Pre University Board 2009 P.U.C Physics, Chemistry, Maths & Biology Mathematics - Question Paper
Code No. 35
Total No. of Questions : 40 ] [ Total No. of Printed Pages : 16
March, 2009
( Kannada and English Versions )
Time : 3 Hours 15 Minutes ] [ Max. Marks : 100
( Kannada Version )
: i) A, B, C, D E aco@ dartrtd
0> dartrt>d t o%.
CO * c _D
ii) - a n 10 oxn>b, dan - b n 20 oxnsb, dan -
c n 40 oXrt>o, dan - d n 20 oxn>o dt 7 _0
dan - e n 10 oxn-t d
_0
- A
X>A 0a d4rt>d t 0 : 10 x 1 = 10
CO _D y oi oi _0
1. 7 30 ,oZ,<dd 5 0o daAart $do Xa XodSoO.
> c * * c
4 x + 2 2x - 3 x + 1
/tX wd, x dddd. XodaoO.
2.
3. ,oXbod Xod&SoO.
ot
AAA
4. 2 i - 3j + 2/c 0/ra aa Xartd XodSoO.
5. x 2 + y 2 + 4x - 2y - k = 0 dJd 4 dod/rt>ad, k do d o ?
6. ddddo a$ ( 2, 3 ) do Jo dort ( 4, 3 ) wd, dd ,&oeddradb. OodoSdoO.
2 cos 1 ( - 1 )
d d ao ?
7. sin
8. 1, ro, ro 2 n >0 OJ do ddort>add, ( 1 - ro + ro 2 ) d d ?
9. x n rtorad3A 3 x sinfr x b,
ot
10. x n rtoraaA 4cos 2 o ,/osft.
+ cos 2x 01
- B
X>Ad)rt>, d/>d)add d4(rt>o J0ft
10 x 2 = 20
11. a = b ( mod m ) WAd, n 0OoD m 0 a = b ( mod n )
0oo ,aft.
|
= 0 0O J>eOft. |
13. G = { 0, 1, 2, 3 } d/do,e 4 rto}ayadSY odo ,odode ? yadra Xzs.
14. x + y = 6 , x + 2y = 4 d)J 0ddo drtO doo 10 dod/
rt>hf\ti d - aoeXdradd XodoaoO.
co ct
AAA A A A
15. 2 i - j + k do Jo 3 i + 4j - k , d/3o Jd Jo$oF& Xrarrt> ,art>3d, d erardd XodaoO.
_0 c
16. atioJ deZrt> doa oJd 5 do Jo 3$rt> doa oJd 4 wAd,
_o * co 7
aerdd ( a > b ) eodtio XodoaoO.
5n
17. cot - 1 x + 2 tan - 1 x = -pr- , SoeXdrad o a.
6
18. , 1 - ii / = 1 d , n Xad dP}>FoXdeo ?
19. y = ( x + >/1 + x 2 ) m W3rt ( V1 + x 2 ) djy - my = 0 0odo ,3.
x
20. y = bea dX,deZrt, doodeZtioo y arferXd drtrX@ oetiodaAdoJrf 0oo J>eO.
e
. /
21. J log e x dx tioo XodoaoO.
e 1
3
d 2 y
2
dx 2 d X< ,aoeXdra {r doo dd/rari
22.
- C
I. odo JOb : 3 x 5 = 15
23. adoy >b, a = 495, b = 675 ,oZ,rt> .,>..
co * * <=i 1 6
XoadoO >rt , 495 ( x ) + 675 ( y ) dd0 iddDO
<=i cv/ v ' CO
j x. y e z x j y n> aedo 0o J>eob. 5
24. Xe doo X>A ,d> ,eXdrart> dOdrt, XoadoO : 5
Cp
3x + y + 2z = 3 2x - 3y - z = - 3 x + 2y + z = 4
25. a) ,oz>w rtra0Y * Sz/ dOddh a * b = yja2 + b2 ,
a, b e K 0o dzabrort , * dOJraedD rtora, ,JraedD rtora ado , d30,do Jeob j>ocr XobabO. 3
b) ( G, * ) , oXo< a doo dzdrfe o oOTfi), ( a - 1 ) 1 = a 0o ,2>pb. 2
AAA AAA
26. a) i - 2j + 3/c J 2i + j + /c ,art> Xe
Xo&ao. 3
A A A A A A A
b) j + 2/c , i - 3j - 2/c - i + 2j ,art>,
sortd ,art>R d&,Jrf 0o JeOb. 2
II. X> Ad)rt> (zd)d,dd 0- 3$ _p : 2 x 5 = 10
27. a) x 2 + y 2 + 2 g 1 x + 2 f 1 y + c 1 = 0 dob
x 2+ y 2 + 2 g 2 x + 2 f 2 y + c 2 = 0 0ddb drtb
oodrt>Adbd aodbb 3
b) 2x 2 + 2y 2 + 2x - 3y + 1 = 0 dbb
x 2 + y 2 -3x + y + 2 = 0 0ddb drt> dbd deod,rt> aodbrt>b ,eO,d ,d>deZrt oodftd 0odb 3e0ft. 2
ot
28. a) y 2 - 4y - 10x + 14 = 0 ddddd od d odbrt> ader3drt >b db3b <dd abeddradd, dodbSbO. 3
cR _0 cR
b) x - 2y + k = 0 ,d>deZ<db x2 +2y2 =12 QeFdX ,3FdeZ</d>rt k dd, dodbSbO. 2
29. a) tan 1 x + tan 1 y + tan 1 z = n WOTrt x + y + z - xyz = 0 0odb 3eOft. 3
b) tan 40 = cot 20 , abedd rad ,d/6 dOddb, dodbSbO. 2
III. X>ft /d)ddd dodo drtn : 3x5=15
30. a) db< 33,aod x n ribrad>ft tan x , ddSft. 3
2 + 3x 2
3 - 2x 2
- 1
b) y = tan
wdrt , dy = _ 2x 4 0od 2
dx 1 + x 4
( 1 - x 2 ) y 2 - xy 1 + p 2 y = 0 0O 3
b) y = x 2 + 7x - 2 di,deZ<do y-Bifi O doi
>oertod <odeZdo ,oeidrado iodoSoO. 2
ot
32. a) x n e 3x f 3 + tan x j 3
V cos x j
b) 4y = x 3 doOo y = 6 - x 2 0ddo di,deZrt> do Xedo ( 2,
2 ) ooy iodoSoO. 2
33. a) x m y n = ( x + y ) m + n , x djy = y 0OO 3
b) x n rtorad>A ---- 2
7 - 6x - x 2 01
34. y 2 = 6x do3o x 2 = 6y 0ddo di,deZrt> do oerarOt iodoSoO.
5
- D
X>A d/di3d 0dd 0 : 2 X 10 = 20
a) oo ood>A dZw XS doo Bd
x 2 y 2
cDoeidraddo - 7-9 = 1 d 6
d a 2 b 2
b 2 + c 2 ab ac
b) ba c 2 + a 2 bc = 4a 2 b 2 c 2 0oo sran). 4 ca cb a 2 + b 2
36. a) cos a + 2 cos p + 3 cos y = 0, sin a + 2 sin p + 3 sin y = 0 WOdrt
i) cos 3a + 8 cos 3p + 27 cos 3y = 18 cos ( a + p + y )
ii) sin 3a + 8 sin 3p + 27 sin 3y = 18 sin ( a + p + y ) 0oo 6 b) [ a + b b + c c + a ] = 2 [ a b c ] 0oo 4
37. a) oo rte> ,Xoart 4n ,o.aoe. rte>
doed aerar drt>o XodaoO doJo rte>
288n c.c. - doed, Xodo&aoO. 6
b) V3 tan x = V2 sec x - 1 , aoeXd ,ddra,w Odrt>o Xodo&aoO. 4
n/4
n
38. a) / log ( 1 + tan x ) dx = 8 log 2 0oo sran). 6
0
b) tan y dy = sin ( x + y ) + sin ( x - y ) X< ,aoeXdra Odo Xodaoo. 4
- E
X>A Job : 1x10=10
39. a.) 3 - i V3 ,oZ6o >doiort>o XodaoO doJo dooood
rtora o Xodo&aoO. 4
CO
a a 2 a 2 a 2 a a 2
b) ( a x b ) = a b - ( a . b ) 0oo sran). 4
c) x 2 + y 2 - 6x - 2y + 5 = 0 doJo x - y + 1 = 0 deZ )rto ;$o {>, Xooaoo. 2
<yJ & CO c
3
y/x + 2
40. a)
4
y/x + 2 + V5 - x
dx d doo OooSoO.
ot
0
b) 0da wdoJrt> ,oJo>Jdob rtod ft/rad, iEOdanoJo 0oo
J>eOft.
4
c) x n ortoradaA sec ( 5x ) 0 ot douft.
2
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. Find the least positive remainder when 7 30 is divided by 5.
4 x + 2
is a symmetric matrix, find x.
2. If
2x - 3 x + 1
3. Define a subgroup.
4. Find the direction cosine of the vector 2 i - 3 j + 2k
5. If the radius of the circle x 2 + y 2 + 4x - 2y - k = 0 is 4 units, then
find k.
6. Find the equation of the parabola if its focus is ( 2, 3 ) and vertex is
( 4, 3 ).
2 cos 1 ( - 1 )
7. Find the value of sin
8. If 1, ro, ro 2 are the cube roots of unity, find the value of ( 1 - ro + ro 2 )
9. Differentiate 3 x sinfr x w.r.t. x.
t T x x /1 - cos 2x
10. Integrate \fa-w.r.t. x.
1 + cos 2x
PART - B
Answer any ten questions :
10 x 2 = 20
11. If a = b ( mod m ) and n is a positive divisor of m, prove that
|
a = b ( mod n ). | |||||||||||||
|
= 0. | ||||||||||||
13. Is G = { 0, 1, 2, 3 } , under modulo 4 a group ? Give reason.
14. Find the equation of two circles whose diameters are x + y = 6 and
x + 2y = 4 and whose radius is 10 units.
15. Find the area of the parallelogram whose diagonals are given by the
AAA A A A
vectors 2 i - j + k and 3 i + 4j - k .
16. Find the eccentricity of the ellipse ( a > b ), if the distance between the directrices is 5 and distance between the foci is 4.
5n
17. Solve cot - 1 x + 2 tan - 1 x = -pr- .
6
18. Find the least positive integer n for which , "j~"y 1 = 1-
19. If y = ( x + V1 + x 2 ) m , prove that ( V1 + x 2 ) djy - my = 0.
x
20. Show that for the curve y = be a the subnormal varies as the square of the ordinate y.
e
J
21. Evaluate J log x dx .
e
1
22. Find the order and degree of the differential equation
3
d 2 y dx 2
PART - C
I. Answer any three questions
23. Find the G.C.D. of a = 495 and b = 675 using Euclid Algorithm.
Express it in the form 495 ( x ) + 675 ( y ). Also show that x and y are not unique where x, y E z. 5
24. Solve the linear equations by matrix method : 5
3x + y + 2z = 3 2x - 3y - z = - 3 x + 2y + z = 4
25. a) On the set of rational numbers, binary operation is defined by
a b = y/a 2 + b 2 , a, b E R, show that is commutative and associative. Also find the identity element. 3
b) If a is an element of the group ( G, ), then prove that
, - 1
( a 1 ) = a. 2
AAA
26. a.) Find the sine of the angle between the vectors i - 2j + 3k
AAA
and 2 i + j + k . 3
A A
b) Show that the vectors j + 2k , i - 3 j - 2k and - i + 2j form the vertices of the vectors of an isosceles triangle. 2
II. Answer any two questions : 2 x 5 = 10 x 2 + y 2 + 2 g 1 x + 2 f 1 y + c 1 = 0 and
x 2 + y 2 + 2 g 2 x + 2 f 2 y + c 2 = 0 to cut orthogonally. 3
b) Show that the radical axis of the two circles
2x 2 + 2y 2 + 2x - 3y + 1 = 0 and
x 2 + y 2 - 3x + y + 2 = 0 is perpendicular to the line joining
the centres of the circles.
2
28. a) Find the ends of latus rectum and directrix of the parabola
y 2 - 4y - 10x + 14 = 0. 3
b) Find the value of k such that the line x - 2y + k = 0 be a tangent to the ellipse x 2 + 2y 2 = 12. 2
29. a) If tan - 1 x + tan - 1 y + tan - 1 z = n, show that
x + y + z - xyz = 0.
3
b) Find the general solution of tan 40 = cot 20.
2
III. Answer any three of the following questions :
3 x 5 = 15
3
2
2 + 3x 2 L 3 - 2x 2
- 1
b) If y = tan
x
30. a) Differentiate tan x w.r.t. x from the first principle.
dy 2x , prove that d = -
dx 1 +
31. a) If y = cos ( p sin 1 x ) , prove that
( 1 - x 2 ) y 2 - xy 1 + p 2 y = 0. 3
b) Find the equation of the normal to the curve y = x 2 + 7x - 2 at
the point where it crosses y-axis. 2
32. a) Integrate e 3x - 3 +o!gan x j w.r.t. x. 3
b) Find the angle between the curves 4y = x 3 and y = 6 - x 2
at ( 2, 2 ). 2
33. a) If x m y n = ( x + y ) m + n , prove that x djy = y. 3 b) Integrate 76x~2 w.r.t. x. 2
34. Find the area between the curves y 2 = 6x and x 2 = 6y. 5
PART - D
Answer any two of the following questions :
2 x 10 = 20
35. a.) Define hyperbola as a locus and hence derive the equation of the
2 2 xy
hyperbola in the form "2 - b""2 = 1. 6
b 2 + c 2 ab ac
ba c 2 + a 2 bc
= 4a 2 b 2 c 2 . 4
b) Show that
ca cb a 2 + b 2
36. a.) If cos a + 2 cos p + 3 cos y = 0, sin a + 2 sin p + 3 sin y = 0,
show that i) cos 3a + 8 cos 3p + 27 cos 3y = 18 cos ( a + p + y )
ii) sin 3a + 8 sin 3p + 27 sin 3y = 18 sin ( a + p + y ).
6
b) Prove that [ a + b b + c c + a ] = 2 [ a b c ] . 4
37. a) The volume of a sphere is increasing at the rate of 4n c.c./sec. Find
the rate of increase of the radius and its surface area when the volume of the sphere is 288n c.c. 6
b) Find the general solution of tan x = sec x - 1. 4
n/4
38. a.) Show that J log ( 1 + tan x ) dx = 8 log 2. 6
0
b) Solve the differential equation
tan y djy = sin ( x + y ) + sin ( x - y ). 4
PART - E
Answer any one of the following questions : 1 x 10 = 10
39. a) Find the cube roots of 3 - i V3 and find their continued product. 4
, c a 2 a 2 a 2 a a 2
b) Show that ( a x b ) = a b - ( a . b ) . 4
c) Find the length of the chord of the circle
x 2 + y 2 - 6x - 2y + 5 = 0 intercepted by the line x - y + 1 = 0. 2
3
y/x + 2
40. a) Evaluate
dx.
4
Vx + 2 + y/5 - x
0
b) Show that among all the rectangles of a given perimeter, the square
has maximum area.
4
c) Differentiate sec ( 5x ) 0 w.r.t. x.
2
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Attachment: |
| Earning: Approval pending. |
