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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

 
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
 
AB  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

http://ebd24.com

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)



 
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



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

 
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

 
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

 
PQ
w`‡K GKK †f±i| awi, †mB †f±i aˆ , aˆ    
PQ

.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    
PQ
(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

 
PQ
iˆ  10 ˆj  18kˆ
 nˆ      
(Ans.)
425
PQ

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


1  ˆi (4  2)  ˆj(4  2)  kˆ (8  8)

ˆj

m 3
10  5 -15



m

mgvb| P  Q  mgvšÍwi‡Ki †ÿÎdj|

ˆi
 
PQ  2





Avevi, P  Q  12  (10 2 )  ( 182 )  P  Q  425

al



 
 P  Q  iˆ  10 ˆj  18kˆ

|e



ˆi
ˆj
 
PQ  4  4

ˆj

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


 
PQ  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
 
PQ
13iˆ  2 ˆj  8kˆ
 nˆ      
237
PQ

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


 



  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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