Uttarakhand Board Class 12 Question Paper 2024 for Maths

UTTRAKHAND BOARD

Question Paper

tud. $*ff, Y".51mft
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Roll No"
No, e'6'prxmte*d 4aa6es : ;*
L28 4*s ii,q.;;;ri
?,02,4
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MATHEMATICS
grffi:3 d ] It qir-gtrr
e\ -' : xCB
T'ime :3 Hours I I e {;ie. 't5.rl i+', r ii{f

ftdql: (i) ilT swl-wi fr g,m 24sw dt qqft ssn" sffi fr t

DireGions: There are in all 24 questions in this questiein Bsp*r" &$i tr,i*:st!cl*g gr*
connpulsory.
(ii) wd Ag fr€fRn 3i6 s{h qru.s s+fu-a dt
Marks alloted to the questions are mentioned against rhtt-,i.
(iii) rd-6 qyq 6t Eqrl.ndo qfr-d. nw v"&f, rtr{ f;frut
Read each question carefully and answer tc the p*int.
(iv) rv{ TiwT t sgfureqlu sr{ * i W sq"E & s-.Ss wtry fu 3rr{ q'eryq fi*;'s" i:o;
ru dl sfr fr-,iim 3nn1 3n<gk6T fr frfu i
Question Nc,. 1 is multiple choice questirrn" F*ur tpti;r: iii{j: fi:!i:it ii:
answerof each partof this questiori. Write cGrrectr;ii1-t.;; .:'r -,:;. -.i ',1",i-.-
bosk.
(v) qvqriezn 1 e;','r*dgu-gEogtfi6Tdt qq{*EqT: t; r:il=jeEtii:tT: rri.i
dt qsqdwl a * t 2ar, * 3i6 & q'wdt srqrfeqT 13 e 1a ':r) -(-ilr .i:iri m;
qr{ dt qsr Hwr 19 t 24 dtr fr ilim b sy{ f;, ffiH-e qqq'f;Gfl 2+ rt'x;;'dn:
grTrrfrd qq{ dt
Each part of Question No. L carries one rnark. Questiarn I'l*" 2l"o i'
are of one mark each. Question ftlo. I to 12 ane,:.if tq*r.-.m rri*ri{$ '-ircr:.
Question No. 13 to 18 are of four marks each" Questiuil i't*. i-{l t*.ili.
are of five marks each, in urhich question i\lo, 24 i9 e$${}/5;:rilrce
based qrestion.
(vi) {fl qrl-qz fr sqn"q-{ olt 1fu@ q* f" mrhrq srir ri
affi qri";r'*rri
q-dTq frqT rqT *t t-S qwT fr trm m fror-i ml -d! sf,'{ affiffir i

There is no overall choiee in this questir:n p;:f,er; h*v*ever, *n ;r'it*iiri:i
choice has been providetl in fevv questioms, Attenrpt" c;r"l!'1, +ir*';1" liit:
given choices in such questions.

lll 9H'e'"{-}.

1 (6) ?;ft sin-1x=y * d-
1

If sin-1x=y, then -
(i) 0<y (ii) -;ryr; (iii) 0.y., (iv)*i"0.5.
(,q) qfr A , 3x: ffi 6T u6 qiEqrTtq of srEi6
d, d) lAdj Al sr qTq d 1 -
Let A be a non-singular square matrix of order 3x3. Then
lAdj Al is equal
to-
(i) lA I (ii) lA '
I (iii) iAi, (iv) 3lAl
(rr) 'x'b qrter sin(cos(xz)) sT erq-otra *rII-
I

Differentiation of sin(cos(x2)) with respect to 'x' wlll i-re -
(i) cos(cos(x2)) (ii) cos(sin 2x)
(iii) 2x cos(sin(x2)) (iv)-2r cos(cos 12) sin xl
(q) * -i,furf r(2)=s ar rtrt t-
#t,n):4x3 'l

I,
oxr(x)= +f - such that f(2)=0, then f(x) is -
ir -q
*+
1 129 ,....j I t}g
(i) x..4--r- u (ii) a L _,;,
^'---o -:_-i (i,i) x"_a
l}g i 7 L29
(rv) x3
x48
(s) L+xZ dx s{w{d-
J 1

I Lt- x2 dx is equal to -
(i) |lr;z. l'"nl[.fr;,.11
(ii) 1 ,, - x273 /z , ,
(iii) *(, + x2)3/2 + c
,?
)__
(iv) +,tl;F r
l*r,"nj[,.",[;])l
(q) qrq ffi Erd fu]ft 3lEcFrT H$=f,{ur b anro 6c{ d 3qftm
+dy ffirtt oi1 u,{qi"*-
1
The nrrmber of arbitrary constants in general solution
of a differential equation
of fourth order are -
(i) 4 (ii) 3 (iii) 2 (iv) 0
428 0Gr)
12l

\

(E) x-3T8r a' rrqi<< d?dr w fr-€ t JTA flft tG-r mi sfrd5{"r d - 1

Equation of a line parallel to x-axis and passing through
the origin is -
x Y z xyz XYZ (iv)
xvz
(i)
00
( ii)
011
(ii i)
100 T=T=1
qFI
(E) qft P(e)=o' s 3it{ P (A n ts) =0.: t A Hqfrqu flfttr-dT P(A/B) o-r fiTIr - 1

If P(B)=3' 5 andP(AnB)=0.3,thenthevalueofconditionalprobability
P(A/B) will be -
(i) 0,6 (ii) 0.2 (iii) o'1s (iv) 0"8
(A) f,en IFKUI (R) & sq fr frkf,
Frtsr,e.T{ qgir r & sil-d dr €urii C, dr mert't Ehl srFroPr<
t
fuqi :rqr t r Frrrftfua f{frTdt (i), (ii), ( iii ) f,sil ( iv) t g{64 Sfl6r
s& r+q 6rBe I

statements labelled
Direction : In next two parts of Question No'-1, there are two
(i), (ii), (iii)
as Assertion (A) and Reason (R.). From the following options
and (iv), select thelr correct answer'
(i) A dsIT n dC r& d aPII R, n of sfr qf@T orar tt
Both A and R are correct and R is the correct explanation
of A'

(ii) A dzir n qfi d qT{ R, n of sfr qlsrT r€} m<n Bt
ffi
Both A and R are correct but R is not the correct explanation
of A'
(iii) A qfr t qrc R rrf,d tl
A is correct but R is incorrect'
(iv) A dqT R atq) rf,d dt
Both A and R are incorrect'
(H) STFr621.q (e) ' f(x)=2y 6KI s-4-fl 6E1"I f : Z'+ Z Uffi tt
iFr<q (R) : qmT f 3{r@rdtD ret tr
1

Assertion(A):Thefunctionf:2.+T,givenbyf(x)=2xisone.one.
Reason (R) Function f is not onto,
(il) srFr6?r{ te) kg n (2i+:i+sri), B(i+zi+:t) 3il{ cfzi-:r<l €todt
rFKUT (R) , I aC l=l AB l+i BC I

Assertion (A) : Points A(-2? + 3j + st<) , e(i +2j +:t1 and c(7i - st1
are

collinear.
Reason (R) : IAC l=l AB l*lgC I ' ,2..
2 {fr y=4 sin x+B cos x t d kd61frsfu +Y=o
#
1

If y=4 sin x+B cos x, then prove that #.Y =0.

t3l I P.T.O.
428 (IGF)

3 eji x=Zat,, 1,.,=tstafr :* rro ffigr I
dv
firnd
; , it x=2at2, y=ata
-'t
4 J*x-+35
-,, a-:Tqrq Srf,'#tsgt
-dxl-l-
I

Evaiuate J;**O*.
d.

# = e'-v &J aTrq6 Ff, EI?I a1fuf
5 gr-dqr(T H#6quT I

rino the general soiution of differential equation j{ = e*-Y .

fr-
x ffii eB qffi HI-f, dfiftu ffi frE x(? + j * r<l um qrr6 sfu et 1

Find the value of x for which x(i + j * [l is a unit vector.
a}fr€rfr (2,3, -4) d (1, 3, -2) o,) Frmri drfr tqsT b ko-:rgqro il-d #ftEt 1

Find the direction ratios of a line joining the points (2,3, -4) ancl (1,3, -Z)

$" 'S{"i-;T
f(x}=f;$s-1x, I . t_t I r t- 1l
r 6T 3iitq dfuut Ss qoq f(x) 6r qk*-{ tfr ffiu
I
t

L 4Z ",lZ ) )
ilraw the grafih of function f(x)=cos-1x, x €
i-
1 1l . Also 'rvi'iie fai'ic*
I'uneti+r: f{x),
I

t_ E,Ji) CIf

s. E:r, F Hr B-.lT -fi'qra e-q -* s *fi.F. of < t eo rfi dl f,rd htks B qf, 6i .*:,.n*e
fins.:E E E6 w*s-sBwil 12 +fr. dI l
The r-acjius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate
al rr'Ehieh the area of the circle is increasing when the radius is i2 cm,
r qix
L(,. J ;,-
t\!" ,- #; .r Jl ffif qFI Efd'mtful
._ rLl
ln "\
p

.^ cix
"-,(x+2)(x+3).
Llinn
I lrlu
i
lr {\

r r., ffiikry tsT'Ieiq} fu ffr 61 6iur nrn ffitsu : r
Find the angie between the following pair of lines :
x ,/ z nll_. x-5 y-2
7"'i=T 3ii{/and *4 =?='Bz-3
:12. ffii qTrefrrffi T{ x 6t qrfuffi-dT 6icq +A kqT rrur *t r 1*.3) 6T qrt srr ffifus p.
i
A ranci'pnr variable x has the following probability distribution. Detenmine the
vnii:+: *f F{x <3).
X ft d
]. Z ?
J 4
f\ l-
2k 4k 3k

42r{ {r{-;F} I4l

13. .+= dtfu fu +A firm. qEFI f : N -+ rs, uffi dPrl3nc'rfiF ffi fr t' 4

f y:.+i, zffr x ftqst
)

I
X-1,qfr xqq*
I
(

Snoiv that f : N + N, glven bY -
X-r3,, rf x is odd
l Xl=
1X - 1, If- x is even
rs both one-one and onto.
3[qrEI /OR
fug ffikg fo A={t,2,3,4,5} fr, n={(a,b) I la-bl Hq t } AT{T trd-fl TitftT R Cr6
T.ur6r <rrisx 8t ast5g fu 11,3,si b mft s+-+qq s-6 (s.t t *riftro dt
Sr,ri/ that the relation R in the set A={L,2,3,4,5} given by R={(a,b) : la:bl is
e ,-on,!, is an equivalence reiation. Show that all the elements of {1,3,5} are
re :ied to each other.
cosx *sinx 0 t
1A
.l -,
F r l-
sinx cosx 0 d d fu-e 61frs fu r(r).F(v)=F(x+v) 4

001 l
,: - sinx 0
ccrr
sinx cosx 0 , show that F(x).F(Y)=F(x+Y).
001
l1 ( 5 fttq qrq h fru Fmfrfrsd q-f,fl srFI frg x=n/z q{ Tidd t' 4

: .' [tosx , qfr x* r Z
n-lX
3 , qfrx=nl2
Eor which value of k, the following given function is continuous at x=n/2'.

; rl= LqI , rf x+nll
n- ZX
3 , tf x=nfZ
3rsEn /oR
'!tc--Iq xin *+(sin x)cosx q;t'x' b gfteT 3ftk'f* mtfrUt
Diriei-errtiate the function x''n *+(sin x)cos
x with respect to 'x'

16 q'qr6"rq Hrf, dftru : 4
Find the integral :

i-x'1-Sin*\
jr c 'i-cosx/
-:- ldx
3Ie[EII /OR

428 ffGr) tsl lP.r.O.

(1 + cos x)dx or q'Fi Hrf, otBuf
"[',o,
,0
rfi
trvaruare I log(1+ cosx)dx.
"0

L7 q tE^+ 6l
^t,; - nl ii+ n+obcisddqrrovfur maotfrs, sd ;
=i*jn r,
b-i+2j+3k 61 4
Find a unit vector perpendicular to each of the vectors
ri + 6l and (6 _ 6) ,
where 6=in j*t , 6=i*2j+3[.
18. frS (r, z, -4) ta,r+qr& 3t{tqnil x-B y + 19 z-1O y:29 z*5
3 -16 3f{ x-15
7 3 I -5
f, * H--d-6 qr dftr tsr ml 6r-flq HqtowT sro ffiut 4
Find the caftesian equation of the line passing through
the point (1,2, _4) and

!E = L# =
perpendicular to each orthe tines
f = =+ ano
1"s" 4ks e6tr,sks G,ef<er.s an-ilfl51!p { 60
=fdi 2ks q51,4ks q,sfkaks ar?r.T6i
=
T&r t e0 *l uo, 2ks G,3lR skg a.lcrf, 6T
Hg. { 70 *r ,""tr 88 am r-da,
-^,
The cost of 4 kgonion, 3 kg wheat and 2 kg rice is
t 60. The cost af 2 kgonion,
4 kg rvheat and 6 kg r-ice is { 90. The cost of 6 kg onion,
2 kg wheat and 3kg
rice is < 70. Find cost of each item per kg by matrix
method.
gerdrl /oR
frsqfrfud qfrffier fr6Tq d ro ffiru,
Snlve the following system of equations :
2 310
-_+___+___A
XYZ
4 6 5 1
XYZ
€,9 20
-+---_=1
XYZ
20. s628 *fr. nd mq d fr g6.d fr tqio fuqr orn t r rm
@ + Erf apn $* * {n Fil-qr
trqf * I d.+tffit fr m6rg fuilfr 6ifi aTRs frs-$ Ef u+Ea mr sfrTfrd'-\#
T-rtrq
*r tr
A wire of length 28 cm is to be cut into two pieces.
one of the pieces is to be
made into a square and the other into a circle, what
should be the length of the
two pieces so that the combined area of the square and
the circle is minimum?
STefc[I /OR

428 0Gr)
t61

tr6 Tfl efl,q u6 flf b qfrq-rqi m.r drr p *, ad p utr gcx dt fua #hs fu rt'6 &t{Erfr'}
mr rfrrr frFmq *, "7q rif mt UsiT qf b anq-b qq1-q{ *t
-Ihe sum of the perimeter of a circle and a square is p, where p is some
eonstant. Prove that the sum of their areas is least when the side of square is
equai to the diameter of the circle,
2t. flfi x2=4y ud keT x=4y--2 * frt fu mr et{Em gra o1ftul 5

Find the area bounded by the curve x2=4y and the line x=4y-2.
zz. frtafrfua 3{?rcnr{ Hfr6$r b frs ftu gu qfrq€r E'r flW qrqt fleTr frfrrE 6f, ilrd ffiru'
\- U*"'2xY =
(l-x2,
' ox
1

l;-- ) Y=o
qfr x=l 5

Find the particular solution satisfying the given conditions for the following
differential equation ;
(1+ x2) zxy : , y=a when x=1
:*. *.
3[emr /oR
frrSrfrq fr E-{d"@T ffi Ekfia< ffi
ftsrrcr ffirffEb Frilfu frri@Tb HqryqT-ft
*I qfr q
H-'{ 2009 rffiq mt aq$wT 1 6,000 eft 3fl-{ sT 20l + C zo,ooo sfr, d ETd ffitfru
q
fr 2o1e fr rtrq'6t Hffiir*rT mn etrfta
The popL.tlation of a village increases continuously at the rate proportional to the
number of its inhabitants present at any time. If the population of the village
was l-6,000 in 2009 and 20,000 in the year 2014, what will be the population
of the village in 2019?
23. 3rffiq frfBr ar<T Jisq srFI z=-SOx+zOy o.r ;TTf,q uH frqfrfua ffittrl b
3r;ilfd f,m q5tfru: g

Determine graphically the minimum value of the objective function z=-50x+20y
subject to the constraints :

2x-y>-5
3x+y>3
2x-3y<LZ
X,Y>0'
24, LrdD mrr€rr;Ir *, ftrsS uer $ fr -set qx rsr*r fu-qT ETcT tr *r mrrqri fr sm $ * erq
qEri tg"*q 3i?rrr-e s6T-{ d Eaft+ A, B slk c E-Ifr gf tt qyfrt A, ts sil{ c
F"cr sdrlffi
ol srl?l: 25o/a,35o/o 3it{ 40olo ust f S qm sqfr dt g{ii fi 6t$ ft qqtq Til mtrrfr aer
qfr t 6ry qeft{t b rorrr 6-r F{er: s, 4 3ft{ 2 qfrpT-d qmr lq-{r6r tffi; dr wtm

428 (rGF) I7t lP.r.O.

4

tieffiTi si frq qmx * firilfrd fuql Erdr t,
L, : ffi $ S em qqftq A HRj q;flqr,rqT *t
L, : rfr $ S coq w?q B 6r-{i $rTqT qqi * t

L. : wi $ fi qN qafrq c 6RlFTr{tT .rfl *t
srE * fu qffiitr 11, Lz riqrT L tr{w-{ sqq-tr sE qft1d *i qfr se;lT ,8, qft 'eiry qq$E 6iq
ffi qeET's qRrTrfrrT fuqT ET* d 3q*m qrrqrcft m srtm erq ft*Tffif-fffl qyff fu srrc
frffis,
di
(i) b gri{ s6Tr6{ ii * nm EN q]qrrrqi fuerflr mrm *, fr m-i*, risk{ A-iir{r qETs
*lfti
Hr+ frt srfuerTr wlr, j
(ii) qet-{ c gxr qr{rei q-re's+t fli 61mkffiT mr *l j
(iii) q-iib gm reiffi ti* uo E-iq qif#-fl frffir-ffir wm B :fk qr trxis".ii:;; em-m ti
ffiet ror mkom t fu q-s arm'r=,st"h B HKr ffirr{fi,rqr Br 3
There is a factory which nranufactures LED hulbs. ln this factory, three cliffei.ent
type of machines A, B and C are instailed for manufacturing LED br-iibs, f"Iachines
A, B ancl C manufactunes 25o/o,35oz'o and 40% of the total procluetion respecilveiy,
Out c,f these no machine is hundred percent efficient and of therr erutputs 5, 4
and 2 E:ercent are respectively defectlve bulbs. The above events are definec! as
ioilours:
l*. : The LED bulb is rnanufactured by machine A.
L, : The LED bulb is manufactured by macliine B
L, : The LED bulb is manufactured by machine C.
Clearly, L1, L2 and l-. events are mutually exclusive and exhaustive. If event
'E' is defined as the event of 'the bulb [s defective', then mftsed on the above
information, answer the following questions:
(i) A bulb is drawrt at random from the total produeticn, then what is the
probabiiity that it is manufactur-ed by the machine A? 1

(ii) What is the probability of manufacturing a defected bulb by rrachirie C? 1
(iii) A bulh is drawn at random from the totai productiern and is found to he
defective. what is the probability that it is manufacturerj by the machine t3?

,* ,r d. * {.

4?8 GCr')
t8I

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