Pre University Board 2008 P.U.C Physics, Chemistry, Maths & Biology " Mathematics " Code 35 for English version see page no 9 - Question Paper
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 )
: i) A, B, C, D E 00 a>rtrt$.
co * c _o
ii) an - a n 10 oxn>b, an - b n 20 oxrt>, an -c n 40 oXrt>o, an - d n 20 oXrt>o o3o
7 _D
_D
- A
X>A 0> 3 On : 10 x 1 = 10
CO _0 y oi oi _0
oo XooaoO.
2.
4321 4322 4323 4324
4. A doo B oort> ,3>eCo i + j + 2k 3 i - 3 j + 2k
_o cp J _ J
P Coo AB Co do,ooro>Ado P ,3 aeCo ,aedb, dodo&SoO.
7 ~ 6 co cp ct
5. ,j5rdod doOo ( a, 0 ) sXeodo oadod ,oeddrado dodoSoO.
6. ( x + 1 ) 2 = - 4 ( y - 3 ) - aCo deZrt dodoSoO.
7. cos - 1 ( sin 330 ) Co dd 0do ?
8. 1, ro, ro 2 >0 dd dort>3Ad, ( 1 + ro - ro 2 ) dd ?
I
10. | e x 1 1 + tan x + dx dd Coo. dodoSoO. cos x + *
- B
X> C/)OTd 0o d)ctn>oct O : 10 x 2 = 20
11. 352 doo 891 d do.,3.. ( G.C.D. ) dodoSoO.
_0 c
1 4
3 2
12.
<A ot
13. dodo ort>o oadod ,odo<) dOdraeCoOTftdorf 0oo srap&.
AAA AAA
14. i + 3j + 2k , 2 i - j + 3k doo i + j + k ,d/3Od o$or& doonrt> droo odo ddo o,rt>o)d oortTOAd d, -
* y co 6 & co
dddo Xodosoo.
* t
15. odo d - dod ( 3, 2 ) doOo deZ x = 1 Wdrt, - ,oeddrado XodoSoO.
2 tan - 1 *\J 1 + XX = V1 - x 2 0odo
16. sin
17. d odo db<X >doeribd doo y = x ,d>deZ<o ded Xeod;do
ni x 2 + y 2 - 4x - 6y + 10 = 0 ddrfo oodA $ea,od dd
ot
18. ( 1 - i ) 9 = 16 - 16i 0odo
19. y = log I 1-COS x ) Wd3>rt, dy = 2 cosec x 0O
a te e V 1 + cos x ! dx
20. y 2 = x dX,deZrt 0>d ,jXd) x-X\X@ 45 (dd Xedrfood/add,
dX;deZo de ododo XodoSoO.
1
21. J x ( 1 - x ) 7 dx d<do XodoSoO.
0
22. ( y - 2) 2 =4a( x+1) ,oeXdrad ddo >dedort$&, , oeXd rad o XodoSoO.
ot
- C
23. a) 756 - dJX >&Xrt> OZS
XodSDO. 3
b) a/bc dD ( a, b ) = 1 a/c 0OD srap&.
2
24. 3x + y + 2z = 3
2x - 3y - z = - 3
x + 2y + z = 4
,DeXdrart> O>drt>D Xe4 do XodSdDO.
5
t cp
25. 4p}roX rtra z de <DDrt> OSD *bla*b = a + b + 3, V a, b E z
0O ro>,z>,ad, d od OJraeD ,oXd< 0Od 5
6 6
26. a) a = i - 2j - 3k , b = 2 i + j - k c = i + 3j
- 2k d, cT n ,dz3oJdOTAdD dJD ,dJS b*
CO _D CO
dJD c de Xdz ,a4o XodSdO. 3
_0 c
AAA AAA
b) 2 i + j + k dJDo i - 2j + 3k 0dd ,d/3OJd
Jd$df& Xrarrt>3A d, ,draoJd Jd$df& erard
* CO 7 _D c
XoDSDO. 2
ii. x> nd)n> <dzdd 0-0 44,rt$rt jo :
2 x 5 = 10
27. a) x 2 + y 2 + 2gx + 2/y + c = 0 4)JX@ ( x 1 , y 1 ) dao>ao
,4fX d. Xod&SdO.
3
<*J j <*J CO t
b) 3x - 4y + 6 = 0 , d> deZrt oro>AdD dJo
x
, DeXdrart >d XodSO.
ot
28. a) 9x 2 + 5y 2 - 36x + 10y - 4 = 0 QeFOd Ooo ><o deZrt> ,Poeddrart>o doOoSoO. 3
ct
14 4
b) <0 = -3- Oo e = 3 doO OdOoi ( Hyperbola x 2 y 2
) w4f dO 9 - 7-9 = 1 ,PoeddraOo doOoSO. 2
a 2 b 2 01
n
29. a.) tan - 1 x + tan - 1 y + tan - 1 z = ,
xy + yz + zx = 1 0O sraQn). 3
2 5 b) sin 2 0 - cos 20 = 4 , Poedd ra 00d0o dooSoO.
2
III. d>A .odo 04 rtrt O : 3x5=15
30. a) x o doOo ax O rt$0 dooSoO. 3
b) y = tan - 1 ( 4 4X 2 ] WOTrt , djy = 4 + 2 0oo ,3Q. 2
2 2
31. a.) y = ( sin - 1 x ) + ( cos - 1 x ) W"3rt ,
( 1 - x 2 ) y 2 - xy 1 - 4 = 0 0O sraQft. 3
b) x = 3 sin 20 + 2 sin 30 Ooo
y = 2 cos 30 - 3 cos 20 ,
dy = - tan 0 0oo sraQ&. 2
dx 2
X
32. a) y = ea d,deZ<d oodeZdo ><dJ deZd rtrX@ ddeddAd d J ,,FdeZ<do o & d ,oZ. wAdoJd
co o <*J cp > _o
0OD ,3>a- 3
n/2
1 f
b) | S|n X . C (S X dx d<d. dodaoO. 2
11 + sin 4 x 01
0
1 2
2 _ 3 tan x
33. a) | -j-~-t- dx d dodaoO. 3
11 + 2 tan x *
dx d dodaO. 2
1
b)
( 1 + e x ) ( 1 _ e x )
34. aeFJ 9x 2 + 16y 2 = 144 d erar od< ao dodao. 5
2 X 10 = 20
35. a) o dd Zk Xa dd jeddra x 2 y 2
o _ 7-9 = 1 d4 o 6 a 2 b 2
|
2 |
36. a) cos a + cos p + cos y = 0 = sin a + sin p + sin y waart
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 ] 0OO 4
37. a) oo rte> 4oed,. berar 8 .,o.oe. /,.oJ w rte>
-e _o
500 n
3 4.,o.oe. rte> na,4) Sodo
drt>o Xodoaoo. 6
ot
b) sin 0 + sin 20 + sin 30 = 0 , oedd Osado dodoaoO. 4
6 ot
n/2
38. a) J 1 , sin x cos x dx = 0O srab. 6
2
cos 2 x n
+ sin x cos x 33
0
b) X> Xo , oedd Osad0 Xoo&SoO :
dy 1 + y 2 2
xy dX = 17X2 1 1 + x + x 2 > 4
- E
x>a dn 3oOn : 1x10=10
39. a) cT + I) + ~c = 0 do3 | cT | =3, | ~b | = 5 do3 | ~c | = 7
Wdrt , a do3o b doa Xedo XodoSoO. 4
7 _D c
b) V3 - i ,o3erar ,oZ6o XodoSdo d)rt>ct wnrod
3d rtodon. 4
J cr>
c) 2 202 ,oZ,oo 11 Ood $ood Xad deddo
6 ot * *
XodoSoO. 2
40. a) odo oXe Xerad aXrar do3o od roa d>3 nd ,oZ,/Ad.
y -o _o cp 6
n
Xerad anerar rtodrorteydd 0ddo roort> doa Xe 0odo
3
3eon. 4
b) J cot 4 ( 3x ) dx doo XodoSoO. 4
( 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 dU- .
J
10. Evaluate | ex 1 1 + tan x + 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 + 3 j + 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 V 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
I f
, , , . sin x . cos x
b) Evaluate | -4- dx. 2
+ sin 4 x
0
33. a.) Evaluate | 23 x dx. 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
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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 | -j-:--dx = = . 6
J 1 + sin x cos x 3yf3
0
b) Find the general solution of the differential equation
dy 1 + y 2 2
xy dx = TT72 ( 1 + x + x 2 ) 4
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 1 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 J cot 4 ( 3x ) dx. 4
c) Differentiate w.r.t. x :
y = log 5 V1 - x 2 . 2
9x = 21 ( mod 30 ) ,aoeXdraX@ dd,- 0dO dO>drt$ ?
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Attachment: |
| Earning: Approval pending. |
