Pre University Board 2010 P.U.C Physics, Chemistry, Maths & Biology Mathematics - Question Paper
Monday, 04 February 2013 06:15Web
Page 1 of 2
Total No. of Questions : 40 ] Code No. 35
Total No. of Printed Pages : 16 ]
March, 2010
MATHEMATICS
( Kannada and English Versions )
Time : 3 Hours 15 Minutes ] [ Max. Marks : 100
( Kannada Version )
: i) A, B, C, D t E 00
0> artrtSo t o%.
CO * c _D
ii) an - a n 10 oxn, an - b n 20 oxn, an -
c n 40 oxn, an - d n 20 oxno
7 _0
an - e n 10 oxndt _0
- A
X>AS 0> t 0
CO _D y oi oi _0
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1. a | b t b | c a | c 0o |
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' 1 |
- 1 |
3 ' |
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'2 |
3 |
1 ' |
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2. |
B + A = |
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Ot B - A = |
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2 |
3 |
4 |
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4 |
2 |
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XoiSO.
ab
3. * a * b = V a, b e Q + 03, >rto ,oZ36
rtra Q + y SS633oSo( (ttjod oSd) XoSO.
__ A A A __ A
4. a = 2 i + j + k b = 3 i + 4j - k 0do a
b doo&aoO.
A A
5. kx 2 + 2 hxy + 4y 2 - 2x + 3y - 7 = 0 ,aoeddra4 oo Jo 4ad
h oJo k n> drt>o dooaoO.
_0 c
6. x = at2 oJ y = 2at ao<o ,oeddraao odo&
J@eoJobR doo.
8. [ 1 + cos 0 + i sin 0 ] 0oo d deod R dooaoO.
9. y = log e e ( 1 + sin x ) wd, dU. or dooaoO,
n/4
10. J sec 2 x dx d (oo dooaoO.
0
- B
d> A4rt>Y /)d 64 r JoO : 10 X 2 = 20
11. 72 d 0"3 d >mdrt> o doDaoO.
CO * & * 0 <=i
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0 ' |
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x 0 |
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12. |
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+ 2 |
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" y |
5 |
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1 - 2 |
= I WAo , I d/ Jd d x o J y n>
CO y CO 2
dn>oJ dooaoO.
ot
3 Code No. 35
13. nyd 10 dY G = { 2, 4, 6, 8 } d XXd db doo
doX ddo (aXd) XodoSoO.
14. 2 i + 3j + 2k doo 4 i + 5j + 3k 0o rt> odo $D&d 0ddo w, OTrt>3dd Xod XodoSoO.
15. x 2 + y 2 - 6x - 2y + 1 = 0 doo x 2 + y 2 + 2x - 8y + 13 = 0 00 0ddo drtb OTd3A ;,5F,od)dodo >eOb.
16. 9x 2 - 4y 2 + 18x - 8y - 31 = 0 Bd d dod Xeodpodod 0 XodoSO.
17. 2 tan - 1 5) + tan - 1 | = 2 0odo ,3pb.
18. x = cos 40 + i sin 40 wd d,
yfx + 1 = 2 cos 2 0 0odo eOb.
Vx
19. x = a 0, y = a wdd,
dy + y = 0 0odo >eOb. dx x
20. odo Xrad) ,d>deZoY 0,dd3rt Bd ddd) s = 4t3 - 6t2 + t - 7 ddz wAd. sndd t = 2 ,Xodort> od Bdd dertdo XodoSoO.
21. I dx dd o dodoSoO.
,aoeddrad {f doJo d,d/rado dodoSoO.
-o y t
I. d> d)rt> /dddd dodo JO
23. 506 doJo 1155 d do.,).. ( G.C.D. ) do dodoSoO doJo >
_0 c _D c
506 (a) + 1155 (b) ( a doJo b dProdrt >o ) ddd ad& n
add } addoed dA 0odo J>eO.
co
24. a) x * 0, y * 0 doJo z * 0 wAdo
_o co
wd d, 1 + "V 1 =
x
1 + X 1 1
1 1 + y 1
1 1 1 + z
25. d3,d ,oZ6rt> on>oct doadod 0 2x2 ,e(o drtr draJdrt rtra M, draJdrt ,od<dY odo dOdJrao ,doo3ododo ,$&. 5
26. a) a = 4 i - j + 3k Do b = - 2 i + j - 2k 00 0-D
ot
3
XoDSDO.
b) a = 5 i + 2j - 3k dJd, b = 4 i + 2j + 5k WOTrt a
,Qj b ,Q Ded o&d/d 4X,ec, XodSdO.
II. X> A4rt> <z4C 0-3 jO : 2x5=10
27. a) ,C>CeZ y = mx + c dd x2 + y 2 = a2 4)JA rXOTrtdD
ey a0 DDr c 2 = a 2 ( 1 + m 2 ) <), XoDSDO. 3
b) 3x 2 + 3y 2 - 9x + 6y - 1 = 0 Jo
2x 2 + 2y 2 - 8x + 16y - 3 = 0 > D>O>X\ D XoDSDO. 2
28. a) aDJ CeZ x + 2 = 0, X, y = 3 dJd 8 Cb
oOk _0 * CO
C (Drt> eXCrartD, Xod&SdO. 3
ot
b) C<D x2 - 3y 2 - 4x - 6y - 11 = 0 eodD ( e )
XoDSDO. 2
29. a) cos - 1 x + cos - 1 y + cos - 1 z = n ,
x 2 + y 2 + z 2 + 2xyz = 1 0oD ,)$&. 3
b) cos 2 0 + cos ( 2 0 ) = 2 , DeXC OCDr XoDSDO.
2
III. JD : 3 x 5 = 15
30. a) 33po x n ,ooo3 sin ( ax ) >4. 3
/ 1 coX ddy dodoSoD. 2
V 1 + cos x J dx *
31. a) y = e m sin 1 x wd,
( 1 - x 2 ) y 2 - xy 1 - m 2 y = 0 0OO ,3&. 3
b) ( 1, 2 ) ooa0Y y 2 = 4x x 2 = 2y - 3 dd,deZrt> do
e4o dodoSoD. 2
32. a) x 3 y 2 = a 5 4d,deZo 4oe /)3d oo aoo><, 4<o dodo&SoD 3rt 4,,2rd 4) x-
CO c *J CO W
ader2 d oJ d3eDd3>Ad3rf 0oo ,3. 3
b) J sec x ( 1 _+x tan x ) dx oo, dodoSoD. 2
f dx
33. a) I -pjT75- o. dodoSoD. 3
J 13 + 12 cos x *
dx
>/8 - 6x - 9x 2
34. ,4/d< a3ao x2 + y 2 =6 4j3o X,eJ;<4o, dodoSoD. 5
- D
X>AS (/dddd dnS t : 2 x 10 = 20
35. a) QeFtoSo odo ododddA dzp dot dd ,&oeXdradS0 x 2 y 2
9 + 7-9 = 1, ( a > b ) dd d 0 XodoSoO. 6
a 2 b 2
2 4
wdd, Xe0 - d,doeodSb. ddieA A_ 1
7 3
XodoSoO. 4 36. a) 03 d Odoeo ,y3>oXrt$rt Sd>eo,d dJ:sesdb) dot sraa&. 6
co d> y ci 0 *
b) ,Q dd (oSo, deA&Xodb
Cp ct
sin ( A - B ) = sin A cos B - cos A sin B 0odo sraQn). 4
37. a) odowotdSo, a 3;&.dd*t d0 otder, dd ,t >t rtOddrteydd
V 6 eJ -0 co _o
do a y/2, oaS i XdAdeXodo teO. 6
b) ( >/3 - 1 ) cos x + ( V3 + 1 ) sin x = V2 0oodd , d/S
d O>ddSx XodoSoO. 4
ot
n
I x tan x
38. a) I -7- dx S (oSo. XodoSoO. 6
sec x + tan x
0
b) doXn > So, aorts,d a3aod ey dby+x2 = x2 ey ,aoeXdrad dO3ddSo, XodoSoO. 4
ot
- E
X>A jDft : 1 x 10 = 10
39. a) V3 - i <b ,0Z dbO XodbroSbD diJb dbrt>b wnaro rtbdbft. 4
y co
b) x 2 + y 2 + 4x - 6y - 9 = 0 d JX@ ( 0, 1 ) aodb><, prftd {d eXdradb, XodbSbD. 4
ot
c) 2x = 2 ( mod 6 ) ,beXdrad dd,,- ,dF,did<Yd dD>drt> ,oZ6 diJb ,dr,dbd dD>ddb XodbSbD. 2
CO c
40. a) a + b + c = 0 dob | a | = 3, | b | = 5, | c | = 7
Wdd, a dbJb b n> do Xedb XodbSbD. 4
7 _0 c
b) J sec 3 ( 2x ) dx db. XodbSbD. 4
c) y = log 5 ( Vsec x ) d, dy. b, XodbSbD. 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 :
1. If a | b and b | c then prove that a | c.
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' 1 |
- 1 |
3 ' |
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'2 |
3 |
1 ' |
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2. |
If B + A = |
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and B - A = |
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2 |
3 |
4 |
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4 |
2 |
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, 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
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
8. Find the amplitude of [ 1 + cos 0 + i sin 0 ]
9. Find djy- if y = log e e
n/4
10. Evaluate J sec 2 x dx.
0
PART - B
Answer any ten questions :
11. Find the sum of all positive divisors of 72.
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' 1 |
0 ' |
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x 0 |
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12. If |
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+ 2 |
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" y |
5 |
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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,
*+i dy 0 2
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
expression is not unique.
1 + x 1 1
1 1 + y 1
= 0 where, x * 0, y * 0 and
1 1 1 + z
z * 0 then show that 1 +
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 .
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.
29. a) If cos 1 x + cos 1 y + cos 1 z = n, then prove that
x 2 + y 2 + z 2 + 2xyz = 1.
b) Find the general solution of the equation
cos 2 0 + cos ( 2 0 ) = 2.
III. Answer any three of the following questions :
30. a) Differentiate sin ( ax ) with respect to x from the first
principle.
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
t, , , | sec x ( 1 + tan x ) , b) Evaluate I -- dx. 2
dx
33. a) Evaluate I -r-5-- . 3
1 13 + 12 cos x
J dx
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
x 2 y a2 + b
2 2 x 2 y 2
+ 72 = 1, ( a > b ) . 6
b) Find A 1 using Cayley-Hamilton theorem if A =
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
38. a) Evaluate | -x tan x- . 6
1 sec x + tan x
0
b) Solve by the method of separation of variables the differential equation e y dy + 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 + b + c = 0 and | a | = 3, | b | = 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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