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XII PUBLIC EXAM - 10 MARK QUESTIONS (192)
1, A¦Ls Utßm A¦dúLôûYL°u TVuTôÓLs úSoUôß A¦ LôQp Øû\«p ¾odL: 1. 2x − y + 3z = 9, x + y + z = 6, x − y + z = 2 2. x − 3y − 8z + 10 = 0, 3x + y = 4, 2x + 5y + 6z = 13 3,
Eg. 1. 8 Ex 1. 2 (5)
¡úWU¬u ®§lT¥ ¾odL:
1 2 1 + − =1 x y z
2 4 1 + + =5 x y z
3 2 2 − − =0 x y z
Ex 1. 4 (9)
4,
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
5,
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)
6,
ùLôÓdLlThÓs[ ANUT¥jRô] úS¬Vf A¦dúLôûY Øû\«p ¾odL : 2x − y + z = 2, 6x − 3y + 3z = 6, 4x − 2y + 2z = 4
NUuTôhÓj
7,
A¦dúLôûY Øû\ TVuTÓj§ x + 2y + z = 2 Gu\ NUuTôÓL°u ùRôÏl©û]j ¾odL,
2x +4y +2z = 4
8,
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
9.
4x + 3y + 6z = 25, x + 5y + 7z = 13, 2x + 9y + z = 1 Gu\ NUuTôÓL°u ùRôÏl× JÚeLûUÜ EûPVRô GuTûR RW Øû\«p LôiL, JÚeLûUÜ EûPVRô«u RW Øû\«p ¾odL, Ex. 1. 5 (1) (i)
10,
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
11,
©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)
12.
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)
13.
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)
XII PUBLIC 24 QPs 10 marks
ùRôÏl©û]
Ex 1.4 (8) x – 2y – z = 0 OBQ
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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
15.
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
2, ùYdPo CVtL¦Rm 1,
JÚ ØdúLôQj§u ÏjÕdúLôÓLs ùYdPo Øû\«p ¨ßÜL,
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)
4.
Sin (A − B) = sin A cos B − cos A sin B G] ùYdPo Øû\«p ¨ì©,
5.
ùYdPo Øû\«p Sin (A + B) = sin A cos B + cos A sin B G] ¨ßÜL,
6.
r r r r r r r r r r r r r rr r rr r a = 2i + 3 j − k , b = −2i + 5k , c = j − 3k , G²p a × (b × c ) = (a.c )b − (a.b )c G] N¬TôodL, Ex. 2. 5 (5)
7,
r r r r r r r a = i + j + k , b = 2i + k , r r r r r r r ( a × b ) × (c × d ) = [ a b d ]
8.
x -1 = y −1 = z + 1 3
−1
Utßm
0
JúW
×s°«p
Nk§dÏm
r r r r r r r r c = 2i + j + k , d = i + j + 2k G²p r r r r r c − [ a b c ] d GuTûRf N¬TôodL,
x -1 = y + 1 = z 1
−1
Utßm
3
Eg. 2. 29
Ex. 2.5 (12)
Eg. 2. 44
x − 2 y −1 −z −1 = = Gu\ úLôÓLs ùYh¥d ùLôsÞm 1 2 1
G]d LôhÓL, úUÛm AûY ùYhÓm ×s°ûVd LôiL, 10.
Ex. 2. 4 (7)
x − 4 y z +1 = = Gu\ úLôÓLs ùYh¥d ùLôsÞm 2 0 3
G]d LôhÓL, úUÛm AûY ùYhÓm ×s°ûVd LôiL,
9.
GuTRû] Eg. 2. 16
(2, -1, -3) Gu\ ×s° Y¯f ùNpYÕm
Ex. 2. 7 (3)
x − 2 y −1 z − 3 = = Utßm 3 2 −4
x −1 y + 1 z − 2 = = Gu\ úLôÓLÞdÏ CûQVô]ÕUô] R[j§u ùYdPo −3 2 2 Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, Eg. 2. 50 11.
(1, 3, 2) Gu\ ×s° Y¯f ùNpYÕm
x +1 y + 2 z + 3 = = Utßm −1 2 3
x − 2 y +1 z + 2 = = Gu\ úLôÓLÞdÏ CûQVô]ÕUô] R[j§u ùYdPo 1 2 2 Ex 2.8 (8) Utßm Lôo¼£Vu NUuTôÓLû[d LôiL,
XII PUBLIC 24 QPs 10 marks
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(-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,
(-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 x + 2 y +1 z − 4 (1, 2, -2) Y¯úVf ùNpXd á¥VÕm = = Gu\ úLôh¥tÏ 3 −2 −4 CûQVôLÜm 2x + 3y + 3z = 8 Gu\ R[j§tÏ ùNeÏjRôLÜm Es[ R[j§u ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, OBQ
14,
15.
x −1 − y z +1 = = Gu\ úLôhûP Es[Pd¡VÕm x - 2y + 3z - 2= 0 Gu\ 2 3 1 R[j§tÏ ùNeÏjRôLÜm NUuTôÓLû[d LôiL,
AûUkR
R[j§u
ùYdPo
Utßm
Lôo¼£Vu OBQ
16,
(− 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
17,
(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)
18.
19.
x − 2 y − 2 z −1 = = Gu\ úLôhûP Es[Pd¡VÕm (− 1. 1. − 1) Gu\ ×s° 2 3 −2 Y¯úVf ùNpXd á¥VÕUô] R[j§u ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, Ex. 2.8 (12) (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 r r r r r r 3i + 4 j + 2k . 2i − 2 j − k
r
r
20,
Utßm 7i + k 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)
21,
ùYhÓjÕiÓ Y¥®p JÚ R[j§u Lôo¼£Vu Øû\«Ûm RÚ®dL,
NUuTôhûP
ùYdPo
Øû\«Ûm Ex. 2.8 (14)
3, LXlùTiLs 1.
P Gàm
×s°
LXlùTi
Uô±
zId
z −1 = 1 dÏ z +i
ϱjRôp Re
¨VUlTôûRûVd LôiL, 2.
Pu
Ex. 3. 2 (8) (iii)
2z +1 = −2 dÏ Pu iz + 1
P Gàm ×s° LXlùTi Uô± zId ϱjRôp Im ¨VUlTôûRûVd LôiL,
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Ex. 3. 2 (8) (i)
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P Guàm ×s° LXlùTi Uô± z Id ϱjRôp P-Cu ¨VUlTôûRûV z −1 π arg Gu\ ¨TkRû]dÏ EhThÓ LôiL, Eg. 3.11 (ii) = z +1
4.
5.
3
a = cos2α + i sin 2α, b = cos2β + i sin 2β Utßm c = cos 2γ + i sin 2γ G²p 1
(i)
abc +
(ii)
a 2b2 + c 2 = 2 cos 2(α + β − γ) G] ¨ì©, abc
abc
= 2 cos (α + β + γ) Ex. 3.4 (10)
α, β GuTûY x2 − 2x + 2 = 0-u êXeLs Utßm cot θ = y + 1 G²p
y + α )n − ( y + β ) n sin (uuuuuuuuuuuuuuuuuuuuuuuu nθ G]d LôhÓL, = uuuuuu n sin θ α −β 6.
Eg. 3. 22
x2 − 2px + (p2 + q2) = 0 Gu\ NUuTôh¥u êXeLs α . β Utßm tan θ = q/(y + p)
y + α )n − ( y + β ) n (uuuuuuuuuuuuuuuuuuuuuuuu nθ G] ¨ßÜL, uuuuuu G²p = q n − 1 sin sin nθ α −β
Ex. 3. 4 (5)
7,
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)
8.
( − 3 − i ) 2/3 u GpXô U§l×Lû[Ùm LôiL,
9.
( 3 + i )2/3 u GpXô U§l×Lû[Ùm LôiL,
10.
(
1
2
−i
3
2
)
3/4
u
GpXô
U§l×Lû[Ùm
ùTÚdLtTXu 1 G]Üm LôhÓL,
Ex. 3. 5 1(iii) Eg. 3. 25
LôiL
Utßm
ARu
U§l×L°u Ex. 3. 5 (5)
11,
¾odL: x4 − x3 + x2 − x + 1 = 0.
12.
x7 + x4 + x3 + 1 = 0 Gu\ NUuTôhûPj ¾odL,
Eg. 3.24
13.
x9 + x5 − x4 − 1 = 0 Gu\ NUuTôhûPj ¾odL,
Eg. 3.23
Ex. 3. 5 4(ii)
4, TÏØû\ TÏØû\ Y¥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, 2. 3. 4. 5. 6.
y2 - 8x + 6y + 9 = 0 y2 - 8x − 2y + 17 = 0 y2 + 8x − 6y + 1 = 0 y2 + 4y + 4x + 8 = 0 x2 − 6x − 12y − 3 = 0 x 2 − 4x + 4y = 0
XII PUBLIC 24 QPs 10 marks
Eg 4.7 (iv) OBQ Ex 4.1 (2 iv) OBQ Ex 4.1 (2 v) OBQ
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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 Eg. 4. 10 HtTÓjRlTÓm G±úLôQm LôiL,
¿sYhPj§tÏ ûUVj ùRôûXj RLÜ. ûUVm. Ï®VeLs. Utßm Ef£LsB¡VYtû\d LôiL, úUÛm ARu Yû[YûWûVd LôiL,
13. 14. 15. 16. 17,
36x2 + 4y2 − 72x + 32y − 44 = 0 16x2 + 9y2 + 32x - 36y = 92 16x2 + 9y2 - 32x + 36y - 92 = 0 9x2 + 25y2 - 18x - 100y - 116 = 0
Eg 4.31 (iv) Ex 4.2 (6 iv) OBQ OBQ
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
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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)
19,
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
20,
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)
21,
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)
22,
JÚ úLô-úLô ®û[VôhÓ ÅWo ®û[VôhÓl T«t£«uúTôÕ AYÚdÏm úLôúLô 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)
23,
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, 24, 25. 26. 27. 28.
12x2 − 4y2 - 24x + 32y - 127 = 0 9x2 − 16y2 - 18x - 64y - 199 = 0 x2 − 4y2 + 6x + 16y - 11 = 0 x2 − 3y2 + 6x + 6y + 18 = 0 9x2 − 7y2 + 36x + 14y + 92 = 0
29.
x - y +4 = 0 Gu\ úSodúLôÓ ¿sYhPm x2 + 3y2 = 12 Ij ùRôÓúLôPôL Es[Õ G] ¨ì©dL, úUÛm ùRôÓm ×s°ûVÙm LôiL, Ex. 4.4 (6)
30.
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)
31,
×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)
XII PUBLIC 24 QPs 10 marks
OBQ Eg 4.56 Ex 4.3 (5 iii) Ex 4.3 (5 iv) OBQ
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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 – I 1,
úYLjRûPûV (Break) ùNÛj§V ©u]o JÚ YôL]m t ®]ô¥L°p ùNpÛm çWm xI x =20 t - 5/3 t2 Gu\ NUuTôhPôp RWlTÓ¡\Õ G²p (i) úYLjRûP ùNÛjRlThP úSWj§p YôL]j§u úYLm (¡,Á,/U¦) (ii) AqYôL]m úRdL ¨ûXdÏ YÚØu AÕ LPkR çWm B¡VYtû\d LôiL, Ex. 5. 1 (3)
2,
JÚ HÜLûQ. RûW«−ÚkÕ ùNeÏjRôL úUpúSôd¡f ùNÛjÕm úTôÕ t úSWj§p ùNpÛm EVWm x GuL, ARu
NUuTôÓ
x = 100t −
25 2 t G²p (i) 2
HÜLûQ«u ùRôPdL §ûNúYLm (ii) HÜLûQ EfN EVWjûR AûPÙm úTôÕ ARu úSWm (iii) HÜLûQ AûPÙm EfN EVWm (iv) HÜLûQ RûWûV AûPÙm úTôÕ ARu §ûN úYLm B¡VYtû\ LôiL, Ex. 5. 1 (1) 3.
14.7 Á, EVWØs[ úUûP«−ÚkÕ JÚ £ßYu JÚ LpûX úUpúSôd¡ G±¡\ôu, LmTj§−ÚkÕ Ntßj Rs° úSoÏjRôL úUpúSôd¡ ùNuß ©u AkR Lp RûWûV AûP¡\Õ, ARu CVdLf NUuTôÓ. ÁhPo Utßm ®]ô¥«p x = 9.8 t − 4.9t2 G²p (i) úUpúSôd¡f ùNpX. Utßm ¸rúSôd¡ YW GÓjÕd ùLôsÞm úSWm GqY[Ü? (ii) Lp RûW«p CÚkÕ úUúXf ùNuß AûPkR A§LThN EVWm Gu]? Eg. 5. 6
4,
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 3 ÁhPWôL CÚdÏm ùTôÝÕ. ¿o UhPj§u EVWm A§L¬dÏm ÅRjûRd LôiL, Eg. 5. 9 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) 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
5.
6.
7,
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
8,
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)
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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)
10,
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)
11.
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
12.
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)
13,
y = x2 Utßm y = (x – 2)2 Gu\ Yû[YûWLs ùYh¥d ùLôsÞm ×s°«p AûYLÞdÏ CûPlThP úLôQjûRd LôiL, Eg. 5. 17
14.
.y2 = x Utßm xy = k Gàm Yû[YûWLs Juû\ùVôuß ùNeÏjRôL ùYh¥d ùLôiPôp. 8k2 = 1 G] ¨ì©dL, Ex. 5.2 (11)
15,
ax 2 + by 2 = 1, a1 x 2 + b1 y 2 = 1 Gu\ Yû[YûWLs Juû\ Juß ùNeÏjRôL ùYh¥d ùLôs[j úRûYVô] ¨TkRû]ûVd LôiL Eg 5.18
16,
U§l× LôiL:
lim π
(tan x)cos x
Ex. 5. 6 (11)
x→ − 2
17, 18.
U§l× LôiL: lim (cot x)sin x
Eg. 5. 34
x→0
3 2 f ( x ) = 2 x + 3 x − 36 x + 10 Gu\ Nôo©u CPgNôokR ùTÚU Utßm £ßU
U§l×Lû[d LôiL,
OBQ
19,
TWYû[Vm y2 = 2x ÁÕ (1.4) Gu\ ×s°dÏ ªL AÚ¡Ûs[ ×s°ûVd LôiL, Eg 5.53
20,
ùLôÓdLlThP JÚ TWlT[®û]d ùLôiP ùNqYLeLÞs NÕWm UhÓúU £ßUf Ñt\[Ü ùTt±ÚdÏm G]d LôhÓL, Ex. 5.10 (3)
21,
ùLôÓdLlThP JÚ Ñt\[®û]d ùLôiP ùNqYLeLÞs NÕWm UhÓúU ùTÚU TWlT[ûYd ùLôi¥ÚdÏm G]d LôhÓL, Ex. 5.10 (4)
22,
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
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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
24.
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
25,
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
26.
r AXÏ BWØs[ AûWYhPj§às ùTÚU A[Ü ùLôsÞUôß YûWVlTÓm ùNqYLj§u TWl×d LôiL, Eg. 5. 54
27,
r BWØs[ YhPj§às YûWVlTÓm ªLl ùT¬V ùNqYLj§u ¿[ ALXeLs Gu]YôL CÚdÏm?
28.
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)
29.
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)
TWlT[Ü ùLôiP Ex. 5.10 (5)
6, YûL ÖiL¦Rm : TVuTôÓLs – II 1.
y = x3 Gu\ Yû[YûWûV YûWL,
2.
y = x3 +1 Gu\ Yû[YûWûV YûWL,
3.
y 2 = 2 x3 Gu\ Yû[YûWûV YûWL,
4,
u=
5,
Ex 6. 2 (1) Eg 6. 9 Eg 6. 10
x y ∂ 2u ∂ 2u Gu\ Nôo×dÏ = GuTûR N¬TôodL, − y 2 x2 ∂x∂y ∂y∂x ∂ 2u ∂ 2u −1 x u = tan Gu\ Nôo×dÏ = GuTûR N¬TôodL, ∂x∂y ∂y∂x y
6,
x ∂ 2u ∂ 2u u = sin Guàm Nôo×dÏ = GuTûR N¬TôodL, ∂x∂y ∂y∂x y
7,
f ( x) =
1 2
x + y2
Gu\ Nôo×dÏ ëX¬u úRt\jûR N¬TôodL,
XII PUBLIC 24 QPs 10 marks
Ex. 6.3 (1) (ii) Ex. 6.3 (1) (iv)
OBQ
Eg 6. 20
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8.
x− y u = sin −1 x+ y
∂u ∂u 1 = tan u G²p ëX¬u úRt\jûRl TVuTÓj§ x + y ∂x ∂y 2
G]d LôhÓL, 9,
Eg 6. 22
x3 + y 3 δu δu +y = sin 2u G²p. x δx δy x− y
ëX¬u úRt\jûRl TVuTÓj§. u = tan −1 G] ¨ì©dL,
10.
Ex. 6.3 (5) (i)
x+ y δu δu 1 u = sin G²p. x + = y x + y δx δy 2 G] ¨ì©dL,
x+ y x+ y cos x + y x + y OBQ
7, ùRôûL ÖiL¦Rm : TVuTôÓLs 1.
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
2.
x2 + y2 = 16 Gu\ YhPj§tÏm y2 = 6x Gu\ TWYû[Vj§tÏm ùTôÕYô] TWlûTd LôiL, Eg 7.29
3.
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
4.
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)
5.
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
6.
x2 y2 + = 1 Gu\ ¿sYhPj§p Es[ CWiÓ ùNqYLXj§tÏ CûPlThP 9 5
TWl©û]d LôiL,
Ex 7.4 (7)
7.
3ay2=x(x−a)2 Gu\ Yû[YûW«u Li¦«u (loop) TWlûTd LôiL,
8.
4y2 = 9x Utßm 3x2 = 16y Gu\ TWYû[VeLÞdÏ CûPlThP TWl©û]d LôiL, Ex 7.4 (9)
9.
y 2 = x Utßm x 2 = y B¡V CÚ TWYû[VeLÞdÏ CûPlThP TWlûTd LôiL,
Eg 7.33
OBQ 10.
y2 = 4x Gu\ TWYû[Vj§tÏm 2x − y = 4 Gu\ úLôh¥tÏm CûPlThP TWl©û]d LôiL, Ex 7.4 (8)
11.
y = x3 Gu\ Yû[YûWdÏm y = x Gu\ úLôh¥tÏm CûPlThP AWeLj§u TWlûTd LôiL, Eg 7.27
12.
y = sin x Utßm y = cos x Gu\ Yû[YûWLs x = 0 Utßm x = π Gu\ úLôÓLs B¡VYtßdÏ CûPúV Es[ AWeLj§u TWlûTd LôiL, Eg 7.30
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BWm ‘r’, ÏjÕWVm ‘h’ EûPV ám©u L]A[ûYd LôÔm ãj§Wj§û] ùRôûL«û]l TVuTÓj§ LôiL, Ex 7.4 (15)
14.
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) Utßm B¡V Øû]Lû[d ùLôiP ØdúLôQj§u TWlT[Ü 0, 0 , 4, 0 4,3 ( ) ( ) ( ) x-AfÑ ÁÕ ÑZtßYRôp HtTÓm ùTôÚ°u L] A]®û]d LôiL, OBQ
15.
x = t2
; y=t−
t3 Gu\ 3
ÕûQVXÏ
NUuTôÓLû[d
x-AfûNl ùTôßjÕ ÑZtßm ãZlThP TWl©u Yû[YûW«Ûs[ §PlùTôÚ°u L] A[ûYd LôiL,
ùLôiP
Li¦Vôp
úTôÕ
HtTÓjÕm OBQ
16.
4y2 = x3 Gu\ Yû[YûW«p x = 0 −ÚkÕ x = 1 YûWÙs[ ®p−u ¿[jûRd LôiL, Eg 7.37
17,
BWm ‘a’ EûPV YhPj§u Ñt\[ûY ùRôûLÂhÓ Øû\«p LôiL, Ex 7.5 (1)
18.
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) 2
19. 20.
2
x 3 y 3 a + a = 1 Gu\ Yû[YûW«u ¿[jûRd LôiL,
Eg 7.38
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
21.
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)
22,
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)
23.
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 NUuTôÓLs 1,
¾odL: (x3 + 3xy2) dx + (y3 + 3x2y) dy = 0
2,
¾odL: dy = x 3 dy + 3 x 2 ydx + sec x (sec x + tan x) dx
3,
¾oÜ LôiL:
( x + y )2
dy = a2 dx
XII PUBLIC 24 QPs 10 marks
Eg 8.14
OBQ Eg 8.7
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( x + y )2
dy =1 dx
4,
¾oÜ LôiL:
5,
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
6,
¾odL: (x2 + y2) dx + 3xy dy = 0
7,
¾odL : (1 − x 2 )
8,
¾odL: (1 + y2) dx = (tan−1 y − x) dy
9.
(1 + 2x 3 )
10,
GkRùYôÚ ×s°«Ûm NônÜ y+2x G]d ùLôiÓ Yû[YûW«u NUuTôÓ y = 2(ex − x − 1) G]d LôhÓL,
11,
¾odL:
Ex 8.2 (7)
Ex 8.3 (5)
dy + 2 xy = x (1 − x 2 dx
Eg 8.18 Eg 8.19
dy + 6x 2 y = cosec 2 x dx
d2y dx 2
−3
dy + 2 y = 2e3 x dx
OBQ B§Y¯VôLf
ùNpÛm Ex 8.4 (9)
; CeÏ x = log2 G²p y = 0 Utßm x = 0 G²p y = 0. Ex 8.5 (6)
12,
¾odL: (D2 − 1) y = cos 2x − 2 sin 2x
13,
¾odL:
(D
2
− 5D + 6 y = sin x + 2e3 x
14,
¾odL:
(D
2
− 2 D + 2 y = sin 2 x + 5
15,
¾odL: (D2 − 6D + 9) y = x + e2x
16,
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
17,
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)
18,
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
Ex 8.5 (11)
)
OBQ
)
OBQ Ex 8.5 (10)
16
UdLs ùRôûL GqY[YôL CÚdÏm? log e = .2070;e.42 = 1.52 13
19,
Ex 8.6 (4)
úW¥Vm (Radium) £ûRÙm UôßÅRUô]Õ. A§p LôQlTÓm A[®tÏ ®¡RUôL AûUkÕs[Õ, 50 YÚPeL°p BWmT A[®−ÚkÕ 5 NRÅRm
XII PUBLIC 24 QPs 10 marks
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www.kalvisolai.com Page 13 £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) 20,
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
21,
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
22,
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 e 21.4 = −0.0426 × 2.303 Utßm log e 21.4 = −0.0945 × 2.303
Eg 8.37
23,
JÚ L§¬VdLl ùTôÚs £ûRÙm UôßÅRUô]Õ. ARu GûPdÏ AûUkÕs[Õ, ARu GûP 10 ª,¡Wôm BL CÚdÏm úTôÕ £ûRÙm Sôù[ôußdÏ 0.051 ª,¡Wôm G²p ARu GûP 10 ¡Wôª−ÚkÕ 5 Ïû\V GÓjÕd ùLôsÞm LôX A[ûYd LôiL, [loge2 = 0.6931]
®¡RUôL UôßÅRm ¡WôUôLd Ex 8.6 (5)
24,
ì 1000 Gu\ ùRôûLdÏ ùRôPof£ áhÓ Yh¥ LQd¡PlTÓ¡\Õ, Yh¥ ÅRmBiùPôußdÏ 4 NRÅRUôL CÚl©u, AjùRôûL GjRû] BiÓL°p Ex 8.6 (2) BWmTj ùRôûLûVl úTôp CÚ UPeLôÏm? (loge2 = 0.6931).
9, R²¨ûX LQd¡Vp 1,
éf£VUt\ LXlùTiL°u LQUô] C − {0} p YûWVßdLlThP f1 (z) = z, f2 (z) = − z, f3 ( z ) =
1 1 , f 4 ( z ) = − ∀z ∈ C − {0}Gu\ Nôo×Ls VôÜm APe¡V z z
LQm {f1, f2, f3, f4} B]Õ Nôo×L°u úNol©u ¸r JÚ GÀ−Vu ÏXm AûUdÏm G] ¨ßÜL, Eg 9.24
2.
1 0 ω 0 ω 2 0 0 1 0 ω 2 0 , , 2 , , 2 , 0 1 0 ω 0 ω 1 0 ω 0 ω
ω 0
Gu\
ùTÚdL−u ¸r JÚ ÏXjûR AûUdÏm G]d LôhÓL, ( ω3=1)
LQm
A¦l Ex 9.4 (6)
3,
(Z7 − {[0]}, .7) JÚ ÏXjûR AûUdÏm G]d LôhÓL,
4.
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)
5.
(Zn, +n) JÚ ÏXm G]d LôhÓL,
XII PUBLIC 24 QPs 10 marks
Eg 9.26
Eg 9.25
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7.
YZdLUô] ùTÚdL−u ¸r 1u nm T¥ êXeLs Ø¥Yô] GÀ−Vu ÏXjûR AûUdÏm G]d LôhÓL, Eg 9.27 x x
x , x ∈ R − {0} Gu\ AûUl©p Es[ A¦Ls VôÜm APe¡V LQm G x
B]Õ A¦lùTÚdL−u ¸r JÚ ÏXm G]d LôhÓL,
8.
Eg 9.21
a 0 , a ∈ R − {0} AûUl©p Es[ GpXô A¦LÞm APe¡V LQm A¦l 0 0
ùTÚdL−u ¸r JÚ GÀ−Vu ÏXjûR AûUdÏm G]d LôhÓL, 9.
G GuTÕ ªûL ®¡RØß Gi LQm GuL,, a * b =
Ex 9.4 (11)
ab ∀a, b ∈ G GàUôß 3
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,
G = {2n / n ∈ Z} Gu\ LQUô]Õ ùTÚdL−u ¸r JÚ GÀ−Vu ÏXjûR AûUdÏm G]d LôhÓL, Ex 9.4 (12)
14.
| 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, ¨LrRLÜl ¨LrRLÜl 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, 1 (3) P (X ≤ x) > BL CÚdL x Cu Áf£ß U§l× LôiL, Eg 10.2 2
3,
JÚ NUYônl× Uô± x Cu ¨LrRLÜ APoj§f Nôo× α α −1 − β x e ; f (x ) = kx 0 ;
x,α,β > 0 Utù \ e¡ Û m
G²p (i) k Cu U§l× LôiL (ii) P(X > 10) LôiL,
XII PUBLIC 24 QPs 10 marks
Ex 10.1 (7)
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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,
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
6,
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 < 1] = 0.3413 7,
Eg 10.29
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)
(ii)
8,
P [0 < z < 0.4] = 0.1554 P [0 < z < 2] = 0.4772
(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, NUYônl× Øû\«p úRokùRÓdLlTÓm NdLWj§u Lôt\ÝjRm 30.5 psi dÏ A§LUôL CÚdL ¨LrRLÜ LôiL, Eg 10.32
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)
9,
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
10,
CVp¨ûXl TWY−u ¨LrRLÜ APoj§f Nôo× f ( x ) = k e−2x G²p k, µ Utßm σ2 Cu U§l× LôiL,
11,
JÚ CVp¨ûXl TWY−u ¨LrRLÜl TWYp f ( x ) = c e− x c, µ, σ2 CYtû\d LôiL,
XII PUBLIC 24 QPs 10 marks
2
+ 3x
2
+ 4x
,−∞
, − ∞ < x < ∞ G²p. Ex 10.5 (8)
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XII PUBLIC EXAM - 6 MARK QUESTIONS (221)
1, A¦Ls Utßm A¦dúLôûYL°u TVuTôÓLs 1.
2.
1 2 Gu\ A¦«u úNolûTd LiÓ. A (adj A) = (adj A) A = | A |. I 3 −5
A=
GuTûRf N¬TôodL,
Ex. 1.1 (2)
−4 −3 −3 0 1 Cu úNol× A¦ A G] ¨ßÜL, A= 1 4 4 3
Ex. 1.1 (8)
(AB)−1 = B−1 A−1 GuTûRf N¬Tôo, 3, 4,
1 1 5 A= 7
A=
2 0 −1 Utßm B = 1 1 2 2 2 −1 Utßm B = 3 −1 1
Eg. 1.6
Ex. 1.1 (5)(i)
5.
3 1 −1 2 −2 0 Gu\ A¦«u úSoUôß A¦ LôiL, 1 2 −1
6.
−1 2 −2 A = 4 −3 4 G²p A = A−1 G]d LôhÓL, 4 −4 5
7.
−2 −3 A= G²p. 5 −6
8.
A, B CWiÓ éf£VUt\ úLôûY A¦Ls G²p, (AB)−1 = B−1 A−1 G] ¨ßÜL,
T
(A ) =(A ) −1
T
−1
Eg. 1.5 (iv)
Ex. 1.1 (10)
GuTûRf N¬TôodL,
OBQ
úSoUôß A¦ LôQp Øû\ Øû\«p ¾odL: ¾odL: 9. 10.
x + y = 3, 2x − y = 7,
2x + 3y = 8 3x − 2y = 11
Eg. 1.7 Ex. 1.2 (1)
12.
1 2 −1 3 2 4 1 −2 3 6 3 −7
14.
3 1 −5 −1 1 −2 1 −5 1 5 −7 2
A¦«u RWm LôiL,
11.
1 2 3 −1 2 4 6 −2 Eg. 1.14 3 6 9 −3
13.
1 −2 3 4 −2 4 −1 −3 Ex. 1.3 (6) −1 2 7 6
XII PUBLIC 24 QPs 6 marks
Ex 1.3 (5)
Eg. 1.16
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15.
3 1 2 0 1 0 −1 0 Ex. 1.3 (3) 2 1 3 0
17.
−2 1 3 4 0 1 1 2 1 3 4 7
OBQ
16.
0 1 2 1 2 −3 0 −1 1 1 −1 0
18.
−4 12 12 4 1 0 4 8 4 4 4 −4 8 0
Ex. 1.3 (4)
OBQ
©uYÚm ANUT¥jRô] ANUT¥jRô] úS¬V NUuTôhÓ ùRôÏl©û] A¦dúLôûY Øû\ Øû\«p ¾odL: 19, 2x + 3y = 8 4x + 6y = 16 Eg 1.17 (2) 20. 4x + 5y = 9 8x + 10y = 18 Ex 1.4 (3) 21. 22. 23. 24.
2x - 3y = 7 2x + 2y + z = 5 x + y + 3z = 4 x + y + 2z = 4
4x - 6y = 14 x−y+z=1 2x + 2y + 6z = 7 2x + 2y + 4z = 8
OBQ Eg 1.18 (3) OBQ Eg. 1.18 (5)
3x + y + 2z = 4 2x + y + z = 10 3x + 3y + 6z = 10
©uYÚm NUuTôÓL°u ùRôÏl©u JÚeLûUÜj JÚeLûUÜj RuûUûVj RW Øû\ Øû\ûVl TVuTÓj§ BWônL, 25. 26.
x+y+z=7 x – 4y +7z = 14
x + 2y + 3z = 18 3x + 8y – 2z = 13
y + 2z = 6 7x – 8y + 26z = 5
Ex. 1.5 (1)(iii) 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
3.
AC Utßm BDI êûX®hPeL[ôLd ùLôiP SôtLWm ABCDu TWl×
a b c = = Guß ¨ì©dL, sin A sin B sin C
1 uuur uuur AC × BD G]d LôhÓL, 2 4.
Eg. 2.26
r r r 4i − 3 j − 2k I ¨ûX ùYdPWôLd ùLôiP ×s° PCp ùNVpTÓm ®ûNLs r r r r r r r r 2i + 7 j , 2i − 5 j + 6k Utßm −i + 2 j − k BÏm, CûYL°u ®û[Ü ®ûN«u r r r §Úl×j §\û] 6i + j − 3k -I ¨ûX ùYdPWôLd ùLôiP Q Gu\ ×s°ûVl ùTôßjÕd LôiL,
r r r r r r b−c c−a
Ex. 2.4 (8)
5.
[a −b
6.
r r r r r r r r r [a + b + c b + c c] = [a b c]
7.
r r r r r r r r r a = i − j , b = j − k , c = k − i G²p
8.
Eg. 2.28
] = 0 G] ¨ßÜL,
OBQ
G] ¨ßÜL,
r r r r r r b−c c−a
[a −b
OBQ
] LôiL,
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,
XII PUBLIC 24 QPs 6 marks
OBQ
OBQ
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r r r r r r r r r r r r ¨ì©: a x b . c x d + b x c . a x d + c x a . b x d = 0
10.
GpXô ùYdPoLs a, b, c dÏm. a × b, b × c, c × a = a b c G] ¨ßÜL,
11.
(3, − 4, − 2) Gu\ ×s°Y¯f ùNpYÕm 9i + 6 j + 2k Gu\ ùYdPÚdÏ CûQVô]ÕUô] úLôh¥u ùYdPo Utßm Lôo¼£Vu NUuTôÓLû[d LôiL, Ex. 2.6 (6)
(
)(
) (
)(
) (
ur ur r
)(
r r r r r r
r
12.
r r=
r r
)
r r r
(i − j ) + t ( 2i − j + k ) Utßm
r r=
r rr
r
r r
Ex 2.5 (10) 2
Eg. 2.38
r
r r
r
r
( 2i + j + k ) + s ( 2i − j + k ) Gu\ CûQ
úLôÓL°u CûPlThP çWjûRd LôiL,
13.
r r=
r
r
r
r
r
r
(3i + 5 j + 7k ) + t (i − 2 j + k ) Utßm
r r=
Eg. 2.42
r r
r
r
r
JÚ R[j§p AûUVôR úLôÓLs G]d LôhÓL,
14.
r r=
(
r r r r i − j + t 2i + k Utßm
) (
)
r r=
(
r
(i + j + k ) + s ( 7i + 6 j + 7k ) GuT] Ex. 2.7 (2)
r r r r r 2i − j + s i + j − k Gu\ CÚ úLôÓLs JúW
) (
)
R[ AûUVôd úLôÓLs G]d Lôh¥. AYt±tÏ CûPlThP çWjûRÙm LôiL, Eg. 2.43 15.
16.
(3, − 1, − 1), (1, 0, − 1) Utßm (5, − 2, − 1) Gu\ ×s°Ls JÚ úLôhPûUl ×s°Ls G]d LôhÓL, Eg. 2.46 r r=
(
r r r r r r 2i + j − 3k + t 2i − j − k Gu\ úLôÓm x − 2y + 3z + 7 = 0 Gu\ R[Øm
) (
)
Nk§d¡u\ ×s°ûVd LôiL,
17.
Ex. 2.9 (4)
r r r 2i − j + 3k 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)
18,
(1. −1. 1)I ûUVUôLÜm U§lûT BWjûRd NUuTôÓLû[j RÚL,
19.
20.
r r r r r − (i + j + 2k = 5 Gu\ úLô[j§u BWj§tÏ NUUô] ùLôiP
úLô[j§u
ùYdPo
Utßm
Lôo¼£Vu Ex. 2.11 (3)
(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
r r r 2i + 6 j − 7 k Utßm
r r r 2i − 4 j + 3k 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
21.
r r r r r r 2i + 6 j − 7 k Utßm −2i + 4 j − 3k 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)
22,
r2 r r r r r − r.(8i - 6 j +10k ) - 50 = 0
Gu\ ûUVjûRÙm BWjûRÙm LôiL,
XII PUBLIC 24 QPs 6 marks
ùYdPo
NUuTôhûPÙûPV
úLô[j§u Eg. 2.65
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úLô[j§u ®hPm. úUtTWl©p HúRàm JÚ ×s°«p HtTÓjÕm úLôQm ùNeúLôQm G]d LôhÓ, Ex. 2.11 (6)
3, LXlùTiLs ©uYÚm NUuTôhûP ¨û\Ü ùNnÙm x Utßm y-«u ùUn U§l×Lû[d LôiL,
1,
x 2 + 3x + 8 + (x + 4)i = y(2 + i)
Ex. 3.1 (4) (iii)
2,
(1 − i) x + (1 + i) y = 1 − 3i
3.
(− 8 − 6i) -Cu YodL êXeLs LôiL,
4.
(− 7 + 24i) -Cu YodL êXeLs LôiL,
5,
LXlùTiL°u ØdúLôQ NU²−ûV Gݧ ¨ì©,
Page 124
6.
z1 Utßm z2 Gu\ CÚ LXlùTiLÞdÏ (i) | z1 z2 | = | z1 |.| z2 | (ii) arg (z1.z2) = arg z1 + arg z2 GuTûR ¨ì©,
Page 127
7,
Ex. 3.1 (4) (i) Ex. 3.2 (2) Eg. 3.16
( a1 + ib1 )( a2 + ib2 ) ...( an + ibn ) = A + iB G²p ¨ì©: ( i ) ( a12 + b12 )( a2 2 + b2 2 ) ...( an 2 + bn 2 ) = A2 + B 2 b1 −1 b2 −1 bn −1 B + tan + ... + tan = kπ + tan , k ∈ Z . A a1 a2 an
( ii ) tan −1
8,
ÑÚdÏL:
( cos θ + i sin θ )
4
(sin θ + i cos θ )5
Eg. 3.19
P Guàm ×s° LXlùTi Uô± z Id ϱjRôp P-Cu ¨VUlTôûRûV ©uYÚm ¨TkRû]dÏ EhThÓ LôiL, 9.
2z −1 = z − 2
OBQ
10.
z − 3i = z + 3i
OBQ
11.
3z − 5 = 3 z + 1
OBQ
12.
z +1 Re =0 z −i
OBQ
13.
(7 + 9i), (− 3 + 7i), (3 + 3i) Gàm LXlùTiLs BoLu R[j§p JÚ ùNeúLôQ ØdúLôQjûR AûUdÏm G] ¨ßÜL, Eg. 3.15
14,
LXlùTi R[j§p (10 + 8i), (− 2 + 4i), (-11 + 31i) Gàm LXlùTiLs AûUdÏm ØdúLôQm JÚ ùNeúLôQ ØdúLôQm G] ¨ßÜL, Ex. 3.2 (4)
15.
ùUnùVi ÏQLeLû[d ùLôiP P(x) = 0 Gu\ TpÛßl×d úLôûYf NUuTôh¥u LXlùTi êXeLs CûQùVi CWhûPVôLjRôu CPmùTßm G] ¨ßÜL, Page 140
16.
3 + i I JÚ ¾oYôLd ùLôiP x4 − 8x3 + 24x2 − 32x + 20 = 0 Gàm NUuTôh¥u ©\ ¾oÜLû[d LôiL, Ex. 3.3 (1)
XII PUBLIC 24 QPs 6 marks
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1 + 2i JÚ êXUôLd ùLôiP x4 − 4x3 + 11x2 − 14x + 10 = 0 Gàm NUuTôh¥u ¾oÜLû[d LôiL, Ex. 3.3 (2)
18.
(1+i) I JÚ ¾oYôLd ùLôiP x4 + 4 = 0 Gàm NUuTôh¥u ¾oÜLû[d LôiL, OBQ
19.
(1 - i) I JÚ ¾oYôLd ùLôiP x3 - 4x2 +6x – 4= 0 Gàm NUuTôh¥u ¾oÜLû[d LôiL, OBQ
20.
m n x = cos α + i sin α, y = cos β + i sin β G²p x y +
1 = 2 cos (mα + nβ) G] x yn m
¨ì©,
Ex. 3.4 (9) n
21.
1 + sin θ + i cos θ π π = cos n − θ + i sin n − θ n GuTÕ ªûL ØÝ Gi G²p 1 + sin θ − i cos θ 2 2 G] ¨ì©dL, Eg. 3.20
22.
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)
23,
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,
24.
(1 + cos θ + i sin θ)n + (1+cos θ − i sin θ)n = 2n + 1 cosn
θ 2
cos
Ex. 3.4 (3) (i)(ii)
nθ G] ¨ßÜL, 2 Ex. 3.4 (4) (iii)
25.
n GuTÕ ªûL ØÝ Gi G²p (1 + i)n + (1 − i)n = 2
n+2 2
cos
nπ G] ¨ì©, 4 Ex. 3.4 (4) (i)
26.
n +1 n ∈ N G²p (1 + i 3 )n + (1 − i 3 )n = 2 cos
27,
x+
28,
¾odL : x4 +4 = 0.
nπ G] ¨ßÜL, 3
Ex. 3.4 (4) (ii)
1 1 1 = 2cosθ G²p (i) x n + n = 2cosnθ (ii) x n − n = 2isinnθ x x x
Ex. 3.4 (7) 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 A§TWYû[Vj§u NUuTôhûPd LôiL,
XII PUBLIC 24 QPs 6 marks
3 G²p
Eg. 4.36
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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 ùRôÓúLôÓL°u NUuTôÓLû[d LôiL,
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,
A§TWYû[Vj§u HúRàm JÚ ×s°«−ÚkÕ ARu ùRôûXj ùRôÓúLôÓL°u ùNeÏjÕj çWeL°u ùTÚdÏjùRôûL JÚ Uô±− Gußm ARu U§l×
13,
ûUVm −2,
a 2b 2 G]Üm LôhÓL, a 2 + b2
YûWVjRdL CÚ Ex. 4.4 (4) (iii)
Eg. 4.68
−3 −2 Utßm 1, Gu\ ×s°Y¯f ùNpÛm §hPf ùNqYL 2 3
A§TWYû[Vj§u NUuTôÓ LôiL,
Eg. 4.69
14,
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)
15,
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
16,
JÚ ùNqYL A§TWYû[Vj§tÏ YûWVlThP ùRôÓúLôh¥u ùRôÓ×s° ùRôûXjùRôÓ úLôÓLÞdÏ CûPlThP TôLj§û] CÚ NUUôLl ©¬dÏm G]d LôhÓL, Eg 4.71
17,
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.
EÚû[ Y¥®Xô] JÚ EúXôLj Õi¥p Es[ 4 ª,Á, ®hPØm 12 ª,Á, BZØm ùLôiP JÚ Õû[«û] ÁiÓm A§LlTÓjR ARu ®hPm 4,12 ª,ÁhPWôL A§L¬dLlTÓ¡\Õ, CRu ®û[YôL Õû[jÕ GÓdLlThP EúXôLj§u úRôWôV A[ûYd LôiL, Eg. 5. 25
10.
1 Gu\ Nôo×dÏ ùUdXô¬u ®¬Ü LôiL, 1+ x
11.
log e (1 + x) Gu\ Nôo×dÏ ùUdXô¬u ®¬Ü LôiL,
12.
tan x,
13.
t Gu\ úSWj§p. R Gu\ ªuRûP«û]Ùm. L Gu\ çiÓ ªuú]ôhPj§û]Ùm. úUÛm E Gu\ Uô\ô ªu CVdÏ ®ûN«û]Ùm ùLôiP
−π π
ªuNôWm i-u NUuTôÓ i =
1
U§l× LôiL: lim cos ecx − x →0 x
XII PUBLIC 24 QPs 6 marks
Eg. 5. 28 (2) Ex. 5. 5 (4)
− Rt E L 1 − e G²p. R ªLf £±VRôL CÚdÏm úTôÕ R
iI LôQ ELkR ãj§WjûRd LôiL, 14,
Ex. 5. 5 (3)
Eg. 5. 36 Eg. 5. 33
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www.kalvisolai.com Page 23 lim x cos x
15,
U§l× LôiL:
16.
lim x x −1 u U§l©û]d LôiL,
17.
x sin x Gu\ Nôo×
18.
f(x) = 20 − x − x2 Gu\ Nôo©u Hßm ApXÕ C\eÏm CûPùY°Lû[d LôiL, Ex. 5.7 (5) (i)
19.
f(x) = x3 − 3x + 1 Gu\ Nôo©u Hßm ApXÕ C\eÏm CûPùY°Lû[d LôiL, Ex. 5.7 (5) (ii)
20.
x > 0 YôLÜm. n > 1 G]Üm CÚl©u (1 + x)n > 1 + nx G] ¨ì©dL,
21.
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
22.
JÚ SLÚm YôL]j§u RûP (F)Cu NUuTôÓ F = 5/x + 100x G²p. RûP«u £ßU U§lûTd LôiL, Ex. 5. 10 (6)
23,
CWiÓ GiL°u áÓRp 100 AqùYiL°u ùTÚdÏj ùRôûL ùTÚU U§lTôL ¡ûPdL AqùYiLs Gu]YôL CÚdL úYiÓm? Ex. 5. 10 (1)
24,
CWiÓ ªûL GiL°u ùTÚdÏj ùRôûL 100. AqùYiL°u áÓRp £ßU U§lTôL ¡ûPdL AqùYiLs Gu]YôL CÚdL úYiÓm? Ex. 5. 10 (2)
25.
y = x3 − 3x + 2 Gu\ Nôo©tÏ Yû[Ü Uôt\l ×s°Ls CÚl©u AYtû\d LôiL, Eg. 5. 65
26.
f ( x ) = (x-1) 3 Gu\ Nôo× GkR CûPùY°L°p GuTûRÙm Utßm Yû[Ü Uôtßl ×s°Lû[Ùm LôiL,
27.
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)
28.
y = sinx, x ∈ (0, 2π) Gu\ Yû[YûW«u Yû[Ü Uôtßl ×s°Lû[d LôiL, Eg. 5. 66
OBQ
x →0 +
1
Ex. 5.6 (10)
x →1
[ 0, π ] p
K¬VpTt\Õ G] ¨ì©dL,
1
Ex. 5.7 (4) (iii)
Eg 5.45,
ϯÜ
AûP¡\Õ Ex. 5. 11 (1)
6, YûL ÖiL¦Rm : TVuTôÓLs – II 1,
YûLÂhûPl
TVuTÓj§
1 10.1
u
U§lûT
úRôWôVUôL
CWiÓ
RNU
CPeLÞdÏ §ÚjRUôL LôiL,
Ex 6.1 (3) (ii)
2,
YûLÂhûPl TVuTÓj§
Ex 6.1 (3) (i)
3,
YûLÂhûPl TVuTÓj§
4,
YûLÂhûPl TVuTÓj§ (1.97)6 dÏ úRôWôV U§l×d LôiL,
XII PUBLIC 24 QPs 6 marks
36.1 dÏ úRôWôV U§l×d LôiL, 3
65 dÏ úRôWôV U§l×d LôiL,
Eg 6.3 Ex 6.1 (3) (iv)
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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
6,
JÚ Gi¦u nm T¥ êXm LQd¡PlTÓm úTôÕ HtTÓm NRÅRl ©ûZ úRôWôVUôL. AkR Gi¦u NRÅRl ©ûZ«u
1 UPeÏ BÏm G]d LôhÓL, n Eg 6.7
7,
∂u u = log (tan x + tan y + tan z) G²p Σ sin 2x = 2 G] ¨ì©, ∂x
Eg 6.15
8.
U = (x − y) (y − z) (z − x) G²p Ux + Uy + Uz = 0 G]d LôhÓL,
Eg 6.16
9.
z = ye x Gu\ Nôo©p x = 2t Utßm y = 1 − t GàUôß CÚl©u
10.
w=
11.
w = x + 2y + z2 Gu\ Nôo©p x = cos t ; y = sin t ; z = t G²p
12.
x = u2 − v2, y = 2uv Guß CÚdÏUôß w = x2 + y2 G] YûWVßdLlTÓ¡\Õ G²p
2
x x2 + y2
Gu\ Nôo©p x = cos t ; y = sin t G²p
dw dt
dz LôiL, Eg 6.17 dt
I LôiL, Ex 6.3 (3)(iii) dw dt
I LôiL,Eg 6.19
∂ω ∂ω Utßm Id LôiL, ∂u ∂v
Ex 6.3 (4) (ii)
∂u ∂u x G²p x +y = 3u G]d LôhÓL, ∂x ∂y y
13.
u = xy 2 sin
14.
u GuTÕ x, yCp n-Bm T¥ NUlT¥jRô] NôoTô«u x
Ex 6.3 (5) (ii) ∂ 2u
∂x
2
+y
∂ 2u
∂y
2
= (n − 1)
∂u G] ∂y
¨ßÜL, 15.
Eg 6.21
V= Zeax + by Utßm Z B]Õ x, y-Cp nBm T¥ NUlT¥jRô] NôoTô«u x
∂V ∂V +y = ∂x ∂y
( ax
+ by + n ) V G] ¨ßÜL,
Ex 6.3 (5) (iv)
7, ùRôûL ÖiL¦Rm : TVuTôÓLs 2π
1.
∫ sin 0
9
x 4 dx u U§l× LôiL, π
2,
U§l©ÓL:
Eg 7.15 (iii)
2
∫ sin
4
x cos 2 x dx
Eg 7.16
0
π
3,
YûWVßjR ùRôûL«u Ti©û]l TVuTÓj§ U§l×d LôiL:
∫ π
−
3
4,
YûWVßjR ùRôûL«u Ti©û]l TVuTÓj§ U§l×d LôiL:
2
∫ 0
x sin xdx Eg 7.7 2
xdx x + 3− x
Ex 7.2 (8)
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5,
1
∫ log x − 1 dx
U§l©ÓL:
Ex 7.2 (7)
0
π
6,
YûWVßjR ùRôûL«u Ti©û]l TVuTÓj§ U§l×d LôiL:
2
∫ log ( tan x ) dx 0
Eg 7.11 π 3
7,
∫ 1+ π
U§l©ÓL:
dx
Eg 7.12
cot x
6
π 3
8,
∫ 1+ π
U§l©ÓL:
dx
Ex 7.2 (10)
tan x
6
9.
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
10.
y = sin 2x Gu\ Yû[YûW. x = 0, x = π Utßm x-AfÑ B¡VYt\ôp AûPTÓm AWeLj§u TWl©û]d LôiL, Eg 7.24
11.
x2 a
2
+
y2 b2
=1
(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 4
1,
¾odL: x dy = ( y + 4 x5 e x ) dx
2,
¾odL: ( x 2 + y 2 ) dy = xy dx
3,
¾odL: x 2
4,
¾odL:
dy + y cot x = 2 cos x dx
Eg 8.17
5.
¾odL:
dy + 2 y tan x = sin x dx
Eg 8.21
6,
¾odL:
dy 4x 1 + 2 y= 2 dx x + 1 ( x + 1) 2
Ex 8.4 (2)
7.
¾odL: (1+ x 2 )
dy + 2xy = cosx dx
Ex 8.4 (4)
8,
¾odL:
dy = y 2 + 2 xy ; x = 1 G²p y = 1 dx
dy +y=x dx
XII PUBLIC 24 QPs 6 marks
Eg 8.8 Ex 8.3 (3) Ex 8.3 (4)
Ex 8.4 (1)
© reserved with N. MAHALAKSHMI PGT (Mathematics)
www.kalvisolai.com Page 26 dy + xy = x dx
9,
¾o:
10,
¾odL : (D2 + 1) y = 0, CeÏ x = 0 G²p y = 2. úUÛm
11,
¾odL : (2D 2 + 5D + 2)y = e
Ex 8.4 (6)
2
x=
π 2
G²p y = − 2.
1 − x 2
Ex 8.5 (5)
Eg 8.28
1 − x 3
12,
¾odL : (3D + 4D + 1)y = 3e
13,
¾odL:
(D
2
+ 4 D + 13 y = cos 3x
14,
¾odL:
(D
2
+ 5 D + 4 y = sin 5 x
15,
¾odL:
(D
2
− 2 D − 3 y = sin x cos x
16,
¾odL:
(D
2
− 3D + 2 y = x
17,
¾odL:
(D
2
− 4D + 1 y = x2
18,
¾odL:
(D
2
+ 14 D + 49 y = e −7 x + 4
19,
¾odL:
( 3D
20.
JÚ Ï°of£VûPÙm ùTôÚ°u ùYlT¨ûX A[Ü T B]Õ Ïû\Ùm Uôß ÅRm T−S Gu\ ®j§VôNj§tÏ ®¡RUôL AûUkÕs[Õ, CeÏ S GuTÕ Ñtßl×\j§u ¨ûXVô] ùYlT ¨ûXVôÏm, BWmTj§p T = 150°C G²p t úSWj§p Ï°of£VûPÙm ùTôÚ°u ùYlT¨ûXûVd LôiL, Eg 8.36
Ex 8.5 (14)
)
Eg 8.30
)
OBQ
)
Ex 8.5 (8)
)
Eg 8.32
)
Eg 8.33
)
2
Ex 8.5 (3)
)
+ D − 14 y = 13e 2 x + 10e x
OBQ
9, R²¨ûX LQd¡Vp 1.
∼ ( ∼ p ) ∧ ( ∼ q ) Gu\ átßdÏ ùUn AhPYûQ AûUdL,
2.
( p ∨ q ) ∧ ( r ) dϬV ùUn AhPYûQûV AûUdL,
3.
( p ∧ q ) ∨ ( r ) u ùUn AhPYûQûV AûUdL,
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 G]d LôhÓL,
Ex 9.3 (2)
8.
p ↔ q ≡ (( ∼ p) ∨ q) ∧ (( ∼ q) ∨ p) G]d LôhÓL,
Ex 9.3 (4)
XII PUBLIC 24 QPs 6 marks
Eg 9.4 (iv) Eg 9.6 Ex 9.2 (10)
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p ↔ q ≡ (p → q) ∧ (q → p) G]d LôhÓL,
Ex 9.3 (3)
10.
p → q Utßm q → p NUô]Ut\ûY G]d LôhÓL,
Ex 9.3 (6)
11.
[(∼ p) ∨ ( ∼ q )] ∨ p JÚ ùUnûU G]d LôhÓL,
12.
(( ∼ p) ∨ q) ∨ (p ∧ ( ∼ p)) JÚ ùUnûUVô GuTRû] ùUn AhPYûQûVd ùLôiÓ ¾oUô²dL, OBQ
13.
(p ∧ ( ∼ q)) ∨ (( ∼ p) ∨ q) Gu\ átß ùUnûUVô. ØWiTôPô GuTRû] ¾oUô²dL, Ex 9.3 (1) (iii)
14.
( p ∧ q) → ( p ∨ q) GuTÕ JÚ ùUnûU G]d LôhÓL,
15,
((∼ q) ∧ p) ∧ q JÚ ØWiTôÓ G]d LôhÓL,
16,
ùUn AhPYûQûVd ùLôiÓ ©uYÚm átß ùUnûUVô ApXÕ ØWiTôPô G]j ¾oUô²dLÜm: (p ∧ ( ∼ p)) ∧ (( ∼ q) ∧ p) Ex 9.3 (1) (v)
17,
YûWVßdLlThP ϱÂh¥u T¥ (Z5 − {[0]}, .5) JÚ ÏXm G] ¨ì©,
18,
1Cu 4Bm T¥ êXeLs JÚ Ø¥Yô] GÀ−Vu ÏXjûR ùTÚdL−u ¸r AûUdÏm G]d LôhÓL, Eg 9.15
19,
Eg 9.10 (i)
Ex 9.3 (7) Eg 9.10 (ii)
OBQ
1 0 −1 0 1 0 −1 0 0 1 , 0 1 , 0 −1 , 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,
20.
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
21.
(Z, +) JÚ Ø¥Yt\ GÀ−Vu ÏXm G] ¨ßÜL,
22.
(Z6, +6) Gu\ ÏXj§u JqùYôÚ Eßl©u Y¬ûN«û]d LôiL,
OBQ
23.
(Z7-{[0]} .7) Gu\ ÏXj§u JqùYôÚ Eßl©u Y¬ûN«û]d LôiL,
OBQ
24,
ÏXj§u ¿dLp ®§Lû[ Gݧ ¨ßÜL,
25.
G JÚ ÏXm GuL, a, b ∈ G G²p (a * b)− 1= b−1 * a−1 G] ¨ì©, (ApXÕ) ApXÕ) ÏXj§u ©u §Úl×ûL ®§ Gݧ ¨ì©, Page 182
Eg 9.12
Page 181
10, ¨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×) ùLôÓdLlThÓs[Õ, X 0 1 2 3 4 5 6 7 8 P(X = x) a 3a 5a 7a 9a 11a 13a 15a 17a
XII PUBLIC 24 QPs 6 marks
¸úZ
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www.kalvisolai.com Page 28 (i) a-u U§l× LôiL, (ii) P(x < 3) (iii) P(3 < x < 7) CYtû\d LôiL,, 3,
Ex 10.1 (4)
JÚ NUYônl× Uô±«u ¨LrRLÜ APoj§f Nôo×
k(1 − x 2 ) f (x) = 0
0 < x <1
G²p. Utù \ e¡Ûm (i) k u U§lûTd LôiL, (ii) NUYônl× Uô±«u TWYp NôoûTd LôiL, 4,
cx(1 − x)3 ; f (x) = 0 ;
Eg 10.6
0 < x <1
Ex 10.1 (6)
Utù \ e¡Ûm
Gu\ Nôo× JÚ ¨LrRLÜ APoj§ Nôo× G²p(i) c (ii) P x <
1 LôiL, 2
5,
JÚ ¨Lrf£«u ùYt±«u ¨LrRLÜ p úUÛm úRôp®«u ¨LrRLÜ q G²p ØRp ùYt± ùT\ úNôRû]L°u Gi¦dûL«u G§oTôojRûXd LôiL, Eg 10.12
6,
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× Eg 10.14 CXôTj ùRôûL VôÕ?
7,
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 Ex 10.2 (3) AmUôQYu ùTßm U§lùTi¦u G§oTôol× Gu]?
8,
JÚ NUYônl× Uô±«u ¨LrRLÜ ¨û\f Nôo× ¸úZ ùLôÓdLlThÓs[Õ :
x P(X = x)
0 0.1
1 0.3
2 0.5
3 0.1
Y = X2 + 2X G²p Y Cu NWôN¬ûVÙm TWYtT¥ûVÙm LôiL,
9.
1 ; f (x ) = 24 0 ;
Ex 10.2 (6)
−12 ≤ x ≤ 12 U t ù \ e ¡ Û m
Gu\ ¨LrRLÜ APoj§f Nôo×dÏ NWôN¬ Utßm TWYtT¥ LôiL, Ex 10.2 (7) (i)
10,
ùRôPof£Vô] NUYônl× Uô± X-u ¨LrRLÜ APoj§f Nôo× 3 x ( 2 − x ); f (x ) = 4 0 ;
0 < x < 2 Utù \ e¡Ûm
G²p NWôN¬ûVÙm. TWYtT¥ûVÙm LôiL,
11.
αe−αx ; f (x) = 0 ;
Eg 10.15
x>0
Utù \ e¡Ûm Gu\ ¨LrRLÜ APoj§f Nôo×dÏ NWôN¬ Utßm TWYtT¥ LôiL, Ex 10.2 (7) (ii)
XII PUBLIC 24 QPs 6 marks
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12.
xe − x ; f (x) = 0 ;
x>0 Ut ù \ e¡Ûm
Gu\ ¨LrRLÜ APoj§f Nôo×dÏ NWôN¬ Utßm TWYtT¥ LôiL, Ex 10.2 (7)(iii) 13,
©uYÚm ¨LrRLÜ APoj§f Nôo©tÏ NWôN¬ûVÙm. TWYtT¥ûVÙm LôiL,
3e−3x ; f (x) = 0 ; 14,
0
Eg 10.16
Utù \ e¡Ûm
JÚ DÚßl×l TWY−u Uô± Xu NWôN¬ 2. §hP ®XdLm
2
G²p. ¨LrRLÜf
3
NôoûTd LôiL,
Eg 10.17
15,
JÚ TLûP CÚØû\ EÚhPlTÓ¡\Õ, ARu úUp Es[ Gi Jtû\lTûP GiQôL CÚjRp ùYt±VôLd LÚRlTÓ¡\Õ, ùYt±«u ¨LrRLÜl TWY−u NWôN¬ Utßm TWYtT¥ûVd LôiL, Ex 10.2 (1)
16,
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 Eg 10.13 TWYûXd LôiL, úUÛm NWôN¬. TWYtT¥ B¡VYtû\d LôiL,
17,
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)
18,
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
19,
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)
20,
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
21,
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)
22,
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)
23,
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)
24,
JÚ CVp¨ûXl TWY−u ¨LrRLÜ APoj§f Nôo× f ( x ) = k e −2x k, µ Utßm σ CYtû\d LôiL,
XII PUBLIC 24 QPs 6 marks
2
+ 4x −2
, G²p OBQ
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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 Ex 10.5 (3) LôiL, [P (0 < z < 1.44 = 0.4251]
26,
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)
27,
300 UôQYoL°u EVWeLs CVp¨ûXl TWYûX Jj§Úd¡\Õ, CRu NWôN¬ 64,5 AeÏXeLs, úUÛm §hP ®XdLm 3,3 AeÏXeLs, GkR EVWj§tÏd Ex 10.5 (6) ¸r 99% UôQYoL°u EVWm APe¡«ÚdÏm?
28,
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)
************
NO SUBSTITUTE FOR HARD WORK
XII PUBLIC 24 QPs 6 marks
***********
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XII PUBLIC EXAM - 3 MARK QUESTIONS (64)
1, A¦Ls Utßm A¦dúLôûYL°u TVuTôÓLs 1.
−1 2 1 −4 Gu\ A¦«u úSoUôß A¦ûVd LôiL,
2.
A JÚ éf£VUt\ úLôûY A¦Vô«u (AT) −1 = (A− 1) T GuTûR ¨ßÜL, Page 5 (2)
Eg.1.5 (i)
2, ùYdPo CVtL¦Rm ∧ ∧
1.
a,b GuT] CÚ AXÏ ùYdPoLs Utßm AYt±u CûPlThP úLôQm θ G²p sin
θ 1 ∧ ∧ = a− b G] ¨ì©, 2 2
Eg. 2.9
2,
r r r r r r r r r r r r a = i + j + 2k , b = 3i + 2 j − k G²p a + 3b . 2a − b Id LôiL, Ex. 2.1 (2)
3,
HúRàm Ko ùYdPo r dÏ r = r.i i + r. j j + r.k k G] ¨ßÜL,
4.
r r r 2i − j + k .
(
r r r r i − 3 j − 5k ,
r
urr r
)(
ur r r
)
ur r r
( ) ( ) ( )
r r r −3i + 4 j + 4k Gu\
ùYdPoLs
ØdúLôQj§u TdLeL[ôL AûUÙm G] ¨ì©,
5.
ùNeúLôQ Eg. 2.11
r r r r r r r r r 3i − 2 j + k , i − 3 j + 5k , 2i + j − 4k GuTûY JÚ ùNeúLôQ ØdúLôQjûR EÚYôdÏm G]d LôhÓL,
6.
Ko
Eg. 2.6
Ex. 2.1 (12)
r r r 2i − 2 j + k Gàm 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.
r r2 rr2 r2 r2 r r a, b GuT] CWiÓ ùYdPoLs G²p a × b + a.b = a b G] ¨ßÜL,
8.
r r urr r r a = 13, b = 5, Utßm a.b = 60, G²p a × b Id LôiL,
Eg. 2.20 Eg. 2.22
9.
r r r r r r 4i − j + 3k , −2i + j − 2k Gàm ùYdPoLÞdÏ ùNeÏjRô]Õm Gi A[Ü 6 EûPVÕUô] ùYdPoLû[d LôiL, Eg. 2.21
10.
r r r 3i + 2 j − 4k 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
r
11.
r
r
B(5, 2, 4) Gu\ ×s° Y¯f ùNVpTÓm ®ûN 4i + 2 j + k u ùYdPo §Úl×j
r
r
r
§\u A(3, − 1, 3) Gu\ ×s°ûVl ùTôßjÕ i + 2 j − 8k G]d LôhÓL, Ex. 2.4 (9)
XII PUBLIC 24 QPs 3 marks
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r r r r r r r −12i + λ k . 3 j − k . 2i + j − 15k Gu\ ùYdPoLû[ Øû]l×s°L[ôLd ùLôiP
CûQLWj§iUj§u L] A[Ü 546 G²p λ-Cu U§l× LôiL,
13.
Ex. 2.5 (2)
r r r r r r r r r a, b, c GuT] JÚR[ AûU ùYdPoLs G²p a + b, b + c, c + a GuTûYÙm
JÚR[ AûU ùYdPoLs BÏm, CRu UßRûXÙm EiûU GuTRû]d LôhÓL, Ex. 2.5 (1)
14.
r r rr rr rr r r r x.a = 0, x.b = 0, x.c = 0 úUÛm x ≠ 0 G²p a , b , c JúW R[ AûU && ùYdPoLs G]d LôhÓL, Eg. 2.34
15,
r r r r r r r r r r a × b × c + b × c × a + c × a × b = 0 G]d LôhÓL,
16,
GkR JÚ a -dÏm i × a × i + j × a × j + k × a × k = 2a G] ¨ì©, Ex. 2.5 (9)
17.
r r r r r r r = ( 5i − 7 j ) + µ ( −i + 4 j + 2k )
(
)
(
r
18.
)
r
(
)
r r
r
( )
(
r r
r
)
2
3
6
Utßm
x +1 =
)
r
r r r r r r = ( −2i + k ) + λ ( 3i + 4k )
Utßm úLôÓL°u CûPlThPd úLôQm LôiL,
x-1 = y + 1 = z − 4
(
r r
Ex. 2.5 (6)
Gu\ Ex. 2.6 (9)
y+2 z−4 = 2 2
CûPlThPd úLôQm LôiL,
Gu\ úLôÓL°u Ex. 2.6 (8)
19.
(3, 2, − 4), (9, 8, − 10) Utßm (λ, 4, − 6) JúW úLôhPûUl ×s°Ls G²p λ-Cu U§l× LôiL, Eg. 2.47
20.
2x − y + z = 4 Utßm x + y + 2z = 4 Gu\ R[eLÞdÏ CûPlThP úLôQm LôiL, Eg. 2.60
21.
2x + y - z = 9 Utßm x + 2y + z = 7 Gu\ R[eLÞdÏ CûPlThP úLôQm LôiL, Ex. 2.10 (1) (i)
22.
r r r r r r r r r. 2i + λ j − 3k = 10 Utßm r. λ i + 3 j + k = 5 Gu\ R[eLs ùNeÏjÕ
(
)
(
)
G²p λ LôiL,
Ex. 2.10 (3)
23,
r r r r r r r r = (i + 2j − k) + µ(2i + j + 2k) Gu\ úLôh¥tÏm R[j§tÏm CûPlThP úLôQm LôiL,
24.
x2 + y2 + z2 + 4x − 8y + 2z = 5 Gu\ úLô[j§u ûUVm. BWm LôiL, Ex. 2.11 (5) (iii)
25.
r2 r r r r r − r.(4i + 2 j - 6k ) - 11 = 0 Gu\ úLô[j§u ûUVm. BWm LôiL, Ex. 2.11 (5) (iv)
26.
r r r r r. (3i − 2 j + 6k) = 0 Gu\ Eg. 2.61
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)
XII PUBLIC 24 QPs 3 marks
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3, LXlùTiLs 1.
1 u ùUn. LtTû]l TϧLû[d LôiL, 1+ i
2,
ÑÚdÏL:
3,
−1 + i 3 −1 − i 3 ω = 1 G²p. + = − 1 G] ¨ßÜL, 2 2
4,
GpXô U§l×Lû[Ùm LôiL, ( i )
Ex. 3.1 (2)(i)
(cos 2θ - i sin 2θ)7 (cos 3θ + i sin 3θ)−5 (cos 4θ + i sin 4θ)12 (cos 5θ - i sin 5θ) −6 5
Ex 3.4 (1)
5
3
1
3
Ex. 3.5 (3)(ii)
Ex. 3.5 (1)(i)
5, YûL ÖiL¦Rm : TVuTôÓLs – I 1.
f(x) = 1 − x 2 , −1 ≤ x ≤ 1 Gu\ Nôo×dÏ úWô−u úRt\jûRl TVuTÓj§. c u U§l× LôiL, Eg. 5.21 (i)
2,
f ( x ) = 4x 3 -9x,
3,
f(x) = sin x, 0 ≤ x ≤ π Gàm Nôo×dÏ úWô−u úRt\jûRf N¬TôdL,
4,
©uYt±tÏ úWô−u úRt\jûRf N¬TôdL: f(x) = sin2 x, 0 ≤ x ≤ π
Eg. 5.22 (iv)
5,
©uYt±tÏ úWô−u úRt\jûRf N¬TôdL: f(x) = | x |, −1 ≤ x ≤ 1
Eg. 5.22 (iii)
6,
©uYt±tÏ úWô−u úRt\jûRf N¬TôdL: f(x) = x3 − 3x + 3; 0 ≤ x ≤ 1 Eg. 5.22 (i)
7.
ex Gu\ Nôo×dÏ ùUdXô¬u ®¬Ü LôiL,
8,
U§l× LôiL: lim
−3 2
x →∞
≤ x ≤
3 2
Gàm Nôo×dÏ úWô−u úRt\jûRf N¬TôdL, Ex. 5.3 (1) (iv) Ex. 5.3 (1)(i)
Eg. 5. 28 (1)
x2
Eg. 5.32
ex sin
2 x
9,
U§l× LôiL: lim
10.
R p ex §hPUôL Hßm Nôo× G] ¨ì©dL,
11,
x3/5 (4 − x) u UôߨûX GiLû[d LôiL,
12.
y = 2 − x2 Gu\ Yû[YûW«u Ï¯Ü (Ï®Ü)-u NôoTLjûRd LôiL,
x →∞
Ex 5.6 (5)
1 x
Ex. 5.7 (1) Eg. 5.47 Eg. 5.59
6, YûL ÖiL¦Rm : TVuTôÓLs – II 1,
YûLÂÓ dy LôiL, úUÛm ùLôÓdLlThP x Utßm dx-u U§l×dÏ dy-u U§l× Ex 6.1 (2) (ii) LQd¡ÓL, y = x4 − 3x3+ x −1, x = 2, dx = 0.1.
XII PUBLIC 24 QPs 3 marks
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2,
u=
x 2 + y 2 G²p. x
∂u ∂u +y = u G]d LôhÓL, ∂x ∂y
Ex 6.3 (2) (i)
7, ùRôûL ÖiL¦Rm : TVuTôÓLs U§l× LôiL: 1,
∫
3.
∫
π
2
0
dx
1
0
5.
∫
7.
∫
π
sin x dx Eg 7.1 1 + cos 2 x
4− x 2
0
1
−1
Ex 7.1 (5)
2
sin 6 x dx Ex 7.3 (2) (i)
3− x log dx 3+ x
2.
∫
π /2
∫
a
4.
∫
π /4
6.
0
e3x cos x dx
Ex 7.1 (11)
a 2 − x 2 dx
Eg 7.3
0
−π /4
x 3 cos3 x dx
Ex 7.2 (2)
Eg 7.6
8, YûLdùLÝf NUuTôÓLs 1, 2,
y = e2x (A + Bx) Gu\ NUuTôh¥tLô] YûLdùLÝf NUuTôhûP AûUdL,
¾odL:
(D
2
Eg 8.2 (i) Eg 8.24
)
+ D +1 y = 0
9, R²¨ûX LQd¡Vp 1.
(p ∨ q) ∧ (∼ q) Gu\ átßdÏ ùUn AhPYûQ AûUdL,
2,
p ∧ (∼ p) JÚ ØWiTôÓ G] ¨ì©,
3.
JÚ ÏXj§u NU² Eßl× JÚûUj RuûU YônkRÕ,
4,
JÚ ÏXj§p JqùYôÚ Eßl×m JúW JÚ G§oUû\ûVl ùTt±ÚdÏm,
Eg 9.4 (iii) Eg 9.9 (ii)
10, ¨LrRLÜl TWYp 1,
©uYÚTY] ¨LrRLÜ APoj§f NôoTô G] N¬TôodLÜm:
2x 9 f (x) = 0 2.
0≤x≤3 Utù \ e¡Ûm
Ex 10.1 (5) (a)
1π F(x) = + tan −1 x , −∞ < x < ∞ GuTÕ JÚ NUYônl× Uô± X Cu TWYp π 2 Nôo× G²p P(0 ≤ x ≤ 1) I LôiL,
XII PUBLIC 24 QPs 3 marks
Eg 10.7
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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
************
NO SUBSTITUTE FOR HARD WORK
XII PUBLIC 24 QPs 3 marks
***********
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XII MATHS COME BOOK MODEL QPs 10 MARK QUESTIONS (9) 3, LXlùTiLs 1.
P Guàm ×s° LXlùTi Uô± z Id ϱjRôp P-Cu ¨VUlTôûRûV z −1 π arg Gu\ ¨TkRû]dÏ EhThÓ LôiL, Ex. 3.2 (8) (v) = z +3
2
5, YûL 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: lim
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 y = e− x . GkR CûPùY°L°p ϯÜ. 쨆 AûP¡\Õ GuTûRÙm. Yû[Ü Uôtßl ×s°Lû[Ùm LôiL, Eg. 5.64
xsin x
x →0 +
Eg. 5.35
2
7, ùRôûL ÖiL¦Rm : TVuTôÓLs 1,
Yû[YûW y2 = x Utßm y = x − 2 Gu\ úLôh¥]ôp AûPTÓm TWl©û]d LôiL, Eg 7.28
8, YûLdùLÝf NUuTôÓLs 1.
dx + xdy = e − y sec 2 y dy ¾oÜ LôiL,
2.
− 3x (1 − x ) dy dx
3.
¾o:
3
(D
************
2
2
Ex 8.4 (7)
y = sec 2 x Gu\ YûLdùLÝf NUuTôh¥û]j ¾odL:
)
− 13D + 12 y = x + 5e x
OBQ
NO SUBSTITUTE FOR HARD WORK
XII COME BOOK 10 marks
OBQ
***********
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XII MATHS COME BOOK MODEL QPs 6 MARK QUESTIONS (21) 1, A¦Ls Utßm A¦dúLôûYL°u TVuTôÓLs −1 2 G²p. A (adj A) = (adj A) A = | A | . I2 GuTûRf N¬TôodL, Eg. 1.3 1 −4
1,
A=
2,
úSoUôß A¦ LôQp Øû\«p ©uYÚm úS¬V NUuTôhÓj ùRôÏl©û]j ¾odLÜm, 7x + 3y = -1, 2x + y = 0. Ex. 1.2 (2)
1,
Jo AûWYhPj§p Es[ úLôQm JÚ ùNeúLôQm, CRû] ùYdPo Øû\«p ¨ì©dL, Eg. 2.14
2, ùYdPo CVtL¦Rm 3, LXlùTiLs 1.
z1, z2 Gu\ HúRàm CÚ LXlùTiLÞdÏ (i)
z z1 = 1 z2 z2
z1 = arg z1 − arg z2 G]d LôhÓL, z2
(ii) arg
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
5, YûL ÖiL¦Rm : TVuTôÓLs – I 1,
2,
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) 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
U§l×Lû[d LôiL,,
XII COME BOOK 6 marks
f ( x) =
x x +1
ÁlùTÚ
[1, 2]
ùTÚU
Utßm
Áf£ß
£ßU
Ex. 5. 9 (2) (v)
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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
2.
∂w ∂w Utßm Id LôiL, ∂r ∂θ
Ex 6.3 (4)(i)
x = u + v, y = u − v Guß CÚdÏUôß w = sin−1 (xy) G] YûWVßdLlTÓ¡\Õ G²p. ∂w ∂w Utßm Id LôiL, ∂u ∂v
Ex 6.3 (4)(iii)
7, ùRôûL ùRôûL ÖiL¦Rm : TVuTôÓLs 1
1,
U§l©ÓL :
∫ xe
−4 x
Eg 7.17 (ii)
dx
0
2.
3,
y2 = 4ax Gu\ TWYû[Vj§tÏm ARu ùNqYLXj§tÏm CûPlThP TWl©û]d LôiL, Ex 7.4 (6) x2 a2
+
y2 b2
=1
(a > b > 0) Gu\ ¿sYhPm HtTÓjÕm TWl©û]. Ït\fûNl
ùTôßjÕf ÑZt±]ôp HtTÓm §PlùTôÚ°u L] A[Ü LôiL,
Eg 7.35
8, YûLdùLÝf NUuTôÓLs 1.
x-Af£u ÁÕ ûUVm Utßm KWXÏ YûLdùLÝf NUuTôhûP AûUdL,
2,
¾odL: 3e x tany dx + (1+ e x )sec2 y dy = 0
3,
¾odL:
BWm
ùLôiP
YhPj
ùRôÏl©u Ex 8.1 (4) Eg 8.4
dy y y = + tan dx x x
Eg 8.12
10, ¨LrRLÜl ¨LrRLÜl 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 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
************
NO SUBSTITUTE FOR HARD WORK
XII COME BOOK 6 marks
***********
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XII MATHS COME BOOK MODEL QPs 3 MARK QUESTIONS (13)
2, ùYdPo CVtL¦Rm
1.
r r r r r r a, b, c GuT] JußdùLôuß ùNeÏjÕ ùYdPoLs G²p a b c = abc
G]d LôhÓ, CRu UßRûXÙm EiûU G]Üm LôhÓL,
2.
r r r 2r + (3i − j + 4k ) = 4 Gu\ úLô[j§u ûUVm. BWm LôiL,
Ex. 2.5 (3) Ex. 2.11 (5) (ii)
5, YûL ÖiL¦Rm : TVuTôÓLs – I 1,
2
y + 2 x + 1 = 0 Gu\ TWYû[Vj§tÏ (-1. 1) Gu\ ×s°«p ùRôÓúLôh¥u
NUuTôÓ LôiL,
OBQ
2,
y = e x . y = e− x Yû[YûWLÞdÏ CûPlThP úLôQjûRd LôiL,
OBQ
3.
f(x) = 2x3 − 5x2 − 4x + 3,
1 ≤ x ≤ 3 úWô−u úRt\jûRl TVuTÓj§. cCu 2
U§l×Lû[d LiÓ©¥dL,
Eg. 5.21 (iii)
6, YûL ÖiL¦Rm : TVuTôÓLs – II 1,
x y
y
x y ∂u ∂u u = e sin + e x cos G²p. x + y = 0 G]d LôhÓL, y x ∂x ∂y
Ex 6.3 (2) (ii)
8, YûLdùLÝf NUuTôÓLs 1,
¾odL:
(D
2
)
+ 6D + 9 y = 0
Eg 8.23
9, R²¨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, ¨LrRLÜl ¨LrRLÜl TWYp 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)
XII COME BOOK 3 marks
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XII STANDARD MATHEMATICS CLASSIFICATION OF QUESTIONS (Q.Nos) TôPm 1 (10) 33 úLs®Ls Eg 1.4 Ex 1.1 (3) Ex 1.1 (7) Ex 1.1 (9) Ex 1.1 (6) Eg 1.8 Ex 1.2 (3) Ex 1.2 (4) Ex 1.2 (5) Eg 1.19 Ex 1.4 (10) Ex 1.4 (9) Eg 1.18 (1) Ex 1.4 (4) Ex 1.4 (6) Eg 1.18 (2) Ex 1.4 (5) Ex 1.4 (7) Eg 1.18 (4) Ex 1.4 (8) Eg 1.21 Eg 1.23 Eg 1.22 Ex 1.5 (1 i) Eg 1.24 Ex 1.5 (1 ii) Eg 1.25 Ex 1.5 (1 v) Ex 1.5 (2) Ex 1.5 (3) Eg 1.26 Eg 1.27 Eg 1.28
TôPm 1 (6) 33 úLs®Ls Eg 1.3 Ex 1.1 (2) Ex 1.1 (8) Eg 1.6 Ex 1.1 (5 i) Eg 1.5 (iv)
Ex 1.1 (4 i) Ex 1.1 (4 ii) Ex 1.1 (4 iii) Ex 1.1 (4 iv) Ex 1.1 (4 v) Ex 1.1 (10) (AB)−1 = B−1 A−1
Eg 1.7 Ex 1.2 (1) Ex 1.2 (2) Eg 1.13 Ex 1.3 (1) Eg 1.12 Eg 1.14 Ex 1.3 (5) Ex 1.3 (6) Eg 1.16 Ex 1.3 (2) Ex 1.3 (3) Eg 1.15 Ex 1.3 (4) Eg 1.17 (2) Ex 1.4 (3) Eg 1.18 (3) Eg 1.18 (5) Ex 1.5 (1 iii) Ex 1.5 (1 iv)
TôPm 1 (3) 16 úLs®Ls Ex 1.1 (1 i) Eg 1.1 Eg 1.2 Ex 1.1 (1 ii) Ex 1.1 (1 iii) Eg 1.5 (i) Eg 1.5 (ii) Eg 1.5 (iii) Ex 1.1 (5 ii) (AT) −1 = (A− 1) T
Eg 1.11 Eg 1.17 (1) Ex 1.4 (1) Eg 1.17 (3)
Ex 1.4 (2) Eg 1.20
TôPm 2 (10) 20 úLs®Ls Eg 2.16 Eg 2.17 Ex 2.2 (4) Ex 2.4 (7) Eg 2.29 Ex 2.5 (5) Ex 2.5 (12) Eg 2.44 Ex 2.7 (3) Eg 2.50 Ex 2.8 (8) Ex 2.8 (9) Ex 2.8 (7) Ex 2.8 (10) Eg 2.51 Ex 2.8 (11) Ex 2.8 (12) Eg 2.52 Ex 2.8 (13) Ex 2.8 (14)
TôPm 2 (6) 73 úLs®Ls Ex 2.6 (4) Eg 2.12 (i) Eg 2.12 (ii) Eg 2.12 (iii) Eg 2.13 (i) Eg 2.13 (ii) Eg 2.13 (iii) Eg 2.14 Eg 2.15 Ex 2. 2 (1) Ex 2. 2 (3) Ex 2. 2 (2) Ex 2. 2 (8) Ex 2.3 (9) Eg 2.25 Eg 2.24
Eg 2.27 Eg 2.28 Eg 2.26 Ex 2.4 (6) Ex 2.4 (5) Ex 2.4 (8) Ex 2.4 (10) Eg 2.33 Ex 2.5 (4) Eg 2.36 Ex 2.5 (7) Ex 2.5 (8) Ex 2.5 (10) Ex 2.5 (11) Eg 2.37 Eg 2.38 Eg 2.39 Ex 2.6 (6) Eg 2.40 Ex 2.6 (7) Eg 2.42 Ex 2.7 (1 i) Ex 2.7 (1 ii) Ex 2.7 (2) Ex 2.9 (2) Eg 2.43 Eg 2.45 Ex 2.7 (4) Eg 2.46 Ex 2.7 (5) Ex 2.8 (5) Ex 2.8 (6) Eg 2.49 Ex 2.8 (4) Eg 2.48 Ex 2.8 (1) Ex 2.8 (15 i) Ex 2.8 (15ii) Eg 2.53 Eg 2.54 Eg 2.55 Ex 2.9 (6) Eg 2.56 Eg 2.57
XII Maths Classification of Questions (Q. Nos)
Ex 2.9 (1) Eg 2.58 Ex 2.9 (3) Eg 2.59 Ex 2.9 (4) Eg 2.62 Ex 2.11 (1) Ex 2.11 (3) Eg 2.63 Eg 2.64 Ex 2.11 (2) Eg 2.65 Ex 2.11 (6)
TôPm 2 (3) 58 úLs®Ls Ex 2.1 (5) Ex 2.1 (6) Ex 2.6 (3) Ex 2.6 (2 i) Ex 2.6 (2 ii) Eg 2.8 Eg 2.10 Eg 2.9 & Ex 2.1(7)
Ex 2.1 (8) Ex 2.1 (2) Eg 2.6 Eg 2.11 Ex 2.1 (13) Ex 2.1 (12) Ex 2.2 (7) Ex 2.2 (6) Eg 2.20 Eg 2.22 Ex 2.3 (8) Ex 2.3 (10) Ex 2.3 (7) Eg 2.23 Ex 2.3 (3) Eg 2.21 Ex 2.3 (4) Ex 2.4 (4) Eg 2.30 Ex 2.4 (2)
Ex 2.4 (3) Ex 2.4 (1) Eg 2.31 Ex 2.4 (9) Eg 2.32 Ex 2.5 (2) Ex 2.5 (3) Ex 2.5 (1) Eg 2.34 Eg 2.35 Ex 2.5 (6) Ex 2.5 (9) Ex 2.6 (5) Eg 2.41 Ex 2.6 (9) Ex 2.6 (8) Eg 2.47 Ex 2.7 (6) Ex 2.10 (1 iii) Eg 2.60 Ex 2.10 (1 i) Ex 2.10 (1 ii) Ex 2.10 (2) Ex 2.10 (3) Ex 2.10 (5) Eg 2.61 Ex 2.10 (4) Ex 2.11 (5 iii) Ex 2.11 (5 iv) Ex 2.11 (4)
TôPm 3 (10) 16 úLs®Ls
Eg 3.11 (i) Ex 3.2 (8 iii) Ex 3.2 (8 i) Eg 3.11 (ii) Ex 3.2 (8 v) Ex 3.4 (10) Eg 3.22 Ex 3.4 (5) Ex 3.4 (8) Ex 3.4 (6) Ex 3.5 (1 iii)
N. MAHALAKSHMI PGT (Mathematics)
www.kalvisolai.com Page 41 Eg 3.25 Ex 3.5 (5) Ex 3.5 (4 ii) Eg 3.24 Eg 3.23
TôPm 3 (6) 47 úLs®Ls úLs®Ls Ex 3.1 (1 ii) Ex 3.1 (1 iv) Ex 3.1 (4 i) Ex 3.1 (4 ii) Ex 3.1 (4 iii) Ex 3.2 (2) Eg 3.16 Ex 3.1 (5) P. No. 124 P. No. 125 P. No. 127 P. No. 128 Eg 3.10 Eg 3.9 (ii) Eg 3.9 (i) Eg 3.9 (iii) Ex 3.4 (2) Eg 3.19 Ex 3.2 (8 iv) Ex 3.2 (8 ii) Ex 3.2 (7) Eg 3.14 Ex 3.2 (5) Eg 3.13 Eg 3.15 Ex 3.2 (4) P. No. 140 Eg 3.17 Ex 3.3 (1) Ex 3.3 (2) Ex 3.3 (3) Ex 3.4 (9) Eg 3.20 Ex 3.4 (3 iii,iv) Ex 3.4 (3 i,ii) Ex 3.4 (3 v) Ex 3.4 (4 iii) Ex 3.4 (4 iv) Ex 3.4 (4 i) Eg 3.21
Ex 3.4 (4 ii) Ex 3.4 (7) Ex 3.5 (2) Ex 3.5 (3 i) Ex 3.5 (3 iii) Ex 3.5 (1 ii) Ex 3.5 (4 i)
TôPm 3 (3) 19 úLs®Ls Ex 3.1 (1 i) Eg 3.4 (iv) Ex 3.1 (3) Ex 3.1 (2 i) Ex 3.1 (2 ii) Eg 3.5 Ex 3.2 (3) Eg 3.6 (ii) Eg 3.8 Ex 3.2 (1) Ex 3.2 (6 i) Ex 3.2 (6 ii) Ex 3.2 (6 iii) Ex 3.2 (6 iv) Eg 3.18 Ex 3.4 (1) Eg 3.12 Ex 3.5 (3 ii) Ex 3.5 (1 i)
TôPm 4 (10) 28 úLs®Ls Eg 4.7 (iv) Ex 4.1 (2 iv) Ex 4.1 (2 v) Eg 4.7 (v) Ex 4.2 (6 ii) Eg 4.31 (iv) Ex 4.2 (6 iv) Eg 4.56 Ex 4.3 (5 iii) Eg 4.57 Ex 4.3 (5 iv) Eg 4.13 Eg 4.12 Eg 4.14 Ex 4.1 (5) Eg 4.8
Eg 4.10 Eg 4.32 Ex 4.2 (10) Eg 4.33 Ex 4.2 (9) Ex 4.2 (8) Ex 4.2 (7) Eg 4.35 Ex 4.4 (5) Ex 4.4 (6) Ex 4.5 (2 ii) Ex 4.6 (3)
TôPm 4 (6) 104 úLs®Ls
Eg 4.7 (iii) Ex 4.1 (2 iii) Eg 4.9 Ex 4.1 (3) Eg 4.11 Ex 4.1 (4) Eg 4.16 Eg 4.17 Eg 4.25 Ex 4.2 (1 ix) Eg 4.15 Ex 4.2 (1 ii) Eg 4.26 Eg 4.18 Ex 4.2 (1 iii) Eg 4.24 Ex 4.2 (1 viii) Eg 4.20 Ex 4.2 (1 iv) Eg 4.21 Ex 4.2 (1 v) Eg 4.23 Ex 4.2 (1 vii) Eg 4.22 Ex 4.2 (1 vi) Eg 4.27 Ex 4.2 (3) Eg 4.28 (iii) Eg 4.29 Ex 4.2 (4 ii) Ex 4.2 (4 iv) Eg 4.30 (iii) Ex 4.2 (5 iii)
Ex 4.2 (5 iv) Eg 4.31 (iii) Eg 4.34 Ex 4.3 (1 i) Eg 4.36 Eg 4.37 Ex 4.3 (1 iii) Ex 4.3 (1 ii) Ex 4.3 (1 iv) Eg 4.38 Eg 4.39 Ex 4.3 (1 v) Eg 4.43 Ex 4.3 (1 viii) Eg 4.42 Ex 4.3 (1 vii) Eg 4.52 Eg 4.41 Eg 4.44 Ex 4.3 (1 vi) Eg 4.45 Ex 4.3 (1 ix) Eg 4.53 Ex 4.3 (4) Eg 4.58 Eg 4.48 Ex 4.3 (2 iii) Eg 4.51 Ex 4.3 (3 ii) Eg 4.54 Eg 4.55 Ex 4.4 (1 i) Ex 4.4 (1 ii) Ex 4.4 (1 iii) Eg 4.61 Ex 4.4 (1 iv) Ex 4.4 (1 v) Ex 4.4 (2 i) Ex 4.4 (2 ii) Ex 4.4 (2 iii) Ex 4.4 (2 iv) Ex 4.4 (3 i) Ex 4.4 (3 iv) Ex 4.4 (3 ii) Ex 4.4 (3 iii) Eg 5.16 Ex 4.4 (4 i) Eg 4.59
XII Maths Classification of Questions (Q. Nos)
Eg 4.62 Ex 4.4 (4 ii) Ex 4.4 (4 iii) Eg 4.64 Ex 4.5 (1 ii) Eg 4.65 Ex 4.5 (2 i) Eg 4.66 Ex 4.5 (3 iii) Eg 4.67 Eg 4.68 Ex 4.6 (2 i) Ex 4.6 (2 ii) Eg 4.69 Ex 4.6 (1) Ex 4.6 (5) Ex 4.6 (4) Ex 4.6 (6 i) Ex 4.6 (6 ii) Ex 4.6 (6 iii) Eg 4.70 Eg 4.71 Ex 4.6 (7)
TôPm 4 (3) 22 úLs®Ls Eg 4.1 (i) Eg 4.1 (iii) Ex 4.1 (1 i) Eg 4.1 (ii) Ex 4.1 (1 ii) Eg 4.2 (i) Ex 4.1 (1 iii) Ex 4.1 (1 iv) Eg 4.2 (ii) Eg 4.3 Ex 4.1 (1 v) Eg 4.6 Ex 4.1 (1 viii) Eg 4.5 Ex 4.1 (1 vii) Eg 4.4 Ex 4.1 (1 vi) Ex 4.1 (1 ix) Ex 4.2 (1 i) Eg 4.40 Eg 4.60 Ex 4.5 (1 i)
TôPm 5 (10) 44 úLs®Ls Ex 5.1 (3) Ex 5.1 (1) Eg 5.6 Eg 5.9 Ex 5.1 (9) Ex 5.1 (8) Eg 5.7 Eg 5.8 Ex 5.1 (6) Ex 5.1 (5) Eg 5.14 Eg 5.15 Eg 5.13 Ex 5.2 (10) Eg 5.20 Ex 5.2 (7) Ex 5.2 (5) Eg 5.17 Ex 5.2 (11) Eg 5.18 Eg 5.35 Ex 5.6 (11) Eg 5.34 Eg 5.48 (a) Ex 5.9 (3 iii) Eg 5.51 Ex 5.9 (3 iv) Ex 5.9 (3 v) Ex 5.9 (3 vi) Eg 5.53 Ex 5.10 (3) Ex 5.10 (4) Eg 5.52 Eg 5.55 Eg 5.57 Eg 5.56 Eg 5.58 Eg 5.54 Ex 5.10 (5) Ex 5.11 (5) Eg 5.63 Ex 5.11 (4) Eg 5.64 Ex 5.11 (6)
N. MAHALAKSHMI PGT (Mathematics)
www.kalvisolai.com Page 42 TôPm 5 (6) 96 úLs®Ls Ex 5.1 (4) Ex 5.1 (2) Ex 5.1 (7) Eg 5.10 Eg 5.11 Eg 5.12 Ex 5.2 (1 i) Ex 5.2 (1 ii) Ex 5.2 (1 iii) Ex 5.2 (1 iv) Ex 5.2 (2) Ex 5.2 (3) Ex 5.2 (4) Ex 5.2 (6) Ex 5.2 (9) Ex 5.2 (8) Eg 5.19 Eg 5.22 (v) Eg 5.23 Ex 5.3 (2) Ex 5.4 (1 i) Eg 5.24 Ex 5.4 (1 v) Ex 5.4 (1 ii) Ex 5.4 (1 iii) Ex 5.4 (1 iv) Eg 5.26 Ex 5.4 (2) Eg 5.27 Ex 5.4 (3) Eg 5.25 Ex 5.5 (3) Ex 5.5 (2) Eg 5.28 (2) Eg 5.28 (3) Ex 5.5 (4) Eg 5.36 Ex 5.6 (2) Eg 5.31 Eg 5.30 Ex 5.6 (6) Ex 5.6 (8) Eg 5.33 Ex 5.6 (9) Ex 5.6 (10) Ex 5.6 (13)
Ex 5.6 (12) Ex 5.7 (3 v) Ex 5.7 (3 iv) Eg 5.41 Eg 5.43 Ex 5.7 (4 i) Ex 5.7 (4 ii) Eg 5.37 Ex 5.7 (4 iv) Ex 5.7 (4 iii) Eg 5.39 Ex 5.7 (5 i) Ex 5.7 (5 v) Ex 5.7 (5 vi) Ex 5.7 (5 ii) Eg 5.38 Ex 5.7 (5 iii) Ex 5.7 (5 iv) Eg 5.40 Eg 5.42 Eg 5.45 Ex 5.8 (1 i) Ex 5.8 (1 ii) Eg 5.46 Ex 5.8 (1 iii) Ex 5.8 (1 iv) Ex 5.9 (1 iii) Ex 5.9 (1 iv) Ex 5.9 (1 vi) Ex 5.9 (1 v) Ex 5.9 (2 i) Ex 5.9 (2 ii) Ex 5.9 (2 iv) Eg 5.48 Ex 5.9 (2 iii) Ex 5.9 (2 v) Ex 5.9 (2 vi) Ex 5.9 (2 vii) Ex 5.9 (3 i) Ex 5.9 (3 ii) Eg 5.49 Eg 5.50 Ex 5.10 (6) Ex 5.10 (1) Ex 5.10 (2) Eg 5.62 Eg 5.65 Ex 5.11 (1)
Ex 5.11 (3) Eg 5.66
TôPm 5 (3) 34 úLs®Ls Eg 5.2 Eg 5.21 (iii) Ex 5.3 (1 iv) Eg 5.21 (i) Eg 5.21 (ii) Eg 5.22 (vi) Ex 5.3 (1 i) Eg 5.22 (iv) Eg 5.22 (ii) Eg 5.22 (iii) Ex 5.3 (1 iii) Eg 5.22 (i) Ex 5.3 (1 ii) Eg 5.28 (1) Ex 5.5 (1 i) Ex 5.6 (4) Ex 5.6 (1) Ex 5.6 (3) Ex 5.6 (7) Eg 5.32 Ex 5.6 (5) Ex 5.7 (3 i) Ex 5.7 (3 ii) Ex 5.7 (1) Ex 5.7 (2) Ex 5.7 (3 iii) Eg 5.44 Ex 5.9 (1 i) Ex 5.9 (1 ii) Eg 5.47 Ex 5.11 (2) Eg 5.59 Eg 5.60 Eg 5.61
TôPm 6 (10) 11 úLs®Ls Ex 6.1 (3 iii) Ex 6.2 (1) Eg 6.9 Eg 6.10 Ex 6.3 (1 iii) Ex 6.3 (1 ii)
Ex 6.3 (1 iv) Eg 6.18 Eg 6.20 Eg 6.22 Ex 6.3 (5 i)
TôPm 6 (6) 26 úLs®Ls Eg 6.2 Ex 6.1 (3 ii) Ex 6.1 (3 i) Eg 6.3 Ex 6.1 (3 iv) Ex 6.1 (4) Ex 6.1 (5) Eg 6.6 Eg 6.5 Eg 6.7 Ex 6.3 (1 i) Eg 6.15 Eg 6.16 Eg 6.17 Ex 6.3 (3 i) Ex 6.3 (3 ii) Ex 6.3 (3 iii) Eg 6.19 Ex 6.3 (3 iv) Ex 6.3 (4 ii) Ex 6.3 (4 i) Ex 6.3 (4 iii) Ex 6.3 (5 ii) Ex 6.3 (5 iii) Eg 6.21 Ex 6.3 (5 iv)
TôPm 6 (3) 9 úLs®Ls úLs®Ls Ex 6.1 (2 i) Ex 6.1 (2 ii) Ex 6.1 (2 iii) Ex 6.1 (2 iv) Ex 6.1 (2 v) Eg 6.8 Eg 6.4 Ex 6.3 (2 i) Ex 6.3 (2 ii)
XII Maths Classification of Questions (Q. Nos)
TôPm 7 (10) Ex 7.2 (4) 23 úLs®Ls Eg 7.34 Eg 7.25 Ex 7.4 (4) Eg 7.29 Eg 7.31 Ex 7.4 (7) Eg 7.33 Eg 7.32 Ex 7.4 (9) Eg 7.28 Ex 7.4 (8) Eg 7.26 Eg 7.27 Eg 7.30 Ex 7.4 (15) Eg 7.37 Ex 7.5 (1) Ex 7.5 (2) Eg 7.38 Eg 7.39 Ex 7.5 (3) Ex 7.5 (4) Eg 7.40
TôPm 7 (6) 54 úLs®Ls Ex 7.1 (6) Ex 7.1 (8) Ex 7.1 (9) Ex 7.1 (4) Ex 7.3 (4 i) Eg 7.17 (ii) Ex 7.1 (10) Eg 7.17 (i) Ex 7.1 (3) Ex 7.1 (7) Eg 7.13 Eg 7.14 Ex 7.3 (1 i) Ex 7.3 (1 ii) Eg 7.15 (iii) Ex 7.3 (3 ii) Eg 7.15 (iv) Eg 7.16 Ex 7.3 (3 i) Eg 7.8
Ex 7.2 (5) Eg 7.7 Eg 7.10 Ex 7.2 (9) Ex 7.2 (8) Ex 7.2 (7) Ex 7.2 (3) Eg 7.9 Eg 7.11 Eg 7.12 Ex 7.2 (10) Eg 7.18 Ex 7.4 (1 i) Eg 7.19 Ex 7.4 (1 ii) Eg 7.20 Eg 7.21 Ex 7.4 (2 i) Ex 7.4 (2 ii) Ex 7.4 (5) Eg 7.22 Eg 7.23 Ex 7.4 (3) Eg 7.24 Ex 7.4 (10) Ex 7.4 (6) Ex 7.4 (16) Ex 7.4 (12) Ex 7.4 (11) Ex 7.4 (14) Ex 7.4 (13) Eg 7.36 Eg 7.35
TôPm 7 (3) 18 úLs®Ls Ex 7.1 (1) Ex 7.1 (2) Eg 7.1 Eg 7.2 Ex 7.1 (11) Eg 7.4 Ex 7.1 (12) Ex 7.1 (5) Eg 7.3 Eg 7.15 (i) Ex 7.3 (2 ii)
N. MAHALAKSHMI PGT (Mathematics)
www.kalvisolai.com Page 43 Ex 7.3 (2 i) Eg 7.15 (ii) Eg 7.5 Ex 7.2 (6) Ex 7.2 (2) Ex 7.2 (1) Eg 7.6
TôPm 8 (10) 25 úLs®Ls Eg 8.14 Eg 8.7 Ex 8.2 (7) Eg 8.10 Ex 8.3 (5) Eg 8.13 Eg 8.15 Ex 8.4 (5) Eg 8.18 Ex 8.4 (7) Ex 8.4(3)/ Eg 8.19
Ex 8.4 (9) Ex 8.5 (6) Ex 8.5 (11) Ex 8.5 (10) Eg 8.39 Ex 8.6 (3) Ex 8.6 (4) Ex 8.6 (1) Eg 8.34 Eg 8.35 Eg 8.37 Ex 8.6 (5) Ex 8.6 (2) Eg 8.38
TôPm 8 (6) 55 úLs®Ls Ex 8.1 (4) Ex 8.1 (2 ix) Eg 8.2 (iii) Ex 8.1 (2 iv) Ex 8.1 (3) Ex 8.2 (3) Eg 8.3
Eg - Example
Eg 8.4 Eg 8.5 Ex 8.2 (5) Ex 8.2 (1) Ex 8.2 (2) Eg 8.6 Ex 8.2 (4) Eg 8.9 Ex 8.2 (8) Eg 8.8 Ex 8.3 (3) Ex 8.2 (6) Eg 8.11 Eg 8.12 Ex 8.3 (1) Ex 8.3 (4) Ex 8.3 (2) Eg 8.16 Eg 8.17 Eg 8.21 Ex 8.4 (2) Ex 8.4 (4) Ex 8.4 (1) Eg 8.20 Ex 8.4 (6) Ex 8.4 (8) Ex 8.3 (6) Ex 8.5 (5) Eg 8.25 Ex 8.5 (1) Ex 8.5 (2) Eg 8.26 Eg 8.27 Eg 8.28 Ex 8.5 (14) Eg 8.29 Ex 8.5 (9) Eg 8.31 Eg 8.30 Ex 8.5 (13) Ex 8.5 (8) Eg 8.32 Eg 8.33 Ex 8.5 (7) Ex 8.5 (4)
Ex 8.5 (3) Ex 8.5 (12) Eg 8.36
TôPm 8 (3) 11 úLs®Ls Ex 8.1 (2 i) Eg 8.2 (iv) Ex 8.1 (2 ii) Ex 8.1 (2 v) Ex 8.1 (2 vi) Eg 8.2 (i) Eg 8.2 (ii) Ex 8.1 (2 vii) Eg 8.22 Eg 8.23 Eg 8.24
TôPm 9 (10) 15 úLs®Ls Eg 9.24 Ex 9.4 (6) Eg 9.26 Ex 9.4 (9) Eg 9.25 Eg 9.27 Eg 9.21 Ex 9.4 (11) Ex 9.4 (5) Eg 9.18 Eg 9.23 Ex 9.4 (8) Eg 9.22 Ex 9.4 (12) Ex 9.4 (7)
TôPm 9 (6) 31 úLs®Ls Eg 9.4 (iv) Ex 9.2 (9) Eg 9.6 Ex 9.2 (10) Eg 9.5 Eg 9.7 Ex 9.3 (5)
Ex 9.3 (2) Ex 9.3 (4) Ex 9.3 (3) Ex 9.3 (6) Ex 9.3 (1 ii) Ex 9.3 (1 iv) Eg 9.10 (i) Eg 9.11 Ex 9.3 (1 iii) Ex 9.3 (7) Eg 9.10 (ii) Ex 9.3 (1 i) Ex 9.3 (1 v) Eg 9.14 Eg 9.15 Ex 9.4 (4) Eg 9.20 Eg 9.19 Eg 9.12 Eg 9.13 Eg 9.16 Eg 9.17 ¿dLp ®§ ©u §Úl×ûL ®§
TôPm 9 (3) 16 úLs®Ls Eg 9.2 Ex 9.2 (1) Ex 9.2 (3) Eg 9.4 (i) Ex 9.2 (2) Eg 9.4 (ii) Ex 9.2 (6) Ex 9.2 (4) Eg 9.4 (iii) Ex 9.2 (5) Ex 9.2 (8) Ex 9.2 (7) Ußl©u Ußl× Eg 9.8 Eg 9.9 (i) Eg 9.9 (ii) e u Y¬ûN 1 e JÚûUj RuûU
P No - Page Number
a-1 JÚûUj RuûU -1 -1
(a ) = a
TôPm 10 (10) 12 úLs®Ls úLs®Ls Eg 10.3 Eg 10.2 Ex 10.1 (7) Eg 10.10 Ex 10.4 (5) Eg 10.26 Eg 10.29 Eg 10.32 Ex 10.5 (5) Eg 10.30 Eg 10.31 Ex 10.5 (8)
TôPm 10 (6) 40 úLs®Ls Ex 10.1 (1) Eg 10.1 Ex 10.1 (2) Ex 10.1 (3) Ex 10.1 (4) Eg 10.4 Eg 10.6 Ex 10.1 (6) Eg 10.8 Ex 10.1 (10) Eg 10.5 Ex 10.1 (8) Eg 10.11 Eg 10.12 Ex 10.2 (5) Eg 10.14 Ex 10.2 (3) Ex 10.2 (6) Ex 10.2 (7 i) Eg 10.15 Ex 10.2 (7 ii) Ex 10.2 (7 iii) Eg 10.16 Eg 10.17 Ex 10.2 (1)
Ex 10.2 (4) Eg 10.13 Ex 10.3 (4) Eg 10.18 Ex 10.3 (5) Eg 10.25 Ex 10.4 (3) Ex 10.4 (1) Eg 10.23 Eg 10.24 Ex 10.4 (4) Ex 10.5 (3) Ex 10.5 (4) Ex 10.5 (6) Ex 10.5 (7)
TôPm 10 (3) 19 úLs®Ls Ex 10.1 (5 a) Ex 10.1 (5 b) Ex 10.1 (9) Eg 10.9 Eg 10.7 Ex 10.2 (2) Eg 10.20 Ex 10.3 (1) Eg 10.19 Eg 10.21 Ex 10.3 (2) Ex 10.3 (3) Ex 10.3 (6) Eg 10.22 Ex 10.4 (2) Eg 10.27 Ex 10.5 (1) Eg 10.28 Ex 10.5 (2)
10 M : 227 6 M : 559(469) 3 M : 226(204) Total: 1012(900) Marks: 160
Ex - Exercise
NO SUBSTITUTE FOR HARDWORK XII Maths Classification of Questions (Q. Nos) N. MAHALAKSHMI PGT (Mathematics)
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XII MATHEMATICS - WINNING STRATEGY BLUE PRINT: Chapter
10 M
6M
1M
TOTAL MARKS
1
1
2
4
26
2
2
2
6
38
3
1
2
4
26
4
3
1
4
40
5
2
2
4
36
6
1
1
2
18
7
2
1
4
30
8
2
1
4
30
9
1
2
4
26
10
1
2
4
26
TOTAL
16
16
40
296
CLASSIFICATION OF QUESTIONS: Lesson
Models
10 M
6M
3M
Total
CW
HW
OBQ
1M (Text)
1M (COME)
1
34
33
33
16
82
34
48
6
19
23
2
58
20
73
58
151
69
82
7
35
39
3
30
16
47
19
82
33
49
6
29
46
4
62
70 (25) 61
84 (17) 113
38
92
34
154 (42) 174
10
51
104 (14) 96
22
5
28 (28) 44
4
44
40
6
17
11
26
9
46
20
26
2
15
44
7
34
23
54
18
95
35
60
4
19
23
8
36
25
55
11
91
34
57
10
26
19
9
13
15
31
20
66
25
41
4
20
39
10
25
12
40
19
71
31
40
1
26
15
Total
360
227
559 (469)
226 (204)
1012 (900)
412 (367)
600 (533)
54
271
380
Highlight FORMULAE and IMPORTANT STEPS in Mathematics note-book with a highlighter pen for easy REVISION. XII Maths Winning Strategy
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STATISTICS (as on SEP 2013): 1. 2. 3. 4. 5. 6.
Public Exam Questions COME book Questions Objective type (Public Exam) Objective type (COME book model) OBQ chapters (10 marks) OBQ chapters (6 marks)
: : : : : :
192 + 221 + 64 = 477 9 + 21 + 13 = 43 389 34 1245678 1 2 3 5 8 9 10
TIME MANAGEMENT DURING THE EXAMINATION: Type of Question
Part III Part II Part I Verification
No. of Questions
Minutes per question
Minutes
Time
8 5 1 -
80 50 40 10 180
11.35 12.25 1.05 1.15
10 10 40 TOTAL
QUESTION PAPER SCHEME:
Objective type questions: 40 (30 - textbook, 10 - COME book) 3 marks - at most 6 questions will be asked. 6M, 10M - 9 questions can be selected from the first 14 questions. 6M, 10M - OBQ - 2 questions - not compulsory. 6M, 10M - Questions 55 & 70 - compulsory. Questions 55 & 70 - either or type : (a) from chapters 1-5 & (b) chapters 6 - 10
EXAMINATION TIPS:
First do 10 marks, then 6 marks and lastly 1 mark. Draw 40 lines and write 1 mark questions, 10 answers per page. Put the Question numbers correctly. Do not attend Excess questions. Anyways it will be found out, while posting. Draw the figures wherever necessary, as they have the stage marks. Be careful while turning over the pages. Confusing letters: 2 - z, 1 - i, 1 - l, 6 - b, 5 - s. 9 - q, 9 - a, u - v In LHS = RHS type of questions, always check both the values are same. Put RHS = 0 in the Cartesian equation in “Vector Algebra - Planes”. Check the Cartesian equation by substituting the point. Check the point of intersection. It is either (x1, y1, z1) or (x2, y2, z2). Do not forget to put square for (x+3) or (y-2) in analytical questions. Draw parabola, ellipse or hyperbola in full page. Use finger in truth table problems. Write the result lastly in truth table questions, as it carries 1 mark. Analytical (center, foci, vertices) - 100 % sure Truth table - 100% sure Ex 9.4 & related examples - 100% sure Ex 5. 1 (or) Ex 5.10 application problems - 90% sure Ex 7.5 & related examples - 90% sure Ex 8.5 & related examples - 90% sure Ex 8.6 & related examples - 90% sure
XII Maths Winning Strategy
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