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XII Maths Questions for slow learners (Q. Nos) N. MAHALAKSHMI

PGT (Mathematics)

CHAPTERS FOR SLOW LEARNERS

CHAPTER TYPE NO OF QUESTIONS MARKS

1 6 marks 33 2 x 6 = 12

3 marks 16

2 10 marks 20 2 x 10 = 20

4 10 marks 28 3 x 10 = 30

9 10 marks 15 1 x 10 = 10

6 marks 35 2 x 6 = 12

3 marks 16

10 10 marks 12 1 x 10 = 10

TOTAL 175 94

ONE MARK (271) (30)

ASSIGNMENTS FOR SLOW LEARNERS

1. A¦Ls. A¦dúLôûYLs (6 marks) JUNE


3. ùYdPo CVtL¦Rm JULY

4. TWYû[Vm AUGUST

5. ¿sYhPm SEPTEMBER

6. A§TWYû[Vm. ùRôÓúLôÓ. ùRôûXjùRôÓúLôÓ OCTOBER

7. ùUn AhPYûQ NOVEMBER

8. ÏXeLs (10 marks) NOVEMBER


10. ¨LrRLÜ (10 marks) DECEMBER

TESTS FOR SLOW LEARNERS

CHAPTER WISE (1, 2, 4, 9, 10) JANUARY

OLD QUESTION PAPERS (1, 2, 4, 9, 10) FEBRUARY

NO SUBSTITUTE FOR HARDWORK

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XII Maths Questions for slow learners (Q. Nos) N. MAHALAKSHMI

PGT (Mathematics)

XII STANDARD MATHEMATICS

QUESTIONS FOR SLOW LEARNERS (Q. Nos)

TôPm TôPm TôPm TôPm 1 (6)1 (6)1 (6)1 (6) Ex 1.1 (1 iii) Ex 4.2 (6 ii) TôPm 9 (6)TôPm 9 (6)TôPm 9 (6)TôPm 9 (6) Ex 9.2 (3)

33 33 33 33 úLs®LsúLs®LsúLs®LsúLs®Ls Eg 1.5 (i) Eg 4.31 (iv) 35353535 úLs®LsúLs®LsúLs®LsúLs®Ls Eg 9.4 (i)

Ex 1.1 (8) Eg 1.5 (ii) Ex 4.2 (6 iv) Eg 9.4 (iv) Ex 9.2 (2)

Eg 1.3 Eg 1.5 (iii) Eg 4.56 Ex 9.2 (9) Eg 9.4 (ii)

Ex 1.1 (2) Ex 1.1 (5 ii) Ex 4.3 (5 iii) Eg 9.6 Ex 9.2 (6)

Eg 1.5 (iv) (AT) −1 = (A− 1) T Eg 4.57 Ex 9.2 (10) Ex 9.2 (4)

Ex 1.1 (4 i) Eg 1.11 Ex 4.3 (5 iv) Eg 9.5 Eg 9.4 (iii)

Ex 1.1 (4 ii) Eg 1.17 (1) Eg 4.13 Eg 9.7 Ex 9.2 (5)

Ex 1.1 (4 iii) Ex 1.4 (1) Eg 4.12 Ex 9.3 (5) Ex 9.2 (8)

Ex 1.1 (4 iv) Eg 1.17 (3) Eg 4.14 Ex 9.3 (2) Ex 9.2 (7)

Ex 1.1 (4 v) Ex 1.4 (2) Ex 4.1 (5) Ex 9.3 (4) Ußl©u Ußl×

Ex 1.1 (10) Eg 1.20 Eg 4.8 Ex 9.3 (3) Eg 9.8

(AB)−1 = B−1 A−1 Eg 4.10 Ex 9.3 (6) Eg 9.9 (i)

Eg 1.6 TôPm 2TôPm 2TôPm 2TôPm 2 (10)(10)(10)(10) Eg 4.32 Ex 9.3 (1 ii) Eg 9.9 (ii)

Ex 1.1 (5 i) 20 úLs®Ls20 úLs®Ls20 úLs®Ls20 úLs®Ls Ex 4.2 (10) Ex 9.3 (1 iv)

Eg 1.7 Eg 2.16 Eg 4.33 Eg 9.10 (i) TôPm TôPm TôPm TôPm 10 (10)10 (10)10 (10)10 (10) Ex 1.2 (1) Eg 2.17 Ex 4.2 (9) Eg 9.11 12121212 úLs®LsúLs®LsúLs®LsúLs®Ls

Ex 1.2 (2) Ex 2.2 (4) Ex 4.2 (8) Ex 9.3 (1 iii) Eg 10.3

Eg 1.13 Ex 2.4 (7) Ex 4.2 (7) Ex 9.3 (7) Eg 10.2

Ex 1.3 (1) Eg 2.29 Eg 4.35 Eg 9.10 (ii) Ex 10.1 (7)

Eg 1.12 Ex 2.5 (5) Ex 4.4 (5) Ex 9.3 (1 i) Eg 10.10

Eg 1.14 Ex 2.5 (12) Ex 4.4 (6) Ex 9.3 (1 v) Ex 10.4 (5)

Ex 1.3 (5) Eg 2.44 Ex 4.5 (2 ii) Eg 9.14 Eg 10.26

Ex 1.3 (6) Ex 2.7 (3) Ex 4.6 (3) Eg 9.15 Eg 10.29

Eg 1.16 Eg 2.50 Ex 9.4 (4) Eg 10.32

Ex 1.3 (2) Ex 2.8 (8) TôPm TôPm TôPm TôPm 9 (10)9 (10)9 (10)9 (10) Eg 9.20 Ex 10.5 (5)

Ex 1.3 (3) Ex 2.8 (9) 15151515 úLs®LsúLs®LsúLs®LsúLs®Ls Eg 9.19 Eg 10.30

Eg 1.15 Ex 2.8 (7) Eg 9.24 Eg 9.12 Eg 10.31

Ex 1.3 (4) Ex 2.8 (10) Ex 9.4 (6) Eg 9.13 Ex 10.5 (8)

Eg 1.17 (2) Eg 2.51 Eg 9.26 Eg 9.16

Ex 1.4 (3) Ex 2.8 (11) Ex 9.4 (9) Eg 9.17

Eg 1.18 (3) Ex 2.8 (12) Eg 9.25 Eg 9.28

Eg 1.18 (5) Eg 2.52 Eg 9.27 Eg 9.29

Ex 1.5 (1 iii) Ex 2.8 (13) Eg 9.21 Eg 9.30

Ex 1.5 (1 iv) Ex 9.4 (11) Ex 9.4 (10) Ex 9.4 (5) ¿dLp ®§

TôPm TôPm TôPm TôPm 1 (3)1 (3)1 (3)1 (3) TôPm TôPm TôPm TôPm 4 (10)4 (10)4 (10)4 (10) Eg 9.18 §Úl×Rp ®§

16161616 úLs®LsúLs®LsúLs®LsúLs®Ls 28282828 úLs®LsúLs®LsúLs®LsúLs®Ls Eg 9.23 10 M : 75

Ex 1.1 (1 i) Eg 4.7 (iv) Ex 9.4 (8) TôPm 9 (3)TôPm 9 (3)TôPm 9 (3)TôPm 9 (3) 6 M : 68

Eg 1.1 Ex 4.1 (2 iv) Eg 9.22 16161616 úLs®LsúLs®LsúLs®LsúLs®Ls 3 M : 32

Eg 1.2 Ex 4.1 (2 v) Ex 9.4 (12) Eg 9.2 Total: 175

Ex 1.1 (1 ii) Eg 4.7 (v) Ex 9.4 (7) Ex 9.2 (1) Marks: 94

Eg - Example P No - Page Number Ex - Exercise

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XII slow learners questions © reserved with N. MAHALAKSHMI

PGT (Mathematics)

CHAPTERS FOR SLOW LEARNERS

CHAPTER TYPE NO OF QUESTIONS MARKS

1 6 marks 33 2 x 6 = 12

3 marks 16

2 10 marks 20 2 x 10 = 20

4 10 marks 28 3 x 10 = 30

9 10 marks 15 1 x 10 = 10

6 marks 35 2 x 6 = 12

3 marks 16

10 10 marks 12 1 x 10 = 10

TOTAL 175 94

ONE MARK (271) (30)

ASSIGNMENTS FOR SLOW LEARNERS



3. ùYdPo CVtL¦Rm (10 marks) JULY

4. TWYû[Vm AUGUST

5. ¿sYhPm SEPTEMBER

6. A§TWYû[Vm. ùRôÓúLôÓ. ùRôûXjùRôÓúLôÓ OCTOBER

7. ùUn AhPYûQ (6 marks & 3 marks) NOVEMBER


9. ÏXeLs (6 marks & 3 marks) NOVEMBER

10. ¨LrRLÜ (10 marks) DECEMBER

TESTS FOR SLOW LEARNERS

CHAPTER WISE (1, 2, 4, 9, 10) JANUARY

OLD QUESTION PAPERS (1, 2, 4, 9, 10) FEBRUARY

NO SUBSTITUTE FOR HARDWORK

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PGT (Mathematics)

XII STANDARD MATHS QUESTIONS FOR SLOW LEARNERS

A¦Ls. A¦dúLôûYLs (33 úLs®Ls) (12(12(12(12))))

1. A =

4 3 3

1 0 1

4 4 3

− − −

Cu úNol× A¦ A G] ¨ßÜL, Ex. 1.1 (8)

A (adj A) = (adj A) A = | A | . I2 GuTûRf N¬TôodL,

2, A = 1 2

1 4

− −

Eg. 1.3 3. A = 1 2

3 5

−

Ex. 1.1 (2)

A¦«u úSoUôß A¦ LôiL,

4.

3 1 1

2 2 0

1 2 1

−

− −

Eg. 1.5 (iv) 5.

1 0 3

2 1 1

1 1 1

− −

Ex. 1.1 4(i)

6, 1 3 7

4 2 3

1 2 1

Ex. 1.1 4(ii) 7.

1 2 2

1 3 0

0 2 1

− − −

Ex. 1.1 4(iii)

8.

8 1 3

5 1 2

10 1 4

− − − − −

Ex. 1.1 4(iv) 9.

2 2 1

1 3 1

1 2 2

Ex. 1.1 4(v)

10. A =

1 2 2

4 3 4

4 4 5

− − − −

G²p A = A−1 G]d LôhÓL, Ex. 1.1 (10)

11. úSoUôßLÞdÏ¬V Y¬ûNUôtß ®§ GÝ§ ¨ßÜL (A(A(A(ApXÕpXÕpXÕpXÕ)))) Page 5

A, B CWiÓ éf£VUt\ úLôûY A¦Ls G²p, (AB)−1 = B−1 A−1

G] ¨ßÜL,

(AB)−1 = B−1 A−1

GuTûRf N¬Tôo,

12, A = 1 2

1 1

Utßm B = 0 1

1 2

−

Eg. 1.6

13, A = 5 2

7 3

Utßm B = 2 1

1 1

− −

Ex. 1.1 (5)(i)

úSoUôß A¦ LôQp Øû\«p ¾odL:

14. x + y = 3, 2x + 3y = 8 Eg. 1.7

15. 2x − y = 7, 3x − 2y = 11 Ex. 1.2 (1)

16. 7x + 3y = − 1, 2x + y = 0 Ex. 1.2 (2)

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PGT (Mathematics)

A¦«u RWm LôiL,

17.

1 1 1

2 3 4

3 2 3

− − −

Eg. 1.13 18.

1 1 1

3 2 3

2 3 4

− − −

Ex. 1.3 (1)

19.

1 1 1 3

2 1 3 4

5 1 7 11

− −

Eg. 1.12 20.

1 2 3 1

2 4 6 2

3 6 9 3

− − −

Eg. 1.14

21.

1 2 1 3

2 4 1 2

3 6 3 7

− − −

Ex 1.3 (5) 22.

1 2 3 4

2 4 1 3

1 2 7 6

− − − − −

Ex. 1.3 (6)

23,

3 1 5 1

1 2 1 5

1 5 7 2

− − − − −

Eg. 1.16 24.

6 12 6

1 2 1

4 8 4

Ex. 1.3 (2)

25,

3 1 2 0

1 0 1 0

2 1 3 0

−

Ex. 1.3 (3) 26.

4 2 1 3

6 3 4 7

2 1 0 1

Eg. 1.15

27.

0 1 2 1

2 3 0 1

1 1 1 0

− − −

Ex. 1.3 (4)

©uYÚm ANUT¥jRô] úS¬V NUuTôhÓ ùRôÏl©û] A¦dúLôûY Øû\«p ¾odL:

28, 2x + 3y = 8 4x + 6y = 16 Eg 1.17 (2)

29. 4x + 5y = 9 8x + 10y = 18 Ex 1.4 (3)

30. 2x + 2y + z = 5 x − y + z = 1 3x + y + 2z = 4 Eg 1.18 (3)

31. x + y + 2z = 4 2x + 2y + 4z = 8 3x + 3y + 6z = 10 Eg 1.18 (5) ©uYÚm NUuTôÓL°u ùRôÏl©u JÚeLûUÜj RuûUûVj RW Øû\ûVl TVuTÓj§ BWônL,

32. x + y + z = 7 x + 2y + 3z = 18 y + 2z = 6 Ex. 1.5 (1)(iii)

33. x − 4y + 7z = 14 3x + 8y − 2z = 13 7x − 8y + 26z = 5 Ex. 1.5 (1)(iv)

A¦Ls. A¦dúLôûYLs (16 úLs®Ls)

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PGT (Mathematics)

úNol× A¦ûVd LôiL,

1. 3 1

2 4

− = −

A Ex. 1.1 (1)(i) 2.

=

a bA

c d Eg. 1.1

3.

1 1 1

1 2 3

2 1 3

= − −

A Eg. 1.2 4.

1 2 3

0 5 0

2 4 3

=

A Ex. 1.1 (1)(ii)

5.

2 5 3

3 1 2

1 2 1

=

A Ex. 1.1 (1)(iii)

A¦«u úSoUôß A¦ LôiL,

6. 1 2

1 4

− = −

A Eg. 1.5 (i) 7. 2 1

4 2

− = −

A Eg. 1.5 (ii)

8. cos sin

sin cos

= −

Aα α

α α Eg. 1.5 (iii)

9. A = 5 2

7 3

Utßm B = 2 1

1 1

− −

G²p. (AB)T = BT A

T G] N¬Tôo, Ex. 1.1 (5)(i)

10. A JÚ éf£VUt\ úLôûY A¦Vô«u (AT)

−1 = (A

− 1)

T GuTûR ¨ßÜL, Page 5

11.

1 2 3

2 4 6

5 1 1

− − − −

Gu\ A¦«u RWm LôiL, Eg. 1.11

©uYÚm ANUT¥jRô] NUuTôhÓj ùRôÏl©û] A¦dúLôûY Øû\«p ¾odL:

12. x + y = 3 2x + 3y = 7 Eg 1.17 (1)

13. 3x + 2y = 5 x + 3y = 4 Ex 1.4 (1)

14. x − y = 2 3y = 3x − 7 Eg 1.17 (3)

15. 2x + 3y = 5 4x + 6y = 12 Ex 1.4 (2)

16. ©uYÚm NUT¥jRô] úS¬Vf NUuTôÓLû[ A¦dúLôûY Øû\«p ¾odLÜm, x + y + 2z = 0 2x + y − z = 0 2x + 2y + z = 0 Eg 1.20

ùYdPo CVtL¦RmùYdPo CVtL¦RmùYdPo CVtL¦RmùYdPo CVtL¦Rm (20 úL(20 úL(20 úL(20 úLs®Ls)s®Ls)s®Ls)s®Ls) (20)(20)(20)(20) 1, JÚ ØdúLôQj§u ÏjÕdúLôÓLs JúW ×s°«p Nk§dÏm GuTRû]

ùYdPo Øû\«p ¨ßÜL, Eg. 2. 16 2, ùYdPo Øû\«p cos (A - B) = cos A cos B + sin A sin B G] ¨ßÜL, Eg. 2.17 3, ùYdPo Øû\«p cos (A + B) = cos A cos B − sin A sin B G] ¨ßÜL, Ex. 2. 2 (4)

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PGT (Mathematics)

4. Sin (A − B) = sin A cos B − cos A sin B G] ùYdPo Øû\«p ¨ì©, Ex. 2. 4 (7)

5. ùYdPo Øû\«p Sin (A + B) = sin A cos B + cos A sin B G] ¨ßÜL, Eg. 2. 29

6. 2 3a i j k= + −rr rr

, 2 5b i k= − +r rr

, 3c j k= −rrr

, G²p ( ) ( . ) ( . )a b c a c b a b c× × = −r r rr r r r r r

G] N¬TôodL, Ex. 2. 5 (5)

7, a i j k= + +rr rr

, 2b i k= +r rr

, 2c i j k= + +rr rr

, 2d i j k= + +r rr r

G²p

( ) ( ) [ ] [ ]a b c d a b d c a b c d× × × = −r r r r r rr r r r r r

GuTûRf N¬TôodL, Ex. 2.5 (12)

8. 1 1

3 1 0

y z= − = +

−

x - 1 Utßm 4 1

2 0 3

x y z− = = +uuuu�uuuu� u� Gu\ úLôÓLs ùYh¥d ùLôsÞm

G]d LôhÓL, úUÛm AûY ùYhÓm ×s°ûVd LôiL, Eg. 2. 44

9. 1

1 1 3

y z= + =

−

x -1 Utßm 2 1 1

1 2 1

x y z− = − = − −uuuuu�uuuu� uuuu� Gu\ úLôÓLs ùYh¥d ùLôsÞm

G]d LôhÓL, úUÛm AûY ùYhÓm ×s°ûVd LôiL, Ex. 2. 7 (3)

10. (2, -1, -3) Gu\ ×s° Y¯f ùNpYÕm - 2 = - 1 - 3

3 2 4

x y z=

−

uuuu� uuuu�uuuu� Utßm

- 1 = 1 - 2

2 3 2

+ =

−

uuuu�uuuu� uuuuuu�x y z Gu\ úLôÓLÞdÏ CûQVô]ÕUô] R[j§u

ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, Eg. 2. 50

11. (1, 3, 2) Gu\ ×s° Y¯f ùNpYÕm 1 = 2 3

2 1 3

x y z+ + = +

−

uuuuuu� uuuuuu�uuuuuu� Utßm

- 2 = 1 2

1 2 2

x y z+ = +uuuuuu�uuuu� uuuuuu� Gu\ úLôÓLÞdÏ CûQVô]ÕUô] R[j§u

ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, Ex. 2.8 (8) 12, (-1. 3. 2) Gu\ ×s° Y¯f ùNpYÕm x + 2y + 2z =5 Utßm 3x + y +2z = 8 B¡V

R[eLÞdÏ ùNeÏjRô]ÕUô] R[j§u ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, Ex. 2.8 (9)

13. - 2 = - 2 - 1

2 3 3

= uuuu�uuuu� uuuu�x y z Gu\ úLôhûP Es[Pd¡VÕm

+ 1 = - 1 + 1

3 2 1

= uuuuu�uuuuu� uuuu�x y z

Gu\ úLôh¥tÏ CûQVô]ÕUô] R[j§u ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, Ex. 2.8 (7)

14. A(1, − 2, 3) Utßm B (− 1, 2, − 1) Gu\ ×s°Ls Y¯úVf ùNpXdá¥VÕm - 2 = + 1 - 1

2 3 4

= uuuu�uuuu� uuuuu�x y z Gu\ úLôh¥tÏ CûQVô]ÕUô] R[j§u ùYdPo

Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, Ex. 2.8 (10)

15, (− 1. 1. 1) Utßm (1. − 1. 1) B¡V ×s°Ls Y¯úVf ùNpXd á¥VÕm

x + 2y + 2z = 5 Gu\ R[j§tÏ ùNeÏjRôL AûUYÕUô] R[j§u ùYdPo Utßm Lôo¼£Vu NUuTôhûPd LôiL, Eg. 2. 51

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PGT (Mathematics)

16, (1. 2. 3) Utßm (2. 3. 1) Gu\ ×s°Ls Y¯úVf ùNpXd á¥VÕm 3x− 2y + 4z − 5 = 0 Gu\ R[j§tÏf ùNeÏjRôLÜm AûUkR R[j§u ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, Ex. 2. 8 (11)

17. 2 2 1

2 3 2

x y z− = − = −−uuuu�uuuu� uuuuu� Gu\ úLôhûP Es[Pd¡VÕm (− 1. 1. − 1) Gu\

×s° Y¯úVf ùNpXd á¥VÕUô] R[j§u ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, Ex. 2.8 (12)

18. (2, 2, − 1), (3, 4, 2) Utßm (7, 0, 6) B¡V ×s°Ls Y¯úVf ùNpXdá¥V R[j§u

ùYdPo Utßm Lôo¼£Vu NUuTôhûPd LôiL, Eg. 2.52

19, 3 4 2 2 2.i j k i j k+ + − −r rr r r r

Utßm 7i k+rr B¡VYtû\ ¨ûX ùYdPoL[ôLd

ùLôiP ×s°Ls Y¯úVf ùNpÛm R[j§u ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, Ex. 2. 8 (13)

20, ùYhÓjÕiÓ Y¥®p JÚ R[j§u NUuTôhûP ùYdPo Øû\«Ûm

Lôo¼£Vu Øû\«Ûm RÚ®dL, Ex. 2.8 (14)

TTTTWYû[VmWYû[VmWYû[VmWYû[Vm (10 (10 (10 (10 úLs®Ls)úLs®Ls)úLs®Ls)úLs®Ls) (10)(10)(10)(10) TWYû[Vj§u AfÑ. Øû]. Ï®Vm. CVdÏYûW«u NUuTôÓ. ùNqYLXj§u NUuTôÓ. ùNqYLXj§u ¿[m B¡VYtû\d LôiL, úUÛm ARu Yû[YûWûV YûWL, 1, y

2 - 8x + 6y + 9 = 0 Eg 4.7 (iv)

2. y2 + 8x− 6y + 1 = 0 Ex 4.1 (2 iv)

3. x2 − 6x − 12y − 3 = 0 Ex 4.1 (2 v)

4. 2x 2x + 8y +17 0− = Eg 4.7 (v)

5, JÚ Yôp ®iÁu (comet) B]Õ ã¬Vû]f (sun) Ñt± TWYû[Vl TôûR«p

ùNp¡\Õ, Utßm ã¬Vu TWYû[Vj§u Ï®Vj§p AûU¡\Õ, Yôp ®iÁu ã¬V²−ÚkÕ 80 ªp−Vu ¡,Á, ùRôûX®p AûUkÕ CÚdÏm úTôÕ Yôp ®iÁû]Ùm ã¬Vû]Ùm CûQdÏm úLôÓ TôûR«u AfÑPu π/3 úLôQj§û] HtTÓjÕUô]ôp (i) Yôp ®iÁ²u TôûR«u NUuTôhûPd LôiL (ii) Yôp ®iÁu ã¬VàdÏ GqY[Ü AÚ¡p YWØ¥Ùm GuTûRÙm LôiL, (TôûR YXÕ×\m §\l×ûPVRôL ùLôsL), Eg. 4. 13

6, RûWUhPj§−ÚkÕ 7,5Á EVWj§p RûWdÏ CûQVôL ùTôÚjRlThP JÚ

ÏZô«−ÚkÕ ùY°úVßm ¿o RûWûVj ùRôÓm TôûR JÚ TWYû[VjûR HtTÓjÕ¡\Õ, úUÛm CkR TWYû[Vl TôûR«u Øû] ÏZô«u Yô«p AûU¡\Õ, ÏZôn UhPj§tÏ 2,5 Á ¸úZ ¿¬u TônYô]Õ ÏZô«u Øû] Y¯VôLf ùNpÛm ¨ûX ÏjÕdúLôh¥tÏ 3 ÁhPo çWj§p Es[Õ G²p ÏjÕdúLôh¥−ÚkÕ GqY[Ü çWj§tÏ AlTôp ¿Wô]Õ RûW«p ®Ým GuTûRd LôiL, Eg. 4.12

7, JÚ ùRôeÏ TôXj§u Lm© YPm TWYû[V Y¥®Ûs[Õ, ARu TôWm

¡ûPUhPUôL ºWôL TW®Ùs[Õ, AûRj RôeÏm CÚ çiLÞdÏ CûPúVÙs[ çWm 1500 A¥, Lm© YPjûRj RôeÏm ×s°Ls ç¦p RûW«−ÚkÕ 200 A¥ EVWj§p AûUkÕs[], úUÛm RûW«−ÚkÕ Lm© YPj§u RôrYô] ×s°«u EVWm 70 A¥. Lm©YPm 122 A¥ EVWj§p RôeÏm LmTj§tÏ CûPúV Es[ ùNeÏjÕ ¿[m LôiL, Eg. 4. 14

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PGT (Mathematics)

8, JÚ ùRôeÏ TôXj§u Lm© YPm TWYû[V Y¥®Ûs[Õ, ARu ¿[m 40 ÁhPo BÏm, Y¯lTôûRVô]Õ Lm© YPj§u ¸rUhPl ×s°«−ÚkÕ 5 ÁhPo ¸úZ Es[Õ, Lm© YPjûRj RôeÏm çiL°u EVWeLs 55 ÁhPo G²p. 30 ÁhPo EVWj§p Lm© YPj§tÏ JÚ ÕûQ Rôe¡ áÓRXôLd ùLôÓdLlThPôp AjÕûQjRôe¡«u ¿[jûRd LôiL, Ex. 4. 1 (5)

9, JÚ W«púY TôXj§u úUp Yû[Ü TWYû[Vj§u AûUlûTd ùLôiÓs[Õ,

AkR Yû[®u ALXm 100 A¥VôLÜm AqYû[®u Ef£l×s°«u EVWm TôXj§−ÚkÕ 10 A¥VôLÜm Es[Õ G²p. TôXj§u Uj§«−ÚkÕ CPl×\m ApXÕ YXl×\m 10 A¥ çWj§p TôXj§u úUp Yû[Ü GqY[Ü EVWj§p CÚdÏm G]d LôiL, Eg. 4.8

10, JÚ WôdùLh ùY¥Vô]Õ ùLôÞjÕmúTôÕ AÕ JÚ TWYû[Vl TôûR«p

ùNp¡\Õ, ARu EfN EVWm 4 Á-I GhÓmúTôÕ AÕ ùLôÞjRlThP CPj§−ÚkÕ ¡ûPUhP çWm 6 Á ùRôûX®Ûs[Õ, Cß§VôL ¡ûPUhPUôL 12 Á ùRôûX®p RûWûV YkRûP¡\Õ G²p ×\lThP CPj§p RûWÙPu HtTÓjRlTÓm G±úLôQm LôiL, Eg. 4. 10

¿sYhPm¿sYhPm¿sYhPm¿sYhPm (10 (10 (10 (10 úLs®Ls)úLs®Ls)úLs®Ls)úLs®Ls) (10)(10)(10)(10) ¿sYhPj§tÏ ûUVj ùRôûXj RLÜ. ûUVm. Ï®VeLs. Utßm Ef£LsB¡VYtû\d LôiL, úUÛm ARu Yû[YûWûVd LôiL,

1, 2 2 4 8 - 16 68 0+ − − =x y x y Ex 4.2 (6 ii)

2. 36x2 + 4y

2 − 72x + 32y − 44 = 0 Eg 4.31 (iv)

3. 16x2 + 9y

2 + 32x - 36y = 92 Ex 4.2 (6 iv)

4, JÚ Yû[Ü AûW-¿sYhP Y¥Yj§p Es[Õ, ARu ALXm 48 A¥. EVWm 20

A¥, RûW«−ÚkÕ 10 A¥ EVWj§p Yû[®u ALXm Gu]? Eg. 4. 32 5, JÚ TôXj§u Yû[Yô]Õ AûW ¿sYhPj§u Y¥®p Es[Õ, ¡ûPUhPj§p

ARu ALXm 40 A¥VôLÜm ûUVj§−ÚkÕ ARu EVWm 16 A¥VôLÜm Es[Õ G²p ûUVj§−ÚkÕ YXÕ ApXÕ CPl×\j§p 9 A¥ çWj§p Es[ RûWl×s°«−ÚkÕ TôXj§u EVWm Gu]? Ex. 4. 2 (10)

6, JÚ ÖûZÜ Yô«−u úUtáûWVô]Õ AûW-¿sYhP Y¥Yj§p Es[Õ, CRu

ALXm 20A¥ ûUVj§−ÚkÕ ARu EVWm 18 A¥ Utßm TdLf ÑYoL°u EVWm 12 A¥ G²p HúRàm JÚ TdLf ÑY¬−ÚkÕ 4 A¥ çWj§p úUtáûW«u EVWm Gu]YôL CÚdÏm? Eg. 4. 33

7, ã¬Vu Ï®Vj§−ÚdÏUôß ùUodÏ¬ ¡WLUô]Õ ã¬Vû] JÚ ¿sYhPl

TôûR«p Ñt± YÚ¡\Õ, ARu AûW ùShPf£u ¿[m 36 ªp−Vu ûUpLs BLÜm ûUVj ùRôûXj RLÜ 0,206 BLÜm CÚdÏUô«u (i) ùUodÏ¬ ¡WLUô]Õ ã¬VàdÏ ªL AÚLôûU«p YÚmúTôÕ Es[ çWm (ii) ùUodÏ¬ ¡WLUô]Õ ã¬VàdÏ ªLj ùRôûX®p CÚdÏmúTôÕ Es[ çWm B¡VYtû\d LôiL, Ex. 4.2 (9)

8, JÚ ¿sYhPl TôûR«u Ï®Vj§p éª CÚdÏUôß JÚ ÕûQdúLôs Ñt±

YÚ¡\Õ, CRu ûUVj ùRôûXj RLÜ ½ BLÜm éªdÏm ÕûQd úLôÞdÏm CûPlThP Áf£ß çWm 400 ¡úXô ÁhPoLs BLÜm CÚdÏUô]ôp éªdÏm ÕûQdúLôÞdÏm CûPlThP A§LThN çWm Gu]? Ex. 4. 2 (8)

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PGT (Mathematics)

9, JÚ úLô-úLô ®û[VôhÓ ÅWo ®û[VôhÓl T«t£«uúTôÕ AYÚdÏm úLô-úLô Ïf£LÞdÏm CûPúVÙs[ çWm GlùTôÝÕm 8Á BL CÚdÏUôß EQo¡\ôo, Aq®Ú Ïf£LÞdÏ CûPlThP çWm 6Á G²p AYo KÓm TôûR«u NUuTôhûPd LôiL, Ex. 4.2 (7)

10, JÚ NUR[j§u úUp ùNeÏjRôL AûUkÕs[ ÑY¬u ÁÕ 15Á ¿[Øs[ JÚ

H¦Vô]Õ R[j§û]Ùm ÑYt±û]Ùm ùRôÓUôß SLokÕ ùLôiÓ CÚd¡\Õ G²p. H¦«u ¸rUhP Øû]«−ÚkÕ 6Á çWj§p H¦«p AûUkÕs[ P Gu\ ×s°«u ¨VUlTôûRûVd LôiL, Eg. 4. 35

A§TWYû[VmA§TWYû[VmA§TWYû[VmA§TWYû[Vm. ùRôÓúLôÓ. ùRôûXjùRôÓúLôÓ. ùRôÓúLôÓ. ùRôûXjùRôÓúLôÓ. ùRôÓúLôÓ. ùRôûXjùRôÓúLôÓ. ùRôÓúLôÓ. ùRôûXjùRôÓúLôÓ (8 (8 (8 (8 úLs®Ls)úLs®Ls)úLs®Ls)úLs®Ls) (10)(10)(10)(10) A§TWYû[Vj§u ûUVj ùRôûXj RLÜ. ûUVm. Ï®VeLs. Ef£Ls B¡VYtû\d LôiL, úUÛm ARu Yû[YûWûV YûWL, 1, 9x2

− 16y2 - 18x - 64y - 199 = 0 Eg 4.56

2. x2 − 4y

2 + 6x + 16y - 11 = 0 Ex 4.3 (5 iii)

3. 9x2 − 16y

2 + 36x + 32y + 164 = 0 Eg 4.57

4. x2 − 3y

2 + 6x + 6y + 18 = 0 Ex 4.3 (5 iv)

5. x - y +4 = 0 Gu\ úSodúLôÓ ¿sYhPm x2 + 3y

2 = 12 Ij ùRôÓúLôPôL Es[Õ

G] ¨ì©dL, úUÛm ùRôÓm ×s°ûVÙm LôiL, Ex. 4.4 (6)

6. 5x + 12y = 9 Gu\ úSodúLôÓ A§TWYû[Vm x2 − 9y

2 = 9 Ij ùRôÓ¡\Õ G]

¨ì©dL, úUÛm ùRôÓm ×s°ûVÙm LôiL, Ex. 4.4 (5)

7, ×s° (2. 0) Y¯VôLf ùNpÛm JÚ A§TWYû[Vj§u ûUVm (2. 4) BÏm,

CRu ùRôûXj ùRôÓúLôÓLs x + 2y − 12 = 0 Utßm x − 2y + 8 = 0 Gu\ úLôÓLÞdÏ CûQVôL CÚl©u. AqY§TWYû[Vj§u NUuTôhûPd LôiL,

Ex. 4. 5 (2) (ii) 8. x + 2y − 5 = 0 I JÚ ùRôûXj ùRôÓúLôPôLÜm. (6. 0) Utßm (− 3. 0) Gu\

×s°Ls Y¯úV ùNpXdá¥VÕUô] ùNqYL A§TWYû[Vj§u NUuTôÓ LôiL, Ex. 4. 6 (3)

ùUn AhPYûQ (20 úLs®LsúLs®LsúLs®LsúLs®Ls) (6)(6)(6)(6) átßdÏ ùUn AhPYûQ AûUdL,

1, ( ) ( )p q∧ � � � 2 ( ) ( )∨ ∨p q r

3. ( ) ( )p q r∨ ∧ 4. ( ) ( )p q r∧ ∨

5. ( ) ( )p q r∧ ∨ �

6. ( ) ( ) ( )p q p q∨ ≡ ∧� � � G]d LôhÓL, Eg 9.7

7. ( ) ( ) ( )p q p q∧ ≡ ∨� � � G]d LôhÓL, Ex 9.3 (5)

8, ( )→ ≡ ∨�p q p q G]d LôhÓL, Ex 9.3 (2)

9. p ↔ q ≡ (( � p) ∨ q) ∧ (( � q) ∨ p) G]d LôhÓL, Ex 9.3 (4)

10. p ↔ q ≡ (p → q) ∧ (q → p) G]d LôhÓL, Ex 9.3 (3)

11. p → q Utßm q → p NUô]Ut\ûY G]d LôhÓL, Ex 9.3 (6)

ùUnûU G]d LôhÓL, 12. ( ) ( ( ))∨ ∨ ∨�p q p q 13. ( ( ))∨ ∨ �q p q

14. [( p) ] p( )q∨ ∨� � 15. (( � p) ∨ q) ∨ (p ∧ ( � q))

16. (p ∧ ( � q)) ∨ (( � p) ∨ q) 17. ( ) ( )p q p q∧ → ∨

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PGT (Mathematics)

ØWiTôÓ G]d LôhÓL, 18, [( ) ]∧ ∧� q p q 19. [( ) ]∧ ∧� p q p

20. (p ∧ ( � p)) ∧ (( � q) ∧ p)

ùUn AhPYûQ (16 úLs®LsúLs®LsúLs®LsúLs®Ls)

átßdÏ ùUn AhPYûQ AûUdL, 1, ( )∨ �p p 2. p ∨ ( � q)

3. ( )∨� p q 4. ( � p) ∨ ( � q)

5. ( ) ( )∧� �p q 6. � (( � p) ∧ q)

7. ( ( ))∨� �p q 8. ( ) ( )∨ ∨ �p q p

9. (p q) ( q)∨ ∧ � 10. (p q) ( q)∧ ∨ �

11. (p ∧ q) ∧ ( � q) 12. (p q) [ (p q)]∧ ∨ ∧�

13. JÚ át±u Ußl©u Ußl× Adátú\VôÏm G]d LôhÓL, 14. ‘Yô]j§u ¨\m ¿Xm’ Gu\ át±tÏ ( p) p≡� � GuTRû]f N¬TôodL, Eg 9.8

15, p ( p)∨ � JÚ ùUnûU G] ¨ì©, Eg 9.9 (i)

16, p ( p)∧ � JÚ ØWiTôÓ G] ¨ì©, Eg 9.9 (ii)

ÏXeLs (15 úLs®LsúLs®LsúLs®LsúLs®Ls) (10)(10)(10)(10)

1, éf£VUt\ LXlùTiL°u LQUô] C − {0} p YûWVßdLlThP f1 (z) = z,

f2 (z) = − z, 3

1( )f z

z= , 4

1( )f z

z= − z∀ ∈ C − {0}Gu\ Nôo×Ls VôÜm APe¡V

LQm {f1, f2, f3, f4} B]Õ Nôo×L°u úNol©u ¸r JÚ GÀ−Vu ÏXm AûUdÏm G] ¨ßÜL, Eg 9.24

2. 2 2

2 2

0 01 0 0 10 0, , , , ,

0 1 1 00 0 0 0

ω ωω ω

ω ω ω ω

Gu\ LQm A¦l

ùTÚdL−u ¸r JÚ ÏXjûR AûUdÏm G]d LôhÓL, ( ω3=1) Ex 9.4 (6) 3, (Z7 − {[0]}, .7) JÚ ÏXjûR AûUdÏm G]d LôhÓL, Eg 9.26

4. 11u UhÓdÏ LôQlùTt\ ùTÚdL−u¸r {[1], [3], [4], [5], [9]} Gu\ LQm JÚ GÀ−Vu ÏXjûR AûUdÏm G]d LôhÓL, Ex 9.4 (9)

5. (Zn, +n) JÚ ÏXm G]d LôhÓL, Eg 9.25 6, YZdLUô] ùTÚdL−u ¸r 1u nm T¥ êXeLs Ø¥Yô] GÀ−Vu ÏXjûR

AûUdÏm G]d LôhÓL, Eg 9.27

7. x x

x x

, x ∈ R − {0} Gu\ AûUl©p Es[ A¦Ls VôÜm APe¡V LQm G

B]Õ A¦lùTÚdL−u ¸r JÚ ÏXm G]d LôhÓL, Eg 9.21

8. 0

0 0

a

, a ∈ R − {0} AûUl©p Es[ GpXô A¦LÞm APe¡V LQm A¦l

ùTÚdL−u ¸r JÚ GÀ−Vu ÏXjûR AûUdÏm G]d LôhÓL, Ex 9.4 (11)

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PGT (Mathematics)

9. G GuTÕ ªûL ®¡RØß Gi LQm GuL,, a * b = 3

ab,a b G∀ ∈ GàUôß

YûWVßdLlThP ùNV− *u ¸r JÚ ÏXjûR AûUdÏm G]dLôhÓL, Ex 9.4 (5) 10. (Z, *) JÚ Ø¥Yt\ GÀ−Vu ÏXm G]d LôhÓL, CeÏ * B]Õ a * b = a + b + 2

GàUôß YûWVßdLlThÓs[Õ, Eg 9.18 11, 1 I R®W Ut\ GpXô ®¡RØß GiLÞm APe¡V LQm G GuL, G p * I

a * b = a + b − ab, ,a b G∀ ∈ GàUôß YûWVßlúTôm, (G, *) JÚ Ø¥Yt\ GÀ−Vu ÏXm G]d LôhÓL, Eg 9.23

12. −1 I R®W Ut\ GpXô ®¡RØß GiLÞm Es[Pd¡V LQm G B]Õ GpXô ,a b G∀ ∈ a * b = a + b + ab GàUôß YûWVßdLlThP ùNV− *-Cu ¸r JÚ

GÀ−Vu ÏXjûR AûUdÏm G]d LôhÓL, Ex 9.4 (8)

13, 2 G {a b / a, b Q}= + ∈ GuTÕ áhPûXl ùTôßjÕ JÚ Ø¥Yt\

GÀ−Vu ÏXm G]d LôhÓL, Eg 9.22

14, 2 nG { / n Z}= ∈ Gu\ LQUô]Õ ùTÚdL−u ¸r JÚ GÀ−Vu ÏXjûR

AûUdÏm G]d LôhÓL, Ex 9.4 (12)

15. | z | = 1 GàUôß Es[ LXlùTiLs VôÜm APe¡V LQm M B]Õ LXlùTiL°u ùTÚdL−u ¸r JÚ ÏXjûR AûUdÏm G]d LôhÓL, Ex 9.4 (7)

ÏXeLs (15 úLs®LsúLs®LsúLs®LsúLs®Ls) (6)(6)(6)(6)

1. 1Cu 3Bm T¥ êXeLs JÚ Ø¥Yô] GÀ−Vu ÏXjûR ùTÚdL−u ¸r AûUdÏm G]d LôhÓL, Eg 9.14

2, 1Cu 4Bm T¥ êXeLs JÚ Ø¥Yô] GÀ−Vu ÏXjûR ùTÚdL−u ¸r

AûUdÏm G]d LôhÓL, Eg 9.15

3, 1 0

0 1

, 0 1

1 0

Gu¡\ A¦Ls. A¦L°u ùTÚdL−u ¸r JÚ ÏXjûR

AûUdÏm G] ¨ßÜL, Ex 9.4 (4)

4, 1 0

0 1

, 1 0

0 1

−

, 1 0

0 1

−

, 1 0

0 1

− −

B¡V SôuÏ A¦LÞm APe¡V Eg 9.20

LQm A¦lùTÚdL−u ¸r JÚ GÀ−Vu ÏXjûR AûUdÏm G]d LôhÓL, 5. 2 × 2 Y¬ûN ùLôiP éf£VUt\ úLôûY A¦Ls VôÜm Ø¥Yt\ GÀ−Vu

ApXôR ÏXjûR A¦ ùTÚdL−u ¸r AûUdÏm G]d LôhÓL, Eg 9.19

6. (Z, +) JÚ Ø¥Yt\ GÀ−Vu ÏXm G] ¨ßÜL, Eg 9.12

7. (R − {0}, .) Ø¥Yt\ GÀ−Vu ÏXm G]d LôhÓL, Eg 9.13

8. (C, +) B]Õ JÚ Ø¥Yt\ GÀ−Vu ÏXm G] ¨ßÜL, Eg 9.16

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PGT (Mathematics)

9, éf£VUt\ LXlùTiL°u LQm. LXlùTiL°u YZdLUô] ùTÚdL−u ¸r JÚ GÀ−Vu ÏXm G]d LôhÓL, Eg 9.17

10. G = {1, − 1, i, − i} GuL, (G, .) ÏXj§p Es[ JqùYôÚ Eßl×dÏm Y¬ûNûVd

LôiL, Eg 9.28 11, YZdLUô] ùTÚdL−u ¸r G = {1, ω, ω2} Gu\ ÏXj§p JqùYôÚ Eßl©u

Y¬ûNûVd LôiL, Eg 9.29 12, (Z4, +4) Gu\ ÏXj§Ûs[ JqùYôÚ Eßl×dÏm Y¬ûNûVd LôiL, Eg 9.30

13, (Z5 − {[0]}, .5) Gu\ ÏXj§p Es[ JqùYôÚ Eßl×dÏm Y¬ûNûVd LôiL, Ex 9.4 (10) 14, ÏXj§u ¿dLp ®§Lû[ GÝ§ ¨ßÜL, Page 181

15. G JÚ ÏXm GuL, a, b ∈ G G²p (a * b)− 1= b−1 * a−1 G] ¨ì©, (ApXÕ) ApXÕ) ApXÕ) ApXÕ)

ÏXj§p G§oUû\«u ÁRô] §Úl×Rp (Y¬ûNUôtß) ®§ GÝ§ ¨ì©, Page 182

¨LrRLÜ¨LrRLÜ¨LrRLÜ¨LrRLÜl TWYpl TWYpl TWYpl TWYp (1(1(1(12222 úLs®Ls)úLs®Ls)úLs®Ls)úLs®Ls) (10)(10)(10)(10) 1, JÚ ùLôsLXj§p 4 ùYsû[ Utßm 3 £Yl×l TkÕLÞm Es[], 3

TkÕLû[ JqùYôu\ôL GÓdÏm úTôÕ. £Yl× ¨\lTkÕL°u Gi¦dûL«u ¨LrRLÜl TWYp (¨û\fNôo×) LôiL, (i) §ÚmT ûYdÏm Øû\«p (ii) §ÚmT ûYdLô Øû\«p Eg 10.3

2, JÚ NUYônl× Uô± X-Cu ¨LrRLÜ ¨û\fNôo× TWYp ©uYÚUôß Es[Õ :

X 0 1 2 3 4 5 6

P(X = x) k 3k 5k 7k 9k 11k 13k

(1) k-Cu U§l× LôiL, (2) P(X < 4), P(X ≥ 5) P(3< X ≤ 6) U§l× LôiL,

(3) P (X ≤ x) > 1

2 BL CÚdL x Cu Áf£ß U§l× LôiL, Eg 10.2

3, JÚ NUYônl× Uô± x Cu ¨LrRLÜ APoj§f Nôo×

1 x

kx e ; x, , 0f (x)0 ; ù \e¡ÛmUt

αα− β

β − α >=

G²p (i) k Cu U§l× LôiL (ii) P(X > 10) LôiL, Ex 10.1 (7)

4, IkÕ YVÕûPV JÚ EVokR YûL Sô«u ØÝ BÙhLôXm JÚ NUYônl×

Uô±VôÏm, ARu TWYp Nôo× (úNol×)

2

0 x 5

F(x) 251 x 5

x

≤

= − >

G²p 5 YVÕûPV Sôn (i) 10 BiÓLÞdÏ úUp

(ii) 8 BiÓLÞdÏd Ïû\YôL (iii) 12−ÚkÕ 15 BiÓLs YûW E«o YôrYRtLô] ¨LrRLÜ LôiL, Eg 10.10

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PGT (Mathematics)

5, JÚ SLWj§p YôPûL Yi¥ KhÓ]oL[ôp HtTÓm ®TjÕL°u Gi¦dûL Tôn^ôu TWYûX Jj§Úd¡\Õ, CRu TiT[ûY 3 G²p. 1000 KhÓSoL°p (i) JÚ YÚPj§p JÚ ®TjÕm HtTPôUp (ii) JÚ YÚPj§p êuß ®TjÕLÞdÏ úUp HtTÓjÕm KhÓ]oL°u Gi¦dûLûVd LôiL, [e−3

= 0.0498] Ex 10.4 (5) 6, JÚ úTÚkÕ ¨ûXVj§p. JÚ ¨ªPj§tÏ Esú[ YÚm úTÚkÕL°u

Gi¦dûL Tôn^ôu TWYûXl ùTt±Úd¡\Õ G²p. λ = 0.9 G]d ùLôiÓ. (i) 5 ¨ªP LôX CûPùY°«p N¬VôL 9 úTÚkÕLs Esú[ YW (ii) 8 ¨ªP LôX CûPùY°«p 10dÏm Ïû\YôL úTÚkÕLs Esú[ YW (iii)11 ¨ªP LôX CûPùY°«p Ïû\kRThNm 14 úTÚkÕLs Esú[ YW. ¨LrRLÜ LôiL, Eg 10.26

7, CVp¨ûX Uô± X u NWôN¬ 6 Utßm §hP ®XdLm 5 BÏm, (i) P(0 ≤ X ≤ 8) (ii) P( | X − 6 | < 10) B¡VYtû\d LôiL, P [0 < z < 1.2] = 0.3849 P [0 < z < 0.4] = 0.1554 P [0 < z < 2] = 0.4772 Eg 10.29

8, SÅ] £tßkÕL°p ùTôÚjRlTÓm NdLWeL°−ÚkÕ NUYônl× Øû\«p

úRokùRÓdLlTÓm NdLWj§u Lôt\ÝjRm CVp¨ûXl TWYûX Jj§Úd¡\Õ, Lôt\ÝjR NWôN¬ 31 psi. úUÛm §hP ®XdLm 0.2 psi G²p (i) (a) 30.5 psi dÏm 31.5 psidÏm CûPlThP Lôt\ÝjRm

(b) 30 psi dÏm 32 psi dÏm CûPlThP Lôt\ÝjRm G] CÚdÏmT¥VôL NdLWj§û] úRokùRÓdL ¨LrRLÜ LôiL,

(ii) NUYônl× Øû\«p úRokùRÓdLlTÓm NdLWj§u Lôt\ÝjRm 30.5 psi

dÏ A§LUôL CÚdL ¨LrRLÜ LôiL, P [0 < z < 2.5] = 0.4938 Eg 10.32

9, JÚ Ï±l©hP Lpí¬«p 500 UôQYoL°u GûPLs JÚ CVp¨ûXl TWYûX Jj§ÚlTRôLd ùLôs[lTÓ¡\Õ, Cru NWôN¬ 151 TÜiÓL[ôLÜm §hP ®XdLm 15 TÜiÓL[ôLÜm Es[] G²p (i) GûP 120 TÜiÓdÏm 155 TÜiÓdÏm CûPúVÙs[ UôQYoLs (ii) GûP 185 TÜiÓdÏ úUp ¨û\Ùs[ UôQYoL°u Gi¦dûL LôiL, Ex 10.5 (5) P [0 < z < 2.067] = 0.4803, P [0 < z < 0.2667] = 0.1026, P [0 < z < 2.2667] = 0.4881

10, JÚ úRo®p 1000 UôQYoL°u NWôN¬ U§lùTi 34 Utßm §hP ®XdLm 16

BÏm, U§lùTi CVp¨ûXl TWYûX ùTt±Úl©u (i) 30−ÚkÕ 60 U§lùTiLÞd¡ûPúV U§lùTi ùTt\ UôQYoL°u Gi¦dûL (ii) Uj§V 70% UôQYoLs ùTßm U§lùTiL°u GpûXLs CYtû\d LôiL, Eg 10.30 P [0 < z < 0.25] = 0.0987, P [0 < z < 1.63] = 0.4484, P [0 < z < 1.04] = 0.35

11, CVp¨ûXl TWY−u ¨LrRLÜ APoj§f Nôo× ( )2

2x 4xf x k e− += , − ∞ < x < ∞

G²p k, µ Utßm σ2 Cu U§l× LôiL, Eg 10.31

12, JÚ CVp¨ûXl TWY−u ¨LrRLÜl TWYp ( )2

x 3xf x c e− += , − ∞ < x < ∞ G²p.

c, µ, σ2 CYtû\d LôiL, Ex 10.5 (8)

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