3. 65/3/B 3 P.T.O.
IÊS> A
SECTION A
àíZ g§»`m 1 go 6 VH$ àË`oH$ àíZ H$m 1 A§H$ h¡ &
Question numbers 1 to 6 carry 1 mark each.
1. {ZåZ AdH$b g_rH$aU Ho$ {bE BgH$s H$mo{Q> d KmV H$m `moJ$b kmV H$s{OE :
0
dx
yd
dx
d
4
2
2
Find the sum of the degree and the order for the following differential
equation :
0
dx
yd
dx
d
4
2
2
2. {ZåZ AdH$b g_rH$aU H$m g_mH$bZ JwUH$ kmV H$s{OE :
x log x
dx
dy
+ y = 2 log x
Find the integrating factor of the following differential equation :
x log x
dx
dy
+ y = 2 log x
3. {ZåZ{b{IV Amì`yh g_rH$aU go (x + y) H$m _mZ kmV H$s{OE :
2
3y7
5x
+
21
43
=
1415
67
Find the value of (x + y) from the following matrix equation :
2
3y7
5x
+
21
43
=
1415
67
4. 65/3/B 4
4. `{X |
a |= a h¡, Vmo {ZåZ H$m _mZ kmV H$s{OE :
|
a ×
^
i |
2
+ |
a ×
^
j |
2
+ |
a ×
^
k |
2
If |
a |= a, then find the value of the following :
|
a ×
^
i |
2
+ |
a ×
^
j |
2
+ |
a ×
^
k |
2
5. aoIm
4
4z
5
3y
3
2x
Ed§ g_Vb 2x – 2y + z – 5 = 0 Ho$ ~rM H$m H$moU
kmV H$s{OE &
Find the angle , between the line
4
4z
5
3y
3
2x
and the plane
2x – 2y + z – 5 = 0.
6. g{Xe
a = 3
^
i + x
^
j VWm
b = 2
^
i +
^
j + y
^
k nañna b§~dV² h¢ & `{X |
a |=|
b | h¡,
Vmo y H$m _mZ kmV H$s{OE &
The vectors
a = 3
^
i + x
^
j and
b = 2
^
i +
^
j + y
^
k are mutually
perpendicular. If |
a |=|
b |, then find the value of y.
IÊS> ~
SECTION B
àíZ g§»`m 7 go 19 VH$ àË`oH$ àíZ Ho$ 4 A§H$ h¢ &
Question numbers 7 to 19 carry 4 marks each.
7. _mZ kmV H$s{OE :
xtanx1
dxx
Find :
xtanx1
dxx
8. 65/3/B 8
A trust caring for handicapped children gets < 30,000 every month from
its donors. The trust spends half of the funds received for medical and
educational care of the children and for that it charges 2% of the spent
amount from them, and deposits the balance amount in a private bank to
get the money multiplied so that in future the trust goes on functioning
regularly. What percent of interest should the trust get from the bank to
get a total of < 1,800 every month ?
Use matrix method, to find the rate of interest. Do you think people
should donate to such trusts ?
18. x H$m _mZ kmV H$s{OE, {OgHo$ {bE
sin [cot–1(x + 1)] = cos (tan–1 x).
AWdm
{ZåZ H$mo {gÕ H$s{OE :
2 sin–1
5
3
– tan–1
31
17
=
4
Find the value of x, if
sin [cot–1(x + 1)] = cos (tan–1 x).
OR
Prove the following :
2 sin–1
5
3
– tan–1
31
17
=
4
19. `{X a + b + c ≠ 0 h¡, VWm 0
bac
acb
cba
h¡, Vmo gma{UH$m| Ho$ JwUY_mªo H$m à`moJ
H$aHo$ {gÕ H$s{OE {H$ a = b = c.
If a + b + c ≠ 0 and 0
bac
acb
cba
, then using properties of
determinants, prove that a = b = c.