1(sin

= θθ+

sin)cos1(

= θθ+

θ sincos

sin1

= cosecθ + cotθ= RHS

16. Þ‹º¬øèO™ ðJŸC ªêŒ¶ 𣼃èœ.

i) sec2θ + cosec2θ = sec2θ cosec2θ vi) )cos1(sinsincos1 2

θ+θθ−θ+

= cotθ

vii) secθ (1-sinθ) (secθ + tanθ) = 1

8. Ü÷Mò™Ü÷Mò™Ü÷Mò™Ü÷Mò™Ü÷Mò™

1. å¼ F‡ñ «ï˜õ†ì ༬÷J¡ Ýó‹ 7 ªê.e ñŸÁ‹ àòó‹ 20 ªê.e âQ™ Üî¡ i) õ¬÷ðóŠ¹ ii)ªñ£ˆîŠ¹øŠðóŠ¹ ÝAòõŸ¬ø‚ 裇è. ( π = 22/7 â¡è)b˜¾ :

Þƒ° r = 7 ªê.e ñŸÁ‹ h = 20 ªê.ei) õ¬÷ðóŠ¹ = 2 πrh

= 2 x 722

x 7 x 20

= 880 ê.ªê.eii) ªñ£ˆîŠ¹øŠðóŠ¹ = 2πr (h+r)

= 2 x 722

x 7 x (20+7)

= 44 x 27= 1188 ê.ªê.e

2. å¼ F‡ñ «ï˜õ†ì ༬÷J¡ Ýó‹ 14 ªê.e ñŸÁ‹ àòó‹ 8 ªê.e âQ™ Üî¡ õ¬÷ðóŠ¹ ñŸÁ‹ªñ£ˆîŠ ¹øŠðóŠ¬ð‚ 裇è.

b˜¾:Þƒ° r = 14 ªê.e, h = 8 ªê.e

õ¬÷ðóŠ¹ = 2 πrh

= 2 x 722

x 14 x 8

= 704 ê.ªê.eªñ£ˆîŠ¹øŠðóŠ¹ = 2πr (h+r)

= 2 x 722

x 14 x (8+14)

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43

= 2 x 722

x 14 x 22

= 1936 ê.ªê.e3. å¼ F‡ñ ༬÷J¡ Ýó‹ 14 ªê.e Üî¡ àòó‹ 30 ªê.e âQ™, Üš¾¼¬÷J¡ èù Ü÷¬õ‚

裇è.

b˜¾:Þƒ°.t r = 14 ªê.e, h = 30 ªê.e

༬÷J¡ èù Ü÷¾ = πr2h

= 722

x 14 x 14 x 30

= 18480 ªê.e3

4. å¼ ñ¼ˆ¶õñ¬ùJ½œ÷ «ï£ò£O å¼õ¼‚° Fùº‹ 7 ªê.e M†ìºœ÷ ༬÷ õ®õ A‡íˆF™õ®„ê£Á õöƒèŠð´Aø¶. ÜŠð£ˆFóˆF™ 4 ªê.e àòóˆFŸ° õ®„ê£Á å¼ «ï£ò£O‚° õöƒèŠð†ì£™.250 «ï£ò£OèÀ‚° õöƒèˆ «î¬õò£ù õ®„ê£P¡ èù Ü÷¬õ‚ 裇è.

b˜¾:

Þƒ° 2r= 7 ªê.e, h = 4 ªê.e

∴ r = 27

ªê.e

å¼ «ï£ò£O‚° «î¬õò£ù õ®„ê£P¡ èù Ü÷¾ = πr2 h

= 722

x 27

x 27

x 4

= 154 ªê.e3

250 «ï£ò£O‚° «î¬õò£ù õ®„ê£P¡ èùÜ÷¾ = 250 x 154

= 38500 ªê.e3 = 100038500

L

= 38.5 L†ì˜ 1 litre = 1000 ªê.e3

5. 62.37 è.ªê.e èùÜ÷¾ ªè£‡ì å¼ F‡ñ «ï˜õ†ì ༬÷J¡ àòó‹ 4.5 ªê.e âQ™ Üš¾¼¬÷J¡Ýóˆ¬î‚ 裇è.

b˜¾:

Þƒ° h = 4.5 ªê.e

àϬ÷J¡ èù Ü÷¾ = 62.37 è.ªê.e i.e. πr2 h = 62.37 ªê.e3

r2 = h37.62

π

= 62.37 x 5.4

1227 ×

= 4.41

r = 41.4 = 2.1 ªê.e

6. Þó‡´ «ï˜õ†ì ༬÷èO¡ ÝóƒèO¡ MAî‹ 2 : 3, «ñ½‹ àòóƒèO¡ MAî‹ 5 : 3 âQ™.ÜõŸP¡ èù Ü÷¾èO¡ MAîˆ¬î‚ è£‡è.

b˜¾:

Þƒ° r1 : r2 = 2 : 3 ñŸÁ‹ h1 : h2 = 5 : 3

r1 = 2x , r2 = 3x , h1 = 5y ñŸÁ‹ h2 = 3y

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44

èùÜ÷¾èO¡ MAî‹ = πr12 h1 : 2πr2

2 h2

= r12 h1 : r2

2 h2

= 2x x 2x x 5y : 3x x 3x x 3y

= 20 : 27

7. å¼ F‡ñ «ï˜õ†ì‚ ËH¡ Ýó‹ ñŸÁ‹ ꣻòó‹ º¬ø«ò 35 ªê.e ñŸÁ‹ 37 ªê.e âQ™ ËH¡

õ¬÷ðóŠ¹ ñŸÁ‹ ªñ£ˆîŠ¹øŠðóŠ¬ð‚ 裇è, ( π = 722

â¡è)

b˜¾:Þƒ°, r = 35 ªê.e, l = 37 ªê.e

õ¬÷ðóŠ¹ = πrl

= 722

x 35 x 37

= 4070 ê.ªê.eªñ£ˆîŠ¹øŠðóŠ¹ = πr (l+r)

= 722

x 35 (37 + 35)

= 722

x 35 x 72

= 7920 ê.ªê.e8. å¼ F‡ñ «ï˜õ†ì‚ËH¡ Ü®„²Ÿø÷¾ 236 ªê.e ñŸø‹ Üî¡ ê£»òó‹ 12 ªê.e âQ™, ܂ËH¡

õ¬÷ŠóŠ¬ð‚ 裇è.b˜¾ :

Þƒ°, ËH¡ Ü®„²Ÿø÷¾ = 236 ªê.e, l = 12 ªê.e

ie 2πr = 236 ªê.e

∴ πr = 118 ªê.e

ËH¡ õ¬÷ðóŠ¹ = πrl= 118 x 12= 1416 cm2

9. ñóˆFù£ô£ù å¼ F‡ñ‚ ËH¡ Ü®„²Ÿø÷¾ 44 ªê.e ñŸÁ‹ Üî¡ àòó‹ 12 ªê.e âQ™ÜˆF‡ñ‚ ËH¡ èù Ü÷¬õ‚ 裇è.b˜¾:

ËH¡ Ü®„²Ÿø÷¾ = 44 e ñŸÁ‹ h = 12 e ie 2πr = 44

πr = 22

r = π22

= 22

722 ×

r = 7 e

ËH¡ èùÜ÷¾ = 31

πr2 h

= 31

x 722

x 7 x 7 x 12

= 616e2

10. å¼ «ï˜õ†ì‚ËH¡ èùÜ÷¾ 216π è.ªê.e ñŸÁ‹ ܂ËH¡ Ýó‹ 9 ªê.e âQ™ Üî¡ àòóˆ¬î‚裇è.

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45

b˜¾:ËH¡ èùÜ÷¾ = 216π è.ªê.e, r = 9 ªê.e

i.e. 31

πr2 h = 216π

31

π x 9 x 9 x h = 216π

h = 993216

××

= 8

= 8 ªê.e11. 14 ªê.e ð‚è Ü÷¾èœ ªè£‡ì å¼ èù„ê¶óˆF™ Þ¼‰¶ ªõ†®ªò´‚èŠð´‹ I芪ðKò ËH¡

èù Ü÷¬õ‚ 裇è.b˜¾:

Þƒ° èù„ê¶óˆF¡ ð‚è Ü÷¾ =14 ªê.e

∴ËH¡ Ýó‹ = 2

14 = 7 ªê.e

ËH¡ àòó‹ = 14 ªê.e

∴ËH¡ èùÜ÷¾ = 31

πr2 h

= 31

x 722

x 7 x 7 x 14

= 718.67 ªê.e3

12. å¼ Þ¬ì‚è‡ì õ®Mô£ù õ£OJ¡ «ñŸ¹ø ñŸÁ‹ Ü®Š¹ø Ýóƒèœ º¬ø«ò 15 ªê.e ñŸÁ‹ 8 ªê.e.

«ñ½‹, Ýö‹ 63 ªê.e âQ™, Üî¡ ªè£œ÷÷¬õ L†ìK™ 裇è. ( π=722

)

b˜¾:Þƒ° R = 15ªê.e, r = 8 ªê.e ñŸÁ‹ h = 63ªê.e

õ£OJ¡ èùÜ÷¾ = 31

πh (R2 + r2 + Rr)

= 31

x 722

x 63 x (152 + 82 + 15 x 8)

= 31

x 722

x 63 x (225 + 64 + 120)

= 31

x 722

x 63 x 409

= 26994 è.ªê.e

= 100026994

L

= 26.994 L13. 7 e àœM†ìºœ÷ å¼ àœkìŸø «è£÷ˆFÂœ à†¹øñ£è å¼ ê˜‚èv ió˜ «ñ£†ì£˜ ¬ê‚AO™

ê£èê‹ ªêŒAø£˜. ܉î ê£èê ió˜ ê£èê‹ ªêŒò‚ A¬ìˆF´‹ àœkìŸø‚ «è£÷ˆF¡ à†¹øŠðóŠ¬ð‚

裇è. ( π=722

)

b˜¾:Þƒ°, M†ì‹ = 7e r = 7/2 e

ê£èê‹ ªêŒò A¬ìˆF´‹ ðóŠ¹ = 4πr2

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46

= 4 x 722

x 27

x 27

= 154 ê.e14. ܬó‚«è£÷ õ®õ A‡íˆF¡ î®ñ¡ 0.25 ªê.e Üî¡ à†¹ø Ýó‹ 5 ªê.e âQ™ Ü‚ A‡íˆF¡

ªõOŠ¹ø õ¬÷ðóŠ¬ð‚ 裇è. ( π=722

)

b˜¾:

Þƒ° w = 0.25 ªê.e , r = 5 ªê.e ∴R = r + w

= 5 + 0.25 = 5.25 ªê.e

∴ªõOŠ¹ø õ¬÷ðóŠ¹ = 2πR2

= 2 x 722

x 5.25 x 5.25

= 173.25 ê.ªê.e15. 98.56 ê.ªê.e ¹øŠð󊹂 ªè£‡ì å¼ F‡ñ‚ «è£÷ˆF¡ Ýóˆ¬î‚ 裇è.

b˜¾ : Þƒ°, ¹øŠðóŠ¹ = 98.56è.ªê.e

ie 4πr2 = 98.56

4 x 722

x r2 = 98.56

r2 = 224756.98

××

= 7.84

r = 84.7 = 2.8

r = 2.8 ªê.e

16. 8.4 ªê.e M†ì‹ ªè£‡ì å¼ «è£÷õ®õ F‡ñ à«ô£è âP°‡®¡ èù Ü÷¬õ‚ 裇è. (π=722

)

b˜¾:2r = 8.4 ªê.e r = 4.2 ªê.e

à«ô£è âP°‡®¡ èùÜ÷¾ = 34

πr3

= 34

x 722

x 4.2 x 4.2 x 4.2

= 310.464 è.ªê.e17. å¼ àœkìŸø «è£÷ˆF¡ ªõO ñŸÁ‹ àœ Ýóƒèœ º¬ø«ò 12 ªê.e ñŸÁ‹ 10 ªê.e âQ™

Ü‚«è£÷ˆF¡ èù Ü÷¬õ‚ 裇è.b˜¾: Þƒ° R = 12 ªê.e r = 10 ªê.e

èùÜ÷¾ = 34

π (R3 - r3)

= 34

x 722

(123 - 103)

= 34

x 722

( 1728 - 1000)

= 34

x 722

x 728

= 3050 .66 è.ªê.e

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47

18. æ˜ Ü¬ó‚«è£÷ˆF¡ èùÜ÷¾ 1152π è.ªê.e. âQ™, Üî¡ õ¬÷ðóŠ¹ è£‡è.

b˜¾:

Þƒ° ܬó‚«è£÷ˆF¡ èùÜ÷¾ = 1152π

ie 32

πr3 = 1152 π

r3 = 2

31152×

= 1728

r = 31728 = 12 ªê.e

õ¬÷ðóŠ¹ = 2πr2 = 2 x π x 144 = 288πê.ªê.e.

11. ¹œOJò™¹œOJò™¹œOJò™¹œOJò™¹œOJò™

1. 43, 24, 38, 56, 22, 39, 45 i„² ñŸÁ‹ i„²‚ªè¿ 裇è.

L = 56, S = 22i) i„² = L - S = 56 - 22 = 34

= 34

ii) i„²‚ªè¿ = SLSL

+−

= 7834

= 0.436

2. 59, 46, 30, 33, 27, 40, 52,35, 29 i„² ñŸÁ‹ i„²‚ªè¿ 裇è.L = 59 , S = 23

b˜¾ = L - S= 59 - 23 = 36

i„²‚ªè¿ = SLSL

+−

= 8236

= 0.443. å¼ ¹œO Mõó ªî£°ŠH¡ eŠªð¼ ñFŠ¹ 7.44, i„² 2.26, âQ™ e„CÁ ñFŠ¹ â¡ù?

i„² = L - S i„² = 2.26 , L = 7.44

2.26 = 7.44 - S S = 7.44 - 2.26

= 5.18

4. å¼ ¹œO MõóˆF¡ e„CÁ ñFŠ¹ 12, i„² 59, âQ™ eŠªð¼ ñFŠ¹ â¡ù?

i„² = L - S i„² = 59 and S = 12

59 = L - 12 L = 59 + 12 = 71

5. å¼ ¹œO MõóˆF¡ I芪ðKò ñFŠ¹ 3.84 A.A. i„² 0,46 A.A. e„CÁ ñFŠ¹ â¡ù?

i„² = L - S L = 3.84 i„² = 0.46

0.46 = 3.84 - S S = 3.84 - 0.46 = 3.38 Kg

6. ºî™ 10 Þò™ â‡èO¡ F†ìMô‚è‹ è£‡è.

ºî™ n Þò™ â‡èO¡ F†ìMô‚è‹ = 12

1n2 − ; n = 10

= 12

1102 − =

121100 −

= 1299

~ 2.87

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48

7. ºî™ 13 Þò™ â‡èO¡ F†ìMô‚è‹ â¡ù?

ºî™ n Þò™ â‡èO¡ F†ì Mô‚è‹ = 12

1n2 −; n = 13

= 2

1132 − =

121169 −

= 12

168 = 14 ~ 3.74

8. å¼ ¹œO MõóˆF¡ ñ£Á𣆴‚ ªè¿ 57, F†ìMô‚è‹ 6.84 âQ™ Æ´ êó£êK 裇è.

C.V = 100×σx

% , C.V. = 57, σ = 6.84

57 = 10084.6 ×x

x = 57

684= 12

9. n = 10 x = 12 2x = 1530 ñ£Á𣆴‚ ªè¿ èí‚A´è.

Mô‚è õ˜‚è êó£êKσ2 = 22

)(n

xx −

= 10

1530 - (12)2

= 153 - 144 = 9

σ = 9 = 3

C.V. = 100×σx

%

C.V. = 123

x 100 = 25%

12. Gè›îè¾Gè›îè¾Gè›îè¾Gè›îè¾Gè›îè¾

1. ºî™ Þ¼ð¶ Þò™ â‡èOL¼‰¶ å¼ º¿ ⇠«î˜‰ªî´‚èŠð´Aø¶, ܉î â‡ å¼ ðè£ â‡ ÝèÞ¼‚è Gè›îè¾ ò£¶?

S = {1, 2, 3, ..... 20}, ie. n (S) = 20ðè£ â‡ A = {2, 3, 5, 7, 11, 13, 17, 19}, n (A) = 8

P(A)= )S(n)A(n

= 208

=52

2. 35 ªð£¼†èœ ÜìƒAò ÃÁ å¡P™ 7 ªð£¼œ °¬ø𣴬ìòù, °¬øð£ìŸø ªð£¼÷£è Þ¼‚èGè›îè¾ ò£¶?

n(S) = 35

°¬ø𣴠à¬ìŒ¬õ = 7

°¬øð£ìŸø¬õ = 35 −−−−− 7 = 28, n(A) = 28

P(A)= 3528

= 54

3. å¼ õ°ŠH™ 35 ñ£íõ˜èO™ 20 «ð˜ ݇èœ, 15 «ð˜ ªð‡èœ, «î˜‰ªî´‚èŠð†ì å¼ ñ£íõ˜ (i)Ýí£è (ii) ªð‡í£è Þ¼‚è Gè›îè¾ è£‡è.

n(S) = 35i) ݇ : A, n(A) = 20

P(A) = 3520

= 74

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49

ii) ªð‡ : B, n(B) = 15

P(B) = 3515

= 73

4. å¼ °PŠH†ì ï£O™ ñ¬ö õ¼õîŸè£ù Gè›îè¾ 0,76, ñ¬ö õó£ñ™ Þ¼‚è Gè›îè¾ ò£¶?

ñ¬ö õ¼õ A âù¾‹, ñ¬ö õó£ñ™ Þ¼‚è A âù¾‹ ªè£œè.

P(A) = 0.76 (P(A) + P( A ) = 1)

P( A ) = 1 −−−−− 0.76 = 0.24

5. êñõ£ŒŠ¹ º¬øJ™ ê£î£óí õ¼ìˆF™ 53 ªõœO‚ Aö¬ñèœ Þ¼‚è Gè›îè¾ ò£¶?

ê£î£óí õ¼ì‹ = 365 = 52 õ£óƒèœ + å¼ ï£œ

52 õ£óƒèœ = 52 ªõœO‚Aö¬ñèœ

1 ï£O™ {Fƒèœ, ªêšõ£Œ, ¹î¡, Mò£ö¡, ªõœO, êQ, ë£JÁ}

n(S) = 7

A = {ªõœO}, n(A) = 1, P(A) = 71

6. êñõ£ŒŠ¹ º¬øJ™ ªï†ì£‡®™ 53 ªõœO‚Aö¬ñ Þ¼‚è Gè›îè¾ ò£¶?

ªï†ì£‡´ = 366 = 52 õ£óƒèœ + Þ¼ èœ

52 õ£óƒèœ = 52 ªõœO‚Aö¬ñèœ

2 ï£†èœ â¡ð¶ = {(ë£, F) (F, ªê) (ªê,¹) (¹,M) (M,ªõ) (ªõ,ê) (ê,ë£)}

n (S) = 7

A = {(Mò£, ªõœO (ªõœO, êQ)}

n (A) = 2

P(A) = 72

7. 1&100 õ¬ó àœ÷ Y†®™ 10 Ý™ õ°ð´‹ ⇠޼‚è Gè›îè¾ è£‡.

n(S) = 100

10 Ý™ õ°ð´‹ ⇠: A = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100}

n(A) = 10

P(A) = 10010

= 101

8. å¼ Yó£ù ðè¬ì Þó‡´ º¬ø ༆ìŠð´Aø¶. ºè â‡ Ã´î™ 9 A¬ì‚è Gè›îè¾?

S = {(1, 1) (1, 2) (1,3) (1, 4) (1,5) (1,6) (2, 1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3, 2) (3, 3) (3, 4) (3,5), (3,6), (4, 1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6, 5) (6, 6)},

n(S) = 36

ºè â‡ Ã´î™ 9 : A = {(3, 6) (4, 5) (5, 4) (6, 3)}

n(A) = 4

P(A) = 364

= 91

9. 12 ï™ô º†¬ìèÀì¡ 3 Ü¿Aò º†¬ìèœ, å¼ º†¬ì Ü¿Aòî£è Þ¼‚è Gè›îè¾ ò£¶?

n(S) = 12 + 3 = 15

Ü¿Aò º†¬ì: A n(A) = 3

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50

P(A) = 153

= 51

10. Þ¼ ï£íòƒèœ ²‡´‹ «ð£¶, ÜFèð†êñ£è å¼ î¬ô A¬ì‚è Gè›îèM¬ù‚ 裇è. S = {HH, HT, TH, TT} n(S) = 4

ÜFèð†ê å¼ î¬ô : A ; A = {HT, TH, TT}n(A) = 3

P(A) = 43

11. 1 ºî™ 6 ⇠õ¬ó 6 ªõœ¬÷ Gø ð‰¶, 7 ºî™ 10 õ¬ó 4 CõŠ¹ Gø ð‰¶ å¼ ð‰¶ â´‚èŠð´Aø¶,i) Þó†¬ì ⇠ªè£‡ì ð‰¶ ii) ªõœ¬÷ Gø ð‰¶

n(S) = 6 + 4 = 10Þó†¬ì ⇠: A = {2, 4, 6, 8, 10} ; n(A) = 5

P(A) = 105

= 21

ªõœ¬÷Š ð‰¶ : B = {1, 2, 3, 4, 5, 6} ; n(B) = 6

P(B) = 106

= 53

12. 1&20 õ¬ó àœ÷ º¿ â‡, ܉î ⇠4¡ ñìƒè£è Þ¼‚è Gè›îè¾ â¡ù? n(S) = 20

4¡ ñ샰 : A = {4, 8, 12, 16, 20}; n(A) = 5

P(A) = 205

= 41

13. Í¡Á ðè¬ìèœ å«ó «ïóˆF™ ༆ìŠð´õF™ Í¡P½‹ å«ó ⇠Ýè Þ¼‚è Gè›îè¾?

S = {(1, 1, 1) ...... (6,6, 6)} n(S) = 6 x 6 x 6 = 216

A = {(1, 1, 1) (2, 2, 2) (3, 3, 3) (4, 4, 4) (5, 5, 5) (6, 6, 6) n(A) = 6

P(A) = 216

6=

361

14. 52 Y†´èœ 膮L¼‰¶ å¼ Y†´ i) 輊¹ Þó£ê£ ii) v«ð´ Þ¼‚è Gè›îè¾?n(S) = 52

i) 輊¹ Þó£ê£: A; n(A) = 2; P(A) = 522

= 261

ii) v«ð´ : B; n(B) = 13; P(B) = 5213

= 41

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