xii maths come book model qps 3 mark questions … · xii maths come book model qps 3 mark...
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XII COME BOOK 3 marks © reserved with Miss N. Mahalakshmi
PGT (Mathematics)
XII MATHS COME BOOK MODEL QPs 3 MARK QUESTIONS (14)
2, 2, 2, 2, ùYdPo CVtL¦RmùYdPo CVtL¦RmùYdPo CVtL¦RmùYdPo CVtL¦Rm
1. ,ar
,br
crGuT] JußdùLôuß ùNeÏjÕ ùYdPoLs G²p a
r br
c
r = abc
G]d LôhÓ, CRu UßRûXÙm EiûU G]Üm LôhÓL, Ex. 2.5 (3)
2. 2 (3 4 ) 4r i j k+ − + =r r r
Gu\ úLô[j§u ûUVm. BWm LôiL, Ex. 2.11 (5) (ii)
5, YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs – I
1, 22 1 0y x+ + = Gu\ TWYû[Vj§tÏ (-1. 1) Gu\ ×s°«p ùRôÓúLôh¥u
NUuTôÓ LôiL, OBQ
2, xy e= . x
y e−= Yû[YûWLÞdÏ CûPlThP úLôQjûRd LôiL, OBQ
3. f(x) = 2x3 − 5x2
− 4x + 3, 1
2 ≤ x ≤ 3 úWô−u úRt\jûRl TVuTÓj§. cCu
U§l×Lû[d LiÓ©¥dL, Eg. 5.21 (iii)
6, YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs – II 1, YûLÂÓ dy LôiL, úUÛm ùLôÓdLlThP x Utßm dx-u U§l×LÞdÏ dy-u
U§l×Lû[ LQd¡ÓL, y = x4 − 3x3+ x −1, x = 2, dx = 0.1. Ex 6.1 (2) (ii)
2, u = sin cos
x y
y xx y
e ey x
+ G²p. u u
x yx y
∂ ∂+
∂ ∂= 0 G]d LôhÓL, Ex 6.3 (2) (ii)
8, 8, 8, 8, YûLdùLÝf NUuTôÓLsYûLdùLÝf NUuTôÓLsYûLdùLÝf NUuTôÓLsYûLdùLÝf NUuTôÓLs 1, ¾odL: ( )2 6 9 0+ + =D D y Eg 8.23
9, 9, 9, 9, R²¨ûX LQd¡VpR²¨ûX LQd¡VpR²¨ûX LQd¡VpR²¨ûX LQd¡Vp 1, ©uYÚm átßdÏ ùUn AhPYûQûV AûUdL : p ∨ ( � q) Ex 9.2 (1) 2, ©uYÚm átßdÏ ùUn AhPYûQûV AûUdL : ( � p) ∨ ( � q) Eg 9.4 (i)
3. ©uYÚm átßdÏ ùUn AhPYûQûV AûUdL : � (( � p) ∧ q) Eg 9.4 (ii)
4, ©uYÚm átßdÏ ùUn AhPYûQûV AûUdL : (p ∧ q) ∧ ( � q) Ex 9.2 (8)
10,10,10,10, ¨Lr¨Lr¨Lr¨LrRLÜl TRLÜl TRLÜl TRLÜl TWYpWYpWYpWYp 1, 5 ØVt£LÞs[. JÚ DÚßl×l TWY−u NWôN¬ Utßm TWYtT¥«u áÓRp 4,8
G²p TWYûXd LôiL, Eg 10.20
2. “JÚ DÚßl×l TWY−u NWôN¬ 6. Utßm §hP ®XdLm 3”, Cdátß ùUnVô
ApXÕ RY\ô? ®Y¬, Ex 10.3 (1)
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PGT (Mathematics)
XII PUBLIC EXAM - 3 MARK QUESTIONS (51)
1, 1, 1, 1, A¦Ls Utßm A¦dúLôûYL°uA¦Ls Utßm A¦dúLôûYL°uA¦Ls Utßm A¦dúLôûYL°uA¦Ls Utßm A¦dúLôûYL°u TVuTôÓLs TVuTôÓLs TVuTôÓLs TVuTôÓLs
1. 1 2
1 4
− −
Gu\ A¦«u úSoUôß A¦ûVd LôiL, Eg.1.5 (i)
2. A JÚ éf£VUt\ úLôûY A¦Vô«u (AT)
−1 = (A
− 1)
T GuTûR ¨ßÜL, Page 5 (2)
2, 2, 2, 2, ùYdPo CVtL¦RmùYdPo CVtL¦RmùYdPo CVtL¦RmùYdPo CVtL¦Rm 1. a,b
∧ ∧
GuT] CÚ AXÏ ùYdPoLs Utßm AYt±u CûPlThP úLôQm θ G²p
1
2 2sin a b
∧ ∧θ= − G] ¨ì©, Eg. 2.9
2, 2a i j k= + +rr rr
, 3 2b i j k= + −r rr r
G²p ( ) ( )3 . 2a b a b+ −r r r r
Id LôiL, Ex. 2.1 (2)
3, HúRàm Ko ùYdPo rr dÏ ( ) ( ) ( ). . .r r i i r j j r k k= + +
r urr r urr r urr r G] ¨ßÜL, Eg. 2.6
4. 2i j k− +rr r. 3 5i j k− −
rr r, 3 4 4i j k− + +
rr rGu\ ùYdPoLs Ko ùNeúLôQ
ØdúLôQj§u TdLeL[ôL AûUÙm G] ¨ì©, Eg. 2.11
5. 3 2i j k− +rr r
, 3 5i j k− +rr r
, 2 4i j k+ −rr rGuTûY JÚ ùNeúLôQ ØdúLôQjûR
EÚYôdÏm G]d LôhÓL, Ex. 2.1 (12)
6. 2 2i j k− +rr rGàm ùYdPÚdÏ CûQVô]Õm GiQ[Ü 5 EûPVÕUô] ®ûN.
JÚ ÕLû[ (1, 2, 3) Gu\ ×s°«p CÚkÕ (5, 3, 7) Gu\ ×s°dÏ SLojÕUô«u Aq®ûN ùNnÙm úYûXûVd LQd¡ÓL, Ex. 2.2 (6)
7. ,a br r
GuT] CWiÓ ùYdPoLs G²p 2 2 2 2
.a b a b a b× + =r r r r r r
G] ¨ßÜL,
Eg. 2.20
8. 13,a =r
5,b =r
Utßm . 60,a b =urr
G²p a b×r r
Id LôiL, Eg. 2.22
9. 4i j 3k− +r r r
, 2i j 2k− + −r r r
Gàm ùYdPoLÞdÏ ùNeÏjRô]Õm Gi A[Ü 6 EûPVÕUô] ùYdPoLû[d LôiL, Eg. 2.21
10. 3 2 4i j k+ −rr r
Gu\ ùYdPWôp RWlTÓm ®ûNVô]Õ (1. − 1. 2) Gu\ ×s°«p ùNÛjRlTÓ¡\Õ, (2, -1, 3) Gu\ ×s°ûVl ùTôßjÕ ®ûN«u §Úl×j§\u LôiL, Eg. 2.31
11. B(5, 2, 4) Gu\ ×s° Y¯f ùNVpTÓm ®ûN 4 2+ +rr r
i j k u ùYdPo §Úl×j
§\u A(3, − 1, 3) Gu\ ×s°ûVl ùTôßjÕ 2 8+ −rr r
i j k G]d LôhÓL, Ex. 2.4 (9)
12. 12 3 2 15. .i k j k i j kλ− + − + −r r rr r r r
Gu\ ùYdPoLû[ Øû]l×s°L[ôLd ùLôiP
CûQLWj§iUj§u L] A[Ü 546 G²p λ-Cu U§l× LôiL, Ex. 2.5 (2)
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13. ,ar
,br
crGuT] JÚR[ AûU ùYdPoLs G²p ,a b+
r r ,b c+r r
c a+r r
GuTûYÙm JÚR[ AûU ùYdPoLs BÏm, CRu UßRûXÙm EiûU GuTRû]d LôhÓL, Ex. 2.5 (1)
14. . 0,x a =r r
&& . 0,x b =
rr . 0x c =
r r úUÛm 0x ≠
rr G²p ,a
r ,br
cr
JúW R[ AûU ùYdPoLs G]d LôhÓL, Eg. 2.34
15, GkR JÚ ar-dÏm ( ) ( ) ( ) 2i a i j a j k a k a× × + × × + × × =
r r r r r r r r r r G] ¨ì©, Ex. 2.5 (9)
16. 5 7 4 2( ) ( )r i j i j kµ= − + − + +rr r r rr
Utßm 2 3 4( ) ( )r i k i kλ= − + + +r rr rr
Gu\ úLôÓL°u CûPlThPd úLôQm LôiL, Ex. 2.6 (9)
17. 1 4
2 3 6
y z= + = −x-1
Utßm
2 41
2 2
y zx
+ −+ = =
Gu\ úLôÓL°u
CûPlThPd úLôQm LôiL, Ex. 2.6 (8)
18. (3, 2, − 4), (9, 8, − 10) Utßm (λ, 4, − 6) JúW úLôhPûUl ×s°Ls G²p λ-Cu U§l× LôiL, Eg. 2.47
19. 2x − y + z = 4 Utßm x + y + 2z = 4 Gu\ R[eLÞdÏ CûPlThP úLôQm LôiL, Eg. 2.60 20. 2x + y - z = 9 Utßm x + 2y + z = 7 Gu\ R[eLÞdÏ CûPlThP úLôQm LôiL, Ex. 2.10 (1) (i)
21. ( ). 2 3 10r i j kλ+ − =r r r r
Utßm ( ). 3 5r i j kλ + + =r r r r
Gu\ R[eLs ùNeÏjÕ
G²p λ LôiL, Ex. 2.10 (3)
22. 2
.(4 + 2 - 6 ) - 11 0r r i j k− =r r r r r
Gu\ úLô[j§u ûUVm. BWm LôiL, Ex. 2.11 (5) (iv)
23. x2 + y2
+ z2 − 3x − 2y + 2z − 15 = 0 Gu\ úLô[j§u ®hPm AB Utßm A-Cu
BVjùRôûXLs (− 1, 4, − 3) G²p B-Cu BVjùRôûXLû[d LôiL, Ex. 2.11 (4)
3, LXlùTiLsLXlùTiLsLXlùTiLsLXlùTiLs
1. 1
1 i+ u ùUn. LtTû]l TϧLû[d LôiL, Ex. 3.1 (2)(i)
2, 3 1ω = G²p.
5 5
1 i 3 1 i 31
2 2
− + − −+ = −
G] ¨ßÜL, Ex. 3.5 (3)(ii)
5, 5, 5, 5, YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs – I 1. f(x) = 2
1 x− , −1 ≤ x ≤ 1 Gu\ Nôo×dÏ úWô−u úRt\jûRl TVuTÓj§. c Cu U§l× LôiL, Eg. 5.21 (i)
2, f(x) = sin x, 0 ≤ x ≤ π Gàm Nôo×dÏ úWô−u úRt\jûRf N¬TôdL, Ex. 5.3 (1)(i) 3, ©uYt±tÏ úWô−u úRt\jûRf N¬TôdL: f(x) = x3
− 3x + 3; 0 ≤ x ≤ 1 Eg. 5.22 (i)
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4. ex Gu\ Nôo×dÏ ùUdXô¬u ®¬Ü LôiL, Eg. 5. 28 (1)
5, U§l× LôiL: limx→∞
2
x
x
e Eg. 5.32
6. R p ex §hPUôL Hßm Nôo× G] ¨ì©dL, Ex. 5.7 (1)
7, x3/5
(4 − x) u UôߨûX GiLû[d LôiL, Eg. 5.47
8. y = 2 − x2 Gu\ Yû[YûW«u Ï¯Ü (Ï®Ü)-u NôoTLjûRd LôiL, Eg. 5.59
7, 7, 7, 7, ùRôûL ÖiL¦Rm : TVuTôÓLsùRôûL ÖiL¦Rm : TVuTôÓLsùRôûL ÖiL¦Rm : TVuTôÓLsùRôûL ÖiL¦Rm : TVuTôÓLs U§l× LôiL:
1 2
20
sin xdx
1 cos x
π
+∫ Eg 7.1 2. /2
3x
0e cos x dx
π
∫ Ex 7.1 (11)
3 1
20 4 −∫
dx
x Ex 7.1 (5) 4.
a2 2
0a x dx−∫ Eg 7.3
5. 2 6
0sin x dx
π
∫ Ex 7.3 (2) (i) 6. /4
3 3
/4x cos x dx
π
−π∫ Ex 7.2 (2)
7. 1
1
3 xlog dx
3 x−
−
+ ∫ Eg 7.6
8, 8, 8, 8, YûLdùLÝf NUuTôÓLsYûLdùLÝf NUuTôÓLsYûLdùLÝf NUuTôÓLsYûLdùLÝf NUuTôÓLs 1, 2xy = e (A + Bx) Gu\ NUuTôh¥tLô] YûLdùLÝf NUuTôhûP AûUdL,
Eg 8.2 (i)
2, ¾odL: ( )2D + D +1 y = 0 Eg 8.24
9, 9, 9, 9, R²¨ûX LQd¡VpR²¨ûX LQd¡VpR²¨ûX LQd¡VpR²¨ûX LQd¡Vp 1. (p q) ( q)∨ ∧ � Gu\ átßdÏ ùUn AhPYûQ AûUdL, Eg 9.4 (iii) 2, p ( p)∧ � JÚ ØWiTôÓ G] ¨ì©, Eg 9.9 (ii)
10,10,10,10, ¨LrRLÜl TWYp¨LrRLÜl TWYp¨LrRLÜl TWYp¨LrRLÜl TWYp 1, ©uYÚTY] ¨LrRLÜ APoj§f NôoTô G] N¬TôodLÜm:
2x 9 0 x 3f (x)
0 ù \e¡ÛmUt
≤ ≤=
Ex 10.1 (5) (a)
2. 11F(x) tan x
2
−π = +
π , x−∞ < < ∞ GuTÕ JÚ NUYônl× Uô± X Cu TWYp
Nôo× G²p P(0 ≤ x ≤ 1) I LôiL, Eg 10.7 3, JÚ DÚßl×l TWY−u NWôN¬ Utßm TWYtT¥«u ®j§VôNm 1 BÏm, úUÛm
AYt±u YodLeL°u ®j§VôNm 11 G²p n Cu U§l× LôiL, Eg 10.21 4, JÚ RûP RôiÓRp TkRVj§p JÚ ®û[VôhÓ ÅWo 10 RûPLû[j RôiP
úYiÓm, JÚYo JqùYôÚ RûPûVj RôiÓY§u ¨LrRLÜ 5/6 G²p. AYo CWi¥tÏm Ïû\Yô] RûPLû[ ÅrjÕY§u ¨LrRLÜ LôiL, Ex 10.3 (6)
5, Tôn^ôu TWYûX TVuTÓj§. ¨LrRL®u áÓRp Juß G] ¨ßÜL, Eg 10.22
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XII MATHS COME BOOK MODEL QPs
6 MARK QUESTIONS (30)
1, 1, 1, 1, A¦Ls Utßm A¦dúLôûYL°uA¦Ls Utßm A¦dúLôûYL°uA¦Ls Utßm A¦dúLôûYL°uA¦Ls Utßm A¦dúLôûYL°u TVuTôÓLsTVuTôÓLsTVuTôÓLsTVuTôÓLs
1, A = 1 2
1 4
−
− G²p. A (adj A) = (adj A) A = | A | . I2 GuTûRf N¬TôodL, Eg. 1.3
2, úSoUôß A¦ LôQp Øû\«p ©uYÚm úS¬V NUuTôhÓj ùRôÏl©û]j
¾odLÜm, 7x + 3y = -1, 2x + y = 0. Ex. 1.2 (2)
3, ©uYÚm ANUT¥jRô] NUuTôhÓj ùRôÏl©û] A¦dúLôûY«û]l
TVuTÓj§j ¾odL: x + y + 2z = 4, 2x + 2y + 4z = 8, 3x + 3y + 6z = 10. Eg. 1.18 (5)
2, 2, 2, 2, ùYdPo CVtL¦RmùYdPo CVtL¦RmùYdPo CVtL¦RmùYdPo CVtL¦Rm
1, Jo AûWYhPj§p Es[ úLôQm JÚ ùNeúLôQm, CRû] ùYdPo Øû\«p
¨ì©dL, Eg. 2.14 2, úLô[j§u ®hPm. úUtTWl©p HúRàm JÚ ×s°«p HtTÓjÕm úLôQm
ùNeúLôQm G]d LôhÓ, Ex. 2.11 (6)
3, 3, 3, 3, LXlùTiLsLXlùTiLsLXlùTiLsLXlùTiLs 1. z1, z2 Gu\ HúRàm CÚ LXlùTiLÞdÏ
(i) 11
2 2
zz
z z= (ii) 1
2
argz
z
= arg z1 − arg z2 G]d LôhÓL, Page 128
2, 2i, 1 + i, 4 + 4i Utßm 3 + 5i Gàm LXlùTiLs BoLu R[j§p JÚ
ùNqYLjûR EÚYôdÏm G]d LôhÓL, Eg. 3.14
3. (2+ 3 i) I JÚ ¾oYôLd ùLôiP x4
- 4x2 + 8x + 35 = 0 Gàm NUuTôhûPj ¾o,
Eg. 3.17 4, cos α + cos β + cos γ = 0 = sin α + sin β + sin γ G²p
cos 3α + cos 3β + cos 3γ = 3 cos (α + β + γ) Utßm sin 3α + sin 3β + sin 3γ = 3 sin (α + β + γ) G]d LôhÓL, Ex. 3.4 (3) (i)(ii)
5. (1 + cos θ + i sin θ)n + (1+cos θ − i sin θ)n
= 2n + 1 cosn (θ / 2) cos
2
nθ G] ¨ßÜL,
Ex. 3.4 (4) (iii)
4, 4, 4, 4, TÏØûTÏØûTÏØûTÏØû\\\\ Y¥YdL¦RmY¥YdL¦RmY¥YdL¦RmY¥YdL¦Rm 1, A§TWYû[Vj§u HúRàm JÚ ×s°«−ÚkÕ ARu ùRôûXj
ùRôÓúLôÓL°u ùNeÏjÕj çWeL°u ùTÚdÏjùRôûL JÚ Uô±− Gußm
ARu U§l× 2 2
2 2
a b
a b+G]Üm LôhÓL, Eg. 4.68
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2, xy = c2 Gu\ ùNqYL A§TWYû[Vj§u HúRàm JÚ ×s°«p YûWVlTÓm
ùRôÓúLôÓ x, y AfÑdL°p ùYhÓm ÕiÓLs a, b G]Üm Cl×s°«p ùNeúLôh¥u ùYhÓm ÕiÓLs p, q G]Üm CÚl©u ap + bq = 0 G]d LôhÓL,
Eg. 4.70
5, 5, 5, 5, YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs – I
1, JÚ ØdúLôQj§u CWiÓ TdLeL°u ¿[eLs Øû\úV 4Á. 5Á BÏm, Utßm AYt±tÏ CûPlThP úLôQ A[®u Hßm ÅRm ®]ô¥dÏ 0,06 úW¥Vu G²p. ¨ûXVô] ¿[eLû[ EûPV AkR TdLeLÞdÏ CûPúV úLôQ A[Ü π/3BL CÚdÏm úTôÕ. ARu TWl©p HtTÓm Ht\ ÅRm LôiL, Ex 5. 1 (7)
2, JÚ Ï°o NôR]l ùTh¥«−ÚkÕ GÓjÕ. ùLô§dÏm ¿¬p ûYjR EPu JÚ
ùYlTUô² −19°CC−ÚkÕ 100° BL Uô\ 14 ®]ô¥Ls GÓjÕd ùLôs¡\Õ, CûP«p HúRàm JÚ úSWj§p TôRWNm N¬VôL 8.5°C/sec. Gu\ ÅRj§p Hß¡\Õ Guß Lôi©dL, Eg. 5.27
3. f(x) = 2x3 + x2
−20x Gu\ Nôo©u Hßm Utßm C\eÏm CûPùY°Lû[d LôiL, Eg. 5.38 4. xCu GkR U§l©tÏ f(x) = 2x3
− 15x2 + 36x + 1 Gu\ Nôo× Hßm úUÛm GkR
U§l©tÏ C\eÏm? úUÛm GkRl ×s°L°p Nôo©u Yû[YûWdÏ YûWVlTÓm ùRôÓúLôÓLs x AfÑdÏ CûQVôL CÚdÏm? Eg. 5.42
5, ùLôÓdLlThP CûPùY°dÏ fCu ÁlùTÚ ùTÚU Utßm Áf£ß £ßU
U§l×Lû[d LôiL,, ( )1
xf x
x=
+ [1, 2] Ex. 5. 9 (2) (v)
6, YûL ÖiL¦Rm : TVuTôÓLs 6, YûL ÖiL¦Rm : TVuTôÓLs 6, YûL ÖiL¦Rm : TVuTôÓLs 6, YûL ÖiL¦Rm : TVuTôÓLs – II 1. x = r cos θ, y = r sin θ Guß CÚdÏUôß w = log (x2
+ y2) G] YûWVßdLlTÓ¡\Õ
G²p w
r
∂
∂ Utßm
w
θ
∂
∂ Id LôiL, Ex 6.3 (4)(i)
2. x = u + v, y = u − v Guß CÚdÏUôß w = sin−1 (xy) G] YûWVßdLlTÓ¡\Õ G²p.
w
u
∂
∂Utßm
w
v
∂
∂ Id LôiL, Ex 6.3 (4)(iii)
7, 7, 7, 7, ùRôûL ÖiL¦Rm : TVuTôÓLsùRôûL ÖiL¦Rm : TVuTôÓLsùRôûL ÖiL¦Rm : TVuTôÓLsùRôûL ÖiL¦Rm : TVuTôÓLs
1, U§l©ÓL : 1
4
0
xxe dx
−
∫ Eg 7.17 (ii)
2. y2 = 4ax Gu\ TWYû[Vj§tÏm ARu ùNqYLXj§tÏm CûPlThP TWl©û]d
LôiL, Ex 7.4 (6)
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3, 2 2
2 21
x y
a b+ = (a > b > 0) Gu\ ¿sYhPm HtTÓjÕm TWl©û]. Ït\fûNl
ùTôßjÕf ÑZt±]ôp HtTÓm §PlùTôÚ°u L] A[Ü LôiL, Eg 7.35
8, 8, 8, 8, YûLdùLÝf NUuTôÓLsYûLdùLÝf NUuTôÓLsYûLdùLÝf NUuTôÓLsYûLdùLÝf NUuTôÓLs
1. x-Af£u ÁÕ ûUVm Utßm KWXÏ BWm ùLôiP YhPj ùRôÏl©u
YûLdùLÝf NUuTôhûP AûUdL, Ex 8.1 (4)
2, ¾odL: x x 23e tany dx + (1+ e )sec y dy = 0 Eg 8.4
3, ¾odL: tandy y y
dx x x= + Eg 8.12
4, ¾odL: ( )2 2 3 sin cosD D y x x− − = Ex 8.5 (8)
10,10,10,10, ¨Lr¨Lr¨Lr¨LrRLÜl TWYpRLÜl TWYpRLÜl TWYpRLÜl TWYp 1, Su\ôLd LûXdLlThP 52 ºhÓdL[Pe¡V ºhÓdLh¥−ÚkÕ CÚ ºhÓLs
§ÚmT ûYdÏm Øû\«p GÓdLlTÓ¡u\], Hv (ace) ºhÓL°u Gi¦dûLdÏ NWôN¬Ùm. TWYtT¥Ùm LôiL, Ex 10.2 (4)
2, JÚ ùRô¯tNôûX«p EtTj§VôÏm RôrlTôsL°p 20% Ïû\ÙûPVûYVôL
Es[], 10 RôrlTôsLs NUYônl× Øû\«p GÓdLlTÓm úTôÕ N¬VôL 2 RôrlTôsLs Ïû\ÙûPVûYVôL CÚdL (i) DÚl×l TWYp (ii) Tôn^ôu TWYp êXUôL ¨LrRLÜ LôiL, [e−2
= 0.1353]. Ex 10.4 (3)
3, JÚ RÓl× F£«u TdL ®û[Yôp Tô§dLlTÓYRtLô] ¨LrRLÜ 0,005
BÏm, 1000 SToLÞdÏ RÓl× F£ úTôÓm ùTôÝÕ (i) A§LThNm 1 STo Tô§dLlTP ii) 4. 5 ApXÕ 6 SToLs Tô§dLlTP ¨LrRLÜ LôiL, [e−5
= 0.0067]
Eg 10.24 4, úTôo ÅWoL°u LôX¦L°u BÙhLôXm CVp¨ûXl TWYûX Jj§Úd¡\Õ,
CkRl TWY−u NWôN¬ 8 UôRUôLÜm. §hP®XdLm 2 UôRUôLÜm AûU¡\Õ, 5000 úNô¥ LôX¦Ls A°dLlThP úTôÕ. GjRû] úNô¥Lû[ 12 UôReLÞdÏs[ôL Uôt\lTP úYiÓùU] G§oTôodLXôm? Ex 10.5 (4)
************ NO SUBSTITUTE FOR HARD WORK ***********
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XII PUBLIC EXAM - 6 MARK QUESTIONS (184)
1, A¦Ls Utßm A¦dúLôûYL°u TVuTôÓLs
1. A =
4 3 3
1 0 1
4 4 3
− − −
Cu úNol× A¦ A G] ¨ßÜL, Ex. 1.1 (8)
2. A = 1 2
3 5
− Gu\ A¦«u úNolûTd LiÓ. A (adj A) = (adj A) A = | A | . I
GuTûRf N¬TôodL, Ex. 1.1 (2)
3.
3 1 1
2 2 0
1 2 1
−
− −
Gu\ A¦«u úSoUôß A¦ LôiL, Eg. 1.5 (iv)
(AB)−1 = B−1 A−1
GuTûRf N¬Tôo,
4, A = 1 2
1 1
Utßm B = 0 1
1 2
−
Eg. 1.6
5, A = 5 2
7 3
Utßm B = 2 1
1 1
− −
Ex. 1.1 (5)(i)
6. A =
1 2 2
4 3 4
4 4 5
− −
− −
G²p A = A−1 G]d LôhÓL, Ex. 1.1 (10)
7. 2 3
5 6A
− − =
− G²p. ( ) ( )
11
TT
A A−
− = GuTûRf N¬TôodL, OBQ
8. A, B CWiÓ éf£VUt\ úLôûY A¦Ls G²p, (AB)−1 = B−1
A−1 G] ¨ßÜL,
úSoUôß A¦ LôQp Øû\«p ¾odL:
9. x + y = 3, 2x + 3y = 8 Eg. 1.7
10. 2x − y = 7, 3x − 2y = 11 Ex. 1.2 (1) A¦«u RWm LôiL,
11.
1 2 3 1
2 4 6 2
3 6 9 3
−
− −
Eg. 1.14 12.
1 2 1 3
2 4 1 2
3 6 3 7
−
− −
Ex 1.3 (5)
13.
1 2 3 4
2 4 1 3
1 2 7 6
− − − − −
Ex. 1.3 (6) 14.
3 1 5 1
1 2 1 5
1 5 7 2
− −
− − −
Eg. 1.16
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15.
3 1 2 0
1 0 1 0
2 1 3 0
−
Ex. 1.3 (3) 16.
0 1 2 1
2 3 0 1
1 1 1 0
− − −
Ex. 1.3 (4)
17.
2 1 3 4
0 1 1 2
1 3 4 7
−
OBQ
©uYÚm ANUT¥jRô] úS¬V NUuTôhÓ ùRôÏl©û] A¦dúLôûY Øû\«p ¾odL:
18, 2x + 3y = 8 4x + 6y = 16 Eg 1.17 (2)
19. 4x + 5y = 9 8x + 10y = 18 Ex 1.4 (3) 20. 2x - 3y = 7 4x - 6y = 14 OBQ
21. 2x + 2y + z = 5 x − y + z = 1 3x + y + 2z = 4 Eg 1.18 (3) ©uYÚm NUuTôÓL°u ùRôÏl©u JÚeLûUÜj RuûUûVj RW Øû\ûVl TVuTÓj§ BWônL, 22. x + y + z = 7 x + 2y + 3z = 18 y + 2z = 6 Ex. 1.5 (1)(iii)
23. x – 4y +7z = 14 3x + 8y – 2z = 13 7x – 8y + 26z = 5 Ex. 1.5 (1)(iv)
2, ùYdPo CVtL¦Rm
1. KÚ Nôn NÕWj§u êûX ®hPeLs Juû\ Juß ùNeÏjRôL ùYh¥d ùLôsÞm GuTRû] ùYdPo Øû\«p ¨ßÜL, Eg. 2.15
2, ùYdPo Øû\«p sin sin sin
a b c
A B C= = Guß ¨ì©dL, Eg. 2.28
3. 4 3 2i j k− −r r r
I ¨ûX ùYdPWôLd ùLôiP ×s° PCp ùNVpTÓm ®ûNLs
2 7i j+r r
, 2 5 6i j k− +r r r
Utßm 2i j k− + −r r r
BÏm, CûYL°u ®û[Ü ®ûN«u
§Úl×j §\û] 6 3i j k+ −r r r
-I ¨ûX ùYdPWôLd ùLôiP Q Gu\ ×s°ûVl ùTôßjÕd LôiL, Ex. 2.4 (8)
4. [a b−r r
b c−r r
c a−r r
] = 0 G] ¨ßÜL, OBQ
5. A(1, 2, 3), B(3, -1, 2), C(−2, 3, 1), D(6, -4, 2) B¡V ×s°Ls JúW R[j§p AûUÙm G]d LôhÓL, OBQ
6 .GpXô ùYdPoLs , ,ur ur ra b cdÏm.
2, , × × × =
r r r r r r r r ra b b c c a a b c G] ¨ßÜL, Eg. 2.38
7. (3, − 4, − 2) Gu\ ×s°Y¯f ùNpYÕm 9 6 2i j k+ +r r r
Gu\ ùYdPÚdÏ CûQVô]ÕUô] úLôh¥u ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL,
Ex. 2.6 (6)
8. ( ) ( ) 2= − + − +r r r r rrr i j t i j k Utßm ( ) ( )2 2= + + + − +
r r r r r rrr i j k s i j k Gu\ CûQ
úLôÓL°u CûPlThP çWjûRd LôiL, Eg. 2.42
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9. ( ) ( )3 5 7 2r i j k t i j k= + + + − +r r r r r r r
Utßm ( ) ( ) s 7 6 7r i j k i j k= + + + + +r r r r r r r
GuT]
JÚ R[j§p AûUVôR úLôÓLs G]d LôhÓL, Ex. 2.7 (2)
10. ( ) ( ) 2r i j t i k= − + +r r r r r
Utßm ( ) ( )2 r i j s i j k= − + + −r r r r r r
Gu\ CÚ úLôÓLs JúW
R[ AûUVôd úLôÓLs G]d Lôh¥. AYt±tÏ CûPlThP çWjûRÙm LôiL, Eg. 2.43
11. (3, − 1, − 1), (1, 0, − 1) Utßm (5, − 2, − 1) Gu\ ×s°Ls JÚ úLôhPûUl ×s°Ls G]d LôhÓL, Eg. 2.46
12. ( ) ( )2 3 2r i j k t i j k= + − + − −r r r r r r r
Gu\ úLôÓm x − 2y + 3z + 7 = 0 Gu\ R[Øm
Nk§d¡u\ ×s°ûVd LôiL, Ex. 2.9 (4)
13. 2 3i j k− +r r r
Gàm ¨ûX ùYdPûW EûPV ×s°ûV ûUVUôLÜm 4 AXÏLû[ BWUôLÜm ùLôiP úLô[j§u ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, Ex. 2.11 (1)
14, (1. −1. 1)I ûUVUôLÜm r (i j 2k 5− + + =r r r r
Gu\ úLô[j§u BWj§tÏ NUUô]
U§lûT BWjûRd ùLôiP úLô[j§u ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[j RÚL, Ex. 2.11 (3)
15. (5. 5. 3) Gu\ ×s° Y¯f ùNpYÕm (1. 2. 3) ûUVUôLÜm AûUÙm úLô[j§u
ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, Eg. 2.63
16. 2 6 7i j k+ −r r r
Utßm 2 4 3i j k− +r r r
Gàm ùYdPoLû[ ¨ûX ùYdPoL[ôLd ùLôiP ×s°Ls Øû\úV A, B, CRû] CûQdÏm ×s°Lû[ ®hPUôLd ùLôiP úLô[j§u NUuTôÓ RÚL, Eg. 2.64
17. 2 6 7i j k+ −r r r
Utßm 2 4 3i j k− + −r r r
Gàm ¨ûX ùYdPoLû[ÙûPV ×s°Ls Øû\úV A, B BÏm, CYtû\ CûQdÏm úLôhûP ®hPUôLd ùLôiP úLô[j§u ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL. úUÛm ûUVm Utßm BWm LôiL, Ex. 2.11 (2)
18, 2
.(8 - 6 +10 ) - 50 0r r i j k− =r r r r r
Gu\ ùYdPo NUuTôhûPÙûPV úLô[j§u ûUVjûRÙm BWjûRÙm LôiL, Eg. 2.65
3, LXlùTiLs
1, ©uYÚm NUuTôhûP ¨û\Ü ùNnÙm x Utßm y-«u ùUn U§l×Lû[d
LôiL, 2x + 3x + 8 + (x + 4)i = y(2 + i) . Ex. 3.1 (4) (iii)
2. (− 8 − 6i) -Cu YodL êXeLs LôiL, Ex. 3.2 (2)
3. (− 7 + 24i) -Cu YodL êXeLs LôiL, Eg. 3.16 4, LXlùTiL°u ØdúLôQ NU²−ûV Gݧ ¨ì©, Page 124
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5. z1 Utßm z2 Gu\ CÚ LXlùTiLÞdÏ (i) | z1 z2 | = | z1 |.| z2 | (ii) arg (z1.z2) = arg z1 + arg z2 GuTûR ¨ì©, Page 127
6, ÑÚdÏL: ( )
4
5
cos sin
(sin cos )
i
i
θ θ
θ θ
+
+ Eg. 3.19
P Guàm ×s° LXlùTi Uô± z Id ϱjRôp P-Cu ¨VUlTôûRûV ©uYÚm ¨TkRû]dÏ EhThÓ LôiL,
7. 2 1 2z z− = − OBQ
8. 3 3z i z i− = + OBQ
9. 3 5 3 1− = +z z OBQ
10. 1
Re 0z
z i
+ =
− OBQ
11. (7 + 9i), (− 3 + 7i), (3 + 3i) Gàm LXlùTiLs BoLu R[j§p JÚ ùNeúLôQ ØdúLôQjûR AûUdÏm G] ¨ßÜL, Eg. 3.15
12, LXlùTi R[j§p (10 + 8i), (− 2 + 4i), (-11 + 31i) Gàm LXlùTiLs AûUdÏm ØdúLôQm JÚ ùNeúLôQ ØdúLôQm G] ¨ßÜL, Ex. 3.2 (4)
13. ùUnùVi ÏQLeLû[d ùLôiP P(x) = 0 Gu\ TpÛßl×d úLôûYf NUuTôh¥u LXlùTi êXeLs CûQùVi CWhûPVôLjRôu CPmùTßm G] ¨ßÜL, Page 140
14. 3 + i I JÚ ¾oYôLd ùLôiP x4 − 8x3
+ 24x2 − 32x + 20 = 0 Gàm NUuTôh¥u ©\
¾oÜLû[d LôiL, Ex. 3.3 (1)
15. 1 + 2i JÚ êXUôLd ùLôiP x4 − 4x3
+ 11x2 − 14x + 10 = 0 Gàm NUuTôh¥u
¾oÜLû[d LôiL, Ex. 3.3 (2)
16. (1+i) I JÚ ¾oYôLd ùLôiP x4 + 4 = 0 Gàm NUuTôh¥u ¾oÜLû[d LôiL,
OBQ 17. (1 - i) I JÚ ¾oYôLd ùLôiP x
3 - 4x
2 +6x – 4= 0 Gàm NUuTôh¥u ¾oÜLû[d
LôiL, OBQ
18. x = cos α + i sin α, y = cos β + i sin β G²p 1m n
m nx y
x y+ = 2 cos (mα + nβ) G]
¨ì©, Ex. 3.4 (9)
19. n GuTÕ ªûL ØÝ Gi G²p 1 sin cos
cos sin1 sin cos 2 2
+ + = − + −
+ −
ni
n i ni
θ θ π πθ θ
θ θ
G] ¨ì©dL, Eg. 3.20
20. cos α + cos β + cos γ = 0 = sin α + sin β + sin γ G²p cos 2α + cos 2β + cos 2γ = 0
Utßm sin 2α + sin 2β + sin 2γ = 0 G]d LôhÓL, Ex. 3.4 (3)(iii)(iv)
21. n GuTÕ ªûL ØÝ Gi G²p (1 + i)n + (1 − i)
n =
2
22n+
cos 4
nπ G] ¨ì©,
Ex. 3.4 (4) (i)
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22. n ∈ N G²p (1 + i 3 )n + (1 − i 3 )
n =
12n+ cos
3
nπ G] ¨ßÜL, Ex. 3.4 (4) (ii)
23, 1
x + = 2cosθx
G²p (i) n
n
1x + = 2cosnθ
x (ii) n
n
1x = 2isinnθ
x− Ex. 3.4 (7)
24, ¾odL : x4
+4 = 0. Ex. 3.5 (4) (i)
4, TÏØû\ Y¥YdL¦Rm 1, JÚ CÚfNdLW YôL]j§u ØLl× ®[d¡p Es[ ©W§T−lTôu JÚ
TWYû[V AûUl©p Es[Õ, ARu ®hPm 12 ùN,Á. BZm 4 ùN,Á G²p ARu Af£p Gq®Pj§p Tp©û] (bulb) ùTôÚj§]ôp ØLl× ®[dÏ ªLf £\kR Øû\«p J°ûVj RWØ¥Ùm G]d LQd¡ÓL, Eg. 4.9
2, JÚ ¿sYhPj§u Ï®VeLs (2. 1). (−2. 1) Utßm ùNqYLXj§u ¿Xm 6 G²p
ARu NUuTôhûPd LôiL, Eg. 4.24
3, CVdÏYûW 2x + y – 1 = 0, Ï®Vm (1, 2) úUÛm ûUVj ùRôûXjRLÜ 3 G²p A§TWYû[Vj§u NUuTôhûPd LôiL, Eg. 4.36
4, ûUVm: (0. 0); AûWd ÏßdLf£u ¿[m 6; e = 3, úUÛm ÏßdLfÑ. y-AfÑdÏ
CûQVôL Es[Õ, Cq®TWeLÞdϬV A§TWYû[Vj§u NUuTôhûPd LôiL, Ex. 4.3 (1) (iii)
5, ûUVm (2. 1) úUÛm JÚ Ï®Vm (8. 1) G]Üm CRtùLôjR CVdÏYûW x = 4
G]Üm EûPV A§TWYû[Vj§u NUuTôhûPd LôiL, Eg. 4.43 6, x2
+ 2x − 4y + 4 = 0 Gu\ TWYû[Vj§tÏ (0, 1) Gu\ ×s°«p ùRôÓúLôÓ. ùNeúLôÓ CYt±u NUuTôÓLs LôiL, Ex. 4.4 (1) (iii)
7, (1, 2)−ÚkÕ 2x2
− 3y2 = 6 Gu\ A§TWYû[Vj§tÏ YûWVjRdL CÚ
ùRôÓúLôÓL°u NUuTôÓLû[d LôiL, Ex. 4.4 (4) (iii) 8. 3x2
− 5xy − 2y2 + 17x + y + 14 = 0 Gu\ A§TWYû[Vj§u ùRôûXj
ùRôÓúLôÓL°u R²jR²f NUuTôÓLû[d LôiL, Eg. 4.64 9. 2x + 3y − 8 = 0 Utßm 3x − 2y + 1 = 0 GuTYtû\j ùRôûXj ùRôÓúLôÓL[ôLÜm.
(5, 3) Gu\ ×s° Y¯VôLf ùNpÛm A§TWYû[Vj§u NUuTôhûPd LôiL, Ex. 4.5 (2) (i)
10, 4x2 − 5y2
− 16x + 10y + 31 = 0 Gu\ A§TWYû[Vj§u ùRôûXj ùRôÓúLôÓLLÞdÏ CûPlThP úLôQjûRd LôiL, Ex. 4.5 (3) (iii)
11, 3x2
− 5xy − 2y2 + 17x + y + 14 = 0 Gu\ A§TWYû[Vj§u ùRôûXj
ùRôÓúLôÓLÞdÏ CûPlThP úLôQjûRd LôiL, Eg. 4.67 12, JÚ §hP ùNqYL A§TWYû[Vj§u Øû]Ls (5. 7) Utßm (− 3.− 1) BLÜm
CÚl©u. ARu NUuTôhûPÙm. ùRôûXj ùRôÓúLôÓL°u NUuTôÓLû[Ùm LôiL, Ex. 4.6 (4)
13, ùNqYL A§TWYû[Vj§u HúRàm JÚ ×s°«PjÕ YûWVlTÓm ùRôÓúLôÓ.
ùRôûXj ùRôÓúLôÓLÞPu AûUdÏm ØdúLôQj§u TWl× JÚ Uô±− G] ¨ßÜL, Ex. 4.6 (7)
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5, YûL ÖiL¦Rm : TVuTôÓLs – I
1. KWXÏ ¨û\ÙûPV JÚ ÕLs t ®]ô¥ úSWj§p HtTÓjÕm CPlùTVof£ x = 3 cos (2t – 4) G²p. 2 ®]ô¥L°u Ø¥®p ARu ØÓdLm Utßm ARu CVdL Bt\p (K.E.) ØR−VYtû\d LôiL, Ex. 5.1 (2)
2. y = x3 Gàm Yû[YûWdÏ (1,1) Gu\ ×s°«p YûWVlTÓm ùRôÓúLôÓ.
ùNeúLôÓ B¡VYt±u NUuTôÓLû[d LôiL, Eg. 5. 10
3. y = x2 – x – 2 Gàm Yû[YûWdÏ (1,− 2) Gu\ ×s°«p YûWVlTÓm ùRôÓúLôÓ.
ùNeúLôÓ B¡VYt±u NUuTôÓLû[d LôiL, Eg. 5. 11
4. 2x2 + 4y2
= 1 Utßm 6x2 – 12y2= 1 Gàm Yû[YûWLs Juû\ Juß ùNeÏjRôL
ùYh¥d ùLôsÞm G]d LôhÓL, Ex. 5.2 (8) 5. f(x) = x3
Gu\ Nôo©tÏ [−2,2] Gu\ CûPùY°«p XôdWôg£«u CûPU§l×j úRt\jûR N¬TôodLÜm, Eg. 5. 24
6. f(x) = x3
− 5x2 − 3x , [1,3] Gu\ Nôo×dÏ ùXdWôg£«u CûPU§l×j úRt\jûR
N¬TôodLÜm, Ex. 5.4 (1) (v)
7. f(x) = 2x3 + x2
− x - 1 , [0, 2] Gu\ Nôo×dÏ ùXdWôg£«u CûPU§l×j úRt\jûR N¬TôodLÜm, Ex. 5.4 (1) (iii)
8, U§Vm 2,00 U¦dÏ JÚ £tßk§u úYLUô² 30 ûUpLs/U¦ G]Üm 2,10
U¦dÏ úYLUô² 50 ûUpLs/U¦ G]Üm LôhÓ¡\Õ, 2,00 U¦dÏm 2,10 U¦dÏm CûPlThP HúRô JÚ NUVj§p ØÓdLm N¬VôL 120 ûUpLs/U¦2 BL CÚk§ÚdÏm G]d LôhÓL, Ex. 5. 4 (3)
9. 1
1 x+ Gu\ Nôo×dÏ ùUdXô¬u ®¬Ü LôiL, Ex. 5. 5 (3)
10. log e (1 + x) Gu\ Nôo×dÏ ùUdXô¬u ®¬Ü LôiL, Eg. 5. 28 (2)
11. tan x, 2
π− < x <
2
π Gu\ Nôo×dÏ ùUdXô¬u ®¬Ü LôiL, Ex. 5. 5 (4)
12, U§l× LôiL: 0
limx→
1
cosecxx
−
Eg. 5. 33
13, U§l× LôiL: cos
0lim→ +
x
xx OBQ
14. 1
limx→
1
1xx − u U§l©û]d LôiL, Ex. 5.6 (10)
15. f(x) = 20 − x − x2 Gu\ Nôo©u Hßm ApXÕ C\eÏm CûPùY°Lû[d LôiL, Ex. 5.7 (5) (i)
16. f(x) = x3
− 3x + 1 Gu\ Nôo©u Hßm ApXÕ C\eÏm CûPùY°Lû[d LôiL, Ex. 5.7 (5) (ii)
17. f(x) = x3 − 3x2
+ 1 , − ½ ≤ x ≤ 4 Gu\ Nôo©u ÁlùTÚ ùTÚUm Utßm Áf£ß £ßU U§l×Lû[d LôiL, Eg. 5. 48
18, CWiÓ GiL°u áÓRp 100 AqùYiL°u ùTÚdÏj ùRôûL ùTÚU
U§lTôL ¡ûPdL AqùYiLs Gu]YôL CÚdL úYiÓm? Ex. 5. 10 (1)
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19, CWiÓ ªûL GiL°u ùTÚdÏj ùRôûL 100. AqùYiL°u áÓRp £ßU U§lTôL ¡ûPdL AqùYiLs Gu]YôL CÚdL úYiÓm? Ex. 5. 10 (2)
20. y = x3
− 3x + 2 Gu\ Nôo©tÏ Yû[Ü Uôt\l ×s°Ls CÚl©u AYtû\d LôiL, Eg. 5. 65
21. ( )1
3f x (x-1)= Gu\ Nôo× GkR CûPùY°L°p Ï¯Ü AûP¡\Õ
GuTûRÙm Utßm Yû[Ü Uôtßl ×s°Lû[Ùm LôiL, Ex. 5. 11 (1)
22. f(x) = 2x3 + 5x2
− 4x Gu\ Nôo× GkR CûPùY°L°p Ï¯Ü AûP¡\Õ GuTûRÙm Utßm Yû[Ü Uôtßl ×s°Lû[Ùm LôiL, Ex. 5. 11 (3)
23. y = sinx, x ∈ (0, 2π) Gu\ Yû[YûW«u Yû[Ü Uôtßl ×s°Lû[d LôiL,
Eg. 5. 66
6, YûL ÖiL¦Rm : TVuTôÓLs – II
1, YûLÂhûPl TVuTÓj§ 36.1 dÏ úRôWôV U§l×d LôiL, Ex 6.1 (3) (i)
2, YûLÂhûPl TVuTÓj§ 3 65 dÏ úRôWôV U§l×d LôiL, Eg 6.3
3. JÚ R² FN−u ¿[m l Utßm ØÝ AûXÜ úSWm T G²p T = k l (k GuTÕ Uô±−). R² FN−u ¿[m 32.1 ùN,Á−ÚkÕ 32.0 ùN,Á,dÏ Uôßm úTôÕ. úSWj§p HtTÓm NRÅRl ©ûZûV LQd¡ÓL, Eg 6.5
4, u = log (tan x + tan y + tan z) G²p Σ sin 2x u
x
∂
∂= 2 G] ¨ì©, Eg 6.15
5. U = (x − y) (y − z) (z − x) G²p Ux + Uy + Uz = 0 G]d LôhÓL, Eg 6.16
6. 2
xz ye= Gu\ Nôo©p x = 2t Utßm y = 1 − t GàUôß CÚl©u
dz
dtLôiL, Eg 6.17
7. w = 2 2
x
x y+ Gu\ Nôo©p x = cos t ; y = sin t G²p
dw
dt I LôiL, Ex 6.3 (3)(iii)
8. w = x + 2y + z2 Gu\ Nôo©p x = cos t ; y = sin t ; z = t G²p dw
dt I LôiL,Eg 6.19
9. x = u2 − v2, y = 2uv Guß CÚdÏUôß w = x2
+ y2 G] YûWVßdLlTÓ¡\Õ G²p
u
ω∂
∂ Utßm
v
ω∂
∂ Id LôiL, Ex 6.3 (4) (ii)
10. 2 x
u = xy siny G²p
u ux + y = 3u
x y
∂ ∂
∂ ∂ G]d LôhÓL, Ex 6.3 (5) (ii)
11. u GuTÕ x, yCp n-Bm T¥ NUlT¥jRô] NôoTô«u 2 2
2 2( 1)
∂ ∂ ∂+ = −
∂∂ ∂
u u ux y n
yx y G]
¨ßÜL, Eg 6.21
12. V= Zeax + by
Utßm Z B]Õ x, y-Cp nBm T¥ NUlT¥jRô] NôoTô«u
( )V V
x y ax by n Vx y
∂ ∂+ = + +
∂ ∂ G] ¨ßÜL, Ex 6.3 (5) (iv)
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PGT (Mathematics)
7, ùRôûL ÖiL¦Rm : TVuTôÓLs
1. 2
9
0
sin4
xdx
π ∫ u U§l× LôiL, Eg 7.15 (iii)
2, YûWVßjR ùRôûL«u Ti©û]l TVuTÓj§ U§l×d LôiL:2
2
sinx xdx
π
π−
∫ Eg 7.7
3, YûWVßjR ùRôûL«u Ti©û]l TVuTÓj§ U§l×d LôiL:3
03
xdx
x x+ −∫
Ex 7.2 (8)
4, U§l©ÓL: 1
0
1log 1
−
∫ dxx
Ex 7.2 (7)
5, YûWVßjR ùRôûL«u Ti©û]l TVuTÓj§ U§l×d LôiL: ( )2
0
log tan x dx
π
∫
Eg 7.11
6, U§l©ÓL: 3
6
1 cot
dx
x
π
π+∫ Eg 7.12
7, U§l©ÓL: 3
6
1 tan
dx
x
π
π+∫ Ex 7.2 (10)
8. y = 2x + 4 Gu\ úLôÓ y = 1, y = 3 Gu\ úLôÓLs Utßm y-AfÑ B¡VYt\ôp AûPTÓm AWeLj§u TWl©û]d LôiL, Eg 7.22
9. 2 2
2 21
x y
a b+ = (a > b > 0) Gu\ ¿sYhPm HtTÓjÕm TWl©û]. ùShPfûNl
ùTôßjÕf ÑZt±]ôp HtTÓm §PlùTôÚ°u L] A[Ü LôiL, Ex 7.4 (14)
8, YûLdùLÝf NUuTôÓLs
1, ¾odL: 4
5( 4 )xx dy y x e dx= + Eg 8.8
2, ¾odL: 2 2( )x y dy xy dx+ = Ex 8.3 (3)
3, ¾odL: 2 22
dyx y xy
dx= + ; x = 1 G²p y = 1 Ex 8.3 (4)
4, ¾odL: cot 2cosdy
y x xdx
+ = Eg 8.17
5. ¾odL: 2 tan x = sin x dy
ydx
+ Eg 8.21
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PGT (Mathematics)
6, ¾odL: 2 2 2
4 1
1 ( 1)
dy xy
dx x x+ =
+ + Ex 8.4 (2)
7. ¾odL: dy2(1+ x ) + 2xy = cosxdx
Ex 8.4 (4)
8, ¾odL: dy
y xdx
+ = Ex 8.4 (1)
9 ¾o: dy
xy xdx
+ = Ex 8.4 (6)
10, ¾odL : 1
x2 2(2D 5D 2)y e
−
+ + = Eg 8.28
11, ¾odL: ( )2 4 13 cos3+ + =D D y x Eg 8.30
12, ¾odL: ( )2 5 4 sin 5+ + =D D y x OBQ
13, ¾odL: ( )23 2− + =D D y x Eg 8.32
14, ¾odL: ( )2 714 49 4xD D y e
−+ + = + Ex 8.5 (3)
15, ¾odL: ( )2 23 14 13 10x xD D y e e+ − = + OBQ
9, R²¨ûX LQd¡Vp
1. ( ) ( )p q∧ � � � Gu\ átßdÏ ùUn AhPYûQ AûUdL, Eg 9.4 (iv)
2. ( ) ( )p q r∨ ∧ dϬV ùUn AhPYûQûV AûUdL, Eg 9.6
3. ( ) ( )p q r∧ ∨ u ùUn AhPYûQûV AûUdL, Ex 9.2 (10)
4. ( ) ( )p q r∧ ∨ � dϬV ùUn AhPYûQûV AûUdL, Eg 9.5
5. ( ) ( ) ( )p q p q∨ ≡ ∧� � � G]d LôhÓL, Eg 9.7
6, ( ) ( ) ( )p q p q∧ ≡ ∨� � � G]d LôhÓL, Ex 9.3 (5)
7. p ↔ q ≡ (( � p) ∨ q) ∧ (( � q) ∨ p) G]d LôhÓL, Ex 9.3 (4)
8. p ↔ q ≡ (p → q) ∧ (q → p) G]d LôhÓL, Ex 9.3 (3)
9. p → q Utßm q → p NUô]Ut\ûY G]d LôhÓL, Ex 9.3 (6)
10. [( p) ] p( )q∨ ∨� � JÚ ùUnûU G]d LôhÓL, Eg 9.10 (i)
11. (( � p) ∨ q) ∨ (p ∧ ( � p)) JÚ ùUnûUVô GuTRû] ùUn AhPYûQûVd
ùLôiÓ ¾oUô²dL, OBQ
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XII PUBLIC 19 QPs 6 marks © reserved with Miss N. Mahalakshmi
PGT (Mathematics)
12. (p ∧ ( � q)) ∨ (( � p) ∨ q) Gu\ átß ùUnûUVô. ØWiTôPô GuTRû] ¾oUô²dL, Ex 9.3 (1) (iii)
13. ( ) ( )p q p q∧ → ∨ GuTÕ JÚ ùUnûU G]d LôhÓL, Ex 9.3 (7)
14, (( q) p) q∼ ∧ ∧ JÚ ØWiTôÓ G]d LôhÓL, Eg 9.10 (ii) 15, ùUn AhPYûQûVd ùLôiÓ ©uYÚm átß ùUnûUVô ApXÕ ØWiTôPô
G]j ¾oUô²dLÜm: (p ∧ ( � p)) ∧ (( � q) ∧ p) Ex 9.3 (1) (v)
16, YûWVßdLlThP ϱÂh¥u T¥ (Z5 − {[0]}, .5) JÚ ÏXm G] ¨ì©, OBQ
17, 1Cu 4Bm T¥ êXeLs JÚ Ø¥Yô] GÀ−Vu ÏXjûR ùTÚdL−u ¸r AûUdÏm G]d LôhÓL, Eg 9.15
18, 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,
19. 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, (CeÏ A¦«u Eßl×Ls VôÜm RIf úNokRûY) Eg 9.19
20. (Z, +) JÚ Ø¥Yt\ GÀ−Vu ÏXm G] ¨ßÜL, Eg 9.12
21. (Z6, +6) Gu\ ÏXj§u JqùYôÚ Eßl©u Y¬ûN«û]d LôiL, OBQ 22. (Z7-{[0]} .7) Gu\ ÏXj§u JqùYôÚ Eßl©u Y¬ûN«û]d LôiL, OBQ 23, ÏXj§u ¿dLp ®§Lû[ Gݧ ¨ßÜL, Page 181
24. 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 ®§«û] Gݧ ¨ì©, Page 182
10,10,10,10, ¨LrRLÜl TWYp¨LrRLÜl TWYp¨LrRLÜl TWYp¨LrRLÜl TWYp 1, êuß TLûPLû[ JÚ Øû\ ÅÑm ùTôÝÕ 6 -Ls ¡ûPlTRtLô] ¨LrRLÜl
TWYûXd LôiL, Ex 10.1 (1) 2, JÚ R²jR NUYônl× Uô± X-u ¨LrRLÜl TWYp (¨û\fNôo×) ¸úZ
ùLôÓdLlThÓs[Õ, X 0 1 2 3 4 5 6 7 8
P(X = x) a 3a 5a 7a 9a 11a 13a 15a 17a
(i) a-u U§l× LôiL, (ii) P(x < 3)
(iii) P(3 < x < 7) CYtû\d LôiL,, Ex 10.1 (4)
3, 3cx(1 x) ; 0 x 1
f (x)0 ;
− < <= ù \e¡ÛmUt
Ex 10.1 (6)
Gu\ Nôo× JÚ ¨LrRLÜ APoj§ Nôo× G²p(i) c (ii) 1
P x2
<
LôiL,
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PGT (Mathematics)
4, JÚ ºWô] TLûPûV ûYjÕ JÚ ®û[VôhÓ ®û[VôPlTÓ¡\Õ, JÚYÚdÏ TLûP«u úUp 2 ®ÝkRôp ì.20 CXôTØm. TLûP«u úUp 4 ®ÝkRôp ì.40 CXôTØm. TLûP«u úUp 6 ®ÝkRôp ì, 30 CZl×m AûP¡\ôo, úYß GkR Gi ®ÝkRôÛm CXôTúUô CZlúTô ¡ûPVôÕ, AYo AûPÙm G§oTôol× CXôTj ùRôûL VôÕ? Eg 10.14
5, JÚ ÖûZÜj úRo®p JÚ UôQYu GpXô 120 úLs®LÞdÏm ®ûPV°dL
úYiÓm, JqùYôÚ úLs®dÏm SôuÏ ®ûPLs Es[], JÚ N¬Vô] ®ûPdÏ 1 U§lùTi ùT\Ø¥Ùm, RY\ô] ®ûPdÏ 1/2 U§lùTi CZdL úS¬Óm, JqùYôÚ úLs®dÏm NUYônl× Øû\«p ®ûPV°jRôp AmUôQYu ùTßm U§lùTi¦u G§oTôol× Gu]? Ex 10.2 (3)
6, JÚ NUYônl× Uô±«u ¨LrRLÜ ¨û\f Nôo× ¸úZ ùLôÓdLlThÓs[Õ :
x 0 1 2 3
P(X = x) 0.1 0.3 0.5 0.1
Y = X2 + 2X G²p Y Cu NWôN¬ûVÙm TWYtT¥ûVÙm LôiL, Ex 10.2 (6)
7. 1; 1 2 x 1 2
f ( x ) 2 4
0 ;
− ≤ ≤
= ù \ e ¡ Û mU t
Gu\ ¨LrRLÜ APoj§f Nôo×dÏ NWôN¬ Utßm TWYtT¥ LôiL, Ex 10.2 (7) (i) 8, ùRôPof£Vô] NUYônl× Uô± X-u ¨LrRLÜ APoj§f Nôo×
3x ( 2 x ) ; 0 x 2
f ( x ) 4
0 ; ù \ e ¡ Û mU t
− < <
=
G²p NWôN¬ûVÙm. TWYtT¥ûVÙm LôiL, Eg 10.15
9. xe ; x 0
f (x)0 ; ù \e¡ÛmUt
−αα >=
Gu\ ¨LrRLÜ APoj§f Nôo×dÏ NWôN¬ Utßm TWYtT¥ LôiL, Ex 10.2 (7) (ii)
10. x
xe ; x 0f (x)
0 ;
− >= ù \ e¡ÛmUt
Gu\ ¨LrRLÜ APoj§f Nôo×dÏ NWôN¬ Utßm TWYtT¥ LôiL, Ex 10.2 (7)(iii) 11, ©uYÚm ¨LrRLÜ APoj§f Nôo©tÏ NWôN¬ûVÙm. TWYtT¥ûVÙm LôiL,
3x3e ; 0 xf (x)
0 ; ù \e¡ÛmUt
− < < ∞=
Eg 10.16
12, JÚ DÚßl×l TWY−u Uô± Xu NWôN¬ 2. §hP ®XdLm 2
3
G²p. ¨LrRLÜf
NôoûTd LôiL, Eg 10.17 13, JÚ ùLôsLX²p 4 ùYsû[Ùm 3 £Yl×l TkÕLÞm Es[], §ÚmT
ûYdÏUôß NUYônl× Øû\«p êuß Øû\ TkÕLû[ Ju\u ©u Ju\ôL GÓdÏm úTôÕ ¡ûPdÏm £Yl×l TkÕL°u Gi¦dûL«u ¨LrRLÜl TWYûXd LôiL, úUÛm NWôN¬. TWYtT¥ B¡VYtû\d LôiL, Eg 10.13
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PGT (Mathematics)
14, JúW NUVj§p 4 SôQVeLs ÑiPlTÓ¡u\], (a) N¬VôL 2 RûXLs (b) Ïû\kRThNm 2 RûXLs (c) A§LThNm 2 RûXLs ¡ûPdL ¨LrRLÜ LôiL, Ex 10.3 (4)
15, JÚ ú_ô¥l TLûPLs 10 Øû\ EÚhPlTÓ¡u\], CÚ TLûPLÞm JúW Gi
LôhÓYûR ùYt± G]d ùLôiPôp (i) 4 ùYt±Ls (ii) éf£V ùYt± - CYt±u ¨LrRLÜ LôiL, Eg 10.18
16, JÚ Ï±l©hP úRo®p. úRof£ ùTt\YoL°u NRÅRm 80 BÏm, 6 SToLs
úRoÜ Gݧ]ôp. Ïû\kRThNm 5 SToLs úRof£ ùT\ ¨LrRLÜ LôiL, Ex 10.3 (5)
17, JÚ Tôn^ôu TWY−p P(X = 2) = P(X = 3) G²p P(X =5) I LôiL, [e−3
= 0.050 G]d ùLôÓdLlThÓs[Õ]. Eg 10.25
18, JÚ Tôn^ôu Uô± Xu NWôN¬ 4 BÏm, (i) P(X ≤ 3) (ii) P(2 ≤ X < 5) LôiL,
[e−4 = 0.0183]. Ex 10.4 (1)
19, JÚ L§¬VdLl ùTôÚ°−ÚkÕ Bp*Tô ÕLsLs NWôN¬VôL 20 ¨ªP LôX
CûPùY°«p 5 G] EªZlTÓ¡\Õ, Tôn^ôu TWYûXl TVuTÓj§ ϱl©hP 20 ¨ªP CûPùY°«p (i) 2 EªZpLs (ii) Ïû\kRThNm 2 EªZpLÞdLô] ¨LrRLûYd LôiL, [e−5
= 0.0067]. Ex 10.4 (4)
20, JÚ CVp¨ûXl TWY−u ¨LrRLÜ APoj§f Nôo× ( )2
2x 4x 2f x k e− + −= , G²p
k, µ Utßm σ CYtû\d LôiL, OBQ 21, AùU¬dL LiPj§p ù_h ®Uô]j§p TVQm ùNnÙm JÚ STo Lôvªd
L§¬VdLj§]ôp Tô§dLlTÓYÕ JÚ CVp¨ûX TWYXôÏm, CRu NWôN¬ 4.35
m rem BLÜm. §hP ®XdLm 0.59 m rem BLÜm AûUkÕs[Õ, JÚ STo 5.20 m
rem dÏ úUp Lôvªd L§¬VdLj§]ôp Tô§dLlTÓYôo GuTRtÏ ¨LrRLÜ LôiL, [P (0 < z < 1.44 = 0.4251] Ex 10.5 (3)
22, 300 UôQYoL°u EVWeLs CVp¨ûXl TWYûX Jj§Úd¡\Õ, CRu NWôN¬
64,5 AeÏXeLs, úUÛm §hP ®XdLm 3,3 AeÏXeLs, GkR EVWj§tÏd ¸r 99% UôQYoL°u EVWm APe¡«ÚdÏm? Ex 10.5 (6)
23, JÚ Ts°«u 800 UôQYoLÞdÏd ùLôÓdLlThP §\]ônÜj úRo®u
U§lùTiLs CVp¨ûXl TWYûX Jj§Úd¡\Õ, 10% UôQYoLs 40 U§lùTiLÞdÏd ¸úZÙm. 10% UôQYoLs 90 U§lùTiLÞdÏ úUÛm ùTß¡\ôoLs, 40 U§lùTiLÞdÏm 90 U§lùTiLÞdÏm CûPúV U§lùTiLs ùTt\ UôQYoL°u Gi¦dûLûVd LôiL, Ex 10.5 (7)
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XII COME BOOK 10 marks © reserved with Miss N. Mahalakshmi
P.G. T. (Mathematics)
XII MATHS COME BOOK MODEL QPs
10 MARK QUESTIONS (16)
2, 2, 2, 2, ùYdPo CVtL¦RmùYdPo CVtL¦RmùYdPo CVtL¦RmùYdPo CVtL¦Rm
1, ùYdPo Øû\«p cos (A - B) = cos A cos B + sin A sin B G] ¨ßÜL, Eg. 2.17 2, (-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)
3, 3, 3, 3, LXlùTiLsLXlùTiLsLXlùTiLsLXlùTiLs 1, P Guàm ×s° LXlùTi Uô± z Id ϱjRôp P-Cu ¨VUlTôûRûV
1arg
1 3
z
z
π− =
+ Gu\ ¨TkRû]dÏ EhThÓ LôiL, Eg. 3.11 (ii)
2. P Guàm ×s° LXlùTi Uô± z Id ϱjRôp P-Cu ¨VUlTôûRûV
1arg
3 2
z
z
π− =
+ Gu\ ¨TkRû]dÏ EhThÓ LôiL, Ex. 3.2 (8) (v)
5, 5, 5, 5, YûL ÖiL¦Rm : TVuTôÓLsYûL ÖiL¦Rm : TVuTôÓLsYûL ÖiL¦Rm : TVuTôÓLsYûL ÖiL¦Rm : TVuTôÓLs - I
1, JÚ ®ûN CÝlTôu êXm ùNÛjRlTÓm LÚeLp _p−Ls. ¨ªPjÕdÏ 30
L,A¥ ÅRm úU−ÚkÕ ¸úZ ùLôhPlTÓmúTôÕ AûY ám× Y¥YjûRd ùLôÓd¡\Õ, GkúSWj§Ûm Adám©u ®hPØm. EVWØm NUUôLúY CÚdÏUô]ôp. ám©u EVWm 10 A¥VôL CÚdÏm úTôÕ EVWm Gu] ÅRj§p EVo¡\Õ GuTûRd LôiL, Ex. 5.1 (9)
2, U§l× LôiL: 0
limx→ +
sin xx Eg. 5.35
3, JÚ ®YNô« ùNqYL Y¥YUô] YVÛdÏ úY−«P úYi¥Ùs[Õ, AqYV−u
JÚ TdLj§p Bß Juß úSodúLôh¥p KÓ¡\Õ, AlTdLj§tÏ úY− úRûY«pûX, AYo 2400 A¥dÏ úY−«P LÚ§Ùs[ôo, AqYûL«p ùTÚU TWlT[Ü ùLôsÞUôß Es[ ¿[. ALX A[ÜLs Gu]? Eg. 5.52
4, Lô³Vu Yû[YûW 2
xy e
−= . GkR CûPùY°L°p ϯÜ. 쨆 AûP¡\Õ GuTûRÙm. Yû[Ü Uôtßl ×s°Lû[Ùm LôiL, Eg. 5.64
6, 6, 6, 6, YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs – II
1, 2 2
x yu
y x= − Gu\ Nôo×dÏ
2 2u u
x y y x
∂ ∂=
∂ ∂ ∂ ∂ GuTûR N¬TôodL, Ex. 6.3 (1) (ii)
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P.G. T. (Mathematics)
7, 7, 7, 7, ùRôûL ÖiL¦Rm : TVuTôÓLsùRôûL ÖiL¦Rm : TVuTôÓLsùRôûL ÖiL¦Rm : TVuTôÓLsùRôûL ÖiL¦Rm : TVuTôÓLs
1, Yû[YûW y2 = x Utßm y = x − 2 Gu\ úLôh¥]ôp AûPTÓm TWl©û]d LôiL,
Eg 7.28
8, 8, 8, 8, YûLdùLÝf NUuTôÓLsYûLdùLÝf NUuTôÓLsYûLdùLÝf NUuTôÓLsYûLdùLÝf NUuTôÓLs
1. 2secydx xdy e y dy
−+ = ¾oÜ LôiL, Ex 8.4 (7)
2. ( )3 2 21 3 sec
dyx x y x
dx− − = Gu\ YûLdùLÝf NUuTôh¥û]j ¾odL: OBQ
3. ¾o: ( )2 13 12 5 xD D y x e− + = + OBQ
9, 9, 9, 9, R²¨ûX LQd¡VpR²¨ûX LQd¡VpR²¨ûX LQd¡VpR²¨ûX LQd¡Vp
1. 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)
2. 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
10,10,10,10, ¨LrRLÜl TWYp¨LrRLÜl TWYp¨LrRLÜl TWYp¨LrRLÜl TWYp
1, 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, Eg 10.32
************ NO SUBSTITUTE FOR HARD WORK ***********
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PGT (Mathematics)
XII PUBLIC EXAM - 10 MARK QUESTIONS (163)
1, A¦Ls Utßm A¦dúLôûYL°uA¦Ls Utßm A¦dúLôûYL°uA¦Ls Utßm A¦dúLôûYL°uA¦Ls Utßm A¦dúLôûYL°u TVuTôÓLs TVuTôÓLs TVuTôÓLs TVuTôÓLs 1, úSoUôß A¦ LôQp Øû\«p ¾odL:
2x − y + 3z = 9, x + y + z = 6, x − y + z = 2 Eg. 1. 8 2, ¡úWU¬u ®§lT¥ ¾odL:
1 2 11
x y z+ − =
2 4 15
x y z+ + =
3 2 20
x y z− − = Ex 1. 4 (9)
3, JÚ ûT«p ì, 1. Utßm ì, 2. Utßm ì,5 SôQVeLs Es[], ìTôn 100
U§l©tÏ ùUôjRm 30 SôQVeLs Es[], AqYô\ô«u JqùYôÚ YûL«Ûm Es[ SôQVeL°u Gi¦dûLûV LôiL, Eg .1.19
4, JÚ £±V LÚjRWeÏ Aû\«p 100 SôtLô−Ls ûYlTRtÏ úTôÕUô]
CPØs[Õ, êuß ùYqúY\ô] ¨\eL°p SôtLô−Ls YôeL úYi¥Ùs[Õ, (£Ll×. ¿Xm Utßm TfûN), £Ll× YiQ SôtLô−«u ®ûX ì,240. ¿XYiQ SôtLô−«u ®ûX ì.260. TfûNYiQ SôtLô−«u ®ûX ì.300, ùUôjRm ì.25.000 U§l×s[ SôtLô−Ls YôeLlThPÕ, AqYô\ô«u JqùYôÚ YiQj§Ûm YôeLjRdL SôtLô−L°u Gi¦dûLdÏ Ïû\kRThNm êuß ¾oÜLû[d LôiL, Ex. 1. 4 (10)
5, A¦dúLôûY Øû\ TVuTÓj§ x + 2y + z = 2 2x +4y +2z = 4 x – 2y – z = 0
Gu\ NUuTôÓL°u ùRôÏl©û]j ¾odL, OBQ
6, RW Øû\«û]l TVuTÓj§ 2x + 5y + 7z = 52, x + y + z = 9, 2x + y − z = 0 Gu\
NUuTôÓL°u ùRôÏl× JÚeLûUY] G] ¨ì©jÕ. ¾oÜ LôiL, Eg. 1.22 7, RW Øû\«û]l TVuTÓj§ x + y + z = 6, x + 2y + 3z = 14, x + 4y + 7z = 30 Gu\
NUuTôÓL°u ùRôÏl× JÚeLûUY] G] ¨ì©jÕ. ¾oÜ LôiL, Eg. 1.24 8, ©uYÚm NUuTôÓj ùRôÏl× JÚeLûUÜ EûPVRô GuTûR BWônL, AqYôß
JÚeLûUÜ EûPVRô«u ARû]j ¾odLÜm (RW Øû\ûVl TVuTÓjRÜm): x + y − z = 1 2x + 2y − 2z = 2 − 3x − 3y + 3z = − 3 Ex. 1. 5 (1) (v) 9. λ-u GpXô U§l×LÞdÏm ©uYÚm NUuTôhÓj ùRôÏl©u ¾oÜLû[j
RWj§û]l TVuTÓj§ BWônL, x + y + z = 2, 2x + y −2z = 2, λx + y + 4z = 2
Ex. 1. 5 (2)
10. k-u GmU§l×LÞdÏ ©uYÚm NUuTôhÓj ùRôÏl× kx + y + z = 1, x + ky + z =1,
x + y + kz = 1 (i) JúW JÚ ¾oÜ (ii) JußdÏ úUtThP ¾oÜ (iii) ¾oÜ CpXôûU ùTßm? Ex. 1.5 (3)
11. λ, µ-Cu GmU§l×LÞdÏ x + y + z = 6, x + 2y + 3z = 10, x + 2y + λz = µ Gu\ NUuTôÓLs (i) VôùRôÚ ¾oÜm ùTt±WôÕ (ii) JúW JÚ ¾oûY ùTt±ÚdÏm (iii)
Gi¦dûLVt\ ¾oÜLû[l ùTt±ÚdÏm GuTRû] BWônL, Eg. 1.26
12. µ u GmU§l©tÏ x + y + 3z = 0, 4x + 3y + µz = 0, 2x + y + 2z = 0 Gu\ ùRôÏl©tÏ (i) ùY°lTûPj ¾oÜ (ii) ùY°lTûPVt\ ¾oÜ ¡ûPdÏm? (RWj§û]l TVuTÓj§) Eg. 1.28
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PGT (Mathematics)
2, ùYdPo CVtL¦RmùYdPo CVtL¦RmùYdPo CVtL¦RmùYdPo CVtL¦Rm
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, Ex. 2. 2 (4)
3. Sin (A − B) = sin A cos B − cos A sin B G] ùYdPo Øû\«p ¨ì©, Ex. 2. 4 (7)
4. ùYdPo Øû\«p Sin (A + B) = sin A cos B + cos A sin B G] ¨ßÜL, Eg. 2. 29
5. 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)
6, 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)
7. 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
8. 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)
9. (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 10, (-1. -2. 1) Gu\ ×s° Y¯f ùNpYÕm x + 2y + 4z +7= 0 Utßm 2x - y + 3z +3= 0
B¡V R[eLÞdÏ ùNeÏjRôLÜm Es[ R[j§u ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, OBQ
11, (1, 2, -2) Y¯úVf ùNpXd á¥VÕm + 2 = + 1 - 4
3 2 4
=
− −
uuuu�uuuuu� uuuuu�x y z Gu\ úLôh¥tÏ
CûQVôLÜm 2x + 3y + 3z = 8 Gu\ R[j§tÏ ùNeÏjRôLÜm Es[ R[j§u ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, OBQ
12, (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)
13, (− 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)
14. 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)
15. (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
16, 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)
17, ùYhÓjÕiÓ Y¥®p JÚ R[j§u NUuTôhûP ùYdPo Øû\«Ûm
Lôo¼£Vu Øû\«Ûm RÚ®dL, Ex. 2.8 (14)
3, LXlùTiLsLXlùTiLsLXlùTiLsLXlùTiLs
1. P Gàm ×s° LXlùTi Uô± zId ϱjRôp 2 1
Im 21
z
iz
+ = −
+ dÏ P-Cu
¨VUlTôûRûVd LôiL, Ex. 3. 2 (8) (i) 2. a = cos2α + i sin 2α, b = cos2β + i sin 2β Utßm c = cos 2γ + i sin 2γ G²p
(i) 1
abcabc
+ = 2 cos (α + β + γ)
(ii) 2 2 2a b c
abc
+ = 2 cos 2(α + β − γ) G] ¨ì©, Ex. 3.4 (10)
3. α, β GuTûY x2 − 2x + 2 = 0-u êXeLs Utßm cot θ = y + 1 G²p
( ) ( )
n ny yα β
α β
+ − +
−
uuuuuuuuuuuuuuuuuuuuuuuu� = sin sinn
nθθ
uuuuuu� G]d LôhÓL, Eg. 3. 22
4. x2 − 2px + (p2
+ q2) = 0 Gu\ NUuTôh¥u êXeLs α . β Utßm tan θ = q/(y + p)
G²p ( ) ( )
n ny yα β
α β
+ − +
−
uuuuuuuuuuuuuuuuuuuuuuuu� = q n − 1 sin sinn
nθθ
uuuuuu� G] ¨ßÜL, Ex. 3. 4 (5)
5, x2 − 2x + 4 = 0-u êXeLs α Utßm β G²p αn
− βn = i2n + 1
sin nπ/3 A§−ÚkÕ α9
− β9 -u U§lûT ùTßL, Ex. 3. 4 (6)
6. 2/3( 3 )i+ u GpXô U§l×Lû[Ùm LôiL, Eg. 3. 25
7. ( )3/4
312 2i− u GpXô U§l×Lû[Ùm LôiL Utßm ARu U§l×L°u
ùTÚdLtTXu 1 G]Üm LôhÓL, Ex. 3. 5 (5)
8, ¾odL: x4 − x3
+ x2 − x + 1 = 0. Ex. 3. 5 4(ii)
9. x7 + x4
+ x3 + 1 = 0 Gu\ NUuTôhûPj ¾odL, Eg. 3.24
10. x9 + x5
− x4 − 1 = 0 Gu\ NUuTôhûPj ¾odL, Eg. 3.23
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PGT (Mathematics)
4, TÏØûTÏØûTÏØûTÏØû\\\\ Y¥YdL¦RmY¥YdL¦RmY¥YdL¦RmY¥YdL¦Rm 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− 2y + 17 = 0 OBQ
3. y2 + 8x− 6y + 1 = 0 Ex 4.1 (2 iv)
4. y2 + 4y + 4x + 8 = 0 OBQ
5. x2 − 6x − 12y − 3 = 0 Ex 4.1 (2 v)
6. 2x 4x + 4y 0− = OBQ
7, 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
8, 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
9, 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
10, 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)
11, 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
12, 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, CߧVôL ¡ûPUhPUôL 12 Á ùRôûX®p RûWûV YkRûP¡\Õ G²p ×\lThP CPj§p RûWÙPu HtTÓjRlTÓm G±úLôQm LôiL, Eg. 4. 10
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XII PUBLIC 19 QPs 10 marks © reserved with Miss N. Mahalakshmi
PGT (Mathematics)
¿sYhPj§tÏ ûUVj ùRôûXj RLÜ. ûUVm. Ï®VeLs. Utßm Ef£LsB¡VYtû\d LôiL, úUÛm ARu Yû[YûWûVd LôiL,
13. 36x2 + 4y
2 − 72x + 32y − 44 = 0 Eg 4.31 (iv)
14. 16x2 + 9y
2 - 32x + 36y - 92 = 0 OBQ
15. 9x2 + 25y2
- 18x - 100y - 116 = 0 OBQ 16, 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
17, 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)
18, 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
19, ã¬Vu Ï®Vj§−ÚdÏUôß ùUodϬ ¡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) ùUodϬ ¡WLUô]Õ ã¬VàdÏ ªL AÚLôûU«p YÚmúTôÕ Es[ çWm (ii) ùUodϬ ¡WLUô]Õ ã¬VàdÏ ªLj ùRôûX®p CÚdÏmúTôÕ Es[ çWm B¡VYtû\d LôiL, Ex. 4.2 (9)
20, 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) 21, 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)
22, 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[Vj§u ûUVj ùRôûXj RLÜ. ûUVm. Ï®VeLs. Ef£Ls B¡VYtû\d LôiL, úUÛm ARu Yû[YûWûV YûWL,
23, 12x2 − 4y2
- 24x + 32y - 127 = 0 OBQ
24. 9x2 − 16y2 - 18x - 64y - 199 = 0 Eg 4.56
25. x2 − 4y2 + 6x + 16y - 11 = 0 Ex 4.3 (5 iii)
26. x2 − 3y2
+ 6x + 6y + 18 = 0 Ex 4.3 (5 iv)
27. 5x + 12y = 9 Gu\ úSodúLôÓ A§TWYû[Vm x2 − 9y2
= 9 Ij ùRôÓ¡\Õ G] ¨ì©dL, úUÛm ùRôÓm ×s°ûVÙm LôiL, Ex. 4.4 (5)
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PGT (Mathematics)
28, ×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)
29. 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)
5, YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs – I 1, JÚ ¿o¨ûXj ùRôh¥Vô]Õ RûX¸Zôn ûYdLlThP JÚ úSoYhP ám©u
Y¥®p Es[Õ, ARu BWm 2 ÁhPo. ARu BZm 4 ÁhPo BÏm, ¨ªPj§tÏ 2 L,ÁhPo ÅRm ùRôh¥«p ¿o TônfNlTÓ¡\Õ, ùRôh¥«p ¿¬u BZm 3ÁhPWôL CÚdÏm ùTôÝÕ. ¿o UhPj§u EVWm A§L¬dÏm ÅRjûRd LôiL,
Eg. 5. 9
2. JÚ ØdúLôQj§u CWiÓ TdL A[ÜLs Øû\úV 12Á. 15Á Utßm CYt±u CûPlThP úLôQj§u Hßm ÅRm ¨ªPj§tÏ 2° G²p ¨ûXVô] ¿[eLs ùLôiP TdLeLÞdÏ CûPlThP úLôQm 60° BL CÚdÏm úTôÕ. ARu êu\ôYÕ TdLj§u ¿[m GqY[Ü ®ûWYôL A§L¬dÏm GuTûRd LôiL, Ex. 5. 1 (8)
3. 10 ÁhPo ¿[Øs[ JÚ H¦ ùNeÏjRô] ÑY¬p NônjÕ ûYdLlThÓs[Õ, H¦«u A¥lTdLm ÑYt±−ÚkÕ ®X¡f ùNpÛm ÅRm 1 Á/®]ô¥ G²p. H¦«u A¥lTdLm ÑYt±−ÚkÕ 6 Á ùRôûX®p CÚdÏm úTôÕ. ARu Ef£ GqY[Ü ÅRj§p ¸rúSôd¡ C\eÏm GuTûRd LôiL, Eg. 5. 7
4, JÚ £tßkÕ A B]Õ U¦dÏ 50 ¡,Á, úYLj§p úUt¡−ÚkÕ ¡ZdÏ úSôd¡f
ùNp¡\Õ, Utù\ôÚ £tßkÕ B B]Õ U¦dÏ 60 ¡,Á, úYLj§p YPdÏ úSôd¡f ùNp¡\Õ, CûY CWiÓm NôûXLs Nk§dÏm CPjûR úSôd¡f ùNp¡u\], NôûXLs Nk§dÏm Øû]«−ÚkÕ £tßkÕ A B]Õ 0,3 ¡,Á, çWj§Ûm £tßkÕ B B]Õ 0,4 ¡,Á, çWj§Ûm CÚdÏmúTôÕ Juû\ Juß ùSÚeÏm úYL ÅRjûRd LQd¡ÓL, Eg. 5. 8
5, SiTL−p A Gu\ LlTp. B Gu\ LlTÛdÏ úUtÏl ×\UôL 100 ¡,Á, çWj§p
Es[Õ, LlTp A B]Õ U¦dÏ 35 ¡,Á, úYLj§p ¡ZdÏ úSôd¡f ùNp¡\Õ, LlTp BB]Õ U¦dÏ 25 ¡,Á, úYLj§p YPdÏ úSôd¡f ùNp¡u\Õ G²p. UôûX 4.00 U¦dÏ CWiÓ LlTpLÞdÏm CûPlThP çWm GqY[Ü úYLUôL Uôßm GuTûRd LôiL, Ex. 5.1 (6)
6, JÚ ØdúLôQj§u ÏjÕVWm 1 ùN,Á / ¨ªPm ÅRj§p A§L¬dÏm úTôÕ.
ARu TWl× 2 N,ùN,Á, / ¨ªPm Gàm ÅRj§p A§L¬d¡\Õ, ÏjÕVWm 10 ùN,Á, BLÜm TWl× 100 N,ùN,Á BLÜm CÚdÏm úTôÕ ØdúLôQj§u A¥lTdLm Gu] ÅRj§p Uôßm GuTûRd LôiL, Ex. 5. 1 (5)
7, x =acos3
θ; y = a sin3θ Gàm ÕûQ AXÏ NUuTôÓLû[d ùLôiP Yû[YûWdÏ θ’Cp YûWVlTÓm ùNeúLôh¥u NUuTôÓ x cos θ – y sin θ = a cos 2θ G]d LôhÓL, Ex. 5. 2 (10)
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PGT (Mathematics)
8. x = a cos4θ, y = a sin4θ, 0 ≤ θ ≤π/2 Gu\ ÕûQ AXÏ NUuTôÓLû[d ùLôiP Yû[YûWdÏ YûWVlThP GkRùYôÚ ùRôÓúLôÓm HtTÓjÕm BV AfÑj ÕiÓL°u áÓRp a G]d LôhÓL, Eg. 5. 20
9. y = x3 Gu\ Yû[YûW«u ÁÕs[ JÚ ×s° P GuL, PCp YûWVlThP
ùRôÓúLôPô]Õ Yû[YûWûV UßT¥Ùm Q Cp Nk§dÏUô]ôp. QCp ùRôÓúLôh¥u NônÜ. PCp Es[ NônûYl úTôp 4 UPeÏ G]d LôhÓL, Ex. 5. 2 (7)
10, y = x2
Utßm y = (x – 2)2 Gu\ Yû[YûWLs ùYh¥d ùLôsÞm ×s°«p
AûYLÞdÏ CûPlThP úLôQjûRd LôiL, Eg. 5. 17
11. .y2 = x Utßm xy = k Gàm Yû[YûWLs Juû\ùVôuß ùNeÏjRôL ùYh¥d
ùLôiPôp. 8k2 = 1 G] ¨ì©dL, Ex. 5.2 (11)
12, U§l× LôiL:
2
lim→ −x
π cos(tan ) x
x Ex. 5. 6 (11)
13, U§l× LôiL: 0
limx→
sin(cot )
xx Eg. 5. 34
14. 3 2
( ) 2 3 36 10f x x x x= + − + Gu\ Nôo©u CPgNôokR ùTÚU Utßm £ßU U§l×Lû[d LôiL, OBQ
15, ùLôÓdLlThP JÚ TWlT[®û]d ùLôiP ùNqYLeLÞs NÕWm UhÓúU
£ßUf Ñt\[Ü ùTt±ÚdÏm G]d LôhÓL, Ex. 5.10 (3) 16, ùLôÓdLlThP JÚ Ñt\[®û]d ùLôiP ùNqYLeLÞs NÕWm UhÓúU
ùTÚU TWlT[ûYd ùLôi¥ÚdÏm G]d LôhÓL, Ex. 5.10 (4) 17, JÚ ÑYùWôh¥«u úUp Utßm A¥«u KWeLs 6 ùN,Á UtßU ARu TdL
KWeLs 4 ùN,Á, BÏm, AfÑYùWôh¥«p AfN¥dLlThP YôNLeL°u TWl× 384 ùN,Á2 G] YûWVßdLlThPôp ARu TWl× £ßU A[Ü ùLôsÞUôß Es[ ¿[ ALXeLû[d LôiL, Eg. 5. 55
18, JÚ ê¥«hP NÕW A¥lTôLm ùLôiÓs[ (L]f ùNqYLj§u) ùTh¥«u
ùLôs[[Ü 2000 L,ùN,Á. AlùTh¥«u A¥lTôLm Utßm úUpTôLj§tLô] êXl ùTôÚhL°u ®ûX JÚ N,ùN,ÁdÏ ì, 3 Utßm ARu TdLeLÞdLô] êXl ùTôÚhL°u ®ûX JÚ NÕW ùN,Á,dÏ ì, 1,50. êXl ùTôÚhL°u ®ûX £ßU A[Ü ùLôsÞUôß Es[ ùTh¥«u ¿[m. EVWm LôiL, Eg. 5. 57
19. a AXÏ BWØs[ úLô[j§às ùTÚU A[Ü ùLôsÞUôß LôQlTÓm ám©u ùLôs[[Ü. úLô[j§u ùLôs[[®u 8/ 27 UPeÏ G]d LôhÓL, Eg. 5. 56
20, 3¡,Á, ALXj§p úSWôL KÓm Bt±u JÚ LûW«p P Gu¡\ ×s°«p JÚYo ¨t¡u\ôo, AYo ¿úWôhP §ûN«p. LûW«u G§oTdLm 8 ¡,Á, ùRôûX®Ûs[ Q ûY úSôd¡ úYLUôLf ùNuß AûPV úYi¥Ùs[Õ, AYo TPûL úSWôL G§oj§ûN RdÏ Kh¥f ùNuß Ae¡ÚkÕ QdÏ K¥fùNpXXôm ApXÕ QdÏ úSWôL TPûL Kh¥f ùNpXXôm ApXÕ Q Utßm RdÏ CûPúVÙs[ SdÏ Kh¥f ùNuß Ae¡ÚkÕ QdÏ K¥f ùNpXXôm AYo TPÏ Kh¥f ùNpÛm úYLm 6 ¡,Á/U¦. KÓm úYLm 8 ¡,Á/U¦ G²p QûY úYLUôLf ùNu\ûPV AYo TPûL GeúL LûW úNodL úYiÓm? Eg. 5.58
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PGT (Mathematics)
21. r AXÏ BWØs[ AûWYhPj§às ùTÚU A[Ü ùLôsÞUôß YûWVlTÓm ùNqYLj§u TWl×d LôiL, Eg. 5. 54
22, r BWØs[ YhPj§às YûWVlTÓm ªLl ùT¬V TWlT[Ü ùLôiP ùNqYLj§u ¿[ ALXeLs Gu]YôL CÚdÏm? Ex. 5.10 (5)
23. f(x) = x4 − 6x2 Gu\ Nôo× GkR CûPùY°L°p Ï¯Ü AûP¡\Õ GuTûRÙm
Utßm Yû[Ü Uôtßl ×s°Lû[Ùm LôiL, Ex. 5. 11 (4) 24. y = 12x2
− 2x3 − x4
Gu\ Nôo× GkR CûPùY°L°p Ï¯Ü AûP¡\Õ GuTûRÙm Utßm Yû[Ü Uôtßl ×s°Lû[Ùm LôiL, Ex. 5. 11 (6)
6, YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs YûL ÖiL¦Rm : TVuTôÓLs – II
1. y = x3 Gu\ Yû[YûWûV YûWL, Ex 6. 2 (1)
2. y = x3 +1 Gu\ Yû[YûWûV YûWL, Eg 6. 9
3. 2 32y x= Gu\ Yû[YûWûV YûWL, Eg 6. 10
4, 1tanx
uy
− =
Gu\ Nôo×dÏ
2 2u u
x y y x
∂ ∂=
∂ ∂ ∂ ∂ GuTûR N¬TôodL, Ex. 6.3 (1) (iv)
5, 2 2
1( )f x
x y=
+ Gu\ Nôo×dÏ ëX¬u úRt\jûR N¬TôodL, Eg 6. 20
6. 1sin
x yu
x y
− −
= +
G²p ëX¬u úRt\jûRl TVuTÓj§ 1
tan2
u ux y u
x y
∂ ∂+ =
∂ ∂
G]d LôhÓL, Eg 6. 22
7, ëX¬u úRt\jûRl TVuTÓj§. 3 3
1tanx y
ux y
− +=
− G²p. sin 2
u ux y u
x y
δ δ
δ δ+ =
G] ¨ì©dL, Ex. 6.3 (5) (i)
7, ùRôûL ÖiL¦Rm : TVuTôÓLsùRôûL ÖiL¦Rm : TVuTôÓLsùRôûL ÖiL¦Rm : TVuTôÓLsùRôûL ÖiL¦Rm : TVuTôÓLs 1. x2
+ y2 = 16 Gu\ YhPj§tÏm y2
= 6x Gu\ TWYû[Vj§tÏm ùTôÕYô] TWlûTd LôiL, Eg 7.29
2. y = x2 − x − 2 Gu\ Yû[YûW x = − 2, x = 4 Gu\ úLôÓLs Utßm x-AfÑ
B¡VYt\ôp AûPTÓm AWeLj§u TWlûTd LôiL, Eg 7.25
3. y = 3x2 − x Gu\ Yû[YûW x-AfÑ x = − 1 Utßm x = 1 Gu\ úLôÓL[ôp
AûPTÓm AWeLj§u TWl©û]d LôiL, Ex 7.4 (4) 4. y = x2
− 2x − 3 Gu\ Yû[YûW x = − 3, x = 5 Gu\ úLôÓLs Utßm x-AfÑ B¡VYt\ôp AûPTÓm Tϧ«u TWl× LôiL, OBQ
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PGT (Mathematics)
5. 2 2
19 5
x y+ = Gu\ ¿sYhPj§p Es[ CWiÓ ùNqYLXj§tÏ CûPlThP
TWl©û]d LôiL, Ex 7.4 (7)
6. 3ay2=x(x−a)2 Gu\ Yû[YûW«u Li¦«u (loop) TWlûTd LôiL, Eg 7.33 7. 4y2
= 9x Utßm 3x2 = 16y Gu\ TWYû[VeLÞdÏ CûPlThP TWl©û]d LôiL,
Ex 7.4 (9)
8. 2y x= Utßm 2
x y= B¡V CÚ TWYû[VeLÞdÏ CûPlThP TWlûTd LôiL, OBQ
9. y = x3
Gu\ Yû[YûWdÏm y = x Gu\ úLôh¥tÏm CûPlThP AWeLj§u TWlûTd LôiL, Eg 7.27
10. x = a (2t − sin 2t), y = a (1 − cos 2t) Gu\ YhP EÚsYû[ (cycloid)«u JÚ Yû[®tÏm. x-Af£tÏm CûPúVÙs[ AWeLj§u TWlûTd LôiL, Eg 7.34
11, BWm ‘r’, ÏjÕWVm ‘h’ EûPV ám©u L]A[ûYd LôÔm ãj§Wj§û]
ùRôûL«û]l TVuTÓj§ LôiL, Ex 7.4 (15)
12. y=0, x=4 Utßm 3x-4y = 0 Gu\ NUuTôÓLû[ TdLeL[ôLd ùLôiP
ØdúLôQj§u TWlûT x-AfûNl ùTôßjÕ ÑZtßYRôp HtTÓm §PlùTôÚ°u L]A[Ü LôiL, (ApXÕ)ApXÕ)ApXÕ)ApXÕ)
( )0,0 , ( )4,0 Utßm ( )4,3 B¡V Øû]Lû[d ùLôiP ØdúLôQj§u TWlT[Ü
x-AfÑ ÁÕ ÑZtßYRôp HtTÓm ùTôÚ°u L] A]®û]d LôiL, OBQ
13. 2x t= ;
3
3
ty t= − Gu\ ÕûQVXÏ NUuTôÓLû[d ùLôiP Li¦Vôp
ãZlThP TWl©u x-AfûNl ùTôßjÕ ÑZtßm úTôÕ HtTÓjÕm Yû[YûW«Ûs[ §PlùTôÚ°u L] A[ûYd LôiL, OBQ
14. 4y2
= x3 Gu\ Yû[YûW«p x = 0 −ÚkÕ x = 1 YûWÙs[ ®p−u ¿[jûRd
LôiL, Eg 7.37 15, BWm ‘a’ EûPV YhPj§u Ñt\[ûY ùRôûLÂhÓ Øû\«p LôiL, Ex 7.5 (1)
16. x = a (t − sin t), y = a (1 − cos t) Gu\ Yû[YûW«u ¿[j§û] t = 0 ØRp t = π
YûW LQd¡ÓL, Ex 7.5 (2)
17.
2 2
3 31
x y
a a
+ =
Gu\ Yû[YûW«u ¿[jûRd LôiL, Eg 7.38
18. y = sin x Gu\ Yû[YûW x = 0, x = π Utßm x-AfÑ B¡VYt\ôp HtTÓm TWl©û] x-Af£û]l ùTôßjÕ ÑZtßm úTôÕ ¡ûPdÏm §PlùTôÚ°u
Yû[TWl× 2π [ 2 + log (1 + 2 )] G] ¨ßÜL, Eg 7.39
19. y2 = 4ax Gu\ TWYû[Vj§u ARu ùNqYLXm YûW«Xô] TWl©û] x-Af£u
ÁÕ ÑZtßmúTôÕ ¡ûPdÏm §PlùTôÚ°u Yû[TWlûTd LôiL, Ex 7.5 (3)
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PGT (Mathematics)
20, BWm r AXÏLs Es[ úLô[j§u ûUVj§−ÚkÕ a Utßm b AXÏLs ùRôûX®p AûUkR CÚ CûQVô] R[eLs úLô[jûR ùYhÓmúTôÕ CûPlTÓm Tϧ«u Yû[TWl× 2π r (b − a) G] ¨ßÜL, C§−ÚkÕ úLô[j§u Yû[TWlûT YÚ®,(b > a). Ex 7.5 (4)
21. x = a (t + sin t), y = a (1 + cos t) Gu\ YhP EÚs Yû[ (cycloid) ARu A¥lTdLjûRl (x-AfÑ) ùTôßjÕ ÑZtßYRôp HtTÓm §Pl ùTôÚ°u Yû[lTWlûTd LôiL, Eg 7.40
8, YûLdùLÝf NUuTôÓLYûLdùLÝf NUuTôÓLYûLdùLÝf NUuTôÓLYûLdùLÝf NUuTôÓLssss 1, ¾odL: (x3
+ 3xy2) dx + (y3 + 3x2y) dy = 0 Eg 8.14
2, ¾oÜ LôiL: ( )2 2+ =
dyx y a
dx Eg 8.7
3, ¾oÜ LôiL: ( )2
1dy
x ydx
+ = Ex 8.2 (7)
4, JÚ ØlT¥l TpÛßl×d úLôûY x = − 1 Gàm úTôÕ ùTÚU U§l× 4 BLÜm x = 1
Gàm úTôÕ £ßU U§l× 0 BLÜm CÚl©u. AdúLôûYûVd LôiL, Eg 8.10
5, ¾odL: (x2 + y2) dx + 3xy dy = 0 Ex 8.3 (5)
6, ¾odL : 2 2(1 ) 2 (1
dyx xy x x
dx− + = − Eg 8.18
7, ¾odL: (1 + y2) dx = (tan−1 y − x) dy Eg 8.19
8. 3 2 2dy
(1 + 2x ) + 6x y = cosec xdx
OBQ
9,, GkRùYôÚ ×s°«Ûm NônÜ y+2x G]d ùLôiÓ B§Y¯VôLf ùNpÛm Yû[YûW«u NUuTôÓ y = 2(ex
− x − 1) G]d LôhÓL, Ex 8.4 (9)
10, ¾odL: 2
3
23 2 2 xd y dy
y edxdx
− + = ; CeÏ x = log2 G²p y = 0 Utßm x = 0 G²p y = 0.
Ex 8.5 (6)
11, ¾odL: (D2 − 1) y = cos 2x − 2 sin 2x Ex 8.5 (11)
12, ¾odL: ( )2 35 6 sin 2 xD D y x e− + = + OBQ
13, ¾odL: ( )2 2 2 sin 2 5D D y x− + = + OBQ
14, ¾odL: (D2
− 6D + 9) y = x + e2x Ex 8.5 (10) 15, ÖiÔ«oL°u ùTÚdLj§p. Tôd¼¬Vô®u ùTÚdLÅRUô]Õ A§p
LôQlTÓm Tôd¼¬Vô®u Gi¦dûLdÏ ®¡RUôL AûUkÕs[Õ, ClùTÚdLjRôp Tôd¼¬Vô®u Gi¦dûL 1 U¦ úSWj§p ØmUPeLô¡\Õ G²p IkÕ U¦ úSW Ø¥®p Tôd¼¬Vô®u Gi¦dûL BWmT ¨ûXûVd Lôh¥Ûm 35 UPeLôÏm G]d LôhÓL, Eg 8.39
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16, ùYlT ¨ûX 15°C Es[ JÚ Aû\«p ûYdLlThÓs[ úR¿¬u ùYlT ¨ûX 100°C BÏm, AÕ 5 ¨ªPeL°p 60°C BL Ïû\kÕ ®Ó¡\Õ, úUÛm 5 ¨ªPm L¯jÕ úR¿¬u ùYlT ¨ûX«û]d LôiL, Ex 8.6 (3)
17, JÚ SLWj§p Es[ UdLs ùRôûL«u Y[of£ÅRm AkúSWj§p Es[ UdLs
ùRôûLdÏ ®¡RUôL AûUkÕs[Õ, 1960Bm Bi¥p UdLs ùRôûL 1,30,000 G]Üm 1990Cp UdLs ùRôûL 1,60,000 BLÜm CÚl©u 2020Bm Bi¥p
UdLs ùRôûL GqY[YôL CÚdÏm? .4216log .2070;e 1.52
13e
= =
Ex 8.6 (4)
18, úW¥Vm (Radium) £ûRÙm UôßÅRUô]Õ. A§p LôQlTÓm A[®tÏ ®¡RUôL
AûUkÕs[Õ, 50 YÚPeL°p BWmT A[®−ÚkÕ 5 NRÅRm £ûRk§Úd¡\Õ G²p 100 YÚP Ø¥®p Á§«ÚdÏm A[Ü Gu]? [A0 I BWmT A[Ü G]d ùLôsL]. Ex 8.6 (1)
19, JÚ CWNôV] ®û[®p. JÚ ùTôÚs Uôt\m AûPÙm Uôß ÅRUô]Õ t úSWj§p Uôt\UûPVôR AlùTôÚ°u A[®tÏ ®¡RUôL Es[Õ, JÚ U¦ úSW Ø¥®p 60 ¡WôØm Utßm 4 U¦ úSW Ø¥®p 21 ¡WôØm ÁRªÚkRôp. BWmT ¨ûX«p. AlùTôÚ°u GûP«û]d LôiL, Eg 8.34
20, JÚ Ye¡Vô]Õ ùRôPo áhÓ Yh¥ Øû\«p Yh¥ûVd LQd¡Ó¡\Õ,
ARôYÕ Yh¥ ÅRjûR AkRkR úSWj§p AN−u Uôß ÅRRj§p LQd¡Ó¡\Õ, JÚYWÕ Ye¡ CÚl©p ùRôPof£Vô] áhÓ Yh¥ êXm BiùPôußdÏ 8% Yh¥ ùTÚÏ¡\Õ G²p. AYWÕ Ye¡«Úl©u JÚ YÚP LôX A§L¬l©u NRÅRjûRd LQd¡ÓL, [e.08
≈ 1.0833 GÓjÕd ùLôsL,]Eg 8.35
21, JÚ C\kRYo EPûX UÚjÕYo T¬úNô§dÏm úTôÕ. C\kR úSWjûRúRôWôUôL
LQd¡P úYi¥Ùs[Õ, C\kRY¬u EP−u ùYlT ¨ûX LôûX 10.00
U¦V[®p 93.4°F G] ϱjÕd ùLôs¡\ôo, úUÛm 2 U¦ úSWm L¯jÕ ùYlT ¨ûX A[ûY 91.4°F G]d Lôi¡\ôo, Aû\«u ùYlT ¨ûX A[Ü (¨ûXVô]Õ) 72°F G²p. C\kR úSWjûRd LQd¡ÓL,(JÚ U²R EP−u NôRôWQ ExQ ¨ûX 98.6°F G]d ùLôsL).
19.4 26.6
log 0.0426 2.303 log 0.0945 2.30321.4 21.4
e e
= − × = − ×
Utßm Eg 8.37
22, JÚ L§¬VdLl ùTôÚs £ûRÙm UôßÅRUô]Õ. ARu GûPdÏ ®¡RUôL
AûUkÕs[Õ, ARu GûP 10 ª,¡Wôm BL CÚdÏm úTôÕ £ûRÙm UôßÅRm Sôù[ôußdÏ 0.051 ª,¡Wôm G²p ARu GûP 10 ¡Wôª−ÚkÕ 5 ¡WôUôLd Ïû\V GÓjÕd ùLôsÞm LôX A[ûYd LôiL, [loge2 = 0.6931] Ex 8.6 (5)
23, ì 1000 Gu\ ùRôûLdÏ ùRôPof£ áhÓ Yh¥ LQd¡PlTÓ¡\Õ, Yh¥
ÅRmBiùPôußdÏ 4 NRÅRUôL CÚl©u, AjùRôûL GjRû] BiÓL°p BWmTj ùRôûLûVl úTôp CÚ UPeLôÏm? (loge2 = 0.6931). Ex 8.6 (2)
9, R²¨ûX LQd¡VpR²¨ûX LQd¡VpR²¨ûX LQd¡VpR²¨ûX LQd¡Vp 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
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PGT (Mathematics)
2, (Z7 − {[0]}, .7) JÚ ÏXjûR AûUdÏm G]d LôhÓL, Eg 9.26
3. 11-u UhÓdÏ LôQlùTt\ ùTÚdL−u¸r {[1], [3], [4], [5], [9]} Gu\ LQm JÚ GÀ−Vu ÏXjûR AûUdÏm G]d LôhÓL, Ex 9.4 (9)
4. (Zn, +n) JÚ ÏXm G]d LôhÓL, Eg 9.25
5, YZdLUô] ùTÚdL−u ¸r 1u nm T¥ êXeLs Ø¥Yô] GÀ−Vu ÏXjûR AûUdÏm G]d LôhÓL, Eg 9.27
6. 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)
7. 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)
8. (Z, *) JÚ Ø¥Yt\ GÀ−Vu ÏXm G]d LôhÓL, CeÏ * B]Õ a * b = a + b + 2
GàUôß YûWVßdLlThÓs[Õ, Eg 9.18
9, 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
10. −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)
11. | 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)
10,10,10,10, ¨Lr¨Lr¨Lr¨LrRLÜl TWYpRLÜl TWYpRLÜl TWYpRLÜl TWYp 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
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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, 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)
5, 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 < 1] = 0.3413 P [0 < z < 2] = 0.4772 Eg 10.29
6, 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) GûP120 TÜiÓdÏm 155 TÜiÓdÏm CûPúVÙs[ UôQYoLs
(ii)GûP185 TÜiÓdÏ úUp ¨û\Ùs[ UôQYoL°u Gi¦dûL LôiL,
P [0 < z < 2.067] = 0.4803, P [0 < z < 0.2667] = 0.1026,
P [0 < z < 2.2667] = 0.4881 Ex 10.5 (5)
7, JÚ úRo®p 1000 UôQYoL°u NWôN¬ U§lùTi 34 Utßm §hP ®XdLm 16
BÏm, U§lùTi CVp¨ûXl TWYûX ùTt±Úl©u (i) 30C−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
8, 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
9, 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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