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Pre University Board 2008 P.U.C Physics, Chemistry, Maths & Biology " Mathematics " code 35 For English version see page no 9 - Question Paper

Tuesday, 05 February 2013 01:05Web

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Code No. 35

Total No. of Questions : 40 ]    [ Total No. of Printed Pages : 15

March, 2008

MATHEMATICS

( Kannada and English Versions )

Time : 3 Hours 15 Minutes ]    [ Max. Marks : 100

( Kannada Version )

: i)    A, B, C, D doJ E 00    a>rtrt$d.

0> amrtrt>o J q&.

CO    *    c    _D

ii) an - a n 10 oxn>b, an - b n 20 oxnsb, an -

c n 40 oXrt>o, an - d n 20 oxn>o doJo 7 _0

an - e n 10 oxndoJ d.

_0

- A

X>A 0> d4,rt>o J 0 :    10 x 1 = 10

CO    _D y oi    oi    _0

1. 2x + 5 = x + 4 ( mod 5 ) Xd 3J, X dP}3FoX x XodoSoO.

5 - x 2y - 8 03


2. A =


a d/jx wd, x do Jo y n>    ?


3. a * b = y-, * ,dJrao 0oo


4.

5.    x 2 + y 2 + 2gx + 2fy + c = 0 4)4 0dD dn>DR ,&F,co Cb4 >0 D ?

6.    y2 = I2x -4Cd 3$o X ort> aerdrt>0

~y    co

doDaDO.

7.    tan ( tan - 1 3 ) + sec - 1 { sec - 1 ( - 2 ) } d Cd ?

8.    i nyd >deD4DR ( Multiplicative inverse ) doO.

9.    x Dr sbOD y = f( x ) adod 4 43d6    rtD}>od

sin x

- B

X> A4)n>, C/4)3d &o Dr O :

10 X 2 = 20


11. z De ,4 3F0rt,D    ( Congruence modulo ) m ,004) z De

,0043AdDrf    a = b ( mod m ), z De

, 004>AdD 0OD ,3.

2001 2004

12. CdDD doDaO :

ot

13.    0 ,oXo< ( G, * ) (o V a e G, a - 1    = a wd, ( G, * ) (o 0

eS(o* ,oXo< 0o JeOb.

14.    Xi + 2j - k i - 3j + 2k    dd <0 ,Qert>3Ad X

(DD XoDSO.

ot

15. x = 3 + 2 cos 0    y = 1 + 2 sin 0    ,oeXdrart$o 4)J

b erardy XodSO.

_0    c

16. 4, Xa = 10, J@eoJ ( e ) = 2 wrort, 34dd(o (Hyperbola) 2 2 - hr

x 2 y 2

,oeXdrad, o - 7-9 = 1 d4 XodSO.


a 2 b 2

17. tan 1 = sin 1 2 + cos 1 +1 wd , x    ?

2    2V2

18. e 1 + i n/3 + e 1 - i n/3 = e 0O ,2>ab.

19. 1 aa j + 3 bj = 2 wd , ( a, b ) )0y djX"    ?

20.    ( 1, 3 ) d x 3 + xy + y 2 = 13 deZ n 4,fX d XodSO.

21.    I --o dx    ?

sin 2 x cos 2 x

eXdrax db.

- C

I. d>Adrt> </ddhdd dodo d4tn>o o&

3 X 5 = 15


23. 39744 - 0dh dhd h&drt> ,oZ, doo 0dh dhd

r-5    4    (b,    4    8\    _0    r-5    4    (b.

h&drt > d n >o dodoaoO. _0

5

a 2 + bc

a

1

24. a)

b 2 + ca

b

1

= - 2 ( a - b ) ( b - c ) ( c - a ) 0odo

c 2 + ab

c

1

SShn).    3

b) d;dod a<dododoJ x doo y n> dd dodoaoO : x + 2y = 7

4x - 5y = 2.    2

25. a) rto}h5hd hdedjd ( Multiplication modulo ) ped

H = { 1, 2, 4 } 7 0oodo G = { 1, 2, 3, 4, 5, 6 } 7 ,0d0< d,0dDd0dD    3

b) ,0d0< dod didodd adddhAd 0odo

A    A

26. a)

2

1    - j + Xk , 4 i + 2j + 9k , 5 i + j + 14k doo 3 i +

A    A

2    j + 7k 00 ,art>o h<o, ,do odort>    & 3 ,art>hAd d,

U    cp CO 7

X d d<doo dodoaoO.    3

b) 2 i - j + 2k dd as, dod add ,ad. dodbaoo.

'    J    zh CO    ci

II. d> A4)n> C/4)3d 0do    O :    2x5=10

27. a) x 2 + y 2 - 6y + 1 = 0 Do x2 + y2 -4y+1 = 0 srtDR <o43A $e,D4 dDo 3x + 4y + 5 = 0 ,d> deZCD Ded XeopoD4)> 4)o ,Deddra4DR doD&SDO.    3

b) ( 4, 2 ) DD ( - 5, 7 ) 8O0rt>D 43,, Drt>3AdD4 4)

, oedd d doD&SDO.    2

ot

x 2 y 2

28. a) y = mx + c ,d>deZCDD 2 - 7-9 = 1 d4CDd, (

a 2 b 2

Hyperbola ) ;,2Fd43)rtCey3rtD a0DR doDSDO.    3

b) y 2 - 8x - 32 = 0 d4<CD ( Parabola ) 3$CdD doDSDO. 2

- , la(a + b + c)    - , lb(a + b + c)    - , I c(a + b + c)

29. tan 1 V bc-+ tan 1 V ca-+ tan 1 V ab-

0 0OD ,3.    5

III. d> C/4)3d odo    O :    3x5=15

30. a) D< po x n ,000J, cosec ( ax ) >, a4a.    3

Code No. 35    6

31. a) ex + e y = e x + y wd- , djy- = - e y - x 0odo ,3.    2

b) x = tan 1 "\J 1 + t , y = cos 1 ( 4t 3 - 3t ) wdd, djy- = 6

0oo ,3n).    3

32. a) y = sin 2 -j cot - 1 yj 1 + j dd, Hx = - 1 0od ,3. 3

r .

sin x

b) I -j-:- dx d (do. Xodoo0.    2

1 + sin x

rI


cos x

33. a) I :2--:-- dx d <doy Xodoo0.    3

2 sin 2 x + 3 sin x + 4

b) J , dx d (doo Xodoo0.    2

yjx 2 - 4

x 2 y 2

34. ,d/x< a3od 25 + 9 = 1 erdjd Xod Xodoaoo. 5

- D

X>A /d)3d 0do J0 :    2 x 10 = 20

x 2    y 2

35. a) erdJd dZo $. - ,aoeXdrado a"2    + b"2" = 1

wdr ddd XodoaoO.    6

2 3

wdd, y3>s0-3>saD< djdoeodo, d<d>eA, A


b) A =


2 5

36. a) 0> XdeD ,&53oXrt$rt a ddd* ddoed

'    co    ZJ

doJ. ,iQ%.

6


b


a


c


b) {>, aodo


sin A sin B sin C


0od ,a dd    ,>&. 4


cp    CO


37. a) a    dJd nod erard)> wJd oJrtrJrteydd w w<Jd) iXd>AdeXo srap&.    6

b) ( >/3 + 1 ) cos 0 + ( V3 - 1 ) sin 0 = 2 0ODd ,d/t <odd2.

Xodoao.

4


n/2

J


dx


38. a)


= V5 log


6


sin x + cos x


V2 + 16

V2 - 1


0

b) dbc" = ( x + y - 1 ) 2 d X< , oeXd rad aa.

4


- E

X>A    o dn Jo :

1 X 10 = 10


39. a) 1 + i ,oZ dortb, Xodoao. d)rt>cf WTOrora*

J, Jeo.

4


y CO

b) x 2 + y 2 - 8x - 6y = 0 d* J doJ x - 7y - 8 = 0 deZ d)rt$od

dod {>. dx Xodoao.

4


J    6    CO    ct

c) 7 123 , oZ.o axx ,) (aa ,3) oXdb, Xodoao.

2


O    Cp    Cp    c

Code No. 35    8

40. a) | ~a | = 13, | ~b | = 19 doJ I + ~b | = 24 wd d,    | ~a - ~b |

<do d ao ?    4

b)    J tan 4 x dx d<doo XodoSoO.    4

c)    y = log Vcos x wd d , djy d oo Xodosoo.    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 integer x satisfying 2x + 5 = x + 4 ( mod 5 ).

5 - x 2y - 8

2. If A =


is scalar matrix, find x and y.

03

3. If a * b = 3ab , then prove that * is associative.

4.    Define co-planar vectors.

5.    Write the condition for the circle x 2 + y 2 + 2gx + 2fy + c = 0 touches both axes.

6.    Find the co-ordinates of the end points of length of the latus rectum of the parabola y 2 = 12x.

7.    Find the value of tan ( tan - 1 3 ) + sec - 1 { sec ( - 2 ) } .

8.    Write the multiplicative inverse of i.

9.    Define the differential coefficient of a continuous function y = f ( x ) w.r.t. x.

10.    Evaluate [ 1-cs x dx.

sin 2 x

PART - B

Answer any ten questions :

10 x 2 = 20


11. The relation Congruence modulo m' is an equivalence relation on z or

prove that a = b ( mod m ) is an equivalence relation on z.

2001 2004 2007 2010

12. Evaluate


13. If in a group ( G, * ) V a e G, a 1 = a, then prove that ( G, * ) is an

Abelian group.

AAA    AAA

14. If the vectors X i + 2j - k and i - 3j + 2k are orthogonal, find X.

15. Find the area of the circle whose parametric equations are

x 2 y 2

16. Find the equation of the hyperbola in the form a""2 - 2 = 1. Given

that transverse axis = 10, and eccentricity ( e ) = 2.

17. Find x if tan 1 = sin 1 1 + cos 1 +1 .

2 2>/2

18. Prove that e 1 + i n/3 + e 1 i n/3 = e.

19. If 1 a J + [ b ) = 2, then find 7 at ( a, b ).

20. Find the length of the sub-tangent to the curve x 3 + xy + y 2 =13 at ( 1, 3 ).

J


21. Evaluate | -2-2 dx.

sin 2 x cos 2 x

22. Form the differential equation by eliminating the parameter c.

sin 1 x + sin 1 y = c.

PART - C

I. Answer any three questions :    3 x 5 = 15

23. Find the number of all positive divisors and the    sum of all positive

24. a) Show that

2 ( a - b ) ( b - c ) ( c - a ).


3


b) Find the values of x and y according to Cramer's rule :

x + 2y = 7

4x - 5y = 2.

2


25. a.) Prove that the set H = { 1, 2, 4 } 7 is a sub-group of the group

G = { 1, 2, 3, 4, 5, 6 } 7 under multiplication modulo 7. 3

b) Prove that the identity element of a group is unique.

2


26.    a.) If the vectors i - j + Xk , 4 i + 2j + 9k , 5i + j + 14k

AAA

and 3 i + 2j + 7k are the position vectors of the four coplanar points, find X.    3

A    A    A

b) Find the unit vector in the direction of 2 i - j + 2k .    2

II. Answer any two questions :    2 x 5 = 10

27.    a) Find the equation of the circle which cuts the two circles

x 2 + y 2 - 6y + 1 = 0 and x 2 + y 2 - 4y + 1 = 0 orthogonally and whose centre lies on the line 3x + 4y + 5 = 0.    3

b) Find the equation of the circle having ( 4, 2 ) and ( - 5, 7 ) as end points of the diameter.    2

28. a) Find the condition for the line y = mx + c to be a tangent to the

2 2

hyperbola 0, - b_2 = 1.    3

b) Find the focus of the parabola y 2 - 8x - 32 = 0.    2

29. Prove that

V


c ( a + b + c ) ab


0 5


, , a ( a + b + c ) tan - 1 \l-bC- + tan '


b ( a + b + c )    - 1

ca-+ tan - 1


III. Answer any three of the following questions :    3 x 5 = 15

30. a.) Differentiate cosec ( ax ) w.r.t. x from the first principle. 3

b) Differentiate sin x with respect to log x.    2

31. a.) If ex + e y = ex + y prove that = - e y x .    2

b) If x = tan 1 "\J 1 + t , y = cos 1 ( 4t 3 - 3t ) , prove that

dy

3


dx

>/


dy

, prove that dx"


+ x x


1


32. a) If y = sin


3


cot


sin x

2


J 2 sin 2 x + 3 sin x + 4

b) Evaluate J , 2 - dx.    2

x

yjx 2 - 4


2 2 xy

34. Find the area of the ellipse 25- + = 1 by integration method. 5

PART - D

Answer any two of the following questions :    2 x 10 = 20

35. a) Define an ellipse. Derive the equation of the ellipse in the standard

form

2    2

4 +     = 1. 6

a 2    b 2

, find A 1 by Cayley-Hamilton theorem.    4

2 3 2 5


b) If A =


36. a) State and prove DMoivres theorem for rational index.    6

b) Prove that the sine rule

a    b    c

:t = :tt = :77 by vector method.    4

sin A sin B    sin C J

37. a) Prove that the greatest size rectangle that can be inscribed in a circle of radius a is a square.    6

b) Find the general solution of

J


dx 1 38. a) Prove that I :- = = log

J cin v* x pno v*    #7T o


6


3 r~ 6

y2 + 1

sin x + cos x    y/2    1 y/2 - 1 4

0


b) Solve the differential equation djy = ( x + y - 1 ) 2 .    4

PART - E

Answer any one of the following questions :    1 x 10 = 10

39.    a.) Find the cube roots of 1 + i and represent the Argand diagram. 4

b)    Find the length of the chord intercepted by the circle

x 2 + y 2 - 8x - 6y = 0 and the line x - 7y - 8 = 0.    4

c)    Find the digit in the unit place of 7 123 .    2

40.    a) If | cT | = 13, | ~b | = 19, | ~a + ~b | = 24, find | ~a - ~b |. 4

b) Find J tan 4 x dx .    4

c) If y = log Vcos x , find djy .    2

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You are here: PAPER Pre University Board 2008 P.U.C Physics, Chemistry, Maths & Biology " Mathematics " code 35 For English version see page no 9 - Question Paper