Pre University Board 2008 P.U.C Physics, Chemistry, Maths
Karnataka second MATHEMATICS June 2008 Kan & ENG version is in Pdf Format Check tat beneath.
Total No. of Questions : 40 ] [ Total No. of Printed Pages : 16
Code No. 35
June, 2008
( Kannada and English Versions )
Time : 3 Hours 15 Minutes ] [ Max. Marks : 100
( Kannada Version )
0> t o%.
CO * c _D
ii) - a n 10 oxn>o, - b n 20 oxnsb, -
c n 40 oxn>o, - d n 20 oxn>o
7 _0
>mrt - e n 10 oxn-t
_0
- A
X>A 0> t 0 : 10 x 1 = 10
CO _D y oi oi _0
XoiSiO.
2.
4321 4322 4323 4324
4. A oo B ,a>ae(o i + j + 2k 3 i - 3 j + 2k
_o cp J _ J
wnart, P (oo AB o o.oos>Ado P ,a aeo ,QeSo Xodo&SoO.
7 ~ 6 co cp t
5. y-&X\X@ ,5FdoS oOo ( a, 0 ) SXeodS oadoS ,oeXdraSo XooSoO.
6. ( x + 1 ) 2 = - 4 ( y - 3 ) - a<o deZrt XooSoO.
7. cos - 1 ( sin 330 ) o dd 0o ?
8. 1, ro, ro 2 rt >o XX ort>aAQd, ( 1 + ro - ro 2 ) dd 0oj ?
9. y = e + x woart, djy Xodo&SoO.
10. | e x 1 1 + tan x + dx dd <oo. XooSoO. cos x + *
x>nS)n> o : 10 x 2 = 20
11. 352 oo 891 - o.,a.B>. ( G.C.D. ) So XooSoO.
_0 c
1 4
3 2
12.
<A
13. odo ort>o oadoS ,oXo<S OSraeos>Adorf 0oo srap&.
AAA AAA
14. i + 3j + 2k , 2 i - j + 3k doJo i + j + k ,d/3oJd Jo$or& doonn> $>>do odo dod<o oJ,rt>,d oort>3A d, -
* y co 6 & co
dod&SoO.
* ct
15. oo d - do ( 3, 2 ) do Jo a<oJ deZ x = 1 - ,Poeddrado dodoSoO.
2 tan - 1 *\J 1 + XX = V1 - x 2 0oo
16. sin
17. do> odoP do>d >doeribd do Jo y = x ,d>deZo doed deodo
art& x 2 + y 2 - 4x - 6y + 10 = 0 d* Jdo odA $eQ,od d*Jo
ct
18. ( 1 - i ) 9 = 16 - 16i 0oo
19. y = log I 1-COS X ) d = 2 cosec x 0oo
a te e V 1 + cos x ! dx
20. y 2 = x dd,deZrt 0> ,dd x-d\d@ 45 od dedo&od/Sdd,
dd;deZo doe ododrfo, dodoSoO.
1
21. J x ( 1 - x ) 7 dx d <oo dodoSoO.
0
22. ( y - 2) 2 =4a( x+1) ,Poeddrad Pdedort,
- C
23. a) 756 - ddX &Xrt> ,oZ6>0r dod B)rt> ddd0
XodoSO. 3
b) a/bc ddo ( a, b ) = 1 a/c 0odo srap&.
2
24. 3x + y + 2z = 3
2x - 3y - z = - 3
x + 2y + z = 4
,oeXdrart> d0adrt>0 XeS dd ood XodoSoO.
5
ct cp
25. dProX rtra z doe <00rt> dOSO * d a * b = a + b + 3, V a, b e z
0odo d>,z>,add, do odo doddraeo ,oXo< 0odo 5
6 6
26. a) a = i - 2j - 3k , b = 2 i + j - k dbd c = i + 3j
- 2k wAd d, cT n ,d/oddd)Add dd ,dddS bT
CO -D Cn
dodo c dxS aXd/ ,QSdd, XodoSoO. 3
_0 ct
AAA AAA
b) 2 i + j + k dodo i - 2j + 3k d 0ddo ,d/3Odd ddr&d XrarrtAd d, ,dz3Odd ddr&d erardd,
* CO 7 _0 ct
XodoSoO. 2
ii. x> nd)n> /d)ddd 0-0 dSnn do :
2 x 5 = 10
27. a) x 2 + y 2 + 2gx + 2/y + c = 0 ddX ( x 1 , y 1 ) ddodood
3
<*J j <*J CO c
b) 3x - 4y + 6 = 0 , d> deZrt oodAdd dd
x
ct
28. a) 9x 2 + 5y 2 - 36x + 10y - 4 = 0 SerSd 3$ od adod deZrt> eddrartrfo dod&SoO. 3
ot
14 4
b) 3$o = -3- dodo_o e = 3 doS dSdo ( Hyperbola x 2 y 2
) 2 - 7-0 = 1 ,oeddraSo dod&SoO. 2
a 2 b 2 01
n
29. a) tan - 1 x + tan - 1 y + tan - 1 z = 3rt ,
xy + yz + zx = 1 0O ,3. 3
2 5 b) sin 2 0 - cos 20 = 4 , oedd ra ,3/ OdSo dodoSdoO.
2
III. d>A d/S)3d .odo ,4,nn JoO : 3x5=15
30. a) x ot doOJo ax aSo do drto dodoSdoO. 3
b) y = tan - 1 ( 4 4X 2 ] W3rt , djy = 4 + 2 0oo ,3. 2
2 2
31. a.) y = ( sin - 1 x ) + ( cos - 1 x ) W3rt ,
( 1 - x 2 ) y 2 - xy 1 - 4 = 0 0O ,3. 3
b) x = 3 sin 20 + 2 sin 30 dodo
y = 2 cos 30 - 3 cos 20 W3rt ,
dy = - tan 0 0oo ,3an. 2
dx 2
X
32. a) y = ea X,deZ<D odeZ<DD a< deZ<D drtrX@ >eDOTAD dJ ,>FdeZ<DD oD A d ,oZ. wAdD
co o <*J cp > _o
0OD A. 3
n/2
1 f
b) | S|n X . C (S X dx dD. XoDSDO. 2
11 + sin 4 x 01
0
1 2
2 _ 3 tan x
33. a) | -j-~-t- dx d D. XoDSDO. 3
11 + 2 tan x *
1
b)
( 1 + e x ) ( 1 _ e x )
dx d <D, XoDSDO. 2
34. aers 9x 2 + 16y 2 = 144 d A/rar oX< >ao XoDSDO. 5
2 x 10 = 20
35. a) oD oDro>A dD ot6Z6 XS dDo d ,DeXdraD
x 2 y 2
o _ 7-9 = 1 d oD ad&A. 6
a 2 b 2
|
2 |
i) cos 2a + cos 2p + cos 2y = 0
sin 2a + sin 2p + sin 2y = 0
ii) cos 2 a + cos 2 p + cos 2 y = 2
sin 2 a + sin 2 p + sin 2 y = 33 0oo
6
c c c c c c c c c 2
b) [ a x b b x c c x a ] = [ a b c ] 0O 4
37. a) aoo rte> doed,. erar 8 .,o.oe. /,.o o,a w rte>
-e _o
500 n
nad 3 d-,o.oe. rte> doo nad Sodod
drt>o dodoSoO. 6
ot
b) sin 0 + sin 20 + sin 30 = 0 , oedd ,ad/, dOsaddo dodoSoO. 4
6 ot
n/2
38. a) J 1 , sin x cos x dx = 0o 6
2
cos 2 x n
+ sin x cos x 33
0
b) X> dd< , oedd ,ad/w dOsaddo dodoSoO :
dy 1 + y 2 2
xy dX = 17X2 1 1 + x + x 2 > 4
- E
X>A tp : 1x10=10
39. a) cT + I) + ~c = 0 dit | cT | =3, | ~b | = 5 dit | ~c | = 7
, a dot b do Xedi XodiSiO. 4
7 _D c
b) V3 - i ,o3erar ,oZ6i XodiSdi d)rt>ct nrora*
td rtidi. 4
y co
c) 2 202 ,oZ,2i 11 Ood $iod Xad deddi
6 ct ct
XodiSiO. 2
40. a) odi oXe Xerad Xrar dit odo dt d ,oZ,</Ad.
y -o -o cp 6
n
Xerad erar rtOdroneydd 0ddo roirt> do Xe 0odi
3
teO. 4
b) J cot 4 ( 3x ) dx d(i XodiSiO. 4
c) y = log 5 VT - x 2 x n , oopd ot d X. 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. Find the number of incongruent solutions of 9x = 21 ( mod 30 ).
4321 4322
2. Evaluate
4323 4324
3. In a group ( Z 6 , + mod 6 ) , find 2 + 6 4 1 + 6 3 1 .
4. Find the position vector of the point P which is the mid-point AB where
AAA AAA
the position vectors of A and B are i + j + 2k and 3 i - 3j + 2k .
5. Find the equation to a circle whose centre is ( a, 0 ) and touching the y-
axis.
6. Find the equation to directrix of ( x + 1 ) 2 = - 4 ( y - 3 ).
8. If 1, ro, ro 2 are the cube roots of unity, find the value of ( 1 + ro - ro 2 )
9. If y = e + x , find d- .
I
10. Evaluate | ex 1 1 + tan x j dx.
cos x
PART - B
Answer any ten questions.
10 x 2 = 20
11. Find the G.C.D. of 352 and 891.
1 4
3 2
12. Find the characteristic roots of the matrix
13. Prove that a group of order three is Abelian.
14. Find the volume of the parallelopiped whose co-terminus edges are the
A A A A A A
AAA
vectors i + 3j + 2k , 2 i - j + 3k and i + j + k .
15. Find the equation to the parabola whose focus is ( 3, 2 ) and its directrix
is x = 1.
16. Prove that
- x
17. Find the equation of a circle passing through the origin, having its centre
on the line y = x and cutting orthogonally the circle x 2 + y 2 _ 4x _ 6y + 10 = 0.
18. Prove that ( 1 _ i ) 9 = 16 _ 16i.
19. If y = log ( 1-cos x + , then prove that = 2 cosec x.
a e ( 1 + cos x ! dx
20. Find the point on the curve y 2 = x the tangent at which makes an angle
of 45 with the x-axis.
1
21. Evaluate J x ( 1 _ x ) 7 dx.
0
22. Form the differential equation by eliminating the arbitrary constant ( y _ 2 ) 2 = 4a ( x + 1 ).
I. Answer any three questions : 3 x 5 = 15
23. a) Find the number of positive divisors and sum of all such positive divisors of 756. 3
b) If a/ bc and ( a, b ) = 1, then prove that a/c. 2
24. Solve by matrix method :
3x + y + 2z = 3 2x - 3y - z = - 3
x + 2y + z = 4. 5
25. Prove that the set z of integers is an Abelian group under binary operation * defined by a * b = a + b + 3, V a, b E z. 5
26. a) If a = i - 2j - 3k , b = 2 i + j - k and c = i +
A A __
3j - 2k , find a unit vector perpendicular to a and in the same plane on b and c . 3
b) Find the area of a parallelogram whose diagonals are the vectors
AAA AAA
2 i + j + k and i - 2j + 3k . 2
II. Answer any two questions : 2 x 5 = 10
27. a.) Find the length of the tangent from the point ( x 1 , y 1 ) to the
circle x 2 + y 2 + 2gx + 2fy + c = 0. 3
b) Find the equations of tangent to the circle
x 2 + y 2 - 2x - 4y - 4 = 0, which are perpendicular to
9x 2 + 5y 2 - 36x + 10y - 4 = 0.
b) Find the equation to the hyperbola in the standard form
3
2 2 x 2 y 2
14
Or - b""2 = 1> given that length of latus rectum = -3- and
4
2
e = 3 .
29. a) If tan - 1 x + tan - 1 y + tan - 1 z = 2 > prove that
xy + yz + zx = 1.
3
2
b) Find the general solution of sin 2 0 - cos 20 = 4
III. Answer any three of the following questions :
3 x 5 = 15
3
- 1 1 4x , dy 4 1 1 -2 + > prove that dT = -
2
dx 4 + x :
4 - x
30. a) Differentiate ax w.r.t. x by first principles. b) If y = tan
31. a) If y = ( sin 1 x ) + ( cos 1 x ) , prove that
( 1 - x 2 ) y 2 - xy 1 - 4 = 0.
3
b) If x = 3 sin 20 + 2 sin 30, and
y = 2 cos 30 - 3 cos 20
dy
prove that dx =
. 0 - tan 2
2
a
square of the ordinate and subtangent is constant. 3
n/2
1 f
, , , . sin x . cos x
b) Evaluate I -4- dx. 2
+ sin 4 x
0
33. a) Evaluate | 23 tanx x. 3
+ 2 tan x
dx. 2
1
b) Evaluate
( 1 + e x ) ( 1 - e x )
34. Find the area of the ellipse 9x 2 + 16y 2 = 144 by integration. 5
PART - D
Answer any two of the following questions : 2 x 10 = 20
|
35. a) Define hyperbola as a locus and derive the standard equation of the | ||||||||||||||||||||||||
| ||||||||||||||||||||||||
36. a) If cos a + cos p + cos y = 0 = sin a + sin p + sin y , prove that
i) cos 2a + cos 2p + cos 2y = 0
sin 2a + sin 2p + sin 2y = 0
ii) cos 2 a + cos 2 p + cos 2 y = 2
2 2 2 3 sin 2 a + sin 2 p + sin 2 y = 2 . 6
b) Prove that [ a x b b x c c x a ] = [ a b c ] . 4
37. a.) The surface area of a sphere is increasing at the rate of 8 sq.cm/sec.
Find the rate at which the radius and the volume of the sphere are
500 n
increasing when the volume of the sphere is 3 c.c. 6
b) Find the general solution of sin 0 + sin 20 + sin 30 = 0. 4
n/2
1 2
1 cos x n
38. a) Prove that I -j-:--dx = = . 6
J 1 + sin x cos x 3yf3
0
b) Find the general solution of the differential equation
Answer any one of the following questions : 1 x 10 = 10
39. a) If a + b + c = 0 and | a | = 3, | b | = 5 and | c | = 7,
find the angle between a and b . 4
b) Find the cube roots of a complex number V3 - i and represent them in argand diagram. 4
c) Find the remainder when 2 202 is divided by 11 ( least positive remainder ). 2
40. a) The sum of the lengths of a hypotenuse and another side of a right
angled triangle is given. Show that the area of the triangle is
n
maximum when the angle between these sides is 3 . 4
b) Evaluate | cot 4 ( 3x ) dx. 4
c) Differentiate w.r.t. x :
y = log 5 V1 - x 2 . 2
9x = 21 ( mod 30 ) ,oeXdraX@ d,d 0>drt$ ?
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Attachment: |
| Earning: Approval pending. |
