HSC Model questions 2nd part
- 1. Model Questions (Suggestion); Higher Mathematics 2nd
Paper
1| ‡h †Kvb `yBwU cÖ‡kœi DËi `vIt 5 2=10
1. K) ev¯Íe msL¨vt
1. gvb wbY©q Kit
i) -16 + 3+-1 - 4- 3 - -1 - 7 ii) -2--6
iii) -1 - 8 + 3 - 1 *** iv) 2 - 6 - 1 - 9
v) 3 - 5 + 7 - 12 vi) -3 - 5
*** vii) 13 + -1 - 4 - 3 - -8 DËit i) 7, ii) 4, iii) 11, iv) 4, v) 7, vi) 8, vii) 7
2. wb‡æi AmgZv¸‡jv cig gvb wPýe¨ZxZ cÖKvk Kit
*** i) x - 2< 5 *** ii) 2x - 3< 7
iii) x - 3 < 7 iv) x < 3
*** v) 5
13
1
x
(x ≠
3
1
) vi) 2x + 4 < 8
DËit i) -3 < x < 7, ii) -5 < x < 2, iii) -4 < x < 10, iv) -3 < x < 3,
v)
3
1
5
2
x ev,
15
4
3
1
x vi) -6 < x < 2
3. wb‡æi AmgZv¸‡jv cig gvb wP‡ýi mvnv‡h¨ cÖKvk Kit
i) 4 < x < 10 *** ii) -2 < x < 6
*** iii) -7 < x < -1 ** iv) 2 x 3
*** v) -1 <2x - 3 < 5 vi) -5 < x < 7
vii) -2 < 3 - x < 8 viii) -8 x 2
DËit i) x - 7< 3, ii) x - 2< 4, iii) x + 4 < 3, iv) x - 5 3,
v) 2x - 5< 3 vi) x - 1< 6, vii) x < 5, viii) x + 3 5.
4. wb‡æi AmgZv¸‡jv mgvavb Ki Ges mgvavb †mU msL¨v‡iLvq †`LvIt
*** i) 3x + 2< 7 * ii) 2x + 1< 3
*** iii) 2x - 5 < 3 * iv) 3x - 4 < 2
v) 2x + 5 < 1 *** vi) 2
53
1
x
vii) 2x + 3 > 9 viii) x < 4
ix) 2x - 5 < 1 x) 2x + 4 < 6.
DËit i) { x R: -3 < x <
3
5
} ii) { x R: -2 < x < 1}
iii) { x R: 1 < x < 4} iv) { x R:
3
2
< x < 1}
v) { x R: -3 < x < -2} vi) { x R:
2
3
< x <
3
5
ev
3
5
< x <
6
11
}
vii) { x R: -6 > x > 3} viii) { x R: -4 < x < 4}
ix) { x R: 2 < x < 3} x) { x R: -5 < x < 1}
5. *** i) x - 1 <
10
1
n‡j †`LvI †h, x2 - 1 <
100
21
ii) x - 1 <
2
1
n‡j †`LvI †h, x2 - 1 <
4
5
iii)x - 1 < 2 n‡j †`LvI †h, x2 - 1 < 8
6.** i) a, b R n‡j, †`LvI †h, ab=ab
*** ii) hw` a, b R nq, Z‡e cÖgvY Ki †h, a + ba+b
* iii) a, b R n‡j, cÖgvY Ki †h, a - ba+b
** iv) a, b R n‡j, cÖgvY Ki †h, a - ba-b
7.** i) ‡`LvI †h, 2 GKwU Ag~j` msL¨v|
*** ii) ‡`LvI †h, 3 GKwU Ag~j` msL¨v|
** iii) ‡`LvI †h, 5 GKwU Ag~j` msL¨v|
8. i) cÖgvY Ki †h, aa 2
Ges
22
aa
ii) cÖgvY Ki †h, x < a n‡j, -a < x < a ( †hLv‡b a > 0).
iii) ‡`LvI †h, -a a a †hLv‡b a †h †Kvb ev¯—e msL¨v|
iv) hw` a, b R nq, Z‡e †`LvI †h, -(a + b) = -a - b Ges (-a)b = -(ab).
* v) hw` a, b R nq, Z‡e †`LvI †h, (ab)-1 = a-1b-1 (a ≠ 0, b ≠ 0), Ges (-a)(-b) = ab.
* vi) cÖgvY Ki †h, hw` a R nq Z‡e a.0 = 0.
*** vii) hw` a, b, c R, ac = bc Ges c 0 nq, Z‡e cÖgvY Ki †h, a = b.
*** viii) hw` a, b, c R Ges a+b = a+c nq, Z‡e cÖgvY Ki †h, b = c.
ix) hw` a < b Ges b < c nq, Z‡e †`LvI †h, a < c.
** x) hw` a < b nq, Z‡e †`LvI †h, a + c < b + c Ges hw` a > b nq,
Z‡e †`LvI †h a + c > b + c, †hLv‡b a, b, c ev¯—e msL¨v|
1
- 2. 1| (L) RwUj msL¨v (i)
1. gWyjvm I Av¸©‡g›U wbY©q Kit
i) i31 ii) i31 DËit i) 2,
3
2
ii) 2,
3
2. eM©g~j wbY©q Kit
*i) 7 - 30 2 ***ii) 168 ***iii) i2 iv) i2 v) 42 2
xi
vi) )1(2 2
xix
DËit i) )235( i ii) )31( i iii) )1( i iv) )1( i
v) )22(
2
1
xix vi) )}1()1{(
2
1
xix
3. cÖgvY Ki t i) )1(
2
1
ii * ii) )1(
2
1
ii
iii) 2 ii
4. gvb wbY©q Kit
**i) 3
1 ii) 3
1 ***iii) 3
i ***iv) 3
i ***v) 4
81
vi) 4
169 ***vii) 6
64
DËit i) 1, )31(
2
1
ii) -1, )31(
2
1
iii) i , )3(
2
1
i
iv) i , )3(
2
1
i v) )1(
2
3
i vi) )1(
2
26
i vii) i2 , )3( i
5.***i) ( iba )( idc ) = iyx n‡j †`LvI †h, ( iba )( idc ) = iyx
*** ii) iyxiba 3
n‡j cÖgvY Ki †h, iyxiba 3
*** iii) iyxiba 3
n‡j cÖgvY Ki †h,
y
b
x
a
yx )(4 22
iv) hw` 122
ba nq, Z‡e †`LvI †h, x Gi GKwU ev¯—e gvb iba
ix
ix
1
1
mgxKiY‡K
wm× K‡i, GLv‡b a I b ev¯—e msL¨v|
*** v) idcibayx :: n‡j
†`LvI †h, 0)()(2)( 222222
ybaxybdacxdc
*** vi) ip 12 n‡j cÖgvY Ki †h, 01246
ppp
vii) ip 12 n‡j cÖgvY Ki †h, 01246
ppp
viii) ix 23 Ges iy 23 n‡j, †`LvI †h, 2322
yxyx
** ix) hw` n
n
n
xaxaxaax 2
210)1( nq, Z‡e †`LvI †h,
2
531
2
420 .....)(......)( aaaaaa = naaaa 210
6.* i) †`LvI †h, 1
iyx
iyx
*** ii) iyxz Ges 212 zz n‡j cÖgvY Ki †h, 122
yx
** iii) iyxz n‡j 2088 zz Øviv wb‡`©wkZ mÂvic‡_i mgxKiY wbY©q Ki| DËit
900259 22
yx
iv) iyxz n‡j 35 z e„‡Ëi e¨vmva© I †K›`ª wbY©q Ki|DËit (5, 0) Ges 3
** v) iyxz n‡j 2088 zz Øviv wb‡`©wkZ mÂvic‡_i mgxKiY wbY©q Ki|
DËit 900259 22
yx
1| (M) RwUj msL¨v ( )
1. GK‡Ki GKwU KvíwbK Nbgyj n‡j, †`LvI †h,
i) 4)1()1( 2222
ii) 4)1()1( 242242
* iii) 8)1)(1)(1( 222
*** iv) 9)1)(1)(1)(1( 10842
*** v) xyyxyxyx 6)()()( 22222
2Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-1
- 3. *** vi) 16)31()31( 44
** vii) 16)1)(1)(1)(1( 16884422
viii) n2).....1)(1)(1( 84422
Drcv`K ch©š— = n2
2
2.***i) hw` GK‡Ki GKwU KvíwbK Nbgyj nq Ges hw` qpx , 2
qpy ,
qpz 2
nq, Z‡e †`LvI †h, pqzyx 6222
ii) hw` 0)()()( 222222
cbacbacba nq Zvn‡j †`LvI
†h, ca ev, )(
2
1
cab
iii) hw` 0)()( 3232
cbacba nq, Z‡e †`LvI †h, )(
2
1
cba ev,
)(
2
1
acb ev, )(
2
1
bac .
*** iv) hw` 0 zyx Ges GK‡Ki GKwU KvíwbK Nbgyj nq Z‡e †`LvI †h,
xyzzyxzyx 27)()( 3232
v) GK‡Ki GKwU KvíwbK Nbgyj n‡j, †`LvI †h, ))(( 2222
babababa
vi) GK‡Ki GKwU KvíwbK Nbgyj n‡j, †`LvI †h,
))()((3 22333
cbacbacbaabccba
vii) hw` n
n
n
xpxpxppxx 2
2
2
210
2
)1( nq, Z‡e †`LvI †h,
1
630 3
n
ppp .
*** viii) cÖgvY Ki †h, 2
2
31
2
31
nn
hLb n Gi gvb 3 Øviv wefvR¨ Ges
ivwkwU = 1 , hLb nAci †Kvb cyY© msL¨v nq|
ix) hw` yxa , yxb , yxc 2
nq, Z‡e cÖgvY Ki †h,
)(3 33333
yxcba
x) hw` )31(
2
1
a Ges )31(
2
1
b nq Z‡e †`LvI †h,
04224
bbaa
cÖkœ 2| ‡h †Kvb `yBwU cÖ‡kœi DËi `vIt 5 2=10
2| (K) eûc`x I eûc`x mgxKiY(g~‡ji cÖK…wZ I g~j-mnM m¤úK©)
1. i) `yBwU g~‡ji †hvMdj k~b¨ n‡j, 0369164 23
xxx mgxKiYwU mgvavb Ki|
ii) g~j¸wj ¸‡YvËi cÖMgb †kªYxfy³ n‡j, 02452263 23
xxx mgxKiYwU mgvavb Ki|
** iii) 013175 23
xxx mgxKiYwUi GKwU g~j 1 n‡j Aci g~j`ywU wbY©q Ki|
*** iv) `yBwU g~‡ji AbycvZ 3 t 4 n‡j, 024222 23
xxx mgxKiYwU mgvavb Ki|
v) GKwU g~j Avi GKwUi wظY n‡j 045631424 23
xxx mgxKiYwU mgvavb Ki|
vi) `yBwU g~‡ji †hvMdj 5 n‡j, 06133 23
xxx mgxKiYwU mgvavb Ki|
vii) 01087 23
xxx mgxKi‡Yi GKwU g~j 31 n‡j mgxKiYwU mgvavb Ki|
viii) GKwU gyj i1 n‡j 0410105 234
xxxx mgxKiYwU mgvavb Ki|
ix) GKwU gyj i1 n‡j 02254 234
xxxx mgxKiYwU mgvavb Ki|
DËit i)
2
3
,
2
3
, 4 ii)
3
2
, 2 , 6 iii) i32 , i32 iv)
2
3
, 2 , 4
v)
4
3
,
2
3
,
3
5
vi)
3
2
, )135(
2
1
vii) 31 , 31 , 5
viii) 1 , 2 , i1 , i1 ix) 21 , 11
2. i) 03
rqxx mgxKi‡Yi g~j¸wj a , b , c n‡j 222
)()()( baaccb Gi
gvb wbY©q Ki|
ii) 03
rqxx mgxKi‡Yi g~j¸wj a , b , c n‡j ))()(( cbabacacb
Gi gvb wbY©q Ki|
*** iii) 023
rqxpxx mgxKi‡Yi g~j¸wj a , b , c n‡j 222
111
cba
Gi gvb wbY©q
Ki|
* iv) 0123 23
xx mgxKi‡Yi g~j¸wj , , n‡j 2 Gi gvb wbY©q Ki|
** v) 023
rqxpxx mgxKi‡Yi g~j¸wj a , b , c n‡j 222222
111
baaccb
Gi
gvb wbY©q Ki|
*** vi) 023
rqxpxx mgxKi‡Yi g~j¸wj , , n‡j 3 Gi gvb wbY©q Ki|
3Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-1
- 4. DËit i) q6 ii) r8 iii) 2
2
2
r
prq
iv) 1 v) 2
2
2
r
qp
vi) rppq 33 3
3. i) p , q g~j` n‡j, †`LvI †h, 0)()(2)( 2222222
qpxqpxqp mgxKi‡Yi
g~j¸wj g~j` n‡e|
* ii) hw` a , b , c g~j` Ges a +b + c = 0 nq, Zvn‡j †`LvI †h,
0)()()( 2
cbaxbacxacb mgxKi‡Yi g~j¸wj g~j` n‡e|
*** iii) ‡`LvI †h, ba bv n‡j 0)(22 222
baxbax mgxKi‡Yi g~j¸‡jv ev¯—e n‡Z
cv‡i bv|
*** iv) k Gi gvb KZ n‡j 04)2()1( 2
xkxk mgxKi‡Yi gyj¸‡jv ev¯Íe I mgvb n‡e?
DËit 10, 2
** v) hw` 086 222
bacabxxa mgxKi‡Yi g~j`ywU mgvb nq, Z‡e cÖgvY Ki †h,
xbxac 22
4)1( mgxKi‡Yi g~j`ywUI mgvb n‡e|
vi) k Gi gvb KZ n‡j 32)3(2)1( 2
kxkxk ivwkwU GKwU c~Y© eM© n‡e| Dt 3, -2
* vii) ‡`LvI†h, 22222
2)( bkhkxxah ivwkwU GKwU c~Y©eM© n‡e hw` 12
2
2
2
b
k
a
h
nq|
*** viii) a , b ev¯—e n‡j †`LvI †h, baxbabx 23)(22 2
mgxKi‡Yi gyj¸wj ev¯Íe n‡e;
hw` mgxKiYwUi GKwU g~j AciwUi wظY nq, Zvn‡j cÖgvY Ki †h, ba 2 A_ev, ba 114
4. i) hw` 02
cbxax mgxKi‡Yi GKwU g~j AciwUi e‡M©i mgvb nq Zvn‡j cÖgvY Ki †h,
abcbacca 3322
* ii) hw` 02
cbxax mgxKi‡Yi GKwU g~j AciwUi e‡M©i mgvb nq Zvn‡j cÖgvY Ki †h,
33
)()( bcabac .
*** iii) 0)2(627 2
pxx mgxKi‡Yi GKwU g~j AciwUi e‡M©i mgvb n‡j p Gi gvb wbY©q
Ki| DËit 6, -1
*** iv) hw` 02
qqxpx mgxKi‡Yi g~j`ywUi AbycvZ m t n nq, Z‡e †`LvI †h,
0
p
q
m
n
n
m
.
v) hw` 02
cbxax mgxKi‡Yi g~j`ywUi AbycvZ 3 t 4 nq, Zvn‡j cÖgvY Ki †h,
acb 4912 2
.
vi) hw` 011
2
1 cxbxa mgxKi‡Yi g~j`ywUi AbycvZ 022
2
2 cxbxa mgxKi‡Yi
g~j`ywUi Abycv‡Zi mgvb n‡j, †`LvI †h,
22
2
2
11
2
1
ca
b
ca
b
.
vii) hw` 02
qpxx mgxKi‡Yi GKwU g~j AciwUi e‡M©i mgvb nq, Z‡e †`LvI †h,
0)13( 23
qpqp
*** viii) hw` 02
qpxx mgxKi‡Yi g~j`yBwU µwgK c~Y© msL¨v nq, Zvn‡j cÖgvY Ki †h,
0142
qp
*** ix) hw` 02
qpxx mgxKi‡Yi g~j `ywUi cv_©K¨ 1 nq, Z‡e cÖgvY Ki †h,
222
)21(4 qqp
*** x)
qxpx
111
mgxKi‡Yi g~j `ywUi AšÍi r n‡j, p ‡K q Ges r Gi gva¨‡g cÖKvk Ki|
DËit 22
42 rqqp
* xi) k Gi gvb KZ n‡j 0)13(3)3( 22
kkxxk mgxKi‡Yi g~j`ywU ci¯úi Dëv n‡e?
DËit 4, -1
*** xii) 02
cbxax mgxKi‡Yi g~jØq , n‡j, cÖgvY Ki †h,
22
2
22 2
)()(
ca
acb
baba
*** xiii) hw` 02
cbxx Ges 02
bcxx mgxKi‡Yi g~j¸wji g‡a¨ †Kej GKwU aªye‡Ki
cv_©K¨ _v‡K, Z‡e cÖgvY Ki †h, 04 cb .
* xiv) 02
cbxax mgxKi‡Yi g~j`ywUi AbycvZ r n‡j †`LvI †h,
ac
b
r
r 22
)1(
xv) 02
cbxax mgxKi‡Yi g~j؇qi AbycvZ 4 t 5 n‡j cÖgvY Ki †h, acb 2120 2
4Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-2
- 5. 2| (L) eûc`x I eûc`x mgxKiY (g~j-mnM m¤úK© I mgxKiY MVb)
1.***i) hw` 012
qxpx Ges 012
pxqx mgxKiY `ywUi GKwU mvaviY g~j _v‡K, Zvn‡j
†`LvI †h, 01 qp .
*** ii) hw` 062
kkxx Ges 022
kxx mgxKiY `ywUi GKwU mvaviY g~j _v‡K Zvn‡j
k Gi gvb¸wj wbY©q Ki| DËit 0, 3, 8
*** iii) hw` 02
cbxax Gi GKwU g~j 02
abxcx Gi GKwU g~‡ji wظY nq, Zvn‡j
†`LvI †h, ca 2 A_ev, 22
2)2( bca
* iv) hw` 02
cbxax Ges 02
abxcx mgxKiY `ywUi GKwU mvaviY g~j _v‡K, Zvn‡j
†`LvI †h, bac .
* v) 02
qpxx Ges 02
pqxx mgxKiY `ywUi GKwU mvaviY g~j _vK‡j †`LvI †h,
Zv‡`i Aci `ywU gyj 02
pqxx mgxKi‡Yi g~j n‡e|
vi) ‡h kZ© mv‡c‡¶ 011
2
1 cxbxa Ges 022
2
2 cxbxa mgxKiY `ywUi GKwU g~j
mvaviY n‡Z cv‡i Zv wbY©q Ki| DËit 2
122112211221 )())(( acaccbcbbaba
vii) ‡h kZ© mv‡c‡¶ 011
2
1 cxbxa Ges 022
2
2 cxbxa mgxKiY `ywUi `ywU g~jB
mvaviY n‡Z cv‡i Zv wbY©q Ki| DËit
2
1
2
1
2
1
c
c
b
b
a
a
viii) 02
qpxx Ges 02
pqxx mgxKiY `ywUi GKwU gvÎ mvaviY g~j _vK‡j cÖgvY
Ki †h, 01 qp
2.***i) 0164 2
xx mgxKi‡Yi g~j`ywU , n‡j
1
Ges
1
g~j wewkó mgxKiYwU
wbY©q Ki| DËit 025304 2
xx
ii) 02
cbxax mgxKi‡Yi g~jØq , n‡j 2
Ges 2
g~jwewkó mgxKiYwU
wbY©q Ki| DËit 03)2( 232223
caabcbacxacbabaxa
*** iii) 02
cbxax mgxKi‡Yi g~jØq , n‡j Giƒc mgxKiY wbY©q Ki hvi g~jØq
1
Ges
1
. DËit 0)()( 22
acxacbcax
iv) 02
abxax mgxKi‡Yi g~jØq , n‡j ba Ges ba g~jØq Øviv MwVZ
mgxKiYwU wbY©q Ki| DËit 022
abxx
* v) 02
baxx mgxKi‡Yi g~jØq , n‡j 2
)( Ges 2
)( g~jwewkó mgxKiYwU
wbY©q Ki DËit 04)2(2 2422
baaxbax
vi) GKwU mgxKiY wbY©q Ki hvi g~j`yBwU h_vµ‡g 014317 2
xx mgxKi‡Yi g~j؇qi
†hvMdj I ¸Yd‡ji mgvb n‡e| DËit 042289289 2
xx
*** vii) Ggb GKwU mgxKiY wbY©q Ki hvi g~j`ywU h_vµ‡g 02 222
baaxx mgxKi‡Yi g~j`ywUi
mgwó I Aš—id‡ji cig gvb n‡e| DËit 04)(22
abxbax
viii) hw` 0)(
4
1 222
baaxx mgxKi‡Yi g~j `ywU , nq, Z‡e cªgvY Ki †h,
0)(2
abxbax mgxKi‡Yi g~j`ywU + I - n‡e|
ix) 02
rqxpx mgxKi‡Yi g~jØq , n‡j 3
1
Ges 3
1
g~jwewkó mgxKiY wbY©q Ki|
DËit 0)3( 3223
axacbbxc
x) hw` 02
pqxpx mgxKi‡Yi g~j`ywU , nq, Zvn‡j cÖgvY Ki †h,
2
))(( pqpqp Ges qp , qp g~jwewkó mgxKiYwU wbY©q Ki|
DËit 022
pqxx
*** xi) hw` 02
cbxax mgxKi‡Yi g~jØq , nq, Zvn‡j 0422
abxcx mgxKi‡Yi
g~j`ywU , Gi gva¨‡g cÖKvk Ki| DËit
2
,
2
** xii) hw` 02
cbxax mgxKi‡Yi g~jØq , nq, Zvn‡j
0)2()1( 22
xacbxac Gi g~j`ywU , Gi gva¨‡g cÖKvk Ki| DËit
,
5Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-2
- 6. 2| (M) wØc`x Dccv`¨
1.***i)
11
2 3
2
x
x Gi we¯Z…wZ‡Z 10
x Gi mnM wbY©q Ki| DËit 47
32330
* ii)
15
2 3
x
a
x Gi we¯Z…wZ‡Z 18
x Gi mnM wbY©q Ki| DËit 4
110565a
iii)
10
2 2
x
y
x Gi we¯Z…wZ‡Z 8
x Gi mnM wbY©q Ki| DËit 4
3360y
* iv) 78
)1()1( xx Gi we¯Z…wZ‡Z 7
x Gi mnM wbY©q Ki| DËit 35
2.***i)
11
2
4
1
2
x
x Gi m¤cÖmvi‡Y x ewR©Z c`wUi gvb wbY©q Ki|
*** ii)
12
3 1
2
x
x Gi we¯Z…wZ †_‡K x ewR©Z c`wUi gvb wbY©q Ki|
*** iii)
10
6
1
2
x
x Gi we¯Z…wZ‡Z x ewR©Z c`wU †ei Ki Ges Gi gvb wbY©q Ki|
*** iv)
18
2
1
x
x
Gi we¯Z…wZ †_‡K aª‚eK c`wU †ei Ki Ges Gi gvb wbY©q Ki|
* v)
6
2
2 1
2
x
x Gi m¤cÖmvi‡Y x ewR©Z c`wUi gvb wbY©q Ki|
vi)
10
3
2
2
1
2
x
x Gi we¯Z…wZ‡Z x ewR©Z c`wU †ei Ki Ges Gi gvb wbY©q Ki|
vii)
10
2
3
4
x
y
y
x
Gi we¯Z…wZ‡Z y ewR©Z c`wU †ei Ki Ges Gi gvb wbY©q Ki|
viii)
15
2
2
3
x
x Gi we¯Z…wZ‡Z x ewR©Z c` Ges c`wUi gvb wbY©q Ki|
*** ix)
n
x
x
2
1
Gi we¯Z…wZ‡Z x ewR©Z c` Ges Zvi gvb wbY©q Kit
*** x) p Ges q abvZ¥K c~Y© msL¨v n‡j, qp
x
x )
1
1()1( Gi we¯Z…wZ‡Z x gy³ c`wUi gvb wbY©q
Ki|
xi) qp
x
x )
4
1
1()41( Gi we¯Z…wZi †Kvb c`wU me©`v x gy³ _vK‡e?
DËit i)495 ii) 1760 iii) 6 Zg c` =
27
28
iv) DËit 13 Zg c` = 18564
v) 924 vi) 840 vii) 7 Zg c`,
32
105 4
x
viii) 6 Zg c`, 115
321001
ix)
!!
)!2(
)1(
nn
nn
x)
!!
)!(
qp
qp
xi) )1( q Zgc`
3. i) hw` 5
)2( xa Gi we¯Z…wZ‡Z 3
x Gi mnM 320 nq, Zvn‡j a Gi gvb KZ n‡e?
DËit 2a
*** ii)
n
x
2
3 Gi we¯Z…wZ‡Z 7
x I 8
x Gi mnM `yBwU mgvb n‡j, (n N), nGi gvb wbY©q Ki|
iii) 34
)34( x Gi we¯Z…wZ‡Z µwgK `yBwU c‡`i mnM mgvb n‡j G c` `yBwUi x Gi NvZ wbY©q Ki|
DËit 19
x , 20
x
iv) n
x)1( Gi we¯Z…wZ‡Z )1( r Zg c‡`i mnM, )3( r Zg c‡`i mn‡Mi mgvb n‡j, †`LvI
†h, 22 nr (n N),
* v) 14
)1( x Gi we¯Z…wZ‡Z )1( r Zg c‡`i mnM, )13( r Zg c‡`i mn‡Mi mgvb n‡j r Gi
gvb wbY©q Ki| DËit 4
vi) n
x)1( Gi we¯Z…wZ‡Z wZbwU µwgK c‡`i mn‡Mi AbycvZ 1 t 7 t 42 n‡j n Gi gvb wbY©q
Ki| DËit 55
4.***i) 44
)1( x Gi we¯Z…wZ‡Z 21 Zg c` I 22 Zg c` `yBwU mgvb n‡j, x Gi
gvb wbY©q Ki| DËit
8
7
6Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-2
- 7. *** ii) hw` n
xa )3( Gi we¯Z…wZ‡Z cÖ_g wZbwU c` h_vµ‡g b , bx
2
21
I 2
4
189
bx nq, Zvn‡j a ,
b Ges n Gi gvb †ei Ki| DËit 2a , 7n , 7
2b
* iii) n
x)1( Gi we¯Z…wZ‡Z hw` a, b, c, d h_vµ‡g 6ô, 7g, 8g, 9g c` nq, Z‡e cÖgvY Ki †h,
c
a
dbc
acb
3
4
2
2
.
5. i)
10
2
2
1
3
x
x Gi we¯Z…wZ‡Z ga¨c`wU wbY©q Ki| DËit 5
5
10
5
2
3
xC
** ii) n N n‡j
12
n
a
x
x
a
Gi we¯Z…wZ †_‡K ga¨c` (c`¸‡jv) wbY©q Ki|
DËit )1( n Zg c` =
x
a
Cn
n 12
, )2( n Zg c` =
a
x
Cn
n
1
12
iii) †`LvI †h,
n
x
x
2
1
Gi we¯Z…wZi ga¨c`wU n
n
n
)2(
!
)12........(5.3.1
.
*** iv) †`LvI †h,
n
x
x
2
1
Gi we¯Z…wZi ga¨c`wU n
n
n
)2(
!
)12........(5.3.1
.
6.***i) ‡`LvI †h, 2
1
41
x Gi we¯Z…wZ‡Z r
x Gi mnM 2
)!(
)!2(
r
r
.
*** ii) ‡`LvI †h, 2
1
21
x Gi we¯Z…wZ‡Z )1( r Zg c‡`i mnM r
r
r
2)!(
)!2(
2
.
7.***i)
)21)(1(
1
xx
Gi we¯Z…wZ‡Z r
x Gi mnM wbY©q Ki| DËit 12 1
r
*** ii) cÖgvY Ki †h, 12
)651(
xx Gi we¯Z…wZ‡Z n
x Gi mnM 11
23
nn
.
iii)
)1)(1( bxax
x
Gi we¯Z…wZ‡Z n
x Gi mnM wbY©q Ki| DËit
ba
ba nn
iv) ‡`LvI †h,
x
x n
1
)1(
Gi we¯Z…wZ‡Z n
x Gi mnM n
2 (n N).
v) 2
1
32
)4321( xxx Gi we¯Z…wZ‡Z r
x Gi mnM wbY©q Ki| DËit 1
8.***i) hw` 32
xxxy nq, Zvn‡j †`LvI †h, ....432
yyyyx
*** ii) hw` .....432
xxxxy nq, Zvn‡j x †K y Gi kw³i DaŸ©µg avivq cÖKvk Ki|
DËit ....432
yyyyx
*** iii) hw` ....432 32
xxxy nq, Z‡e †`LvI †h, .....
16
5
8
3
2
1 32
yyyx
iv) cÖgvY Ki †h, ....)321....)(1( 22
xxxx =
....)5.44.33.22.1(
2
1 32
xxx
v) n N Ges n
n
n
xcxcxccx ...)1( 2
210 n‡j cÖgvY Ki †h,
1
531420 2.........
n
cccccc .
vi) 8x n‡j
2
1
8
1
x
†K x Gi kw³i DaŸ©µgvbymv‡i cÂg c` ch©šÍ we¯Ívi Ki Ges †`LvI
†h,
2
3
....
32
5
.
24
3
.
16
1
.
8
1
24
3
.
16
1
.
8
1
16
1
.
8
1
8
1
1
DËit ....
2
.
32
5
.
24
3
.
16
1
.
8
1
2
.
24
3
.
16
1
.
8
1
2
.
16
1
.
8
1
2
.
8
1
1 4
4
3
3
2
2
xxxx
7Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-2
- 8. 3| (‡hvMvkªqx ‡cÖvMÖvg) ‡h †Kvb GKwU cÖ‡kœi DËi `vIt 5 1=5
3| (K) Group A
1.***i) ‡hvMvkªqx ‡cÖvMÖvg wK? †hvMvkªqx ‡cÖvMÖv‡gi myweav I kZ©vejx Av‡jvPbv Ki|
*** ii) ÒAvaywbK Drcv`b I e›Ub e¨ve¯’vq †hvMvkªqx †cÖvMÖvg GKwU Acwinvh© nvwZqviÓ| e¨vL¨v Ki|
* iii) wKfv‡e †hvMvkªqx ‡cÖvMÖv‡gi mgm¨v MVb Kiv nq? Zv we¯ÍvvwiZfv‡e eY©bv Ki|
2. wbæwjwLZ †hvMvkªqx †cÖvMÖvg‡K †jLwP‡Îi mvnv‡h¨ mgvavb K‡i cÖvwš—K we›`y wbY©q Ki Ges m‡e©v”PKiY Kit
*** i) z = 4x + 6y; kZ©t x + y = 5, x 2, y 4, x,y 0
*** ii) z = 3x + 4y; kZ©t x + y 7, 2x +5y 20, x,y 0
*** iii) z = 2x + 3y; kZ©t x + 2y 10, x + y 6, x 4., x,y 0
* iv) z = 2x + y; kZ©t x + 2y 10, x + y 6, x – y 2, x – 2y 10, x,y 0
*** v) z = 12x + 10y; kZ©t 2x + y 90, x + 2y 80, x + y 50, x 0, y 0.
** vi) z = 5x + 7y; kZ©t x + y 4, 3x + 8y 24, 10x + 7y 35, x 0, y 0.
vii) z = 3x + 2y; kZ©t x + y 1, y – 5x 0, 5y – x 0, x - y -1, x + y 6,
x 3, x,y 0.
* viii) z = 3x + y; kZ©t 2x + y 8, 2x + 3y 12, x 0, y 0.
* ix) z = 3x + 4y; kZ©t x + y 450, 2x + y 600, x 0, y 0.
** x) z = 3x + 2y; kZ©t 2x + y 8, 2x + 3y 12, x 0, y 0.
*** xi) z = 12x + 10y; kZ©t 2x - y 90, x - 2y 80, x - y 50, x,y 0.
DËit i) 26 ii) 23 iii) 16 iv) 10
v) 580 vi) 24.8 vii) 15 viii) 12
ix) 1800 x) 13 xi) 380
3. wbæwjwLZ †hvMvkªqx †cÖvMÖvg‡K †jLwP‡Îi mvnv‡h¨ mgvavb K‡i cÖvš—we›`y wbY©q Ki Ges me©wbæKib Kit
*** i) a) z = 2y – x
b) z = - x +y
Dfq †¶‡ÎB kZ©t 3y – x 10, x + y 6, x – y 2, x,y 0.
*** ii) a) z = 2x – y
b) z = 4x - y
Dfq †¶‡ÎB kZ©t x + y 5, x + 2y 8, 4x + 3y 12, x,y 0.
*** iii) z = 2x – y kZ©t x + y 5, x + 2y 8, x,y 0.
iv) z = 3x + 5y kZ©t x 2y + 2, x 6 – 2y, y x, x 6.
* v) z = 4x + 6y kZ©t x + y = 5, x 2, y 4, x,y 0.
vi) z = 3x1 + 2x2, kZ©t x1 + 2x2 4, 2x1 + x2 4, x1 + x2 5, x1, x2 0.
DËit i) (a) -2 (b) -2 ii) (a) 1 (b) 5 iii) -5 iv) 16 v) 20 vi)
3
20
4.***i) A I B `yB cÖKv‡i Lv‡`¨i cÖwZ †KwR‡Z †cÖvwUb I †k¦Zmv‡ii cwigvb I Zvi g~j¨ wbæiƒct
Lv`¨ ‡cÖvwUb cÖwZ †KwR ‡k¦Zmvi cÖwZ †KwR cÖwZ †KwRi g~j¨
A 8 10 40 UvKv
B 12 6 50 UvKv
‣`wbK b~¨bZg cÖ‡qvRb 32 22
me‡P‡q Kg Li‡P Kxfv‡e •`wbK Lv`¨ cª‡qvRb †gUv‡bv hv‡e Zv wbY©q Ki|
*** ii) A I B `yB ai‡Yi Lv‡`¨i cÖwZ wK‡jv‡Z †cÖvwUb I d¨vU wbæiƒct
Lv`¨ ‡cÖvwUb d¨vU wK‡jv cÖwZ g~j¨
A 1 3 2 UvKv
B 3 2 3 UvKv
‣`wbK b~¨bZg cÖ‡qvRb 9 12
me‡P‡q Kg Li‡P Kxfv‡e •`wbK Lv`¨ cª‡qvRb †gUv‡bv hv‡e Zv wbY©q Ki|
iii) F1 I F2 `yB ai‡Yi Lv‡`¨i cÖwZ wK‡jv‡Z wfUvwgb C I D cvIqv hvq wbæiƒct
Lv`¨ wfUvwgb C wfUvwgb D wK‡jv cÖwZ g~j¨
F1 2 3 5 UvKv
F2 5 6 3 UvKv
‣`wbK b~¨bZg cÖ‡qvRb 50 60
me‡P‡q Kg Li‡P ‣`wbK wfUvwgb C I D Gi Pvwn`v Kxfv‡e †gUv‡bv hv‡e Zv wbY©‡qi Rb¨ GKwU
†hvMvkªqx †cÖvMÖvg mgm¨v •Zix Ki|
* iv) GK e¨w³ X I Y `yB ai‡bi Lv`¨ MÖnY K‡i| wZb ai‡Yi cywó N1, N2, N3 Gi cwigvb, Lv‡`¨i g~j¨ I
cywói •`wbK me©wbæ cÖ‡qvRb wbæiƒct
`vg
X Y ‣`wbK b~¨bZg
cÖ‡qvRb1.00 UvKv 3.00 UvKv
N1 30 12 60
N2 15 15 60
N3 6 18 36
‡hvMvkªqx †cÖvMÖv‡gi mvnv‡h¨ Lv‡`¨i Ggb GKwU mgš^q wbY©q Ki, hv me©wbæ Li‡P H e¨w³i •`wbK
cÖ‡qvRb †gUv‡e|
(v) wb‡æi cÖ`Ë ZvwjKv †_‡K mgvavb †ei Ki Ges me©wbæ e¨‡q cÖ‡qvRbxq cywó mgwš^Z Lv‡`¨i m‡e©vrK…ó
mgš^q Kit
cÖwZ GK‡Ki g~j¨ (UvKv)
Lv`¨-A Lv`¨-B b~¨bZg GKK cÖ‡qvRb
1.00 2.00
cywó-I 20 8 40
cywó-II 10 10 40
cywó-III 4 12 24
DËit i) A cÖKvi 1 †KwR, B cÖKvi 2 †KwR| ii)A cÖKv‡ii †KwR Ges B cÖKv‡ii , †gvU LiP UvKv|
iii) F1 – 5, F2 – 8. iv) X cÖKvi 3 †KwR, Y cÖKvi 1 †KwR|
v) A cÖKvi 3 GKK, B cÖKvi 1 GKK|
8Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-3
- 9. 3| (L) Group B
1.***i) GK e¨w³ 500 UvKvi g‡a¨ Kgc‡¶ 6 Lvbv MvgQv Ges 4 Lvbv †Zvqv‡j wKb‡Z Pvq| cÖwZLvbv MvgQvi
`vg 30 UvKv Ges cÖwZLvbv †Zvqv‡ji `vg 40 UvKv| cÖ‡Z¨K cÖKv‡ii KZLvbv wRwbm wKb‡j †m cÖ`Ë
kZ©vax‡b me©v‡c¶v †ewk msL¨K wRwbm wKb‡Z cvi‡e?
*** ii) GKwU †jvK me©vwaK 100 UvKv e¨q K‡i K‡qKLvbv _vjv I Møvm wKb‡Z Pvb| cÖwZwU _vjv I Møv‡mi g~j¨
h_vµ‡g 12 UvKv I 8 UvKv| Aš—Zt 1 Lvbv _vjv I 8 wUi †ewk Møvm wZwb wKb‡eb bv| Dc‡iv³
UvKvq wZwb †Kvb cÖKv‡ii KZK¸‡jv wRwbm wKb‡j GK‡Î me©vwaK msL¨K wRwbm wKb‡Z cvi‡eb?
iii) GKwU †jvK me©vwaK 500 UvKv e¨‡q K‡qKwU Kvc I †cøU wKb‡Z Pvb| cÖwZ Kv‡ci `vg 30 UvKv I
†cø‡Ui `vg 20 UvKv| Ab~¨b 3 wU †cøU I AbwaK 6 wU Kvc †Kbvi k‡Z© H UvKvq †Kvb cÖKv‡ii
KZ¸‡jv wRwbm wKb‡j wZwb †gvU me©vwaK wRwbl wKb‡Z cvi‡eb?
iv) GK e¨w³ Zvi evMv‡b Kgc‡¶ 12wU bvi‡K‡ji Pviv Ges 4wU Av‡gi Pviv jvMv‡Z Pvb| cÖwZwU
bvi‡K‡ji Pviv I Av‡gi Pvivi g~j¨ h_vµ‡g 20 UvKv Ges 30 UvKv| H e¨w³ 600 UvKvi †ekx e¨q
bv K‡i cÖ‡Z¨K cÖKv‡ii KZ¸‡jv Pviv wKb‡Z cv‡ib hv‡Z †gvU Pvivi msL¨v me©vwaK nq?
*** v) GK e¨w³ 1200 UvKvi gv‡Qi †cvbv wKb‡Z Pvq| 100 iyB gv‡Qi †cvbvi `vg 60 UvKv Ges 100
KvZj gv‡Qi †cvbvi `vg 30 UvKv n‡j, wZwb †Kvb gv‡Qi KZ †cvbv wKb‡Z cvi‡eb hvi †gvU msL¨v
me©vwaK 3000 nq?
DËit i) MvgQv 10 Lvbv, ‡Zvqv‡j 5 Lvbv| ii) _vjv 3 Lvbv, Møvm 8 wU|
iii) 6 Uv Kvc, 16 Uv ‡cøU| iv) bvi‡Kj Pviv 18 wU, Av‡gi Pviv 8 wU|
v) iyB 1000, KvZj 2000|
2.***i) GKRb dj we‡µZv Av½yi I Kgjv wgwj‡q 500 UvKvi dj wKb‡e| wKš‘ ¸`vgN‡i 12 wUi AwaK ev·
ivL‡Z cv‡i bv| GK ev· Kgjvi `vg 50 UvKv Ges GK ev· Av½y‡ii `vg 25 UvKv| †m cÖwZ ev·
Kgjv I Av½yi h_vµ‡g 10 UvKv I 6 UvKv jv‡f weµq K‡i| †jvKwU †h cwigvY dj †K‡b Zvi meB
wewµ n‡h hvq| Kgjv I Av½yi KZ¸‡jv µh Ki‡j †m m‡e©v‛P jvf Ki‡Z cvi‡e?
*** ii) GKRb e¨emvqx Zvi †`vKv‡bi Rb¨ †iwWI Ges †Uwjwfkb wg‡j 100 †mU wKb‡Z cv‡ib| †iwWI †mU
I †Uwjwfkb †mU cÖwZwUi µh g~j¨ h_vµ‡g 40 Wjvi I 120 Wjvi| cÖwZ †iwWI I †Uwjwfkb †m‡U
jvf h_vµ‡g 16 Wjvi I 32 Wjvi| m‡e©v‛P 10400 Wjvi wewb‡qvM K‡i m‡e©v‛P KZ jvf wZwb
Ki‡Z cv‡ib?
iii) GKRb K…lK avb Ges M‡gi Pvl Ki‡Z wM‡q †`L‡jb †h cÖwZ †n±i Rwg‡Z avb I Mg Pv‡li LiP
h_vµ‡g 1200 UvKv Ges 800 UvKv| cÖwZ †n±i Rwg‡Z avb I Mg Pv‡li Rb¨ h_vµ‡g 4 Rb I 6
Rb K‡i kªwg‡Ki cÖ‡qvRb nq| m‡e©v‛P 26 Rb kªwgK wb‡qvM K‡i Ges 4800 UvKv wewb‡qvM K‡i
m‡e©v‛P KZ †n±i Rwg wZwb Pvl Ki‡Z cvi‡eb?
DËit i) Kgjv 8 ev·, Av½yi 4 ev·| ii) 2880 Wjvi iii) 5 ‡n±i|
3. i) GKwU cvYxq •Zixi KviLvbvq `yBwU kvLv I Ges II Gi Df‡qB A, B Ges C wZb cÖKv‡ii cvbxq
†evZjRvZ K‡i| kvLv `yBwUi •`wbK Drcv`b ¶gZv wbæiƒct
kvLv A cÖKv‡ii cvbxq B cÖKv‡ii cvbxq C cÖKv‡ii cvbxq
I 3000 1000 2000
II 1000 1000 6000
A, B I C cÖKv‡ii cvbx‡qi gvwmK Pvwn`v h_vµ‡g 24000, 16000 Ges 48000 †evZj| I Ges II
kvLvi •`wbK Kvq© cwiPvjbvq e¨q h_vµ‡g 600 UvKv I 400 UvKv| gv‡m †Kvb kvLv KZ w`b Pvjy
ivL‡j Zv me©wbæ Kvh© cwiPvjbvi e¨‡q cvbx‡qi gvwmK Pvwn`v c~iY Ki‡Z cvi‡e? me©wbæ e¨q KZ?
ii) GKwU cÖwZôvb Zv‡`i Drcvw`Z A I B cb¨ n‡Z GKK cÖwZ h_vµ‡g 3 UvKv I 4 UvKv jvf K‡i|
cÖwZwU cb¨ M1 I M2 †gwk‡b •Zix nq| A cb¨wU M1 I M2 †gwk‡b •Zix‡Z h_vµ‡g 1 wgwbU I 2
wgwbU mgq jv‡M Ges B cb¨wU M1 I M2 †gwk‡b h_vµ‡g 1 wgwbU I 1 wgwb‡U •Zix nq| cÖwZ Kv‡Ri
w`‡b M1 †gwkb me©vwaK 7
2
1
N›Uv I M2 †gwk‡b me©vw©aK 10 N›Uv e¨envi Kiv hv‡e| A I B cb¨ wK
cwigvb •Zix Ki‡j me©vwaK jvf n‡e? †hvMvkªqx †cÖvMÖv‡gi GKwU g‡Wj •Zix Ki|
iii) GKwU dvg© `yBwU cb¨ †Uwej I †Pqvi •Zix K‡i| A I B ‡gwk‡bi mvnv‡h¨ cb¨ `ywU‡K cÖwµqvRvZ Kiv nq| A
†gwkb 60 N›Uv ch©š— I B †gwkb 48 N›Uv ch©š— KvR Ki‡Z m¶g| GKwU †Uwej •Zix Ki‡Z A †gwk‡b 4
N›Uv Ges B †gwk‡b 2 N›Uv mgq jv‡M| cÖwZ †Uwe‡j 8 UvKv Ges cÖwZ †Pqv‡i 6 UvKv gybvdv n‡j me©vwaK
gybvdv cvIqvi Rb¨ KqLvbv †Uwej I KqLvbv †Pqvi •Zix Ki‡Z n‡e Zv wbY©q Ki|
DËit i) I cÖwZgv‡m 4 w`b, II cªwZgv‡m 12 w`b, me©wbæ e¨q 7200 UvKv|
ii) A – 150, B – 300. iii) 12 †Uwej, 6 †Pqvi|
9Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-3
- 10. 4| ‡h †Kvb `yBwU cÖ‡kœi DËi `vIt 25=10
4| (K) cive„Ë (Parabola)
1. wb‡Pi cÖwZwU cive„‡Ëi kxl©we›`y, Dc‡K›`ª, Dc‡Kw›`ªK j¤^, A¶‡iLv Ges w`Kv‡¶i mgxKiY wbY©q Ki|
* )i 01812103 2
yxy DËit )2,3( ; )2,
6
13
( ;
3
10
; 2y ;
6
23
x
* )ii 023282
yxy DËit )1,3( ; )1,1( ; 8 ; 1y ; 5x
)iii 07822
xyx DËit )
2
9
,4( ; )4,4( ; 2 ; 4x ; 5y
* )iv 0410155 2
yxx
DËit )
40
61
,
2
3
(
; )
40
41
,
2
3
(
; 2 ; 0
2
3
x ; 0
40
81
y
** )v 05643 2
xyx DËit )2,1( ; )
3
5
,1(
;
3
4
; 01 x ; 073 y
*** )vi 0592305 2
yxx
DËit )7,3( ; )
10
71
,3(
;
5
2
; 03 x ; 06910 y
*** )vii 06822
xyx DËit )5,4( ; )
2
9
,4(
; 2 ; 4x ; 0112 y
)viii 0242
yxx DËit )2,2( ; )
2
3
,2( ; 2 ; 02 x ; 052 y
* )ix 05643 2
xyx DËit )2,1( ; )
3
5
,1(
;
3
4
; 01 x ; 073 y
)x yxy 882
DËit )4,2( ; )4,0( ; 8 ; 04 y ; 04 x
* )xi 0592305 2
yxx
DËit )7,3( ; )
10
71
,3(
;
5
2
; 03 x ; 06910 y
** )xii 06822
xyx DËit )5,4( ; )
2
9
,4(
; 2 ; 4x ; 0112 y
)xiii )1(42
yx DËit )1,0( ; )0,0( ; 4 ; 0x ; 2y
)xiv 582
xy DËit )0,
8
5
(
; )0,
8
11
( ; 8 ; 0y ;
5
18
x
* )xv )3(22
xy DËit )0,3( ; )0,
2
5
(
; 2 ; 0y ; 072 x
2.*** pxy 42
cive„ËwU )2,3( we›`y w`‡q Mgb Ki‡j Gi Dc‡Kw›`ªK j‡¤^i •`N©¨ Ges Dc‡K‡›`ªi ¯’vbvsK
wbY©q Ki| DËit
3
4
; )0,
3
1
(
3. Giƒc cive„‡Ëi mgxKiY wbY©q Ki hvit
*** )i Dc‡K›`ª )2,8( Ges w`Kv‡¶i mgxKiY 92 yx
DËit 02592116)2( 2
yxyx
*** )ii Dc‡K›`ª )1,1( Ges w`Kv‡¶i mgxKiY 143 yx , Zvi A‡¶iI mgxKiY wbY©q Ki|
DËit 0494244)34( 2
yxyx ; 0143 xy
*** )iii Dc‡K›`ª )1,1( Ges w`Kv‡¶i mgxKiY 01 yx , cive„‡Ëi A‡¶i mgxKiY Ges
Dc‡Kw›`ªK j‡¤^i •`N©¨ I Gi mgxKiY wbY©q Ki|
DËit 0362)( 2
yxyx ; 02 yx , 0 yx , 2
)iv Dc‡K›`ª )0,(a Ges w`Kv‡¶i mgxKiY 0cx DËit )2)((2
caxcay
* )v Dc‡K›`ª )0,2( Ges w`Kv‡¶i mgxKiY 02 x DËit xy 82
)vi Dc‡K›`ª )4,0( Ges w`Kv‡¶i mgxKiY 04 y DËit 0162
yx
4.***i) †h cive„‡Ëi Dc‡K›`ª (3,4) Ges kxl© (0,0) Zvi w`Kv‡¶i mgxKiY wbY©q Ki|
DËit 3x+4y+25=0
ii) †h cive„‡Ëi Dc‡K›`ª (-1,1) Ges kxl© (2,-3) Zvi A¶ I w`Kv‡¶i mgxKiY wbY©q Ki|
DËit 4x+3y+1=0, 3x-4y-43=0
** iii) †h cive„‡Ëi Dc‡K›`ª (0,0) Ges kxl© (-2,-1) Zvi w`Kv‡¶i mgxKiY wbY©q Ki|
DËit 2x+y+10=0
5. Giƒc cive„‡Ëi mgxKib wbY©q Ki hvit
i) Dc‡K›`ª (-1,3) Ges kxl© (4,3) we›`y‡Z| DËit y2
+20x-6y-71=0
ii) Dc‡K›`ª (-6,-3) Ges kxl© (-2,1) we›`y‡Z| DËit (x-y)2 +38x+26y+41=0
** iii) Dc‡K›`ª (2,5) Ges x = 4 †iLvwU Gi kxl© we›`y‡Z ¯úk© K‡i| DËit y2 - 10y+8x-7=0
6. Giƒc cive„‡Ëi mgxKib wbY©q Ki hvit
* i) kxl© (2,3) Ges w`Kv‡¶i mgxKiY y = 6; Gi Dc‡Kw›`ªK j‡¤^i •`N©¨I wbY©q Ki|
10Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-4
- 11. DËit x2 – 4x+12y-32=0, 12
* ii) kxl© (3,1) Ges w`Kv‡¶i mgxKiY 4x+3y-5=0; DËit (3x-4y)2 – 190x-80y+625=0
* iii) kxl© (4,3) Ges w`Kv‡¶i mgxKiY y = 7; DËit x2 – 8x+16y-32=0
7.* i) Giƒc cive„‡Ëi mgxKiY wbY©q Ki hvi kxl© (4,-3) we›`y‡Z Aew¯’Z Ges w`Kv¶ x-A‡¶i mgvšÍivj
Ges hv (-4,-7) we›`y w`‡q AwZµg K‡i| DËit x2 – 8x+16y+64=0
* ii) GKwU cive„‡Ëi mgxKiY wbY©q Ki hvi kxl©we›`y (4,-3) we›`y‡Z Aew¯’Z, Dc‡Kw›`ªK j‡¤^i •`N©¨ 4
Ges A¶wU x-A‡¶i mgvš—ivj| DËit (y+3)2 = 4(x-4)
*** iii) y = ax2 + bx + c cive„ËwUi kxl© (-2,3) we›`y‡Z Aew¯’Z Ges GwU (0,5) we›`y w`‡q AwZµg
K‡i| a, b, c-Gi gvb wbY©q Ki| DËit a =
2
1
, b = 2, c = 5
8.***i) y2 = 8x cive„‡Ëi Dcwiw¯’Z †Kvb we›`yi Dc‡Kw›`ªK `~iZ¡ 8; H we›`yi ¯’vbvsK wbY©q Ki|
DËit 34,6(
*** ii) y2 = 16x cive„‡Ëi Dcwiw¯’Z †Kvb we›`yi Dc‡Kw›`ªK `~iZ¡ 6; H we›`yi ¯’vbvsK wbY©q Ki|
DËit 24,2( )
iii) y2 = 9x cive„˯’ †Kvb P we›`yi †KvwU 12 n‡j H we›`yi Dc‡Kw›`ªK `~iZ¡ wbY©q Ki|
DËit 18
4
1
9. i) y2 = 12x cive„‡Ëi kxl©we›`y I Dc‡Kw›`ªK j‡¤^i abvZ¥K w`‡Ki cÖvš—we›`yi ms‡hvRK †iLvi mgxKiY
wbY©q Ki| DËit y = 2x.
* ii) ‡`LvI †h, 0 nmylx †iLvwU axy 42
cive„ˇK ¯úk© Ki‡e hw` 2
ln am nq|
iii) cive„‡Ëi Av`k© mgxKiY wbY©q Ki|
4| (L) Dce„Ë (Ellipse)
10. wb‡Pi Dce„˸wji cÖwZwUi Dr‡Kw›`ªKZv, Dc‡K›`ª, Dc‡Kw›`ªK j‡¤^i •`N©¨ Ges Dc‡Kw›`ªK j‡¤^i I w`Kv‡¶i
mgxKiY wbY©q Kit (i – iv).
* i) 225259 22
yx DËit
5
4
, )0,4( ,
5
18
, 4x , 254 x
** ii) 1243 22
yx DËit
2
1
, )0,1( , 3 , 1x , 4x
*** iii) 4001625 22
yx DËit
5
3
, )3,0( ,
5
32
, w`Kv‡¶i mgxKib
3
25
y
*** iv) 4002516 22
yx DËit
5
3
, )0,3( ,
5
32
, 3x
3
25
x
* v) 144169 22
yx Dce„‡Ëi Dc‡K›`ªØ‡qi ¯’vbvsK Ges w`Kv¶Ø‡qi mgxKiY wbY©q Ki|
DËit )0,7( ;
7
16
x
** vi) 132 22
yx Dce„‡Ëi Dc‡Kw›`ªK j‡¤^i •`N©¨ Ges Dc‡K›`ª `ywUi ¯’vbvsK wbY©q Ki|
DËit 2
3
2
, )0,
6
1
(
** vii) 145 22
yx Dce„‡Ëi w`Kv¶ `yBwUi mgxKiY wbY©q Ki| DËit
2
5
y
*** viii) 01101654 22
yxyx Dce„‡Ëi Dc‡K›`ª `yBwU, Dc‡Kw›`ªK j‡¤^i •`N©¨,
Dr‡Kw›`ªKZv Ges w`Kv‡¶i mgxKiY wbY©q Ki|
DËit )1,3( , )1,1( ,
5
8
,
5
1
, 07 x , 03 x
ix) †`LvI †h, 01101654 22
yxyx mgxKiYwU GKwU Dce„Ë wb‡`©k K‡i; Gi
Dr‡Kw›`ªKZv, †K›`ª, Dc‡K›`ª, Dc‡Kw›`ªK j‡¤^i •`N©¨ Ges w`Kv‡¶i mgxKiY wbY©q Ki| DËit
5
1
,
)1,2( , )1,3( I )1,1( ,
5
8
, 03 x
11Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-4
- 12. * x) †`LvI †h, 03095 22
xyx mgxKiYwU GKwU Dce„Ë wb‡`©k K‡i; Gi Dc‡K›`ª `ywUi
¯’vbvsK wbY©q Ki| DËit )0,5( , )0,1(
* xi) †`LvI †h, 01282 22
yxyx mgxKiYwU GKwU Dce„Ë wb‡`©k K‡i; Gi
Dr‡Kw›`ªKZv, †K›`ª Ges Dc‡K›`ª `ywUi ¯’vbvsK wbY©q Ki|
DËit
2
1
e , †K›`ª )1,2( , Dc‡K›`ª )3,2( I )1,2(
2.***i) p Gi gvb KZ n‡j 14 22
ypx Dce„ËwU )0,1( we›`y w`‡q hv‡e? Dce„ËwUi Dc‡K‡›`ªi
¯’vbvsK, Dr‡K›`ªZv Ges A¶Ø‡qi •`N©¨ wbY©q Ki|
DËit 1p ,
2
3
, )0,
2
3
( ; e„nrA¶ 2 , ¶z`ª A¶ 1
*** ii) p Gi gvb KZ n‡j, 1
25
22
y
p
x
Dce„ËwU )4,6( we›`y w`‡q AwZµg Ki‡e? Dce„ËwUi
Dr‡Kw›`ªKZv Ges Dc‡K‡›`ªi Ae¯’vb wbY©q Ki| DËit 100p ,
2
3
, )0,35(
iii) 1
25
22
p
yx
Dce„ËwU )6,4( we›`y w`‡q AwZµg K‡i| p Gi gvb, Dr‡Kw›`ªKZv Ges Dc‡K‡›`ªi
Ae¯’vb wbY©q Ki| DËit 100p ,
2
3
e , )35,0(
* iv) p Gi gvb KZ n‡j, 804 22
pyx Dce„ËwU )4,0( we›`y w`‡q AwZµg Ki‡e? Dce„ËwUi
A¶Ø‡qi •`N©¨ Ges Dr‡Kw›`ªKZv wbY©q Ki| DËit 5p ,
5
1
e , 54 , 8
3. GKwU Dce„‡Ëi mgxKiY wbY©q Ki hvit
*** i) Dc‡K›`ª )4,3( , w`Kv¶ 02 yx Ges Dr‡Kw›`ªKZv
3
1
.
DËit 044614010417217 22
yxyxyx
* ii) Dc‡K›`ª )3,2( , w`Kv¶ 07 yx Ges Dr‡Kw›`ªKZv
3
1
.
DËit 0292210255 22
yxxyyx
* iii) Dc‡K›`ª )1,1( , Dr‡Kw›`ªKZv
2
1
Ges w`Kv‡¶i mgxKiY 03 yx
DËit 071010277 22
yxxyyx
iv) Dc‡K›`ª )1,1( , w`Kv¶ 02 yx Ges Dr‡Kw›`ªKZv
2
1
; Gi Dc‡Kw›`ªK j¤^I wbY©q Ki|
DËit 041212233 22
yxxyyx , 4
* v) Dc‡K›`ª )2,0( , Dr‡Kw›`ªKZv
2
1
Ges w`Kv‡¶i mgxKiY 04 y , Zvi Dr‡Kw›`ªK j‡¤^i
•`N©¨I wbY©q Ki| DËit 02434 22
yyx ; 6
vi) Dc‡K›`ª )1,2( , Dr‡Kw›`ªKZv
3
1
Ges wbqvg‡Ki mgxKiY 32 yx
DËit 066244841411 22
yxxyyx
vii) Dc‡K›`ª )3,2( , wbqvg‡Ki mgxKiY 032 yx Ges Dr‡Kw›`ªKZv
3
1
DËit 0186847241411 22
yxxyyx
viii) Dc‡K›`ª g~jwe›`y,
5
4
e Ges w`Kv¶ 2x . DËit 6425649 22
yxx
4. Dce„‡Ëi cÖavb A¶ `yBwU‡K x I y -A¶ we‡ePbv K‡i Giƒc Dce„‡Ëi mgxKiY wbY©q Ki hvit
*** i) Dr‡Kw›`ªKZv
3
1
Ges Dc‡Kw›`ªK j‡¤^i •`N©¨ 8 . DËit 1
1881
4 22
yx
ii) Dr‡Kw›`ªKZv
3
2
Ges Dc‡Kw›`ªK j‡¤^i •`N©¨ 5 . DËit 4053620 22
yx
iii) Dc‡Kw›`ªK j‡¤^i •`N©¨ 8 Ges Dr‡Kw›`ªKZv
2
1
DËit 642 22
yx
* iv) Dc‡K‡›`ªi ¯’vbvsK )0,3( Ges Dr‡Kw›`ªKZv
3
1
; Dce„‡Ëi w`Kv‡¶i mgxKiYI wbY©q Ki|
DËit 64898 22
yx , 27x
12Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-4
- 13. * v) Dc‡K‡›`ªi ¯’vbvsK )4,0( Ges Dr‡Kw›`ªKZv
5
4
; DËit 1
259
22
yx
* vi) Dc‡K›`ªØq )0,2( Ges e„nr A¶ 8 GKK| DËit 1
1216
22
yx
vii) Dc‡K›`ªØq )1,1( , )2,2( Ges e„nr A‡¶i •`N©¨ 8 GKK| DËit 1
23
2
16
22
yx
* viii) e„nr A¶ 12 Ges Dr‡Kw›`ªKZv
3
1
. DËit 1
3236
22
yx
ix) Dc‡K›`ªØq )0,1( Ges Dc‡Kw›`ªK j‡¤^i •`N©¨ 3 GKK| DËit 1243 22
yx
x) Dr‡Kw›`ªKZv
5
4
Ges )5,
3
10
( we›`yMvgx| DËit 225259 22
yx
* xi) Dc‡K›`ªØ‡qi `~iZ¡ 8 Ges w`Kv¶Ø‡qi ga¨Kvi `~iZ¡ 18 . DËit 18095 22
yx
*** xii) hv )2,2( Ges )1,3( we›`yMvgx| Gi Dr‡Kw›`ªKZvI wbY©q Ki|
DËit 3253 22
yx ,
5
2
* xiii) hv )6,1( Ges )0,3( we›`yMvgx| DËit 2743 22
yx
xiv) hv )4,2( Ges )2,5( we›`yMvgx| DËit 5632 22
yx
5. i) ‡h Dce„‡Ëi Dc‡Kw›`ªK j¤^ e„nr A‡¶i A‡a©K Zvi Dr‡K›`ªZv KZ? DËit
2
1
ii) †Kvb Dce„‡Ëi ¶z`ª A‡¶i •`N©¨ Zvi Dc‡K›`ªØ‡qi `~i‡Z¡i mgvb Ges Dc‡Kw›`ªK j¤^ 10 ; Dce„ËwUi
Dr‡Kw›`ªKZv I mgxKib wbY©q Ki| DËit
2
1
; 1002 22
yx
iii) †Kvb Dce„‡Ëi GKwU Dc‡K›`ª I Abyiƒc w`Kv‡¶i ga¨Kvi `~iZ¡ 16 Bw Ges Dr‡Kw›`ªKZv
5
3
;
Dce„‡Ëi cÖavb A¶ `yBwUi •`N©¨ wbY©q Ki| DËit 30 Bw I 24 BwÂ|
** iv) cÖgvY Ki †h, 5 xy mij‡iLvwU 144169 22
yx Dce„ˇK ¯úk© K‡i| ¯úk© we›`yi
¯’vbvsK wbY©q Ki| DËit
5
9
,
5
16
4| (M) Awae„Ë (Hyparabola)
1.** i) 01996418169 22
yxyx Awae„‡Ëi †K›`ª, kxl©we›`y, Dc‡K›`ª Ges Dr‡Kw›`ªKZv
wbY©q Ki| DËit (1,-2); (5,-2); (-3,-2); (6,-2); (4,-2);
4
5
e
** ii) 1
1625
22
yx
Awae„ËwUi kxl©we›`y, Dr‡Kw›`ªKZv Ges Dc‡K›`ª wbY©q Ki|
DËit )0,5( ;
5
41
e ; )0,41(
*** iii) †`LvI †h, 28 22
yx Awae„‡Ëi w`Kv‡¶i mgxKiY 43 x Ges Dc‡Kw›`ªK j‡¤^i •`N©¨
22
1
.
** iv) 1
25144
22
yx
Awae„‡Ëi Dc‡K‡›`ªi ¯’vbvsK Ges Dr‡Kw›`ªKZv wbY©q Ki|
DËit )0,13( ;
12
13
e
*** v) 823 22
xyx Awae„‡Ëi A‡¶i •`N©¨, Dr‡Kw›`ªKZv Ges †K‡›`ªi ¯’vbvsK wbY©q Ki|
DËit 6 ; 32 ;
3
2
e ; (1,0)
*** vi) 1
169
22
yx
Awae„‡Ëi Dc‡K›`ª `yBwUi ¯’vbvsK I w`Kv¶ `yBwUi mgxKiY wbY©q Ki|
DËit )0,5( ;
* vii) 144169 22
yx Awae„‡Ëi kxl©, Dc‡K›`ª Ges Dr‡K›`ªZv wbY©q Ki|
DËit )0,4( ; )0,5( ;
4
5
** viii) 4001625 22
yx Awae„‡Ëi ‡K›`ª, Dc‡K›`ª Ges Dr‡Kw›`ªKZv wbY©q Ki|
DËit (0,0); )0,41( ;
4
41
* ix) 4002516 22
yx Awae„ËwUi kxl©we›`y Ges Dc‡K›`ª wbY©q Ki| Dt )0,5( ; )0,14(
13Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-4
- 14. x) 2054 22
xy Awae„‡Ëi Dr‡K›`ªZv, Dc‡K›`ª I w`Kv‡¶i mgxKiY wbY©q Ki|
DËit;
√
)3,0( ;
xi) 03649 22
yx Awae„‡Ëi kxl©, Dc‡K›`ª, bvwfj¤^ I wbqvg‡Ki mgxKiY wbY©q Ki|
DËit )3,0( ; )13,0( ; 13y ;
√
xii) 06379 22
yx Awae„‡Ëi Dc‡K‡›`ªi Ae¯’vb I w`Kv‡¶i mgxKiY wbY©q Ki|
DËit )4,0( ; 94 y
2. Awae„‡Ëi mgxKiY wbY©q Ki hvi-
*** i) Dc‡K›`ª (1,1), w`Kv‡¶i mgxKiY 2x + y =1 Ges Dr‡Kw›`ªKZv 3
DËit 07421227 22
yxxyyx
*** ii) Dc‡K›`ª (1,-8), Dr‡Kw›`ªKZv 5 Ges w`Kv¶ 3x-4y=10.
DËit 02255024114 22
xxyyx
3. i) GKwU Awae„‡Ëi A¶ `yBwU ¯’vbvs‡Ki A¶ eivei| Awae„ËwU (-2,1) Ges (-3,-2) we›`yMvgx n‡j Zvi
mgxKiY wbY©q Ki| DËit 3x2 -5y2 = 7
** ii) GKwU Awae„Ë (6,4) I (-3,1) we›`y w`‡q AwZµg K‡i| Gi †K›`ª g~jwe›`y‡Z Ges Avo A¶ x
A¶ eivei n‡j Zvi mgxKiY wbY©q Ki| DËit 1
436
5 22
yx
iii) g~jwe›`y‡Z †K›`ª wewkó GKwU Awae„Ë (4,0) Ges (5, 2.25) we›`y w`‡q AwZµg K‡i; Awae„ËzwUi
Avo A¶ x A¶ eivei Aew¯’Z n‡j Zvi mgxKiY wbY©q Ki|
DËit 9x2 -16y2 =144
4. Awae„‡Ëi A¶ `yBwU‡K ¯’vbvs‡Ki A¶ a‡i Ggb GKwU Awae„‡Ëi mgxKiY wbY©q Ki hvi-
i) Dc‡K›`ª )13,0( Ges AbyewÜ A‡¶i •`N©¨ 24 GKK| DËit 1
14425
22
xy
* ii) Dc‡K›`ª `yBwUi `~iZ¡ 16 Ges Dr‡Kw›`ªKZv 2 DËit x2 - y2 = 32
5. i) Dc‡K›`ª `yBwUi ¯’vbvsK (4,2) I (8,2) Ges Dr‡Kw›`ªKZv 2 n‡j, Awae„‡Ëi mgxKiY wbY©q Ki|
DËit 1
3
)2(
1
)6( 22
yx
** ii) GKwU Awae„‡Ëi mgxKiY wbY©q Ki hvi Avo A¶ y A¶ eivei, AbyewÜ A¶ x -A¶ eivei,
kxl©Ø‡qi `~iZ¡ 2 Ges Dr‡Kw›`ªKZv 2 DËit 122
xy
5| ‡h †Kvb `yBwU cÖ‡kœi DËi `vIt 5 2=10
5| (K) wecixZ w·KvYwgwZK dvskb
1. cÖgvY Kit
*** i)
2
11
tan
5
2
cos
5
4
sin 111
ii)
48
1
tan2
7
1
tan
5
1
tan2 111
iii)
5
3
cos
2
1
9
2
tan
4
1
tan 111
iv) 2tan
3
2
cos
3
1
sin 111
v)
11
27
tan
3
5
cot
5
4
cos 111
* vi)
29
2
tan
2
5
cos
5
13
sec 111
ec
vii)
11
27
tan
5
3
sin
3
5
cot 111
* viii)
)5cos3(cot4 11
ec ***ix)
)3cot
5
1
(sin4 11
*** x)
2
13
sec
23
2
tan 11
xi)
632
16
cos
3
2
cos 11
*** xii)
2
)2cos(sin)sin2(sin 11
* xiii) xxx 2)3(cossin)cos(sin 11
xiv) 21
12)sin2sin( xxx
2. cÖgvY Kit
i)
7
1
tan
3
1
tan2 11
=
4
ii)
x
x
ecx
2
1
cos
2
1
tan
2
11
iii) )
2
1
tan4sin()
7
1
tan2cos( 11
*** iv) )tantan(tan2)tan2tan( 3111
xxx
14Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-4
- 15. v) 0)(cottan)(cottan)2tan
2
1
(tan 3111
AAA
* vi) )2(sintan}tan)12{(tan}tan)12{(tan 111
*** vii) xxxec 1111
tan)cottantan(costan2
viii) xxx
)3tan(cot)2(tancot 11
* ix)
cos
cos
cos
2
tantan2 11
ba
ab
ba
ba
* x)
x
x
x
x
x
1
2
sin
2
1
1
1
cos
2
1
tan 111
*** xi) 2
1
2
2
1
2
11
1
2
tan
1
1
cos
1
2
sintan2
x
x
x
x
x
x
x
xii)
xy
yx
yx
1
tantantan 111
3. cÖgvY Kit
*** i) 2tan
3
1
tan
5
3
sin
2
1
5
1
cos 1111
** ii)
29
28
tan2cot
13
5
cos
2
1
5
3
sin 1111
4. cÖgvY Kit
*** i) 15)3(cotcos)2(tansec 1212
ec
** ii) 25)3(sectan)4(tansec 1212
iii)
36
13
2)2(tancos)3(cotsec 1212
ec
iv) 1)3(cotsec3)
2
1
(tancos 1212
ec
** v)
9
2
)
3
1
(sincos)
3
1
(cossin 1212
5. cÖgvY Kit
i)
4
3
4
3
tansincoscot 11
**ii) xx 11
tansincoscot
** iii) xx 11
sincottancos ***iv) xx 11
costancotsin
*** v)
y
xy
y
x 22
11 2
sectancossin
vi)
4
3
4
3
costancotsin 11
vii) 2
2
11
2
1
tancoscotsin
x
x
x
viii) 2
2
11
2
1
cotsintancos
x
x
x
6.* i) x
b
b
a
a 1
2
2
1
2
1
tan2
1
1
cos
1
2
sin
n‡j †`LvI †h,
ab
ba
x
1
* ii)
z
z
ec
y
y
x
2
1
cos
2
1
1
1
sec
2
1
tan
2
1
2
2
11
n‡j
†`LvI †h, xyzzyx
iii)
2
sinsin 11
yx n‡j †`LvI †h,
***a) 122
yx b) 111 22
xyyx
iv) hw`
zyx 111
tantantan nq, Z‡e †`LvI †h, xyzzyx
v) hw`
zyx 111
coscoscos nq, Z‡e †`LvI †h, 12222
xyzzyx
*** vi) hw`
b
y
a
x 11
coscos nq, Z‡e †`LvI †h,
2
2
2
2
2
sin
cos2
b
y
ab
xy
a
x
*** vii) hw` )sincos()cossin( nq, Z‡e †`LvI †h,
4
3
sin
2
1 1
viii) CBA , 2tan 1
A Ges 3tan 1
B n‡j †`LvI †h,
4
C
15Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-5
- 16. 5| (L) w·KvYwgwZK mgxKiYt (mvaviY mgvavb)
mgvavb Kit
1. i) 2cottan 22
ii) 2tan
sin1
cos
iii) 04sec32tan2
** iv) tan3tansec 22
v) 2sin2cossin ** vi) 3cotcos ec
vii) 2tancot viii) sin2tan1tansin2
* ix) 0cos32sin 22
DËit i)
4
n ii)
3
2
n vi)
3
2
n vii)
8
)14(
n
iii) n2 , hLb 3sec iv)
4
n , n hLb
2
1
tan
v)
4
n , hLb n Gi gvb k~b¨ A_ev †Rvo msL¨v
viii)
4
n ,
6
)1(
n
n ix)
2
)12(
n ,
3
2
n ,
3
2
2
n
2. *** i) cosx + 3 sinx = 2 * ii) 1cossin iii)
2
1
cossin
iv) 1sin2cos ***v) 3cos3sin xx *vi) 1sincos xx
DËit i)
12
7
2
n ,
12
2
n ii) n2 ,
2
)14(
n iii)
12
7
2
n ,
12
2
n
iv) n2 , 22 n hLb
5
1
cos v) n2 ,
3
)16(
n vi) n2 ,
2
)14(
n
3. * i)
2
5
cos
2
5
sin
4
1
cos4sin
* ii) cot2x = cosx + sinx.
iii) cos2x + sinx = 1 *** iv) 4sin7coscos
* v) 5sin3cos7cos ** vi) 5coscos3cos2
* vii) sin4cos37sin *** viii) cosx + sinx = cos2x + sin2x.
ix) cosx + sinx = cos2x – sin2x. x) cos6x + cos4x = sin3x + sinx
xi) cosx + cos2x + cos3x = 0 ** xii) sinx + sin2x + sin3x = 0
xiii) 2coscos13sin2sinsin
xiv)
4
3
3cossin3sincos 33
xxxx
DËit i)
6
)1(
3
nn
ii)
4
n ,
2
)1(
2
nn
,
2
15
sin
iii) n ,
6
)1(
n
n iv)
4
n
,
18
)1(
3
nn
v)
5
n
,
12
7
)1(
2
nn
vi)
6
)12(
n ,
8
n
vii)
8
)12(
n ,
9
)1(
3
nn
viii) n2 ,
6
)14(
n
ix)
3
2 n
,
2
2
n x)
2
)12(
n ,
6
)14(
n ,
14
)14(
n
xi)
4
)12(
n ,
3
2
n xii)
2
n
,
3
2
2
n
xiii)
2
)12(
n ,
3
2
2
n ,
6
)1(
n
n xiv)
8
)14(
n
4.** i) tanx + tan2x + tanxtan2x = 1. **ii) 3 (tanx+tan2x)+tanxtan2x = 1
iii) 32tantan32tantan
iv) tanx + tan3x = 0 ***v) 1tan2tan
vi) tanx + tan2x + tan3x = 0
* vii) tanx + tan2x + tan3x = tanxtan2xtan3x.
DËit i)
12
)14(
n ii)
18
)16(
n iii)
9
)13(
n
iv)
4
n
v)
6
n vi)
3
n
,
2
11
tnan
vii)
3
n
16Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-5
- 17. 5| (M) w·KvYwgwZK mgxKiY (we‡kl mgvavb)t
1. * i)
2
tan22
2
sec2 xx
; hLb 00
3600 x
ii) 2cos2sin ; hLb -2 2
*** iii) 5)cos(sin4 2
; hLb -2 < < 2
*** iv) sec2tancot ; hLb -2 < < 2
* v) tan)31(tan31 2
; 00
3600
* vi) cos2sin21cossin4 ; 00
1800
vii) 22sinsin2 22
xx ; hLb - < x <
viii) cos3sin2 2
; hLb -2 2
DËit i)
4
,
4
3
ii)
2
,
2
3
iii)
3
5
,
3
,
3
,
3
5
iv)
6
11
,
6
7
,
6
,
6
5
v) 450, 300, 2250, 2100.
vi)
6
,
3
2
,
6
5
vii)
4
,
2
,
4
3
viii)
3
,
3
5
2. * i) 2cossin xx ; hLb - < x <
*** ii) 1sincos3 xx ; hLb -2 < x < 2
*** iii) 2cossin3 ; hLb -2 < < 2
* iv)
2
1
sincos ; hLb - < <
DËit i)
4
ii)
2
3
,
6
,
2
,
6
11
iii)
3
4
,
3
2
iv)
12
7
,
12
3. * i) 22sec4sec ; hLb 0 < < 3600.
* ii) xxxx 3cos5cos7cos9cos ; hLb
4
< x<
4
.
*** iii) 13cos2coscos4 xxx ; hLb 0 < x <
iv) 13sinsin2 ; hLb 20
DËit i) 180, 900, 540, 2700, 1260, 1620, 1980, 2340, 3060, 3420.
ii) 0,
12
,
6
iii)
8
,
3
,
8
3
,
3
2
,
8
5
,
8
7
iv)
4
,
6
,
4
3
,
6
7
,
6
5
,
4
5
,
6
11
,
4
7
17Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-5
- 18. 6| w¯’wZwe`¨vt 5+5=10
[K I L Gi g‡a¨ Dccv`¨ I AsK wewbgq n‡e]
6| (K) ej ms‡hvRb I wefvRb Ges mgwe›`ye‡ji fvimvg¨ (Dccv`¨+AsK)
Dccv`¨t
1.***i) e‡ji mvgvšÍwiK m~ÎwU eY©bv Ki| GK we›`y‡Z †Kv‡Y wµqvkxj `yBwU e‡ji jwäi gvb I w`K wbY©q
Ki|
*** ii) ‡Kvb wbw`©ó w`‡K GK we›`yMvgx `yBwU e‡ji j¤^vs‡ki exRMwYZxq mgwó GKB w`‡K G‡`i jwäi
j¤^vs‡ki mgvb- cÖgvY Ki|
2.*** i) e‡ji wÎfyR m~ÎwU eY©bvmn cÖgvY Ki|
A_ev, cÖgvY Ki, hw` GKwU we›`y‡Z Kvh©iZ wZbwU e‡ji gvb I w`K GKB µ‡g M„wnZ †Kvb wÎfy‡Ri wZbwU evû Øviv
wb‡`©k Kiv hvq, Z‡e Zviv mvg¨ve¯’vq _vK‡e|
*** ii) jvwgi Dccv`¨wU eY©bvmn cÖgvY Ki|
iii) cÖgvY Ki †h, †Kvb we›`y‡Z wµqviZ wZbwU GKZjxq e‡ji cÖ‡Z¨KwUi gvb Aci `ywUi wµqv‡iLvi Aš—
M©Z †Kv‡Yi mvB‡bi mgvbycvwZK n‡j Ges †KvbwUB Aci `ywUi jwäi mgvb bv n‡j, Zviv mvg¨ve¯’v m„wó
Ki‡e|
(mgm¨vejx)
1. ‡Kvb KYvi Dci wµqviZ `yBwU e‡ji jwä Zv‡`i GKwUi mv‡_ mg‡KvY Drcbœ K‡i Ges AciwUi GK
Z…Zxqvsk nq| †`LvI †h, ej؇qi gv‡bi AbycvZ 3 t 22
2. *** ‡Kvb we›`y‡Z wµqviZ 3P Ges 2P gv‡bi `yBwU e‡ji jwä R; cÖ_g ejwUi gvb wظY Ki‡j jwäi gvbI
wظY nq| ej؇qi Aš—M©Z †KvY wbY©q Ki| DËit 1200.
3. * †Kvb we›`y‡Z wµqviZ `yBwU e‡ji e„nËg I ¶y`ªZg jwäi gvb h_vµ‡g F Ges G. cÖgvY Ki †h, ej؇qi
wµqv‡iLvi ga¨eZx© †Kv‡Yi gvb n‡j Zv‡`i jwäi gvb
2
sin
2
cos 2222
GF n‡e|
4. *** mggv‡bi `yBwU ej †Kvb we›`y‡Z 2 †Kv‡Y wµqviZ _vK‡j †h jwä Drcbœ nq Zv Zviv 2 †Kv‡Y wµqviZ
_vK‡j †h jwä nq Zvi wظY| cÖgvY Ki †h, cos = 2cos.
5. †Kvb we›`y‡Z wbw`©ó †Kv‡Y wµqviZ P I Q ej؇qi jwä 3 Q Ges Zv P e‡ji w`‡Ki mv‡_ 300 †KvY
Drcbœ K‡i| †`LvI †h, P = Q A_ev P = 2Q.
6. *** †Kv‡Y wµqviZ P, Q gv‡bi ej؇qi jwä (2m + 1)
22
QP , D³ †KvYwU 900 - n‡j jwäi
gvb nq (2m - 1)
22
QP | cÖgvY Ki †h,
1
1
tan
m
m
.
7. *** i) P + Q Ges P - Q ejØq 2 †Kv‡Y wµqvkxj Ges Zv‡`i jwä Zv‡`i Aš—M©Z †Kv‡Yi mgwØLÛK
†iLvi mv‡_ †KvY Drcbœ K‡i| †`LvI †h Ptan = Qtan
ii) P + Q Ges P - Q ejØq †Kv‡Y wµqviZ| Zv‡`i jwä Zv‡`i AšÍM©Z †Kv‡Yi mgwØLÛ‡Ki mv‡_
2
†KvY Drcbœ K‡i| cÖgvY Ki †h, P t Q =
2
tan
t
2
tan
.
8. ‡Kvb we›`y‡Z P Ges 2P gv‡bi `yBwU ej wµqvkxj| cÖ_gwU‡K wظY K‡i wØZxqwUi gvb 8 GKK e„w× Ki‡j
jwäi w`K AcwiewZ©Z _v‡K| P Gi gvb wbY©q Ki| DËit P = 4 GKK
9. P I Q ej؇qi Aš—M©Z †KvY ; ej`yBwUi Ae¯’vb wewbgq Ki‡j Zv‡`i jwä hw` †Kv‡Y m‡i hvq Z‡e
†`LvI †h,
2
tan
2
tan
QP
QP
.
10. * O we›`y‡Z wµqviZ P I Q ej؇qi jwä R; GKwU mij‡iLv Zv‡`i wµqv‡iLv¸‡jv‡K h_vµ‡g L, M, N
we›`y‡Z †Q` Ki‡j, cÖgvY Ki †h,
ON
R
OM
Q
OL
P
11. †Kv‡Y †njv‡bv OA Ges OB evû eivei wµqvkxj h_vµ‡g P I Q ej؇qi jwä R ejwU OA Gi w`‡Ki
mv‡_ †KvY Drcbœ K‡i| cwieZ©b n‡q /
n‡j Zv‡`i jwä R/ ejwU OA Gi w`‡Ki mv‡_ /
†KvY
Drcbœ K‡i| n‡j, †`LvI †h,
)sin(
)sin(
/
/
R
R
12. ABC wÎfy‡Ri mgZ‡j Aew¯’Z O GKwU we›`y| BC, CA, AB evû¸‡jvi ga¨we›`y h_vµ‡g D, E, F n‡j cÖgvY
Ki †h, OD, OF Ges ED Øviv m~wPZ ej¸‡jvi jwä OB Øviv m~wPZ n‡e|
13. ***ABC wÎfy‡Ri CA I CB evû eivei wµqviZ `yBwU e‡ji gvb cosA I cosB Gi mgvbycvwZK| cÖgvY
Ki †h, Zv‡`i jwäi gvb sinC Gi mgvbycvwZK Ges Zvi w`K C ‡KvY‡K
2
1
(C + B – A) Ges
2
1
(C
+ A – B) As‡k wef³ K‡i|
14. * †Kvb we›`y‡Z wµqviZ P – Q, P, P + Q gv‡bi wZbwU e‡ji w`K GKB µgvbymv‡i †Kvb mgevû wÎfy‡Ri
evû¸‡jvi mgvš—ivj| Zv‡`i jwä wbY©q Ki|
DËit 3Q GKK, P – Q Gi w`‡Ki mv‡_ 2100 †Kv‡Y|
1
18Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-6
- 19. 15. ABC wÎfy‡Ri BC , CA, AB evûi mgvš—ivj w`‡K P gv‡bi wZbwU mgvb ej †Kvb we›`y‡Z
wµqviZ Av‡Q| cÖgvY Ki †h, Zv‡`i jwä CBAP cos2cos2cos23 .
16. ** †Kvb we›`y‡Z wµqviZ P I Q gv‡bi `yBwU e‡ji jwä R Ges P Gi w`K eivei R Gi j¤^vsk Q n‡j,
cÖgvY Ki †h, ej `yBwUi AšÍM©Z †KvY = 2sin-1
Q
P
2
Ges PQPQR 222
.
17.*** †Kvb we›`y‡Z wµqviZ P I Q gv‡bi `yBwU e‡ji jwä Zv‡`i AšÍM©Z †KvY‡K GK-Z…Zxqvs‡k wef³ K‡i| †`LvI
†h, Zv‡`i AšÍM©Z †Kv‡Yi cwigvb 3cos-1
Q
P
2
Ges jwäi gvb
Q
QP 22
(P > Q).
mewe›`y e‡ji fvimvg¨t
18. ABC wÎfy‡Ri j¤^‡K›`ª O we›`y n‡Z BC, CA, AB evûi Dci j¤^ eivei h_vµ‡g wµqvkxj P, Q, R ej
wZbwU mg¨ve¯’vq _Kv‡j, cÖgvY Ki †h, P : Q : R = sinA : sinB : sinC
19. GKZjxq wZbwU ej †Kvb we›`y‡Z wµqviZ n‡q fvimvg¨ m„wó Ki‡Q| Zv‡`i gvb 1, 3 , 2 Gi mgvbycvwZK
n‡j, Zviv G‡K Ac‡ii mv‡_ wK †KvY Drcbœ K‡i wbY©q Ki|
DËit 1500, 1200, 900.
20.*** 4P Ges 3P ej `yBwU O we›`y‡Z wµqvkxj Ges 5P Zv‡`i jwä| †Kvb †Q`K Zv‡`i wµqv‡iLv‡K h_vµ‡g
L, M, N we›`y‡Z †Q` K‡i Z‡e †`LvI †h,
ONOMOL
534
.
21.*** P, Q, R ej wZbwU †Kvb wÎfz‡Ri A, B, C kxl© we›`y n‡Z h_vµ‡g Zv‡`i wecixZ evûi j¤^vwfgyLx w`‡K
wµqviZ †_‡K fvimvg¨ m„wó K‡i‡Q| cÖgvY Ki †h, P : Q : R = a : b : c
22. ABC Gi AšÍt‡K›`ª I n‡Z IA, IB, IC eivei h_vµ‡g P, Q, R ej wZbwU wµqviZ †_‡K fvimvg¨ m„wó
K‡i‡Q| cÖgvY Ki †h,
*** i) P : Q : R = cos
2
A
: cos
2
B
: cos
2
C
ii) P2 : Q2 : R2 = a(b + c – a) : b(c + a – b) : c(a + b – c)
23. mgvb •`‡N©¨i wZbwU GKZjxq mij‡iLv OA, OB, OC hw` O we›`yMvgx †Kvb mij‡iLvi GKB cv‡k¦© Aew¯’Z
bv nq Ges P, Q, R ej wZbwU hw` D³ †iLv¸‡jv eivei Ggb fv‡e wµqviZ _v‡K †h,
OAB
R
OCA
Q
OBC
P
Zvn‡j †`LvI †h, P, Q, R ejwZbwU fvimvg¨ m„wó Ki‡e|
24. ABCD e„˯’ PZzfy©‡Ri AB, AD eivei h_vµ‡g X I Y ejØq wµqviZ Av‡Q| C n‡Z A Gi w`‡K CA
eivei wµqviZ Z ejwUi Øviv Zv‡`i mgZv i¶v Kiv n‡j, †`LvI †h,
BD
Z
CB
Y
CD
X
25.** GKB Abyf~wgK †iLvi c GKK `~i‡Z¡ Aew¯’Z `yBwU we›`y‡Z l GKK `xN© GKwU mi‚ iwki cÖvš—Øq evav
Av‡Q| Aev‡a Szjv‡bv W GKK IRb wewkó GKwU e¯‘‡K enb K‡i Ggb GKwU gm„b IRb wenxb AvsUv H
iwk¥i Dci w`‡q Mwo‡q hv‡‛Q| †`LvI †h, iwki Uvb
22
2 cl
lW
26.*** ACB myZvwUi `yB cÖvš— GKB Abyf~wgK †iLv¯’ A I B we›`y‡Z Ave× Av‡Q| myZvwUi C we›`y‡Z W
IR‡bi GKwU e¯‘‡K wMU w`‡q evav n‡q‡Q| ABC wÎfy‡Ri evû¸‡jvi •`N©¨ a , b , c Ges Zvi †¶Îdj
n‡j, †`LvI †h, myZvwUi CAAs‡ki Uvb )(
4
222
bac
c
Wb
27. i¤^mvK…wZ GKLvbv mylg cv‡Zi GKwU avi f~wgZ‡ji mgvš—ivj I GKwU †KvY 1200; i¤^mwUi †K›`ª n‡Z
KY© eivei P I Q ejØq wµqviZ †_‡K Zv‡K Lvovfv‡e †i‡L‡Q; P >Q n‡j †`LvI †h, 22
3QP .
28.***P, Q ejØq h_vKª‡g GKwU bZ mgZ‡ji •`N¨© I f~wgi mgvšÍiv‡j †_‡K cÖ‡Z¨‡KB GKKfv‡e gm~Y Z‡ji
Dci¯’ W IR‡bi e¯— enb Ki‡Z cv‡i| cÖgvY Ki †h, 222
111
WQP
.
29.** GKwU †njv‡bv mgZ‡ji f~wg I •`‡N¨©i mgvšÍiv‡j h_vµ‡g wµqvkxj `yBwU c„_K ej P I Q-Gi cÖ‡Z¨‡K
GKvKx W IR‡bi †Kvb e¯‘‡K mgZ‡ji Dci w¯’ifv‡e a‡i ivL‡Z cv‡i| cÖgvY Ki †h, W =
22
QP
PQ
.
30.* l ‣`N©¨ wewkó GKwU myZvi GKcÖvš— †Kvb Lvov †`Iqv‡j AvUKv‡bv Av‡Q Ges Zvi Aci cÖvš— a e¨vmva©
wewkó GKwU mylg †Mvj‡Ki Dci¯’ †Kvb we›`y‡Z mshy³ Av‡Q| †MvjKwUi IRb W n‡j †`LvI †h, myZvwUi
Uvb
2
2
)(
lal
laW
.
31.*** †Kvb we›`y‡Z wµqviZ P, Q, R ej wZbwU fvimvg¨ m„wó K‡i‡Q| P I Q-–Gi Aš—M©Z †KvY P I R-Gi
Aš—M©Z †Kv‡Yi wظY n‡j cÖgvY Ki †h, R2 = Q(Q – P).
19Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-6
- 20. A_ev-6| (L) mgvšÍivj ej (Dccv`¨+AsK)
Dccv`¨)
1.***†Kvb eo e¯‘i Dci wµqvkxj `yBwU Amgvb I m`„k mgvšÍivj e‡ji jwä I Zvi cÖ‡qvM we›`y wbY©q Ki|
2.***†Kvb Ro e¯‘i Dci wµqvkxj `yBwU Amgvb I Am`„k mgvšÍivj e‡ji jwe×i gvb, w`K I wµqvwe›`y wbY©q
Ki|
mgm¨vejx
1.* i) GKwU †mvRv mylg i‡Wi GKcÖv‡šÍ 10 †KwR IR‡bi GKwU e¯‘ Szjv‡bv n‡j, H cÖvšÍ n‡Z 1 wgUvi `~‡i
Aew¯’Z GKwU LyuwUi Dci iWwU Abyf~wgKfv‡e w¯’i _v‡K| LyuwUi Dci Pv‡ci cwigvY 30 †KwR IRb
n‡j iWwUi IRb I •`N©¨ wbY©q Ki| DËit 3 wgUvi, 20 †KwR
** ii) GKwU †jvK GKwU mylg jvwVi GKcÖv‡šÍ GKwU †evSv Kuv‡a enb Ki‡Q| †evSvwUi IRb W Ges †jvKwU
Kvua n‡Z †evSvwUi I †jvKwUi nv‡Zi `~iZ¡ h_vµ‡g a I x n‡j †`LvI †h Zvi Kuv‡ai Dci Pvc W
x
a
1 n‡e|
iii) 20 ‡m.wg. `xN© GKwU nvév AB jvwV `yBwU ‡c‡i‡Ki Aew¯’Z| †c‡iK `yBwUi `~iZ¡ jvwVi •`‡N©¨i
A‡a©K| A I B we›`y‡Z 2W I 3W †KwR IRb Szwj‡q jvwVUv‡K Ggbfv‡e ¯’vcb Kiv nj †hb
†c‡iK `yBwUi Dci mgvb Pvc c‡o, †c‡iK `yBwwUi Ae¯’vb wbY©q Ki|
DËit B n‡Z 3 †m.wg. `~‡i GKwU LyuwU Ges A n‡Z 7 †m.wg. `~‡i Aci LyuwU|
*** iv) GKwU nvjKv `‡Ûi GKcÖvšÍ n‡Z 2, 8, 6 wg. `~i‡Z¡ Aew¯’Z wZbwU we›`y‡Z h_vµ‡g P, Q, R
gv‡bi wZbwU mgvšÍivj ej wµqv Ki‡Q| `ÛwU fvimvg¨ Ae¯’vq _vK‡j †`LvI †h, P : Q : R =
1 : 2 : 3.
2.***i) ‡Kvb wÎfz‡Ri †K․wYK we›`y¸wj‡Z h_vµ‡g P, Q, R gv‡bi wZbwU mggyLx mgvšÍivj ej wµqviZ Av‡Q|
G‡`i jwä H wÎfz‡Ri fi‡K‡›`ª wµqviZ n‡j †`LvI †h, P = Q = R.
** ii) ABC wÎf~‡Ri A, B, C †K․wYK we›`y‡Z h_vµ‡g P, Q, R gv‡bi wZbwU mggyLx mgvš—ivj ej
wµqviZ Av‡Q| Zv‡`i jwä H wÎf~‡Ri AšÍt‡K‡›`ª wµqviZ n‡j †`LvI †h,
(i) P : Q : R = SinA : sinB : sinC (ii)
c
R
b
Q
a
P
*** iii) ABC wÎf~‡Ri A, B, C †K․wYK we›`y‡Z h_vµ‡g P, Q, R gv‡bi wZbwU mggyLx mgvš—ivj ej
wµqviZ Av‡Q| Zv‡`i jwä H wÎf~‡Ri j¤^‡K›`ªMvgx n‡j, cÖgvY Ki †h,
(i) (ii) P(b2 + c2 – a2) = Q(c2 + a2 – b2) = R(a2 + b2 – c2)
iv) P, Q, R gv‡bi wZbwU mggyLx mgvš—ivj ej ABC wÎf~‡Ri †K․wYK we›`y‡Z wµqv Ki‡Q| Zv‡`i
mvaviY we›`y hvB †nvKbv †Kb, Zv‡`i jwä hw` me©`vB H wÎf~‡Ri cwi‡K‡›`ª wµqviZ nq, Z‡e cÖgvY
Ki †h, (i)
C
R
B
Q
A
P
2sin2sin2sin
.
(ii) P : Q : R = acosA : bcosB : ccosC.
*** v) O we›`ywU ABC wÎf~‡Ri cwi‡K›`ª Ges AO eivei P gv‡bi ejwU wµqv Ki‡Q| †`LvI †h, B I C
we›`y‡Z wµqviZ P e‡ji mgvš—ivj AskK؇qi AbycvZ sin2B : sin2C.
3.*** i) †`LvI †h, P I Q `yBwU mgvšÍivj e‡ji Q †K
Q
P2
†Z cwieZ©b K‡i Q Gi mv‡_ ¯’vb cwieZ©b
Ki‡j jwäi Ae¯’vb GKB _v‡K|
*** ii) `yBwU wecixZgyLx mgvš—ivj ej P I Q (P>Q) Gi cÖ‡Z¨‡Ki gvb hw` mgcwigvb ewa©Z Kiv nq,
Z‡e cÖgvY Ki †h, Zv‡`i jwäi wµqvwe›`y P n‡Z AviI `~‡i m‡i hv‡e|
iii) 12 GKK I 8 GKK gv‡bi `yBwU mggyLx mgvšÍivj ej h_vµ‡g †Kvb Abo e¯‘i A I B we›`y‡Z
wµqv Ki‡Q| Zv‡`i Ae¯’vb wewbgq Kiv n‡j, Zv‡`i jwäi wµqvwe›`y AB eivei KZ`~‡i m‡i hv‡e
Zv wbY©q Ki| DËit
*** iv) P Ges Q `yBwU mggyLx mgvšÍivj ej| P ejwUi wµqv†iLv mgvšÍivj †i‡L Zvi wµqvwe›`y‡K x `~‡i
miv‡j, †`LvI †h, Zv‡`i jwä
QP
Px
`~‡i m‡i hv‡e|
*** v) `yBwU wecixZgyLx mgvš—ivj ej P Ges Q (P>Q) h_vµ‡g AI B we›`y‡Z wµqviZ; P Ges Q Gi
cÖ‡Z¨K‡K x cwigv‡Y e„w× Ki‡j †`LvI †h, Zv‡`i jwäwU d `~i‡Z¡ m‡i hv‡e, hLb .
*** vi) †Kvb Abo e¯‘i A I B we›`y‡Z h_vµ‡g wµqviZ `yBwU mggyLx mgvš—ivj ej P I Q (P>Q) Gi
ci¯ú‡ii Ae¯’vb wewbgq Ki‡j, †`LvI †h,Zv‡`i jwäi wµqv we›`y AB eivei d `~i‡Z¡ m‡i hv‡e,
hLb d = AB.
*** vii) †Kvb e¯‘i Ici wµqviZ `yBwU mggyLx mgvš—ivj ej P I Q Gi mv‡_ GKB mgZ‡j b `~i‡Z¡ `yBwU
mgvb S gv‡bi wecixZgyLx mgvšÍivj ej‡K mshy³ Ki‡j, †`LvI †h, wgwjZ ej¸‡jvi jwä
`~i‡Z¡ m‡i hv‡e|
*** viii) P, Q gv‡bi `yBwU mggyLx mgvšÍivj e‡ji jwä O we›`y‡Z wµqv K‡i| P †K R cwigv‡Y Ges Q †K
S cwigv‡Y e„w× Ki‡jI jwä O we›`y‡Z wµqv K‡i| Avevi P, Q Gi e`‡j Q, R wµqv Ki‡jI jwä
O we›`y‡Z wµqv K‡i| †`LvI †h, S = R -
QP
RQ
2
)(
.
20Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-6
- 21. 7| MwZwe`¨vt 5 5=10
[K I L Gi g‡a¨ Dccv`¨ I AsK wewbgq n‡e]
7| (K) †eM I Z¡iY (Dccv`¨+ AsK)
Dccv`¨
1.* ‡e‡Mi mvgvšÍwiK m~ÎwU †jL Ges ci¯úi †Kv‡Y wµqvkxj `yBwU ‡eM u Ges v Gi jwäi gvb I w`K wbY©q
Ki|
2.***mPivPi ms‡KZgvjvi 2
2
1
ftuts m~ÎwU cÖwZôv Ki|
3.***mPivPi ms‡KZgvjvq cÖgvY Ki †h, fsuv 222
mgm¨vejx (‡eM)
1.***`yBwU †e‡Mi e„nËg jwä G‡`i ¶z`ªZg jwäi n¸Y| †eM؇qi ga¨eZx© †KvY n‡j, jwä‡e‡Mi gvb G‡`i
mgwói A‡a©K nq| †`LvI †h,
)1(2
2
cos 2
2
n
n
2.***i) †mªvZ bv _vK‡j GK e¨w³ 100wgUvi PIov GKwU b`x muvZvi w`‡q wVK †mvRvmywRfv‡e 4 wgwb‡U cvi
nq Ges †mªvZ _vK‡j H GKB c‡_ †m b`xwU 5 wgwb‡U cvi n‡Z cv‡i| †mªv‡Zi MwZ‡eM wbY©q Ki| DËit
15 wgUvi/wgwbU
*** ii) GKRb †jvK t mg‡q GKwU b`x †mvRvmywR cvox w`‡Z cv‡i Ges 1t mg‡q †mªv‡Zi AbyKz‡j mgvb `~iZ¡
AwZµg Ki‡Z cv‡i| kvš— b`x‡Z †jvKwUi †eM uGes †mªv‡Zi †eM v n‡j †`LvI †h, t t 1t =
vu t vu
iii) 550 wgUvi cÖ¯’ GKwU b`xi †mªvZ N›Uvq 3 wKwg †e‡M cÖevwnZ nq| `yBwU †b․Kvi cÖ‡Z¨KwU N›Uvq 5
wKwg †e‡M GKwU †b․Kv ¶z`ªZg c‡_ Ges AciwU ¶z`ªZg mg‡q b`xwU AwZµg Ki‡Z †Pôv Ki‡Q| hw`
Zviv GKB mg‡q hvÎv ïi‚ K‡i Z‡e Zv‡`i Aci cv‡o †cu․Qvevi mg‡qi cv_©K¨ wbY©q Ki|DËit 1
wgwbU 39 †m‡KÛ
3.* ‡Kvb we›`y‡Z wµqviZ uI v †eM؇qi jwä w ; uGi w`K eivei w Gi j¤^vs‡ki gvb v n‡j cÖgvY Ki
†h, †eM `yBwUi Aš—M©Z †KvY
v
uv 1
cos Ges uvvuw 222
4. ‡Kvb e¯‘ KYvq GKB mv‡_ wµqvkxi wZbwU †eM u, v , w ci¯úi , , †Kv‡Y AvbZ| †`LvI †h,
G‡`i jwä 2
1
222
)cos2cos2cos2( wuvwuvwvu
5. GKwU Kbv †Kvb mgZj¯’ GKwU mij‡iLv eivei 3 wgUvi/†m. †e‡M Pj‡Q| 3 †m‡KÛ c‡i Kbvi MwZc‡_i
mv‡_ jwäi w`‡K 4 wg./†m. MwZ ms‡hvM Kiv nj| MwZ ïi‚ nIqvi 5 †m‡KÛ c‡i KbvwU hvÎvwe›`y n‡Z
KZ`~‡i _vK‡e? DËit 17 wgUvi
6.* ‡Kvb e›`i n‡Z GKLvbv RvnvR DËi cwðg w`‡K N›Uvq 15 wK.wg. †e‡M hvÎv Kij| GKB mg‡q GKB ¯’vb
n‡Z Aci GKLvwb RvnvR `w¶Y-cwðg w`‡K N›Uvq 12 wK.wg. †e‡M hvÎv Kij| Zv‡`i †eZvi h‡š¿i MÖnb
kw³i mxgv 500 wK.wg. n‡j KZ¶Y Zviv G‡K Ac‡ii mv‡_ †hvMv‡hvM i¶v Ki‡Z cvi‡e?DËit 02.26
N›Uv
7. `yBwU †ijc_ ci¯úi mg‡Kv‡Y Aew¯’Z| GKwU †ijc‡_ N›Uvq 30 wK.wg. †e‡M Pjgvb GKwU Mvwo mKvj
10 Uvq Rskb AwZµg K‡i| H gyûZ© n‡Z Aci †ijc‡_ N›Uvq 40 wK.wg. †e‡M Pjgvb Avi GKwU Mvox
we‡Kj 3 Uvq Rsk‡b †cu․‡Q| KLb G‡`i `yiZ¡ b~¨bZg wQj Ges ¶z`ªZg `~iZ¡ KZ wQj?
DËit
5
16
N›Uv, 120wK.wg.
8. GKLvbv w÷gvi c~e©w`‡K N›Uvq uwK.wg. †e‡M Ges wØZxq GKLvbv w÷gvi c~e© w`‡Ki mv‡_ †Kv‡Y DËigyLx
w`‡K u2 †e‡M MwZkxj| cÖ_g w÷gv‡i Aew¯’Z †Kvb hvÎxi wbKU wØZxq w÷gv‡ii w`K DËi-c~e© e‡j g‡b
n‡‛Q| †`LvI †h,
4
3
sin
2
1 1
21Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-7
- 22. Z¡iYt
9.** GKwU ey‡jU †Kvb †`Iqv‡ji wfZi 2 †m.wg. XyKevi ci A‡a©K †eM nvivq| ey‡jUwU †`Iqv‡ji wfZi Avi
KZ `~i XyK‡e? DËit
3
2
†m.wg.
10. GKwU evN 20 wgUvi `~‡i GKwU nwiY‡K †`L‡Z †c‡q w¯’i Ae¯’v n‡Z 3 wgUvi/eM©-‡m‡KÛ Z¡i‡Y nwiYwUi
cðv‡Z †`․ovj| nwiYwU 13 wgUvi/†m‡KÛ mg‡e‡M †`․ov‡Z _vK‡j KZ¶Y c‡i KZ `~‡i wM‡q evNwU
nwiY‡K ai‡Z cvi‡e? DËit 10 †m‡KÛ, 150wgUvi
11. GK e¨w³ Zvi 50 wgUvi mvg‡b w¯’ive¯’v n‡Z mylg Z¡i‡Y GKwU evm Qvo‡Z †`‡L mg‡e‡M †`․ov‡Z jvMj
Ges GK wgwb‡U †Kvb iK‡g evmwU ai‡Z cvij| †jvKwUi †eM I ev‡mi Z¡iY wbY©q Ki|
DËit
3
5
wg./†m.
36
1
wg./†m2.
12.***mgZ¡i‡Y PjšÍ †Kvb we›`y 1t , 2t , 3t mg‡q h_vµ‡g mgvb mgvb µwgK `~iZ¡ AwZµg K‡i| cÖgvY Ki †h,
321321
3111
tttttt
.
13.***‡Kvb mij‡iLvq mgZ¡i‡Y PjšÍ †Kvb we›`yi Mo‡eM avivevwnK 1t , 2t , 3t mg‡q h_vµ‡g 1v , 2v , 3v
n‡j, †`LvI †h,
32
21
32
21
tt
tt
vv
vv
.
14.***‡Kvb mij‡iLvq f mgZ¡i‡Y PjšÍ GKwU KYv t mg‡q s `yiZ¡ I cieZx© t/ mg‡q s/ `yiZ¡ AwZµg K‡i|
†`LvI †h, f = 2
t
s
t
s
/(t + t/)
15.* u Avw`‡e‡M PjšÍ KYv cÖ`Ë `~i‡Z¡i A‡a©K f mylg Z¡i‡Y Ges Aewkó A‡a©K f1 mylgZ¡i‡Y Mgb K‡i| †`LvI
†h †kl †e‡Mi gvb GKB n‡e hw` KYvwU mg¯— `~iZ¡
2
1
(f + f1) mgZ¡i‡Y Mgb K‡i|
16.***i) GKLvbv †ijMvwo GK †÷kb n‡Z †Q‡o 4 wgwbU ci 2 wK‡jvwgUvi `~‡i Aew¯’Z Aci ‡÷k‡b _v‡g|
MvwoLvbv Zvi MwZc‡_i cÖ_gvsk x mgZ¡i‡Y Ges wØZxqvsk y mgg›`‡b Pj‡j cÖgvY Ki †h,
4
11
yx
.
*** ii) w¯’ive¯’v n‡Z mij‡iLvq Pjš— GKwU e¯‘KYv cÖ_‡g x mgZ¡‡Y I c‡i y mgg›`‡b P‡j| KYvwU hw`
t mg‡q s `~iZ¡ AwZµg K‡i, Z‡e †`LvI †h,
yxs
t 11
2
2
.
* iii) GKLvbv †ijMvwo GK †÷kb n‡Z †Q‡o Ab¨ †÷k‡b wM‡q _v‡g| MvwoLvbv MwZc‡_i cÖ_gvsk f
mgZ¡i‡Y Ges c‡i †eªK cÖ‡qvM K‡i Acivsk f1 mgg›`‡b P‡j| †÷kb `yBwUi `~iZ¡ ÔaÕ n‡j ‡`LvI
†h, MvwoLvbv GK ‡÷kb n‡Z Aci †÷k‡b †c․Qvi mgq
1
1 )(2
ff
ffa
n‡e|
17. GKwU KYv wbw`©ó †e‡M hvÎv K‡i mgZ¡i‡Y 3 †m‡K‡Û 81 †m.wg. `~iZ¡ AwZµg Kij| AZtci Z¡i‡Yi wµqv
eÜ n‡q †Mj Ges KYvwU cieZx© 3 †m‡K‡Û 72 †m.wg. `~iZ¡ AwZµg Kij| KYvwUi Aw`‡eM I Z¡iY wbY©q
Ki| DËit 30 †m.wg./†m.; 2 †m.wg./†m2.
18.* GKLvbv †ijMvwo GK †÷kb n‡Z †Q‡o Ab¨ †÷k‡b wM‡q _v‡g| MvwoLvbv Zvi MwZc‡_i cÖ_g
m
1
Ask
mgZ¡i‡Y, †kl
n
1
Ask mgg›`‡b Ges Aewkóvsk mg‡e‡M P‡j| cÖgvY Ki †h, m‡e©v‛P †eM I Mo‡e‡Mi
AbycvZ
nm
11
1 : 1.
19. `yBwU KYv GKB mij‡iLvq h_vµ‡g a Ges b mgZ¡i‡Y Pj‡Q| H mij‡iLvi †Kvb wbw`©ó we›`y n‡Z hLb
Zv‡`i `yiZ¡ x I y ZLb Zv‡`i †eM h_vµ‡g u Ges v; †`LvI †h, Zviv `yBev‡ii AwaK wgwjZ n‡Z cv‡i
bv| hw` Zviv `yBevi wgwjZ nq Z‡e Zv‡`i wgwjZ nevi mg‡qi cv_©K¨
))((2)(
2 2
bayxvu
ba
20.***i) GKB jvB‡b `yBLvbv †ijMvwo ci¯ú‡ii w`‡K h_vµ‡g u1 Ges u2 †e‡M AMÖmi n‡‛Q| Zv‡`i `~iZ¡
hLb x, ZLb G‡K Aci‡K †`L‡Z †cj| †eªK cÖ‡qv‡M Drcbœ m‡e©v‛P g›`b h_vµ‡g f1 I f2 n‡j,
†`LvI †h, `yN©Ubv Gov‡bv m¤¢e n‡e hw`, xfffufu 211
2
22
2
1 2 nq|
* ii) GKB jvB‡b GKLvbv G·‡cÖm Mvwo Ab¨ GKLvbv gvjMvwo‡K AwZµg Ki‡Q| Zv‡`i †eM hLb
h_vµ‡g u1 Ges u2 ZLb x `~iZ¡ n‡Z G‡K Aci‡K †`L‡Z cvq| Mvwo `yBLvbvi m‡e©v‛P Z¡iY I
m‡e©v‛P g›`b h_vµ‡g f1 Ges f2 n‡j †`LvI †h, `yN©Ubv Gov‡bv †KejgvÎ m¤¢eci n‡e hw` (u1 –
u2)2 = 2(f1 + f2)x nq|
22Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-7
- 23. A_ev-
7| (L) gva¨vKl©‡Yi cÖfv‡e e¯‘i Dj¤^ MwZ Ges cÖ‡¶cKt (Dccv`¨+AsK)
Dccv`¨
1.***GKwU e¯‘KYv u ‡e‡M Ges Abyf~wgi mv‡_ †Kv‡Y wbw¶ß nj| e¯‘ KYvwUi me©vwaK D‛PZv, me©vwaK D‛PZvq
†cu․Qvi mgq, wePiYKvj, Abyf~wgK cvjøv I me©vwaK Abyf~wgK cvjøv wbY©q Ki|
2.* cÖgvY Ki †h, cÖw¶ß e¯‘i wePiYKvj Dnvi me©vwaK D‛PZvq DVvi mgqKv‡ji wظY|
3.***cÖgvY Ki †h, evqyk~b¨ ¯’v‡b cÖw¶ß e¯‘ KYvi MwZc_ GKwU cive„Ë|
4.***†`LvI †h, evqynxb ¯’v‡b †Kvb cÖw¶ß e¯‘i MwZc‡_i mgxKiY y = xtan
R
x
1
5.* ‡`LvI †h, f~wgi D‛P †Kvb ¯’v‡bi wbw`©ó we›`y n‡Z Abyf~wg‡K evqyk~b¨ ¯’v‡b wbw¶ß e¯‘KYvi wePiY c_ GKwU
cive„Ë|
mgm¨vejx
gva¨vKl©‡Yi cÖfv‡e e¯‘i Dj¤^ MwZt
1. 10wg./†m. †e‡M DaŸ©Mvgx †Kvb †ejyb n‡Z cwZZ GK UzKiv cv_i 10 †m. c‡i gvwU‡Z coj| hLb cv_‡ii
UzKiv cwZZ nq, ZLb †ejy‡bi D‛PZv KZ? DËit 390 wgUvi
2.** 49 wg./†m. †e‡M GKwU ej‡K Lvov Dc‡ii w`‡K wb‡¶c Kiv nj Ges 2 †m. c‡i GKB we›`y n‡Z GKB
†e‡M Aci GKwU ej wb‡¶c Kiv nj| †Kv_vq Ges KLb Zviv wgwjZ n‡e?
DËit wØZxq ejwU wb‡¶c Kivi 4 †m. c‡i 6.117 wgUvi D‛PZvq
3.***GKwU UvIqv‡ii Pzov n‡Z GKLÛ cv_i x wgUvi wb‡P bvgvi ci Aci GKLÛ cv_i Pzovi y wgUvi wbP n‡Z
†d‡j †`Iqv nj| hw` Df‡qB w¯’ive¯’v n‡Z c‡o Ges GKB m‡½ f~wg‡Z cwZZ nq, Z‡e †`LvI †h,
UvIqv‡ii D‛PZv
x
yx
4
)( 2
wgUvi|
4.* 5.4 †m‡KÛ hveZ mg‡e‡M Lvov Dc‡ii w`‡K DVevi ci GKwU †ejyb n‡Z GKwU fvix e¯‘ c‡o †Mj| hw`
e¯‘wU 7 †m‡K‡Û f~wg‡Z c‡o Z‡e †ejy‡bi MwZ‡eM Ges KZ DPz n‡Z e¯‘wU c‡owQj Zv wbY©q Ki| DËit
88.20 wgUvi/†m; 96.93 wgUvi
5.* mg‡e‡M Lvov DaŸ©Mvgx GKwU D‡ovRvnvR n‡Z GKwU †evgv †Q‡o †`Iqvq 5 †m. c‡i Zv gvwU‡Z c‡o|
†evgvwU hLb gvwU‡Z c‡o ZLb D‡ovRvnv‡Ri D‛PZv wbY©q Ki|
DËit 5.122 wgUvi
6.* GKwU KYv u wg./†m. †e‡M Lvov Dc‡ii w`‡K wb‡¶c Kiv nj Ges t ‡m. c‡i H GKB we›`y n‡Z GKB
Avw`‡e‡M Aci GKwU KYv Dc‡ii w`‡K wb‡¶c Kiv nj| cÖgvY Ki †h, Zviv
g
tgu
8
4 222
D‛PZvq
wgwjZ n‡e|
7.***Lvov Dc‡ii w`‡K wbw¶ß GKwU cv_i 1t Ges 2t mg‡q f~wgi h D‛PZvq Ae¯’vb Ki‡j †`LvI †h,
212 tgth |
8.*** Lvov Dc‡ii w`‡K wbw`©ó †e‡M wbw¶ß GKwU KYv t ‡m. mg‡q h D‛PZvq D‡V Ges AviI t1 ‡m‡KÛ c‡i
f~wg‡Z †c․Qvq, Z‡e cÖgvY Ki †h h =
2
1
gtt1.
9.* GKwU k~b¨ K~‡ci g‡a¨ GKwU cv_‡ii UzKiv †Q‡o †`Iqvi ci Zv 19.6 wg./†m. †e‡M K~‡ci Zj‡`‡k cwZZ
nq| UzKivwU †Q‡o †`Iqvi 2
35
2
†m. c‡i cv_iwUi cZ‡bi kã †kvbv †Mj, k‡ãi †eM wbY©q Ki| DËit
343 wgUvi/†m.
10. GKwU k~b¨ K~‡c GKwU cv_‡ii UzKiv †djv nj Ges 5.3 †m. c‡i UzKivwUi K~‡ci Zj‡`‡k cZ‡bi kã
†kvbv †Mj| k‡ãi †eM 327 wg./†m. Ges 81.9g wg./†m.2 n‡j K~‡ci MfxiZv wbY©q Ki|DËit 5.54
wgUvi
11.* f~wg n‡Z gy2 wg./†m. †e‡M Lvov Dc‡ii w`‡K cÖw¶ß GKwU i‡KU Zvi e„nËg D‛PZvq D‡V we‡ùvwiZ
nj| i‡K‡Ui cÖ‡qvM we›`y Ges Zv n‡Z f~wg eivei x wgUvi `~‡i f~wg‡Z Aew¯’Z Aci GKwU we›`y‡Z
we‡ùvi‡Yi kã Avm‡Z †h mgq jv‡M Zv‡`i Aš—i
n
1
†m.| †`LvI †h, k‡ãi MwZ‡eM
)( 22
yyxn wg./†m.|
12. GKwU k~b¨ K~‡ci g‡a¨ GKwU wXj †djvi t ‡m. c‡i K~‡ci Zj‡`‡k wXj covi kã †kvbv †Mj| hw` k‡ãi
†eM v Ges Ky‡ci MfxiZv h nq, Z‡e cÖgvY Ki †h,
*** i) (2h – gt2) v 2 + 2hgt v = h2g
* ii) Kz‡ci MfxiZv =
)1(2
2
v
gt
gt
, h Gi Zzjbvq v GZ e„nr †h
2
v
h
†K AMÖvn¨ Kiv hvq|
** iii) t =
v
h
g
h
2
cÖ‡¶cKt
13.* i) GKRb †L‡jvqvo 3.5 wg. D‛PZv n‡Z f~wgi mv‡_ 300 †Kv‡Y 9.8 wg./†m. †e‡M GKwU ej wb‡¶c
K‡i Ges Aci GKRb †L‡jvqvo 2.1 wgUvi DuPy‡Z Zv a‡i †d‡j| †L‡jvqvo `yRb KZ`~‡i wQj?DËit
44.10 wgUvi
ii) GKRb †L‡jvqvo 2 wg. D‛PZv n‡Z f~wgi mv‡_ 300 †Kv‡Y 20 wg./†m. †e‡M GKwU ej wb‡¶c Ki‡j
Aci GKRb †L‡jvqvo 1 wgUvi DuPy‡Z Zv a‡i †d‡j| †L‡jvqvo `yRb KZ`~‡i wQj?DËit 37 wgUvi
23Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-7
- 24. * iii) `yB wgUvi Dci n‡Z 50 wg./†m. †e‡M Ges Abyf~wgK Z‡ji mv‡_ 300 †Kv‡Y wbw¶ß GKwU wµ‡KU ej
f~wg n‡Z 1 wgUvi Dc‡i GKRb †L‡jvqvo a‡i †d‡j| †L‡jvqvo؇qi `~iZ¡ wbY©q Ki|DËit 219
wgUvi
* iv) 80 wg. DuPz GKwU cvnv‡oi P~ov n‡Z GKLÛ cv_i 128 wg./†m. †e‡M Ges Abyf~wgK Z‡ji mv‡_ 300
†Kv‡Y wbw¶ß nj| cv_i LÛwU cvnv‡oi cv`‡`k n‡Z KZ`~‡i f~wg‡Z co‡e Zv wbY©q Ki| DËit
48.1492 wgUvi
v) 60 wg. DuPz GKwU cvnv‡oi P~ov n‡Z GKLÛ cv_i 40 wg./†m. †e‡M Ges Abyf~wgK Z‡ji mv‡_ 300
†Kv‡Y wbw¶ß nj| GwU cvnv‡oi cv`‡`‡k KZ`~‡i wM‡q f~wg‡Z co‡e?
DËit 963.210 wgUvi
vi) GKRb •egvwbK 5000 wgUvi Dci w`‡q N›Uvq 250 wK.wg. †e‡M D‡o hvIqvi mgq GKwU †evgv
bvwg‡q w`j| †m †h ¸nvq AvNvZ Ki‡Z Pvq †mB ¸nv n‡Z Zvi Abyf~wgK `yiZ¡ KZ nIqv cÖ‡qvRb?
DËit 2218 wgUvi
14.* i) GKwU e¯‘ GKB †e‡M Abyf~wgKZ‡ji mv‡_ `yBwU wfbœ †Kv‡Y cÖw¶ß n‡q GKB Abyf~wgK cvj- v R
AwZµg K‡i| hw` Zvi ågYKv‡j t1 Ges t2 nq Z‡e †`LvI †h, R =
2
1
gt1t2.
*** ii) hw` †Kvb cÖw¶ß e¯‘i `yBwU MwZc‡_ e„nËg D‛PZv H Ges H1 nq Z‡e †`LvI †h, 14 HHR .
iii) †Kvb wbw`©ó †e‡Mi Rb¨ e„nËg cvjøv D n‡j †`LvI †h, R = Dsin2 Ges G n‡Z cÖgvY Ki †h,
†Kvb Abyf~wgK cvjøv R Gi Rb¨ mvaviYZ `yBwU mÂvic_ _v‡K| DcwiD³ `yBwU mÂvic‡_ jä e„nËg
D‛PZv h1, h2 n‡j †`LvI †h, D = 2(h1 + h2).
15.***i) u Avw`‡e‡M cÖw¶ß †Kvb KYv KZ…©K jä e„nËg D‛PZv H n‡j †`LvI †h, Zvi Abyf~wgK cvjøv
H
g
u
HR
2
4
2
* ii) ‡Kvb Abyf~wgK Z‡ji Dci¯’ GKwU we›`y n‡Z GKwU KYv u †e‡M Ges †Kv‡Y cÖw¶ß nj| Zvi
cvjøv R Ges e„nËg D‛PZv H n‡j cÖgvY Ki †h,16gH2 – 8u2H + gR2 = 0
16. I ( > ) †Kv‡Y `ywU e¯‘ wbw¶ß n‡jv| Giv h_vµ‡g 1t I 2t mg‡q GKB Abyf~wgK cvj- v
AwZµg Ki‡j, cÖgvb Ki †h,
)sin(
)sin(
2
2
2
1
2
2
2
1
tt
tt
.
17.* i) GKwU ¸wj 9.8 wg. `~‡i Aew¯’Z 2.45 wg. D‛P GKwU †`Iqv‡ji wVK Dci w`‡q Abyf~wgKfv‡e P‡j
hvq| ¸wjwUi cÖ‡¶c‡e‡Mi gvb I w`K wbY©q Ki|
DËit 5.15 wg/†m.;
2
1
tan 1
** ii) GKwU wµ‡KU ej‡K f~wg †_‡K wb‡¶c Kiv n‡j GwU 100 MR `~‡i f~wg‡Z wd‡i Av‡m| Gi
wePiYc‡_i me©vwaK D‛PZv 56
4
1
dzU n‡j Gi wePiYKvj I cÖ‡¶cY †Kv‡Yi gvb wbY©q Ki| DËit
4
15
†m‡KÛ,
3
8
cot 1
18.* i) GKwU e¯‘‡K Abyf~wg‡Ki mv‡_ 600 †Kv‡Y Ggbfv‡e cÖ‡¶c Kiv nj †hb 7 wgUvi e¨eav‡b Aew¯’Z
3.5 wg. D‛P `yBwU †`Iqv‡ji wVK Dci w`‡q P‡j hvq| e¯‘wUi Abyf~wgK cvj- v wbY©q Ki| DËit
37 wgUvi
ii) GKwU e¯‘‡K u ‡e‡M Abyf~wgK Z‡ji mv‡_ †Kv‡Y Ggbfv‡e wb‡¶c Kiv nj †hb 2a e¨eav‡b
Aew¯’Z a cwigvb D‛PZv wewkó `yBwU †`Iqv‡ji wVK Dci w`‡q P‡j hvq| cÖgvY Ki †h,
2
cot2
aR .
* iii) GKwU wµ‡KU ej‡K AvNvZ Ki‡j Zv wb‡¶c we›`y †_‡K h_vµ‡g b Ges a `~i‡Z¡ Aew¯’Z a Ges
b D‛PZv wewkó `yBwU †`Iqvj †Kv‡bv iK‡g AwZµg K‡i| †`LvI †h, Gi cvj- v R
ba
baba
22
.
19.* i) GKwU Lvov †`Iqv‡ji cv`‡`k n‡Z f~wg eivei x `~i‡Z¡ ‡Kvb we›`y n‡Z 450 †Kv‡Y GKwU e¯‘ wb‡¶c
Kiv nj| Zv †`Iqv‡ji wVK Dci w`‡q †Mj Ges †`Iqv‡ji Aci cvk¦©¯’ y `~i‡Z¡ wM‡q gvwU‡Z coj|
†`LvI †h, †`Iqv‡ji D‛PZv
yx
xy
.
** ii) hw` †Kvb cÖw¶ß e¯‘ t mg‡q Zvi MwZc‡_i Dci¯’ P we›`y‡Z †c․‡Q Ges t1mg‡q P n‡Z cÖ‡¶c
we›`yMvgx Abyf~wgK Z‡j wd‡i Av‡m, Z‡e †`LvI †h, Z‡ji Dc‡i P Gi D‛PZv
2
1
gtt1.
* iii) ‡Kvb cÖw¶ß e¯‘ Zvi cÖ‡¶c we›`y n‡Z x Abyf~wgK `~i‡Z¡ Ges y Lvov `~i‡Z¡ Aew¯’Z †Kvb we›`y AwZµg
K‡i| e¯‘wUi Abyf~wgK cvj- v R n‡j †`LvI †h, cÖ‡¶c †KvY
xR
R
x
y
.tan 1
20.* i) GKwU UvIqv‡ii kxl© j¶¨ K‡i e›`yK n‡Z wbw¶ß GKwU †evgv UvIqv‡ii ga¨we›`y‡Z AvNvZ Ki‡j,
†`LvI †h UvIqvi‡K AvNvZ Kivi mgq †evgvwU Abyf~wg‡K P‡j|
ii) GKwU e¯‘ 2.39 wg./†m. †e‡M f~wgi mv‡_ 0
30 †Kv‡Y wbw¶ß nj| KZ mgq c‡i e¯‘wU wb‡¶c w`‡Ki
m‡½ j¤^fv‡e Pj‡e? GB mg‡q Gi †eM KZ n‡e? DËit 8 .†m.; 9.67 wg./†m.
24Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-7
- 25. 8| ‡h †Kvb `yBwU cÖ‡kœi DËi `vIt 5 2=10
8| (K) we¯Ívi cwigvct
1. wb‡Pi MYmsL¨v wb‡ekb n‡Z Mo e¨eavb wbY©q Kit
‡kÖwY 0-10 10-20 20-30 30-40 40-50
MYmsL¨v 3 7 11 15 5
DËit 9.59 (cÖvq)
2. msL¨v¸wji cwiwgZ e¨eavb wbY©q Kit DËit √
3. wb‡Pi MYmsL¨v wb‡ek‡bi cwiwgZ e¨eavb wbY©q Kit
‡kÖwYe¨vwß 20-30 30-40 40-50 50-60 60-70 70-80
MYmsL¨v 8 10 15 10 9 5
DËit 14.98 (cÖvq)
4. wb‡Pi Dcv‡Ëi Rb¨ cwiwgZ e¨eavb I †f`v¼ wbY©q Kit
cÖvß b¤^i 31-40 41-50 51-60 61-70 71-80 81-90 91-100
QvÎmsL¨v 6 8 10 12 5 7 2
DËit 16.72, 279.558 (cÖvq)
5. wb‡Pi Z_¨ n‡Z cwiwgZ e¨eavb I †f`vsK wbY©q Kit
mvßvwnK Avq 10-20 20-30 30-40 40-50 50-60 60-70
kÖwgK msL¨v 5 10 15 20 10 5
DËit 13.368, 178.70 (cÖvq)
6. wb‡Pi Dcv‡Ëi Rb¨ cwiwgZ e¨eavb I †f`v¼ wbY©q Kit
b¤^i 10 20 30 40 50 60 70
QvÎmsL¨v 4 5 10 25 10 6 4
DËit 14.68, 215.50 (cÖvq)
7. cwiwgZ e¨eav‡bi myweav I Amyweav¸wj wjL|
8. cwiwgZ e¨eavb I †f`v‡¼i msÁv `vI| wb‡Pi Z_¨ n‡Z cwiwgZ e¨eavb I †f`v¼ wbY©q Ki|
‡kÖwYe¨vwß 20-24 25-29 30-34 35-39 40-44 45-49
RbmsL¨v 7 10 15 12 10 5
DËit 7.44, 55.35 (cÖvq)
9. wb‡P Øv`k †kÖwYi QvÎ-Qvw·`i D”PZi MwY‡Z cÖvß b¤^‡ii mviwY †`Iqv nj| cÖvß b¤^‡ii cwiwgZ e¨eavb I
Mo e¨eavb wbY©q Ki|
b¤^i 20-24 25-29 30-34 35-39 40-44 45-49
QvÎ-QvÎx 7 10 15 13 9 6
DËit 7.38, 16.694 (cÖvq)
10. wb‡Pi Dcv‡Ëi Rb¨ cwiwgZ e¨eavb I †f`v¼ wbY©q Kit
‡kÖwY 6-15 16-25 26-35 36-45 46-55 56-65 66-75 76-85
MYmsL¨v 10 20 30 40 50 60 70 80
DËit 19.72, 388.89 (cÖvq)
11. wb‡P DcvË n‡Z cwiwgZ e¨eavb wbY©q Kit
†kÖwY e¨vwß 200-300 300-400 400-500 500-600 600-700 700-800
MYmsL¨v 12 18 36 24 10 8
DËit 134.63 (cÖvq)
12. wb‡P DcvË n‡Z cwiwgZ e¨eavb wbY©q Kit
b¤^i 5-9 10-14 15-19 20-24 25-29 30-34
QvÎ 15 30 55 17 10 3
DËit 5.759 (cÖvq)
13. ‡f`vsK wK? wb‡Pi Z_¨ n‡Z cwiwgZ e¨eavb I †f`vsK wbY©q Kit
gvwmK Avq 5-9 10-14 15-19 20-24 25-29 30-34
kÖwgK msL¨v 15 30 55 17 10 3
DËit 6.11, 37.33 (cÖvq)
14. wb‡Pi msL¨v¸wji cwiwgwZ e¨eavb †f`v¼ wbY©q Kit
6, 10, 9, 12, 21, 24, 25, 15, 16, 22.
DËit 6.39, 40.80 (cÖvq)
15. wb‡Pi Dcv‡Ëi Rb¨ cwiwgZ e¨eavb I †f`v¼ wbY©q Kit
cÖvß b¤^i 51-60 61-70 71-80 81-90 91-100
QvÎmsL¨v 10 15 20 12 3
DËit 11.119, 123.632 (cÖvq)
16. wb‡Pi wb‡ek‡bi cwiwgZ e¨eavb Kit
x 10 13 25 30 37 42 45
f 3 7 8 15 10 5 2
DËit 10 (cÖvq)
17. B¯úvnvwb wek¦we`¨vjq K‡j‡Ri 100 Rb Qv‡Îi D”Pv †m.wg.) wb‡¤œi mviYx‡Z cÖ`Ë n‡jv:
D‛PZv 141-150 151-160 161-170 171-180 181-190
QvÎ-QvÎx 5 16 56 19 4
DËit 8.43, 70.99 (cÖvq)
18. Find variance for the data set 11, 13, 15, ........, 25.
Ans: 21
19. 50 Rb Qv‡Îi eq‡mi Mo 22 eQi I cwiwgZ e¨eavb 4 eQi| wKš‘ 2 Rb Qv‡Îi eqm 25 I 24 eQ‡ii
¯’‡j h_vµ‡g 13 I 11 †jLv nq| Zv‡`i eq‡mi Mo I cwiwgZ e¨eavb wbY©q Ki|
DËit 22.5 eQi; 3.46 eQi
20. a) cÖ_g n msL¨K †Rvo ¯^vfvweK msL¨vi †f`vsK wbY©q Ki| DËit
b) cÖ_g n msL¨K we†Rvo ¯^vfvweK msL¨vi †f`vsK wbY©q Ki| DËit
25Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-8
- 26. 8| (L) m¤¢ve¨Zv(Dccv`¨)
1. m¤¢ve¨Zvi ms‡hvM m~Î (Additional law of Probability):
*** i) eR©bkxj ev wew‛Qbœ NUbvi †¶‡Î m¤¢ve¨Zvi ms‡hvM m~ÎwU eY©bv I cÖgvY Ki|
** ii) AeR©bkxj NUbvi †¶‡Î m¤¢veZvi ms‡hvM m~ÎwU eY©bv I cÖgvY Ki|
2. m¤¢ve¨Zvi ¸Yb m~Î (Multiplication law of Probability):
* i) `yBwU ¯^vaxb NUbvi †¶‡Î m¤¢ve¨Zvi ¸Yb m~ÎwU eY©bv `vI I cÖgvY Ki|
ii) `yBwU Aaxb NUbvi †¶‡Î m¤¢ve¨Zvi ¸Yb m~ÎwU eY©bv `vI I cÖgvY Ki|
3. kZ©vaxb m¤¢ve¨Zv (Conditional Probability):
‡Kvb bgybv RM‡Z A I B `yBwU NUbv Ges P(B) > 0 n‡j, B NUbvwU NUvi kZ©vax‡b A NUbvwU NUvi
m¤¢ve¨Zv
P(B)
B)P(A
B
A
P
cÖgvY Ki|
4. m¤¢ve¨Zvi c~iK m~Î (Complementary theorem of Probalily):
m¤¢ve¨Zvi c~iK m~ÎwU eY©bv Ki I cÖgvY `vI|
8| (M) m¤¢ve¨Zv(mgm¨vejx/AsK)
1. i) 52 Lvbv Zv‡mi c¨v‡K‡U 4wU †U°v Av‡Q| wbi‡c¶ fv‡e †h †Kvb GKLvbv Zvm †U‡b †U°v bv cvIqvi
m¤¢ve¨Zv KZ?
ii) 52 Lvbvi GK c¨v‡KU Zvm n‡Z niZ‡bi ivRv (K) mwi‡q ivLv nj| Aewkó Zvm¸‡jv fvj K‡i
Zvmv‡bv nj| wbi‡c¶fv‡e GKwU Zvm Uvb‡j †mUv niZb nIqvi m¤¢ve¨Zv wbY©q Ki|
* iii) 52 Lvbv Zv‡mi c¨v‡KU n‡Z GKLvbv Zvm •`efv‡e DVv‡bv nj| ZvmwU (a) jvj †U°v (b) jvj A_ev
†U°v nIqvi m¤¢ve¨Zv KZ?
iv) 52 Lvbv Zv‡mi c¨v‡KU †_‡K †hgb Lywk †U‡b avivevwnKfv‡e 4 Lvbv †U°v cvIqvi m¤¢ve¨Zv wbY©q Ki|
v) 52 Lvbv Zv‡mi c¨v‡KU n‡Z wZbLvbv Zvm Uvbv n‡jv| wZbwU ZvmB ivRv nIqvi m¤¢ve¨Zv KZ?
vi) 52 wU Zv‡mi GKwU c¨v‡KU n‡Z •`efv‡e GKwU Zvm wb‡j Zv i‚BZb ev ivRv nevi m¤¢ve¨Zv wbY©q Ki|
2.***i) `yBwU Q°v GK‡Î wb‡¶c Kiv n‡j Zv‡`i bgybv‡¶Î •Zix Ki Ges `ywU Qq IVvi m¤¢ve¨Zv wbY©q Ki|
*** ii) GKwU Q°v I `yBwU gy`ªv GK‡Î wb‡¶c Kiv n‡j Zv‡`i bg~bv‡¶ÎwU •Zix Ki Ges Q°vq 4 IVvi
m¤¢ve¨Zv wbY©q Ki|
iii) GKwU mylg gy`ªv cici 3 evi Um Kiv nj| cÖwZwU U‡mB cÖ_g †nW cvIqvi k‡Z© 2 ev Z‡ZvwaK †nW
cvIqvi m¤¢vebv KZ? †Kvb kZ© Av‡ivc Kiv n‡j 2 ev Z‡ZvwaK †nW cvIqvi m¤¢vebv KZ?
iv) ‡Kvb cix¶‡Y GKB mv‡_ GKwU bxj Ges GKwU jvj Q°v wb‡¶c Kiv nj| hw` x bxj QKvq cÖvß
†dvUvi msL¨v Ges y jvj Q°vq cÖvß †dvUvi msL¨v wb‡`©k K‡i Zvn‡j Dcv`vb (x, y) e¨envi K‡i
NUbRMZ s wbY©q Ki|
v) GKwU gy`ª cici wZb evi Um Kiv nj| ch©vqµ‡g gy`ªvwUi †nW Ges †Uj cvevi m¤¢ve¨Zv wbY©q Ki|
vi) GKwU Q°v I `yBwU gy`ªv GK‡Î wb‡¶c Kiv n‡j Zv‡`i bgybv‡¶ÎwU •Zwi Ki Ges (a) 2 `ywU †nW I
†Rvo msL¨v (b) Q°vq 4 cvevi m¤¢ve¨Zv wbY©q Ki|
3.***i) P(A) =
3
1
, P(B) =
4
3
, A I B ¯^vaxb n‡j P(A B) wbY©q Ki|
* ii) hw` P(A) =
3
1
, P(B) =
4
3
nq, Z‡e P(A B) KZ? †hLv‡b A I B ¯^vaxb|
*** iii) P(A B) =
3
1
, P(A B) =
6
5
, P(A) =
2
1
n‡j P(B), P(B/), P(A/) wbY©q Ki| A I B wK
¯^vaxb?
iv) P(A) =
2
1
, P(B) =
5
1
Ges P
B
A
=
8
3
n‡j, P
A
B
wbY©q Ki|
** v) GKRb Qv‡Îi evsjvq cv‡mi m¤¢ve¨Zv
3
2
; evsjv I A¼ `yBwU wel‡q cv‡mi m¤¢ve¨Zv
45
14
Ges `ywUi †h
†Kvb GKwU‡Z cv‡mi m¤¢ve¨Zv
5
4
n‡j, Zvi As‡K cv‡mi m¤¢ve¨Zv KZ?
** vi) GKRb cix¶v_x©i evsjvq †dj Kivi m¤¢ve¨Zv
5
1
; evsjv Ges Bs‡iwR `yBwU‡Z cv‡mi m¤¢ve¨Zv
4
3
Ges
`yBwUi †h †Kvb GKwU‡Z cv‡mi m¤¢ve¨Zv
8
7
n‡j, Zvi †Kej Bs‡iwR‡Z cv‡mi m¤¢ve¨Zv KZ?
*** vii) A I B Gi GKwU A‡¼i mgvavb Ki‡Z cvivi m¤¢ve¨Zv h_vµ‡g
3
1
Ges
4
1
| Zviv GK‡Î A¼wU
mgvav‡bi †Póv Ki‡j A¼wUi mgvavb wbY©‡qi m¤¢ve¨Zv KZ?
viii) ‡Kvb evwYwR¨K cÖwZôv‡bi wZbwU c‡`i Rb¨ GKRb cÖv_x© Av‡e`b K‡i‡Q| H wZbwU c‡` cÖv_x© msL¨v
h_vµ‡g 3, 4, 2 n‡j H cÖv_x©i Aš—Z GKwU c‡` PvKwi cvIqvi m¤¢ve¨Zv KZ?
*** ix) MwYZ I cwimsL¨vb wel‡q 200 Rb cix¶v_x©i g‡a¨ 20 Rb cwimsL¨v‡b Ges 40 Rb MwY‡Z †dj
K‡i| Dfq wel‡q `kRb †dj K‡i‡Q| wbi‡c¶fv‡e GKRb Qv·K evQvB Ki‡j Zvi cwimsL¨v‡b cvm
I MwY‡Z †dj nIqvi m¤¢ve¨Zv wbY©q Ki|
** x) GKwU K‡j‡R GKv`k †kªYxi 40 Rb Qv‡Îi wfZi 20 Rb dzUej †L‡j, 25 Rb wµ‡KU †L‡j Ges
10 Rb dzUej I wµ‡KU †L‡j| Zv‡`i ga¨ n‡Z •`ePq‡b GKRb‡K wbe©vPb Kiv nj| hw` †Q‡jwU
dzUej †L‡j, Z‡e Zvi wµ‡KU †Ljvi m¤¢ve¨Zv KZ?
* xi) 10 †_‡K 30 ch©š— msL¨v n‡Z †h †Kvb GKwU‡K B‛QvgZ wb‡j H msL¨vwU †g․wjK A_ev 5 Gi
¸wYZK nevi m¤¢ve¨Zv wbY©q Ki|
26Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-8
- 27. xii) †Kvb Rwi‡c †`Lv †Mj 70% †jvK B‡ËdvK c‡o, 60% ‡jvK msev` c‡o Ges 40% †jvK Dfq
cwÎKv c‡o| wbi‡c¶fv‡e evQvB Ki‡j GKRb †jv‡Ki B‡ËdvK ev msev` covi m¤¢ve¨Zv wbY©q Ki|
4.***i) GKwU ev‡· wewfbœ AvKv‡ii 6wU mv`v ej, 7wU jvj ej Ges 9wU Kv‡jv ej Av‡Q| Gjv‡g‡jvfv‡e
GKwU ej Zz‡j †bIqv nj| ejwU jvj ev mv`v nIqvi m¤¢ve¨Zv KZ?
** ii) GKwU e¨v‡M 5wU mv`v, 7wU jvj Ges 8wU Kv‡jv ej Av‡Q| G‡jv‡g‡jvfv‡e 3 wU ej Zz‡j †bIqv nj|
ej¸‡jv jvj ev mv`v nIqvi m¤¢ve¨Zv KZ?
iii) GKwU ev‡· 4wU jvj, 5wU bxj Ges 7wU mv`v is Gi ej Av‡Q| •`ePq‡b GKwU e‡ji jvj ev mv`v
nIqvi m¤¢ve¨Zv KZ?
** iv) GKwU e¨v‡M 4wU mv`v Ges 5wU Kv‡jv ej Av‡Q| GKRb †jvK wbi‡c¶fv‡e wZbwU ej DVv‡jb| wZbwU
ejB Kv‡jv nIqvi m¤¢ve¨Zv wbY©q Ki|
* v) GKwU _wj‡Z 3wU mv`v Ges 2wU Kv‡jv ej Av‡Q| Aci GKwU _wj‡Z 2wU mv`v Ges 5wU Kv‡jv ej
Av‡Q| wbi‡c¶fv‡e cÖ‡Z¨K _wj n‡Z GKwU K‡i ej †Zvjv nj| `yBwU e‡ji g‡a¨ Aš—Z GKwU mv`v
nIqvi m¤¢ve¨Zv wbY©q Ki|
vi) GKwU e¨v‡M 5wU mv`v, 7wU jvj Ges 8wU Kv‡jv ej Av‡Q| hw` wewbgq bv K‡i GKwU K‡i ci ci
PviwU ej Zz‡i †bIqv nq, Z‡e me¸‡jv ej mv`v nIqvi m¤¢ve¨Zv KZ?
vii) GKwU ev‡· 5wU jvj I 10wU mv`v gv‡e©j Av‡Q| GKwU evjK †hgb Lywk Uvb‡j cÖwZev‡ii `ywU wfbœ
is‡Oi gv‡e©j cvIqvi m¤¢ve¨Zv KZ?
viii) GKwU e¨v‡M wZbwU Kv‡jv Ges 4wU mv`v ej Av‡Q| •`efv‡e GKwU K‡i 2wU ej Zz‡j †bIqv nj; wKš‘
cÖ_gwU DVv‡bvi ci Zv Avi e¨v‡M ivLv nj bv| wØZxq ejwU mv`v nIqvi m¤¢ve¨Zv KZ?
ix) GKwU ev‡· 5wU jvj I 4wU mv`v wµ‡KU ej Ges Aci GKwU ev‡· 3wU jvj I 6wU mv`v wµ‡KU ej
Av‡Q| cÖ‡Z¨K ev· n‡Z GKwU e‡j ej DVv‡bv n‡j `yBwU e‡ji g‡a¨ Kgc‡¶ GKwU jvj nIqvi
m¤^ve¨Zv wbY©q Ki|
* x) `yBwU GKB iKg ev‡·i cÖ_gwU‡Z 4wU mv`v I 3wU jvj Ges wØZxqwU‡Z 3wU mv`v I 7wU jvj ej
Av‡Q| mgm¤¢e Dcv‡q GKwU ev· wbe©vPb Kiv nj| H ev· n‡Z wbi‡c¶fv‡e GKwU ej Uvbv n‡jv,
ejwU mv`v nIqvi m¤¢ve¨Zv wbY©q Ki| hw` ejwU mv`v nq Zvn‡j cÖ_g ev· †_‡K wbe©vwPZ nIqvi
m¤¢ve¨Zv KZ?
xi) GKwU e¨v‡M 1wU UvKv I 3wU cqmv, wØZxq e¨v‡M 2wU UvKv I 4wU cqmv Ges Z…Zxq e¨v‡M 3wU UvKv I
1wU cqmv Av‡Q| jUvwii gva¨‡g GKwU e¨vM evQvB K‡i m¤úyY© wbi‡c¶fv‡e GKwU gy`ªv D‡Ëvjb Ki‡j
†mwU UvKv nIqvi m¤¢ve¨Zv wbY©q Ki|
* xii) GKwU ev‡· 10 wU bxj I 15wU jvj gv‡e©j Av‡Q| GKwU evjK †hgb Lywk Uvb‡j cÖwZev‡i `ywU (a)
wfbœ es‡qi (b) GKB is‡qi gv‡e©j nIhvi m¤¢ve¨Zv KZ?
* xiii) `yBwU _wji GKwU‡Z 5wU jvj Ges 3wU Kv‡jv ej Av‡Q| Aci _wj‡Z 4wU jvj I 5wU Kv‡jv ej
Av‡Q| h_vm¤¢e Dcv‡q GKwU _wj wbe©vPb Kiv nj Ges Zv †_‡K `yBwU ej †Zvjv n‡j GKwU jvj, GKwU
Kv‡jv nIqvi m¤¢ve¨Zv wbY©q Ki|
xiv) GKwU e¨v‡M 7wU jvj Ges 5wU mv`v ej Av‡Q| wbi‡c¶ fv‡e 4wU ej †Zvjv n‡j 2wU jvj I 2wU mv`v
nIqvi m¤¢ve¨Zv wbY©q Ki|
DËimg~nt
1. i)
13
12
ii)
17
4
iii)
26
1
,
13
7
iv)
270725
1
2. i)
36
1
ii)
6
1
iii)
8
3
,
2
1
iv)
4
1
v)
8
1
,
6
1
3. i)
6
5
ii)
4
1
iii)
3
2
,
3
1
,
2
1
, A I B ¯^vaxb|
iv)
20
3
v)
9
4
vi)
40
33
vii)
2
1
viii)
4
3
ix)
20
3
x)
2
1
xi)
21
11
xii)
10
9
4. i)
22
13
ii)
76
3
iii)
16
11
iv)
42
5
v)
7
5
vi)
969
1
vii)
21
10
viii)
7
4
ix)
27
19
x)
140
61
,
61
40
xi)
9
4
xii)
2
1
,
2
1
xiii)
504
275
xiv)
33
14
msKj‡b-
‡gvt Avãyi iDd
cÖfvlK(MwYZ)
nvwKgcyi gwnjv wWMÖx K‡jR
m¤úv`bvq-
‡gvt AvRvnvi Avjx
cÖfvlK(MwYZ)
KvUjv wWMÖx K‡jR|
27Model Questions (Suggestion); Higher Mathematics 2nd
PapercÖkœ-8
