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cÖ_g c‡Îi As‡Ki mgvavb
First Paper Mathematics Solution
02| ‡f±i (Vector)
(2)(m) (3)(2) (5)(10) 2m 6 50 0
- 56
m
m - 28 (Ans.)
2
5| A 2ˆi 2ˆj kˆ I B 6ˆi 3ˆj 2kˆ n‡j A I B
1| hw` A 3ˆi ˆj 2kˆ I B 2ˆi 3ˆj kˆ nq Z‡e
|A B| KZ?
ˆi ˆj
kˆ
A B 3 1 2
(6)(2) ( 3)(2) (2)(1) 12 6 2 8
(Ans.)
3| A 5ˆi 2ˆj-3kˆ I B 15ˆi aˆj-9kˆ | a Gi gvb KZ
n‡j A I B ci¯úi mgvšÍivj n‡e?
A I B ci¯úi mgvšÍivj n‡e hw` A B 0 nq|
ˆj kˆ
2 3
15 a -9
A B ˆi( 18 3a) ˆj(-45 45) kˆ(5a-30)
A B ˆi (3a 18) kˆ(5a 30)
A B (3a 18) 2 (5a 30) 2
cÖkœg‡Z, A I B ci¯úi mgvšÍivj n‡j A B 0
Sh
d
M
@
ht
(3a 18) 2 (5a 30) 2 0
ig
(3a 18) 2 (5a 30) 2 0 [Dfq c¶ ‡ K eM K‡ i]
op
yr
32 (a 6) 2 5 2 (a 6) 2 0
(a 6) 2 (32 5 2 ) 0
(a 6) 2 0 Dfq c¶ ‡ K (32 5 2 ) Øviv fvM K‡ i
a 6 (Ans.)
C
.c
o
24
(-1)(2) 12-6-2 4
2
2
2
A 2 2 (1) 9 3
B 6 2 (3) 2 2 2 36 9 4 49 7
θ Cos
1
A .B
4
4
-1
Cos 1
Cos
(3)(7)
21
AB
79 .02 (Ans.)
6. ‡f±i B 6ˆi 3ˆj 2 kˆ Gi Dci ‡f±i
A 2ˆi 2ˆj kˆ Gi j¤^ Awf‡ÿc wbY©q Ki|
A.B AB cos
A.B
A Gi j¤^ Awf‡ÿc, A cos
B
ah
ˆi
AB 5
A .B A x B x A y B y A z B z (2)(6) (2)(-3)
bd
2| hw` A 6ˆi 3ˆj 2kˆ I B 2ˆi 2ˆj kˆ nq Z‡e
A.B KZ?
A.B A x B x A y B y A z B z
|e
ˆi ( 1 6 ) ˆj(3 4 ) kˆ (9 2 ) 5 ˆi 7 ˆj 11kˆ
| A B | 52 (7) 2 112 25 49 121 13 . 96(Ans.)
A.B AB cos
A
.B
A I B Gi ga¨eZ©x †KvY, θ Cos 1
AB
al
1
m
3
Ja
2
Gi ga¨eZ©x †KvY wbY©q Ki|
m
‡gv: kvn Rvgvj
mnKvix Aa¨vcK (c`v_©weÁvb wefvM )
we G Gd kvnxb K‡jR ‡ZRMuvI, XvKv
‡dvb: +8801670856105, +88029125630, +88029115369
E-mail: sjamal59@gmail.com
4| A 2ˆi 3ˆj-5kˆ I B mˆi 2ˆj - 10kˆ | m Gi
gvb KZ n‡j A I B ci¯úi j¤^ n‡e|
A I B ci¯úi j¤^ n‡j A . B 0 n‡e|
A .B Ax Bx Ay By Az Bz 0
A .B A x B x A y B y A z B z
( 2)(6) ( 2)(-3) (1)(2) 12-6 2 8
B 6 2 ( 3) 2 2 2 36 9 4 49 7
A.B 8
A Gi j¤^ Awf‡ÿc =
7
B
7| A 3ˆi 2ˆj kˆ , B ˆi 2ˆj 3kˆ I C ˆi ˆj 2kˆ
n‡j cÖgvY Ki †h, A.(B C) (A B).C
ˆi ˆj kˆ
B C 1 2 3
1 1 2
ev, B C ˆi (4 3) ˆj(2 3) kˆ(1-2) 7ˆi-5ˆj -kˆ
L.H.S A . (B C)
(3)(7) (2)(-5) (1)(-1) 21 10 1 10
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02| ‡f±i (Vector)
10| hw` A A x ˆi A y ˆj A z kˆ I B Bx ˆi B y ˆj Bz kˆ nq
Z‡e †`LvI †h, A.B A x B x A y B y A z B z
A . B (A x ˆi A y ˆj A z kˆ) . ( Bx ˆi B y ˆj B z kˆ)
A . B A x B x ( ˆi .ˆi ) A x B y ( ˆi .ˆj) A x B z (ˆi .kˆ)
kˆ
GLb, A B 3 2 1
1 2 -3
ev, A B = ˆi ( 6 2 ) ˆj( 9 1) kˆ ( 6 - 2 )
8 ˆi 10 ˆj 4 kˆ
R.H.S ( A B ) . C (- 8 )(1) (10 )(1)
A y B x (ˆj.ˆi ) A y B y (ˆj.ˆj) A y B z (ˆj.kˆ)
A z B x (kˆ. ˆi ) A z B y (kˆ.ˆj) A z B z (kˆ.kˆ )
A . B Ax Bx (1) A x By (0) Ax Bz (0)
.c
o
( 4 )( 2 ) - 8 10 8 10
L. H. S R.H.S
A_©vr, A.(B C) (A B).C ( Pr oved.)
Ay Bx (0) A y By(1) A y Bz (0)
Az Bx (0) Az By (0) Az Bz (1)
A . B Ax Bx 0 0 0 Ay By 0 0 0 Az Bz
A . B Ax Bx A y By A z Bz ( Pr oved)
24
8| A ˆi 3ˆj 2kˆ , B ˆi 2ˆj kˆ I C 2ˆi 3ˆj 4kˆ
n‡j cÖgvY Ki †h, (B C) A B A C A
(B C) (1 2) ˆi (2 3)ˆj ( 1 4)kˆ 3ˆi ˆj 3kˆ
1
3
2
11| hw` A 2ˆi 4ˆj 5kˆ I B ˆi 2ˆj 3kˆ ‡f±i ؇qi jwä
†f±‡ii mgvšÍivj GKK †f±i wbY©q Ki|
R A B 2 ˆi 4 ˆj 5 kˆ ˆi 2 ˆj 3k
al
L.H.S (B C) A 3 1 3
2
d
2
M
3
Sh
ˆi(4 3) ˆj(2 1) kˆ(3-2) 7ˆi 3ˆj kˆ
ˆi ˆj kˆ
C A 2 3 4
1
iˆ( 6 12) ˆj(4-4) kˆ(6 3) 18ˆi 9kˆ
ig
ht
@
R.H.S B A C A 7ˆi 3ˆj kˆ 18ˆi 9kˆ
11 ˆi 3ˆj 10kˆ
A_©vr, (B C) A B A C A ( Pr oved.)
9| A 9ˆi ˆj-6kˆ Ges B 4ˆi 6ˆj 5kˆ †f±i `ywUi
Mybdj wbY©q K‡i †`LvI †h, Giv ci¯ú‡ii Dci j¤^|
op
yr
R 3ˆi 6ˆj 2kˆ
R
R Gi mgvšÍivj GKK †f±i aˆ
R
R 32 6 2 (2) 2 9 36 4 49 7
ah
ˆi ( 2 9) ˆj(6 3) kˆ(9 1) 11 ˆi 3ˆj 10kˆ
ˆi ˆj kˆ
Avevi, B A 1 2 1
1 3
|e
kˆ
m
ˆj
Ja
ˆi
m
ˆj
bd
ˆi
A.B A x B x A y B y A z B z
12| GKB we›`y‡Z wµqvkxj `ywU mgvb gv‡bi †f±‡ii ga¨eZ©x †KvY KZ
n‡j G‡`i jwäi gvb †h †Kvb GKwU †f±‡ii mgvb n‡e?
R2=P2+Q2+2PQCos
ev, X2= X2+X2+2X.X.Cos
GLv‡b,
ev, X2- X2-X2=2X2Cos
awi, ‡f±i, P=Q=X
ev, -X2=2X2Cos
jwä, R= X
X2
ev , Cos
AšÍf
© ³
~ †KvY,
2X 2
1
ev, Cos
2
1
ev , Cos 1
2
120 (Ans.)
C
(9)(4) (1)( 6) ( 6)(5) 36 6 30 0
A . B ABCosθ 0
wKš‘ A 0, B 0 CosθC 0
ev , Cos θ Cos 90 θ 90
AZGe A I Β ci¯ú‡ii Dc‡ii j¤^|
R 3ˆi 6ˆj 2kˆ 3 ˆ 6 ˆ 2 ˆ
aˆ
i j k (Ans.)
7
7
7
7
R
13| Ae¯’vb †f±i r x ˆi y ˆj z kˆ †K e¨eKjb K‡i wKfv‡e †eM
I Z¡iY cvIqv hvq?
Avgiv Rvwb,
dr
‡eM, v
dt
http://ebd24.com
2
02| ‡f±i (Vector)
d 2x ˆ d2 y ˆ d2z ˆ
i 2 j 2 k
(Ans.)
dt 2
dt
dt
14| P t 2 ˆi t ˆj ( 2 t 1) kˆ I Q 5tˆi tˆj t 3 kˆ
d
d
(P. Q) ? (P Q) ?
dt
dt
2
P.Q ( t )(5t ) ( t )(t ) (2t 1)( t 3 )
P.Q 5t 3 t 2 2t 4 t 3
P.Q 2 t 4 4t 3 t 2
d
d
(P.Q) (2 t 4 4t 3 t 2 )
dt
dt
d
(P.Q) 8t 3 12t 2 2t
dt
ˆi
ˆj
kˆ
P Q t 2 t 2t 1
16| †Kvb GKwU KYvi Ae¯’vb †f±i
†eM V wbY©q Ki|
LHS A B A.B
al
2
17| cÖgvY Ki t A B A.B
Ja
m
2
Ae¯’vb K‡i Zvi Dj¤^w`‡K GKwU GKK †f±i wbY©q Ki|
Avgiv Rvwb,
`ywU †f±‡ii µm ¸bdj †f±i `ywU Øviv MwVZ mgZ‡ji Dci j¤^
nq| †mB j¤^ †f±‡ii mgvšÍivj GKK †f±iB n‡e mgZ‡ji Dj¤^
2
A 2 B2
2
2
2
ˆAB sin AB cos
ˆ 2 A 2 B 2 sin 2 A 2 B 2 cos 2
1. A 2 B 2 sin 2 A 2 B 2 cos2
A 2 B 2 sin 2 cos 2
A 2 B 2 .1 A2 B 2 L.H .S R.H .S (Proved)
ah
Sh
d
M
@
ht
ig
op
yr
15| P 2ˆi 3ˆj 4kˆ, Q ˆi - 2ˆj 3kˆ ‡f±i Øq †h Z‡j
C
|e
d
(P Q) (4 t 3 4 t 1)ˆi (5t 4 20 t 5)ˆj
dt
(3t 2 10 t )kˆ
(Ans.)
ˆi ˆj
kˆ
P Q 2 3 4 ˆi (9 8) ˆj(6 4) kˆ (4 3)
1 2 3
bd
24
dr
Avgiv Rvwb, V
dt
d
V [(3.5ms 1 )t 4.2m]ˆi [5.3ms 1 t]ˆj
dt
V 3.5ˆi 5.3ˆj (Ans.)
4
2
2
5
d
d ˆi ( t 2t t ) ˆj(10t 5t t )
( P Q)
dt
dt
kˆ (t 3 5t 2
d
d
(P Q) ( t 4 2t 2 t )ˆi
dt
dt
d
d
( t 5 10 t 2 5t )ˆj ( t 3 5t 2 ) kˆ
dt
dt
PQ
w`‡K GKK †f±i| awi, †mB †f±i aˆ , aˆ
PQ
.c
o
r [(3.5ms 1 )t 4.2m] ˆi [5.3ms 1 ]ˆj n‡j
3
(ˆi 10ˆj 7kˆ)
( Ans.)
150
m
aˆ
5t t
t
P Q ˆi ( t 4 2 t 2 t ) ˆj(10 t 2 5t t 5 )
kˆ ( t 3 5t 2 )
3
P Q ˆi 10ˆj 7kˆ
P I Q ‡h Z‡j Aew¯’Z Zvi Dj¤^ w`‡K †f±i
( P Q)
(ˆi 10ˆj 7kˆ)
aˆ
PQ
(1) 2 (10) 2 (7) 2
d
(x ˆi y ˆj z kˆ)
dt
dx ˆ dy ˆ dz ˆ
i
j k
(Ans.)
dt
dt
dt
dv d dx ˆ dy ˆ dz ˆ
Avevi, Z¡iY a
i
j k
dt dt dt
dt
dt
18| P ˆi 2ˆj kˆ Ges Q 3ˆi 6ˆj 3kˆ n‡j †`LvI †h, P I
Q ci¯úi mgvšÍivj|
P I Q ci¯úi mgvšÍivj n‡e hw` P Q 0 nq|
ˆi ˆj kˆ
P Q 1 2 1 ˆi (6 6) ˆj(3 3) kˆ (6 6)
3 6 3
P Q ˆi (0) ˆj(0) kˆ (0) 0 0 0 0
P Q 0 P Q 0 P I Q ci¯úi mgvšÍivj| (cÖgvwYZ)
19| P 2iˆ 3ˆj 4kˆ Ges Q 2ˆi ˆj 3kˆ ‡f±i Øq †h Z‡j
Aew¯’Z Zvi Dj¤^w`‡K GKwU GKK †f±i wbY©q Ki|
Avgiv Rvwb,
`ywU †f±‡ii µm ¸bdj †f±i `ywU Øviv MwVZ mgZ‡ji Dci j¤^ nq| †mB
j¤^ †f±‡ii mgvšÍivj GKK †f±iB n‡e mgZ‡ji Dj¤^ w`‡K GKK
P Q
†f±i| awi, †mB †f±i nˆ , nˆ
P Q
http://ebd24.com
02| ‡f±i (Vector)
GLb,
2 2 1
Sh
P Q 6ˆi 6ˆj
P Q 6 2 6 2 72 8.49 GKK (Ans.)
24
bd
PQ
iˆ 10 ˆj 18kˆ
nˆ
(Ans.)
425
PQ
23| hw` A GKwU †f±i nq, Z‡e cÖgvY Ki †h,
A ( A. iˆ)iˆ ( A. ˆj ) ˆj ( A.kˆ)kˆ
mgvavb:
awi, A Ax iˆ Ay ˆj Az kˆ
A. iˆ Ax iˆ.iˆ Ay iˆ. ˆj Az iˆ.kˆ Ax
Abyiƒc fv‡e †`Lv‡bv hvq A. ˆj Ay Ges A. kˆ Az
GLb, Ax , Ay I Az Gi gvb ewm‡q cvB,
A Ax iˆ Ay ˆj Az kˆ
A ( A. iˆ)iˆ ( A. ˆj ) ˆj ( A.kˆ)kˆ cÖgvwYZ|
ht
24| ‡`LvI †h,
A ( x 2 y 4 z )iˆ ( 2 x 3 y z ) ˆj (4 x y 2 z ) kˆ GKwU
AN~Y©bkxj †f±i|
ig
@
M
d
21| P 2ˆi mˆj-3kˆ I Q 10ˆi 5ˆj-15kˆ | m Gi gvb
KZ n‡j P I Q ci¯úi mgvšÍivj n‡e?
P I Q ci¯úi mgvšÍivj n‡e hw` P Q 0 nq|
ah
kˆ
1 ˆi (4 2) ˆj(4 2) kˆ (8 8)
ˆj
kˆ
m 3
10 5 -15
m
mgvb| P Q mgvšÍwi‡Ki †ÿÎdj|
ˆi
PQ 2
Avevi, P Q 12 (10 2 ) ( 182 ) P Q 425
al
P Q iˆ 10 ˆj 18kˆ
|e
ˆi
ˆj
PQ 4 4
ˆj
kˆ
2 1 iˆ(4 3) ˆj (4 6) kˆ(6 12)
6 3 2
Ja
22| A 2iˆ 2 ˆj kˆ Ges B 6iˆ 3 ˆj 2kˆ `yÕwU ‡f±i ivwk|
G‡`i j¤^ Awfgy‡L GKwU GKK †f±i wbY©q Ki|
Avgiv Rvwb,
`ywU †f±‡ii µm ¸bdj †f±i `ywU Øviv MwVZ mgZ‡ji Dci j¤^ nq| †mB
j¤^ †f±‡ii mgvšÍivj GKK †f±iB n‡e mgZ‡ji Dj¤^ w`‡K GKK
iˆ
PQ 2
20| P 4ˆi 4ˆj kˆ Ges Q 2ˆi 2ˆj kˆ †f±iØq
GKwU mgvšÍwi‡Ki `ywU mwbœwnZ evû wb‡`©k Ki‡j Gi †ÿÎdj wbY©q
Ki|
Avgiv Rvwb, `ywU †f±i GKwU mgvšÍwi‡Ki `ywU mwbœwnZ evû wb‡`©k
Ki‡j H mgvšÍwi‡Ki †ÿÎdj n‡e †f±i `ywUi µm ¸Yd‡ji gv‡bi
2
(m 1) 0 Dfq c¶ ‡ K ( 15 10 ) Øviv fvM K‡ i
m 1 (Ans.)
P Q
†f±i| awi, †mB †f±i nˆ , nˆ
P Q
2 ˆ
8
13 ˆ
nˆ
i
j
( Ans.)
237
237
237
2
m
P Q 13ˆi 2ˆj 8kˆ
Avevi, P Q 132 2 2 8 2 P Q 237
PQ
13iˆ 2 ˆj 8kˆ
nˆ
237
PQ
4
2
.c
o
ˆi ˆj kˆ
P Q 2 3 4 ˆi (9 4) ˆj(6 8) kˆ (2 6)
2 1 3
P Q ˆi ( 15m 15) ˆj(-30 30) kˆ( 10-10m)
C
op
yr
P Q ˆi ( 15m 15) kˆ( 10 10m)
P Q ( 15m 15) 2 ( 10 10m)2
cÖkœg‡Z, P I Q ci¯úi mgvšÍivj n‡j, P Q 0
( 15m 15) 2 ( 10 10m)2 0
mgvavb: A AN~Y©bkxj †f±i n‡j Gi Kvj© Aek¨B k~b¨ n‡e|
ˆj
iˆ
kˆ
A
x
y
z
x 2 y 4 z 2x 3 y z 4x y 2z
( 15m 15) 2 ( 10 10m) 2 0 [Dfq c¶ ‡ K eM K‡ i]
152 (m 1) 2 102 (m 1) 2 0
(m 1) 2 ( 152 102 ) 0
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02| ‡f±i (Vector)
25| b Gi gvb KZ gv‡bi Rb¨ †f±i
5
v ( x 3 y )iˆ (by z ) ˆj ( x 2 z )kˆ mwjbqWvj n‡e|
mgvavb: †Kvb †f±i mwjbqWvj n‡e hw` Gi WvBfvi‡RÝ k~b¨ nq|
.c
o
m
.v iˆ ˆj
.
y z
x
{( x 3 y )iˆ (by z ) ˆj ( x 2 z )kˆ}
.v ( x 3 y) (by z ) ( x 2 z )
x
y
z
.v 1 b 2
.v b 1
kZ©vbymv‡i, .v b 1 0
b 1 DËi: b=1
C
op
yr
ig
ht
@
M
d
Sh
ah
Ja
m
al
|e
bd
24
A iˆ ( 4 x y 2 z ) (2 x 3 y z )
z
y
+ ˆj ( x 2 y 4 z ) (4 x y 2 z )
x
z
+ kˆ (2 x 3 y z ) ( x 2 y 4 z )
y
x
A iˆ(-1 + 1) + ˆj(4 - 4) kˆ(2 - 2)
A 0
A GKwU AN~Y©bkxj †f±i cÖgvwYZ|
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