evÕ¦e msLÅv I AmgZv 1
cÉ^g AaÅvq
evÕ¦e msLÅv I AmgZv
cixÞvq Kgb ˆcGZ AviI ckÉ ² I mgvavb
ckÉ ² 1 g(y) = y 1 ‰es h(x) = x + 1 x + 2y = 8
K. ‰KwU ˆmU wjL hvi mywcÉgvg ‰i A¯¦fzÆÚ nGjI ev, xy = 1 ............. (iii)
8+4
Bbwdgvg A¯¦fÆzÚ bq| 2 ‰es x + y = 5
L. 3 nGj ˆ`LvI ˆh, |g(y) g(y + 2)| < 39 . 4 ev, x + y = 1 ............. (iv)
|g(y)| < 5 25 5 5
M. g(x) + h(2y 8) 0 ‰es h(x) + g(y) 5 ˆjLwPò AâGbi RbÅ Avbyf„wgK ˆiLv eivei XOX ˆK x-
AmgZvàGjvi mgvavb ˆmGUi ˆjLwPò Aâb KGiv| 4 AÞ ‰es DjÁ¼ ˆiLv eivei YOY ˆK y-AÞ aGi QK
1 bs cGÉ ki² mgvavb KvMGRi cÉwZ 5 eMÆ = 1 ‰KK aGi
K awi, S1 = {x: x ‰es 3 < x 8} A(8, 0), C(0, 4),
‰LvGb, S1 ‰i mywcÉgvg = 8 S1
B(0, 5) ‰es D(5, 0) we±`yàwj Õ©vcb Kwi|
‰es S1 ‰i Bbwdgvg = 3 S1
S1 ˆmUwUi mywcÉgvg ‰i A¯¦fzÆÚ nGjI Bbwdgvg ‰i A¯¦fzÆÚ x + 2y = 8 ˆjLwPGòi ˆh cvGk gƒjwe±`y Zvi wecixZ cvGkÆi¼
bq|
mKj we±`yi RbÅ (i) AmgZvwU mZÅ|
x + y = 5 ˆjLwPGòi ˆh cvGk gƒjwe±`y ˆmB cvGk¼Æi mKj
L ˆ`Iqv AvGQ, g(y) = y 1 we±`yi RbÅ (ii) mZÅ|
kZÆgGZ, |g(y)| < 3 Y
5
ev, |y 1| < 3 ............. (i) B(0, 5)
5
Avevi, |y + 1| = |y 1 + 2| = |(y + 2) 1| = |f(y + 2)|
(i) bs ‰i DfqcGÞ 2 ˆhvM Kwi, C(0, 4)
|y 1| + 2 < 3 +2
5
ev, |y 1 + 2| < 13
5
ev, 13 ............. (ii)
|y + 1| < 5
(i) bs I (ii) bs àY KGi cvB, X
|y + 1| |y 1| < 13 3 X O D(5, 0) A(8, 0)
5 5
ev, |(y + 1) (y 1)| < 39
25
|g(y) g(y + 2)| < 39 (ˆ`LvGbv nGjv) ckÉ ² 2 `k† ÅKÍ-¸ 1: (x) = 2x – 5, g(x) = x + 5
25
M ˆ`Iqv AvGQ, h(x) = x + 1 `k† ÅKÍ-¸ 2: x y
Q R
g(y) = y 1 P = x + y, = = 1; x, y
‰Lb, g(x) + h(2y 8) 0 K. 1 < 2x 3 < 5 AmgZvwUGK ciggvb wPGn×i mvnvGhÅ
ev, x 1 + 2y 8 + 1 0
ev, x + 2y 8 0 ............. (i) cKÉ vk KGiv| 2
‰es h(x) + g(y) 5
ev, x + 1 + y 1 5 L. |g(x)| |(x)| AmgZvwU mgvavb Ki ‰es mgvavb ˆmU
ev, x + y 5 ............. (ii)
c`É î AmgZvàwjGK mgZv aGi ˆjLwPò Aâb Kwi ‰es evÕ¦e msLÅvGiLvq ˆ`LvI| 4
M. cgÉ vY KGiv ˆh, |P| |Q| + |R|. 4
2 bs cÉGki² mgvavb
mgvavGbi mÁ¿veÅ ‰jvKv wbYÆq Kwi| K 1 < 2x 3 < 5
Avgiv cvB, ev, 1 + 3 < 2x < 5 + 3
ev, 2 < 2x < 8
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò ‰Kv`k-«¼v`k ˆkwÉ Y
2+8 x0
2 =5
[ ]ev, 2 5 < 2x 5 < 8 5 wbGYÆq mgvavb : x 10 A^ev x 0
msLÅvGiLvq mgvavb ˆmU :
ev, 3 < 2x 5 < 3
|2x 5| < 3 (Ans.)
L |g(x)| |(x)| 3 2 1 0 1 2 3 4 5 6 7 8 9 10
ev, |x + 5| |2x 5|
ev, (x + 5)2 (2x 5)2 M ˆ`Iqv AvGQ, P = x + y
ev, x2 + 10x + 25 4x2 20x + 25
ev, 3x2 + 30x 0 xy
ev, x2 + 10x 0 Q =R =1
ev, x2 10x 0; [ 1 «¼viv àY KGi]
ev, x(x 10) 0 ............. (i) ev, x =1 ‰es y =1
(i) bs AmgZv mZÅ nGe hw`I ˆKej hw` x ‰es (x 10) ‰i Q R
`By wUB abvñK A^ev `yBwUB FYvñK nq A^ev ˆhGKvGbv ‰KwUi
Q=x R=y
‰Lb, (|x| + |y|)2 = |x|2 + 2|x| |y| + |y|2
= x2 + 2|xy| + y2 [ |x|2 = x2, |y|2 = y2, |x| |y| = |xy|]
ev, (|x| + |y|)2 x2 + 2xy + y2 [ |xy| xy]
gvb 0 nq| ev, (|x| + |y|)2 (x + y)2
‰Lb, abvñK gvGbi RbÅ,
ev, (|x| + |y|)2 (|x + y|)2
x 0 ‰es (x 10) 0 A^Ævr, x 10
ev, |x + y|2 (|x| + |y|)2
x 10 ev, |x + y| |x| + |y|
Avevi, FYvñK gvGbi RbÅ, |P| |Q| + |R| (cÉgvwYZ)
x 0 ‰es (x 10) 0 A^Ævr x 10
DËi ms‡KZmn m„Rbkxj cÖkœ
ckÉ ² 3 f(a) = 3a + 1. M. `k† ÅKÍ-¸ 2 ‰i AmgZv«¼Gqi hyMcr ˆjLwPò Aâb Ki|4
A K. D`vniYmn evÕ¦e msLÅvi ciggvGbi msæv `vI|
2 Dîi: K. ( , 2]
( )L. 1 5, a 1 ˆK ciggvb wPn× eÅwZZ cÉKvk cÉk²5 f(a) = a 1.
f(a) 3
| | A K. | f(x) | 3 ˆK ciggvb wPn× eÅwZZ cKÉ vk Ki| 2
Ki| 4 L. mgvavb Ki: | f(x) | = | 3f(x) 1 | 4
M. a = 2 nGj, ˆ`LvI ˆh, | f(a) | ‰KwU Agƒj` msLÅv| 4 M. {2f(x) 1} {f(x)}2 < 0 ‰i mgvavb Ki| 4
f(x) + 2
Dîi: L. 2 a 4 ; a 1
5 15 3 Dîi: K. L. 35
2 x 4; 2,4
cÉk²4 `†kÅKÍ-¸ 1: x, y, z , xz = yz ‰es z 0.
`k† ÅKÍ-¸ 2: 2x 3y > 1 ‰es 2x + 3y 7. M. 1 < x < 3 x 1; mKj x ‰i RbÅ|
2,
B K. x + 1 0 AmgZvi mgvavb Ki| 2
2 4
L. `k† ÅKÍ-¸ 1 eÅenvi KGi cÉgvY Ki ˆh, x = y
cixÞvq Kgb ˆcGZ AviI cÉk² I mgvavb
cÉk² 1 g(y) = y 1 ‰es h(x) = x + 1wkLbdj-4, 5 I 10 cÉ`î AmgZvàwjGK mgZv aGi ˆjLwPò Aâb Kwi ‰es
K. ‰KwU ˆmU wjL hvi mywcÉgvg ‰i A¯¦fzÆÚ nGjI mgvavGbi mÁ¿veÅ ‰jvKv wbYÆq Kwi|
Bbwdgvg A¯¦fÆzÚ bq| 2 Avgiv cvB,
L. 3 nGj ˆ`LvI ˆh, |g(y) g(y + 2)| < 39 . 4 x + 2y = 8
|g(y)| < 5 25
ev, xy = 1 ............. (iii)
M. g(x) + h(2y 8) 0 ‰es h(x) + g(y) 5 8+4
AmgZvàGjvi mgvavb ˆmGUi ˆjLwPò Aâb KGiv|4 ‰es x + y = 5
1 bs cÉGk²i mgvavb ev, x y = 1 ............. (iv)
5 +5
K awi, S1 = {x: x ‰es 3 < x 8}
‰LvGb, S1 ‰i mywcÉgvg = 8 S1 ˆjLwPò AâGbi RbÅ Avbfy „wgK ˆiLv eivei XOX ˆK
‰es S1 ‰i Bbwdgvg = 3 S1 x-AÞ ‰es DjÁ¼ ˆiLv eivei YOY ˆK y-AÞ aGi QK
S1 ˆmUwUi mywcÉgvg ‰i A¯¦fzÆÚ nGjI Bbwdgvg ‰i A¯¦fzÆÚ
bq| KvMGRi cÉwZ 5 eMÆ = 1 ‰KK aGi A(8, 0), C(0, 4),
B(0, 5) ‰es D(5, 0) we±`yàwj Õ©vcb Kwi|
L ˆ`Iqv AvGQ, g(y) = y 1 x + 2y = 8 ˆjLwPGòi ˆh cvGk gƒjwe±`y Zvi wecixZ
kZÆgGZ, 3 cvGki¼Æ mKj we±`yi RbÅ (i) AmgZvwU mZÅ|
5
|g(y)| < x + y = 5 ˆjLwPGòi ˆh cvGk gƒjwe±`y ˆmB cvGki¼Æ mKj
ev, |y 1| < 3 ............. (i) we±`yi RbÅ (ii) mZÅ|
5
Avevi, |y + 1| = |y 1 + 2| = |(y + 2) 1| = |g(y + 2)| Y
(i) bs ‰i DfqcGÞ 2 ˆhvM Kwi,
|y 1| + 2 < 3 +2 B(0, 5)
5
ev, |y 1 + 2| < 13 C(0, 4)
5
ev, 13 ............. (ii)
|y + 1| < 5
(i) bs I (ii) bs àY KGi cvB,
|y + 1| |y 13 3
1| < 5 5
ev, 39 X
25
|(y + 1) (y 1)| < X O D(5, 0) A(8, 0)
|g(y) g(y + 2)| < 39 (ˆ`LvGbv nGjv)
25
M ˆ`Iqv AvGQ, h(x) = x + 1
g(y) = y 1 cÉk²2 `†kÅK͸-1: (x) = 2x – 5, g(x) = x + 5
‰Lb, g(x) + h(2y 8) 0 `†kÅK͸-2: P = x + y, x = y = 1; x, y wkLbdj-4, 5 I 6
ev, x 1 + 2y 8 + 1 0 Q R
ev, x + 2y 8 0 ............. (i)
‰es h(x) + g(y) 5 K. 1 < 2x 3 < 5 AmgZvwUGK ciggvb wPGn×i
ev, x + 1 + y 1 5
ev, x + y 5 ............. (ii) mvnvGhÅ cÉKvk KGiv| 2
L. |g(x)| |(x)| AmgZvwU mgvavb Ki ‰es mgvavb
ˆmU evÕ¦e msLÅvGiLvq ˆ`LvI| 4
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò
M. cÉgvY KGiv ˆh, |P| |Q| + |R|. 4 Avevi, FYvñK gvGbi RbÅ,
2 bs cÉGki² mgvavb x 0 ‰es (x 10) 0 A^Ævr x 10
K 1 < 2x 3 < 5 x0
ev, 1 + 3 < 2x < 5 + 3
ev, 2 < 2x < 8 wbGYÆq mgvavb : x 10 A^ev x 0
msLÅvGiLvq mgvavb ˆmU :
ev, 2 5 < 2x 5 < 8 5 2 + 8 = 5
2
3 2 1 0 1 2 3 4 5 6 7 8 9 10
ev, 3 < 2x 5 < 3 M ˆ`Iqv AvGQ, P = x + y
|2x 5| < 3 (Ans.) xy
Q =R =1
L |g(x)| |(x)|
ev, |x + 5| |2x 5| ev, x =1 ‰es y =1
ev, (x + 5)2 (2x 5)2 Q R
ev, x2 + 10x + 25 4x2 20x + 25
ev, 3x2 + 30x 0 Q=x R=y
ev, x2 + 10x 0
ev, x2 10x 0; [ 1 «¼viv àY KGi] ‰Lb, (|x| + |y|)2 = |x|2 + 2|x| |y| + |y|2
ev, x(x 10) 0 ............. (i)
(i) bs AmgZv mZÅ nGe hw`I ˆKej hw` x ‰es (x 10) = x2 + 2|xy| + y2
‰i `yBwUB abvñK A^ev `yBwUB FYvñK nq A^ev [ |x|2 = x2, |y|2 = y2, |x| |y| = |xy|]
ˆhGKvGbv ‰KwUi gvb 0 nq| ev, (|x| + |y|)2 x2 + 2xy + y2 [ |xy| xy]
‰Lb, abvñK gvGbi RbÅ, ev, (|x| + |y|)2 (x + y)2
ev, (|x| + |y|)2 (|x + y|)2
ev, |x + y|2 (|x| + |y|)2
ev, |x + y| |x| + |y|
x 0 ‰es (x 10) 0 A^Ævr, x 10 |P| |Q| + |R| (cÉgvwYZ)
x 10
DËi ms‡KZmn m„Rbkxj cÖkœ
cÉk² 3 a, b, c ‰es P = 2.090909 ... L. – 1 < 2f(x) – 5 < 5 AmgZvwUGK ciggvGbi wPGn×i
I Q = 3.274274274 .. wkLbdj-3, 4 mvnvGhÅ cÉKvk Ki| 4
A K. a < b ‰es a, b > 0 nGj ˆ`LvI ˆh 1 > 1 2 M. |g(x)| < 1 nGj ˆ`LvI ˆh, |x2 – 1| < 19 4
a b 9 81
L. cÉgvY Ki ˆh, |a + b| |a| + |b| 4 Dîi: L. |2x 5| < 3
M. ˆ`LvI ˆh P I Q gƒj` msLÅv| 4 cÉk²6 g(x) = 1 ‰es (x) = 3x x2 + 4
1 5x
cÉk²4 A = {x : 16 x2 100} ‰es wkLbdj-1, 6, 7
B = {x : x 0 ‰es x2 5x + 6 < 0} wkLbdj-5, 6 A K. gƒj` I Agƒj` msLÅvi msæv `vI| 2
K. 12 2 3 ...ˆmUwUi Bbwdgvg wbYÆq Ki? 2 L. g(x) ‰i ciggvb 3 ‰i eo bv nGj mgvavb ˆmU
5 10
msLÅvGiLvq ˆ`LvI| 4
L. Sup A ‰es InfA wbYÆq Ki| 4 M. (x) < 0 AmgZvGK ciggvb wPGn×i mvnvGhÅ cÉKvk
M. ˆjLwPGòi mvnvGhÅ B ˆmUwUi mgvavb wbYÆq Ki| 4 KGiv| 4
Dîi: K. 0; L. 10, 4; M. {x : x , 2 < x < 3} Dîi: L.
02 4
cÉk² 5 (x) = x + 1, g(x) = x – 1 wkLbdj-3, 7 M. x 3 5 15 15
K. p, q nGj |p + q| |p| + |q| A 2 2
2 >
ˆhvMvkÉqx ˆcvÉ MÉvg 1
w«¼Zxq AaÅvq
ˆhvMvkÉqx ˆcvÉ MÉvg
cixÞvq Kgb ˆcGZ AviI ckÉ ² I mgvavb
ckÉ ² 1 (i) bovBj miKvwi wfGÙvwiqv KGjGRi AaÅÞ cÉGdmi ˆjLwPò nGZ ˆ`Lv hvq (i)bs ˆiLvi ˆhcvGk gƒjwe±`y ˆmB
mvgv`DÍÏvn gRyg`vi kxZKvGj KGjR KÅvÁ·vGm wkÞKG`i cvGki mKj we±`y, (ii) bs I (iii) bs ˆiLvi ˆhcvGk gƒj we±`y
eÅvWwg´Ÿb ˆLjvi RbÅ mGeÆvœP 4800 UvKv eÅGq wKQy eÅvU I KKÆ AvGQ Zvi wecixZ cvGki mKj we±`yi RbÅ|
wKbGZ ejGjb| cwÉ ZwU eÅvGUi `vg 400 UvKv ‰es KGKÆi `vg 100 400x + 100y 4800, x 4 ‰es y 12 mZÅ|
UvKv| KgcGÞ 4 Lvbv eÅvW I 12 wU KKÆ wZwb wKbGZ Pvb| ˆjLwPò nGZ cvB AmgZvàGjvi mÁ¿veÅ mgvavb ‰jvKv CAB|
(ii) (x) = 2x + 1; g(x) = x – 2 ‰KGò meÆvwaK wRwbm wKbGZ cvivi kGZÆ AfxÓ¡ dvskb
K. ˆhvMvkÉqx ˆcvÉ MÉvg ejGZ wK eyS? 2 zmax = (x + y)
L. wZwb ˆKvb cÉKvGii KZàwj wRwbm wKbGZ cviGeb? 4
M. (x)g(x) 3 nGj, mgvavb KGiv ‰es msLÅvGiLvq ˆ`LvI|4 mÁ¿veÅ mgvavb ‰jvKvi ˆKŒwYK we±`yàwj h^vKGÌ g,
1 bs cÉGki² mgvavb
A(9, 12), C(4, 12), B(4, 32)
K ˆhvMvkÉqx ˆcvÉ MÉvg: meÆwbÁ² wewbGqvGMi wewbgGq mÁ¿veÅ mGeÆvœP
gybvdv ARÆGbi jGÞÅ ˆKvGbv cwiK͸bvGK (i) DGókÅ dvskb ‰Lb, A(9, 12) we±`yGZ z = 9 + 12 = 21
(ii) wm«¬v¯¦ PjK I (iii) kZÆ ev mxgve«¬Zv ‰B wZbwU Z^ÅGK
KÅvbGUvGivwfGPi wbqGg MvwYwZK gGWGj i…c`vb KiGj ˆh C(4, 12) ” z = 4 + 12 = 16
mgvavb ˆhvMÅ MvwYwZK mgmÅv cvIqv hvq ZvGK ˆhvMvkÉqx B(4, 32) ” z = 4 + 32 = 36
ˆcvÉ MÉvg ejv nq|
eÅvU KÌq KiGeb 4wU, KKÆ KÌq KiGeb 32wU (Ans.)
M ˆ`Iqv AvGQ, (x) = 2x + 1 ‰es g(x) = x – 2
L gGb Kwi, wZwb xwU eÅvU ‰es ywU KKÆ KÌq KiGeb| (x) g(x) 3
c`É î kZÆvbymvGi AmgZvàGjv nGjv: 400x + 100y 4800
ev, (2x + 1) (x – 2) 3 ev, 2x2 – 4x + x – 2 – 3 0
x ev, 2x2 – 3x – 5 0 ev, 2x2 – 5x + 2x – 5 0
ev, x(2x – 5) + 1(2x – 5) 0 ev, (2x – 5) (x + 1) 0
y
AmgZvàGjvGK mgZv aGi mgxKiYàGjvi ˆjLwPò Aâb Kwi ev, 2x – 5 (x + 1) 0
‰es mgvavGbi mÁ¿veÅ AbyK„j ‰jvKv ˆei Kwi| 2
AZ‰e, Avgiv cvB, 400x + 100y = 4800
ev, 4x + y = 48 ev, x – 52(x + 1) 0 ... ... (1)
ev, x + y = 1 ... ... (i) (1) bs AmgZvwU mZÅ nGe hw` ‰es ˆKej hw` (x + 1) I
12 48
x = 4 ... ... ... (ii) x – 5 ‰i ‰KwU abvñK I AciwU FYvñK nq|
2
y = 12 ... ... ... (iii)
kZÆ (x + 1) ‰i x – 5 ‰i (x + 1) x – 5
‰Lb, x-AÞ I y AÞ eivei Þz`ËZg eGMiÆ cÉwZ 1 evüi wPn× 2 2
Š`NÆÅGK 2 ‰KK aGi (i), (ii), I (iii) bs mgxKiGYi ˆjLwPò
Aâb Kwi| wPn× ‰i wPn×
(0, 48) x<–1 – – +
– 1 x 5 + – –
2
B(4, 32) 5 + + +
x> 2
myZivs (i) bs mZÅ nGe hw` ‰es ˆKej hw` – 1 x 5 nq|
2
wbGYÆq mgvavb: – 1 x 5
2
(iii) msLÅvGiLv: 3 2 1 0 1 2 5 3 (Ans.)
2
A(9, 12)
(0, 12) C(4, 12) ckÉ ² 2 `k† ÅKÍ-¸ 1: ƒ(x) = x 3, x .
`†kÅK͸-2: ‰KRb eÅemvqx Zvi ˆ`vKvGbi RbÅ mvµv¦ wnK mGeÆvœP 200wU
X O (4, 0) X ˆiwWI I ˆUwjwfkb ˆmU ˆKbvGePv KiGZ cvGib| cÉwZwU ˆiwWI I
(i) ˆUwjwfkb ˆmGUi gƒjÅ h^vKÌGg 80 I 240 Wjvi ‰es jvf h^vKÌGg 40
I 60 Wjvi|
(ii)
Y
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò ‰Kv`k-«¼v`k ˆkwÉ Y
K. 1 ‰i RwUj Nbgƒj wbYÆq Ki| 2 AmgZvàGjvGK mgZv aGi mgxKiYàGjvi ˆjLwPò AsKb Kwi|
L. 0 < ƒ(x) < 4 mgvavb ˆmU wbYÆq Ki ‰es mgvavb ˆmU x + y = 200 x y = 1 ............. (i)
200 + 200
msLÅvGiLvq ˆ`LvI| 4
‰es 80x + 240y = 20800 x + 3y = 260
M. H eÅemvqx hZàGjv ˆmU KÌq KGib Zvi meàGjv wewKÌ nGq
xy
hvq| eÅemvqxi ˆgvU wewbGqvM 20800 Wjvi nGj KZwU ˆiwWI 260 + 260 = 1 ............. (ii)
I ˆUwjwfkb ˆmU KÌq KiGj Zvi jvf mevÆ waK nGe| 4 3
2 bs cÉGki² mgvavb x = 0 ............. (iii)
y = 0 ............. (iv)
K gGb Kwi, 3 1 = x ZvnGj, x3 = 1 Y ˆÕ„j: x I y AÞ eivei 1 Ni = 10 ‰KK
ev, x3 1 = 0 B(0, 200)
ev, (x 1) (x2 + x + 1) = 0
x 1 = 0 A^ev x2 + x + 1 = 0 ( )D0260
‰Lb, x 1 = 0 nGj, x = 1 3
Avevi, x2 + x + 1 = 0 nGj,
x = 1 1 4 = 1 ( 1 i 3)
2
2 P(170, 30)
A(200, 0)
myZivs, ‰KGKi Nbgƒjàwj 1, 1 ( 1 + i 3) C(260, 0)
2
O (0, 0) X
‰es 1 ( 1 i 3) (Ans.)
2
cwÉ ZwU ˆiwWI I ˆUwjwfkGb jvf nq h^vKGÌ g 40 Wjvi I 60
L ˆ`Iqv AvGQ, Wjvi|
ƒ(x) = x 3, x mGeÆvœP jvf z = 40x + 60y
0 < ƒ(x) < 4 mÁ¿veÅ mgvavb ‰jvKvi ˆKŒwYK we±`yàwj h^vKGÌ g O(0, 0),
0<x3<4 ( )A(200, 0), P(170, 30) ‰es D 0 260
3
ev, 0 + 3 < x 3 + 3 < 4 + 3 [cGÉ ZÅK cGÞ 3 ˆhvM KGi]
ev, 3 < x < 7 ‰Lb O(0, 0) we±`yGZ z = 0
mgvavb ˆmU, S = {x : 3 < x < 7} A(200, 0) we±`yGZ z = 40 200 + 60 0 = 8000
mgvavb ˆmU wbGÁ² msLÅvGiLvq ˆ`LvGbv nGjv : P(170, 30) we±`yGZ z = 40 170 + 60 30
0 1 2 3 4 56 7 = 6800 + 1800 = 8600
M gGb Kwi, ˆiwWI KÌq KGib xwU ‰es ˆUwjwfkb ˆmU KÌq ( )D 0 260 we±`GZ z= 40 0 + 60 260 = 5200
KGib ywU 3 3
c`É î kGZÆi AvGjvGK AmgZvàwj wbÁ²i…c :
Õ·Ó¡Zt P(170, 30) we±`yGZ z ‰i mGeÆvœP gvb we`Ågvb|
x + y 200 A^Ævr zmax = 8600
80x + 240y 20800
x 0, y 0 170wU ˆiwWI ˆmU ‰es 30wU ˆUwjwfkb ˆmU KÌq KiGj mGeÆvœP
jvf KiGZ cviGeb|
DËi ms‡KZmn m„Rbkxj cÖkœ
ckÉ ² 3 ‰K MvGgÆ´Ÿm dÅvÙixGZ Š`wbK KgcGÞ 200 wU kvUÆ Dîi: K. zmax = 20x + 50y; L. 21150 UvKv; M. 63wU|
‰es mGeÆvœP 315 wU cÅv´Ÿ ŠZwi nq| cÉwZwU kvUÆ ŠZwiGZ mGeÆvœP 50 ckÉ ² 4 ‰KwU cvbxq ŠZwii KviLvbvq `By wU kvLv I ‰es II ‰i
DfqB A, B ‰es C wZb cKÉ vGii cvbxq ˆevZjRvZ KGi| kvLv
wgwbU ‰es cÉwZwU cÅv´Ÿ ŠZwiGZ mGeÆvœP 100 wgwbU mgq jvGM| `By wUi Š`wbK Drcv`b ÞgZv wbÁ²i…c t
dÅvÙixGZ ˆh cwigvY kwÉ gK I h¯¨cvwZ AvGQ Zv w`Gq MGo Š`wbK kvLv A cKÉ vGii B cKÉ vGii C cKÉ vGii
cvbxq cvbxq cvbxq
mGeÆvœP 45,000 wgwbU mgq kvUÆ I cÅv´Ÿ ŠZwiGZ eÅenvi Kiv mÁ¿e|
cwÉ ZwU kvGUÆ 20 UvKv ‰es cÉwZwU cÅvG´Ÿ 50 UvKv jvf nq|
A K. kvGUÆi msLÅv x ‰es cÅvG´Ÿi msLÅv y aGi AmgZv ‰es I 3,000 1,000 2,000
AfxÓ¡ dvskbwU wbYÆq KGiv| 2 II 1,000 1,000 6,000
L. Š`wbK KZàGjv kvUÆ I cÅv´Ÿ ŠZwi KiGj mGeÆvœP jvf nGe? 4 A, B I C cÉKvGii cvbxGqi gvwmK Pvwn`v h^vKGÌ g 24,000, 16,000
M. cÉwZwU cÅv´Ÿ ŠZwiGZ hw` 10 wgwbU mgq Kg jvGM, ZvnGj ‰es 48,000 ˆevZj| I I II kvLvi Š`wbK KvhÆ cwiPvjbvi eÅq
mGeÆvœP jvGfi kGZÆ KZwU kvUÆ ˆewk evbvGZ nGe? 4 h^vKGÌ g 600 UvKv I 400 UvKv|
ˆhvMvkÉqx ˆcvÉ MÉvg 3
A K. “ˆhvMvkÉqx ˆcvÉ MÉvGgi gƒj jÞÅ bƒÅbZg wewbGqvGMi `Þ A K. AmgZvàGjv wbYÆq KGiv| 2
eÅenvi I mGeÆvœP gybvdv ARÆb” @ eÅvLÅv Ki| 2 L. AmgZvàGjvGK QK KvMGR Õ©vcb KGi mgvavb Ask
L. ˆKvb kvLv KZw`b ˆLvjv ivLv nGe ‰B mgmÅvwUi wPwn×Z KGiv| 4
mgvavGbi RbÅ ‰KwU cƒYÆvã ˆhvMvkÉqx ˆcÉvMÉvg MVb Ki|4 M. mgvavb AçGji ˆKŒwYK we±`yàGjv (0, 0), (20, 80), (100, 0)
M. KviLvbvwUi gvwmK mGeÆvœP eÅq 25600 UvKv nGj cvÉ µ¦ ( )‰es0260 nGj, mÁv¿ eÅ mGeÆvœP jvf wbYÆq KGiv| 4
3
AmgZvwUGK QK KvMGR Õ©vcb KGi mgvavb Açj
Dîi: K. x + y 100, 40x + 120y 10400, x 0 ‰es y 0;
wPwn×Z Ki| 4
M. 2880 Wjvi
cÉk²5 ‰KRb eÅemvqx 10400 Wjvi wewbGqvM KGi Zvi ˆ`vKvGbi
RbÅ ˆiwWI I ˆUwjwfkb wgGj 100 ˆmU wKbGZ cvGib| ˆiwWI ˆmU I
ˆUwjwfkb ˆmU cÉwZwUi KÌqgƒjÅ h^vKÌGg 40 Wjvi I 120 Wjvi| cÉwZ
ˆiwWI I ˆUwjwfkb ˆmGU jvf h^vKÌGg 16 Wjvi I 32 Wjvi|
1
cixÞvq Kgb ˆcGZ AviI cÉk² I mgvavb
cÉk²1 `†kÅKÍ-¸ 1: x + y 7, 2x + 5y 20, x, y 0 cÉ`î AmgZvàGjvGK mgZv aGi cÉvµ¦ mgxKiYàGjvi
`†kÅK͸-2: (x) = x – 1. A 1 I 2 AaÅvGqi mg®¼Gq ˆjLwPò Aâb Kwi ‰es mgvavGbi mÁ¿veÅ AbKy „j ‰jvKv
K. a, b nGj, cÉgvY KGiv ˆh, |a – b| |a| + |b| 2 wbYÆq Kwi|
L. `†kÅKÍ-¸ 2 ˆ^GK 1 AmgZvwUi mgvavb AZ‰e Avgiv cvB,
|3(x) – 2|
> 2 x+y=7
ˆmU msLÅv ˆiLvq ˆ`LvI| 4 ev, xy
7 + 7 = 1 ... ... (1)
M. `†kÅKÍ-¸ 1 Abmy vGi z = 4x + 5y ‰i mGeÆvœP gvb wbYÆq
2x + 5y = 20
KGiv| 4
ev, 2x 5y
1 bs cÉGk²i mgvavb 20 + 20 = 1
K Avgiv Rvwb, | a + b | | a | + | b | ev, xy
DcGivÚ mÁ·GKÆ b ‰i cwieGZÆ b emvGj cvB, 10 + 4 = 1 ... ... (2)
| a + (b) | | a | + | (b) | x = 0 ................ (3)
| a b | | a | + | b | (cÉgvwYZ) [ | (b) | = | b | ] y = 0 ................ (4)
L `†kÅKÍ-¸ 1 ˆ^GK cvB, (x) = x – 1 Y ˆÕ•j: x I y AÞ eivei Þz`Ë eGMÆi
5 evü = 2 ‰KK
1 (0, 7)
|3(x) – 2| > 2
ev, |3(x 1 – 2| > 2
– 1)
ev, 1 > 2 C(0, 4)
–3
|3x – 2| B(5, 2)
ev, 1 5| > 2
|3x –
ev, |3x – 5| < 1 ‰es 3x 5 0 X O A(7, 0) (10, 0) X
2 Y
ev, – 1 < 3x – 5 < 1 ev, x 5
2 2 3
ev, – 1 + 5 < 3x – 5 + 5 < 1 + 5 ˆjLwPò nGZ ˆ`Lv hvq ˆh, mgxKiY (1), (2) ‰i mKj
2 2 we±`y ‰es ‰G`i ˆhcvGk gƒjwe±`y AewÕ©Z ˆmB cvGki
mKj we±`yi RbÅ cÉ`î AmgZvàGjv mZÅ| wPòvbmy vGi,
ev, 9 < 3x < 11 A, B, C h^vKÌGg, (1) I (4), (1) I (2) ‰es (2) I (3)
2 2 ‰i ˆQ` we±`y|
ZvnGj mÁv¿ eÅ mgvavb ‰jvKv nGœQ OABCO hv wPGò
ev, 9 < x < 11 Qvqv ˆNiv ‰jvKv wnmvGe wPwn×Z Kiv AvGQ ‰es mÁ¿veÅ
6 6 mgvavb ‰jvKvi ˆKŒwYK ev cÉvw¯¦K we±`yMGjv h^vKÌGg
3 < x < 11 0(0, 0), A(7, 0), B(5, 2), C(0, 4)
2 6
‰Lb, O(0, 0) we±`yGZ, Z = 4 0 + 5 0 = 0
mgvavb ˆmU, S = x : 3 < x < 11 ‰es x 53 A(7, 0) we±`yGZ, Z = 4 7 + 5 0 = 28
2 6 B(5, 2) we±`yGZ, Z = 4 5 + 5 2 = 30
C(0, 4) we±`yGZ, Z = 4 0 + 5 4 = 20
msLÅv ˆiLv:
3 5 11
236
M ˆ`Iqv AvGQ, AfxÓ¡ dvskb Z = 4x + 5y
‰es mxgve«¬Zvi kZÆmgƒn: x + y 7, 2x +5y 20, x, y 0
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò
Õ·Ó¡Z B(5, 2) we±`yGZ Z ‰i mGeÆvœPgvb cvIqv hvq| AZ‰e Avgiv cvB, x + 2y = 10
Zmax = 30 (Ans.) xy (i)
10 + 5 = 1
cÉk² 2 `†kÅK͸ 1: (x) = |x 2|
‰es x + y = 1 (ii)
`†kÅK͸ 2: z = 2x + 3y kZÆ x + 2y 10, x + y 6, x 4; 6 6
x, y 0 1 I 2 AaÅvGqi mg®¼Gq x=4 (iii)
K. 7 < x < 1 ˆK ciggvb wPGn×i mvnvGhÅ cÉKvk Ki| 2 x=0 (iv)
y=0 (v)
L. (x) < 1 nGj cÉgvY Ki ˆh, (x) (x + 4) < 37 4 Y (iii) ˆÕ•j: x I y AÞ eivei Þz`ËZg eGMÆi cÉwZ
9 81 2 evüi Š`NÆÅ mgvb 1 ‰KK
M. `†kÅK͸ 2 ‰i AvGjvGK ˆjLwPGòi mvnvGhÅ z ‰i mGeÆvœP (ii)
gvb wbYÆq Ki| 4
2 bs cÉGki² mgvavb (i) D C
K 7 < x < 1 B
ev, 7 + 4 < x + 4 < 1 + 4 [4 ˆhvM KGi]
ev, 3 < x + 4 < 3 X O A X
(i)
|x + 4 | < 3 (Ans.) Y (iii) (ii)
L ˆ`Iqv AvGQ, ˆjLwPGò ˆ`Lv hvq ˆh, mgxKiY (i), (ii) I (iii) ‰i
mKj we±`y ‰es ‰G`i ˆh cvGk gƒj we±`y AewÕ©Z ˆmB
(x) = |x 2| cvGki mKj we±`yi RbÅ cÉ`î AmgZvàGjv mZÅ| ˆhLvGb
O(0, 0) nGœQ gƒj we±`y|
(x + 4) = |x + 2| wPòvbmy vGi, A, B, C I D h^vKÌGg (iii) I (v); (ii) I
(x) (x + 4) = |x 2| |x + 2| = |x2 4| (iii); (i) I (ii) ‰es (i) I (iv) ‰i ˆQ` we±`y|
ZvnGj, mÁ¿veÅ mgvavb ‰jvKv nGœQ OABCDO hv wPGò
kZÆvbmy vGi, (x) < 1 Qvqv ˆNiv ‰jvKv wnmvGe wPwn×Z Kiv AvGQ ‰es mÁ¿veÅ
9 mgvavb ‰jvKvi ˆKŒwYK ev cÉvw¯¦K we±`yàGjv h^vKÌGg-
O(0, 0), A(4, 0), B(4, 2), C(2, 4) ‰es D(0, 5)
ev, |x 2| < 1 ‰Lb O(0, 0) we±`yGZ F = 2 0 + 3 0 = 0
9 A(4, 0) we±`yGZ Z = 2 4 + 3 0 = 8
B(4, 2) we±`yGZ Z = 2 4 + 3 2 = 14
ev, 1 < x 2 < 1 C(2, 4) we±`yGZ Z = 2 2 + 3 4 = 16
9 9 D(0, 5) we±`yGZ Z = 2 0 + 3 5 = 15
Õ·Ó¡Z: C(2, 4) we±`yGZ Z ‰i mGeÆvœPgvb cvIqv hvq|
ev, 1 + 2 < x < 1 + 2 AZ‰e mGeÆvœP gvGbi we±`ywU C(2, 4) ‰es
9 9 mGeÆvœP gvb Zmax = 16 (Ans.)
ev, 1 + 18 1 + 18
9 <x< 9
ev, 17 19
9 <x< 9
ev, 289 < x2 < 361 [eMÆ KGi]
81 81
ev, 289 4 < x2 4 < 361 4
81 81
ev, 35 < x2 4 < 37
81 81
ev, 37 < x2 4 < 37 [ 37 < 35
81 81 81 81]
cÉk² 3 `†kÅK͸-1: 2x + 3 x+3
|x2 4| < 37 x3 <x1
81
`†kÅK͸-2: ‰K eÅwÚ 500 UvKvi gGaÅ KgcGÞ 6wU ˆcqviv I
(x) (x 37 (cÉgvwYZ)
. + 4) < 81 4wU Avg wKbGZ Pvb| cÉwZwU ˆcqvivi `vg 30 UvKv I cÉwZwU
M ˆ`Iqv AvGQ, AfxÓ¡ dvskb Z = 2x + 3y AvGgi `vg 40 UvKv| 1 I 2 AaÅvGqi mg®¼Gq
‰es mxgve«¬Zvi kZÆmgƒn: x + 2y 10, x + y 6,
K. S = {x : x2 3x + 2 0} nGj S ˆmU ‰i mywcÉgvg
x 4, x, y 0 I Bbwdgvg wbYÆGq ˆZvgvi wm«¬v¯¦ Kx? 2
cÉ`î AmgZvàGjvGK mgZv aGi cÉvµ¦ mgxKiYàGjvi L. `†kÅK͸-1 ‰i AmgZvwU mgvavb KGi msLÅvGiLvq
ˆjLwPò Aâb Kwi ‰es mgvavGbi mÁ¿veÅ AbKy „j ‰jvKv
wbYÆq Kwi| ˆ`LvI| 4
ˆhvMvkÉqx ˆcvÉ MÉvg 3
M. `†kÅKÍ-¸ 2 ‰i AvGjvGK cÉGZÅK cÉKvGii KZàwj dj M gGb Kwi, ˆcqviv x wU ‰es Avg y wU wKbGZ nGe|
wKbGj wZwb cÉ`î kZÆvaxGb meÆvGcÞv dj wKbGZ AfxÓ¡ dvskb Z = Max (x + y)
cviGeb? 4 mxgve«¬Zvi kZÆmgƒn: x 6
3 bs cÉGki² mgvavb y4
K ˆ`Iqv AvGQ, S = {x : x2 3x + 2 0} 30x + 40y 500
= {x : x2 2x x + 2 0} x 0, y 0
= {x : x(x 2) 1 (x 2) 0}
= {x : (x 2) (x 1) 0} cÉ`î AmgZvàGjvGK mgZv aGi mgxKiYàGjvi ˆjLwPò
= {x : x 2, x 1}
Aâb Kwi ‰es mgvavGbi mÁ¿veÅ AbKy „j ‰jvKv ˆei
mywcÉgvg I Bbwdgvg bvB|
Kwi|
Avgiv cvB, x = 6 (i)
y=4 (ii)
L cÉ`î AmgZv, 2x + 3 x+3 30x + 40y = 500
x3 <x1
xy (iii)
50 + 50 = 1
2x + 3 x + 3
ev, x3 x 1 <0 34
ev, (2x + 3) (x 1) (x + 3) (x 3) <0 x=0 (iv)
y=0 (v)
(x 3) (x 1) Y ˆÕ•j: x I y AGÞ 2 Ni =1 ‰KK
ev, 2x2 2x + 3x 3 (x2 9) <0 (iii) (i)
(x 3) (x 1)
ev, 2x2 + x 3 x2 + 9
(x 3) (x 1) < 0
C(6,8)
ev, x2 + x + 6 A(6,4)
(x 3) (x 1) < 0
1 (ii)
B(113 ,4)
x2 + 1 + 12 12 + 6
2.x.2 2 2
ev, <0
(x 3) (x 1)
x + 12 + 23 X O X
2 4
ev, (x 3) (x 1) < 0 ... ... ... ... (i) Y
x ‰i ˆhGKvGbv gvGbi RbÅ x 12 23 ˆjLwPGò ˆ`Lv hvq (i) ‰es (ii) ‰i mKj we±`y ‰es (i)
2 4
+ + >0 I (ii) ‰i cvGk gƒjwe±`y Zvi wecixZ cvGki mKj we±`yi
(i) bs AmgZv mZÅ nGe hw` ‰es ˆKej hw` (x 3) RbÅ x 6 ‰es y 4 mZÅ| ˆjLwPò nGZ mgvavGbi
‰es (x 1) wecixZ wPn× wewkÓ¡ nq| mÁ¿veÅ AbKy „j ‰jvKv ABCA.
kZÆ (x 1) ‰i (x 3) ‰i (x 1)(x 3) A nGœQ (i) ‰es (ii) ‰i ˆQ`we±`y| A(6, 4)
wPn× wPn× ‰i wPn×
B ” (ii) ” (iii) ” ” B11 31 4
x<1 + C ” (i) ” (iii) ” ” C (6, 8)
1<x<3 + ‰LbA(6, 4) ‰i RbÅ z = 6 + 4 = 10
x>3 + + + B11 13 4 ” ” z = 11.33 + 4 = 15.33
(i) bs mZÅ nGe hw` 1 < x < 3 nq| C (6, 8) ” ” z = 6 + 8 = 14
2 1 0 1 2 3 4 ˆ`Lv hvq, B11 31 4 we±`yGZ z ‰i gvb mGeÆvœP nq hv
wbGYÆq mgvavb {x , 1 < x < 3} (Ans.) ‰KwU f™²vsk| wRwbGmi msLÅv f™²vsk nGe bv|
KvGRB ‰GÞGò x = 11 ‰es y = 4
Ans. ˆcqviv 11wU I Avg 4wU|
4 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò
DËi ms‡KZmn m„Rbkxj cÖkœ
cÉk² 4 Y M. DóxcGK DwÍÏwLZ ˆjLwPGòi EF I GH ˆiLv «¼viv
A wbG`ÆwkZ AsGki RbÅ z ‰i meÆwbÁ² gvb wbYÆq Ki| 4
C Dîi: L. O(0,0), B(40, 0), P11030 31530 I C(0, 30);
G
ED M. 160
9
X O HF B X
Y
wkLbdj-1 I 2 cÉk² 5 `†kÅK͸-1: mxgve«¬Zv: x + 2y 10, x + y 6, x
«v¼ iv wbG`ÆwkZ 4, x, y 0
5x + 6y = 200 ........... (AB) AfxÓ¡ dvskb z = 2x + 3y wkLbdj-4(A-1), 2(A-2), 6(A-3)
2x + 5y = 150 ........... (CD)
5x + 6y = 100 ........... (EF) `†kÅK͸-2: 2x2 < x + 3 ‰KwU AmgZv| A
4x + 3y = 60 ............ (GH)
z = x + y; x, y 0 K. i ‰i Nbgƒj wbYÆq Ki| 2
K. ˆjLwPGò DGÍÏwLZ mijGiLv L. `†kÅK͸-1 nGZ AfxÓ¡ dvskGbi mGeÆvœPKiY Ki| 4
M. `†kÅK͸-2 ˆK ciggvb wPGn×i mvnvGhÅ cÉKvk Ki| 4
AçjàGjvi AmgZv wbYÆq Ki| A 2 Dîi: K. i, i 3 ; L. zmax = 16; M. x 1 < 5
2 4 4
L. AB I CD «v¼ iv wbG`ÆwkZ mgvavb AçGji ˆKŒwYK
we±`yàGjv wbYÆq Ki| 4
RwUj msLÅv 1
Z‡Zxq AaÅvq
RwUj msLÅv
cixÞvq Kgb •cGZ AviI ckÉ ² I mgvavb
ckÉ ² 1 `k† ÅKÍ-¸ I: 3x + 4y 2i ‣es 4ix 3iy + 14 `yBwU ev, x = y = 1
−50 −50 −25
Abye®¬x RwUj msLÅv •hLvGb i2 = 1.
x = y = 1
`†kÅKÍ-¸ II : z ‣KwU RwUj msLÅv •hb z = 8 6 1 2 2
K. `k† ÅKÍ-¸ II nGZ z wbYÆq Ki| 2 x = 2 ‣es y = 2
L. `k† ÅKÍ-¸ II nGZ cgÉ vY Ki •h, z4 + (z)4 ‣KwU evÕ¦e ‣LvGb, Abye®¬x RwUj msLÅv«¼q h^vKÌGg 14 + 2i ‣es 14 2i
msLÅv| 4 [ x = 2 ‣es y = 2]
M. `k† ÅKÍ-¸ I ‣ ewYÆZ Abye®¬x RwUj msLÅv«¼Gqi gaÅeZÆx awi, z1 = 14 + 2i ‣es z2 = 14 2i
•KvY wbYÆq Ki| 4 Arg (z1) = tan1 2 = tan1 1 = 8.13
14 7
1 bs cÉGki² mgvavb
( )Arg (z2) = tan1 142 = tan1 1
K z = 8 6 1 7 = 8.13
= 8 6i [ i = 1] Abye®¬x RwUj msLÅv«¼Gqi gaÅeZÆx •KvY
= 1 6i 9 = Arg(z1) Arg (z2) = 8.13 (8.13)
= 1 6i + 9i2 [ i2 = 1] = 8.13 + 8.13 = 16.26 (Ans.)
ev, z = (1 3i)2 2
cos
z = (1 3i) (Ans.) ckÉ ² 2 z = 3 + + i sin = x + iy
L ‣LvGb, z = 8 6i ‣i Abyew®¬ RwUj msLÅv K. 3 4i ‣i eMgÆ ƒj wbYÆq Ki| 2
z = 8 + i6 L. hw` (z 3i) ‣i AvàÆGg´Ÿ ‣es |z + 6| = 5 nq ZGe z
z2 + (z)2 = ( 8 6i)2 + ( 8 + 6i)2 wbYÆq Ki| 4
= 2{(8)2 + (6i)2}
[ (a + b)2 + (a b)2 = 2(a2 + b2)] M. cgÉ vY Ki •h, 2(x2 + y2) = 3x 1 4
= 2(64 36) = 2 28 2 bs cÉGki² mgvavb
= 56, hv evÕ¦e msLÅv| K −3 − 4i ‣i eMÆgƒj = −3 − 4i
Avevi, z4 + ( z )4 = (z2)2 + {( z )2}2 = 1 4i 4
= 1 2.2i.1 + (2i)2
= (z2 + z2)2 2z2z2 = (1 2i)2
= (1 2i) (Ans.)
= 562 2(zz)2
= 562 2{( 8 6i)( 8 + 6i)}2 L •`Iqv AvGQ, arg (z − 3i) =
= 562 2(64 i236)2
= 562 2 (64 + 36)2 ev, arg (x + iy − 3i) = [ z = x + iy]
= 3136 20000 ev, arg {x + i(y − 3)} =
= 16864, hv evÕ¦e msLÅv (ˆ`LvGbv nGjv) y−3
M •`Iqv AvGQ, `yBwU Abye®¬x RwUj msLÅv h^vKÌGg x
3x + 4y 2i ‣es 4ix 3iy + 14 ( )ev, tan−1 =
‣LvGb, 4ix − 3iy + 14 = 14 + i(4x − 3y)
4ix − 3iy + 14 ‣i Abye®¬x RwUj msLÅv = 14 − i(4x − 3y) ev, y − 3 = tan
x
3x + 4y − 2i = 14 − i(4x − 3y)
ev, y − 3 = 0
DfqcÞ nGZ evÕ¦e I AevÕ¦e Ask mgxK‡Z KGi cvB, x
3x + 4y = 14 ev, 3x + 4y − 14 = 0 ... ... (i)
‣es 4x − 3y = 2 ev, 4x − 3y − 2 = 0 ... ... (ii) ev, y − 3 = 0
(i) I (ii) eRÊàYb KGi cvB,
y=3
x y1
−8 42 = −56 + 6 = −9 −16 ‣es |z + 6| = 5
ev, |x + iy + 6| = 5
ev, |(x + 6) + iy| = 5
ev, (x + 6)2 + y2 = 5
ev, x2 + 12x + 36 + y2 = 25 [eMÆ KGi]
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò ‣Kv`k-«¼v`k •kwÉ Y
ev, x2 + 12x + 36 + 32 = 25 [ y = 3] 3 + cos ‣es y = 5 sin ....... (ii)
ev, x2 + 12x + 20 = 0 x = 5 + 3cos ........ (i) + 3cos
ev, x2 + 10x + 2x + 20 = 0
ev, x(x + 10) + 2(x + 10) = 0 evgcÞ = 2(x2 + y2)
ev, (x + 10) (x + 2) = 0 = 253++3ccooss2 + 5 +s3incos2 [(i) I (ii) bs
x = −2, −10 nGZ]
x = − 2 ‣es y = 3 nGj z = x + iy = 29 + 6cos + cos2 + sin2
(5 + 3cos)2
z = −2 + 3i (Ans.)
= 2 9(5++6c3ocsos+)21
Avevi, x = −10 ‣es y = 3 nGj, z = −10 + 3i (Ans.)
M ‣LvGb, x + iy = 3 + 2 + isin = 2 (10 + 6cos)
cos (5 + 3cos)2
2(3 + cos isin) = 2 2(5 + 3cos)
= (3 + cos + isin)(3 + cos isin) (5 + 3cos)2
6 + 2cos i2sin 4
= (3 + cos)2 i2sin2 = 5 + 3cos
6 + 2cos i2sin WvbcÞ = 3x 1 =3 3 + cos 1
= 9 + 6cos + cos2 + sin2 5 + 3cos
= 2(3 + cos) i2sin [ sin2 + cos2 = 1] 9 + 3cos 5 3cos
9 + 6cos +1 = 5 + cos
= 2(3 + cos) i2sin 4
10 + 6cos 2(5 + 3cos) = 5 + 3cos
= 2(3 + cos) i2sin 2(x2 + y2) = 3x 1 (cÉgvwYZ)
2(5 + 3cos) 2(5 + 3cos)
= 3 + cos + i (sin)
5 + 3cos 5 + 3cos
DËi ms‡KZmn m„Rbkxj cÖkœ
ckÉ ² 3 p(y) = 1 y + y2, 3x + i 1 = 0 ‣KwU RwUj msLÅv| Dîi: K. 1 < x < 1 L. x2 + y2 = 1;
9 6; 36 20
K. 3 + 3i3 RwUj msLÅvwUGK •cvjvi AvKvGi cÉKvk Ki|2 cÉk² 5 `†kÅK͸-1: |2x – 3| < |3x + 1|
L. x41 + 1 1 x14 ivwkwUi gvb wbYÆq Ki| 4 2i)3
3x 3x2 + `†kÅK͸-2: (1 + 4i
•h, 3 –
M. ‣KGKi Kv͸wbK Nbgƒj nGj •`LvI 4
A K. 1 – 2i •K •cvjvi AvK‡wZGZ cÉKvk Ki| 2
p()p(2)p(4) ... ... 2r Drcv`K = 22r.
L. `†kÅK͸ 1 ‣i AmgZvwU mgvavb KGi mgvavb •mU
Dîi: K. 2cos 3; L. 13;
3 i sin msLÅvGiLvq •`LvI| 4
cÉk² 4 ƒ(x, y) = x + iy M. `†kÅK͸ 2 ‣i ivwkwUi eMÆgƒj wbYÆq Ki| 4
K. mgvavb Ki: | 5 – 2 | < 1 2 Dîi: K. 5 (cos + i sin), •hLvGb = tan1 2;
3x
L. mgvavb: x < – 4 A^ev x > 2
L. |ƒ(x 4, y)| + | ƒ(x + 4, y)| = 12 «¼viv wbG`ÆwkZ 5
mçvicG^i mgxKiY wbYÆq Ki| 4 msLÅvGiLv: –4 0 2
5
M. •`LvI •h, ƒ(1, 0) ‣i Nbgƒjàwji RwUj gƒj«¼Gqi
‣KwU AciwUi eGMÆi mgvb| 4
1
cixÞvq Kgb •cGZ AviI cÉk² I mgvavb
cÉk² 1 `†kÅKÍ-¸ 1: p = 7, q = 30 – 2 ‣es g(x) = x. mxgve«¬Zvi kZÆmgƒn : 5x + 15y 45 ev, x + 3y 9
`†kÅK͸-2: •Kvb ‣KwU ‣jvKvq kiYv^ÆxG`i chÆvGjvPbv KGi 15x + 10y 60 ev, 3x + 2y 12
•`Lv hvq ZvG`i wk÷iv wewf®² AcywÓ¡GZ fzMGQ| ZvG`i Lv`Å x, y 0
mieivGni RbÅ F1 I F2 `yB aiGbi Lv`Å wbeÆvPb Kiv nGjv| AmgZvàwjGK Abyi…c mgxKiY AvKvGi cÉKvk KGi cvB,
hvGZ cÉwZ wKGjvGZ wfUvwgb C I wfUvwgb D cÉvwµ¦i cwigvY
x + 3y = 9 x y = 1 ... ... ... ... (i)
wbÁ²i…c: 2 I 3 AaÅvGqi mg®¼Gq 9 +3
Lv`Å wfUvwgb C wfUvwgb D wKGjv cÉwZ gƒjÅ 3x + 2y = 12 x y = 1 ... ... ... ... (ii)
4 +6
F1 5 15 7 UvKv
x = 0 ... ... ... ... (iii)
F2 15 10 14 UvKv
y = 0 ... ... ... ... (iv)
A K. `†kÅKÍ-¸ 1 nGZ p q ‣i gvb wbYÆq Ki| 2
L. `†kÅKÍ-¸ 1 nGZ •`LvI •h, g(i) + g(i) = 2 ; Y (2) •Õ•j : X I Y AGÞ 5 Ni = 1 ‣KK
C(0,6)
•hLvGb i = 1. 4
M. wfUvwgb C I wfUvwgb D ‣i ․`wbK bŃ bZg
cÉGqvRb h^vKÌGg 45 I 60 nGj meGPGq Kg LiGP (3)
․`wbK wfUvwgb C I D ‣i Pvwn`v wKfvGe •gUvGbv (1)
(0,3)
hvGe? 4
1 bs cÉGk²i mgvavb ( )178175
K p – q = 7 – 30 –2 = 7 30 2i B (4)
= 52 2.5.3 2.i + (3 2i)2 X(4) (1) X
= (5 3 2i)2 A(9,0)
O(0,0) (3) (4,0) (2)
Y
p q = (5 3 2i) (Ans.)
L g(i) + g(i) = i + – i [ g(x) = x] •jLwPGò mgxKiYàwj ewmGq cÉvµ¦ ABC AçGji
= 1 . 2i + 2i DciÕ© ‣es Wvb cvk¼ÆÕ© we±`ymgƒn cÉ`î mKj kZÆGK
2 2
1 i2 + 2i + 1 + 1 . 1 + i2 2i mg^Æb KGi weavq H AçjwU mÁ¿veÅ mgvavb Açj| hvi
=
22 •KŒwYK we±`ymgƒn :
[ i2 = 1] A(9, 0), B178 175 ((i) I (ii) ‣i •Q`we±`y) ‣es C(0, 6)
1 (1 + i)2 + 1 (1 i)2
= 22 A(9, 0) we±`yGZ, Z = 7 9 + 14 0 = 63
= 1 (1 + i) 1 (1 i) B178 15 we±`yGZ, Z = 7 18 + 14 15
22 7 7 7
= 1 (1 + i + 1 – i) = 48
2
C(0, 6) we±`yGZ Z = 0 + 6 14 = 84
= 2 = 2
2 Õ·Ó¡Z B we±`yGZ Z meÆwbÁ²
g(i) + g(i) = 2 (ˆ`LvGbv nGjv) F1 ‣i cwigvY = 18 •KwR
7
M awi, ․`wbK F1 cÉKvGii LvG`Åi x •KwR cwigvY cÉGqvRb
‣es F2 ‣i cwigvY = 15 •KwR (Ans.)
F2 ,, ,, y ,, ,, ,, 7
AfxÓ¡ dvskb, Zmin = (7x + 14y)
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò
cÉk² 2 `†kÅKÍ-¸ 1: ‣KwU •`vKvb `yB aiGbi •Ljbv, A Y x I y AÞ eivei 100 ‣KK = 1eMÆNi
‣es B wewKÌ KGi| •`vKvGbi gvwjK cÉwZ ‣KK A ‣es B (0, 2000)
•Ljbv KÌq KGi h^vKÌGg 8 UvKv ‣es 14 UvKv w`Gq| cÉwZ A
‣KK A ‣es B •Ljbv wewKÌGZ jvf h^vKÌGg 2 UvKv ‣es 3 10000
UvKv| •`vKvGbi gvwjGKi cwiK͸bv nGe 2000 ‣i •ewk 7
•Ljbv wewKÌ KiGe bv ‣es 20,000 UvKvi •ewk BbGfÓ¡ KiGe 0( )S
bv|
`†kÅK͸-2: 2x + 3 x+3 1, 2 I 3 AaÅvGqi mg®¼Gq
x3 <x1
A K. z = x + iy nGj |z + i| = |z + 2| «v¼ iv wbG`ÆwkZ
mçvic^ wbYÆq Ki| 2 X O P(2000, 0) X
Y
L. `†kÅKÍ-¸ 1 nGZ cÉwZ gvGm mGeÆvœP KZàGjv •Ljbv
KÌq KiGj mGeÆvœP gybvdv nGe? 4 mÁ¿veÅ mgvavb ‣jvKvi •KŒwYK we±`yàGjv h^vKÌGg
M. `†kÅKÍ-¸ 2 ‣ ewYÆZ AmgZvwUi mgvavb Ki ‣es S0 107000 , A40300 2000 ‣es P(2000, 0)
3
msLÅvGiLvq •`LvI| 4
2 bs cÉGki² mgvavb S0 107000 we±`yGZ, Z=20+ 3 10000
7
K •`Iqv AvGQ, z = x + iy
= 4285.71
‣i Abye®¬x,
z = x iy A40300 20300 we±`yGZ, Z = 2 4000 + 3 2000
z 3 3
‣Lb, |z + i| = |z + 2|
ev, |x + iy + i| = |x iy + 2| = 4666.67
ev, |x + i(1 + y)| = |(x + 2) iy| P(2000, 0) we±`yGZ, Z = 2 2000 + 3 0 = 4000
ev, x2 + (1 + y)2 = (x + 2)2 + ( y)2 Õ·Ó¡Z A40300 2000 (1333.33, 666.67) we±`yGZ
3
[ |x + iy| = x2 + y2 ]
Z ‣i mGeÆvœP gvb we`Ågvb|
ev, x2 + (1 + y)2 = (x + 2)2 + y2 [eMÆ KGi]
•hGnZz •Ljbvi msLÅv f™²vsk nGZ cvGi bv ZvB wZwb
ev, x2 + 12 + 2.1.y + y2 = x2 + 2.x.2 + 22 + y2
1333wU A ‣es 666wU B •Ljbv KÌq KiGeb ‣es ZLb
ev, 1 + 2y = 4x + 4
ev, 4x + 4 2y 1 = 0 jvf, Z = 2 1333 + 3 666 = 4664 UvKv
4x 2y + 3 = 0 hv ‣KwU mijGiLvi mçvic^ M DóxcK-1 AbymvGi, 2x + 3 x+3
x3 <x1
wbG`Æk KGi|
L awi, •`vKvb gvwjK x msLÅK A ‣es y msLÅK B KÌq ev, 2x + 3 x + 3 <0
KGib| x3 x 1
cÉ`î kZÆvbmy vGi AmgZvàGjv nj ev, (2x2 + 3x 2x 3) (x2 9) <0
x + y 2000 (x 3) (x 1)
8x + 14y 20000; x, y 0 ev, 2x2 + x 3 x2 + 9 <0
(x 3) (x 1)
‣es Z = 2x + 3y
AmgZvàGjvGK mgZv aGi mgxKiGYi •jLwPò Aâb ev, x2 + x + 6 <0
Kwi ‣es mgvavGbi mÁ¿veÅ AbKy „j ‣jvKv •ei Kwi: (x 3) (x 1)
x + y = 2000 x y = 1 ... ... ... (i) x2 + 1 .x + 122+ 6 1
2000 + 2000 2.2 4
ev, <0
(x 3) (x 1)
‣es 8x + 14y = 20000
x 12 23
ev, 8x 14y =1 + 2 + 4
20000 + 20000
ev, (x 3) (x 1) < 0 ... ... ... (i)
x y = 1 ... ... ... ... (ii) ‣LvGb, x + 12 + 23 >0
2500 + 10000 2 4
7
x = 0 ... ... ... ... (iii) (x 3) I (x 1) ‣i gGaÅ ‣KwUi wPn× abvñK ‣es
y = 0 ... ... ... ... (iv) AciwUi wPn× FYvñK nGj (i) AmgZvwUi kZÆ wm«¬ KGi|
RwUj msLÅv 3
kZÆ (x 1) (x 3) ‰i (x 3) (x 1) Avgiv cvB, 4x + y = 16 x + y = 1 ... ... ... (i)
‰i wPn× wPn× ‰i wPn× 4 16
x<1 + 4x + 7y = 40 x + y = 1 ... ... ... (ii)
10 40
1<x<3 +
7
x>3 + + +
(i) AmgZvwU mZÅ nGe hw` 1 < x < 3 nq| Y •Õ•j: x I y AÞ eivei
wbGYÆq mgvavb •mU, S = { x : 1 < x < 3} A(0,16) Þz`ËeGMÆi 1 evü = 1 ‣KK
msLÅvGiLv:
1 0 1 2 3 4
cÉk² 3 `†kÅKÍ-¸ 1: ƒ(x) = |x 3|
`†kÅK͸-2: 4x + y 16, 4x + 7y 40, x, y 0
(0,5.714)
1, 2 I 3 AaÅvGqi mg®¼Gq B(3,4)
A K. 2 3 + 2i •K •cvjvi AvKvGi cÉKvk Ki| 2
L. ƒ(x) < 1 nGj •`LvI •h, ƒ(x2 6) < 31 4 C(10,0)
5 25
M. `†kÅKÍ-¸ 2 ‣i AvGjvGK •jLwPGòi mvnvGhÅ X O (0,0) (4,0) X
z = 4x + 2y ‣i meÆwbÁ² gvb wbYÆq Ki| 4 Y
3 bs cÉGki² mgvavb mÁ¿veÅ ‣jvKvi •KŒwYK we±`yàGjv h^KÌGg A(0,16),
K gGb Kwi, z = 2 3 + 2i B(3, 4), C(10,0)
‣LvGb, 2 3 = r cos ‣es 2 = r sin
A we±`yGZ, z = 4x + 2y = 4 0 + 2 16 = 32
r = (2 3)2 + 22 = 16 = 4 B we±`yGZ, z = 4 3 + 2 4 = 20
C we±`yGZ, z = 4 10 + 2 0 = 40
‣es = tan1 2 = tan1 1
2 3 3 =6 zmin = 20 (Ans.)
•hGnZz we±`ywU 2q PZzfÆvGM AewÕ©Z, cÉk² 4 z = x + iy ‣es ‣KGKi ‣KwU RwUj Nbgƒj |
KvGRB gƒLÅ AvàÆGg´Ÿ = 5 wkLbdj-2, 3, 4, I 6
6 =6
K. |2z 1| = |z 2| «v¼ iv wbG`ÆwkZ mçvicG^i mgxKiY
ZvnGj, 2 3 + 2i = r(cos + i sin)
wbYÆq Ki| 2
4cos 5 5 ‣wUB msLÅvwUi •cvjvi AvKvi|
= 6 + i sin 6 ; L. z ‣ x •K 1 «v¼ iv cÉwZÕ©vcb KiGj ‣es a2 + b2 = 1 nGj
L •`Iqv AvGQ, ƒ(x) = |x 3| •`LvI •h, y ‣i ‣KwU evÕ¦e gvb ¯z = a ib mgxKiYGK
z
myZivs ƒ(x2 6) = |x2 6 3| = |x2 9|
wm«¬ KGi| •hLvGb, a, b | 4
‣Lb, |x + 3| = |x 3 + 6|
M. cÉgvY Ki •h, n + 2n = 2, hLb n ‣i gvb 3 «v¼ iv
| x + 3| |x 3| + |6| [ |a + b| |a| + |b|]
wefvRÅ ‣es 1, hLb n Aci •h •KvGbv cƒYÆmsLÅv nq| 4
1 1 4 bs cÉGki² mgvavb
ev, |x + 3| < 5 + 6 [ |x 3| < 5]
|x + 3| < 31 K •`Iqv AvGQ, |2z 1| = |z 2|
5 ev, |2(x + iy) 1| = |x + iy 2|
ev, |(2x 1) + 2iy| = |(x 2) + iy|
|x + 3| |x 3| < 31 1
5 5 ev, (2x 1)2 + (2y)2 = (x 2)2 + y2
ev, (2x 1)2 + 4y2 = (x 2)2 + y2
ev, |x2 – 9| < 31 ev, 4x2 4x + 1 + 4y2 = x2 4x + 4 + y2
25 ev, 3x2 + 3y2 = 3
f(x2 – 6) < 31 (ˆ`LvGbv nGjv) x2 + y2 = 1
25
hv wbGYÆq mçvicG^i mgxKiY| (Ans.)
M AfxÓ¡ dvskb, z = 4x + 2y
mxgve«¬Zvi kZÆmgƒn:
4x + y 16, 4x + 7y 40, x, y 0 L •`Iqv AvGQ, z = x + iy
cÉ`î AmZvàGjvGK mgZv aGi mgxKiYàGjvi •jLwPò ¯z = x iy
Aâb Kwi ‣es mgvavGbi mÁ¿veÅ AbKy „j ‣jvKv wbYÆq
Kwi| x = 1 nGj, z = 1 + iy
‣Lb, ¯z = a ib
z
4 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò
ev, 1 iy = a ib 5 bs cÉGk²i mgvavb
1 + iy
K 1 + 3i
ev, 1 + iy 1 [wecixZKiY KGi]
1 iy = a ib gWzjvm = |1 + 3i| = 12 + ( 3)2 = 2 (Ans.)
ev, 1 + iy –1 + iy 1 a + ib [weGqvRb-•hvRb KGi] AvàÆGg´Ÿ nGj, = tan1 y
1 + iy + 1 iy = 1 + a ib x
ev, 2iy (1 a + ib)(1 + a + ib) = tan1 3
2 = (1 + a ib)(1 + a + ib) 1
ev, (1 + ib – a) (1 + ib + a) = tan1 tan
iy = (1 +a)2 (ib)2 3
(1 + ib)2 a2
= 1 + 2a + a2 i2b2 = 3 (Ans.)
1 + 2ib + i2b2 a2 L •`Iqv AvGQ, z1 = 3 + i, z2 = 3 i
= 1 + 2a + a2 + b2
z12 = ( 3 + i)2 = ( 3)2 + 2 3.i + i2
1 + 2ib (a2 + b2) = 3 + 2 3i 1 [ i2 = 1]
= 1 + 2a + a2 + b2
= 2 + 2 3i
1 + 2ib 1 [ a2 + b2 = 1]
= 1 + 2a + 1 z1z2 = ( 3 + i) ( 3 i)
= ( 3)2 (i)2 = 3 i2 = 3 + 1 = 4
2ib ib
= 2(1 + a) = 1 + a 11 3i 3i 3i
z1 = = 3 i) = 3 i2 =
b 3+i ( 3 + i) ( 4
1+a,
y= hv y ‣i ‣KwU evÕ¦e gvb| (ˆ`LvGbv nGjv) [ i2 = 1]
M cÉ`î ivwk = n + 2n = n + (2)n evgcÞ = z12 + z1z2 + 83
z1
‣LvGb n = 3m nGj, m
3i
cÉ`î ivwk = 3m + (2)3m = 2 + 2 3i + 4 + 8 3 . 4
= (3)m + (3)2m = 2 + 2 3i + 4 + 2 3 ( 3 i)
=1+1=2 = 2 + 2 3i + 4 + 6 2 3i
n = 3m + 1 nGj, = 12 = WvbcÞ
cÉ`î ivwk = ()3m + 1 + (2)3m + 1 z12 + z1z2 + 83 = 12 (ˆ`LvGbv nGjv)
z1
= (3)m. + (3)2m. 2
= + 2 M gGb Kwi, 6 – 64 = x
ev, x6 = 64 [DfqcGÞi NvZGK 6 «¼viv àY KGi]
=1 ev, x6 + 64 = 0
ev, (x2)3 + 43 = 0
n = 3m + 2 nGj, ev, (x2 + 4) (x4 4x2 + 16) = 0
nq x2 + 4 = 0 ev, x2 = 4 = 4i2
cÉ`î ivwk = ()3m + 2 + (2)3m + 2
x = 2i
= (3)m. 2 + (3)2m. 4
= 2 + A^ev, x4 4x2 + 16 = 0
=1
A^Ævr, n ‣i gvb 3 «v¼ iv wefvRÅ nGj, cÉ`î ivwkwU = 2
‣es n ‣i gvb Aci •KvGbv cƒYÆ msLÅv nGj, x2 = 4 16 64
2
ivwkwU = 1 (cÉgvwYZ)
ev, x2 4 48 4 i 48 4 i.4 3
= 2 =2 =2
cÉk²5 x = 6 64, z1 = 3 + i, z2 = 3 i
= 2 i.2 3 = ( 3)2 1 i.2 3
wkLbdj-2, 4, 6 = ( 3)2 2. 3 .i + i2
x2 = ( 3 i)2
K. 1 + 3i ‣i gWzjvm I AvàÆGg´Ÿ wbYÆq Ki| 2
L. •`LvI •h, z12 + z1z2 + 83 = 12 4 x = ( 3 i)
z1 4
x = 2i, z1, z2 (cÉgvwYZ)
M. cÉgvY Ki •h, x = 2i, z1, z2
RwUj msLÅv 5
DËi ms‡KZmn m„Rbkxj cÖkœ
cÉk²6 z = 3 + 2i wkLbdj-2, 3, 5 I 6 M. f21 ‣i Nbgƒjàwj wbYÆq Ki| 4
A AK. z ‣i gWzjvm I AvMÆyGg´Ÿ wbYÆq Ki| 2 Dîi: K. |x| < 5 |x| < 5;
L. cÉgvY Ki •h, z2 + z–z + –z2 = 23 4 L. {x :61 x 13, x 14};
M. 12z – 43 ‣i gvb wbYÆq Ki| 4
Dîi: K. 13, tan–132 msLÅvGiLvq wbÁ²i…c: 011 1
64 3
M. (3 + 4i) M. 1, –1 3i
2 ;
cÉk² 7 `†kÅKÍ-¸ 1: S = {x : 4 x2 81}
`†kÅKÍ-¸ 2: x = 3 + 2i wkLbdj-2, 4 I 6 cÉk² 9 (x) = ax2 + bx + c ‣KwU w«N¼ vZ dvskb
A K. 4 – 16‣i gvb wbYÆq Ki| 2 ‣es z = 1 + 32ii‣KwU RwUj msLÅv|
1 –
L. `†kÅKÍ-¸ 1 nGZ S, supS, infSwbYÆq Ki| 4
K. – 1 + 3i •K r(cos + isin) AvKvGi cÉKvk Ki| 2
M. y = a – ib ‣es y = x¯ nGj a2 + b2‣i gvb wbYÆq Ki| 4
L. z1 mƒò eÅenvi KGi
Dîi: K. 2 (1 i); arg z2 = arg(z1) – arg(z2)
L. S = {x : 2 x 9}
•`LvI •h, tan– 12 + tan– 13 = 3 4
Sup S = 9, inf S = 2; 4.
M. 13; M. ‣KGKi Kv͸wbK Nbgƒj ‣es {()}3 + (2)}3 = 0
cÉk² 8 f(x) = |1 – 4x| ‣KwU ciggvb dvskb| nGj cÉgvY Ki •h, a = 1 (b + c)
K. –2 < 3 – x < 8 •K ciggvb wPGn×i mvnvGhÅ cÉKvk Ki| 2 2
L. mgvavb KGi mgvavb •mU msLÅvGiLvq •`LvI:
ev b = 1 (c + a) ev c = 1 (a + b). 4
2 2
f(1x) 3, •hLvGb, x 14
4 Dîi: K. 2cos 2 + isin 2
3 3
eüc`x I eüc`x mgxKiY 1
PZz^Æ AaÅvq
eüc`x I eüc`x mgxKiY
cixÞvq Kgb ˆcGZ AviI ckÉ ² I mgvavb
ckÉ ² 1 (x) = (x − p) (x − q) + (x − q) (x − r) + (x − r) (x − p) k = 3 nGj, 6x2 + 6x + 7 = 0 ... ... ... ... (iii)
‰es g(x) = (k2 − 3) x2 + 2kx + (2k + 1) `By wU dvskb| (iii) bs mgxKiYwUi gƒj«¼q I
K. p ‰i gvb KZ nGj x2 + px − 6p = 0 mgxKiGYi gƒj«¼q evÕ¦e + = − 6 = − 1 ‰es = 7
6 6
I mgvb nGe| 2
‰Lb, + = 3 + 3 Avevi,
L. (x) cYƒ ÆeMÆ nGj ˆ`LvI ˆh p = q = r 4 2 2 22
= 1
M. mgxKiGYi gƒj«¼q nGj ‰es gƒjwewkÓ¡ ( + )3 − 3 ( + ) 2 2
g(x) = 0 , 2 2 = 22
1
mgxKiY ˆei Ki hLb k = 3 4 (−1)3 − 3. 7 (−1) =7
1 bs cÉGki² mgvavb 6
90 6
( )= 7 2 = 49
K c`É î mgxKiY, x2 + px − 6p = 0 ... ... ... ... (i) 6 6
(i) bs mgxKiGYi wbøvqK =7
D = p2 − 4.1. (−6p) ‰es gƒjwewkÓ¡ mgxKiY
= p2 + 24p 2 2
ˆhGnZz gƒj«¼q evÕ¦e x2 − 2 + 2 x + 2 . 2 = 0
D=0 ev, x2 − 90 x + 6 = 0
49 7
ev, p2 + 24p = 0
ev, p(p + 24) = 0 49x2 − 90x + 42 = 0 (Ans.)
ev, p = 0
A^ev, p + 24 = 0 ckÉ ² 2 x3 + ax2 + bx + c = 0 ‰KwU wòNvZ mgxKiY|
p = – 24 K. evÕ¦e mnMwewkÓ¡ ‰gb ‰KwU mgxKiY wbYÆq Ki hvi `By wU
p ‰i gvb 0 A^ev −24 (Ans.) gƒj 1 – 2 I 1 + i 2
L ˆ`Iqv AvGQ, L. DóxcGKi mgxKiGYi gƒj wZbwU mgv¯¦i cÉMgbfzÚ nGj
(x) = (x − p) (x − q)+(x − q)(x − r)+(x − r)(x − p) ˆ`LvI ˆh, 2a3 – 9ab + 27c = 0 4
= x2 − px − qx + pq + x2 − qx − rx + qr + x2 − rx − px + pr
= 3x2 − 2(p + q + r)x + (pq + qr + pr) M. DóxcGKi mgxKiGYi gƒjòq , , nGj 1 + 1
+ ,
(x) cƒYÆeMÆ nGe hw` (x) «¼viv MwVZ mgxKiGYi gƒj«¼q mgvb nq|
I + 1 gƒjwewkÓ¡ mgxKiYwU wbYÆq Ki| 4
3x2 − 2(p + q + r)x + (pq + qr + pr) = 0 ... ... ... ... (ii)
(ii) bs mgxKiGYi wbøvqK 2 bs cÉGki² mgvavb
D = {−2(p + q + r)}2 − 4.3 (pq + qr + rp) K evÕ¦e mnMwewkÓ¡ mgxKiGYi Agƒj` I RwUj gƒjàwj hyMGj
= 4 (p + q + r)2 − 4 . (3pq + 3qr + 3rp) ^vGK| KvGRB ˆKvb mgxKiGYi `yBwU gƒj 1 – 2 I 1 + i nGj
= 4 (p2 + q2+r2+2pq + 2qr + 2pr −3pq − 3qr − 3rp) Aci gƒj `yBwU nGe 1 + 2 I 1 – i
= 4(p2 + q2 + r2 − pq − qr − rp) mgxKiYwU{x – (1 – 2)}{x – (1 + 2)}{x – (1 + i)}{x
= 2(2p2 + 2q2 + 2r2 − 2pq − 2qr − 2rp)
= 2{(p − q)2 + (q − r)2 + (r − p)2} – (1 – i)} = 0
ˆhGnZz gƒj«¼q mgvb {(x – 1) + 2}{(x – 1) – 2}{(x – 1) – i}{(x – 1) + i} = 0
{(x – 1)2 – 2}{(x – 1)2 + 1} = 0
D=0 {(x – 1)2}2 – 2(x – 1)2 + (x – 1)2 – 2 = 0
(x2 – 2x + 1)2 – (x – 1)2 – 2 = 0
ev, 2{(p − q)2 + (q − r)2 + (r − p)2}= 0 x4 + 4x2 + 1 – 4x3 – 4x + 2x2 – x2 + 2x – 1 – 2 = 0
A^ev, (q − r)2 = 0 ev, q = r x4 – 4x3 + 5x2 – 2x – 2 = 0
‰es (r − p)2 = 0 ev, r = p
p = q = r (ˆ`LvGbv nGjv)
M ˆ`Iqv AvGQ, g(x) = 0 L c`É î mgxKiY: x3 + ax2 + bx + c = 0 ... ... ... ... (i)
ev, (k2 − 3) x2 + 2kx + (2k + 1) = 0 awi, mgv¯¦i cÉMgbfzÚ gƒjòq – , I +
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò ‰Kv`k-«¼v`k ˆkwÉ Y
–+++=–a M c`É î mgxKiY: x3 + ax2 + bx + c = 0 ... ... ... ... (i)
mgxKiGYi gƒjòq , , nGj
ev, 3 = – a
ev, = – a ... ... ... ... (ii) + + = a
3 + + = b ... ... ... ... (ii)
= c
Avevi, ( – ) + ( + ) + ( + )( – ) = b ‰gb ‰KwU mgxKiY MVb KiGZ nGe hvi gƒjòq
ev, 2 – + 2 + + 2 – 2 = b
ev, 32 – 2 = b ... ... ... ... (iii) + 1 + 1 I + 1
,
‰es ( – )( + ) = – c
1
(2 – 2) = – c awi, y = +
( ) – a a2 – 2 =–c y = + 1
3 9
y = – c + 1
a2 – 2 3c
9 = a = 1 – c
y
a2 3c
2 = 9 – a ... ... (iv) 1–c [ˆhGnZz (i) bs ‰i ‰KwU gƒj = x]
x= y
(ii) I (iv) bs nGZ I 2 ‰i gvb (iii) bs mgxKiGY ewmGq
1–c 3 1–c 2
a2 a2 3c ( ) ( )x-‰i gvb (i) bs mgxKiGY ewmGq
3. 9 – 9 + a = b y +a y
1–c
y
2a2 3c ( )+ b +c=0
9 =–a+b
(1 – c)3 + a(1 – c)2y + b(1 – c)y2 + cy3 = 0
2a2 – 3c + ab
9= a cy3 + b(1 – c)y2 + a(1 – c)2y + (1 – c)3 = 0
2a3 = – 27c + 9ab PjK e`j KGi cvB, cx3 + b(1 – c) x2 + a(1 – c)2x + (1 – c)3 = 0
BnvB wbGYÆq mgxKiY|
2a3 – 9ab + 27c = 0 (ˆ`LvGbv nGjv)
DËi ms‡KZmn m„Rbkxj cÖkœ
ckÉ ² 3 A = (k2 − 3) x2 + 3kx + (3k + 1); B = px2 + qx + r L. P(x) = 0 mgxKiYwUi gƒj«¼q I nGj + 1 I +
K. K ‰i gvb KZ nGj A = 0 mgxKiGYi gƒj«¼q ciÕ·i DΟv 1 gƒjwewkÓ¡ mgxKiY wbYÆq Ki| 4
nGe| 2
M. Q(x) = 0 mgxKiGYi ‰KwU gƒj 1 + 3 nGj mgxKiYwU mgvavb
L. A = 0 mgxKiGYi gƒj `wy U , nGj 1 + 2 ‰es Ki| 4
Dîi: K. 6 6, 6 6; L. 12x2 91x + 169 = 0;
1 + 2 gƒj wewkÓ¡ mgxKiY wbYÆq Ki, ˆhLvGb K = − 3 4 M. x = 6, 1 + 3, 1 3
M. B = 0 mgxKiGYi gƒj«¼q a ‰es b nGj ckÉ ² 5 ax2 + bx + b = 0 ‰KwU w«¼NvZ mgxKiGYi gƒj«¼q I
pr (x2 + 1) − (q2 − 2pr)x = 0 mgxKiGYi gƒj«¼qGK a, b K. 4 + 4 ‰i gvb KZ? 2
‰i gvaÅGg cKÉ vk Ki| 4 L. cgÉ vY Ki ˆh, P Q b =0 hLb DóxcGKi
Q+ P+ a
Dîi: K. k = 1 A^ev 4; L. 32x2 + 108x + 57 = 0;
mgxKiGYi gƒj«¼Gqi AbycvZ P t Q| 4
M. b ‰es a
a b M. 1 I 1 gƒjàGjv «¼viv MwVZ mgxKiY wbYÆq Ki| 4
4 4
ckÉ ² 4 P(x) = 1 7x + 12x2 Dîi: K. b4 4ab3 + 2a2b2 ;
a4
Q(x) = x3 8x2 + 10x + 12
M. (25ab 4b2)x2 3abx + a2 = 0
K. k ‰i gvb KZ nGj 5x2 kx + 6 = 0 mgxKiGYi ‰KwU gƒj
AciwUi cuvPàY nGe| 2
cixÞvq Kgb •cGZ AviI cÉk² I mgvavb
cÉk²1 z = 2 3i ‣KwU RwUj ivwk| = {22 ( 3i)2} = (4 + 3) = 7
cÉk²gGZ, 7 = 64
3 I 4 AaÅvGqi mg®¼Gq ev, = 64
7
K. x2 + px 12 = 0 mgxKiGYi ‣KwU gƒj 3 nGj p ‣i gvb
wbGYÆq gƒjòq 2 64
wbYÆq Ki| 2 3i, 2 + 3i, 7 (Ans.)
L. Arg( z ) wbYÆq Ki| 4 cÉk² 2 Z1 = a + ib ‣es Z2 = c + id `yBwU RwUj msLÅv|
M. •Kvb wòNvZ mgxKiGYi ‣KwU gƒj z ‣es gƒjàGjvi
àYdj 64 nGj mgxKiYwUi mgvavb wbYÆq Ki| 4 3 I 4 AaÅvGqi mg®¼Gq
K. hw` x2 px + q = 0 mgxKiGYi gƒj«¼q KÌwgK cƒYÆmsLÅv
1 bs cÉGk²i mgvavb nq ZGe •`LvI •h, p2 4q 1 = 0 2
K •`Iqv AvGQ, x2 + px 12 = 0 mgxKiGYi ‣KwU gƒj 3 L. 3 Z1 = Z2 nGj •`LvI •h, 4cd(c2 d2) = ad + bc 4
M. x : y = Z1 : Z2 nGj cÉgvY Ki •h, 4
32 + p.3 12 = 0
(c2 + d2) x + (a2 + b2) y = 2(bd + ac)
ev, 9 + 3p 12 = 0 y x
ev, 3p = 3
2 bs cÉGki² mgvavb
p = 1 (Ans.)
L z = 2 3i K cÉ`î mgxKiY, x2 – px + q = 0
gWzjvm, r = 22 + ( 3)2 = 7 awi, gƒj `yBwU, , + 1
‣LvGb, = tan1 xy gƒj«¼Gqi •hvMdj, + + 1 = p
ev, 2 = p – 1
tan1 3 p–1
= 2 = 2
= tan1 3 ‣es gƒj«¼Gqi àYdj, ( + 1) = q
2 p –1 p –
ev, 2 2 1 + 1 = q [gvb ewmGq]
•hGnZz we±`ywU 4^Æ PZzfÆvGM AewÕ©Z|
AvàÆGg´Ÿ = ev, p–1 p – 1 + 2 = q
2 2
AvàÆGg´Ÿ tan1 3
= 2 ev, (p – 1) (p + 1) = 4q
Avevi, z = r(cos + i sin) ev, p2 – 1 = 4q
p2 – 4q – 1 = 0 (ˆ`LvGbv nGjv)
= rei
11 L •`Iqv AvGQ, Z1 = a + ib, Z2 = c + id
z = (rei)2 = r2 ei . 2
‣Lb, 3 Z1 = Z2 nGj Avgiv cvB,
z ‣i AvàÆGg´Ÿ = = 1 tan1 3 (Ans.)
2 2 2 3 a + ib = c + id
M •`Iqv AvGQ, wòNvZ mgxKiGYi ‣KwU gƒj z = 2 3i A^Ævr, a + ib = (c + id)3 [DfqcÞGK Nb KGi]
•hGnZz RwUj gƒjàGjv Abey ®¬x hyMj AvKvGi ^vGK ZvB ev, a + ib = c3 + 3c2id + 3ci2d2 + i3d3
Aci gƒjwU nGe = 2 + 3i
awi, Aci gƒjwU = c3 + 3c2di 3cd2 id3
= c3 3cd2 + 3c2di id3
gƒjòGqi àYdj = (2 3i) (2 + 3i) = c3 3cd2 + i(3c2d d3)
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò
‣Lb, evÕ¦e I AevÕ¦e Ask mgxK‡Z KGi cvB, L •`Iqv AvGQ, P(x) = 1 + 1 1
+xc
a = c3 3cd2 ‣es b = 3c2d d3 x a x b
‣Lb, WvbcÞ = ad + bc = cd ac + bd cÉk²gGZ, P(x) = 0
ev, 11 1
xa+xb +xc=0
c3 3cd2 3c2d d3
= cd c + d (x − b) (x − c) + (x − a) (x − c) + (x − a) (x − b)
ev,
c(c2 3d2) d(3c2 d2) (x − a) (x − b) (x − c) =0
c d
= cd + ev, x2 − bx − cx + bc + x2 ax − cx + ac + x2 − ax
− bx + ab = 0 (x − a) (x − b) (x − c)
= cd (c2 3d2 + 3c2 d2) = 4cd(c2 d2)
ev, 3x2 − 2bx – 2cx – 2ax + ab + bc + ca = 0
= evgcÞ
ev, 3x2 − 2x (a + b + c) + (ab + bc + ca) = 0
4cd(c2 d2) = ad + bc (ˆ`LvGbv nGjv)
mgxKiYwUi gƒj«¼q evÕ¦e nGe hw` mgxKiYwUi c†^vqK,
M •`Iqv AvGQ, x a + ib
y = c + id D 0 nq|
ev, x(c + id) = y(a + ib) ‣Lb, c†^vqK, D = { 2(a + b + c)}2 − 4 3 (ab + bc + ca)
ev, cx + idx = ay + iby = 4(a + b + c)2 − 12 (ab + bc + ca)
= 4 {(a + b + c)2 − 3(ab + bc + ca)}
ev, idx iby = ay cx = 4 {a2 + b2 + c2 + 2(ab + bc + ca) − 3 (ab + bc + ca)}
= 4 {a2 + b2 + c2 − (ab + bc + ca)}
ev, i(dx by) = (ay cx) = 4 (a2 + b2 + c2 − ab − bc − ca)
ev, i2(dx by)2 = (ay cx)2 [eMÆ KGi] 1
2
ev, (d2x2 2bdxy + b2y2) = a2y2 2acxy + c2x2 = 4 {(a − b)2 + (b − c)2 + (c − a)2} ... ... ... (i)
ev, d2x2 + 2bdxy b2y2 = a2y2 2acxy + c2x2 [ (a − b)2 + (b − c)2 + (c − a)2 = 2(a2 + b2 + c2
− ab − bc − ca)]
ev, (a2 + b2) y2 + (c2 + d2) x2 = 2(ac + bd) xy
= 2 {(a − b)2 + (b − c)2 + (c − a)2}
ev, (a2 + b2) y2 + (c2 + d2) x2 = 2(ac + bd)
xy xy a, b, c cÉGZÅGKB evÕ¦e nGj AekÅB
[DfqcÞGK xy «v¼ iv fvM KGi] (a − b)2 + (b − c)2 + (c − a)2 0 nGe
(a2 + b2) y + (c2 + d2)yx = 2(ac + bd) (cÉgvwYZ) A^Ævr c†^vqK, D 0
x
p(x) = 0 mgxKiGYi gƒj«¼q meÆ`v evÕ¦e nGe|
cÉk² 3 1 1 1 (ˆ`LvGbv nGjv)
P(x) = x a +xb +xc
gƒj«¼q evÕ¦e I mgvb nq hLb c†^vqK = 0 nq|
wkLbdj-3, 4, 5 I 6 (i) bs nGZ cvB,
K. P(x) . (x a) (x b) (x c) ‣i aË‚eK c` wbYÆq Ki| 2 c†^vqK, D = 4 1 {(a – b)2 + (b – c)2 + (c – a)2}|
2
L. •`LvI •h, 1 1 1 = 0 mgxKiGYi c†^vqK kbƒ Å nGe hw` I •Kej hw` a = b = c nq|
xa + xb + xc
a = b = c bv nGj gƒjàwj mgvb nGZ cvGi bv|
gƒjàGjv evÕ¦e ‣es mgvb nGZ cvGi bv hw` bv a = b = c
(ˆ`LvGbv nGjv)
nq| 4
M P(x) = 0
M. hw` a = 3, b = 2, c = 1 nq ZGe P(x) = 0 mgxKiYwUi
ev, 111
gƒjàwj wbYÆq Ki I ‣G`i ․ewkÓ¡Å weGkÏlY Ki| 4 xa +xb +xc =0
3 bs cÉGki² mgvavb ev, (x b) (x c) + (x c) (x a) + (x a) (x b)
K •`Iqv AvGQ, P(x) = 1 1 1 =0
+xb +xc
x a [DfqcÞGK (x a) (x b) (x c) «v¼ iv àY KGi]
(x a) (x b) (x c) P(x) = 1 + 1 +x 1 ev, 3x2 2(a + b + c)x + ab + bc + ca = 0
x
a x b c a = 3, b = 2, c = 1 ewmGq cvB,
(x a) (x b) (x c) 3x2 12x + 11 = 0
= (x b) (x c) + (x a) (x c) + (x a) (x b) ‣B mgxKiGYi c†^vqK, D
= x2 bx cx + bc + x2 ax cx + ca + x2 ax bx + ab
= 3x2 2ax 2bx 2cx + ab + bc + ca = (12)2 4 3 11
= 3x2 2(a + b + c)x + (ab + bc + ca)
= 144 132 = 12
(x a) (x b) (x c) P(x) eüc`xi a‚eË c`
‣LvGb, c†^vqK cƒYÆeMÆ msLÅv bq
ab + bc + ca (Ans.) myZivs gƒj«¼q Agƒj`|
eüc`x I eüc`x mgxKiY 3
‣Lb, 3x2 12x + 11 = 0
ev, (12) (12)2 4 3 11
x= 23
12 144 132 12 12 12 2 3
=6 =6 =6
12 23 = 2 1
=6 6 3
cÉ`î mgxKiGYi gƒj«¼q 2 1 , gƒj«¼q Agƒj`| (Ans.)
3
DËi ms‡KZmn m„Rbkxj cÖkœ
cÉk² 4 `†kÅKÍ-¸ 1: 2 2b = 2 + 2 1 M. `†kÅK͸ 2 ‣ ewYÆZ mgxKiGYi a ‣es b gƒj` nGj
`†kÅK͸-2: g(x) = (n + 1)x2 x 1 •`LvI •h mgxKiYwUi gƒj«¼q gƒj` nGe| 4
wkLbdj-1, 4 Dîi: K. 196; L. 11x2 – 6x + 1 = 0;
K. 2i ‣i eMÆgƒj wbYÆq Ki| 2
L. (1 b2)2 (1 + b2)2 ‣i gvb wbYÆq Ki| 4 cÉk² 7 Zzwj I wgwj covi RbÅ jvBGeËix nGZ wKQy eB avi
M. g(x) = 0 mgxKiGYi ‣KwU gƒj AciwUi w«¼àY nGj, wbj| ZvG`i KvGQ ^vKv eBGqi msLÅv x2 ax + b = 0.
n ‣i gvb wbYÆq Ki| 4 mgxKiYwUi gƒj «v¼ iv cÉKvk Kiv hvq| wgwji •PGq Zzwj wKQy
Dîi: K. (1+ i); L. i = i; M. 11 eB •ewk wbGqwQj| A wkLbdj-1, 4, 6
1 9;
B K. mgxKiYwUi c†^vqK •ei KGiv| 2
cÉk² 5 `†kÅKÍ-¸ 1: z = x + iy RwUj msLÅv| L. a = 9 I b = 20 nGj Zzwj I wgwji KvGQ ^vKv
wkLbdj-1, 4 eBGqi msLÅv •ei KGiv| 4
`†kÅK͸-2: px2 + qx + r = 0 ‣KwU w«N¼ vZ mgxKiY|
K. 3 i RwUj msLÅvwUi gWzjvm I AvàÆGg´Ÿ KZ? 2 M. wgwj 1 wU eB ZzwjGK covi RbÅ w`Gj ZvG`i KvGQ
L. |z 2| = 3 «v¼ iv wbG`ÆwkZ mçvic^ KxGmi mgxKiY ^vKv eBGqi msLÅv cÉKvGki RbÅ ‣KwU mgxKiY
wbG`Æk KGi? `†kÅK͸-1 nGZ •`LvI| 4 MVb KGiv| 4
M. `†kÅKÍ-¸ 2 ‣ ewYÆZ mgxKiGYi gƒj«¼q I 2 nGj, Dîi: K. a2 − 4b; L. 5 I 4; M. x2 − 9x + 18 = 0
cÉgvY Ki •h,
cÉk² 8 `†kÅK͸-1: z = x + iy
p q3 p
r q =r 4 |z 2| = 3|z + 2| wkLbdj-4, 7, 8
Dîi: K. ; L. e†î; M. p `†kÅK͸-2: x2 + px + q = 0.
6 r
2, ; K. 1 3i ‣i gWzjvm I AvàÆGg´Ÿ wbYÆq Ki| A 2
cÉk² 6 `†kÅK͸ 1: 1 ‣KwU RwUj msLÅv| L. `†kÅK͸-1 ‣ DGÍÏwLZ mçvicG^i mgxKiY •ei
3+i 2 Ki| 4
`†kÅK͸ 2: x2 − 2x = (a − b)2 − 1 ‣KwU w«¼NvZ mgxKiY| M. `†kÅK͸-2 ‣ DGÍÏwLZ mgxKiGYi `ywU gƒj I
wkLbdj-4, 5 I 6 nGj ‣gb ‣KwU mgxKiY wbYÆq Ki hvi gƒj«¼q nGe
B K. p ‣i gvb KZ nGj px2 + 3x + 4 = 0 mgxKiGYi + ‣es . 4
gƒj `yBwU evÕ¦e I mgvb nGe? A 2 Dîi: K. 2, 23; L. x2 + y2 + 5x + 4 = 0;
L. `†kÅK͸ 1 ‣ ewYÆZ RwUj msLÅvwU •h w«¼NvZ M. x2 – ( p2 4q p)x p p2 4q = 0
mgxKiGYi gƒj Zvi mgxKiY wbYÆq Ki| 4
w«¼c`x weÕ¦w‡ Z 1
cÂg AaÅvq
w«¼c`x weÕ¦‡wZ
cixÞvq Kgb •cGZ AviI ckÉ ² I mgvavb
5 15 x 1 = 15C8 37.x7 (1)8 58
x2 5 2 x16
(ii)
( ) ( )ckÉ ² 1 (i)3x– 1 –
K. x-‣i •Kvb gvGbi RbÅ 3x2 + 5x + 4 ‣i weÕ¦w‡ Z ․ea? 2 6435 37 58
(2 + x)2 (3 + x) = x9 (Ans.)
L. (i) ‣ ewYÆZ w«¼c`xwUi gaÅc` wbYÆq Ki| 4 11
x 2 = 1 + x 2
5 5
M. (ii) ‣ ewYÆZ w«¼c`xwUGK x-‣i kwÚi Da»ÆKÌgvbymvGi cçg c` ( ) ( )M (ii) ‣ c`É î ivwk, 1
ch¯Æ ¦ weÕ¦Z‡ Ki ‣es •`LvI •h, ( ) ( )( )111 1 1 1 1 2
2 2 2
– 1 – 11 – 11 – 11 – 3 4 = 1+ 1 5x+ 2 5x2+2 5x3
5 2.52 2.53 23.53 5 2
1 ... ... = 2! 3!
1 bs cGÉ ki² mgvavb ( )( )( )11 1 1 2 1 3
2 2 2
1 11 x 2 ‣i weÕ¦w‡ Z ․ea nGe 2 5x4+ .....
(2 + x)2 = x 2=4 2
( )K 1 + + 4!
( )4 1 + 2
1 .1 1 .1 .3 1 .1 .3 .5
22 x2 222 x3 2 222 x4
| |hw` x = 1 1 . x 2 52 2.3 53 2.3.4 . 54 ... ... ...
2 <1 2 5
ev, |x| < 2 ‣Lb, x = 2 ewmGq cvB,
ev, 2 < x < 2 1 11 22 1 .1 .3 23 1 .1 .3 .5 24
2 .2 52 2 22 53 2 222 54
1 ‣i weÕ¦w‡ Z ․ea ( )1 2 2 = 1 1 .2 . 2.3 . 2.3.4 . − ... ...
1 1 1 x 5 2 5 2
3+x x =3 1+3
( )Avevi, =
( )3 1 + 3
3 1 1 1 .1 1 .1 1 .1
5 5 2 52 2 53 23 53
( )ev, 2 = 1 ......
| |nGe hw` x 1 1 1 .512 1 .513 1 .513 ...... = 3 (ˆ`LvGbv nGjv)
3 <1 5 2 2 23 5
ev, |x| < 3
ev, 3 < x < 3 b n ‣KwU w«¼c`x ivwk|
2x + x3
x ‣i ․ea eÅewa = {2 < x < 2} {3 < x < 3} ( )ckÉ ² 2
=2<x<2 A K. (1 − 5x)−1 •K cçg c` chƯ¦ weÕ¦Z‡ KGiv| 2
= |x| < 2 (Ans.) L. ivwkwUi weÕ‡¦wZGZ aË‚e c`wUi gvb wbYÆq Ki, hLb n = 12 4
( )L (i) nGZ cvB, cÉ`î weÕ¦w‡ Z 5 15 M. n = 10 nGj ivwkwUi weÕ¦‡wZGZ 5 Zg I 6 Zg cG`i mnM
x2
3x mgvb nGj b ‣i gvb wbYÆq KGiv| 4
•hGnZz 15 ‣KwU weGRvo msLÅv| myZivs •gvU c` msLÅv nGe 2 bs cÉGki² mgvavb
16wU| hv ‣KwU •Rvo msLÅv| K (1 − 5x)−1 = 1 + 5x + (5x)2 + (5x)3 + (5x)4 + ....
myZivs gaÅc` ^vKGe `yBwU| = 1 + 5x + 25x2 + 125x3 + 625x4 + .... (Ans.)
15 + 1 15 + 1
2 2 +1
( ) ( )c` `By wU nGjv Zg c` b 12
Zg c` I 2x + x3
( )L DóxcGK cÉ`î ivwkGZ n = 12 ewmGq cvB,
= 8 Zg c` I 9 Zg c`|
b 12 ‣i weÕ¦w‡ ZGZ (r + 1) Zg cG` aË‚e c`
2x + x3
8Zg c` = 15C7 (3x)15 7 x527 ( )awi,
= 15C7 315 7 x15 7 (1)7.57 x14 iGqGQ A^Ævr x0 iGqGQ|
6435 38 57
( ) (r + 1) Zg c` = 12Cr (2x)12r b r
= x6 (Ans.) x3
9Zg c` = 15C8 (3x)15 8 x52 8 = 12Cr 212 − r x12 − r. br
x3r
= 12Cr 212 − r .b r. x12 − r− 3r
= 12Cr 212 − r .b r. x12 − 4r
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò ‣Kv`k-«¼v`k •kwÉ Y
•hGnZz ‣B c`wUGZ x0 iGqGQ, •mGnZz = 10C426x6 b4 = 10C426b4 1
x0 = x12 − 4r ev, 0 = 12 − 4r ev, 4r = 12 r = 3 x12 x6
3 + 1 ev, 4 Zg cG` a˂e c` AvGQ| 5 Zg cG`i mnM = 10c4 26b4
wbGYÆq aË‚e c` = 12C3 212 − 3 b3 = 12C3 29b3 ( )‣es 6 Zg c` = 10C5 (2x)10 − 5 b 5 10C5 25x5 b5
x3 x15
= 220 512 b3 = 112640 b3 (Ans.) =
b 10 = 10C5 25b5 1
2x + x3 . x10
( )M DóxcGK n = 10 ewmGq cvB,
6 Zg cG`i mnM = 10C5 25b5
10‣i weÕ¦‡wZGZ (r + 1) Zg c` = 10Cr(2x)10 −r b kZÆgGZ, 10C4 26b4 = 10C5 25b5
b x3 r
( ) ( )2x + x3 210 2 5
252 3
( ) 5 Zg c` = (4 + 1) Zg c` = 10C4(2x)10 − 4 b4 ev, 210 2 = 252 b ev, b = = (Ans.)
x3
DËi ms‡KZmn m„Rbkxj cÖkœ
x 1 M. 1 nGj •`LvI •h, ivwkwUi weÕ¦w‡ ZGZ Zg
6 2
( )ckÉ ² 3 `†kÅK͸-1: x < 6 nGj 1 2 •K Ea»KÆ ÌwgK avivq n = (r + 1)
weÕ¦Z‡ KGi 1 1 1 1 1 1 3 ... ... = 2 cvIqv hvq| c` ( 1)2rr..a(nr!)r2(2r)!xr 4
6 6. 12 6. 12 . 18 3
Dîi: K. 1 + 10x + 40x2 + 80x3 + 80x4 + 32x5;
2x 1 (1 + x)2
( )`†kÅK͸-2: 1 3 2 I (1 x)3 `yBwU w«¼c`x ivwk| L. b = 3, a = 35 I n = 5;
K. x-‣i •Kvb gvGbi RbÅ `†kÅK͸-2 ‣ ewYÆZ w«¼c`x cÉk²5 (x) = A.B ‰KwU dvskb ˆhLvGb A = (1 2x);
B = (1 4x) ‰es x R.
ivwkwUi weÕ¦w‡ Z Awfmvix? 2
L. `k† ÅKÍ-¸ 1 ‣ ewYÆZ DwÚwUi mZÅZv hvPvB Ki| 4 K.( )A1 wbYÆq KGiv| 2
2
M. cgÉ vY Ki •h, `k† ÅK͸-2 ‣ ewYÆZ w«¼c`x weÕ¦w‡ ZwUi xr ‣i
L. An ‣i cçg cG`i mnM ‣es Bn ‣i Z‡Zxq cG`i mnM
mnM 2r2 + 2r + 1| 4
mgvb nGj n wbYÆq KGiv| 4
Dîi: K. 3 < x < 3 ;
2 2 M. •`LvI •h, [(x)]1 ‣i weÕ¦w‡ Zi ‣i mnM 1
xr 2
cÉk² 4 (a + 2x)n ‣KwU w«¼c`x ivwk •hLvGb a evÕ¦e msLÅv| (4r + 1 2r + 1) 4
A K. a = 1 I n = 5 nGj cÅvmGKGji wòfzGRi mvnvGhÅ Dîi: K. 0; L. 6;
DóxcGKi ivwkwUi weÕ¦w‡ Z wbYÆq Ki| 2
L. ivwkwUi weÕ¦w‡ Zi cÉ^g wZbwU c` b, 10 bx I 40 bx2 nGj
3 9
a, b, n ‣i gvb wbYÆq KGiv| 4
1
cixÞvq Kgb •cGZ AviI cÉk² I mgvavb
cÉk²1 (i) x2 – 2ax + a2 = b2 ‣KwU w«N¼ vZ mgxKiY| 16 ev, (r + 1)! (23 – r)! = 1
r!(24 – r)! 4
(ii) (1 + x)24 ‣KwU exRMvwYwZK ivwk| ev, (r + 1)r! (23 – r)! = 1 [ n! = n(n – 1)!]
r!(24 – r) (23 – r)! 4
4 I 5 AaÅvGqi mg®¼Gq
A K. (ii) ‣i ivwkwUi weÕ¦‡wZGZ gaÅc` wbYÆq Ki| 2 ev, r+1 = 1 ev, 4r + 4 = 24 – r
24 – r 4
L. ‣gb ‣KwU mgxKiY wbYÆq Ki hvi gƒj `yBwU (i) bs
mgxKiGYi gƒj `yBwUi mgwÓ¡ ‣es A¯¦idGji cig ev, 5r = 20
gvb nGe| 4 r=4
M. (ii) ‣i ivwkwUi weÕ¦‡wZGZ `yBwU KÌwgK c` wbYÆq Ki KÌwgK c` `ywU 5-Zg I 6-Zg c` (Ans.)
hvG`i mnGMi AbycvZ 1 t 4 4
1 bs cÉGk²i mgvavb cÉk² 2 (x) = (1 + x + x3)n, n
K (1 + x)24 ivwkwUi NvZ 24 hv •RvomsLÅv| myZivs gaÅc` 4 I 5 AaÅvGqi mg®¼Gq
nGe 1wU|
B K. (1 – x2)6 ‣i weÕ¦‡wZGZ gaÅc` wbYÆq KGiv| A 2
c`É î ivwkwUi weÕ¦‡wZGZ 24 + 1 = (12 + 1) Zg c`| L. n = 1 ‣i RbÅ (x) = 0 mgxKiGYi gƒj wZbwU a, b
2
I c nGj, a2b wbYÆq KGiv| 4
gaÅc` = 24C12 12412 x12 = 2704156x12 (Ans.)
M. n = 9 nGj, (x) ‣i weÕ‡¦wZGZ x5 ‣i mnM wbYÆq KGiv|4
L DóxcK •^GK Avgiv cvB, x2 – 2ax + a2 = b2
ev, x2 – 2ax + a2 – b2 = 0 2 bs cÉGki² mgvavb
gGb Kwi, x2 2ax + a2 b2 = 0 mgxKiGYi gƒj«¼q ,
K (1 – x2)6 ‣i c` msLÅv 7wU
+ = (2a) = 2a
1 weÕ¦‡wZi gaÅc` nGe 26 + 1 Zgc` = (3 + 1) Zgc`
(3 + 1) Zgc` = 6C3 (–x2)3 = –20x6
= a2 b2 = a2 b2
1 L DóxcK •^GK Avgiv cvB,
( )2 = ( + )2 4 = (2a)2 4(a2 b2) (x) = (1 + x + x3)n
= 4a2 4a2 + 4b2 = 4b2
‣Lb, n = 1 ‣i RbÅ (x) = 0
| | = 2b
0 = (1 + x + x3)1
gƒj `yBwU nGe + I | |
ev, 1 + x + x3 = 0 ev, x3 + x + 1 = 0
gƒj«¼Gqi •hvMdj = ( + ) + | | =2a +2b = 2(a + b) mgxKiGYi gƒj wZbwU a, b, c nGj
‣es gƒj«¼Gqi àYdj = ( + ) | | = 2a 2b = 4ab
0
wbGYÆq mgxKiY x2 2(a + b)x + 4ab = 0 a+b+c=1=0
M DóxcK •^GK Avgiv cvB, ab + bc + ca = 1 ‣es abc = – 1
‣Lb a2b = a2b + a2c + b2c + b2a + c2a + c2b
(1 + x)24 ‣KwU exRMvwYwZK ivwk|
= a2b + b2a + a2c + c2a + b2c + c2b
(1 + x)24 ‣i weÕ¦‡wZGZ
= ab(b + a) + ac(a + c) + bc(b + c)
Tr+1 = 24Cr xr ‣es Tr+2 = 24Cr+1 xr+1 = (a + b + c) (ab + bc + ca) – 3abc
= 0.1 – 3(–1) = 3 (Ans.)
kZÆgGZ, 24Cr = 1
24Cr+1 4
ev 24! (r + 1)! (24 – r – 1)! = 1
r!(24 – 24! 4
r)!
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò
M DóxcK •^GK Avgiv cvB, (x) = (1 + x + x3)n + 9C1x3 (1 + 8C1x + 8C2x2 + 8C3x3 + ... ...)
‣Lb, n = 9 nGj, f(x) = (1 + x + x3)9 + 9C2x6 (1 + x)7 + ... ...
‣Lb, (1 + x + x3)9
wbGYÆq x5 ‣i mnM = 9C5 + 9C1 8C2
= {(1 + x) + x3}9
= (1 + x)9 + 9C1(1 +x)8x3 + 9C2(1 + x)7(x3)2 + ... ... = 126 + 9 28 = 126 + 252
= (1 + x)9 + 9C1x3(1 + x)8 + 9C2x6(1 + x)7 + ... ...
= (1 + 9C1x + 9C2x2 + 9C3x3 + 9C4x4 + 9C5x5 + ... ...) = 378 (Ans.)
DËi ms‡KZmn m„Rbkxj cÖkœ
cÉk² 3 P = 5x + 2 ‣KwU w«c¼ `x ivwk| K. w«N¼ vZ ivwki mGeÆvœP ev meÆwbÁ² nIqvi ivwkgvjv
cÉwZcv`b Ki| 2
wkLbdj-4, 5 I 6 L. `†kÅK͸-1 Abhy vqx k ‣i •Kvb gvGbi RbÅ RbÅ x ewRÆZ
K. 3x2 – 1 10 ‣i weÕ¦‡wZGZ gaÅc` wbYÆq KGiv| 2 c` 405 nGe? 4
3x
M. g3 x ‣i weÕ¦‡wZGZ x7 ‣es x8 ‣i mnM mgvb nGj
L. P34 ‣i weÕ¦‡wZGZ `yBwU KÌwgK cG`i mnM mgvb + 2 n
nGj, ‣ c` `yBwUi x-‣i NvZ wbYÆq KGiv| 4 ‣i gvb wbYÆq Ki| 4
M. – 1 weÕ¦‡wZGZ xr ‣i mnM wbYÆq KGi Dîi: L. k = 3; M. n = 55;
P
2-‣i
weÕ¦‡wZwUi PZz^Æ c`wUI •ei KGiv| 4 cÉk² 6 `†kÅK͸-1: 2x2 1 10
2x3
Dîi: K. − 252x5; L. 24 I 25; M. 625 x3;
`†kÅK͸-2 : P = 1 5x + 6x2 wkLbdj-4 I 5
128 2
cÉk²4 2x 110 ‣KwU w«c¼ `x ivwk| wkLbdj-2 I 4 K. (a + 2x)5 ‣i weÕ¦‡wZGZ x3 ‣i mnM 320 nGj, a ‣i gvb
x
+ wbYÆq Ki| 2
A K. w«c¼ `x ivwkwUi weÕ¦‡wZi PZz^Æ c` wbYÆq KGiv| 2 L. `†kÅK͸-1 ‣ cÉ`î weÕ¦‡wZGZ x ewRÆZ c`wU •ei KGi ‣i
L. w«c¼ `x ivwkwUi weÕ¦‡wZi x–6 ‣i mnM wbYÆq KGiv| 4 gvb wbYÆq Ki| 4
M. •`LvI •h, w«c¼ `x ivwkwUi weÕ¦‡wZi gaÅc`wU M. `†kÅK͸-2 ‣i AvGjvGK cÉgvY Ki 1 ‣i weÕ¦‡wZGZ xm
P
x ewRÆZ| 4
Dîi: K. 15360x4; L. 180; ‣i mnM
cÉk²5 `†kÅKÍ-¸ 1: k ‣i •Kvb evÕ¦e gvGbi RbÅ 3m+1 2m+1 4
x k 10 ‣KwU w«c¼ `x ivwk| wkLbdj-2 I 4 Dîi: K. a = 2; L. 5 Zg c`, 840;
x2
`†kÅK͸-2: g(x) = xn •hLvGb n ‣KwU •hvMGevaK cƒYÆmsLÅv|
KwYK 1
lÓ¤ AaÅvq
KwYK
mgvavb
( )cÉk²1e= 4 , P 130 5 , Q(2, 3) +2 M P(x, y)
5 2= 2
K. y2 = 4px cive†î Q we±`yMvgx nGj ‰i DcGK±`Ê wbYÆq Ki| 2 ev, = 4 – 2
L. ˆKvGbv Dce†Gîi AÞ `yBwU Õ©vbvGâi AÞ `yBwUi Ici =2 y=6 Z
A(2,3) S(,)
DrGKw±`ÊKZv e ‰es ‰wU P we±`yMvgx nGj Dce†îwUi ‰es 3 = + 6
2
mgxKiY wbYÆq Ki| 4
M. ‰KwU cive†Gîi kxlÆwe±`y Q ‰es wbqvgK y = 6 nGj = 0.
cive†îwUi mgxKiY wbYÆq Ki| 4
DcGK±`Ê S (2, 0).
awi, cive†îwUi DciÕ© P(x, y) ˆhGKvGbv ‰KwU we±`y|
1 bs cÉGki² mgvavb
K c`É î civeî† , y2 = 4px ............. (i) P (x, y) we±`y nGZ w`KvÞ y – 6 = 0 ‰i
y–6
(i) bs Q(2, 3) we±`yMvgx nGj cvB, jÁ¼ `ƒiZ½ PM nGj, PM = 12 =y–6
32 = 4p.2 PS = (x – 2)2 + (y – 0)2 = (x –2)2 + y2
ev, 9 = 8p civeG† îi msævbymvGi Avgiv Rvwb, PS = PM
p = 9 ev, (x – 2)2 + y2 = y – 6
8
p ‰i gvb (i) bs ‰ ewmGq cvB,
ev, (x – 2)2 + y2 = (y – 6)2 [eMÆ KGi]
9 ev, x2 – 4x + 4 + y2 = y2 – 12y + 36
y2 = 4 . 8 x ........... (ii) ev, x2 – 4x + 4 + y2 – y2 + 12y – 36 = 0
(ii) bs y2 = 4ax ‰i mvG^ Zzjbv KGi cvB, a = 9 x2 – 4x + 12y – 32 = 0
8
hv wbGYÆq civeG† îi mgxKiY| (Ans.)
( ) DcGK±`Ê 9
8 0 (Ans.) ckÉ ² 2 Y
(6,4)
L gGb Kwi, Dce†Gîi mgxKiY, x 2 y2 1
a2 b2 (-3,1)
O
b2 X
‰LvGb, e2 a2 A S(-2,0) S(2,0) A X
= 1
ev, 4 2 = 1 – b2 ev, b2 = 1 16
5 a2 a2 25
Y
ev, b2 9 ev, b3 b = 3a K. y2 4x + 8 = 0 cive†Gîi DcGK±`Ê wbYÆq Ki| 2
a2 = 25 a=5 5
ˆhGnZz Dce†îwU P 10 , 5 we±`y w`Gq AwZKÌg KGi, L. DóxcGKi Dce†îwUi AA = 8 nGj ‰i mgxKiY wbYÆq Ki| 4
3 M. DóxcGK DGÍÏwLZ Awae†Gîi mgxKiY wbYÆq Ki| 4
100 5 1 ev, 100 5 25 1 ev, 9a2 = 225 2 bs cÉGki² mgvavb
9a 2 b2
9a2 9a2 K c`É î mgxKiY, y2 4x + 8 = 0
a=5 ev, y2 = 4x 8
ev, y2 = 4.1.(x 2)
b=3 DcGK±`Ê (a, 0)
ev, X = a, Y = 0
wbGYÆq Dce†Gîi mgxKiY, x2 y2 1 (Ans.) ev, x 2 = 1, y = 0
ev, x = 3, y = 0
25 9
DcGK±`Ê (3, 0) (Ans.)
M gGb Kwi, civeG† îi DcGK±`Ê S (, ) ‰es kxlÆwe±`y Q (2, 3)
‰i AÞGiLv nGe wbqvgK y – 6 = 0 ‰i Ici jÁ¼| L gGb Kwi, P(x, y) wPGòi DceG† îi DcwiÕ© ‰KwU we±`y|
‰Lb, y – 6 = 0 ˆiLvi Ici jÁ¼GiLvi mgxKiY DceG† îi agÆvbymvGi,
x + k = 0 ‰wU (2, 3) we±`yMvgx|
PS + PS = AA
2+k=0
k=–2 ev, (x 2)2 + (y 0)2 + (x + 2)2 + (y 0)2 = 8 [ AA = 8]
ev, (x 2)2 + y2 + (x + 2)2 + y2 = 8
AÞGiLvi mgxKiY, x – 2 = 0 x = 2
Avevi, y – 6 = 0 y = 6
AÞGiLv I wbqvgK ˆiLvi ˆQ`we±`yi Õ©vbvâ Z(2, 6)
kxlÆwe±`y Q (2, 3), ZS ‰i gaÅwe±`y|
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò ‰Kv`k-«¼v`k ˆkwÉ Y
ev, (x 2)2 + y2 = 8 (x + 2)2 + y2 ev, 9 1 = 1 ............... (2)
a2 b2
ev, (x 2)2 + y2 = 64 16 (x + 2)2 + y2 + (x + 2)2 + y2
[DfqcÞGK eMÆ KGi] ˆhGnZz (1) Awae†îwU (6, 4) we±`yMvgx
ev, (x 2)2 (x + 2)2 = 64 16 (x + 2)2 + y2 36 – 16 = 1 ... ... ... (3)
a2 b2
ev, (x + 2)2 (x 2)2 = 16 (x + 2)2 + y2 64
[DfqcÞGK (–1) «¼vivàY KGi] (2) ˆK 4 «¼viv àY KGi, (3) weGqvM KGi cvB,
ev, 4.x.2 = 16 (x + 2)2 + y2 64 [ (a + b)2 (a b)2 = 4ab] 36 – 36 – 4 + 16 = 4 – 1
a2 a2 b2 b2
ev, 8x = 16 (x + 2)2 + y2 64
ev, 12
ev, x = 2 (x + 2)2 + y2 8 b2 = 3
ev, (x + 8)2 ={2 (x + 2)2 + y2 )}2 ev, b2 = 4
ev, x2 + 16x + 64 = 4{(x + 2)2 + y2}
ev, x2 + 16x + 64 = 4(x2 + 4x + 4) + 4y2 b=2
ev, x2 + 16x + 64 = 4x2 + 16x + 16 + 4y2
ev, 3x2 + 4y2 + 16 64 = 0 b ‰i gvb (2) bs mgxKiGY ewmGq cvB,
3x2 + 4y2 − 48 = 0 (Ans.) 9 1 =1
a2 22
ev, 9 1
a2 =1+4
M wPò nGZ ˆhGnZz Awae†Gîi Avo AÞ x-AÞ eivei ‰es ˆK±`Ê ev, 9 5
gƒjwe±`yGZ AewÕ©Z| a2 =4
ev, a2 = 9 4 = 36
5 5
Awae†îwUi mgxKiY,
x2 y2 x2 y2
a2 b2 = 1 ................. (1) (1) nGZ cvB, 36 22 = 1
wK¯§ Awae†îwU (3, 1) ‰es (6, 4) we±`yMvgx [DóxcGKi wPò nGZ] 5
x = – 3 ‰es y = 1 ewmGq (1) mgxKiY nGZ cvB, 5x2 y2 =1 (Ans.)
36 4
(3)2 12
a2 b2 =1
DËi ms‡KZmn m„Rbkxj cÖkœ
ckÉ ² 3 y2 = 4ax ‰KwU civeG† îi mgxKiY| wkLbdj- 3, 4, 9 Dîi: K. 3 L. M. 144.
A K. ‰KwU mij wPGòi mvnvGhÅ cive†î DcÕ©vcb Ki| 2 2 < r < 1; 5 3, 0 ;
L. n = am2 nq ZGe ˆ`LvI ˆh, x + my + n = 0 ˆiLv ckÉ ² 5 `†kÅK͸-1: ‰KwU cive†Gîi AÞGiLv y-AGÞi mgv¯¦ivj|
civeî† wUGK Õ·kÆ KiGe| B 4
M. a = 4 ‰es ˆKvb we±`yi DcGKw±`ÊK `iƒ Z½ 6 nGj H we±`yi `†kÅK͸-2 : ‰KwU Awae†Gîi DrGKw±`ÊKZv 45, DcGK±`Ê (2, 0) ‰es
wbqvgK ˆiLvi mgxKiY 4x – 3y = 2 wkLbdj- 6, 23, 26, 27
Õ©vbvâ wbYÆq Ki| 4 x2 y2
Dîi: M. (2, 4 2) 9 25
B K. + = 1 DceG† îi mvGcGÞ (6, –5) we±`yi AeÕ©vb
ckÉ ² 4 x2 + y2 = 1 ‰KwU Dce†Gîi mgxKiY| wkLbdj- 3, 12, 21 wbYÆq Ki| A 2
p 52 L. `k† ÅKÍ-¸ 1 ‰ ewYÆZ cive†îwU (3, 0), (–3, 0) I (2, 5)
A K. DrGKw±`ÊKZv e = 2r + 3 nGj r ‰i Dci Kx kZÆ AvGivc
KiGj KwYKwU Dceî† nGe? B 2 we±`y w`Gq AwZKgÌ KiGj Dnvi mgxKiY wbYÆq Ki| 4
M. `k† ÅKÍ-¸ 2 ‰ ewYÆZ Awae†Gîi mgxKiY wbYÆq Ki| 4
L. DóxcGK DwÍÏwLZ Dceî† wU (6, 4) we±`yMvgx nGj Dîi: (K) wfZGi ; (L) x2 + y – 9 = 0;
DcGKG±`Êi Õ©vbvâ wbYÆq Ki| 4
M. ‰KwU Awae†Gîi Abye®¬x AGÞi Š`NÆÅ 24 ‰es DcGK±`Ê«¼q (M) 7y2 + 24xy – 48x – 12y + 60 = 0
(0, 13) nGj ˆ`LvI ˆh, p = 144. 4
1
cixÞvq Kgb ˆcGZ AviI cÉk² I mgvavb
cÉk² 1 `†kÅKÍ-¸ 1: `†kÅKÍ-¸ 2: Dce†Gîi msæv ˆ^GK Avgiv cvB,
p(x, y) M Y SP = e ev, SP2 = e2 . MP2
(3, 1) (6, 4) MP
x2 + y2 2x + 2y + 2 = 1 x y+ 22
2 2
C S(1, 1) Z X O X 4(x2+y22x + 2y + 2) = x2 + y2 + 4 2xy + 4x 4y
3x2 + 3y2 + 2xy 12x + 12y + 4 = 0, hv wbGYÆq
Y Dce†Gîi mgxKiY|
wkLbdj-10, 15, 16, 17 I 21 S(1, 1) ˆ^GK MZ-‰i Dci jÁ¼ `iƒ Z½,
K. y2 = 2(x + 3) cive†îwUi DcGK±`Ê ‰es wbqvgK SZ = 1+1 (+12)2= 4 2
12 + =2
ˆiLvi mgxKiY wbYÆq Ki| 2
2
L. DóxcGK DwÍÏwLZ Dce†Gîi mgxKiY I DcGKw±`ÊK DcGKw±`ÊK jGÁ¼i Š`NÆÅ = 2e. SZ = 2. 1 . 2 2 = 4
jGÁ¼i Š`NÆÅ wbYÆq Ki| (e = 1 ) 4 2
2 M wPò nGZ ˆhGnZz Awae†Gîi Avo AÞ x-AÞ eivei ‰es
ˆK±`Ê gƒjwe±`yGZ AewÕ©Z|
M. DóxcGK DGÍÏwLZ Awae†Gîi mgxKiY I AÞ«G¼ qi
Š`NÆÅ wbYÆq Ki| 4 Awae†îwUi mgxKiY,
1 bs cÉGk²i mgvavb
x2 y2 = 1 ................. (1)
K ˆ`Iqv AvGQ, y2 = 2(x + 3) a2 b2
ev, y2 = 4. 1 .(x + 3) wK¯§ Awae†îwU (3, 1) ‰es (6, 4) we±`yMvgx [DóxcGKi wPò
2
nGZ]
‰LvGb, x + 3 = X ‰es y = Y emvGj Avgiv cvB,
x = – 3 ‰es y = 1 ewmGq (1) mgxKiY nGZ cvB,
1
Y2 = 4. 2 .X (3)2 12 =1
a2 b2
BnvGK Y2 = 4aX ‰i mvG^ Zzjbv KGi cvB,
ev, 9 1 = 1 ............... (2)
1 a2 b2
a = 2
ˆhGnZz (1) Awae†îwU (6, 4) we±`yMvgx
1
DcGK±`Ê: X=a ev, X=2 , ‰es Y = 0 36 – 16 = 1 ... ... ... (3)
a2 b2
1
ev, x+3=2 y = 0 (2) ˆK 4 «¼viv àY KGi, (3) weGqvM KGi cvB,
x = – 5 36 – 36 – 4 + 16 = 4 – 1
2 a2 a2 b2 b2
wbGYÆq DcGK±`Ê 5 0 (Ans.) ev, 12 = 3 ev, b2 = 4
2 b2
wbqvgK ˆiLvi mgxKiY, X = 1 b=2
2
b ‰i gvb (2) bs mgxKiGY ewmGq cvB,
ev, x+3 = 1 9 1 =1
2 a2 22
ev, 2x + 6 = 1 ev, 9 1
a2 =1+4
2x + 7 = 0 (Ans.)
ev, 9 5
L gGb Kwi, Dce†îwUi DcGK±`Ê S (1, 1); wbqvgKGiLv, a2 = 4
MZ ‰i mgxKiY x y + 2 = 0 ... (i)
ev, a2 = 9 4 36
5 =5
‰es P(x, y) we±`ywU Dce†Gîi Dci AewÕ©Z|
SP = (x 1)2 + (y + 1)2
SP = x2 + y2 2x + 2y + 2
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò
(1) nGZ cvB, x2 y2 =1 ev, 36 = 1 16
36 22 p 25
5 ev, 36 9
p = 25
5x2 y2 =1 (Ans.)
36 4 ev, 9p = 36 25
Avo AGÞi Š`NÆÅ = 2a = 2 6 12 12 5 ‰KK p = 100 (Ans.)
= = x2 y2
5 5 5 100 + 25 = 1
(Ans.) ‰LvGb, a = 10, b = 5 ‰es a > b
Abey ®¬x AGÞi Š`NÆÅ = 2b = 2 2 = 4 ‰KK (Ans.) b2
a2
cÉk² 2 2x + y = 1 ... ... ... ... (i) DrGKw±`ÊKZv, e = 1 = 1 25 = 3
100 2
x2 y2
S1 (1, 1), e = 3, p + 52 = 1 ... ... ... ... (ii) A (6, 4) (Ans.)
wkLbdj-9, 10, 11, 15, 23 I 24 kxlÆwe±`y«¼Gqi Õ©vbvvâ ( a, 0) ev ( 10, 0) (Ans.)
DcGK±`Ê«G¼ qi Õ©vbvvâ ( ae, 0)
K. y2 = 16x cive†Gîi DciÕ© ˆKvGbv we±`yi DcGKw±`ÊK
ev, 10. 23 0
`ƒiZ½ 6, we±`ywUi Õ©vbvâ wbYÆq Ki| 2 ev, ( 5 3, 0) (Ans.).
L. (ii) mgxKiY hw` A we±`y w`Gq hvq, ZvnGj P ‰i gvb
KZ? ‰QvovI kxlÆwe±`y, DrGKw±`ÊKZv ‰es DcGK±`Ê«¼Gqi M gGb Kwi, P(x, y) Awae†Gîi Ici ˆh ˆKvGbv we±`y|
DcGK±`Ê (1, 1) nGZ P(x, y) we±`yi `ƒiZ½ =
Õ©vbvâ ˆei KGiv| 4
M. wbqvgGKi mgxKiY (i), DcGK±`Ê S1 ‰es DrGKw±`ÊKZv e
(x 1)2 + (y 1)2
nGj, Awae†îwUi mgxKiY wbYÆq Ki| 4
2 bs cÉGki² mgvavb 2x + y 1
wbqvgK nGZ P we±`yi jÁ¼ `iƒ Z½ = 22 + 12
K ˆ`Iqv AvGQ, y2 = 16x (i) 2x + y 1
5
ev, y2 = 4.4.x =
‰GK y2 = 4ax ‰i mvG^ Zzjbv KGi cvB, a = 4 Awae†Gîi msæv nGZ Avgiv cvB,
Avgiv Rvwb, DcGKw±`ÊK `ƒiZ½ = a + x (x 1)2 + (y 1)2 = 2x + y 1
5
6=4+x 3 .
x=2 (x 1)2 + (y 1)2 = 3 2x + y 12
5
(i) bs mgxKiGY x-‰i gvb ewmGq cvB, y2 = 32
y=4 2 5{(x 1)2 + (y 1)2} = 3(2x + y 1)2
wbGYÆq we±`yi Õ©vbvâ (2, 4 2) (Ans.) 7x2 2y2 + 12xy 2x + 4y 7 = 0 ‰wUB wbGYÆq
L x2 y2 = 1, hv (6, 4) we±`yMvgx Awae†Gîi mgxKiY|
p + 25
36 16
p + 25 = 1
DËi ms‡KZmn m„Rbkxj cÖkœ
cÉk² 3 P(x, y) M. 271x2 + 247y2 + 70xy – 2086x + 2802y + 6439 = 0
M y2 = 16x
cÉk² 4 cive†Gîi kxlÆ (0, 2) ‰es DcGK±`Ê (2, 5) A
12 wkLbdj-10 I 25
Z AS
wkLbdj-10, 16 I 20 K. x2 y2 =1 Awae†Gîi DrGKw±`ÊKZv wbYÆq Ki| 2
9 16
K. 4x2 9y2 1 = 0 KwYKwUi AmxgZU ˆiLvi L. DóxcGKi cive†Gîi w`KvGÞi mgxKiY wbYÆq Ki| 4
mgxKiY wbYÆq Ki| 2 M. DóxcGKi DcGK±`ÊwU hw` Dce†Gîi DcGK±`Ê nq,
L. P we±`yi Õ©vbvâ wbYÆq Ki| A 4 DrGKw±`ÊKZv 1 ‰es wbqvgGKi mgxKiY,
M. DóxcGKi wPòwU ‰KwU Dce†Gîi Ask weGkl, 2 4
2SP = PM ‰es Z(–2, 3) I S(3, – 4) nGj x − y + 2 = 0 Dce†Gîi mgxKiY wbYÆq Ki|
Dce†Gîi mgxKiY wbYÆq Ki| 4 Dîi: K. 5 L. 2x + 3y + 7 = 0;
3;
Dîi: K. 2 L.
y = 3 x; (8, 8 2); M. 3x2 + 3y2 + 2xy 20x 36y + 112 = 0
wecixZ wòGKvYwgwZK dvskb I wòGKvYwgwZK mgxKiY 1
mµ¦g AaÅvq
wecixZ wòGKvYwgwZK dvskb
I wòGKvYwgwZK mgxKiY
ckÉ ² 1 A 2
y = cos
C
r 2
=x
Bx x2 + y2
wPò nGZ cos = x = x
r x2 + y2
x2 2 x2 + y2
y2 + p =x
( )K. = cosec1 nGj p ‣i gvb wbYÆq KGiv|2
( )L. •`LvI •h, tan 4 + 2+ tan 4 2= 2 1+ y2 4 x2 + y2
x = 2 x2
M. x + y = 2r nGj •`LvI •h, = A^ev 5 4 ( )y 2
y x x 6 6
=2 1+ x
1 bs cÉGki² mgvavb
K •`Iqv AvGQ, = cosec1 x2 ( ) tan 4 + 2+ tan 4 2= 2 1+ y 2 (ˆ`LvGbv nGjv)
y2 + p x
ev, x2 + p = cosec ev, x2 + p = cosec2 M •`Iqv AvGQ, x + y = 2r
y2 y2 y x x
x 2 BC 2
y AC
= cosec2 [wPò nGZ]
( ) ( )ev, p = cosec2 ev, BC + AC = AB [wPò nGZ]
AC BC 2BC
= cosec2 cot2 = 1. (Ans.)
ev, cot + tan = 2sec
tan4 2+ 4 2 tan tan ev, cos + sin = 2
tan4 + tan2 4 2 sin cos cos
L + tan = +
tan4 ev, cos2 + sin2 2
1 tan 2 1 + tan 4 tan 2 sin.cos = cos
1 tan ev, 1 = 2
1 + tan 2 2 sin.cos cos
= +
cos (2 sin − 1) = 0
1 tan 2 1 + tan 2
nq, 1 A^ev, cos = 0
1 + tan 22 + 1 tan 22 sin = 2
= 1 tan 2 1 + tan 2
ev, sin = = (2n + 1) hLb n
sin6 2
= n + (1)n hLb n
6
2 1 + tan2 2
= n = 0 ewmGq =
6, 2
1 tan2 2 n = 1 ewmGq
= 2 = = 5 ‣es = 3π
6 6 2
1 tan2 2 n >/ 1 KviY <
1 + tan2 ‣es KviY ACB = [wPò nGZ]
2 2 2
2 [wòfzGRi •h •KvGbv •KvGYi gvb AGcÞv eo nGZ cvGi bv]
=
= A^ev 5 (ˆ`LvGbv nGjv)
cos 2. 2 6 6
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò ‣Kv`k-«¼v`k •kwÉ Y
ckÉ ² 2 ƒ(x) = 3 cosx + sinx ‣es g(x) = tan– 1x. 2 1 + x2 – 1 2( 1 + x2 – 1)
1 x
K. a = sin(cos1b) nGj a2 + b2 ‣i gvb •ei Ki| 2 = tan–1 x2 – – + x2 – = tan–1 2 x
x2 + 2 1 1 + x2 – 2
L. p = cosec(g(x)) – tan2 – g(x) nGj •`LvI •h, 2g(p) 1
x2 x2
= g(x) 4 = tan–1 2 ( 1 + x2 – 1) x2
x 2( + x2
M. ƒ(x) = 1 mgxKiYwUi mgvavb Ki •hLvGb 2 x 2 4 1 – 1)
2 bs cÉGki² mgvavb
= tan–1 x = WvbcÞ (ˆ`LvGbv nGjv)
K •`Iqv AvGQ, M •`Iqv AvGQ, ƒ(x) = 3 cosx + sinx
a = sin(cos1b) ƒ(x) = 1
ev, a2 = {sin(cos1b)}2 [DfqcÞGK eMÆ KGi] ev, 3 cos x + sin x = 1
ev, a2 = 1 {cos(cos1b)}2
ev, a2 = 1 b2 ev, 3 11
2 cos x + 2 sin x = 2
a2 + b2 = 1 (Ans.)
[DfqcÞGK ( 3)2 + 12 = 3 + 1 = 2 «¼viv fvM KGi]
L •`Iqv AvGQ, g(x) = tan1x ev, cos x cos + sin x sin = cos
6 6 3
p = cosec (g(x)) tan 2 g(x)
ev, cos x – 6 = cos ev, x – = 2n .
3 6 3
= cosec (tan1 x) tan 2 tan1 x
x = 2n •hLvGb, n ‣i gvb kƒbÅ ev AbÅ
= cosec (tan1 x) cot(tan1x) 3 +6
•`LvGZ nGe, 2g(p) = g(x) •hGKvGbv cƒYÆ msLÅv|
ev, 2 tan1 {cosec (tan1x) cot(tan1x)} = tan1x
evgcÞ = 2tan–1 hLb, n = 0 ZLb, x = = 3 , = , –
3 +6 6 6 2 6
( )cosec cosec–1 1 + x2 1 hLb, n = 1 ZLb,
x x
– cot cot–1 x = 2 + = 2 + + , 2 – + = 5 11
3 6 3 6 3 6 2, 6
[wbGPi wPò nGZ]
hLb, n = –1 ZLb,
1 + x2 – x1
= 2 tan–1 x x = –2 + = –2 + 2 + = – 3 , 13
3 6 3 +6, 3 6 2 6
x 1 + x2
= 2 tan–1 1 + x2 – 1 wbGYÆq mgvavb – 11 – 3
x x = 2 , 6 , 6 , 2 (Ans.)
2 1 + x2 – 1 1
x
= tan–1
1 + x2 – 12
1 – x
DËi ms‡KZmn m„Rbkxj cÖkœ
ckÉ ² 3 x)44, sin−1 1 cos−1 y − cot−1 ‣es K. gvb wbYÆq KGiv: sec2(tan– 14) + tan2(sec – 13) 2
2 L. hw` sin(. BC) = cos(.AC) nq, ZGe •`LvI •h,
g(x) = (1 + Q = x + z
cx2 + bx + a = 0 ‣KwU mgxKiY| ( )= 1 sin– 1 3 4
2 4
K. g(x) ‣i weÕ¦w‡ ZGZ 21Zg I 22Zg c` `By wU mgvb nGj,
M. hw` BC – AC = 1 nq, ZGe – < < mxgvi gGaÅ
x ‣i gvb wbYÆq Ki| 2 2
L. x = 3 y = 5 ‣es z = 2 nGj, •`LvI •h, tanQ = 28 4 ‣i gvbàGjv •ei KGiv| 4
5, 13 29
3 5 7
M. hw` ax2 + bx + c = 0 ‣i ‣KwU gƒj DóxcGK DGÍÏwLZ Dîi: K. 25; L. 2 , 4 ; M. = 12 , 12
mgxKiGYi ‣KwU gƒGji w«¼àY nGj, •`LvI •h, ckÉ ² 5 (x) = tan– 1x ‣es g(x) = sinx
2a − c = 0 A^ev (2a + c)2 = 2b2 4 4
3
Dîi: K. 7 K. sintan1 coscot1 ‣i gvb wbYÆq Ki| 2
8
ckÉ ² 4 A L. {g(x)}2 + g2 x = 5 nGj x ‣i mvaviY mgvavb wbYÆq
4
Ki| 4
M. •`LvI •h, 2f(cosec(tan–1x) – tan(cot–1x)) = (x) 4
1
Dîi: K. 4 ; L. x = 2n ; •hLvGb n
B cos C 41 3
1
cixÞvq Kgb •cGZ AviI cÉk² I mgvavb
cÉk²1 (i) z = x + iy, z1 = 10 + 6i M •`Iqv AvGQ, Q() = cos
(ii) Q() = cos 1, 3 I 7 AaÅvGqi mg®¼Gq 3 Q(A) + Q2 A = 1
K. ciggvb wPn× eÅwZZ cÉKvk Ki: |x 2| 5. 2
L. z2 = 4 + 6i ‣es argzz z1 = nGj cÉgvY Ki ev, 3cosA + cos2 A = 1
z2 4 ev, 3 cos A + sin A = 1
•h, x2 + y2 14x 18y + 112 = 0 4
M. mgvavb Ki: 3 Q(A) + Q2 A = 1, ev, 3 11
2 cos A + 2 sin A = 2
hLb 2 < A < 2. 4 [DfqcÞGK 2 «v¼ iv fvM KGi]
1 bs cÉGk²i mgvavb ev,
cos A cos 6 + sin A sin 6 = cos 3
K x – 2< 5
ev, –5 < x – 2 < 5 ev, cos A –
ev, – 5 + 2 < x – 2 + 2 < 5 + 2 6 = cos 3
– 3 < x < 7 (Ans.) ev, A – = 2n .
6 3
L •`Iqv AvGQ, argzz z1 = A = 2n + ; •hLvGb n ‣i gvb kƒbÅ ev AbÅ
z2 4 3 6
ev, (x + iy) (10 + 6i) •hGKvGbv cƒYÆ msLÅv|
arg (x + iy) (4 + 6i) = 4 hLb, n ZLb, 3 –
3 +6 6 6 2 6
= 0 A = = , = ,
ev, (x 10) + (y 6)i hLb, n= 1 ZLb, A = 2
arg (x 4) + (y 6)i = 4 3 +6
ev, arg{(x 10) + (y 6)i} arg{(x 4) + (y 6)i} = = 2 + 2 – + = 5 11
4 3+6, 3 6 2, 6
ev, tan1 y6 tan1 y 6 = hLb, n = –1 ZLb, x = –2
x 10 x 4 4 3 +6
y6 y6
x 10 x 4
ev, tan1 (y 6)2 = –2 + 3 +6, 2 3 + 6
=4
1 + (x 10)(x 4) 3 13
= – 2 , 6
ev, (y 6)(x 4) (x 10) (y 6) wbGYÆq mgvavb A = , – , 11 , – 3
(x 10) (x 4) + (y 6)2 = 1 2 6 6 2
ev, xy 6x 4y + 24 xy + 6x + 10y 60 =1
x2 14x + 40 + y2 12y + 36 cÉk² 2 DóxcK-1: 4(sin2 + cos ) = 5
ev, 6y 36 = x2 + y2 14x 12y + 76 DóxcK-2: (x + a)n = xn + nxn 1a + n(n 1) xn2a2 + ... + an
2!
ev, x2 + y2 14x 12y + 76 6y + 36 = 0
x2 + y2 14x 18y + 112 = 0 (cÉgvwYZ) 5 I 7 AaÅvGqi mg®¼Gq
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò
A K. mgvavb Ki: 2tan−1 x = sin−11 2a + cos−1 1 − b2 2 (iii) •^GK cvB, n2 n xn a2 = 30375
+ a2 1 + b2 2 x2
L. DóxcK-1 ‣i mgxKiYwU −2 < < 2 eÅewaGZ ev, n2 n 729 102 = 30375
2 n
mgvavb Ki| 4
n2 n 30375 2
M. weÕ¦‡wZi cÉ^g wZbwU msLÅv 729, 7290 ‣es 30375 ev, n2 = 729 100
nGj, a ‣i gvb wbYÆq Ki| 4 ev, n2 n 5
n2 = 6
2 bs cÉGki² mgvavb
K •`Iqv AvGQ, 2tan−1 x = sin−1 2a + cos−1 1 − b2 ev, 6n2 6n = 5n2
+ a2 1 + b2
1 ev, n2 = 6n
ev, 2tan−1 x = 2tan−1 a + 2tan−1 b n = 6 [ n 0]
(i) x6 = 729 x = 3
ev, tan−1 x = tan−1 a + tan−1 b
ev, tan−1 x = tan−1 a+b (iv) a = 10 a = 5 (Ans.)
1 − ab 3 6
a+b cÉk² 3 `†kÅK͸-1: z = x + iy ‣KwU RwUj msLÅv|
x = 1 − ab (Ans.)
`†kÅK͸-2:
A
L 4(sin2 + cos) = 5
ev, 4 sin2 + 4cos 5 = 0 1 sin
ev, 4(1 cos2) + 4cos 5 = 0
ev, 4 cos2 4cos + 5 4 = 0 C
ev, 4cos2 4cos + 1 = 0 B
ev, (2cos 1)2 = 0
ev, 2cos 1 = 0 1, 3 I 7 AaÅvGqi mg®¼Gq
A K. 12 2 3 ... •mUwUi Bbwdgvg wbYÆq Ki? 2
5 10
1
ev, cos = 2 = cos 3 L. |z 8| + |z + 8| = 20 «v¼ iv wbG`ÆwkZ mçvi c^
= 2n ; hLb n ‣i gvb kƒbÅ wbYÆq Ki| 4
3
M. hw` 3 . BC + AC = 1 nq ZGe 2 < < 2
ev AbÅ •hGKvGbv cƒYÆ msLÅv|
mxgvi gGaÅ ‣i gvbàGjv •ei Ki| 4
hLb, n = 0, ZLb, = 3 bs cÉGki² mgvavb
3
hLb, n = 1, ZLb, = 5 , 7 K cÉ`î •mU 12 2 3 ... ... ...
3 3 5 10
hLb, ZLb, 5 73 •mUwUi wbÁ²mxgvi •mU {x : x , x 0}
3
n = 1, = , Bbwdgvg = 0 (Ans.)
wbw`ÆÓ¡ eÅewaGZ ‣i gvbmgƒn : , 5 L •`Iqv AvGQ, z = x + iy
3 3 ‣es |z – 8| + |z + 8| = 20
ev, |x + iy – 8| + |x + iy + 8| = 20
M (x + a)n = xn + nxn1a + n(n1) xn2a2 + ... + an ev, |x – 8 + iy| + |x + 8 + iy| = 20
2! ev, (x – 8)2 + y2 + (x + 8)2 + y2 = 20
ev, (x + 8)2 + y2 = 20 – (x – 8)2 + y2
kZÆgGZ, xn = 729 ... (i)
ev, (x + 8)2 + y2 = {20 – (x – 8)2 + y2}2
nxn1a = 7290 ... (ii)
[DfqcÞGK eMÆ KGi]
n(n 1) xn2a2 = 30375 ... (iii) ev, x2 + 16x + 64 + y2
2!
= 400 + (x – 8)2 + y2 – 40 (x – 8)2 + y2
(ii) •^GK cvB, nxn a = 7290
x ev, x2 + y2 + 16x + 64 = 400+ x2 + 64 – 16x +
ev, n 729 a = 7290 y2 – 40 (x – 8)2 + y2
x
a 10
x = n ... (iv)
wecixZ wòGKvYwgwZK dvskb I wòGKvYwgwZK mgxKiY 3
ev, 32x – 400 = – 40 (x – 8)2 + y2 A K. 4 sin–1 1 + cot–1 3 ‣i gvb wbYÆq Ki| 2
5
ev, 8 (4x – 50) = – 40 (x – 8)2 + y2
L. OA + AA = OB + BB nGj x-‣i gvb wbYÆq
ev, 4x – 50 = – 5 (x – 8)2 + y2
Ki| 4
ev, (4x – 50)2 = 25 {(x – 8)2 + y2} [DfqcÞGK eMÆ KGi]
M. •`LvI •h, sin cos1tan sec1OOAA
ev, 16x2 – 400x + 2500 = 25 (x2 – 16x + 64 + y2 )
ev, 25x2 – 400x + 1600 + 25 y2 – 16x2 + 400x – 2500 = 0 2 OA2 OB2
9x2 25y2 = OA .
ev, 9x2 + 25y2 = 900 ev, 900 + 900 = 1 4
ev, x2 y2 x2 y2 4 bs cÉGki² mgvavb
100 + 36 = 1 102 + 62 = 1
K 4 sin–1 1 cot–1 3
hv Dce†Gîi mgxKiY wbG`Æk KGi| 5 +
M ‣LvGb, AB = 1, AC = sin A = 4 tan–1 1 + tan–1 1 [wPò nGZ]
2 3
BC = cos 11
2+3
cÉ`î mgxKiY, 1 sin = 4.tan–1 5
2
3 . BC + AC = 1 11 1
2.3
ev, 3 cos + sin = 1 1 –
ev,
3 1 1 B «¼viv CfvM KGi] 5
2 +2 =2
cos sin [DfqcÞGK 2 tan–1 6 = 4 tan–1 (1)
5
= 4
6 sin 6 = cos 3
ev, cos cos + sin 6
ev, cos – = (Ans.)
6 = cos 3 = 4.4
ev, – = 2n . L •`Iqv AvGQ,
6 3
OA + AA = OB + BB
•hLvGb n ‣i gvb kƒbÅ ev AbÅ
= 2n 3 + 6 ; ev, OA + AA OB + BB
OA = OA
•hGKvGbv cƒYÆ msLÅv| hLb, n = 0 ZLb,
OA + AA OB + BB
= = 3 , , – ev, OA = OB
3 +6 6 6 =2 6
[OA = OB; ‣KB e†Gîi eÅvmvaÆ]
hLb, n = 1 ZLb, = 2
3 +6 ev, OA AA OB BB
OA + OA = OB + OB
= 2 + + , 2 – + = 5 11
3 6 3 6 2, 6 ev, cosx + sinx = cos2x + sin2x
hLb, n = –1 ZLb, = –2 ev, cos x cos 2x = sin 2x sin x
3 +6
ev, 3x x 3x x
2 sin 2 sin 2 = 2 cos 2 sin 2
= –2 + 3 +6, 2 3 + 6
= – 3 , 13 ev, sin x sin32x – cos32x = 0
2 6 2
wbGYÆq mgvavb = , – , 11 , – 3 nq, 3x 3x A^ev,
2 6 6 2 sin 2 = cos 2
3x
sin 2
cÉk² 4 B ev, 3x = 1 x
A sin 2 = 0
cos 2
2x ev, tan 3x = tan ev, x = n
2 4 2
x
O B A ev, 3x = n + x = 2n (Ans.)
2 4
2n
x = 3 + 6 (Ans.)
wkLbdj-1 I 3
4 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò
M sin cos1 tan sec1OOAA A ‣Lb, f(sin) = 0
= sin cos1tan tan1 OAO2 AOA2 (2sin – 1)(2sin – 3) = 0
2OA2OA2
OA2OA2 2sin – 1 = 0 nGj, 2sin = 1
ev, sin = 1
2
= sin cos1 OAO2 AOA2
ev, sin = sin
6
= sin sin1 2OA2 OA2 A O
OA OA •hLvGb n
= n + (1)n 6
2OA2 OA2 2sin – 3 = 0 nGj,
= OA
sin = 3
2 = sin 3
OA2OA2 = n + (– 1)n •hLvGb n
3,
sin cos1 tan sec1OOAA = 2OA2 OB2 n = 0 nGj, = ,
OA 6 3
[OA = OB; ‣KB e†Gîi eÅvmvaÆ] n = 1 nGj, = 5 – = 2
6, 3 3
(ˆ`LvGbv nGjv)
0 < < mxgvi gGaÅ wbGYÆq mgvavb,
cÉk² 5 m f(x) = ax2 + bx + c = , , 5 , 2
6 3 6 3
b M wPò nGZ cvB, tanA = m A = tan– 1 m
Ba a a,
nA ‣es tanB = n
b
C
n
wkLbdj-1 I 4 B = tan– 1 b
K. sec2(cot– 13) + cosec2(tan– 12) gvb wbYÆq Ki| 2 Avevi, A + – B + C =
2
L. a = 4, b = – 2(1 + 3), c = 3 ‣es 0 < < nGj,
m n 2
f(sin) = 0 mgxKiYwUi mgvavb wbYÆq Ki| 4 tan– 1 a – tan– 1 b + 3 + 2 =
M. C = 2 ‣es 2mn = ab nGj •`LvI •h, tan– 1 m – tan– 1 n = – 2 –
3 a b 3 2
n – m = 3 4 6 4 3
b a 2 =6 =6
5 cÉGki² mgvavb n – m
K sec2(cot– 13) + cosec2(tan– 12) b a
tan– 1
mn = 6
= 1 + tan2tan– 1 13 + 1 + cot2cot– 1 21 1 + ab
= 1 + tan tan– 1 12 + 1 + cot cot– 1 12 m – n
3 2 a b
12 12 1 1 mn = tan 6 [ 2mn = ab]
= 2 + 3 + 2 = 2 + 9 + 4
1 + 2mn
72 + 4 + 9 85
= 36 = 36 (Ans.) n m = 3 1
b a 2 3
L f(x) = ax2 + bx + c 3 n – m = 3 (ˆ`LvGbv nGjv)
b a 2
f(x) = 4x2 + {– 2(1 + 3)} x +
= 4x2 – 2x – 2 3x + 3
= 2x(2x – 1) – 3(2x – 1)
= (2x – 1)(2x – 3)
wecixZ wòGKvYwgwZK dvskb I wòGKvYwgwZK mgxKiY 5
DËi ms‡KZmn m„Rbkxj cÖkœ
cÉk²6 `†kÅKÍ-¸ 1: (x) = sin– 1x L. + = nGj, cÉgvY Ki •h, x2 2xy cos + y2
`†kÅK͸-2: cosx + cos2x + cos3x = 0 wkLbdj-1 I 3 = r2 sin2 4
B K. (x) + (y) = nGj •`LvI •h, x2 + y2 = 1 2 M. (x) ( 2x) ( + 3x) = 1 mgxKiYwUGK
2 4
L. `†kÅKÍ-¸ 2 ‣i mgxKiYwUi mgvavb Ki| 4 mgvavb Ki| [hLb 0 < x < ] 4
M. •`LvI •h, cos tan–1 cot (x) = x 4 Dîi: M. , , 3 , 2 , 5 , 7
8 3 8 3 8 8
Dîi: L. 2n 2 •hLvGb, n
x = (2n + 1)4 ; 3 b2
b2
cÉk²7 cÉk² 8 `†kÅK͸-1: 2 tan1x = sin1 1 2a cos1 1
+ a2 1 +
N P
`†kÅK͸-2: cos cos7 = sin4 wkLbdj-1, 3
r K. mgvavb Ki, 4(sin2 + cos) = 5 2
r
L. `†kÅK͸-1 ‣ ewYZÆ mÁ·KÆwUi AvGjvGK •`LvI •h,
y; (x) = cosx
M Q R ab 4
Lx x = 1 + ab
wkLbdj-1 I 4 M. `†kÅK͸-2 ‣ ewYÆZ mgxKiYwU mgvavb KGiv| 4
A K. •`LvI •h, 4 sin1 1 + cot13 = 2 Dîi: K. = 2n M. x = n , n + (1)n .
5 3 4 3 18
wÕ©wZwe`Åv 1
AÓ¡g AaÅvq
wÕ©wZwe`Åv
ckÉ ² 1 200 ˆm.wg. `xNÆ AB nvjKv `íwUi A I B cÉvG¯¦ h^vKÌGg ev, AC = 3 .200
5
12 ˆKwR I 8 ˆKwR gvGbi `yBwU eÕ§ SzjvGbv AvGQ|
AC = 120 ˆm.wg.
K. ˆKvb we±`yGZ Q gvGbi `By wU mgvb ej 120 ˆKvGY wKÌqviZ|
jwº¬i wKÌqvwe±`y AB eivei mGi hvGe (120 80) ˆm.wg.
‰KB we±`yGZ wKÌqviZ 25N eGji mvnvGhÅ ‰G`iGK fvimvgÅ
= 40 ˆm.wg. (Ans.)
ivLv nq| Q-‰i gvb wbYÆq Ki| 2
M. A
L. eÕ§«¼q Õ©vb wewbgq KiGj jwº¬ KZ `ƒGi mGi hvGe? 4 C O DB
M. ‰K eÅwÚ 100 ˆm.wg. eÅeavGb `íwU eÕ§mn `yB nvZ w`Gq enb
KiGZ Pvb| ˆKvb AeÕ©vGbi RbÅ `By nvGZi Dci cÉhyÚ ej
mgvb nGe? 4 12 R 8
1 bs cÉGki² mgvavb gGb Kwi, C I D we±`yGZ `By nvZ Õ©vcb KGi AB `G´£i A I
B cÉvG¯¦ 12 ˆKwR I 8 ˆKwR IRb Õ©vcb KiGj `yB nvGZi
K ˆhGnZz ej«¼q mgvb ‰es 25N eGji mvG^ ‰KB we±`yGZ wKÌqv Dci mgvb Pvc cGo| myZivs Bnvi CD ‰i gaÅwe±`y O ˆZ
jwº¬ IRb KvR KGi|
KGi fvimvgÅ m†wÓ¡ KGi, KvGRB 25N eGji mvG^ Dfq ejB ‰LvGb, AB = 200 ˆm.wg., CD = 100 ˆm.wg.
mgvb ˆKvY Drc®² KiGe| CO = DO = 100 2 = 50 ˆm.wg.
ˆhGnZz ej«¼Gqi jwº¬ R, O we±`yGZ wKÌqv KGi| mZzivs,
awi, Q I 25N eGji A¯¦MÆZ ˆKvY ,
12.OA = 8.OB
+ + 120 = 360
ev, 3.OA = 2.OB
= 120 Q ev, 3(AC + CO) = 2(BD + DO)
ev, 3AC + 3CO = 2BD + 2DO
120 Q ev, 3AC 2BD = 2DO 3CO
O ev, 3AC 2BD = 2.50 3.50
ev, 3AC 2BD = 100 150
ev, 2BD 3AC = 50 .................... (i)
Avevi, AB = 200
25N ev, AC + CD + BD = 200
ev, AC + BD + 100 = 200
‰Lb jvwgi DccvG`Åi mvnvGhÅ cvB, ev, AC + BD = 100 .................... (ii)
Q 25 Q = 25N 3BD + 3AC = 300 .................... (iii)
sin 120 = sin 120
(i) I (iii) ˆhvM KGi cvB,
L gGb Kwi, 200 ˆm.wg. `xNÆ AB nvjKv `íwUi A I B cvÉ G¯¦
5BD = 350 BD = 70
h^vKGÌ g 12 ˆKwR I 8 ˆKwR gvGbi `By wU mggyLx mgv¯¦ivj
BD ‰i gvb (ii) bs ‰ ewmGq cvB,
IRb ej wKÌqv KiGQ| ‰G`i jwº¬ C we±`yGZ wKÌqv KiGQ|
AC + 70 = 100 AC = 30
12.AC = 8.BC A B
A cvÉ ¯¦ nGZ 30 ˆm.wg. `Gƒ i ‰KwU nvZ ‰es B cÉv¯¦ ˆ^GK
ev, 3AC = 2BC (8) (12) 70 ˆm.wg. `ƒGi Aci nvZwU ^vKGe| (Ans.)
ev, 3AC = 2(AB AC)
ev, 3AC = 2AB 2AC 12 C C 8
ev, 5AC = 2AB
ev, 2
AC = 5 AB
ev, 2 [ AB = 200 ˆm.wg.]
AC = 5 .200
AC = 80 ˆm.wg. ckÉ ² 2 wPò-1:
Avevi, ej«¼q Õ©vb wewbgq KiGj hw` jwº¬ C we±`yGZ wKÌqv
KGi, ZvnGj, 8AC = 12BC C B
ev, 2AC = 3BC R
ev, 2AC = 3(AB AC)
ev, 5AC = 3AB P
2
ev, AC = 3 .AB
5 O PA
OACB ‰KwU mvgv¯¦wiK|
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò ‰Kv`k-«¼v`k ˆkwÉ Y
wPò-2: 3N ev, P + 4P cos = 4P
ev, P = 4P(1 – cos) = 4P.2sin2
2
4 2N ev, sin2 = 1
N 2 8
ev, sin = 1
2
1N 2 2
K. wPò-1 nGZ P ‰es cos ‰i mvGcGÞ jwº¬ ‘R’ ˆei Ki| 2 ev, = sin–1 1
2 2
L. wPò-1 ‰ OC eivei 4P ej wKÌqviZ nGj cgÉ vY Ki ˆh, 2
= 2sin12 1 ˆhLvGb, P eivei R ‰i jÁ¼vsk 4P| 4 = 2sin–1 1 (cÉgvwYZ)
2
22
M. wPò-2 ‰ PviwU ej ‰KwU eGMÆi evü eivei wKÌqvkxj| ZvG`i
M gGb Kwi, 1N, 2N, 3N, 4N gvGbi ejàGjv h^vKÌGg ABCD
jwº¬i gvb, w`K I AeÕ©vb wbYÆq Ki| 4 eMGÆ ÞGòi AB, BC, CD, DA evü eivei wKÌqv KiGQ|
gGbKwi, eMGÆ ÞGòi evüi Š`NÆÅ a.
2 bs cÉGki² mgvavb
K ˆ`Iqv AvGQ, OACB ‰KwU mvgv¯¦wiK hvi OA = P, 2N O E
F
P
OB = 2 B K
C 2 D 3C
N N
P HG 42
2R NN
A 1N B
O PA AB eivei 1N ‰es CD eivei 3N gvGbi mgv¯¦ivj
ej«¼Gqi jwº¬ 2N ejwUI ZvG`i mgv¯¦ivj ‰es OF eivei
‰es OA I OB ‰i gaÅeZxÆ ˆKvY nGj, wKÌqv KGi I Zvi wKÌqvGiLv ewaÆZ AD ˆK E we±`yGZ ‰gbfvGe
ˆQ` KGi ˆh,
jwº¬, R = P2 + P2 + P .cos
2 2.P.2
=
P2 + P2 + P2 cos 3N.DE = 1N.AE = 1N(a + DE)
4
1
4P2 + P2 + 4P2 cos DE = 2a
=4
Avevi, BC eivei 2N ‰es DA eivei 4N ej«¼Gqi jwº¬
5P2 + 4P2 cos
=4 2N ejwU I ZvG`i mgv¯¦ivj ‰es OG eivei wKÌqv KGi I
P2 (5 + 4 cos) Zvi wKÌqvGiLv ewaÆZ BA ˆK G we±`yGZ ‰gbfvGe ˆQ` KGi
=4
ˆh,
P 4N.AG = 2N.BG = 2N(a + AG)
=2
5 + 4 cos (Ans.) AG = a
L C B gGb Kwi, EF ‰es OG ˆiLv«¼q O we±`yGZ ˆQ` KGi| ZvnGj
OF ‰es OG eivei mgGKvGY wKÌqviZ 2N ‰es 2N gvGbi
4P ej«¼Gqi jwº¬i gvb (2N)2 + (2N)2 = 2 2N ‰es Zv
R A FOG ˆKvGYi mgw«¼L´£K OH eivei A^Ævr CA ˆiLvi
mgv¯¦ivGj wKÌqv KiGe| ZvnGj HOG = 45 = OHG.
gGb Kwi, CA ‰es OH mgv¯¦ivj ˆiLv«¼Gqi `iƒ Z½ AK ‰es
OP
ewaÆZ AG ˆiLv OH ˆK H we±`yGZ ˆQ` KGi|
ˆ`Iqv AvGQ, OABC mvgv¯¦wiGKi O we±`yGZ OA eivei P ‰Lb, GH = OG = AE = 3a
gvGbi ej OC eivei 4P gvGbi ej wKÌqviZ ‰es ZvG`i jwº¬ 2
R, OB eivei wKÌqviZ|
3 5
awi, P I 4P ‰i A¯¦MÆZ ˆKvY ‰es P ‰i w`K I R ‰i AH = AG + GH = a + 2a = 2a
w`GKi A¯¦MZÆ ˆKvY |
P ‰i w`GK R ‰i jÁ¼vsk 4P AK = AHsin45 = 5 2
a
Rcos = 4P 4
ZvnGj P ‰i w`K eivei ejàwji jÁ¼vsk wbGq cvB, myZivs jwº¬i gvb 2 2N ‰es Zv 2N I 3N gvGbi ej«¼Gqi
Rcos = Pcos0 + 4P cos wKÌqv ˆiLvi ˆQ`we±`y w`Gq AswKZ KYÆ nGZ H eMGÆ ÞGòi evüi
ev, 4P = P + 4P cos 52 àY `iƒ GZ½ H KGYÆi mgv¯¦ivj mijGiLv eivei wKÌqv
ev, P + 4P cos = 4P 4
KiGe| (Ans.)
wÕ©wZwe`Åv 3
DËi ms‡KZmn m„Rbkxj cÖkœ
ckÉ ² 3 `k† ÅKÍ-¸ 1: 5N gvGbi wZbwU ej ‰KwU we±`yGZ K. ej«¼Gqi e†nîg jwº¬i gvb Þz`ËZg jwº¬i 3 àY| ej«¼Gqi
‰gbfvGe KvhÆiZ ˆh ‰G`i w`K ABC ‰i BC, CA ‰es AB AbycvZ wbYÆq Ki| 2
evüi mgv¯¦ivj| L. jwº¬ hw` e†nîg ejwUi mvG^ 30 ˆKvY Drc®² KGi ZGe
`k† ÅKÍ-¸ 2: ABC ‰i cwiGK±`Ê O ‰es 20N gvGbi ‰KwU ej AO ej«¼Gqi A¯¦fÆyÚ ˆKvY wbYÆq Ki| 4
eivei KvhÆiZ| A M. e†nîg I Þz`ËZg jwº¬ h^vKÌGg S I T nGj cgÉ vY Ki ˆh,
90 ej«¼Gqi jwº¬i gvb S2cos2 + T2 sin2 4
2 2
O
Dîi: K. 2 t 1; L. 120
B 30 60 C
K. ˆKvb we±`yGZ wKÌqviZ wZbwU ej mvgÅveÕ©v m†wÓ¡ KGi| cÉk²5 ‰KB Avbyf„wgK ˆiLv eivei 26 ˆm.wg. `iƒ GZ½ `By wU
ZvG`i 1g I 2qwUi gaÅeZÆx ˆKvY 120 ‰es 2q I AvsUvi mvG^ 34 ˆm.wg. Š`GNÆÅi ‰KwU myZvi cvÉ ¯¦«¼q evauv AvGQ| 12
3qwUi gaÅeZÆx ˆKvY 90 nGj ej wZbwUi AbycvZ wbYÆq ˆKwR IRGbi ‰KwU eÕ§ myZvwUi Dci w`Gq MwoGq PjGZ cvGi|
Ki| 2 A K. wPGòi mvnvGhÅ mgmÅvwU DcÕ©vcb Ki| 2
L. `†kÅK͸-1 ˆ^GK ejàwji jwº¬i gvb wbYÆq Ki| 4 L. mvgÅveÕ©vi myZvwUi Uvb wbYÆq Ki| 4
M. `k† ÅKÍ-¸ 2 ‰ B I C we±`yGZ 20N eGji mgv¯¦ivj M. eÕ§wUGK ‰gb AeÕ©vGb AvUGK ˆ`Iqv nGjv ˆhb myZvwU
Dcvsk«¼Gqi AbycvZ wbYÆq Ki| 4 90 ˆKvY Drc®² KGi| ZvnGj myZvwUi e†nîg I Þz`ËZg
Dîi: K. 2 t 1 t 3; L. 2.588; M. 1 t 1 AsGki Uvb wbYÆq Ki| 4
cÉk²4 `ywU Amgvb I Amgv¯i¦ vj ej ciÕ·i -ˆKvGY ˆKvb Dîi: L. 51 M. 60 N, 144 N
; 13 13
wbw`ÆÓ¡ ‰KwU we±`yGZ wKÌqv KiGQ|
30N
1
cixÞvq Kgb •cGZ AviI cÉk² I mgvavb
cÉk² 1 A L 1g wPGò, R1 C B
•KvGY •njvGbv Zj AB ‣i 90°
B D Ici C we±`yGZ W IRGbi
O eÕ§GK f„wgi mgv¯¦ivGj wKÌqviZ Q
W Q ej wÕ©i AeÕ©vq ivGL| (90)
•njvGbv ZjwUi Ici eÕw§ Ui
R C
Pvc R1 aiv nGj ej R1, Q, W
fvimvgÅ m†wÓ¡ KiGe| A
ABC ‣i cwiGK±`Ê O ‣es AB ‣KwU gm†b Zj hvi D R1 Pvc AB ‣i mvG^ jwÁ¼K w`GK nGe| W
jvwgi mƒòvbyhvqx,
we±`yGZ W IRGbi ‣KwU eÕ§ mvgÅeÕ©vq iGqGQ| wPò-1
6 I 8 AaÅvGqi mg®¼Gq A R1 Q W
sin90 = sin(90 90 ) = sin(90 )
K. 9x2 7y2 + 63 = 0 Awae†Gîi wbqvgGKi mgxKiY
wbYÆq Ki| 2 ev, R1 = Q = W
1 sin cos
L. P I Q ej«q¼ h^vKÌGg AB mgZGji ․`NÆÅ ‣es
W sin cos W W2
f„wgi mgv¯¦ivj •^GK ‣KKfvGe W IRGbi eÕ§GK Q= cos ev, sin = Q ev, cot2 = Q2 ... ... ... (i)
aGi ivLGZ cviGj cÉgvY Ki •h, 1 1 1 4 2q wPGò,
P2 Q2 = W2 •njvGbv ZjwUi ․`GNÆÅi w`GK P
ej wKÌqvkxj W IRGbi eÕ§wUi R2 B P
M. cÉgvY Ki •h, B I C we±`yGZ wKÌqviZ R eGji •njvGbv ZjwUi Ici wÕ©i C
mgv¯¦ivj Ask«¼Gqi AbycvZ sin 2B t sin 2C. 4 90°
1 bs cÉGk²i mgvavb AeÕ©vq ^vGK| ‣ eÕ§i Pvc R2
K •`Iqv AvGQ, Awae†Gîi mgxKiY, aiGj R2, P, W ej wZbwU
9x2 – 7y2 + 63 = 0 fvimvgÅ m†wÓ¡ KiGe ‣es R2 Pvc
ev, 9x2 – 7y2 = – 63 •njvGbv ZGji mvG^ jwÁ¼K w`GK
ev, 7y2 – 9x2 = 63 wKÌqvkxj ^vKGe| A
ev, y2 – x2 = 1 W
9 7
jvwgi mƒòvbyhvqx, wPò-2
ev, y2 – x2
(3)2 ( 7)2 = 1 R2 = P = W
sin(90 + 90 sin90
y2 x2 + ) sin(90 – )
b2 a2
‣GK – = 1 ‣i mvG^ Zzjbv KGi cvB, ev, R2 = P = W
cos sin l
a = 7, b = 3 ‣es Dce†îwUi y-AÞB Avo AÞ|
P = W sin ev, P =W ev, W = cosec
ZvnGj, e = b2 + a2 7+9 4 sin P
b2 = 9 =3
ev, W2 = cosce2 ... ... ... (ii)
wbqvgK •iLvi mgxKiY, b P2
y = e
(ii) bs nGZ (i) bs weGqvM KGi cvB,
3 W2 W2
ev, y = 4 P2 Q2 = cosec2 cot2
3 ev, W2P12 Q12 = 1
4y = 9 (Ans.)
ev, 1 = 1 1 (cÉgvwYZ)
W2 P2 Q2
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò
M gGb Kwi, B I C we±`yGZ wKÌqviZ P eGji mgv¯¦ivj L. `†kÅK͸-1 ‣i RbÅ •MvjKwUi IRb w nGj •`LvI
AskK«¼q Q I R. A^Ævr Q I R eGji jwº¬ P
•h, myZvi Uvb T= w(l + a) 4
A 2al + l2
M. `†kÅK͸-2 ‣i AvGjvGK we±`ywUi Õ©vbvâ wbYÆq
Ki| 4
2 bs cÉGki² mgvavb
O K •`Iqv AvGQ, 5x2 + 4y2 = 1
B DP C x2 y2
1+ 1 =1
54
QR ‣LvGb, 1 1
5 4
a2 = , b2 =
‣Lb, B I C we±`yGZ wKÌqvkxj Q I R ‣i jwº¬ BC 11
a= 5,b=2
•iLvi DciÕ© D we±`yGZ wKÌqv KiGe| A, O ‣es D
‣KB •iLvq AewÕ©Z| P, AD eivei wKÌqvkxj| a<b
AZ‣e Q.BD = R.CD ‣es, Dce†îwUi e†nr AÞ y-AGÞi Ici AewÕ©Z|
ev, Q CD CD/OD myZivs ‣GÞGò,
R = BD = BD/OD ... ... ... (1)
1 1
‣Lb, COD wòfzR nGZ, CD = OD e2 = b2 a2 = 4 5 = 1 4 = 1
sinCOD sinOCD b2 1 20 5
ev, CD sinCOD 4
OD = sinOCD
Avevi, BOD wòfzR nGZ, BD = OD AZ‣e, DrGKw±`ÊKZv, e = 1
sinBOD sinOBD
5
BD sinBOD
OD = sinOBD ... ... ... (2) L •MvjGKi •K±`Ê O| O
we±`yGZ •MvjGKi IRb W
(1) bs mgxKiGY DcGivÚ gvbàGjv ewmGq cvB, wKÌqvkxj| DÍÏÁ¼ •`qvGji A A
we±`y nGZ l ․`GNÆÅi myZv w`Gq
Q CD/OD sinCOD/sinOCD •MvjKwU wPGòi bÅvq C l
R = BD/OD = sinBOD/sinOBD ... ... ... (3) we±`yGZ euvav| T
W
•hGnZz OB = OC = cwieÅvmvaÆ R
C
myZivs OCD = OBD OC = OB = a| B we±`yGZ a
•MvjKwU •`qvjGK Õ·kÆ KGi
sinOCD = sinOBD nGj, ‣es •`qvGji AwfjÁ¼ R
cÉwZwKÌqv R, O we±`yGZ OaB
(3) bs mgxKiY nGZ cvB, Abfy „wgK eivei wKÌqvkxj|
myZvi Uvb, T, OA eivei W
Q sinCOD sin( AOC) sinAOC wKÌqvkxj|
R = sinBOD = sin( AOB) = sinAOB
Avevi, e†Gîi •K±`ÊÕ© •KvY cwiwaÕ© •KvGYi w«¼àY eGj,
AOC = 2B ‣es AOB = 2C
Q sin2B
R = sin2C
Q : R = sin2B : sin2C (ˆ`LvGbv nGjv)
cÉk²2 `†kÅKÍ-¸ 1: l ․`NÆÅwewkÓ¡ ‣KwU myZvi ‣KcÉv¯¦ •hGnZz, T, W I R mvgÅveÕ©v m†wÓ¡ KGi ZvB ejàGjvGK
‣KwU DjÁ¼ •`qvGj AvUKvGbv AvGQ ‣es AbÅcÉv¯¦ a OAB wòfzGRi h^vKÌGg OA, AB I BO evü eivei
eÅvmvaÆwewkÓ¡ ‣KwU mylg •MvjGKi mvG^ mshyÚ AvGQ| A ‣KBKÌGg mƒwPZ KGi jvwgi DccvG`Åi wecixZ cÉwZæv
`†kÅK͸-2: y2 = 8x cive†îÕ© •KvGbv we±`yi •dvKvm `ƒiZ½ 8. nGZ cvB,
6 I 8 AaÅvGqi mg®¼Gq T = W = R A^Ævr l T a = W
OA AB BO + OA2 OB2
K. 5x2 + 4y2 = 1 Dce†Gîi DrGKw±`ÊKZv wbYÆq Ki| 2
ev, T W
l + a = (l + a)2 a2
wÕ©wZwe`Åv 3
ev, l T a = W awi, cive†îÕ© we±`y P(x, y) ‣i •dvKvm `ƒiZ½, SP = 8.
+ l2 + 2al + a2 a2 ‣Lb, y2 = 4ax ‣i mvG^ y2 = 8x ‣i Zyjbv KGi cvB,
T = W(l + a) (ˆ`LvGbv nGjv) a = 2 = AS = AZ
2al + l2
P(x, y) we±`y nGZ AGÞi Ici PK jÁ¼ AuvwK|
M P(x, y) SP = MP = ZK = AZ + AK
M 8=a+xx+2=8x=6
•hGnZy, P(x, y) we±`ywU y2 = 8x ‣i Ici AewÕ©Z|
myZivs y2 = 8.6 = 48 y = 4 3
Z SK myZivs wbGYÆq we±`ywUi Õ©vbvâ (6 4 3)
A
X
DËi ms‡KZmn m„Rbkxj cÖkœ
cÉk² 3 wbGPi wPGò O we±`yGZ KvhÆiZ P, Q ej `yBwUi wkLbdj-4, 9 I 11
jwº¬ R. hv Q eGji mvG^ 50 •KvY Drc®² KGi| P = 15N K. 10 N I P N gvGbi `yBwU ej ciÕ·i 120 •KvGY
‣es P I Q ‣i gaÅeZÆx •KvY 150 wkLbdj-4 I 6 wKÌqvkxj| jwº¬ ej 10N gvGbi eGji mvG^
mgGKvGY ^vKGj P eGji gvb KZ? 2
PR L. `†kÅKÍ-¸ 1 nGZ •`LvI •h, iwki Uvb 5kg-wt. 4
150 M. `†kÅK͸-2 ‣i ej«¼Gqi cÉGZÅKGK 2N cwigvGY
50
e†w«¬ Kiv nGj •`LvI •h, ZvG`i jwº¬ 6 wgUvi
O
Q `ƒiGZ½ mGi hvGe| 4
K. wZbwU mgvb ej mvgÅveÕ©v m†wÓ¡ KiGj ZvG`i gaÅeZÆx Dîi: K. 20N;
•KvY wbYÆq Ki| 2 cÉk² 5 •Kvb we±`yGZ P, Q I R gvGbi wZbwU ej wKÌqviZ|
L. R I Q ‣i gvb wbYÆq Ki| 4 wkLbdj-4, 5 I 9
M. P = Q = 15N nGj, R ‣i gvb wbYÆq Ki ‣es K. •`LvI •h, `ywU mgvb eGji jwº¬ ‣G`i A¯¦MÆZ •KvYGK
•`LvI •h, = 75. 4 mgw«L¼ w´£Z KGi| 2
Dîi: K. = 120; L. R = 9.79N; Q = 19.28 N; L. DóxcGKi cÉ^g `ywU eGji A¯¦MÆZ •KvYGK Z‡Zxq ejwU
P2 Q2
M. 7.76 N, 75 ‣K-Z‡ZxqvsGk wefÚ KiGj •`LvI •h, R = Q
cÉk²4 `†kÅK͸-1: ‣KB Avbfy „wgK •iLvq 8 wgUvi `ƒiGZ½ •hLvGb, (P > Q) 4
M. DóxcGKi 1g I 2q eGji A¯¦MÆZ •KvY 1g I 3q eGji
AewÕ©Z `yBwU we±`yGZ 10 wgUvi `xNÆ ‣KwU mi‚ iwki cÉv¯¦«q¼ A¯¦MÆZ •KvGYi w«à¼ Y nGj ‣es ejòq mvgÅeÕ©vq ^vKGj
euvav AvGQ| AevGa SzjvGbv 6 kg-wt ‣KwU eÕ§GK enb KGi cÉgvb Ki •h, 4
‣gb ‣KwU gm†Y IRbwenxb AvswU H iwki Dci w`Gq MwoGq R2 = Q(Q P)
PjvPj KiGZ cvGi|
`†kÅK͸-2: 10N I 5N gvGbi `yBwU wecixZgyLx mgv¯¦ivj ej
ciÕ·i 15 wgUvi `ƒiGZ½ AewÕ©Z `yBwU we±`yGZ wKÌqviZ|
mgZGj eÕ§KYvi MwZ 1
beg AaÅvq
mgZGj eÕ¦yKYvi MwZ
ckÉ ² 1 `k† ÅKÍ-¸ 1 : ‰KwU eÕ§GK Avbyf„wgGKi mvG^ 60 ˆKvGY M ˆ`Iqv AvGQ, wbGÞcY ˆeM, = 60
wbGÞc Kiv nGjv hv 6 wgUvi eÅeavGb 3 wgUvi DuPz PQ I ST ˆ`qvGji DœPZv, y = 3 wgUvi
ˆ`qvGji Dci w`Gq hvq|
( )Avgiv Rvwb, y = x tan 1 x
u R
PS
( )ev, x
3 = x tan60 1 R
33 x2 3
R
= 60 ev, 3=x 3
O Q 6 wg. T A ev, 3R = Rx 3 x2 3
R
`k† ÅKÍ-¸ 2. Avwmd wÕ©iveÕ©v nGZ mgZ½iGY ‰gbfvGe gUi ev, x2 3 xR 3 + 3R = 0
mvBGKj PvjvGjv ˆh 3 ˆmGK´£ cGi ˆeM nq 9 wgUvi/ˆmGK´£| hv x ‰i w«¼NvZ mgxKiY|
‰fvGe 10 ˆmGK´£ PvjvGbvi ci cieZxÆ 15 wgwbU mgGeGM awi, ‰i `ywU gƒj x1, x2 (x1 > x2)
PvjvGjv| AZtci ˆeËK KGl 3 wgwbU 10 ˆmGK´£ cGi mvBGKj R3 =R
x1 + x2 = 3
^vgvGjv| wkLbdj- 4, 5, 9 I 10
3R
K. 2 wg./ˆm.2 Z½iGY Pjgvb ‰KwU Mvwo hw` 6th ˆmGKG´£ 65 x1x2 = 3
wgUvi `iƒ Z½ AwZKgÌ KGi MvwowUi Avw`GeM KZ? 2 ˆ`qvj `yBwUi eÅeavb, x1 x2 = 6
ev, (x1 x2)2 = 62
L. `†kÅK͸-2 ‰ AvwmGdi M¯e¦ ÅÕ©Gji `ƒiZ½ KZ? 4 ev, (x1 + x2)2 4x1x2 = 36
M. `k† ÅKÍ-¸ 1 ˆ^GK P we±`yGZ cGÉ ÞcGKi ˆeGMi gvb wbYÆq ev, R2 4.3R = 36
Ki| 4 3
1 bs cÉGki² mgvavb
K tth ˆmGKG´£ AwZKvÌ ¯¦ `ƒiZ½, ev, 3R2 12R 36 3 = 0
sth = u + 1 f(2t 1), ‰LvGb, u Avw`GeM ev, 3R2 18R + 6R 36 3 = 0
2
ev, 3R(R 6 3) + 6(R 6 3) = 0
s6th = u + 1 2(2.6 1) ˆ`Iqv AvGQ,
2 Z½iY, f = 2 wg/ˆm2 ev, (R 6 3) ( 3R + 6) = 0
ev, 65 = u + 11 s6th = 65 nq, R 6 3 = 0 ev, 3R + 6 = 0
u = 54 wg/ˆm. (Ans.) R = 6 3m R = 6 hv AMÉnYGhvMÅ|
3
L 1g ˆÞGò, u = 0, t1 = 3 ˆm. v = 9 wgUvi/ˆm. mgZ½iY nGj Avgiv Rvwb, R = u2 sin 2
v = u + t mƒò nGZ cvB g
9 = 0 + .3 =3 wg./ˆm.2
10 ˆmGKG´£ AwZKvÌ ¯¦ `ƒiZ½ ev, 6 u2 sin(2 60)
3 = 9.8
S1 = ut + 1 t2 = 0 + 1 3 102 = 150 wgUvi ev, 6 3 9.8 = u2
2 2 sin 120
Avevi 10 ˆmGK´£ cGi ˆeM, v1 = u + t = 0 + 3 10 = 30 wg./ˆm. ev, 117.6 = u2 u = 10.84 ms1
2q ˆÞGò, t2 = 15 wgwbU = 15 60 = 900 ˆm.
‰Lb, y = (u sin)t 1 gt2
2
AwZKvÌ ¯¦ `ƒiZ½, S2 = v1t2 = 30 900 = 27000 wgUvi|
3q ˆÞGò, t3 = 3 wg. 10 ˆmGK´£ = 3 60 + 10 = 190 ˆm. ev, 3 = (10.84 sin 60)t 1 9.8 t2
2
v1 = 30, v2 = 0 S3 = v1 + v2 t3 ev, 3 = 9.39t 4.9t2
2
ev, 4.9t2 9.39t + 3 = 0
30 + 0
2
( )= 190 = 2850 wgUvi ev, (9.39) (9.39)2 4 4.9 3
t= 2 4.9
AvwmGdi M¯¦eÅÕ©Gji `iƒ Z½ t = 0.40 sec, 1.51 sec
= S1 + S2 + S3 = 150 + 27000 + 2850 myZivs P we±`yGZ ˆcŒu QvGZ mgq 0.40 sec
S we±`yGZ ˆcŒu QvGZ mgq 1.51 sec
= 30000 wg. = 30 wK.wg. (Ans.)
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò ‰Kv`k-«¼v`k ˆkwÉ Y
Avevi, Vx = u cos = 10.84 cos 60 Abyf„wgGKi mvG^ ˆKvY, = 30
g = 980 cms−2 = 9.8ms−2
Vx = 5.42 ms1
meÆvwaK DœPZv, H = uo2 sin2 = (21)2 (sin 30)2
‰es Vy = u sin gt 2g 2 9.8
= 10.84 sin 60 9.8 0.40 = 5.46 ms1 = 5.625 m (Ans.)
myZivs, P we±`yGZ ˆeM, V = vx2 + vy2 = 5.422 + 5.462 meÆvwaK DœPZvq ˆcuŒQvGbvi mgq = u0sin = 21 sin 30 = 1.07s
g 9.8
= 7.69 ms1 (Ans.)
ckÉ ² 2 `k† ÅKÍ-¸ 1 : ˆZvgvi evwoi weovjwU mvgGb ‰KwU Bu`yi
M 2s ci Avbyf„wgK ˆeM, vx = u0 cos = 18.19 ms−1 (Ans.)
ˆ`GL wÕ©iveÕ©v nGZ 0.3 m/s2 mgZ½iGY ‰i cøvGZ ˆ`ŒovGœQ| DÍÏÁ¼ ˆeM, vy = u0 sin − gt
`k† ÅKÍ-¸ 2 : wKÌGKU ˆLjvq wgivR ‰KwU ejGK 21m/s ˆeGM I = 21 sin 30 −
Abyf„wgGKi mvG^ 30 ˆKvGY wbGÞc Kij| [ˆhLvGb, g = 980 9.8 2 =– 9.1 ms−1
cm/s2] 2s ci ˆeM = vx2 + vy2
K. ˆ`o wgwbU ci weovjwUi ˆeM KZ? 2 = (18.19)2 + (–9.1)2 = 20.34 ms−1
AZ‰e, 2s ci ˆeM 20.34 ms−1
L. cÉGÞcKwUi MwZcG^ ejwUi meÆvwaK DœPZv I H DœPZvq Avevi, 2s ci AeÕ©vb y nGj,
ˆcŒQvi mgq wbYÆq Ki| 4
M. 2 ˆm. ci ‰i AeÕ©vb I ˆeM wbYÆq Ki| 4
2 bs cÉGki² mgvavb y = (u0 sin)t − 1 gt2
2
K ‰LvGb, Avw` ˆeM, u = 0 ms−1 = (21 sin 30) 2 − 1 9.8 (2)2 = 1.4 m
mgq, t = 1.5 min = 90s, Z½iY, a = 0.3 ms−2 2
ˆ`o wgwbU ci ˆeM v nGj, 2s ci 1.4 m DÍÏÁ¼ `ƒiZ½ AeÕ©vb KiGe| (Ans.)
v = u + at = 0 + 0.3 90 = 27 ms−1 (Ans.)
L ‰LvGb, wbGÞcY ˆeM, uo = 21 ms−1
DËi ms‡KZmn m„Rbkxj cÖkœ
ckÉ ² 3 ‰K eÅwÚ ‰KwU ej u wg./ˆm. ˆeGM Lvov Dci w`GK QyGo Dîi: K. 2 Bwç; L. 15495 wg./ˆm. ‰es tan–121;
gvij ‰es t ˆmGK´£ ci ‰KB Õ©vb nGZ ‰KB ˆeGM Aci ‰KwU 3
ej ‰KB w`GK QyGo gvij| c^É g ejwU mGeÆvœP H DœPZvq hvq ‰es a
1+x
mGeÆvœP DœPZvq ˆcŒu QvGZ T mgq ˆbq| ( )M. R = W
cÉk²5
K. 2 ˆmGK´£ ci ejwUi ˆeM 16.4 wg./ˆm. nGj Avw`GeM u
KZ? 2
uw
L. H I T ˆK u I g ‰i gvaÅGg cKÉ vk Ki| d
ˆhLvGb g = gvaÅvKlÆYRwbZ Z½iY| 4 90
M. cÉgvY Ki ˆh, ej`yBwU 4u2 - g2t2 wg. DœPZvq wgwjZ nGe| 4
v
8g
Dîi: K. u = 36 wg./ˆm; L. T = u , H = u2; d cÉÕ© wewkÓ¡ b`xGZ ˆbŒKvi ˆeM u, ˆmÉvGZi ˆeM v ‰es jwº¬ ˆeM w.
A K. 10 wgUvi/ˆm. ˆeM Abyf„wgGKi mvG^ 60 ˆKvY Drc®²
g 2g
cÉk²4 `k† ÅKÍ-¸ 1: ‰KwU àwj 9.8 wg. `ƒGi AewÕ©Z 2.45 wgUvi KiGj ˆeMwUGK jÁ¼vsGk wefvRb Ki| 2
DœP ‰KwU ˆ`IqvGji wVK Dci w`Gq Avbyf„wgKfvGe PGj hvq|
`†kÅKÍ-¸ 2: ‰KRb ˆjvK ‰KwU jvwVi ‰KcÉvG¯¦ ‰KwU ˆevSv KuvGa L. ˆbŒKvwU t mgGq ˆmvRvmywR ‰es t1 mgGq ˆmÉvGZi AbyK„Gj
enb KiGQ| ˆevSvwUi IRb w ‰es ˆjvKwUi Kuva nGZ ˆevSvwUi I mgvb `ƒiZ½ cvwo w`Gj ˆ`LvI ˆh, t t t1 = u + v t u v. 4
ˆjvKwUi nvGZi `ƒiZ½ h^vKÌGg a I x. M. d = 100 wg. nGj b`xGZ ˆmÉvZ bv ^vKGj ˆbŒKvwU b`xi
A K. ‰KwU eyGjU ˆKvb ˆ`IqvGji wfZi 2 Bwç XzKevi ci ˆmvRvmywR 4 wgwbGU wK¯§ ˆmÉvZ ^vKGj 5 wgwbGU cvwo
Dnvi AGaÆK ˆeM nvivq| eyGjUwU ˆ`IqvGji wfZi AviI ˆ`q| v ‰i gvb wbYÆq Ki| 4
KZ Bwç XzKGe? 2 Dîi: K. Abyf„wgK eivei 5 wgUvi/ˆm. ‰es DjÁ¼ eivei 5 3
L. `†kÅK͸-1 ‰i àwjwUi cÉGÞc ˆeGMi gvb I w`K wbYÆq Ki|4 wgUvi/ˆm.; M. v = 15 wg./wgwbU|
M. `k† ÅKÍ-¸ 2 ‰i ˆjvKwUi Kvu Gai Dci Pvc wbYÆq Ki| 4
1
cixÞvq Kgb •cGZ AviI cÉk² I mgvavb
cÉk²1 `†kÅK͸-1: 20 •m.wg. `xNÆ AB nvÍ•v `íwU 10 M awi, `´£wU 10 cm eÅeavGb C I D we±`yGZ AewÕ©Z
•cGiK `yBwUi Dci mgvb Pvc«q¼ P I P|
•m.wg. eÅeavGb `yBwU •cGiGKi Dci Avbyf„wgKfvGe
ZvnGj ‣G`i jwº¬ 2P, hv CD ‣i gaÅwe±`y O •Z wKÌqv
AewÕ©Z| A I B we±`yGZ h^vKÌGg 2W ‣es 3W IRb KiGe|
SzjvGbv nj| A
`†kÅK͸-2: ‣KwU kƒbÅ K„Gci gGaÅ ‣KwU cv^Gii UzKiv •QGo OC = OD = 5 cm
•`Iqvi ci Zv 19.6 wg./•m. •eGM K„Gci ZjG`Gk cwZZ nq| mvgÅveÕ©vi RbÅ 2w I 3w IRb«¼Gqi jwº¬ 5w, AekÅB
O •Z Lvov wbÁ²w`GK KvhÆKi|
UzKivwU •QGo •`Iqvi 2 •m.cGi cv^iwUi cZGbi kõ
235
2W AO = 3W BO
•kvbv •Mj| 8 I 9 AaÅvGqi mg®¼Gq
AO BO AO + BO
A K. ‣KwU •ijMvwo 80 wK.wg./N¥Ÿv •eGM Pjvi mgq 3=2 = 5 2P
MvwowUi mgv¯¦ivj cG^ ‣KB w`GK ‣KwU evm 100 20 A C OD B
= 5 = 4 cm
wK.wg./hr •eGM PjGQ| •ij Mvwoi mvGcGÞ evmwUi 5W 3W
AvGcwÞK •eM KZ? 2 AO = 12 cm 2W
L. `†kÅKÍ-¸ 2 nGZ kGõi •eM wbYÆq Ki| 4 BO = 8 cm
M. `†kÅKÍ-¸ 1 ‣i •cGiK `yBwUi •Kvb AeÕ©vGbi RbÅ AC = OA OC = 12 5 = 7 cm
‣G`i Dci Pvc mgvb nGe? 4 BD = OB OD = 8 5 = 3 cm
1 bs cÉGk²i mgvavb •cGiK `yBwU A I B we±`y •^GK h^vKÌGg 7 cm I
K •ijMvwowU 80 wK.wg./N¥Ÿv •eGM PjGQ| ‣i mgv¯¦ivGj 3 cm `ƒiGZ½ Õ©vcb KiGZ nGe|
‣KB w`GK ‣KwU evm 100 wK.wg./N¥Ÿv •eGM PjGQ|
myZivs •ijMvwoi mvGcGÞ MvwowUi AvGcwÞK •eM cÉk² 2 `†kÅK͸-1: •Kvb KYvi Dci wKÌqviZ `yBwU eGji
= (100 – 80) wK.wg./N¥Ÿv = 20 wK.wg/N¥Ÿv (Ans.)
jwº¬ ‣KwU eGji Dci jÁ¼ ‣es ‣i gvb AciwUi gvGbi ‣K
L gGb Kwi, cv^Gii cZbKvj t •m. ‣es K„Gci MfxiZv h wg.
Z‡ZxqvsGki mgvb|
myZivs, v2 = u2 + 2gh mƒò nGZ cvB,
`†kÅK͸-2: e†wÓ¡ 30 wg./•m. •eGM LvovfvGe coGQ| ‣KRb
ev, (19.6)2 = 2 9.8 h
[ u = 0] •ijMvwoi hvòxi KvGQ Zv LvovGiLvi mvG^ 60 •KvGY coGQ
h = 19.6 eGj gGb nq| 8 I 9 AaÅvGqi mg®¼Gq
Avevi, v = u + gt nGZ cvB, A K. ‣KwU Mvox mgZ½iGY 30 km/hour Avw`GeGM 100
ev, 19.6 = 9.8 t km c^ AwZKÌg KGi 50 km/hour P„ov¯¦ •eMcÉvµ¦
ev, 19.6 nq| MvoxwUi Z½iY KZ nGe? 2
t = 9.8
L. `†kÅK͸-2 ‣i •ijMvwoi •eM wbYÆq Ki| 4
t=2
M. `†kÅK͸-1 •^GK •`LvI •h, ej«¼Gqi AbycvZ 2 2 : 3 4
myZivs, K„Gci ZjG`k nGZ kõ DcGi AvmGZ mgq jvGM
2 bs cÉGki² mgvavb
2 – 2 •m.
= 235 2 = 35 K •`Iqv AvGQ, MvwowUi Avw`GeM, u = 30 km/h
kGõi •eM v nGj, s = vt ‣i mvnvGhÅ cvB, AwZKÌv¯¦ `ƒiZ,½ s = 100 km
ev, v 2 = 19.6 •kl •eM, v = 50 km/h
35
MvwowUi Z½iY nGj, v2 = u2 + 2fs
19.6 35
ev, v = 2 = 343 wg./•m. ev, v2 – u2 (50)2 – (30)2
f = 2s = 2 100 = 8
v = 343 wg./•m. (Ans.) Z½iY 8 km/h2 (Ans.)
2 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò
L gGb Kwi, •ijMvwo u •eGM PjGj L gGb Kwi, UvIqviwUi DœPZv = h wgUvi ‣es 3s cGi
eÕ§«q¼ f„wg nGZ x DœPZvq wgwjZ nq|
30 wg./•m. •eGM LvovfvGe co¯¦ 30 wg./•m. 60
h x
e†wÓ¡i •dvUv DjGÁ¼i mvG^ 60 u •ijMvwoi •eM
60
•KvGY coGQ eGj gGb nq|
tan60 = u sin90 †ewÓ¡i •eM h
30 + ucos90
ev, u x
3 = 30
u = 30 3 wg./•m. (Ans.)
M cx^vGMvivm nGZ, Q UvIqviwUi P„ov nGZ co¯¦ eÕ§i 3 •mGKG´£ AwZKÌv¯¦
`ƒiZ½,
Q2 = R2 + P2 Q
R= 3
ev, Q2 = Q2 + P2 h x = 1 g (3)2
3 2
ev, 9Q2 = Q2 + 9P2 P ev, h x = 44.1 wgUvi
ev, 8Q2 = 9P2 Avevi, 28 m/s Avw`GeGM UvIqviwUi cv`G`k nGZ
P2
ev, Q2 = 8 wbwÞµ¦ eÕ§i 3 •mGKG´£ AwZKÌv¯¦ `ƒiZ½ x = ut 1 gt2
9 2
P t Q = 2 2 t 3 (ˆ`LvGbv nGjv) ev, x = 28 3 1 9.8 32
2
cÉk² 3 DóxcK-1: ‣KwU UvIqvGii P„ov •^GK ‣KwU eÕ§
ev, x = 39.9 wgUvi
AevGa coGZ •`qv nj| ‣KB mgGq Aci ‣KwU eÕ§
UvIqviwUi DœPZv = (h x) + x
UvIqvGii cv`G`k •^GK 28m/s •eGM Lvov DcGi wbGÞc Kiv
= 44.1 + 39.9 = 84 wgUvi (Ans.)
nj| Zviv 3 •mGK´£ ci wgwjZ nq|
M gGb Kwi, eÕ§wUi cZbKvj t •mGK´£
DóxcK-2 : 50 wgUvi DuPz ‣KwU wgbvGii P„ov •^GK ‣KLí
1
cv^i 30m/s •eGM ‣es Abfy „wgGKi mvG^ 30 •KvGY wbwÞµ¦ ‣Lb, h = u sint + 2 gt2
nj| wkLbdj-8, 9 ev, 30 1 t2
2
K. wÕ©i AeÕ©v •^GK hvòv KGi ‣KwU KYv 4 •mGKG´£ 24 50 = 30 sin t + 9.8
wgUvi AwZKÌg KGi| KYvwU 7th •mGKG´£ KZ `ƒiZ½ ev, 50 = 15t + 4.9t2
AwZKÌg KiGe? 2 ev, 4.9t2 15t 50 = 0
L. DóxcK-1 : UvIqvGii DœPZv KZ? 4 ev, 15 1205
t = 2 4.9
M. DóxcK-2 : cv^iLíwU KZ Abfy „wgK `ƒiZ½ AwZKÌg
KiGe? 4 t = 5.073
3 bs cÉGki² mgvavb ev, t = 2.012 [hv MÉnYGhvMÅ bq]
cv^i L´£wUi AwZKÌv¯¦ Avbfy „wgK `ƒiZ½
K •`Iqv AvGQ, mgq, t = 4s
4 •mGK´£ mgGq AwZKÌv¯¦ `ƒiZ,½ S = 24 wgUvi = u cost
Avgiv Rvwb, = 30 cos 30 5.073
1 ft2 = 30 3 5.073
2 2
S = ut +
= 131.8 wgUvi (Ans.)
1
ev, 24 = 0 4 + 2 (4)2 cÉk² 4 my±`ieb KGjGRi evwlÆK KÌxov cÉwZGhvwMZvq ekÆv
ev, = 3 wgUvi/•mGK´£2 wbGÞc BGfG´Ÿ Kvgvj 1g nq| •m ekÆvwU Abfy „wgGKi mvG^
KYvwUi 7th •mGKG´£ AwZKÌv¯¦ `ƒiZ,½ •KvGY u •eGM wbGÞc KGi| ekÆvwU mGeÆvœP H DœPZvq DGV
S7th = u + 1 (2t 1) ‣es T mgq ci R `ƒiGZ½ f„wgGZ cwZZ nq| wkLbdj-8
2
= 0 + 1 3 (2 7 1) K. ekÆvwUi meÆvwaK Abfy „wgK cvÍÏv 48.75 wgUvi nGj
2
wbGÞcb •eM KZ? 2
= 19.5 wgUvi (Ans.)
mgZGj eÕ§KYvi MwZ 3
L. ekÆvwU meÆvwaK `ƒiGZ½ cwZZ nIqvi kGZÆ R I H ‣i gGaÅ ev, H = u2 g
R 4g u2
mÁ·KÆ wbYÆq Ki| 4
R
M. •`LvI •h, u = 1 (g2T4 + 1 4 ev, H= 4
2T 4R2)2
ev, R = 4H
4 bs cÉGki² mgvavb
‣wUB wbGYÆq mÁ·KÆ|
u2
K mGeÆvœP cvÍÏv, Rmax = g M cÉ`î kZÆvbmy vGi, T = 2u sin
g
cÉk²gGZ, Rmax = 48.75
1
ev, u2 u sin = 2 gT ... ... (i)
g = 48.75
‣es u2 sin 2
ev, u2 = 48.75 g R = g
ev, u2 = 48.75 9.81 R = u sin . 2u cos = 1 2u cos
g 2 gT . g
u = 21.87 ms1 (Ans.)
L Avgiv Rvwb, Rmax = u2 ucos = R ... ... (ii)
g T
R = Rmax nGe hLb = 45 (i) I (ii) •K eMÆ KGi •hvM Kwi,
meÆvwaK DœPZv H = u2 sin2 u2sin2 + u2cos2 = 21gT2 + R2
2g T
cÉk²gGZ, = 45 ewmGq cvB, ev, u2 (sin2 + cos2) = 14g2 T2 + R2
T2
H = u2 (sin 45)2 ev, u2 = g2T4 + 4R2
2g 4T2
u2 1 2 u2 1 u2 1
2g 2 2g 2 4g 4R2 + g2 T4 (4R2 + g2T4)2
= = = u= 4T2 =
u2 2T
H 4g (ˆ`LvGbv nGjv)
R = u2
g
DËi ms‡KZmn m„Rbkxj cÖkœ
cÉk² 5 `†kÅKÍ-¸ 1: A Dîi: L. t1 : t2 = u1 + u2 : 2 u1 u2
B cÉk²6 `†kÅK͸-1: ‣KwU evN ‣KwU nwiYGK wÕ©iveÕ©v
nGZ 2 wg./•m.2 mgZ½iGY QyUGZ •`GL 10 wg./•m. mgGeGM
•`Œo ÷i‚ Kij|
d wkLbdj-3, 5, 9(A-8)
`†kÅK͸-2: PQR wòfzGRi P, Q, R •KŒwYK we±`yGZ h^vKÌGg
F1, F2, F3 gvGbi wZbwU m`†k mgv¯¦ivj ej wKÌqv KiGQ| A
Ad d C
gGb Ki mvZvi‚i •eM u1 ‣es •mÉvGZi •eM u2. K. jÁ¼vskK Dccv`Å eYÆbv I eÅvLÅv Ki| 2
`†kÅK͸-2: f„wgi mvG^ •KvGY ‣es u •eGM wbwÞµ¦ ‣KwU a L. evNwU meÆvwaK KZ •cQGb ^vKGj nwiYwUGK aiGZ
cviGe? 4
DœPZvi eÕ§ `ywU •`qvj gvò AwZKÌg KiGZ cvGi hvG`i M. `†kÅK͸-2-‣ ewYÆZ ejòGqi jwº¬ wòfzGRi jÁ¼GK±`ÊMvgx
gaÅeZxÆ `iƒ Z½ 2a. wkLbdj-2, 9 nGj, cÉgvY Ki •h, F1cotP = F2cotQ = F3 cotR. 4
Dîi: L. 25 wgUvi
K. MvwYwZK ivwkgvjvmn msæv `vI : wePiYKvj. 2
L. `†kÅKÍ-¸ 1: nGZ t1, t2 ‣i AbycvZ wbYÆq Ki •hLvGb t1 cÉk² 7 h wgUvi Mfxi ‣KwU kƒbÅ Kzqvq ‣KwU cv^i Lí
mvZvi‚ KZK‡Æ b`xwU AB eivei AwZKÌvG¯¦i cÉGqvRbxq
•QGo w`Gj t •mGK´£ cGi Kzqvi ZjG`Gk ‣i cZGbi kõ
mgq ‣es t2, AC eivei `ƒiZ½ AwZKÌvG¯¦i cÉGqvRbxq
•kvbv •Mj| A wkLbdj-5 I 8
mgq| b`x AC cÉevwnZ| 4
K. 29.4ms1 •eGM Lvov IcGii w`GK Zxi QyoGj KZ
M. `†kÅKÍ-¸ 2: nGZ cÉgvY Ki •h, R = •hLvGb
2a cot 2 , R mgq ci Zv f„wgGZ wdGi AvmGe? 2
Avbyf„wgK cvÍÏv. 4 L. t = 3.5s nGj K„Gci MfxiZv wbYÆq Ki|
4 cvGéix m†Rbkxj DœPZi MwYZ w«¼Zxq cò
[kGõi MwZGeM 327 wgUvi/•mGK´£] 4 L. •`IqvjwU f„wg eivei x `ƒiGZ½ AewÕ©Z ‣es
M. kGõi MwZGeM v nGj cÉgvY Ki •h, 4
•`IqvGji Aci cvGk¼Æ y `ƒiGZ½ wMGq gvwUGZ coGj
(2h gt2)v2 + 2hgtv = h2g.
Dîi: K. 6s L. 54.5 wgUvi| cÉgvY Ki •h, •`IqvGji DœPZv xy 4
x+y
cÉk² 8 ‣KwU cv^i LíGK u wg./•m. •eGM Lvov DcGii M. hw` 60 •KvGY wbGÞc Kiv nGZv ZvnGj ejwU 7
w`GK wbGÞc KiGj t1 mgGq h DœPZvq AeÕ©vb KGi| A
wgUvi eÅeavGb AewÕ©Z 3.5 wgUvi DœP `yBwU
•`IqvGji wVK Dci w`Gq PGj •hGZv| eÕ§wUi
wkLbdj-8 I 9 Avbyf„wgK cvÍÏv wbYÆq Ki| 4
K. Lvov DcGii w`GK wbwÞµ¦ eÕ§i Avbfy „wgK `ƒiZ½ kƒbÅ Dîi: M. 7 3 wgUvi|
nq •Kb? 2 cÉk²10 gviwd Avbyf„wgGKi mvG^ 30 •KvGY 40 wgUvi/•mGK´£
L. cv^iwU t2 mgGqI f„wgi h DœPZvq AeÕ©vb KiGj •eGM dzUeGj wKK KiGjb| wkLbdj-2 I 4
cÉgvY Ki •h, h = 1 gt1t2 4 K. gviwdi wKK •bIqv ejwUi e†nîg DœPZv wbYÆq Ki|2
2
L. 2.6 wgUvi DœPZvi •MvjGcvGÓ¡i 5 wgUvi `ƒiZ½ •^GK
M. t •mGK´£ ci ‣KB we±`y nGZ ‣KB •eGM Aci ‣KwU
gviwdi •bIqv wKKwU Kx •Mvj nGe? MvwYwZKfvGe
cv^i Lí wbGÞc Kiv nGj cÉgvY Ki •h, h =
weGkÏlY KGiv| A 4
4u2 g2t2 wg. DuPzGZ Zviv wgwjZ nGe| 4
8g M. gviwd cÉwZ •mGKG´£ 9 wgUvi `ƒiZ½ AwZKÌg KiGj
cÉk² 9 ‣KwU wKÌGKU ejGK Abfy „wgGKi mvG^ 45 •KvGY wKK •bIqv ejwU Kx Avevi aiGZ cviGe?
AvNvZ KiGj •KvGbv •`qvGji wVK Ici w`Gq PGj hvq|
MvwYwZKfvGe weGkÏlY Ki| 4
Dîi: K. 20.4 ms1; L. •Mvj nGe bv;
wkLbdj-10 M. ejwU aiGZ cviGe bv
K. •`LvI •h, wbGÞcY •KvY 45 nGj Abfy „wgK cvÍÏv
mGeÆvœP nGe| 2
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