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Class – XIISubject: Maths

 Time allowed: 3 hrs.

M.M. 100

General Instructions:1. ll !uestions are com"ulsor#.$. The !uestion "a"er consists o% $& !uestions di'ided into three

sections ( ) * C. Section com"rises o% 10 !uestions o% 1 mar+each. Section ) com"rises o% 1$ !uestions o% , mar+s each andsection C Com"rises o% - uestions o% / mar+s each.

3. ll !uestions in Section are to be answered in one word( onesentence or as "er the eact re!uirement o% the !uestion.

,. There is no o'erall choice. owe'er( internal choice has been"ro'ided in , !uestions o% , mar+s each and $ !uestions o% /mar+ each. 2ou ha'e to attem"t onl# one o% the alternati'es in all

such !uestions.. 4se o% calculator is not "ermitted.

SECTION – A

1. 5hat is the number o% binar# o"erations on set 6a( b78$. 5hat is the "rinci"al 'alue o%

Cos91 Cos3

2∏  ; +  Sin91 Sin

3

2∏  ;

  3. I%18

  x

 

 x

2  <

18

6  

6

2  =ind .

,.I% the "oints $( 93; λ  ( 91; and 0(,; are collinear( >nd the 'alueo% λ  . 

. ?'aluate∫ 

e   

   −

2

11

 x x   d

/ ?'aluate ∫  Logx  d

-. 5hat is the an@le between the 'ectors a  and b  withma@nitude 3  and $ res"ecti'el#8 Gi'en a .b  < 3.

A. =ind a unit 'ector in the direction o% a  < 3 i  9 $   j  B / k 

&. I% the Cartesian e!uation o% a line ) is2

12   − x  <

7

4   y−  <

2

1+ z 

5rite the direction ratios o% a line "arallel to ).

10. =or the matri <

6

1

7

5  show that B ; is a s#mmetric

matri.

SECTION – B11. Show that the relation D on the set

< 6 ∈  E: 0 ≤  ≤  1$7( @i'en b#D < 6a(b;: Fa – bF is a multi"le o% ,7

  is an e!ui'alence relation.

1$. Prove tan-1 

−++

−−+

 x x

 x x

11

11  =

4

π  

 -2

1 cos-1x

  OR 

  Sol'e %or :

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  If tan-1 2

1

−

−

 x

 x

+ tan-1 2

1

+

+

 x

 x=

4

π  

 , then find the va!e of "x#$

13 ro'e that

 c

b

cb   %&   +

 c

ac

a

%&   +  

%&   ba

b

a

+

 < $abc aBbBc;3

14. Find ‘a’ and ‘b’ such that the function defined by5 if x ≤  2

  ax+b if 2<x<10 

21 if x ≥ 10

  OR

Heri%# mean 'alue theorem %or the %unction: %; < $ – , 93 in 1( ,J

1. If x   y+1 +   x y   +1   ='

  Prove thatdx

dy=-   2

%1&

1

 x+

  OR 

 I% <a sec3θ   and #<atan3

θ  (

=inddx

dy  at θ   <

3

π  

1/. Show that %cottan&2

'

 x x +∫ π 

  d < π 2

1-.ro'e that the cur'es <#$ and # <+ intersect at ri@ht an@les i% A+$

< 1

1A. Sol'e the %ollowin@ diKerential e!uation

dx

dy  B $# < $lo@

  OR

  Sol'e the diKerential e!uationsec$ tan# d B sec$# tan d# <0

1&. The scalar "roduct o% the 'ector I B j B + with a unit 'ector alon@the sum o% 'ectors $iB,j9+ and λiB$jB3+ is e!ual to one. =ind the

'alue o% λ.

$0. =ind the 'alues o% " so that the lines3

1   x−  <

 p

 y

2

147   − <

2

3− z  and

 p

 x

3

77 −  <

1

5− y <

5

6   z −  are at ri@ht an@les.

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$1. robabilit# o% sol'in@ s"eci>c "roblem inde"endentl# b# and )

are2

1 and

3

1 res"ecti'el#. I% both tr# to sol'e the "roblem

inde"endentl#( >nd the "robabilit# thata. The "roblem is sol'ed.b. ?actl# one o% them sol'es the "roblem.

$$. =orm the diKerential e!uation not containin@ the arbitrar#constants and satis>ed b# the e!uation # < aeb( a and b are arbitrar#constants.

Section –C

/ Mar+s uestions$3. 4sin@ elementar# trans%ormations( >nd the in'erse o% (5herethe matri.

  <

LDSol'e b# matri method.

2(63

42

265

−=++

−=++

=−+

 z  y x

 z  y x

 z  y x

$,. Show that the 'olume o% the lar@est cone that can be inscribed in a

s"here o% radius D is 27

8

 o% the 'olume o% the s"here. 

$. 4sin@ inte@ration >nd the area o% re@ion bounded b# the trian@lewhose 'ertices are1(0; ($($; and 3(1;.

$/. ?'aluate :

∫    +4)

'

%tan1o*&

π  

dx x

  LD?'aluate

  ∫    $ $ B B-; d( as a limit o% sum .

$-. =ind the distance o% the "oint 1( 9$( 3; %rom the "lane – # B E< measured alon@ a line

"arallel to2

 x  <

3

 y  <

6−

 z 

$A Com"an# manu%acturers two t#"es o% to#s and ). To#

re!uires , minutes %or cuttin@ and A minutes %or assemblin@ and Aminutes %or cuttin@. There are 3 hrs. and $0 minutes a'ailable in a da#%or cuttin@ and , hrs. %or assemblin@. The ro>t on a "iece o% to# is Ds.0 and that on to# ) is Ds. /0. ow man# to#s o% each t#"e should bemade dail# to ha'e maimum "ro>t8 Sol'e the "roblem @ra"hicall#

$& . man is +nown to s"ea+ truth 3 out o% , times. e throws a dieand re"orts that it is a si. =ind the "robabilit# that it is actuall# a si.

2 -3 3

2 2 3

3 -2 2