Higher_Math

G·Kwz¬ mf g‡Wj †U÷ I DËigvjv : eûwbev© Pwb

35 G·K¬zwmf g‡Wj †U÷ 01 welq ˆKvW : 1 2 6

mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v c~Y©gvb : 25
[we. `.ª : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î c‡Ö kœi µwgK b¤‡^ ii wecix‡Z c`Ö Ë eYm© sewjZ e„Ëmgn~ n‡Z mwVK/ m‡e©vrK…ó Dˇii eË„ wU ej c‡q›U Kjg Øviv m¤úY~ © fivU Ki|]

1. m¤v¢ ebvi mxgv wb‡Pi †KvbwU? 10. 60 †Kv‡Yi m¤ú~iK †Kv‡Yi A‡a©K 33
†Kv‡Yi gvb KZ? 18. 3 729 Gi gvb KZ?
K 0<p<1 L 0<p1

M 0p<1 N 0p1 K 120 L 60 K3 1

A M 30 N 15 2 L 33

2. 3 ˆm.wg. 4 ˆm.wg. 11. mKvj 8 : 20 Uvq Nwoi NÈvi KuvUv I M 39 1

N 3.39
19. 1 + x126 Gi x ewR©Z c‡`i gvb KZ?
B C wgwb‡Ui Kvu Uvi Aš—MZ© †KvY KZ
n‡e?
wP‡Î BAC5 ˆmG.wig. gvb KZ?
K 140 L 130 K1 L6
K 45 L 60 M 115 N 110 M 7 N 12

M 90 N 120
3. Xvj 3 Ges ( 2, 3) we›`My vgx †iLvi 12. x A‡¶i mgvš—ivj †iLvi mgxKiY 20. (8, 6) we›`y n‡Z x A‡¶i `~iZ¡ KZ?
mgxKiY †KvbwU? K y=0 L x=0 K2 L6

K y = 2x  3 L y = 3x  2 M x=a N y=b M8 N 14
M y = 3x  4 N y = 3x + 3
13. A I B we›`yi Ae¯’vb †f±i h_vµ‡g 21. A

4.  300 †KvYwU †Kvb PZzfv© ‡M a, b n‡j AB = KZ?
Aew¯’Z?
K 1g L 2q M 3q N 4_© K ab L a+b B 45 D
C
5. ABC G B m~²‡KvY n‡j wb‡Pi M ba N 21(a  b) wP‡Î BAC = 20 Ges ACD =

†KvbwU mwVK? 14. S = {x : x  R Ges x2 + 1 = 0} n‡j 45 n‡j ABC Gi gvb KZ?
K AC2 < AB2 + BC2
L BC2 < AB2 + AC2 wb‡Pi †KvbwU mwVK? K 20 L 25
M AB2 > AC2 + BC2
N AB2 < AC2 + BC2 K S= L S=R M 30 N 35
22. Nb‡Ki evûi ˆ`N©¨ 3 †m.wg. n‡j Zvi
M S=N N R\S =  K‡Y©i ˆ`N©¨ KZ?

 DÏxcKwU c‡o 68 bs cÖ‡kœi DËi `vI : 15. f(x) = x x 2 n‡j K 3 2 †m.wg. L 3 3 †m.wg.
x  3y  12 = 0 GKwU mij‡iLvi 
6. M 9 †m.wg. N 27 †m.wg.
7. mgxKiY| i. †Wvg f = R\{2} 23. wÎf‡z Ri cwie„‡Ëi e¨vm 8 †m.wg. n‡j

8. †iLvwUi Xvj KZ? ii. f GK-GK dvskb bewe›`y e‡„ Ëi e¨vmva© KZ †m.wg.?
9.
K3 L 1 iii. f1 (2) = 4 K2 L4
3
wb‡Pi †KvbwU mwVK? M 16 N 32
M 3 N 4 K i I ii L i I iii
†iLvwU x I y A¶‡K h_vµ‡g A I B  DÏxcKwU c‡o 24 I 25 bs cÖ‡kiœ DËi `vI :
M ii I iii N i, ii I iii `yBwU wbi‡c¶ gy`vª GKmv‡_ wb‡¶c
we›`‡y Z †Q` Ki‡j AB = KZ GKK?
16. a > b Ges c < 0 n‡j Kiv n‡jv|
K 16 L 4 10 i. ac > bc
24. †Kv‡bv H bv cvIqvi m¤v¢ ebv KZ?
M8 2 N8 ii. ac < bc
3 1
A¶Øq Øviv Drcbœ wÎfzR‡¶Î OAB iii. a < b K 4 L 4
Gi †¶Îdj KZ eM© GKK? c c
3 1
K 36 L 24 wb‡Pi †KvbwU mwVK? M 8 N 8

M 18 N 12 K i I ii L i I iii 25. Kgc‡¶ 1wU H cvIqvi m¤v¢ ebv KZ?
(a, 0), (0, b) Ges (1, 1) we›`y wZbwU
mg‡iL n‡j wb‡Pi †KvbwU mwVK? M ii I iii N i, ii I iii 3 1
17. y2 = 2x Ges yx = 4 n‡j x = KZ? K 4 L 2

K a+b=1 L a+b=1 K1 L2 M 1 N 3
M4 N2 4 8
M a + b =  ab N a + b = ab

Self test 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN
10 KLMN 11 KLMN 12 KLMN 13 KLMN 14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN
19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN

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1 N 2 M 3 N 4 K 5 K 6 L 7 L 8 L 9 N 10 L 11 L 12 N 13 M
DËigvjv 14 K 15 N 16 M 17 N 18 M 19 K 20 L 21 L 22 L 23 K 24 L 25 K

36 G·K¬zwmf g‡Wj †U÷ 02 welq ˆKvW : 1 2 6

mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v c~Y©gvb : 25
[we. `ª. : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î cÖ‡kiœ µwgK b¤^‡ii wecix‡Z cÖ`Ë eY©msewjZ e„Ëmgn~ n‡Z mwVK/ m‡e©vrK…ó Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]

1. †h‡Kv‡bv kvš— †mU A Gi Rb¨ 9. ejwU Kv‡jv nIqvi m¤v¢ e¨Zv KZ? 19. hw` a, b, x > 0 Ges a  1, b  1¸
(A/(A/A)) = ? 2 1 5 4 ZLb
K A L A M U N  K 9 L 3 M 9 N 9
i. log a a  log b b = 4
2. f (x) = x3 + 3 n‡j f1 (3) = ? 10. ejwU jvj bv nIqvi m¤v¢ e¨Zv KZ?
1 2 7 4 ab
K 0 L 3 M 27 N 30 K 3 L 9 M 9 N 9 ii. log x = log a + log b  log x

3. x 5 = A + B 11. hw` cos  = 1 nq, Zvn‡j ax 3 a2 hLb 3
(x + 1) (x  2) x+1 x2 2 2
iii. = x =
†hLv‡b A Ges B gj~ ` msL¨v| A Gi 1
gvb KZ? i. sec2  = 2 ii. sin2  = 2 wb‡Pi †KvbwU mwVK?

K3 L2 M1 N2 iii. tan2  = 1 K i I ii L i I iii
wb‡Pi †KvbwU mwVK?
 wb‡æi wP‡Îi Av‡jbv‡K 4 I 5 bs K i I ii L i I iii M ii I iii N i, ii I iii
c‡Ö kiœ DËi `vI :
4. M ii I iii N i, ii I 20. f(x) = 1 2 dvskbwUi †Wv‡gb wb‡Pi
A 3x 

iii †KvbwU?

12. 3x + y  5 = 0 mij‡iLvwU x A‡¶i K {x : x   Ges x  0}
B mv‡_ KZ wWwMÖ †KvY Drcbœ Ki‡e?
DC
†KvbwU DB Gi Dci AC Gi j¤^ L {x : x   Ges x  0}
Awf‡¶c? K 30° L 60° M 120° N 150°
M {x : x   Ges x  32}
K AD L DC M DB N CB  wb‡æi Z‡_¨i Av‡jv‡K 13 I 14 bs
cÖ‡kiœ DËi `vI :
5. B = m~²‡KvY n‡j, †KvbwU AC2 Gi 6 †m. wg. e¨vmwewkó GKwU avZe N {x : x   Ges x   32}
6. gvb wb‡`©k K‡i?
K AB2 + BC2  2 BC.BD †MvjK‡K Mwj‡q 3 †m. wg. fw‚ gi 21. hw` (a, 0), (0, b) Ges (1, 1) we›`y
7. L AB2 + BC2  2 BC.CD e¨vmva©wewkó mgeË„ f‚wgK wmwjÛvi wZbwU mg‡iL nq Z‡e wb‡Pi †KvbwU
M AB2 + BC2 + 2 AC.CD ˆZwi Kiv n‡jv| mwVK?
N AB2 + BC2 + 2 AB.AD 13. DrcbKœ …Z wmwjÛv‡ii D”PZv KZ?
K 4 †m. wg. L 6 †m. wg. K a+b=1 L a+b=1
bx2 + ax + c = 0 mgxKi‡Yi g~jØq M a + b =  ab N a + b = ab
M 8 †m. wg. N 12 †m. wg.
ev¯—e, Amgvb Ges gj~ ` n‡e, hw` 14. wmwjÛviwUi eµZ‡ji †¶Îdj KZ 22. hw` cos  + sin  = 2 nq, Z‡e  = ?
i. a2  4bc > 0 Ges Dnv GKwU cY~ ©eM©
bv nq eM© †mw›UwgUvi? K 30° L 45°

ii. a2  4bc > 0 Ges Dnv GKwU c~Y©eM© nq K 12  L 24  M 36  N 42  M 60° N 90°

iii. a2  4bc = 0 nq 15. †Kv‡bv Abµy ‡gi n Zg c` 1  ( 1)n 23. ABC G cosec A + C = ?
2 2
wb‡Pi †KvbwU mwVK?
Ki L ii n‡j, 17 Zg c` KZ? K sec  L cosec 
K1 L0 M1 N2 2 2
M i I ii N i, ii I iii 16. eûc`x P (x) = 2x2  9x + 6 †K (x 
ax = b, by = c Ges cz = a n‡j, xyz = M sec B N cosec B
4) Øviv fvM Ki‡j fvM‡kl KZ n‡e? 2 2
KZ? K 4 L 2 M 1 N  2 24. GKwU wÎf‡z Ri †KvY¸‡jv mgvš—i
K  1 L 0 M 1 N 2 17. GKwU mylg PZz¯—j‡Ki †h‡Kv‡bv cÖMgbfz³ Ges ¶`z Zª g †KvYwU en„ Ëg
8. A, B, C Gi Ae¯’vb †f±i h_vµ‡g a, 2 †Kv‡Yi A‡a©K n‡j, e„nËg †KvYwUi
b, c Ges C we›`y AB †K 5 : 11 av‡ii ˆ`N¨© 2 †m. wg. Ges D”PZv 3 eË„ xq cwigv‡ci gvb KZ?

Abycv‡Z Aš—wef³ Ki‡j c = ? †m. wg. n‡j PZz¯—jKwUi AvqZb KZ? 4 
1 2 9 3
K 5b + 11a L 11b + 5a K 2 Nb †m. wg. L 3 Nb †m. wg. K L
16 16
M 1 Nb †m. wg. N 2 Nb †m. wg.  
M 5b  11a N 11b  5a 18. GKwU 4 †m.wg. e¨v‡mi †MvjK M 2 N 9
16 16

 wb‡æi Z‡_¨i Av‡jv‡K 9 I 10 bs AvK…wZi ej GKwU wmwjÛvi AvK…wZi 25. †f±i †hvR‡bi ms‡hvM wewa †KvbwU?
c‡Ö kiœ DËi `vI : ev‡· wVKfv‡e G‡u U hvq| ev·wUi K u + v = v + u
GKwU e¨v‡M 4 wU jvj, 6 wU mv`v I 8 AvqZb KZ?
wU Kv‡jv ej Av‡Q| ˆ`e fv‡e GKwU K 2  Nb †m. wg. L 4  Nb †m. wg. L (u + v) + w = u + (v + w)
ej cQ›` Kiv n‡j M 8  Nb †m. wg. N 16  Nb †m. wg.
M m (u + v) = mu + mv
N ( u) + u = 0

Self test 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN
10 KLMN 11 KLMN 12 KLMN 13 KLMN 14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN
19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN

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1 K 2 K 3 N 4 L 5 K 6 L 7 M 8 K 9 N 10 M 11 N 12 M 13 K
DËigvjv 14 L 15 M 16 L 17 L 18 N 19 K 20 M 21 N 22 L 23 M 24 K 25 L

37 G·K¬wz mf g‡Wj †U÷ 03 welq ˆKvW : 1 2 6

mgq : 25 wgwbU D”PZi MwYZ eûwbe©vPwb Afx¶v cY~ g© vb : 25
[we. `ª. : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î cÖ‡kœi µwgK b¤‡^ ii wecix‡Z c`Ö Ë eY©msewjZ eË„ mgn~ n‡Z mwVK/ m‡e©vrK…ó Dˇii eË„ wU ej c‡q›U Kjg Øviv m¤úY~ © fivU Ki|]

1. A  B = B Ges A  B n‡j wb‡Pi 10. 2x2  11x + 9 = 0 mgxKi‡Yi  DÏxcKwU c‡o 17 I 18 bs cÖ‡kiœ
†KvbwU mwVK? DËi `vI :
K AB L BA i. GKwU g~j 1 4x  3.2x+2 + 25 = 0 GKwU m~PKxq
mgxKiY Ges 2x = y.
MAB=B NBA ii. wbðvqK 49
iii. gj~ ¸‡jv gj~ `
2. f(x) = 3x + 5, x  R GKwU GK-GK 17. y2  12y = KZ?
3. dvskb n‡j 1(2) Gi gvb KZ? wb‡Pi †KvbwU mwVK?
K i I ii L i I iii K 32 L  32
4. K 1 L1
M3 N5 M ii I iii N i, ii I iii M 16 N  16
18. y Gi gvb KZ?
2  x5  x2  3x2 + x3  2  x 11. 3x+4 = 81 mgxKi‡Y x Gi gvb
K 4, 8 L  4, 8
eûc`xi †KvbwU?
i. gLy ¨c‡`i mnM 2 M  4,  8 N 4,  8
K4 L3 19. (1 + ax)5 Gi we¯—…wZ‡Z x2 Gi mnM
ii. gL~ ¨c` 2x4
iii. gvÎv 3 M2 N0 270 n‡j a Gi gvb KZ n‡e?

wb‡Pi †KvbwU mwVK?  DÏxcKwU c‡o 12 I 13 bs cÖ‡kœi K 3 L3 3
DËi `vI :
K i I ii L i I iii 10 †_‡K ¶`z ªZi †Kvb ¯^vfvweK M 2 N2
msL¨vi e‡M©i mv‡_ 6 †hvM Ki‡j 20. 5C3  8C4 = KZ?
M ii I iii N i, ii I
iii K 70 L 100
†hvMdj H msL¨vi 5 ¸Y A‡c¶v
P(x) = 2x2  7x + 5 n‡j P(1) = KZ? e„nËi| M 700 N 750
21. O(0, 0) we›`y †_‡K A(4, 4) I B(4, y)
K 2 L 1 12. mgm¨vwU AmgZvi gva¨‡g cÖKvk we›`iy `~iZ¡ mgvb n‡j y = KZ?
M0 N1
5. ABC Gi AD ga¨gv BC evû‡K Ki‡j wb‡Pi †KvbwU n‡e?
K4 L3 M2 N0
mgwØLwÛZ Ki‡j wb‡Pi †KvbwU K 5x + 6 > x2 L x2 + 6 > 5x 22. GKwU wÎfz‡Ri evû¸‡jvi ˆ`N¨©
G¨v‡cvwjwbqv‡mi Dccv`¨? M 5x + 6 < x2 N 6 + x2 < 5x
h_vµ‡g 3, 4, I 5 GKK n‡j
K AB2 + AC2 = 2AD 13. msL¨v¸‡jvi m¤¢ve¨ †mU wb‡Pi i. S = 12
L AB2 + AC2 = 2(AD2 + BD2) †KvbwU?
M 2(AB2 + AC2) = AD2 + BD2 ii. wÎfzRwUi †¶Îdj eM© 6 GKK
K {4, 5, 6, 7, 8, 9} iii. wÎfzRwU mg‡KvYx
N AB2 + AC2 = AD2 + BD2
6. wÎfz‡Ri wZb kxl© we›`yMvgx e„ˇK Kx L {4, 5, 6, 7, 8, 9, 10} wb‡Pi †KvbwU mwVK?

 e‡j? M {1, 4, 5, 6, 7, 8, 9} K i I ii L i I iii

K kxle© „Ë L cwieË„ N {1, 4, 5, 6, 7, 8, 9, 10} M ii I iii N i, ii I iii

M Aš—te„Ë N ewnteË„ 14. 0.5 + 0.05 + 0.005 + ....... avivwUi 23.
AmxgZK mgwó KZ?
DÏxcKwU c‡o 7 I 8 bs c‡Ö kœi DËi C
5 9
`vI : K 9 L 5 v

5, 6 I r †m.wg. e¨vmvaw© ewkó wZbwU 3 2 Au B
eË„ ci¯úi ewnt¯úk© Ki‡j Zv‡`i 2 3
M N CB Gi gvb I w`K m~wPZ nq wb‡Pi

†K›`ª¸‡jv †hvM Ki‡j †h wÎfzR 15. wÎf‡z Ri wZbwU †Kv‡Yi Abcy vZ 1 : 2 : †KvbwU Øviv?
cvIqv hvq Zvi cwimxgv 36 †m.wg.| 3 n‡j ¶`z ªZi †KvYwUi e„Ëxq gvb K v + u
7. r = KZ †m.wg.? KZ? L u+v

K 2.64 L5 M vu N uv

M7 N 25 K  L 2 24. GKwU wcÖR‡gi †¶Îdj 6 eM© †m.wg.
3 3 Ges D”PZv 8 †m.wg. n‡j wcÖRgwUi
8. r †m.wg. e¨vmva©wewkó e‡„ Ëi †¶Îdj
KZ eM© †m.wg.? M  N 2 AvqZb KZ Nb †m.wg.?
6 5
K7 L 14  K 24 L 48 M 52 N 72
M 28  N 49   
16. tan2 6  sin2 6 = KZ? 25. Q°vq 3 Øviv wefvR¨ msL¨v Ges g`y vª q
†h‡Kv‡bv wcV cvIqvi m¤v¢ ebv KZ?
9. wb‡Pi †KvbwU 4x  1  x2 = 0 Gi 3 L 213
gj~ ? K 4 K 1 L 1
3 4
K2 3 L2+ 3 1 1
M2 3 N2+2 3 M 12 N  12 M 1 N 5
6 12

Self test 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN
10 KLMN 11 KLMN 12 KLMN 13 KLMN 14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN
19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN

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1 L 2 K 3 K 4 M 5 L 6 L 7 M 8 N 9 M 10 N 11 N 12 L 13 M
DËigvjv 14 K 15 M 16 M 17 L 18 K 19 L 20 M 21 K 22 M 23 N 24 L 25 K

38 G·K¬wz mf g‡Wj †U÷ 04 welq ˆKvW : 1 2 6

mgq : 25 wgwbU D”PZi MwYZ eûwbe©vPwb Afx¶v cY~ g© vb : 25
[we. `.ª : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î cÖ‡kœi µwgK b¤‡^ ii wecix‡Z cÖ`Ë eYm© sewjZ e„Ëmg~n n‡Z mwVK/ m‡e©vrK…ó Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]

1. †Kvb †m‡Ui m`m¨ msL¨v n n‡j Zvi 9. msL¨v `yBwU wK wK? 18. AvqZb KZ?
cKÖ …Z Dc‡m‡Ui msL¨v KZwU n‡e?
K 2n  2 L 2n  1 K 10 Ges 9 L 30 Ges 3 K 301.59 Nb †m.wg.
M 18 Ges 5 N 18 Ges 7 L 300 Nb †m.wg.
M 2n N n2
10. A, B, C we›`iy Ae¯v’ b †f±i h_vµ‡g M 305.59 Nb †m.wg.
2. 2x3 + x2 + ax + 18 eûc`xq GKwU a, b, c, C we›`wy U AB †iLvsk‡K 1 : 2 N 412 Nb †m.wg.
Drcv`K x + 2 n‡j a Gi gvb KZ? Abycv‡Z Aš—wef© ³ Ki‡j †KvbwU 19. cos (330) Gi gvb KZ?

K  15 L 3 mwVK? K23 1
2
M3 N 15 c^ a^ + b^ c^ 2a^ + b^ L
3 3
3. ax2 + bx + c = 0 mgxKi‡Yi gj~ Øq K = L = 1 3
4. 2 2
mgvb n‡j, wb‡Pi †Kvb Z_¨wU mwVK? c^ = a^ + 2b^ c^ = 2a^ + 2b^ M N
5. K b2  4ac > 0 L b2  4ac < 0 3 3
M N 20. †Kvb mg‡KvYx wÎfz‡Ri ga¨gvÎq hw`

M b2  4ac  0 N b2  4ac = 0 11. hw` log x = 313 nq, Z‡e x Gi gvb p, q, r Ges AwZfzR d nq, Z‡e †Kvb
8
hw` cos  = 1 nq Z‡e KZ? m¤úKw© U mwVK?
2 K p2 + q2 + r2 = d2
i. sec2  = 2
K 32 L 16 L p2 + q2 + r2 = 3d2
1
ii. sin2  = 2 M 8 N 64 M 3(p2 + q2 + r2) = 4d2

iii. tan2  = 1 12. xy = yx Ges x = 2y n‡j (x, y) = N 2(p2 + q2 + r2) = 3d2
KZ? x3
wb‡Pi †KvbwU mwVK? 21. x2  9 fMvœ skwUi mgvb KZ?
K (2, 4) L (4, 2)
K i I ii L i I iii 9 x
M (3, 1) N (1, 3) K x +  L x + x2 
M ii I iii N i, ii I iii x2 9 9
13. n(n  1)! Gi gvb †KvbwU?
2x + 1x6 Gi we¯—w… Z‡Z (n  2)! Mx + 9x N x + 1
x2  9 x2  9
Kn L n1
i. c` msL¨v 7  wb‡Pi Z‡_¨i Av‡jv‡K 22 I 23 bs
M n(n  1) N n2 cÖ‡kœi DËi `vI :
ii. n hy³ c` 4_© c`
iii. n g³y c‡`i gvb 160 14. F(x) = x  1 dvskbwUi †Wvg A
†KvbwU?
wb‡Pi †KvbwU mwVK?
K i I ii L i I iii K x<1 L x1

M x>1 N x1

M ii I iii N i, ii I iii 15. e„‡Ëi ewnt¯’ †Kv‡bv we›`y †_‡K H e‡„ Ë 45
6. bewe›`y e„‡Ëi e¨vmva© wÎf‡z Ri KZwU ¯úk©K Avu Kv hvq? D CB

6 cm

cwie¨vmv‡a©i KZ ¸Y? K1 L2 22. BD Gi Dci AC Gi j¤^ Awf‡¶c †KvbwU?

K A‡a©K L wظY M 3 N AmsL¨ K BD L CD M AB N BC

M wZb¸Y N Pvi¸Y  wb‡Pi Z‡_¨i Av‡jv‡K 16  18 bs 23. DC = KZ?
K 2 †m.wg. L 4 †m.wg.
7. mKvj 4.20 Uvq Nwoi NÈvi KuvUv I c‡Ö kiœ DËi `vI :
wgwb‡Ui Kvu Uvi AšM— Z© †KvY KZ n‡e? M 6 †m.wg. N 8 †m.wg.
1

K 140 L 130 h h = 8 cm 24. GKwU NUbv A Gi Rb¨ m¤¢vebvq mxgv
r = 6 cm

M 115 N 110 r †KvbwU?

 wb‡Pi Z‡_¨i Av‡jv‡K 8 I 9 bs 16. †njv‡bv D”PZv KZ? K O < P(A) < 1 L O  P(A)  1
c‡Ö kiœ DËi `vI : K 8 †m.wg. L 9 †m.wg. M O  P(A) < 1 N O < P(A)  1
8. M 10 †m.wg. N 11 †m.wg.
25. A

`By wU abvÍK c~Ym© sL¨vi e‡Mi© Aš—i 19 17. eµZ‡ji †¶Îdj †KvbwU? BC
Ges ¸Ydj 90| K 187.5 eM© †m.wg.
AB †f±‡ii gvb KZ?
msL¨v `ywUi e‡Mi© mgwó KZ? L 188.5 eM© †m.wg.
K AB  AC L AC + BC
K 90 L 181 M 287.5 eM© †m.wg.

M 361 N 181 N 278.5 eM© †m.wg. M AC + CB N AC  CB

Self test 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN
10 KLMN 11 KLMN 12 KLMN 13 KLMN 14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN
19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN

----------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------
1 L 2 M 3 N 4 N 5 N 6 K 7 L 8 L 9 K 10 L 11 K 12 L 13 M
DËigvjv 14 N 15 L 16 M 17 L 18 K 19 N 20 N 21 M 22 N 23 K 24 L 25 M

39 G·Kwz¬ mf g‡Wj †U÷ 05 welq ˆKvW : 1 2 6

mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v c~Y©gvb : 25
[we. `ª. : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î c‡Ö kiœ µwgK b¤^‡ii wecix‡Z c`Ö Ë eY©msewjZ e„Ëmgn~ n‡Z mwVK/ m‡e©vrK…ó Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]

1. (2,  1) we›`My vgx Ges Xvj 2 n‡j, 8. N  Z  Q  R n‡j mvweK© †mU  wb‡Pi Z‡_¨i Av‡jv‡K 1618 bs
mgxKiY †KvbwU? †KvbwU? cÖ‡kiœ DËi `vI :

K y + 2x  5 = 0 L y  2x + 5 = 0 KQ LN x + 2xn †hLv‡b n †Rvo msL¨v|
NZ
M y  2x  5 = 0 N y + 2x + 5 = 0 MR B 16. (r + 1) Zg c` x ewRZ© n‡j, r Gi

2. †Kv‡bv wÎf‡z Ri cwi‡K›`ª fi‡K›`ª I 9. O P gvb KZ?
j¤^we›`y mgš‡^ q MwVZ wÎf‡z Ri
†¶Îdj wb‡Pi †KvbwU? n
A K0 L 2 Mn N 2n
K4 L2
M1 N0 wP‡Î OP = 5 †m. wg., PA = 4 †m. wg. 17. x ewRZ© c`wUi gvb †KvbwU?
AOP Gi †¶Îdj KZ eM© †m. wg.? n
3. a = 2, b = 3 Ges c = 2 n‡j, K ncn 2 L ncn 2n
K3 L6

i. ax2 + bx + c GKwU exRMvwYwZK M9 N 18 2
ii. ivwk
ax2 + bcxy + cy2 cÖwZmg ivwk 10. GKwU B‡Ui avi msL¨v Zvi c„ôZj M ncn 2n n
msL¨vi KZ ¸Y? N ncn 22
L3
K2 2
18. n = 8 n‡j x ewR©Z c`wU KZ?
iii. ax2 + by2 + cz2 Pµ-µwgK ivwk K 968 L 1020
M4 N5 M 1120 N 1168

wb‡Pi †KvbwU mwVK? 11. GKwU _wj‡Z bxj ej 12 wU, mv`v ej 19. A
16 wU I Kv‡jv ej 20 wU Av‡Q| ej
K i I ii L i I iii mv`v bv nIqvi m¤v¢ ebv KZ?

M ii I iii N i, ii I iii 1 1 B CD
16 3
4. A B K L wP‡Î AC = 5 †m. wg., BD = 10 †m. wg.,

PQ M 2 N 3 AD = 4 †m. wg. Ges AD  BD n‡j
3 4 AB = KZ †m. wg.?
DC
12. cos  + cos 16  = KZ? K 76 L 116
ABCD UªvwcwRqv‡gi AC I BD 15 15 M 74 N 116
K‡Yi© ga¨we›`y h_vµ‡g P I Q n‡j,
1 20. xy  y2 = 1 Ges x2  xy = 2 n‡j, x2
PQ = KZ? K 1 L 2  y2 = KZ?
M0
K 1 (DC  AB) L 1 (DC + AB) N1 K3 L3 M4 N6
2 2
0.1. 2. †K Amxg ¸‡bvËi avivq cÖKvk a b
M 1 (AD + BC) N 1 (AD  BC) 13. 21. a > b n‡j †Kvb k‡Z© c > c n‡e?
2 2
Ki‡j avivwUi K c<0 L c=0
 wb‡Pi Z‡_¨i Av‡jv‡K 57 bs c‡Ö kiœ
DËi `vI : i. mvaviY AbycvZ 0.01 M c0 N c>0
5. 22. 4  4 + 4  4 + 4  avivwUi cÖ_g
p = loga (bc), q = logb (ca), r = logc ii. c_Ö g wZbwU c‡`i mgwó 0.121212 10wU c‡`i mgwó KZ?
(ab)
iii. AmxgZK mgwó 4 K 4 L0
1 + q = KZ? 33
M 4 N 40
K loga (abc) L logb (abc) wb‡Pi †KvbwU mwVK?
M loga bc 23. mKvj 9:30 wgwb‡U Nwoi NÈvi KuvUv I
K i I ii L i I iii wgwb‡Ui Kvu Uvi Aš—M©Z †Kvb KZ wWwMÖ?
a
N loga bc M ii I iii N i, ii I iii K 90° L 100°

6. 1 + 1 Gi gvb †KvbwU? 14. x  x + 4 AmgZvi mgvavb wb‡Pi M 105° N 110°
+ + 3 24. O (0, 0), P (4, 0) I Q (0, 4) GKB
1 q 1 r
†KvbwU? mgZ‡j wZbwU we›`|y POQ wÎfRz wU
K 1 L 1+q
1 + q K x  12 L x6 †Kvb ai‡bi?

M q N p M x4 N x3 K mgevû L welgevû
+ +
1 q p 1 15. (2y)y = yx Ges x = 2y n‡j, (x, y) = M ¯’~j‡KvYx N mgwØevû mg‡KvYx

7. r = 0 n‡j, ab = KZ? KZ? 25. †Kv‡bv Nb‡Ki K‡Y©i ˆ`N¨© 5 3 †m.

K0 L1 K (4, 2) L (2, 4) wg. n‡j, Gi AvqZb KZ Nb †m. wg.?
M loga abc N loga bc M (3, 4) N (8, 4)
K 5 L 25 M 75 N 125

Self test 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN
10 KLMN 11 KLMN 12 KLMN 13 KLMN 14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN
19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
-------------------------------------------------------------------------------------- ------------------------------------------------------------ -------------------------------
1 L 2 N 3 K 4 K 5 L 6 N 7 L 8 M 9 L 10 K 11 M 12 M 13 N
DËigvjv 14 L 15 K 16 L 17 N 18 M 19 L 20 L 21 N 22 L 23 M 24 N 25 N

40 G·K¬wz mf g‡Wj †U÷ 06 welq ˆKvW : 1 2 6

mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v c~Yg© vb : 25
[we. `ª. : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î cÖ‡kiœ µwgK b¤‡^ ii wecix‡Z c`Ö Ë eY©msewjZ e„Ëmg~n n‡Z mwVK/ m‡e©vrK…ó Dˇii eË„ wU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]

1. wb‡Pi †KvbwU dvskb bq?  wb‡Pi Z‡_¨i Av‡jv‡K 11 I 12 bs 19. 2x  5y  10 = 0 †iLvwU Øviv A¶‡qi
cÖ‡kiœ DËi `vI : mv‡_ Drcbœ wÎf‡z Ri †¶Îdj KZ?
K {(a, b), (a, c), (b, c), (c, d)} A

L {(1, 2), (2, 3), (3, 4), (4, 5)} Dx K 3 eM© GKK L 5 eM© GKK
y
M {( 1, 1), (1,  1), (2,  2), ( 2, 2)} M 7 eM© GKK N 10 eM© GKK
N {(1, 2), (4, 3), (3, 4), (6, 5)} BC
wb‡Pi †KvbwU cÖwZmg ivwk? 20. g~j we›`y Ges (x1, y1) we›`yMvgx
2. 11. wP‡Î (AC + BD) †KvbwU? mij‡iLvi mgxKiY †KvbwU?
3. K xy  yz + zx L xy + yz  zx
M xy + yz + zx N  xy + yz + zx K BC L BD x1
4. y1
mvweK© †mU U Gi †h‡Kv‡bv Dc‡mU A M DC N AB K y = mx L y = x
Gi Rb¨ A \ (A \ A) Gi gvb †KvbwU?
12. y = 70° n‡j, BDC = KZ? y1
x1
K A LA K 110° L 100° M y = x

M N {0} M 70° N 40° N y  y1 = m (x  x1)

f(x) = 4x  9 n‡j, f1 (5) = KZ? 13. wÎfz‡Ri cwie‡„ Ëi e¨vmva© a n‡j 21. P I Q we›`iy Ae¯v’ b †f±i (a  b) I
x2 2
bewe›`y e„‡Ëi e¨vmva© KZ? (a + b) n‡j, PQ = ?
K1 L3

M 3 N 3 Ka L 2a K 2a L 2b
5 Ma+b N ab
5. x3 + 2x2 + 2x + a Gi GKwU Drcv`K (x M a N a
2 4
+ 1) n‡j, a Gi gvb KZ?  GKwU K¨vcmy‡ji ˆ`N¨© 15 †m. wg.|
14. ax2 + bx + c = 0 wØNvZ mgxKi‡Yi wmwjÛvi AvKw… Zi As‡ki e¨vmva© 3 †m.
K 5 L 1 †jLwPÎ x A¶‡K me©vwaK KZ evi wg.|
M1 N5
wb‡Pi DÏxc‡Ki Av‡jv‡K 6 I 7 bs †Q` Ki‡Z cv‡i? Dc‡ii DÏxc‡Ki Av‡jv‡K 22 I 23
 cÖ‡kiœ DËi `vI : bs c‡Ö kœi DËi `vI :
K1 L2

p(x) = 2x3  5x2 + 6x  3 M 3 N AmsL¨

6. p (x) †K (x  3) Øviv fvM Ki‡j 15. 1, 2 , 53, 4  Abµy gwUi P Zg c` 22. wmwjÛvi AvK…wZi As‡ki c„ôZ‡ji
7. fvM‡kl KZ n‡e? 3 7 †¶Îdj KZ eM© †m. wg.?
8. KZ?
9. K  120 L  30 K 54  L 27 
2p  1 p
M  24 N 24 K p L 2p  1
p (x) Gi GKwU Drcv`K wb‡Pi †KvbwU? M9 N3

K x3 L x+1 M 1 1 N p p 1 23. `yB cÖv‡š—i Aa© †MvjvK…wZ As‡ki
2p   cô„ Z‡ji †¶Îdj KZ eM© †m. wg.?
M x2 N x1
x + y + z = 0 n‡j, x3 + y3 + z3 Gi gvb 16. tan  + sec  = x n‡j sin  = KZ?
K9 L 18 
KZ? x + 1 x  1
L (x  y) (y  z) (z  x) K x  1 L x + 1 M 27  N 36 
K0
M 3xyz N xyz x2  1 x2 + 1 24. †Kvb NUbv E Gi Rb¨ m¤¢vebvi mxgv
mgevû wÎf‡y Ri evûi ˆ`N©¨ 7 †m. wg. M x2 + 1 N x2  1 wb‡Pi †KvbwU?

n‡j ga¨gvi ˆ`N¨© KZ †m. wg.? 17. y = 1  3x Gi wecixZ dvskb K 0 < P(E) < 1
†KvbwU?
K 9.24 L 6.06 L 0  P(E)  1
M 11.13 N 12.12 1
K log3 (1  y) L log3 1  x M 0  P(E) < 1
10. P M 1  3x N 0 < P(E)  1

N 3x  1 25. GKwU wbi‡c¶ g`y ªv wZbevi wb‡¶c
Kiv nj| `yBwU †nW I GKwU †Uj
Q S GRi j¤^ Awf‡¶‡ci ˆ`N¨© 0 18. 1  x42n Gi we¯—w… Z‡Z Z…Zxq c‡`i

i. PQ 7 cvIqvi m¤v¢ ebv KZ?
4
ii. PR2 > PS2 + RS2 iii. PS2 = PQ2 + mnM n‡j n Gi gvb wbY©q Ki| 7 5
RQ2 8 8
K L
wb‡Pi †KvbwU mwVK?
K i I ii L i I iii K7 L8
M ii I iii N i, ii I iii M3 N6 M 3 N 1
8 8

Self test 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN
10 KLMN 11 KLMN 12 KLMN 13 KLMN 14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN
19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN

-------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------
1 K 2 M 3 L 4 K 5 M 6 N 7 N 8 M 9 L 10 K 11 N 12 K 13 N
DËigvjv 14 L 15 L 16 M 17 L 18 L 19 L 20 M 21 L 22 K 23 N 24 L 25 M

41 G·K¬zwmf g‡Wj †U÷ 07 welq ˆKvW : 1 2 6

mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v c~Y©gvb : 25
[we. `.ª : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î c‡Ö kœi µwgK b¤^‡ii wecix‡Z cÖ`Ë eY©msewjZ eË„ mgn~ n‡Z mwVK/ m‡e©vrK…ó Dˇii eË„ wU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]

1. log 3 55 = KZ? 11.  4x = 7 mgxKi‡Yi †jLwPÎ Kxiƒc 20. hw` ax = b nq, hLb a > 0, x  N
n‡e? Z‡e
K3 L2 K x-A‡¶i mgvš—ivj mij‡iLv
1 1 L cive˄
M 2 N 3 i. logab = x
ii. logaab = b
2. F(x) = |x| n‡j, F( 3) Gi gvb wb‡Pi M y A‡¶i mgvš—ivj mij‡iLv
†KvbwU? iii. logab = log3b  loga3
N gj~ we›`My vgx mij‡iLv
K 3 L0 12. GKwU mg‡KvYx wÎfz‡Ri fw‚ g 6 wb‡Pi †KvbwU mwVK?

M3 N 3 †m.wg.| fw‚ g msjMœ †KvY 60| K i I ii L i I iii

3. 1 + xa7 Gi we¯—w… Z‡Z x2 Gi mnM wÎfRz wUi †¶Îdj KZ eM© †m.wg.? M ii I iii N i, ii I iii

KZ? K6 3 L9 21. D(x) abvÍK gvÎvi eûc`x, p  0 n‡j,

( )K 7 a7 ( )L 7 a2 M 12 3 N 18 3 D(x) †K px  q Øviv fvM Ki‡j
0 2 13. f(x) = 3x2 n‡j, f1( 3) = KZ? fvM‡kl
K0 L1
( )M 7 a3 ( )N 7 M 3 N Am½vwqZ K Dqp L Dpq
3 2
4. sin 3A = cos 3A n‡j, A Gi gvb  wb‡Pi Z‡_¨i Av‡jv‡K 14 I 15 bs
KZ? c‡Ö kœi DËi `vI : N Dpq
L 20 `yBwU Q°v GK‡Î wb‡¶c Kiv n‡jv| M D(pq)
K 15
14. Q°vi Dc‡ii wc‡Vi msL¨vi mgwó 12  wb‡Pi Z‡_¨i Av‡jv‡K 22 I 23 bs
M 30 N 40

 wb‡Pi Z‡_¨i Av‡jv‡K 5 I 6 bs nIqvi m¤v¢ ebv KZ? cÖ‡kiœ DËi `vI :
cÖ‡kiœ DËi `vI : 1 1 35 11
Y K 12 L 36 M 36 N 12 †Kv‡bv wÎf‡z Ri wZb evûi ˆ`N¨© 6
†m.wg., 8 †m.wg. I 10 †m.wg.|
15. Q°vi Dc‡ii wc‡V GKB msL¨v bv
A(3, B) nIqvi m¤v¢ ebv KZ? 22. wÎfzRwUi cwie‡„ Ëi e¨vm KZ †m.wg.?

5. O B(2, 0) X K 1 L 35 K5 L6
6 36
6. AB †iLv x A‡¶i mv‡_ KZ wWMÖx M8 N 10
7. †KvY Drcbœ K‡i? 5 1
M 6 N 36 23. wÎfRz wUi
K 30 L 45 M 50 N 60
16. OAC G B, AC Gi ga¨we›`y| hw` i. †¶Îdj 24 eM© †m.wg.
OAB = KZ?
OA = a Ges OB = b nq, Z‡e OC ii. cwie„‡Ëi †¶Îdj 25 eM© †m.wg.
K 20 L 30 M 45 N 50 Gi gvb a I b Gi gva¨‡g cÖKvk

mgevû wÎf‡z Ri cwie„‡Ëi e¨vmva© 2
†m.wg. n‡j, wÎfzRwUi †¶Îdj KZ
eM© †m.wg.? Ki‡j wb‡Pi †KvbwU mwVK? iii. Aš—©e‡„ Ëi †¶Îdj 16 eM© †m.wg.
wb‡Pi †KvbwU mwVK?
K2 3 L3 3 K 2b + a L ba

M4 3 N6 3 M 2b  a N b+a K i I ii L i I iii

8. x3  ax2  9x  5 eûc`xi GKwU 14 M ii I iii N i, ii I iii
9. Drcv`K x  5 n‡j, a Gi gvb KZ?
17. a9 a7 a6Gi gvb KZ? 24. ABCD eË„ ¯’ PZfz yR© hvi C =
K3 L3 1 120| sin A = KZ?
K a L a14 M a7 N a14
M 5 N 9 1
18. A I B †h‡Kv‡bv `By wU †mU n‡j, A\B = 2
cos  127 = KZ? KZ? K0 L

K1 L0 M1 N K AB L AB 1 3
2 2
10. GKwU ¸‡YvËi avivi 1g c` 1 Ges M A  B N A  B M N
2
2 19. hw` x < y nq Z‡e, z Gi FYvÍK gv‡bi
AmxgZK mgwó 5 n‡j, avivwUi Rb¨ wb‡Pi †Kvb m¤úK©wU mwVK? 25. `cy iy 1 : 20 Uvq Nwoi NÈvi Kvu Uv I
x y x y wgwb‡Ui KuvUvi Aš—f©y³ †KvY KZ?
mvaviY Abcy vZ KZ? K z = z L z < z
1 1 3 2 K 80 L 90
K 5 L 4 M 8 N 3 x y x y
M z > z N 2 < z M 100 N 110

Self test 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN
10 KLMN 11 KLMN 12 KLMN 13 KLMN 14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN
19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN

------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------
1 K 2 M 3 L 4 K 5 N 6 L 7 L 8 K 9 L 10 L 11 M 12 N 13 N
DËigvjv 14 L 15 M 16 M 17 K 18 M 19 M 20 N 21 L 22 N 23 K 24 N 25 K

42 G·K¬wz mf g‡Wj †U÷ 08 welq ˆKvW : 1 2 6

mgq : 25 wgwbU D”PZi MwYZ eûwbev© Pwb Afx¶v cY~ ©gvb : 25
[we. `.ª : mieivnK…Z eûwbe©vPwb Afx¶vi DËic‡Î c‡Ö kœi µwgK b¤‡^ ii wecix‡Z c`Ö Ë eYm© sewjZ e„Ëmg~n n‡Z mwVK/ m‡e©vrK…ó Dˇii eË„ wU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]

1. O we›`y g~jwe›`y Ges A we›`yi ¯v’ bv¼ 9. log 8 x = 3 1 n‡j x-Gi gvb KZ? 19. GKwU mg‡KvYx wÎf‡z Ri ga¨gvÎq
2. (cos, sin) n‡j x A‡¶i Ici OA 3 h_vµ‡g d, e I f Ges wÎfRz wUi
3. K 4 L 8 M 16 N 32 AwZfzR c n‡j wb‡Pi †KvbwU mwVK?
†iLvs‡ki j¤^ Awf‡¶c wb‡Pi †KvbwU? [
GKwU m~²‡KvY] 10. GKwU mgevû wÎfz‡Ri cwÖ ZwU ga¨gvi ˆ`N©¨ 3 †m. K d2 + e2 + f2 = c2
wg. n‡j cwÖ ZwU evûi ˆ`N¨© KZ cvÖ q †m. wg.? L 4(d2 + e2 + f2) = 5c2
K x A¶ L OA M cos N sin K 2.59 †m. wg. L 3.46 †m. wg. M 2(d2 + e2 + f2) = 3c2
GKwU wÎfz‡Ri f‚wg, wkit‡KvY I N 2(d2 + e2 + f2) = 2c2
Aci `yB evûi mgwó †`Iqv Av‡Q| M 4.24 †m. wg. N 4.5 †m. wg.
11. A = [ 5, 5] I C = [1, 3] hw` R Gi 20. hw` f(x, y, z) GKwU eûc`x Ges f(x,
GB Dcv‡Ëi wfwˇZ KZwU wÎfRz `yBwU e¨ewa nq, Z‡e A  C y, z) = f(y, z, x) n‡j f m¤ú‡K© wb‡Pi
A¼b m¤e¢ ? K [5, 5] L [4, 8] †KvbwU mwVK?
M [1, 3] N [5, 3[ K mggvwÎK L 3-gvwÎK
K GKwU L `yBwU 12. 1. 2. 3I. †K ¸‡YvËi Abš— avivq cKÖ vk M PµµwgK N aª“ec` = 0
M wZbwU N PviwU
mvweK© †mU U-Gi `yBwU Dc‡mU A Ges B 21. wb‡Pi †KvbwU msÁvwqZ?
Ki‡j, mvaviY Abcy vZ KZ nq?
n‡j K 0.1 L 0.01 M 0.001 N 0.0001
13. A I B we›`iy Ae¯v’ b †f±i h_vµ‡g a 2
i. A\B = A  B I b Ges AB †iLvsk C we›`‡y Z m : n K cosec L cot 3

ii. ( A  B) = A  B Abcy v‡Z ewnwe©f³ n‡j, C we›`iy M tan 5 N sec 3
2 2
iii. A  B = A, hLb B  A Ae¯’vb †f±i wb‡Pi †KvbwU?
wb‡Pi †KvbwU mwVK?
K i I ii L ii I iii ma + nb mb + na 22. XvKv †_‡K Lyjbvq ev‡m hvIqvi
m + n m + n m¤¢vebv P(K) Ges Lyjbv †_‡K
M i I iii N i, ii I iii K L ivRkvnx‡Z †U‡ª b hvIqvi m¤¢vebv

4. †h †Kv‡bv eË„ ¯’ PZfz y©‡Ri †¶‡Î M ma  nb N mb  na P(R) n‡j; XvKv †_‡K Lyjbvq ev‡m
c‡Ö hvR¨ wb‡Pi †Kvb Dccv`¨? m  n m  n
14. (1 + x)8 GB eûc`xi we¯—w… Z‡Z Ges Ljy bv †_‡K ivRkvnx‡Z †U‡ª b
K wc_v‡Mviv‡mi Dccv`¨ ga¨c‡`i mnM KZ? hvIqvi m¤¢vebv KZ?
L G¨v‡cv‡jvwbqv‡mi Dccv`¨
M U‡jwgi Dccv`¨ K 88CC64 L 8(C8C54 K P(K) + P(R) L P(K)  P(R)
M N + 8C3)/2
N eªþ¸‡ßi Dccv`¨ 15. x + y  0 AmZvi †jLwPÎ wb‡Pi †KvbwU? M P(K)P(R) N P(K)
P(R)
 DÏxcKwU c‡o 5 I 6 bs c‡Ö kœi DËi K 45 L 45 23. ev¯—e PjK wewkó wb‡Pi †Kvb

`vI : x 5Ges MN mgxKi‡Yi mgvavb cÖwµqvq cvÖ ß
45 g~j¸wji ïw×cix¶v K‡i †`Lv
g1 (x) =  g1Gi wecixZ `iKvi?
3 45
dvskb = g K x2  5x + 6 = 0L 2x+7 = 4x+2
 wb‡Pi Z‡_¨i Av‡jv‡K 16 I 17 bs
c‡Ö kiœ DËi `vI :
5. g(5) Gi gvb KZ? A I B we›`yi ¯’vbv¼ h_vµ‡g (cos, 0)
6. K 0 L 10 M 20 N 30 M 2x + 9  x  4 = x  1
7. Ges (0, sin)|
†Wvg g1 = wb‡Pi †KvbwU? 16. AB †iLvs‡ki ˆ`N¨© KZ GKK? N 5x  15 = 0
8.
KR L R\{0} K 0 GKK L 1 GKK 24. †Kvb ˆ`e cix¶‡Yi bgybv †¶Î S
M R\{3} N R\{5} Ges H cix¶‡Yi GKwU NUbv A
11 1 M sin GKK N cos GKK (†hLv‡b, S A)- n‡j, A-Gi
ax = by = cz Ges abc = 1 n‡j, m¤v¢ ebvi mxgv †`Lv‡bv n‡q‡Q wb‡Pi
wb‡Pi †KvbwU mZ¨? 17.  = 45 n‡j, AB mij‡iLv x- A‡¶i FYvKÍ †Kvb AmgZvq?
w`‡Ki mv‡_ KZ wWMxÖ †KvY Drcbœ K‡i?
K x + y + z = 0 L xy + yz + xz = 0 K 135L 45 M  45 N  45
M xyz = 0 N xyz = 1 K  1  P(A)  1 L  1 < P (A) 
18. A 1
ABC-Gi †¶‡Î
i. C m~²‡KvY n‡j, AB2 > AC2 + M 0  P(A) < 1 N 0 < P(A)  1
CB
BC2 25. GKwU †jvnvi wb‡iU †Mvj‡Ki e¨vmva©
ii. C mg‡KvY n‡j, AB2 =AC2 + BC2 ABC †f±i wÎf‡y Ri †¶‡Î wb‡Pi †KvbwU mwVK? 6 †m. wg.| Gi †jvnv †_‡K 8 †m.wg.
iii. C ¯’~j‡KvY n‡j, AB2 < AC2 + BC2 D”PZv Ges 3 †m. wg. e¨vmv‡a©i KZwU
wb‡Pi †KvbwU mwVK? K AB2 = BC2 + CA2 wbi‡U wmwjÛvi ˆZwi Kiv hv‡e?
L AB + BC + CA = 0
K i L ii M BC + CA > AB K 2wU L 4wU

M iii N i, ii I iii N AB= BC+ CA = 0 M 8wU N 16wU

Self test 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN
10 KLMN 11 KLMN 12 KLMN 13 KLMN 14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN
19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN

-------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------
1 M 2 L 3 N 4 M 5 K 6 K 7 K 8 K 9 N 10 L 11 K 12 M 13 N
DËigvjv 14 L 15 M 16 L 17 K 18 L 19 M 20 M 21 L 22 M 23 M 24 N 25 L

43 G·K¬wz mf g‡Wj †U÷ 09 welq ˆKvW : 1 2 6

mgq : 25 wgwbU D”PZi MwYZ eûwbe©vPwb Afx¶v c~Yg© vb : 25
[we. `.ª : mieivnK…Z eûwbev© Pwb Afx¶vi DËic‡Î c‡Ö kiœ µwgK b¤‡^ ii wecix‡Z c`Ö Ë eY©msewjZ e„Ëmgn~ n‡Z mwVK/m‡ev© rKó… Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤úY~ © fivU Ki|]

1. ƒ(x) = 2xx31, x   1 n‡j, ƒ( 2) 10. 1 +  14 + 1 +  116 + .........  2 †m. wg. e¨vmva© wewkó GKwU avZe
2 2 8 KwVb †MvjK Mwj‡q 2 †m. wg. e¨vmva©
avivwUi AmxgZK mgwó KZ? wewkó GKwU mge„Ëfw‚ g‡K wmwjÛvi
Gi gvb KZ? 1 ˆZwi Kiv n‡jv|
5 5
K  3 L  K1 L 1
2
1 5 Dc‡ii Z‡_¨i Av‡jv‡K 18 I 19 bs
M 5 N 3 M 1 N 1 c‡Ö kiœ DËi `vI :
3 4
2. hw` P(x) = 5x3 + 6x2  2ax  6 †K
3. (x  2) Øviv fvM Ki‡j fvM‡kl 6 11.  = 360 n‡j 18. †Mvj‡Ki c„ôZ‡ji †¶Îdj KZ?
4. nq, Z‡e a Gi gvb KZ?
5. i. cos   6n = 3 K 12.51 eM© †m. wg.
K 5.5 L 7 M 13 N 14.5 2
 L 16.76 eM© †m. wg.
GKwU mgevû wÎfz‡Ri ewnt¯’ ii. cot   n6 = 3
†KvY¸‡jvi †hvMdj KZ? M 33.51 eM© †m. wg.

K 90 L 180 M 270 N 360 iii. tan   4n = 1 N 50.27 eM© †m. wg.

mgevû wÎfz‡Ri evûi ˆ`N¨© 5 †m. wg. wb‡Pi †KvbwU mwVK? 19. wmwjÛv‡ii D”PZv KZ?
n‡j Zvi ga¨gvi ˆ`N©¨ KZ?
K 2.50 †m. wg. L 4.33 †m. wg. K i I ii L i I iii K 0.7 †m. wg. L 2 †m. wg.
M 5 †m. wg. N 8.66 †m. wg.
2 †m. wg. e¨vmva© wewkó e„‡Ëi †K›`­ M ii I iii N i, ii I iii M 2.67 †m. wg. N 8 †m. wg.
n‡Z 5 †m. wg. †Kv‡bv wbw`ó© we›`‡y Z
Aw¼Z ¯úk©‡Ki ˆ`N¨© KZ? 12.  520 †KvYwU †Kvb PZfz v© ‡M Ae¯’vb 20. GKwU wbi‡c¶ Q°v GKevi wb‡¶c
K 3 †m. wg. L 4.6 †m. wg. Ki‡e? Kiv n‡j 5 Gi Kg Ges †gŠwjK
M 21 †m. wg. N 29 †m. wg. K 1g L 2q msL¨v covi m¤v¢ ebv KZ?

wb‡Pi Z‡_¨i Av‡jv‡K 6 I 7 bs M 3q N 4_© K 1 L 1
c‡Ö kœi DËi `vI : 5 3
13. 15 x10 x8
A †KvbwU? x4 Gi mij gvb M 1 N 3
2 4
K x15 Lx
21. GKwU g`y ªv‡K 4 evi wb‡¶c Kiv n‡j
1 bgybv we›`yi msL¨v KZ?
M x15 N1
K4 L8
BDC 14. (1 + 3x)5 Gi x2 Gi mnM KZ n‡e?
M 16 N 32
wP‡Î ABC-G AB = AC = 6 cm K 80 L 90
ADC = 90 Ges BC = 4 cm. 22. hw` n (A) = 5, n (B) = 6 Ges A  B
6. AD Gi ˆ`N¨© KZ? M 170 N 270 =  nq, Z‡e n (A  B) = ?
7.
8. K 2 cm L 2 2 cm 15. A (a, b), B (b, a) Ges C 1a b1 K7 L8

9. M 3 2 cm N 4 2 cm mg‡iL n‡j (a + b) Gi gvb †KvbwU? M 9 N 11

ABC Gi †¶Îdj KZ? K0 L 1 23. x3 + 2x2  5x  6 ivwkwUi Drcv`K
2 wb‡Pi †KvbwU?
K 4 2 cm2 L 6 2 cm2
M1 N 7 K x4 L x1
M 8 2 cm2 N 10 2 cm2 4
16. A (2, 3), B (5, 6), C ( 1, 4) kxlw© e›`y
2px  1 = 2qpx  2 Gi mgvavb †KvbwU? ABC wÎf‡z Ri †¶Îdj KZ? M x+2 N x+3

K p LP K 5 eM© GKK L 6 eM© GKK 24. 4x  1  x2 = 0 mgxKi‡Yi wbðvqK
2 †KvbwU?
M 7 eM© GKK N 12 eM© GKK
M  p N 2 17. B K2 2 L3
2 p M 12 N 17
a (x + b) < c Ges a < 0 n‡j wb‡Pi
†KvbwU mwVK? 25. hw` log 8x = 331 nq, Z‡e x Gi gvb
OA KZ?
wP‡Î OB + BA + AO = KZ?
K x < c  b L x < c + b K  OA L OA
a a

M x > c  b N x > c + b M AO + AO N AO + OA K8 L 16
a a
M 32 N 64

Self test 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN
10 KLMN 11 KLMN 12 KLMN 13 KLMN 14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN
19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN

-------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------
1 N 2 M 3 N 4 K 5 L 6 N 7 M 8 N 9 M 10 M 11 K 12 M 13 L
DËigvjv 14 L 15 K 16 L 17 N 18 N 19 M 20 L 21 M 22 N 23 N 24 M 25 M

44 G·K¬wz mf g‡Wj †U÷ 10 welq ˆKvW : 1 2 6

mgq : 25 wgwbU D”PZi MwYZ eûwbe©vPwb Afx¶v c~Yg© vb : 25
[we. `.ª : mieivnKZ… eûwbe©vPwb Afx¶vi DËic‡Î c‡Ö kœi µwgK b¤‡^ ii wecix‡Z c`Ö Ë eY©msewjZ eË„ mg~n n‡Z mwVK/m‡e©vrK…ó Dˇii e„ËwU ej c‡q›U Kjg Øviv m¤ú~Y© fivU Ki|]

1. sin2  + cos2  = KZ?  wb‡Pi Z‡_¨i Av‡jv‡K 9 I 10 bs 17. †Kvb †f±‡ii wbw`©ó w`K †bB?
6 3 cÖ‡kiœ DËi `vI : K wecixZ †f±i L kb~ ¨ †f±i

K 1 L 1 7, 8 I r †m. wg. e¨vmva© wewkó wZbwU M mgvb †f±i N GKK †f±i
4 2 e„Ë ci¯úi‡K ewnt¯úk© K‡i| Zv‡`i
†K›`ª mgn~ †hvM Ki‡j †h wÎfzR  wb‡Pi Z‡_¨i Av‡jv‡K 18 I 19 bs
M1 N 3 Drcbœ nq Zvi cwimxgv 42 †m. wg.| c‡Ö kiœ DËi `vI :
2 r = KZ †m. wg.? (x  2y)6 GKwU wØc`x ivwk|

( )2. 33 x+3 9. 18. x3y3 Gi mnM KZ?

= 81 K3 L6 K 60 L 160

K3 L3 M8 N9 M  160 N  192
M6 N9
10. wÎf‡z Ri †¶Îdj KZ eM© †m. wg.? 19. Gi cÂg c` KZ?
 wb‡Pi Z‡_¨i Av‡jv‡K 3 I 4 bs K 40 K  12x5y L 60x4y2
L 80 M  192xy5 N 240x2y4
c‡Ö kœi DËi `vI :
M 81 N 96
11. c (x + a) > b Ges c < 0 n‡j wb‡Pi 20. 2 + 4 + 6 + 8 + ........., avivwU
sin  + cos  = P n‡j, †KvbwU mwVK? i. n Zg c` 2n

3. 0 <  < 90 mxgvi g‡a¨ P = 2 K x > b  a L x < b  a ii. n c‡`i mgwó n(2n + 1)
4. n‡j,  Gi gvb KZ? c c iii. c_Ö g 10wU c‡`i mgwó 110
5.
K  L  M x  a < b N x > b + a wb‡Pi †KvbwU mwVK?
4 3 c c K i I ii L i I iii

M  N 2 12. sin 120 = KZ? M ii I iii N i, ii I iii
2 3
1 1 16x5  3x3  27
 =  n‡j, P Gi gvb KZ? K 2 L 2 21. 4x eûc`xi gL~ ¨ mnM
3
3 KZ?
3+1 3 M1 N 2 K4 L x4
K 2 L 2
 wb‡Pi Z‡_¨i Av‡jv‡K 13 I 14 bs  27
3+2 2 c‡Ö kiœ DËi `vI : M 4 N 4x4
2 3
M N ab = ba nq 22. 2 + 1 + 1 + ------ avivwUi 8Zg c`
2 8
y A‡¶i mgvš—ivj †iLvi mgxKiY 13. baba Gi gvb †KvbwU? KZ?
wb‡Pi †KvbwU?
11
K 212 L 213
K y=0 L 2x = y a b 11
M x=2 N x=0 K  1 L  1 M214 N 215
ab aa

6. AvqZvKvi evMv‡bi cwimxgv 24 wg. M a1  a N a1 + a  wb‡Pi Z‡_¨i Av‡jv‡K 23 I 24bs
Ges cÖ¯’ 3 wg. n‡j evMv‡bi †¶Îdj b b cÖ‡kœi DËi `vI :
KZ e. wg.? GKwU †MvjK GKwU NbK AvK…wZ
14. a = 2b n‡j b Gi gvb KZ? ev‡· wVKfv‡e Gu‡U hvq| †Mvj‡Ki
e¨vm 6 †m. wg.
K 12 L 18 K0 L1
M 27 N 36 M2 N3

7. 1  A + B n‡j, 15. 8 x9 x6 = KZ? 23. †Mvj‡Ki c‡„ ôi †¶Îdj KZ eM©. †m. wg.?
(x + 1) (x  3) x+1 x3 K 113.09 L 28.27
x10

A Gi gvb KZ? K1 Lx M 56.55 N 131.18
M x2 24. Nb‡Ki AbwaK…Z As‡ki AvqZb KZ
K 1 L 1 N 3 Nb. †m. wg.?
4 4 2
K 178.30 L 152.38
M4 N1 16. x2 + 4x  3 mgxKi‡Yi g~‡ji cÖK…wZ M 102.96 N 131.18
wb‡Pi †KvbwU?
8. log381 3 = KZ? K Aev¯—e, Amgvb I g~j` 25. f(x) = 2x Gi †Wvg wb‡Pi †KvbwU?
L ev¯—e, Amgvb I Ag~j` x2
K3 L 7
3 K {x :  x R} L {x : x  2}

M4 N 9 M ev¯—e, mgvb I g~j` M {x : x  R, x  2}
2 N ev¯—e, Amgvb I g~j` N {x : x  R, x = 2}

Self test 1 KLMN 2 KLMN 3 KLMN 4 KLMN 5 KLMN 6 KLMN 7 KLMN 8 KLMN 9 KLMN
10 KLMN 11 KLMN 12 KLMN 13 KLMN 14 KLMN 15 KLMN 16 KLMN 17 KLMN 18 KLMN
19 KLMN 20 KLMN 21 KLMN 22 KLMN 23 KLMN 24 KLMN 25 KLMN
------------------------------------------------------------------------ -------------------------------------------------------------------------- -------------------------------
1 L 2 N 3 K 4 K 5 M 6 M 7 K 8 N 9 L 10 M 11 L 12 N 13 K
DËigvjv 14 M 15 M 16 L 17 L 18 M 19 N 20 L 21 K 22 L 23 K 24 M 25 M