Pre University Board 2011 P.U.C Physics, Chemistry, Maths & Biology Mathematics - Question Paper
Total No. of Questions : 40 ]
Total No. of Printed Pages : 16 ]
Code No. 35
March, 2011
( Kannada and English Versions )
Time : 3 Hours 15 Minutes ] [ Max. Marks : 100
( Kannada Version)
sL/dzSrf: i) ds A, B, C, D E <3020 sodo &2rtrts?d.
o( _D
ii) - Art 10 orteb, &2rt - B rt 20 otfrtsfe, <art - C rt
40 o3rts>, asrt - D rt 20 otfrtsto &rt - E rt
10 o3rts?ci>3 d.
& So3: drt,5 OAi : 10 x 1 = 10
twiCM ircw, -/ ot
CO .V Or Or _A ~
4 3 2 1
2. A =
3. srod sios36rt* rtra R fSQ * S,o>o& a * b = a + b + 5, V a, be.R
4. AB = 3i + 2j + 6fc OA = i - j - 3& add, B 3 oe&So&kdo.
*
5. P ( 1, 1 ) 8oc>&3ocS 3x2 + 3y 2 + 6x + 9y - 2 = 0 4
3oc&SoSok>0.
6. y = mx + c tfdtfdeaJo&i t/2 = 4cuc stedooi rsrsdd, adeFrtsfe wdo&o. .
' IN v'2
n _ j 3 -sin
ti odiSoScOiO.
7. sin
8. e iTt 3 ensso, 2J3rtsS?& tfockSoSo&O. . * t
9. y = sin ( 2 sin ~ 1 x ) add, 55? 3oc&&o8cdO.
I
10. J sec2x dxrf 3oc&SoSo>0.
arart - B
3s oira4roddra srfrts eroO : 10 x 2 = 20
11. a j be *&&,, a z&Q b rts? sk.s3.. ( G.C.D.) 1 add, a | c aock
|
1 |
2 |
0 " |
sSd B = |
2 |
1 - 3 * |
|
. - 3 |
1 V |
- 2 . |
. 2 |
1 1 - |
/
13. ,G = { l, oo, co2},(os$) rtoesud . Eodoroo co2 d <a&es>sjck, tioct&oBc&O.
-A t
14. ( G, * ) sio&wriQ. ( a * b) 1 = b" 1 * a" 1 , V a, be G <aoc& EapSi.
15. 2 i - j + lc , 1 - 3j - 5ic sfc' 3i - 4j - 4& STSrf sSartsrod ZSockrtsb ooso&aesi doerfd skracb orWocfc ;ieO&.
16. ( 5, - 7 ) ( 7, - 5 } aoc&rtsk &>>rts?3fta sci eddrarf? 3od)2ao3iO.
17. sin-1 x+ cos-1 x = - 1 <x< 1 <aod>
18. x - cos A + i sin A, y = cos B + i sin B tsdd, xy - ~ = 2 i sin {A + B )
30C&) ij3?Drj.
V
 |
|
dx2 |
1 - cos X 1 + cos X .
19. y = tan 1
20. y 2 = 3 - 4x sdeidrl rdea3oo, 2x + y - 2 = 0 tfdtfjcJesJrt sSs&ssraodOTcrart, d3,des3o& s&>?)?3 3oc&So&o&0.
9 dx rf s3e3a>5k, 3oc2oao&o.
-6 V
21.
cos x . sin * x
22. 4,2Jrdd Lt3 ,m atoo, dajauaodo&rf ( Origin ) d;xo3 Soadialraertod <3273, ?Jddes3rt sis&jasod zJ&drad? (BejJdSako*' gesteF) disM.
3 x 5 = 15
I. AsSdjrt'tfS, oires$creddj3 5&& gjrtsfc sroD
23. a) 189 S&&, 243 d dDSo d ddFfjds ( G.C.D.) 3od>SoacO>D.
m, ne z wcrori, d? 189 m + 243 ncLodd, zodo&O. 3
b) 2000 iyprarod >a, &z&>zztfTi<$ ijk&j rio<fd6oS>.3odDSoacDO.
1 ab a+b 1 be b + c 1 ca c + a
24. a)
= ( a- b) ( b- c) ( c- a) >oc> 3
b) es&drf sfcaood & ?5d ri&draris? dOdd? 3odD8oao&0 :
x - 2y = 8 doii 2x - y = 7.
25. 0-{-l},-lrf srad 2jysrtoeora s>osJ6rW rtrarofld, * addJ3?S d,a&oo 0 - { - 1 } s&e?3 a*b=a+b+ ab, V a, beQ-{-l} wrori, (-{-l},* ) od: dodiraeo* xio&o >odi 5
5 Code No. 35
26. a) ot = 3 i + j - 2ic, 1? = - f + 3j + 4/c.
"c = 4-2 ? - 6 /c t?dd, "a rts?r1 s&;& "c
u fl ' _0
ooEosrsftdosj 3 tfocfc&aofoO. 3
b) A s ( 2, - 3, 6\) B = ( 3, - 1, - 6 ) sJairtvsfldosS aIj
tfoc&SoSofcO. 2
3s cxSj3srocjdj3 >dc& sris?rt : 2 x 5 = 10
27. a) x2 + y2 + 2g1x + 2/,y + c1 =00
x2 + y2 + 2g 2x + 2 f2y + c 2 = 0 srts* sfcjaercd sS&ietfdrasft
tfoc&SoQotoO sosrtjs <adc& 4 9 aockrttfci
rieo&sj riddeidrt oosoOTAd c3 <aoc> ;leO&. 3
b) ( 2, 1 ) &ocks$ Seodrtos *&;&, 3x + 4y - 5 = 0 zldtfdcslatotf tfj&F'&sS s3dd dras 3oc&&oScooO. 2
28. a) 16x2 + 9y 2 + 32x - 36y - 92 = 0 sto&fcscj 3oc&So80&0. 3
b) y2 = 4ax sddsjoo>d ,sbe)rf (x, , y } J aockaabd edd
( a, 0) rl ctosj dodsj Xj + a wftcfccSock &eOS>. 2
29. a) cos " 1 x - sin ~ 1 % = cos 1 ( x V3 ) zjoedfSS&foj E3Q&. 3
b) tan 2 x - 4 sec x + 5 = 0 d sjOaoadrf?* 3oc&SoQo&0. 2
30. a) siuaoaod x f\ firtoraOTf\ cosec ( 4x) 0d30A).
b) y = (tan x)sln 1 * add, 3oc&SoQo&0.
f 2x ' v 1 + x2 y
1 - x-1 + x:
31. a) cos 1
- l
71 fkrtrasroA sin
e?5 e>d30&. 3
b) y=x(x2-4) d y = 2x2 - 3x - 2 ddeajrts? rfc&arf Jedsidi),
aDdrt3j Iraerfds ( 1, - 3 ) zaodo&C), tfodo&o&o&O.
32. a) y = sin ( m tan 1 x ) fc?CT3ri,
( 1 + x2)y2 + 2x( 1 + x2)y1 + m2y = 0 >odi 3
dx
o--- d &5e3oisS> tfodksctoO.
b)
33. a)
b)
x - 6x + 8 04
X 1
7-0 V/ --5-r dx rf 23dO>?&, Odi2octo0.
(x-2)(x-3) 04
3x
-~j dx & zS&obdy SodiSo&o&O. 1 + 2x4 01
34. 'cktorf arasSSod x2 + y2 = a2 dd 3odiSoacft>0. 5
M
6
| ||||||||||||
|
4 J |
edd, djsfceo&ds erosdoioeA A ~ 1
3o<>&jS'cdO. 4
36. a) nrf ae.stodbecdb sLraorWrl a djseoid, sSscoisSrfj, Wfc do& 6 ( b) tan x + sec x = >/3 0> Trasto dBSoddc tfodi&oSO&O. 4
37. a) 170 3o.<&>e. ad&dod doF&fk 4 <&>?./tfdocs* dertdS, Ssdi), 2oOdi
aedd tfowaod clod 3doSd?&. 8-5 &e. ai.dd 3osod aedaod asrad eroorbd
i) fSdS?rf UUti d WdcadreOk dd s&&
* a -o
ii) Fods?s3 &Scdb 23of3o> dddrf tfodi&o&o&O. 6
b) ?jasj erosJolraedodD, oftsdjde efcs ABC C&,
a b c
V OdD rJa)pn).
sin A sin B sin
xtanx dx _ odD 3?0.
38. a)
sec x + cos x 4
o
b) ( 2y - 1 ) dx - ( 2x + 3 ) dy = 0 #i edO tf&oetfdrad dOSoSdd? tiOU&loQC&Q. 4
39. a) V3 + i rf $>3 skoort's* 0iSoSc0D0. ris tJrraroc 2&d< rbd&.: 4
b) x2 + y2-2x-2y-7 = 0 x2 + y2 + 4x+2y + 1=0 PTOS&fcol 233, d VOd d?k tiotij&oBO&Q. 4
O D < t
c) ?Jdide&?o);io> <Drf)od 32 x 127 x 44 7 Ood A&a3rt srooix>d esfcfc 3oc&Soao&0.
(qJcTO3J&!iS33ftb23e!fo) 2
40. a) odi watod Frao zoarttf sJjss dareftd. ts tso&d
rie3$e33rts33dd ssd: 2&S tfnrcftds&Sock JjseD. 4
b) J tan 4x (ksi s3eJo>? 30C&SoS0>0. 4
c) y = log 6 V cos x add, s3e3cdb tfoc&Soao&O. 2
Instructions: i) The question paper has five Parts * A, B, C, D and B. 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.
Answer all the ten question': 10x1 = 10
1. Find the number of incongruent solutions of 6x = 3 ( mod 15 ).
4 3
, find A + A
2. If A =
L 2 1 J
3. On R [ the set of all real numbers ] an operation * is defined by
a b = a + b + 5, V a, b g R. Examine whether * is a binaiy operation or not.
4. If aIj = 3i + 2 j + 6fc and OA = i - j - 3(c, find the position vector of B.
Code No. 35 10
5. Find the length of the tangent to the circle 3x2 + 3y 2 + 6x + 9y - 2 = 0 from the point P ( 1, 1 ).
6. If y = mx + c touches the parabola y 2 = 4ax, then write the coordinates of the point of contact.
n
3 " Sin
7. Evaluate : sin
8. Find the imaginary part of e171 .
9. If y = sin ( 2 sin 1 x) , find
r
10. Evaluate: J sin2 x dx.
o
PART - B
10 x 2 = 20
Answer any ten questions :
11. If a | be and the G.C.D. of a and b is 1, then prove that a \ c .
|
1 2 O' |
1 CO 1 r-4 (N i | ||
|
12. If A = |
and B = | ||
|
.-3 1 -2. |
.2 1 1 - |
, find AB 1
13. Construct the multiplication table for G = { 1, (fl, oo 2 } , where co is a cube root of unity. Find the inverse of co 2.
14. Prove that, in a group ( G, * ), ( a b ) 1 = b~ 1 * a~ 1 , V a, b e G.
AAA
15. Show that the points whose position vectors are 2 i - j + k ,
i - 3j - 5 /c and 3 i - 4j - 4 fc form a right angled triangle.
16. Find the equation of the circle which is described on the diameter whose end points Eire ( 5, - 7 ) and ( 7, - 5 ).
Tt
17. Prove that sin~ 1 x + cos~ 1 x = '
18. If x = cos A + i sin A, y = cos B + i sin B, then show that
xy - = 2 i sin ( A + B ). y xy
d 2y dx:
1 - cos X 1 + cos X .
19. If y = tan 1
, prove that A 2
= 0.
20.i Find the point on the curve y2 = 3 - 4x, where the tangent is parallel to
the line 2x + y - 2 = 0.
cos 2x
dx.
21. Evaluate :
cos 2 x . sin 2 x
22. Form the differential equation for the family of straight lines passing
through the origin having slope m.
1 ab a+b 1 be b + c 1 ca c + a
= (a-b)(b-c)(c-a). 3
24. a) Prove that
23. a) Find the G.C.D. of 189 and 243. Express it in the form
189 m + 243 n, where m, n e z.
b) Find the number positive divisors of 2000.
b) Solve by Cramers Rule : x- 2y = 8 and 2x- y = 7. . 2
25. Let Q - { - 1 } be the set of rational numbers except - 1 and * be a binary operation on 0 - { - 1 } defined by a * b = a + b + ab, V a, b e 0 - { - 1 } . Show that ( Q - { - 1 } , * ) is an Abelian
group. 5
26. a) Given, that the vectors ~a = 3 i + j - 2fc, = - i + 3 j + 4&
and ~c = 4 i - 2j - 6ic, find a unit vector coplanar with ~a and b but perpendicular to c. 3
b) Find the direction cosines of the vector A, where
2
A = ( 2, - 3, 6 ) and B = ( 3, - 1, - 6 ).
II. Answer any two questions : 2 x 5 = 10
27. a) Derive the equation of the radical axis of two circles
x2 + y2 + 2glx+2f1y + c1 = 0 and x2 + y2 + 2g2x + 2f2y + c2 =0.
Also show that the radical axis of the two circles is perpendicular to the line joining their centres. 3
b) Find the equation of the circle having its centre at ( 2, 1 ) and touching the line 3x + 4y - 5 = 0. 2
28. a) Find the foci of the conic 16x2 + 9y 2 + 32x - 36y - 92 = 0. 3
b) Show that the distance of any point ( x x , y x ) on the parabola y 2 = 4 ax from the focus ( a, 0 ) is x j + a. 2
29. a) Solve: cos- 1 x - sin" 1 x = cos 1 ( x V3 ) . 3 b) Find the general solution of tan 2 x - 4 sec x + 5 = 0. 2
III. Answer any three of the following questions : 3x5=15
30. a) Differentiate cosec ( 4x) with respect to x, using first principles.
3
b) If y = (tan x) sin 1 x , find . 2
[ Turn over
31. a) Differentiate sin ~ 1 1 - x2
1 + x-
2x
cos
b) Find the angle of intersection of the curves y=x(x2-4) and y = 2x2 - 3x- 2 at ( 1, - 3 ). 2
32. a) If y = sin ( m tan - 1 x ) , then show that
(l + x2)2y2 + 2x ( 1 + x2)y1 + m2y = 0. dx
3
2
dx.
b) Evaluate : 33. a) Evaluate : b) Evaluate :
x2 - 6x + 8 '
x- 1 ( x - 2 ) ( x - 3 )
3x
1 + 2x4
dx.
34. Find the area of the circle x2 + y 2 = a 2 by the method of integration. 5
2 x 10 = 20
Answer any two of the following questions :
35. a) Define an ellipse as a locus. Derive its equation in standard form
x
y
a
= 1, ( a > b ) . b) Find A-1 using Cayley-Hamilton theorem if A
10-7 -5 4
36. a) State and prove De Moivres theorem for all rational values of n. 6
b) Find the general solution of tan x + sec x = V3 .
15 Code No. 35
37. sO A man 170 cm tall, walks at the rate of 4 m/sec away from the
source of light which is hung 8-5 m above the horizontal ground.
i) How fast is the length of his shadow increasing ?
ii) How fast is the tip of his shadow moving ? 6
b) Prove for any triangle ABC, the sine rule
by vector method. 4
x tan x , rc2 dx =
38. a) Show that
sec x + cos x 4
b) Solve : ( 2y - 1 ) dx - ( 2x + 3 ) dy = 0. 4
PART - E
Answer any one of the following questions : 1x10=10
39. a) Find the cube roots of V"3 + i. Respresent them on the Argand diagram. 4
b) Find the length of the common chord of the two intersecting circles x2 + y2-2x-2y-7 = 0 and x2 + y2 + 4x+2y+l=0. 4
c) Find the remainder obtained when 32 x 127 x 44 is divided by
7 using the method of congruences.
( The remainder should be least positive ) 2
[ Turn over
40. a) The sum of the four sides of a rectangle is constant: Show that the area of the rectangle is maximum when it is a square. 4
b) Evaluate : J tan 4 x dx.
6x s 3 (drac 15) edra ddd ?jdr?osjDsjo,d >skj d03osdrts?d ?
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Attachment: |
| Earning: Approval pending. |
