Pre University Board 2010 P.U.C Physics, Chemistry, Maths & Biology Mathematics - Question Paper
Total No. of Questions : 40 ] Code No. 35
Total No. of Printed Pages : 16 ]
March, 2010
( Kannada and English Versions )
Time : 3 Hours 15 Minutes ] [ Max. Marks : 100
( Kannada Version )
0> artrt>o t Oft.
CO * c _D
ii) an - a n 10 odn>, an - b n 20 oxrt>o, an -
c n 40 oXrt>o, an - d n 20 odn> oto 7 _0
an - e n 10 oxndot
_D
- A
X>A 0> t 0ft
10 x 1 = 10
CO _0 y oi oi _0
a oo.
|
1. a | b oto b | c WOTrt a | c 0oo ,3ft. | |||||||||||||||||||||||||||
| |||||||||||||||||||||||||||
|
doOSOO. |
ab
3. * 3/O<OO a * b = 2 V a, b e Q + 0 ,oz6
rtra Q + toOc) (ttod 4o44oc)) XodSoO.
oR ( t)'
__ A A A __ A
4. a = 2 i + j + k doJ b = 3 i + 4j - k 0ddo ,Srt>3d3rt a b dod&So.
A A
5. kx 2 + 2 hxy + 4y 2 - 2x + 3y - 7 = 0 ,deddrad) 0 ddd dpdd
h d J k n> rt>d dodSO.
_0 c
6. x = at2 dJ y = 2at dp1 ,oeddraaod odo&d
Jddd, doo.
1
4
1
7. sin
2 cos
5
d dd, dodSdO.
ot
8. [ 1 + cos 0 + i sin 0 ] 0od d deodd d dodSdO.
9. y = log e e ( 1 + sin x ) wdd, du. oi dodSdO,
n/4
>. J
10. J sec 2 x dx dd dodSdO.
0
- B
d> Ad)rt> d/dddd JO : 10 x 2 = 20
11. 72 d 03 dJvd >rndrt> dJ dd, dodoSdO.
CO * S' * _D c
| |||||||||||||||
|
rt>d dodSdO. ot |
= I wAd, I add/ d/Jd wAd d x dJo y rt>
CO y CO 2
3 Code No. 35
13. nyd d/dd 10 d G = { 2, 4, 6, 8 } d XeXd db
dX ddd (aXddd) XodaO.
14. 2 i + 3j + 2k dO 4 i + 5j + 3k 0o rt> od $D&d 0dd w, OTdrt>3dd *|&d Xeod XodaO.
15. x 2 + y 2 - 6x - 2y + 1 = 0 doO x 2 + y 2 + 2x - 8y + 13 = 0 00 0dd drtb ;,5F,d)dod >eOb.
16. 9x 2 - 4y 2 + 18x - 8y - 31 = 0 Bd dddd Xeodpodd d XodoSO.
17. 2 tan - 1 5) + tan - 1 -i2| = 2 0od srab.
18. x = cos 40 + i sin 40 wdd,
yfX + 1 = 2 cos 2 0 0o eOb.
19. x = a 0, y = a wdd,
dy + y = 0 0od >eOb. dx x
20. od Xrad) ,d>deZoY 0,ddirt dd ddd) s = 4t3 - 6t2 + t - 7 d>d/ wAd. sndd t = 2 ,Xodort> od dd dertdd XodSO.
ot
Code No. 35
J logX
3/2
2
d 2 y dx 2
i++dxy
22.
,aoeddra {f do3o dodoSoO.
-o y t
- C
3 x 5 = 15
23. 506 do3o 1155 d do.,).. ( G.C.D. ) do dodoSoO do3o o
_0 c _D c
506 (a) + 1155 (b) ( a do3o b })FOdrt >0 ) dddS ad& )rt
add } addoed d)fio 0oo 3>e0.
co
24. a) x * 0, y * 0 do3o z * 0 Wftdo
-0 CO
wd d, 1 + "V 1 =
x
0 0O 3>e0.
b) X,edod* aododoJ S
2x - 3y = 5
7x - y = 8.
5
1 + X 1 1
1 1 + y 1
1 1 1 + z
= 0
3
2
25. d),d ,oZ6rt> on>ot doadod 0O5n 2 x 2 ,e(o drtF dz)Xrt> rtra M, d/3)Xn> ,odY oo dOdFo ,doo3<ododo 5
26. a) a = 4 i - j + 3k diJi b = - 2 i + j - 2k 00 0Cdi oodftCidoJ dOd/ra 12 oQCid ,Qd0
XodiaiO.
ot
3
b) a = 5 i + 2j - 3k diJi b = 4 i + 2j + 5k Wsrt a
,Qd) b ,Qd died o&id/did dX/ddi XodiaiO.
2
II. X> Ad)rt> /d)addi 0do JO : 2x5=10
27. a) ,C>CeZ y = mx + c tin x2 + y 2 = a2 dJX ,FXdDrtOi
eyad ao <ii c 2 = a 2 ( 1 + m 2 ) Cd <), XodiaiO. 3
b) 3x 2 + 3y 2 - 9x + 6y - 1 = 0 do Ji
2x 2 + 2y 2 - 8x + 16y - 3 = 0 dJrt > di>0>X\d i XodiaiO. 2
28. a) a<iJ CeZ x + 2 = 0, X, y = 3 diJ a$od d 8 Cid
oOk _0 * co
d C d irt> ieXCrarti XodoaiO. 3
ot
b) 3d Cdi x2 - 3y 2 - 4x - 6y - 11 = 0 d J@eoJii ( e ) XodiaiO. 2
29. a) cos - 1 x + cos - 1 y + cos - 1 z = n Wdart ,
x 2 + y 2 + z 2 + 2xyz = 1 0o 3
b) cos 2 0 + cos ( 2 0 ) = 2 , ieXC rad ,ad/6 dOaCdiR XodiaiO.
2
III. X>A /roddi Joft : 3 x 5 = 15
30. a) JJpod x n ,ooftdoJ sin ( ax ) , >4. 3
/ 1 coX ddy XooaoO. 2
- 1
b) y = tan
V 1 + cos x J dx *
31. a) y = e m sin 1 x wd,
( 1 - x 2 ) y 2 - xy 1 - m 2 y = 0 0OO ,)&. 3
b) ( 1, 2 ) aoo>, y 2 = 4x do x 2 = 2y - 3 dX,deZrt>
&dO, XodoaoO. 2
ot
32. a) x 3 y 2 = a 5 dX;deZ<do oe /)ddi odo aodo>, dod d do XodoaO >rt d,drXd d d) x-
co ci syJ CO
ader Xd oJ odaeodaAdoJ. 0odo ,aft. 3
b) J sec x ( 1 -x tan x ) dx 00, XodoaoO. 2
f dx
33. a) I -pj-T75- o. XodoaoO. 3
J 13 + 12 cos x *
dx
>/8 - 6x - 9x 2
34. ,d/X< >Qod x 2 + y 2 = 6 dJd X,e;Jd<do, XodoaoO. 5
- D
35. a) Qerdd odo ododdddA ddzdpb doo dd ,oeXdrado x 2 y 2
9 + 7-9 = 1, ( a > b ) dd d XodoSoO. 6
a 2 b 2
2 4
wdd, Xe - d,o d,doeod. ddieAb .A_ 1
b) A =
7 3
XodoSoO. 4
36. a) 0d d Odoeo ,ydoXrt$rt Qd>eo,d d,d6eodo $b do3o ,dQb. 6
co d> y ci 0 *
b) ,Q dd 3oo d<eAbXodo
Cp
sin ( A - B ) = sin A cos B - cos A sin B 0odo ,dQb. 4
37. a) odowod a &&.dd)3 d oderb, dd rtOdddrteyddd
V 6 eJ -0 co _o
do a y/2, i XddAdeXodo 3eOb. 6 b) ( V3 - 1 ) cos x + ( V3 + 1 ) sin x = V2 0oodd , dd/ d Odddo XodoSoO. 4
n
1 x tan x
38. a) I -7- dx o. Xodo&SoO. 6
J sec x + tan x *
0
b) ddoxn > ortS,od ddQod ey "jy+x2 = x2 ey
,oeXdrad dOdddo XodoSoO. 4
- E
/d))d oo d, 3 : 1x10=10
39. a) V3 - i ao, ,oZ6o dodoSO d3o drt>o
wn)Fod 3, rtodo. 4
y co
b) x 2 + y 2 + 4x - 6y - 9 = 0 dX ( 0, 1 ) aodo>0, F&d {) ,aoeddradb, dodoSoO. 4
ot
c) 2x = 2 ( mod 6 ) ,aoeddrad dd;,d ,dF,dod<Y dO>drt> ,oZ6 d3o ,dF,dod< d dO)ddo dodoSoO. 2
CO c
40. a) a + b + c = 0 doo | a | = 3, | b | = 5, | c | = 7
wdd, a d3o b n> do Xedo dodoSoO. 4
7 _0 c
b) J sec 3 ( 2x ) dx doo dodoSoO. 4
c) y = log 5 ( Vsec x ) wd d, dy dodoSoO. 2
( English Version )
Instructions : i) The question paper has five Parts - A, B, C, D and E. Answer all the parts.
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. If a | b and b | c then prove that a | c.
|
, find A. |
3. Find the identity element in the set of all positive rationals Q + , is
ab
defined by a b = V a, b e Q + .
AAA
A A
4. If the vectors a = 2 i + j + k and b = 3 i + 4j - k , find a b .
5. Find the values of h and k for the equation
kx 2 + 2hxy + 4y 2 - 2x + 3y - 7 = 0 to represent a circle.
6. Write the eccentricity of the conic section represented by the parametric equations x = at 2 and y = 2at.
7. Evaluate sin
1
2 cos
- 1
8. Find the amplitude of [ 1 + cos 0 + i sin 0 ]
9. Find djy if y = log e e
( 1 + sin x )
n/4
10. Evaluate J sec 2 x dx.
0
PART - B
Answer any ten questions :
10 x 2 = 20
11. Find the sum of all positive divisors of 72.
|
'1 |
0 ' |
x 0 | ||
|
12. If |
+ 2 | |||
|
y |
5 |
1 - 2 |
= I where I is the identity matrix, find x
5 * 1 1 - 2J and y.
13. Write the composition table for G = { 2, 4, 6, 8 } under multplication modulo 10 and find the identity element.
14. Find the area of the triangle whose two adjacent sides are determined by
AAA AAA
the vectors 2 i + 3 j + 2k and 4 i + 5j + 3k .
15. Show that the two circles x 2 + y 2 - 6x - 2y + 1 = 0 and
x 2 + y 2 + 2x - 8y + 13 = 0 touch each other externally.
16. Find the centre of the hyperbola 9x 2 - 4y 2 + 18x - 8y - 31 = 0 .
17. Prove that
18. If x = cos 40 + i sin 40, show that yfx + = 2 cos 2 0.
yjx
19. If x = a 0, y = a , show that y = 0.
0 dx x
20. A particle is travelling in a straight line whose distance is given by s = 4t 3 - 6t 2 + t -7 units. Find the velocity of the particle after t = 2 seconds.
21. Evaluate I dx.
x 2
22. Find the order and degree of the differential equation,
dx
PART - C
23. Find the G.C.D. of 506 and 1155 and express it in the form of 506 (a) + 1155 (b) ( where a and b are integers ). Also show that the
5
expression is not unique.
1 + x 1 1
1 1 + y 1
24. a) If
= 0 where, x * 0, y * 0 and
1 1 1 + z
z * 0 then show that 1 +
3
2
b) Solve by Cramers Rule : 2x - 3y = 5, 7x - y = 8.
25. Prove that the set M of all 2 x 2 matrices with elements of real
numbers form an Abelian group with respect to addition of matrices.
5
26. a) Find the vector of magnitude 12 units which is perpendicular to
> A A A
both the vectors a = 4 i - j + 3k and
_> A A A
b = - 2 i + j - 2k . 3
b) If a = 5 i + 2j - 3k and b = 4 i + 2j + 5k are two
vectors, find the projection of a on b .
2
27. a) Derive the condition for the line y = mx + c to be a tangent to
the circle x 2 + y 2= a 2 in the form c 2 = a 2 ( 1 + m 2 ) . 3
b) Find the radical axis of the circles 3x 2 + 3y 2 - 9x + 6y - 1 = 0 and 2x 2 + 2y 2 - 8x + 16y - 3 = 0. 2
28. a) Find the equations of the parabolas whose directrix is x + 2 = 0,
axis is y = 3 and the length of the latus rectum is 8 units. 3
b) Find the eccentricity of the hyperbola
x 2 - 3y 2 - 4x - 6y - 11 = 0.
2
29. a) If cos 1 x + cos 1 y + cos 1 z = n, then prove that
x 2 + y 2 + z 2 + 2xyz = 1.
3
b) Find the general solution of the equation
cos 2 0 + cos ( 2 0 ) = 2.
2
III. Answer any three of the following questions :
3 x 5 = 15
30. a) Differentiate sin ( ax ) with respect to x from the first
principle.
3
Vi
- cos x
2
+ cos x
b) Find if y = tan - 1
Code No. 35 14
31. a) If y = e m sm 1 x , prove that ( 1 - x 2 ) y 2 - xy 1 - m 2 y = 0.
3
b) Find the angle between the curves y 2 = 4x and x 2 = 2y - 3 at the point ( 1, 2 ) . 2
32. a) Find the length of sub-normal to the curve x 3 y 2= a 5 at any
point on it. Also show that length of sub-tangent varies directly
as abscissa at that point. 3
I
I
t, , , | sec x ( 1 + tan x ) , b) Evaluate I -- dx. 2
dx
33. a) Evaluate I -r-5-- . 3
1 13 + 12 cos x
f dx
dx
>/8 - 6x - 9x 2 '
b) Evaluate I , . 2
34. Find the area of the circle x 2 + y 2 = 6 by integration method. 5
PART - D
Answer any two of the following questions : 2 x 10 = 20
35. a) Define an ellipse as a locus and derive its equation in standard form
2
2 2 x 2 y 2
+ 72 = 1, ( a > b ) . 6
b) Find A 1 using Cayley-Hamilton theorem if A =
2 4 7 3
4
36. a) State and prove De Moivres theorem for rational index. 6
b) Prove by vector method
sin ( A - B ) = sin A cos B - cos A sin B. 4
37. a.) Show that rectangle of maximum perimeter which can be inscribed in
a circle of radius a is a square of side a V2 . 6
b) Find the general solution of
( V3 - 1 ) cos x + ( + 1 ) sin x = V2 . 4
J
38. a.) Evaluate | -x tan x- dx. 6
sec x + tan x
0
b) Solve by the method of separation of variables the differential equation e y ddLL + x 2 = x 2 e y . 4
PART - E
Answer any one of the following questions : 1 x 10 = 10
39. a.) Find the cube roots of the complex number V3 - i and express them in an Argand diagram. 4
b) Find the equation of the chord of the circle
x 2 + y 2 + 4x - 6y - 9 = 0 bisected at ( 0, 1 ). 4
c) Find the number of incongruence solutions and the incongruence
solution of 2x = 2 ( mod 6 ). 2
40. a) If a + _ + c = 0 and | a | = 3, | _ | = 5, | c | = 7,
then find the angle between the vectors a and b . 4
b) Find J sec 3 ( 2x ) dx. 4
c) If y = log 5 ( Vsec x ) , find . 2
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Attachment: |
| Earning: Approval pending. |
