Pre University Board 2009-2nd Year P.U.C Physics, Chemistry, Maths & Biology MATHAMETICS (KAN & ENG version) IN Pdf - Question Paper
Code No. 35
Total No. of ques. : 40 ] [ Total No. of Printed Pages : 16
June/July, 2009
MATHEMATICS
( Kannada and English Versions )
Time : three Hours 15 Minutes ] [ Max. Marks : 100
PAPER IS IN PDF FORMAT BELOW
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) A, B, C, D E 00
CO c _D
ii) - a n 10 odrt>, - b n 20 oxn>o, -c n 40 oxn >o, - d n 20 oxn>o doto
7 _0
>mrt - e n 10 oxndot d.
_D
- A
X>A 0> d.!)rt>o, t 0 :
10 x 1 = 10
CO _D y oi oi _0
1. 3x = 2 ( mod 6 ) ,doe&eot rt d0>d<Y- ?
12
2. a <o d,dX, X), ,n >0 ( Direction cosines ) tj- ,75- dot n rt>dd n
33
3. dp}oxn> rtra I * 3/oo Zw a * b = a b , V a, b e I Wftd.
Sooo Od/ 3/o<oe I doeSode 0oo d0e3.
4. A, B rt >o ode dOd/ra ( Order ) dod 0ddo drtr d/SjXrto. | A | = 4, | B | = 5 Wd, | AB | (oo dodoSoO.
5. 0ddo d*S rt> Xeodrt> dodra d>d d wAdo , r , , r 2 rt>o &&.rt>3Ad d, w
d f > co 12 -6 co
d)Sort>i ododo FOTrtesd ( External touch ) (
Condition ) doO.
6. 4x 2 + 9y 2 = 36 d doeSdod (/>4e o d>drt> dSo dodoSoO.
7. sin - 1 ( sin 130 ) (o dd(oo dodoSoO.
ot
( 1 - i + n
8. I 1 + i ) = 1 wrtodoS n dad dProd d(oo dodoSoO.
9. Sj f(x)=|x| WAdd, Lf'( 0 ) dodoSoO.
n/4
10. J ( sin 3 x + cos x ) dx d(oo dodoSoO.
- n/4
- B
(/rodd 0o d4tn>o Son : 10 x 2 = 20
11. ca = cb ( mod m ) wAdo c, m rt >0 ,ded, ,oZ,rt>3dd a = b ( mod m
K co <*A 6 6
) 0oo ,>pn.
cos 0 sin 0
WOTrt , AA 1 (oo , doo ( Symmetric ) d/SXoe
12. A =
3 Code No. 35
13. d,odDdDt ,od< d/dde 5# ( + 5 ) dbD { 1, 2, 3, 4 }
rtrad dodDKde 0od dOeS.
14. Q + ( ,o<Zrt> rtra) rtrad * 00 3/xb<D dZ a * b =
a b
3 , V a, b e Q + Ad. >oS dDD Q + , a - 1 Dr dodDSDO.
A A
15. X i + j + 2k , 2 i - 3j + 4k dDD i + 2j - k rt>0 , dD , QS n >Ad ( Coplanar vectors ), X d d<dDD dodDSDO.
16. ( 0, 0 ), ( 3, 0 ) d2b ( 0, 4 ) n >d sortn>3Acbd ,$D&d dOdd ,DeddradD dodDSDO.
17. S : tan - 1 x = sin - 1 -r - cot - 1 1 .
V2 3
i tan - 1 3
18. 5e 3 0o ,oZ6D d3,d ( Real ) dDD 6 ( Imaginary ) rtrt>D dddA 3, 4 0o >eO.
V' - + I i~
/ i- \
x - 1 - 1 Vx + 1
dy
19. y = sin - 1 - + sec - 1 - wdd, d~ = 0 0odo
\yjx + 1 / Wx - 1 '
20. x m y n = am + n 0o dd,deZ<dD <d/>d) ody 0> dSrdd oD ( Abscissa ) ddrtd 0oD
21. J [ sin ( log x ) + cos ( log x ) ] dx d dDD dodDSDO.
22. y-Bdd Dr dD<aob>< ( Origin ) ,&F,od drt> dd< , Deddrad Dr ( Differential equation ) ddDO.
- C
I. X>Ad)rt>, d/d)3d drt>o Oft : 3 x 5 = 15
23. a) a, b d>ra3FoXrt> dD.,).1. ( GCD ) dD dzn). 275 dDD 726 d
dD.,).. XodDSDO. 3
b) 252 D ,oZ,rt> rtDK dA dD - m&XrtdD,
ct * 6 6 C? ct 44 0
0oD XodDSDO. 2
24. Xe d d dD >Qod : 2x - y = 10
cp 4
x - 2y = 2
S-Dd ,)O Xrt> Xed) yS-Dq d,doedDd0 ( Satisfies ) 0O dOeS. 5
25. V a, b e H, ab - 1 e H W)rt, G , oXo< 6d<, drtrad)d H rtd) G dD d,oXD<dodD ,>a&. dD d<>eA H dDo K n> G dD
* c _D
d , oXD<n>)d d H I K G do d,oXD<doD ,>&. 5
AAA A A A
26. a) a = 2i + j + k , b = i + 2j* - k 0OD Xljrt a n
od)AdDd dDo a , b n> ,dD< dod
0 CO
( coplanar with a and b ) &X ( Unit vector ) dD
XodDSDO. 3
b) a + b + c = 0 Wdd,
II. X> Ad)rt> (radddd 0do drtrt 0 : 2 x 5 = 10
27. a) x 2 + y 2 + 2 g 1 x + 2 f 1 y + c 1 = 0 do:
x 2 + y 2 + 2 g 2 x + 2 f 2 y + c 2 = 0 drt >0 odA $eQ,:d aod 3
b) dd: de)1(rt> d>X\ Xeodd ( 1, 2 ) wAd. od: dd ,:eddrad)
x 2 + y 2 - 2x + 3y = 0 WAdd, Xeod.,d) drt> >n d ddtf doe 0odo does. 2
28. a) 9x 2 + 4y 2 - 18x + 16y - 11 = 0 ( Xeod, d: dd ,>(:d
( Auxiliary ) dd erd XodoSO. 3
b) d d d <( x = 2t 2 , y = 4t ( a(:d ( Directrix ) ,:eddrad:R Xod:S:0. 2
n
29. a) sin - 1 x + sin - 1 y + sin - 1 z = Wd"3rt
x 2 + y 2 + z 2 + 2xyz = 1 0od 3
b) tan 20 tan 0 = 1 deXdrad ,d/6 d0>dd: Xod:a:0. 2
III. X>A (/d)ddd dodo drtrt 0 : 3x5=15
30. a) doaod x n ,oopdo sin 2x : aa ( Differentiate ). 3
b) x n ,oodo adjR : (sin x )logx 2
31. a) cos - 1 ( 4x 3 - 3x ) d: cos - 1 ( 1 - 2x 2 ) rt ,oon)doJ
aSi. 3
b) d XxdeZrt >d y = 6 + x - x 2 do y(x-1) = x+ 2 (2,4)
od: ,.5r,: d 0od: eo. 2
CO vJ 0
( 1 - x 2 ) y 2 - xy 1 + m 2 y = 0 0O 3
32. a) y = sin ( m cos - 1 x ) wd,
-1- dx d<oo XoosoD. 2
b)
x ( x 5 + 1 )
33. a) x n ,000J X0I ( Integrate ) sin x + 18 cos x
3
3 sin x + 4 cos x
b) J \/ -x dx d <oo XooSoD. 2
34. oX< ( Integration ) >ao x 2 + y 2 = 6 dJd erardo
XooSoD. 5
- D
X>A 0- JD : 2 X 10 = 20
x 2 y 2
35. a) y = mx + c ,d>deZ<oo 2 - 7-9 = 1 dddoX,
a 2 b 2 *
,FXdarto S0oo ( Derive ). ,Foodo XodoSoD. aooo defift x - y + 5 = 0 deZrt x 2 y 2
,d/oJdd>rtodoJ 16 - 12" = 1 X4 ,jrXrt> ,oeXdrart>o
XodoSoD. 6
1 a a 2 + bc
b) 1 b b 2 + ca = 2 ( a - b ) ( b - c ) ( c - a ) 0o ,)$&. 4
1 c c 2 + ab
36. a) Q deo d,doeodo dOft. >dd) d, Uora dP}>rodrt>Adod>rt
ddoeidod ,$&. do d<d>eAft
Z 10 - 1
Z = cos 0 + i sin 0 Wd>rt ZT0-1 = i tan 50 0od 6
b) cos 20 = V2 ( cos 0 - sin 0 ) ,oeddrad jd/s dOd>ddo dodoQoO. 4
37. a) odo ne> d <d) ddd dd 4n c.c./sec. Ad. d<d) 288 n
d>rt 3, doed >ft,erarrt>b ddd ddrt dodoQoO. dod
(i) 5 ,dodort > dddS wrtod d,d>., dodo (ii) dd 288 n >ri
v ' co * co 66 _ov,*t
'IxKjX , oopftdod d<d) ddod ddrt dodoQoO. 6
b) ( 1, 5 ) ddo ( 1, 1 ) odort> >$ ortAdod dddodod ,oeddrart>orf dodoQoO. 4
ot
I a
x dx_ = n 2
cos 2 x + b 2 sin 2 x 2ab
38 a) I - . 2 2 v . u 2 2 -- = 0od 6
0
b) xy ( 1 + x 2 ) djy - y 2= 1 d BddK ,oeddrad ( Particular )
d Od>ddo dod&QoO : x = 1, y = 0 0odo dd. 4
- E
d>A /d)d>dd o d, d_pft : 1x10=10
39. a) | + b + c | = | a + b - c | wdd, a + b dodo c rt>
dodra de d o dodoQoO. 4
ot
b) oo >to Xrardod 0 <oXe ,|,$ort>S, ,doQzOTo
*|x$o&d erard) rt0dd>Adot 0oo ,a. 4
1
c) ' 16 cis 2 ) 4 Xod&SoO. 2
40. a) 2 150 x 3 12 x 135 = a ( mod 7 ) wd, a oo 7 0o >A,od3rt rtod Xad eddo XodosoO. 4
* ot
2 2
b) 2 ( x 2 + y 2 ) - 12x - 4y + 10 = 0 doto.
x 2 + y 2 + 5x - 13y + 16 = 0 de)tn> ,d/, { do
XodoSoO. 4
2
yfx
v dx oo, XodoSoO. 2
c)
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. 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 1 .
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 dy- = 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 ,
y = 4t.
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
1
dx.
2
b) Evaluate
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
I 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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