Get documents about "Class XII Mathematics 2012 Question Paper (Set 2)
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This is the Set 2 of the Mathematics question paper of Class XII CBSE (Central Board of Secondary Education) examinations.
Document Sample
I Series : SMAlII m-.t.
Code No. 65/1/2
'ffi;r -.t. ~ ~ ~ snr-~ ~ lf9-~
Roll No. "CK~~ I
Candidates must write the Code on
the title page of the answer-book.
• ::~'\ilfqq;r~fcp~~-'Cf:f.q~~S t I
• ~-'Cf:f .q ~ ~r<l"cnl31R ~ l'["(f ~ ~ ~ ~ snr-~ ~ lf9-~"CK ~ I
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• ~ ~-'Cf:f ~ 11iR ~~ 15 ~q1f ~ m
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~~ 110.15 ~U 10.30~"M>~~~-'Cf:f-wt ~~ 3lCrfu~~ctsnr-
~"CKCf#snr~~ I
• Please check that this question paper containsS printed pages.
• Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
• Please check that this question paper contains 29 questions.
• Please write down the Serial Number of the question before attempting it.
• 15 minutes time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the
question paper only and will not write any answer on the answer script during this period.
TTilrRr
MATHEMATICS
f.Nfftrr '(!1fl{ : 3 ~J [~3icn: 100
Time allowed : 3 hours J [ Maximum marks: 100
'H1i'i10<4~:
(i)
,
rrifI JlF{ 31f.:rcrr4 tf
(ii) "ffl' JlF{ q;;r if 29 JlF{ t \IfT ffr.;PlTa~ if ~ t; 3T, <f[ fMT"?J f lSfTl5 3T if 10 JlF{ t f2rRif
#~ 'l!?fi' t
.#1; CfiT I lSfTl5 <f[ if 12 JlF{ t f2rRif # ~ 'f!.{ff' .#1; CfiT t I lSfTl5"?J if 7 JlF{
t f2rRif # ~ UY .#1;CfiT I t
(iii) lSfTl5
tf
ar if ?1'4t JfFif. * drT( 'l!?fi' ~ 'l!?fi' C{TCf?[ 312F:TTJlF{ cit 3i7Cf~4Cfirtl 3Fjmr fW' -;;rr ~
(iv) vat JlF{q;;{ if ~ ::rift I fiR" 'Jjf 'f!.{ff' 3icfJ ~ 4 rrRl if fMT UY'3fqff ~ 2 rrRl if
3fFr/RCfi ~ t 11[#?1'4t rrRl if # 3fTTlCnT f!CI7tt ~ CfiRT tf
(v) c1wl'J'Hc < * JFffrT cit 3FJ}Tfrr ';fit t I
65/112 1 [P.T.O.
General Instructions :
(i) All questions are compulsory.
(ii) The question paper consists of 29 questions divided into three Sections A, Band C.
Section A comprises of 10 questions of one mark each, Section B comprises of 12
questions of four marks each and Section C comprises of 7 questions of six marks
each.
(iii) All questions in Section A are to be answered in one word, one sentence or as per
the exact requirement of the question.
(iv) There is no overall choice. However, internal choice has been provided in 4
questions of four marks each and 2 questions of six marks each. You have to attempt
only one of the alternatives in all such questions.
(v) Use of calculators is not permitted.
~-ar
SECTION-A
m-.:r ~ 1 ~ 10 (lCf) ~ m-.:r CfJT 1 ~ t I
Question numbers 1 to 10 carry 1 mark each.
/. ~"ffi(f~J (1-x)~dx. 1
Evaluate J (1 - x) ~ dx.
3
~"ffi(f~: J ~dx 1
2
3
Evaluate: J~ dx
2
.i ~(~' ~)(~~-!)=(=~ ~)t mxCfJT~~ I 1
.1-3) -9
2 3)( -2· 4 (-4 6),write the value
If ( 5 7 = x ofx.
~ a = Ai + J + 4k CfJTb = 2i + 6J + 3k <rt~ 4 ffitt m 'A' CfJT~"ffi(f ~ I 1
Find 'A' when the projection ofa. = Ai + J + 4k on b = 2i + 6J + 3k is 4 units.
65/1/2 2
1
If a line has direction ratios 2, -1, -2, then what are its direction cosines?
J
l1RTN "ll ~ (a, b) E N ~ ~ ~ m:3W:lRT ~ *, a * b = B.-n. .(a, b) mr ~ ~ I
5 * 7 'ffi(l~ I 1
Let * be a 'binary' operation on N given by a * b = LCM (a, b) for all a, b E N. Find
5 * 7.
7. 1
''-\ I.
'""
Write the principal value of cos-{1) - 2 sin-{ -1).
. [cos e sin o ] . [sin o -cos e ]
t/ 8. WR~:cose . e +sme 1
-sm cose cose sin o
[ cos o sin e ] . [sin e -cos o ]
Simplify: cos e . e +sme
-sm cose cose sin e
/9. 1
~ '=' ~~ ~
a = 1- 2J, b = 21 - 3J, c = 21 + 3k.
'='~ ~ 1\
Find the sum of the following vectors:
--+- ~ '='~ ~ ,0;.-1- ~ 1\
a = 1- 2J, b = 21 - 3J, c = 21 + 3k.
5 3 8
/0. ~~= 2 1 ~m~a32
1 2 3 ° Cf1T~~ I 1
5 3 8
If ~= 2 1 , write the cofactor of the element a32.
1 2 3 °
65/1/2 3 [P.T.O.
~-~
SECTION -B
~~11 ~22 (fCI?~~~4 I ~t
Question numbers 11 to 22 carry 4 marks each.
/11. ~ a, b ~ C~ -Q:U~ t fcp Ia I = 5, Ib I = 12 ~ lei = 13 t ~ a+ s+ C = 0t
.....,:+-+ -+ ~ -+ -+ -+ ~
01 a . b+b .c + C . a cpr llR -mn q)1I"l~ I 4
-+-t-+ .' -+ -+ -+ -+ -+ -+ -+
If a, b, c are three vectors such that I a 1= 5, I b 1= 12 and I c I = 13, and a + b + C = 0,
-+ -+ -+ -+ -+ -+
find the value of a . b + b . c + C . a.
~
12. J'"18f<51Psl(1 cnT
B41Cfi{OI ~ ~ : 4
/ 2X2~-2xy+y2=0.
Solve the following differential equation:
2x2~_ 2xy + y2 = O.
¥.3. ~ ~ ~ ~ ~ ~ ~ ~ ~
fcp"Cfi11 "Cfi11
~ fern I'Uf4Cfi\'11 ~
3lR CfiT 80%
~m? 4
Flowmany times must a man toss a fair coin, so that the probability of having at least
one head is more than 80% ?
14. ~ (cosx)Y = (cos y)X t or ~ -mn ~ I 4
~
/ ~siny=xsin(a+y)t or~~fcp~dx=sin2~a+y)t
sma
I
If(cosx)Y = (cos y)X, find~.
OR
.. ~ sin2(a + y)
If sm y = x sm(a + y), prove that dx =
, .
sma
Y llRf A ~IR - {3} _ B ~IR - {I} t- I <nR'If: A 4 B"l't f(x) ~ (~=;) r~t
m
TR fcrcw ~ I~ fcpf~ ~ 3i1iۤI~Cfi I ~:
t f-1 -mn ~ I 4
Let A = IR - {3} and B = IR - {I}. Consider the function f: A ~ B defined by
f(x) = t =~). Show that f is one-one and onto and hence find f~1•
65/1/2 4
16. m~fq;-tan-{1 ~o:i~;)=~-~;x E (-~,~). 4
~
m~ sin-I(~) + sin-I(l) = cos-1(36)
" 17 5 85 .
Prove that tan
-I(1 cosx
+ sin
)_ZE.-..-:~2' x (_ZE.
x - 4 E 2'
ZE.)
2 .
OR
'5
Prove that S1O-1 (17) + S1O-1 (3) = cos" (36) .
." 8 . 85
4
~CfiTm~-V49.5 ~f"1CfiC:
CfiT (approximate) 11R~~ I
Find the point on the curve y = .J - llx + 5 at which the equation of tangent is y = x-II.
OR
Using differentials, find the approximate value of-V49~5.
- \
18. 11R~~: J sin x sin2x sin 3x dx 4
Evaluate: J sin x sin 2x sin 3x dx
OR
Evaluate: ""f (1 _ x)(1
2 + x2) dx
65/1/2 5 [P.T.O.
cI m<fU ,,,,I ij;- 'J"1'l'lT <!if m 'Ii{ f.,,,,f,,,f,,, " ei\ fu;;;, ~ : 4
III
a b c =(a-b)(b-c)(c-a)(a+b+c)
a3 b3 c3
Using properties of determinants, prove the following:
III
a b c =(a-b)(b-c)(c-a)(a+b+c)
a3 b3 c3
'./ ~ y = 3 cos(log x) + 4 sin(log x) ~, ill ~ fcp 4
d 2 d
x2Q:Y.+x~+y=O
dx2 dx
If y = 3 cos(log x~ + 4 sin(lo g x), show that
~=Y=~(f!ITx+2=Y.::.l=z+1
1 2 3 -3 2 5'
Findthe equation of the line passing through the point (-1,3, -2) and perpendicular
to the lines
~ = Y = ~ and x + 2 = Y.::.l = z + 1
1 2 3 -3 2 5'
4
(x + 1) ~ = 2e-Y - 1; Y= 0 ~ x = O.
Find the particular solution of the following differential equation:
~ '
(x 1- 1) dx = 2e-Y - 1; Y= 0 when x = O.
65/1/2 6
~-~
SECTION -C
>w-1rf&IT 23 ~ 29 ncn~ >w-1 *" 6 ~ t I
Question numbers 23 to 29 carry 6 marks each.
s-:~ Cflf wWT CR f•• :<ifMf(5l('1~
p fP:IICf){OI ~ CfiT ~ ~ :
x- y + 2z = 7
3x + 4 Y - 5z = -5
2x-y+3z=12 6
~
~ ~'* *" 'Wlfrr mT, f.,8fMf@('1 ~ Cflf ~"ffi(f ~ :
(-i ~ n
Using matrices, solve the following system of linear equations:
x-y+ 2z = 7
3x + 4 Y - 5z = -5
2x-y+3z= 12
OR
Using elementary operations, find the inverse of the following matrix:
( -~ ~
3 1 1
~]
~ ~
3fY 1i:\Cf) Jffi" ~ *"
t I ~ ~ 1c.i f.:r:ITur .q ~ A 1IT 1 era Jffi"
Cflf f.:r:ITur Cf)"@
~ B 1IT 3 tft Cf?n1 Cf)8T ~ t ~ ~ ~ ~ f.:rGtur .q 3 tft ~ A 1IT ~ 1 *"
cR:r ~ B 1ITCf?n1 Cf)8T ~ t I cw1c.i ~ ~ ~ ~ 17.50 ~ ~ 1IT~ ~ ~ 7
(1l'l"f Cfi1"ffiff~ I <:tK ~ ~ Cflf ~ wWT 12 tft ~ ~ ill 1c.i ~ ~ -q: ~
~*"~~3('q1~('1 ~~fcp~(1l'l"fm?
34{1Cff1 CfiT ~ ~ >lffin:R ww:rr GRTCR m1f>CfiT ~ ~ ~ ~ I 6
A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and
3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A
and 1 hour on machine B to produce a packa~f bolts. He earns a profit of ~ 17.50
per package on nuts and ~ 7 per pac~ag~ of bolts. How many packages of each
should be produced each day so as to maximize his profits if he op~rates his
machines for at the most 12 hours a day? Form the above as a linear programming
problem and solve it graphically.
65/112 7 [P.T.O.
~,m A(3, -1,2), B(5, 2, 4) ~ C(-l, -1,6) rnr Rmfuf W!mYf fiT ~41Cf){OI
C ~ ~ I
~W!mYfcn1~P(6, 5, 9) ~~~~ I 6
Find the equation of the plane determined by the points A(3, -1, 2), B(5, 2, 4) and
C(-l, -1, 6) and hence find the distance be~een the plane and the point P(6, 5, 9).
/ ftR;-~flI;- [UtanX+-VcotX)dx~--rz.~ 6 -
~
3
J (2x2 + 5x)dx CfiT 1=IR <WIT ciTWilT ~~"# ~ ~ I
1
nl4
Prove that J (-V tan x + -V cot x )dx = -{2 .~
o
OR
3
Evaluate J (2x2 + 5x)dx as a limit ofa·sum.
1
27/~fcp~~~~~,FmCfiT~~W:rr%-~~~tciT~
;7 ~~~oqm~~%- I 6
Show that the height of a closed right circular .cylinder of given surface and
.;aximum volume, is equal to the diameter. of its base. . . .
/ ~ ~ ~ ~ mrr %-I ~ ~ 5 <IT 6 ciT~ >W{f m t "ill ~ ~ fucfc)? cnT ~
3ۤ1(1(11
// ~ ~ ~ ciT~ ~ ~ %-I ~ ~ 1, 2, 3 <IT 4 ciT~ >W{f ~ t "ill
3ۤ1(1(11 %-
~p, ~~~ ~ ~ ciT~ ~ ~ %-I ~~
3ۤ1(1(11 %-
~ ~ itcP~ fern >W{f ~
m
t "ill ~ rnr ~ ~ "!1ffi o/ 1; 2, 3 <IT 4 m1<f ciT~if<!l Cf)(1 %-?
1
I~ 6
A girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the
number of heads. If she gets 1, 2, 3 or 4, she tosses a coin two times and notes the
number of heads obtained. If she obtained exactly two heads, what is the probability
that she threw 1, 2, 3 or 4 with the die?
I
2.~. ! fcW.T~m~R8~,m~ftlt~CfiT~~~: 6
_
3x y - 3 = 0, 2x + Y - 12 = 0, x - 2y - 1 = °
mg the method of integration, fmd the area of the region bounded by the following
lines:
3x - y - 3 = 0, 2x + Y - 12 = 0, x - 2y - 1 = °
65/1/2 8
