Pre University Board 2009 P.U.C Physics, Chemistry, Maths
Total No. of Questions : 40 ] [ Total No. of Printed Pages : 16
Code No. 35
June/July, 2009
( Kannada and English Versions )
Time : 3 Hours 15 Minutes ] [ Max. Marks : 100
( Kannada Version )
: i) d, dXdo A, B, C, D doJ E 00 aco@ dartrtd. 0a >dartrt>o J 0ft.
CO * c _D
ii) - A n 10 odrt>3, - B n 20 odrt>3, -c n 40 odrt>o, dan - d n 20 oxn>o doJo
7 _D
- e n 10 oxndoJ d.
_D
- A
X>A 0a J 0ft :
10 x 1 = 10
CO _0 y oi oi _0
1. 3x = 2 ( mod 6 ) ,doe&edoJn d0ad<Y. ?
12
2. a do d,dX," X, ( Direction cosines ) 75- ,75- doJo n rt>add n
33
d doo XodoSoO.
3. dP}aroXn> rtra I (do, * 3/dodo aZ a * b = a b , V a, b e I wAd. 3/dodoo qjdzrf 3/dodoe da I doe<,de 0odo doeft.
4. A, B ode dOd/ra ( Order ) dod 0dd drtr d/;drt>b. | A | = 4, | B | = 5 add, | AB | dodSoO.
5. 0ddo d rt> deod,rt> dodra dad d aAd, r , , r 2 n> &&.rt>3Ad d, a
a) o ,J/ co 12 y 6 co
drtb odapd d)6 ,Fd3rtde5ad ( External touch ) aod(o0 ( Condition ) d(oO.
6. 4x 2 + 9y 2 = 36 d doeSdod (/dde od dadrt> dado dodoaDO.
7. sin - 1 ( sin 130 ) ( d (oy dodoSoO.
ot
( 1 - i + n
8. I i + i I = 1 artodo n dad d dP}Fod d (oo dodtfoO.
9. 4) f ( x ) = | x | ahdd, L f1 ( 0 ) dodoaoO.
n/4
10. J ( sin 3 x + cos x ) dx d (o0 dodoaoO.
- n/4
- B
d> Ad)rt> (/d)ddda dnO o : 10 x 2 = 20
11. ca = cb ( mod m ) aAd c, m rt >i ,aded, ,oZ,rt>add a = b ( mod m
K co <66
) 0od srap&.
cos 0 sin 0
12. A =
sin 0 cos 0
adn , AA 1 (o , ( Symmetric ) d/ddbe
0od does.
3 Code No. 35
13. d,odD<dD ,od< d/dde 5# ( + 5 ) dbD { 1, 2, 3, 4 }
rtrad) d,odD<de 0od dOeft.
14. Q + ( >rt ,o<Zrt> rtra) rtrad * 00 3/xb<D dZ a * b =
a b
3 , V a, b e Q + wAd. >o dDDo Q + a - 1 D dodDaDO.
A A
15. X i + j + 2k , 2 i - 3j + 4k dD i + 2j - k rt >0 , dD< , rt Add ( Coplanar vectors ), X d d<DD dodDaDO.
16. ( 0, 0 ), ( 3, 0 ) do ( 0, 4 ) rt >d ort rt >3Acbd $D&d dOdd ,DeddradD dodDaDO.
17. : tan - 1 x = sin - 1 -r - cot - 1 1 .
V2 3
i tan - 1 3
18. 5e 3 0o ,oZ,w<D d,d ( Real ) dD 6 ( Imaginary ) rtrt>D djdDdA 3, 4 0o 3eOft.
V' - + / i~
/ i- \
x - 1 - 1 Vx + 1
dy
19. y = sin - 1 - + sec - 1 - wd d, d~ = 0 0o
\yjx + 1 / \y/x - 1 / dx
20. x m y n = am + n 0o dd,deZD </d)d &orio> 0> drdd) odD $D&dA ( Abscissa ) ddrtorf 0odD ,a&.
21. J [ sin ( log x ) + cos ( log x ) ] dx dDD dodDaDO.
22. y-d\d Dr dD<aodb>< ( Origin ) ,&F,od drt> dd< , Deddrad Dr ( Differential equation ) ddDO.
- C
I. X>Ad)rt>0, Ob : 3 x 5 = 15
23. a) a, b d>})FoXrt> do.,).. ( GCD ) do 275 doo 726 d
do.,).B>. XodoSoO. 3
b) 252 O ,oZ,rt> rtora ddo - )&Xrt>do,
Ct * 6 6 cp ci 44 0
0oo XodoSoO. 2
24. Xe d iao Sb : 2x - y = 10
cp 4
x - 2y = 2
dod ,io Xrt> Xed) yS-oq deodo ( Satisfies ) 0oo dOeSjb. 5
25. V a, b e H, ab - 1 e H wirt, G , oXo< 6d<, drtradid H rtd G d,oXododo ,iab. b, d<>eAb H do3o K n> G
4 c _D
d , oXo<nd d H I K Xrai G d,oXododo ,ib. 5
AAA A A A
26. a) a = 2i + j + k , b = i + 2j* - k 0O Xljrt a n
od)Adod do3o a , b n> ,do3<0 dod
0 CO
( coplanar with a and b ) &X ( Unit vector ) do
XodoSoO. 3
b) a + b + c = 0 wd,
a x b = b x c = c x a 0oo ,)b. 2
II. X> Ad)rt>S d/>d)ddd 0do drtrt Ob : 2 x 5 = 10
27. a) x 2 + y 2 + 2 g 1 x + 2 f 1 y + c 1 = 0 doo
x 2 + y 2 + 2 g 2 x + 2 f 2 y + c 2 = 0 drt >0 oddA $eQ,d aoddDD ad)&b. 3
b) dDdD d)1(rt> dDOT Xeodd ( 1, 2 ) wAd. odD dd ,DeXdrad) x 2 + y 2 - 2x + 3y = 0 WAdd, Xeod,d) drt> >n ddd ddrt ddoe 0odo dOeSjb. 2
28. a) 9x 2 + 4y 2 - 18x + 16y - 11 = 0 (do Xeod, dDD dd ,SddDX
( Auxiliary ) dd brdd XodDSdDO. 3
b) d d d <dD x = 2t 2 , y = 4t dD adDd ( Directrix ) ,oeddrad0 XodDSdDO. 2
n
29. a) sin - 1 x + sin - 1 y + sin - 1 z = Wddrt
x 2 + y 2 + z 2 + 2xyz = 1 0od srab. 3
b) tan 20 tan 0 = 1 ,aoeXdrad ,d/6 dO>ddD XodDSdDO. 2
III. X>A dradddd dodo drtrt Ob : 3x5=15
30. a) dDKQod x rt ,oobdo sin 2x Dr adjab ( Differentiate ). 3
b) x n ,oobdo adjab : (sin x )logx 2
31. a) cos - 1 ( 4x 3 - 3x ) d D cos - 1 ( 1 - 2x 2 ) rt ,oobdoJ
adb. 3
b) d X,deZrt >dd y = 6 + x - x 2 doo y(x-1) = x+ 2 rt>D (2,4)
odoa S d 0odD 3eOb. 2
CO vJ 0
( 1 - x 2 ) y 2 - xy 1 + m 2 y = 0 0O 3
32. a) y = sin ( m cos - 1 x ) wd,
-1- dx dtioo dooaoO. 2
b)
x ( x 5 + 1 )
33. a) x n ,000J d<0& ( Integrate ) sin x + 18 cos x
3
3 sin x + 4 cos x
b) J \/ -x dx d tioo dooaoO. 2
34. d< ( Integration ) >ao x 2 + y 2 = 6 4)J eraro dOOaOO. 5
- D
d>A JO : 2 X 10 = 20
x 2 y 2
35. a) y = mx + c ,d>deZtioo 2 - 7-9 = 1 3dtiod,
a 2 b 2 *
,FdOTrto 0tioO ( Derive ). <y ,fooo
dooaoO. aotioO tieA x - y + 5 = 0 deZrt
x 2 y 2
,/oJdrortooJ 16 - 12" = 1 dr@ ,Fdrt> ,oeddrart>o
dooaoO. 6
1 a a 2 + bc
b) 1 b b 2 + ca = 2 ( a - b ) ( b - c ) ( c - a ) 0o ,)$&. 4
1 c c 2 + ab
36. a) Q deo d,doe(Ddo don. Jd) , Uora dProXrtAdodrt
d,doe(Ddo ,an. do defin
Z 10 - 1
Z = cos 0 + i sin 0 Wdrt ZT0-1 = i tan 50 0odo srapn. 6
b) cos 20 = V2 ( cos 0 - sin 0 ) ,oeXdrad Jd/S dO3ddO XodoQoO. 4
37. a) odo ne>d d<d) ddd dd 4n c.c./sec. aAd. d<d) 288 n
adrt 3, doed &njex>Frt> ddod ddrt>o XodoQoO. dod
(i) 5 ,Xodort > ddd and d.3D., doJo (ii) dd 288 n adri
v ' co * co 66 _Dv'*i
, oopndoJ dd<d) ddod ddrt>o XodoQoo. 6
b) ( 1, 5 ) dJo ( 1, 1 ) aodOrt>o <on>Adod ddd<(od
,oeXdrart>o XodoQoO. 4
ot
I a
x dx_ = n 2
cos 2 x + b 2 sin 2 x 2ab
38 a) I - . 2 2 u 2 2 -- = 0od nin). 6
0
b) xy ( 1 + x 2 ) djx- - y 2= 1 d dXo ,oeXdrad aarj ( Particular ) d 0>ddo XodoSoO : x = 1, y = 0 0od X.d. 4
- E
X>A (/d)ddd o d, Jon : 1x10=10
39. a) | a + b + c | = | a + b - c | add, a + b doJo c rt>
dodra Xe d o XodoQoO. 4
ot
b) odo aar >Jdo drardod 0a, <oXe drt>, doadao xdd ft/rard) rtOddaAdoJd 0odo ,aaft. 4
1
c) ' 16 cis 2 ) 4 XodoadoO. 2
40. a) 2 150 x 3 12 x 135 = a ( mod 7 ) wdd, a dorfo, 7 0od da>A,odart ftrtod dad d do dodoadoo. 4
* ot
2 2
b) 2 ( x 2 + y 2 ) - 12x - 4y + 10 = 0 do Jo
x 2 + y 2 + 5x - 13y + 16 = 0 dJrt ,ad/, {d d do
dodoaoo. 4
2
\fx
v dx d doo, dodoadoO. 2
c)
V2 - x + yfx
ii) Part - A carries 10 marks, Part - B carries 20 marks, Part - C carries 40 marks, Part - D carries 20 marks and Part - E carries 10 marks.
PART - A
Answer all the ten questions : 10 x 1 = 10
1. 3x = 2 ( mod 6 ) has no solution. Why ?
2. If direction cosines of a are 3 , and n, find n.
3. On I ( the set of all integers ), and operation * is defined by a * b = a b ,
V a, b G I. Examine whether * is binary or not on I.
4. A and B are square matrices of the same order and | A | = 4, | B | = 5. Find | AB |.
5. Given two circles with radii r 1 , r 2 and having d as the distance between their centres, write the condition for them to touch each other externally.
6. Find the sum of the focal distances of any point on 4x 2 + 9y 2 = 36.
7. Evaluate sin - 1 ( sin 130 ) .
8. Find the least positive integer n for which '
9. Given the function f ( x ) = | x |, find L f 1 ( 0 ) .
n/4
10. Evaluate J ( sin 3 x + cos x ) dx .
- n/4
Answer any ten questions : 10 x 2 = 20
11. If ca = cb ( mod m ) and c, m are relatively prime then prove that a = b ( mod m )
, verify that AA 1 is symmetric.
cos 0 sin 0 sin 0 cos 0
12. For the matrix A =
13. Define a semigroup. Examine whether { 1, 2, 3, 4 } is a semigroup under addition modulo 5 ( + 5 ) .
14. On Q + ( set of all +ve rationals ) , an operation * is defined by ab
a * b = -3- , V a, b e Q + . Find the identity element and a ~ 1 in Q + .
A A A A A A A AA
15. If X i + j + 2k , 2 i - 3j + 4k and i + 2j - k are coplanar, find X.
16. Find the equation of the circumcircle of the triangle formed by ( 0, 0 ), ( 3, 0 ) & ( 0, 4 ).
17. Solve tan 1 x = sin 1 - cot 1 i- .
V2 3
t - 1 4
i tan 3
18. Show that the real and imaginary parts of 5e are 3, 4
respectively.
19. If y = sin 1 - + sec 1 vll T , prove that = 0.
yfx + 1 ) & yfx - 1 dx
20. At any point on the curve x m y n = a m + n , show that the subtangent varies as the abscissa of the point.
21. Evaluate J [ sin ( log x ) + cos ( log x ) ] dx.
22. Form the differential equation of the family of circles touching y-axis at origin.
I. Answer any three questions : 3 x 5 = 15
23. a) Define GCD of two integers a and b. Find the GCD of 275 and 726. 3
b) Find the number of positive divisors of 252 by writing it as the product of primes ( prime power factorisation ). 2
24. Solve by matrix method : 2x - y = 10
x - 2y = 2
Also, verify that the coefficient matrix of this system satisfies Cayley-Hamilton theorem. 5
25. Prove that a non-empty subset H of a group G, is a subgroup of G, if
V a, b e H, ab - 1 e H. Hence prove that, if H and K are subgroups of a group G then H I K also, is a subgroup of G. 5
A A A A A A
26. a) Given a = 2i + j + k , b = i + 2j - k , find a unit
vector perpendicular to a and coplanar with a and b . 3
b) If a + b + c = 0 , prove that a x b = b x c = c x a .
2
II. Answer any two questions : 2 x 5 = 10
27. a.) Derive the condition for the two circles
2 + y 2 + 2 g 1 x + 2 f 1 y + c 1 = 0 and
x
x 2 + y 2 + 2 g 2 x + 2 f 2 y + c 2 = 0 to cut each other orthogonally. 3
b) ( 1, 2 ) is the radical centre of three circles. One of the circles is x 2 + y 2 - 2x + 3y = 0. Examine whether the radical centre lies inside or outside all the circles. 2
9x 2 + 4y 2 - 18x + 16y - 11 = 0, find its centre and the area
of its auxiliary circle.
3
b) Obtain the equation of the directrix of the parabola x = 2t 2 ,
2
29. a.) If sin - 1 x + sin - 1 y + sin - 1 z = 2 , prove that
x 2 + y 2 + z 2 + 2xyz = 1.
b) Find the general solution of tan 20 tan 0 = 1.
2
3
2
III. Answer any three of the following questions :
3 x 5 = 15
30. a.) Differentiate sin 2x w.r.t. x from first principle.
3
2
b) Differentiate ( sin x ) log x w.r.t x.
31. a) Differentiate cos 1 ( 4x 3 - 3x ) w.r.t. cos 1 ( 1 - 2x 2 ) . 3
- 1
b) Show that the curves y = 6 + x - x 2 and y ( x - 1 ) = x + 2
touch each other at ( 2, 4 ).
2
32. a) If y = sin ( m cos - 1 x ) , prove that
( 1 - x 2 ) y 2 - xy 1 + m 2 y = 0.
3
b) Evaluate
dx.
2
x ( x 5 + 1 )
1
33. a) Integrate 1- w.r.t. x. 3
3 sin x + 4 cos x
b) Evaluate J \l 1 + x dx. 2
34. Find the area of x 2 + y 2 = 6 by integration. 5
PART - D
Answer any two of the following questions :
2 x 10 = 20
35.
a) Derive a condition for y = mx + c to be a tangent to the hyperbola
2 2 xy
0~2 - b""2" = 1. Also, find the point of contact. Using the condition
2 2 xy
derived, find the equations of tangents to yg - = 1, which are
parallel to x - y + 5 = 0.
6
b) Prove that 1
a 2 + bc b 2 + ca c 2 + ab
a
1 b 1 c
= 2 ( a - b ) ( b - c ) ( c - a ).
4
a) State De Moivres theorem. Prove it for positive and negative integral
Z 10 - 1
= i tan 50 if
indices. Using it prove that -
Z + 1
Z = cos 0 + i sin 0. 6
b) Find the general solution of cos 20 = V2 ( cos 0 - sin 0 )
4
37. a) The volume of a sphere increases at the rate of 4n c.c./sec. Find the rates of increase of its radius and surface area when its volume is 288 n c.c. Also find (i) the change in volume in 5 secs, (ii) rate of increase of volume w.r.t. radius when the volume is 288 n c.c. 6
b) Obtain the equations of parabolas having ( 1, 5 ) and ( 1, 1 ) as ends of the latus rectum. 4
0
b) Find the particular solution of xy ( 1 + x 2 ) dx _ y 2 = 1, given
38. a.) Prove that
that, when x = 1, y = 0.
PART - E
Answer any one of the following questions : 1 x 10 = 10
39. a.) If |a + b + c | = |a + b - c | , find the angle between a + b and c . 4
b) Among all right-angled triangles of a given hypotenuse, show that the triangle which is isosceles has maximum area. 4
n
c) Find the fourth roots of 16 cis . 2
J a
2
x dx
6
2 cos 2 x + b 2 sin 2 x 2ab '
2 , dy
4
40. a) If 2 150 x 3 12 x 135 = a ( mod 7 ), find the least positive remainder when a is divided by 7. 4
b) Given the circles 2 ( x 2 + y 2 ) - 12x - 4y + 10 = 0 and
x 2 + y 2 + 5x - 13y + 16 = 0, find the length of their common chord. 4
2
&- dx. 2
c) Evaluate
y/2 - x + yfx
0
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Attachment: |
| Earning: Approval pending. |
