Uttarakhand Board Class 12 Question Paper 2023 for Mathematics

Uttarakhand Board
Question Paper
CLASS 12

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2023
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MATHEMATICS
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1.. (6') cos- ,[r ) gw qrr
lE )6T t_
1

Principat vatue or .or-,[ral
i,

3n
(i) (,i)
7t
1t
4 4 (iii) (iv)
TI

4 3
(q) 3nqr A aEr B to
$rt il q-trq dlt ildd crfr_
1
Matrices A and B wi, be inverse
of each other onry if-
(i) AB=BA (ii) AB=BA=o
(iii) AB=0, BA=I
(iv) AB=BA=I
(rI) sin(x2) rFI ,x, b vrter
$TsdT d_
1
Differentiation of sin(x2)
with respect to ,x, is_

(i) cos(x2) (ii) 2x sin(x2) (iii) 2x
cos(x2) (iv) cos(2x)

(q) crfr f1a +
b - x) : f(x) , d jt* f(x) dx 6r qH d_ I
I

If f(a+b-x)=f(x), tfren f(x) dx is equal to_
Jox

(i) o (ii) a+b
f rro * x)dx
(iii)
b-a
2 f rrrl a* (iv) --t'f(x) dx

(s) 3rn6tr. r",#_3#*y=o
616)Ed_
1

The order of the differential
equation Zr,
*_
dx' 3 +
dx
+ y = 0 is:
(i) 2 (ii) 1 ( iii) 0 (iv) 3
428(rGT)
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(u) Hftrvr i = i + i - zt b @otsrfl d 1

Direction cosines of the vector i = i * i - Zt ut"-
/.r rL -zl
(i) {r
-l\ (ii) (1, 1,_2)
[6,6,6)
(t 1 -2)
(iii) ("[0, Jo, -2,f6) (iv)
[6'G'G)
(u) Hqcrf, e'.(i + i - r?l = 2 dE rtrqerYo<oTt-
The Cartesian equation of the plane r. (i + i - t?l = 2 is-

(i) x+Y-z=O (ii) x+Y-z=2
(iii) x+Y-z=f (iv)x+Y+z+2=0
g)
(H) qfr n sfu s ffiir sf,iTs d, Ed srftrorrE P(A) = 6' 3 f,erT P(B) = 0' 6 d fr p (a n
6-r qH drn- 1

If A and B are independent events, where probabilities P(A)=0'3 and
I P(B)=9,6, then vaiue of P(A n B) will be-

(i) 1 (ii) o" 18
(iii) 0.e (iv) 0'01

2. cos (sec-l x+cosec-l x), lxl>1 61 ql-f Sffl dfret 1

Find the value of cos (sec-l x+cosec-1x), lxl>1'

3. qft ffi slqd fr t+ sa-rqE d, dl E-s6l ritrd drftd mr d nCIft de 1

If a matrix has 14 elements, what are the possible orders it can have?

4. uft v+sinY=cosx d avlax are alBul 1

Find dyldx, if Y+sinY=cosx.

s. r1o; Tcqrd d x gorEqj b f{fiq t q.rq gr-f, 3nq sqdiC n(*)=3x2+36x+s t rdr tl
trd 3{rq f,lcI fiBu ffir x=1s *l 1

product is
The total revenue in Rupees received from the sale of x units of a
given by R(x)=3x2+36x+5. Find the marginal revenue when x=15'

428(rcr) I3l [P'r'o'

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qg1 tan-1x tF[ 'x' b grta $TIIFFFI aa drhgt 1
1+ x'
- tan-1x
Integrate the function with respect to 'x'
, * *,
7 $qroaffiSnc#fre ' 1

Find the integral :

J{+"" + 2)dx.
8 qfr a =2i -sj+k af{ b = 3i+ j+3k dt a.6 o'rqrqErdotBst 1

If A = Zi -lj+lt and 6 = 3?* j*3[, then calculate a,b.

9 x-3TH b vqrf,{ den Tf,-k€ t sr+ aT-fr tler 6r ir$r6<oT sTa o1frsl 1

Find the equation of a line parallel to x-axis and passing through the origin.

10, qRfq riqrciib *ryao q ti uo fuffinfu riftq 'a' fu q.6r{ t vRqfr6 6-
axb=a*ab; a, beQ
aqif$T fu '*' T d oq frFr+q t q fr srEffidt 2

In a set of rational numbers Q, a binary operation 'x' is defined as follows:
axb=a*ab; a, beQ
Show that 'x' is neither commutative nor associative.

gerfl / oR

fue dfrs fr Erffkd ri@T3ii b vgaa R ii R={(a, b):a<b} dRT qfu{rft-c €Eitr
R wgFrT ud fromo tr

Show that the relation R in set of real numbers R defined as R={(a, b):a<b},
is reflexive and transitive.

11. t sFu{Tm flrf, dhs hdi f(x)=2*:-3x2-36x 6KI tr(fl EFT f ffii'qn tl 2

Find the intervals in which the function f given by f(x) =2x3-3x2-36x is increasing.

L2 . " .or' x dx tFI qFI Hrf, olBg
JJ
f 2

rn/2
Evaluate
Jo cos'x dx .
428(rcr) t4I

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13. A Hrdrmi 3x-6y+22=10 3il-{ 2x+2y-22=15 b frs mr 6\ur ffi dfret 2

Find the angle between the two planes 3x-6y+22=10 and 2x+2y-22=t5'

14. u6 dffi C too sd f;, kg'ii 10 ZEgtF f;t s oeq b q{f fi t, m qft s6 b zfrrytr
q 6Yl d qrftrdaT ffirt? 2

In a box containing 100 bulbs, 10 are defective. What is the probability that out
of a sample of 5 bulbs, none is defective?

1s. f(x)=4va36KIrdif,H?l=Tf : R-+ REKkdRdiBu'f fudotfrsfuf g5uoftqf,1 I
o,qftrdms?Flarodfrul a

consider the function f : R -+ R given by f(x)=4x+3' show that f is inveftible'
Find the inverse function of f.

3[e[dr / oR

lgd61tutu-
Prove that-

,un-'I A+t-fi-; )"1-,
l----cos-x 1
-_ <x<1.
1+x+ 1-x )42 .,12

t-r 2 0l I-ol
16. xbfueulqbftruF 2 rl Iz ,l lrl=o tr
0 4

Ir 0 ,i L..l
L)

[r z ol 0
For what value of x , I z rl lz o 1l 2 =0?
[roz) X

tr. qfr x=a(cos t+t sin t) 3ltt y=61sin t-t cos t), d # ar+ 6Bgt 4

d,Y
If x=a(cos t+t sin t) and y=315in t-t cos t), find d*,

428 (rGT) tsl I P.T.O.

d

xtanx
1B
I sec x+tan x dx o.r qFi srd olhur
4

rvatuate xtanx
f sec x+tan x dx
3IeE[/9p

r 3x-2
J (x +rfx + 3) d* or qrf, Hrd otBur
3x -2
rvatuate J
(x + 1)2 (x+3) dx

1e' qrfr a=i*j** sik 5=2i-j+s[
a] 2 d-b den Ia,br 6TqTqgroot*ut
If e=i* j*[ ana b=2?_ j+:ti +
thenfindthevalueof 2a_b
and la*b1.
20' ts*n F=(i+i)+ )'(2i-j*rir
sS< r =Qi* j-(l+p(3i-sj+z()
b frd bt I
q-{d,T gfi ara atBsr
4
Find the shortest distance
between the rines F
= (i + i) + ),(zi _j * (l and
r =ei * j-rtl+p(3i_sj+z().

SrerEI/OR

ss B€ b frfu6 sra 6tBu
Hd fugg,T A(3, 4,1) 3f{ B(s,
xY-dm of mnrft dr
1, 6) 6} m ilfr tesr
Find the coordinates of
the point where the rine through
and B(5, 1, 6) crosses the the points A(3, 4, r)
Xy_plane.

27 ffiftH furq oi snq.a frf$ * 6d 6tBu_
Solve the foilowing system 5
of rinear equations, using
matrix method_
x+y*z=6
x+32=11
x-2y*z=O
428 ocr)

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22. fud dttss B r Bsr b Tild b 3irfa Ekrq 3rr{kFT b riq gsrq qi$ fi *,r$
}t *r
Show that the altitude of the right circular cone of maximum volume that can
be inscribed in a sphere of radius r is
$.
3terEr/OR

qrtrtq y2=4ax b frE (at2, 2at) q< Hqf ts[ 3fu 3Tffrdd e s+6iq wra mtfrsi
Find the equations of the tangent and normal to the parabola y2=4ax at the
point (at2, 2at).

23. Ad€fr
+.+= 1 Et tqt zx+:y=6 t ftrt mg d-* a;r strhr f,rd ettsut 5

Find the area of the smaller region bounded bythe ellipse
+.+= 1 and the line
2x+3Y=6.

24. 3i?tcbd Hftd$l (x+y)dy+(x*y)dx=0 6T BfrIE Em f,t?T ffiu, fuo, E:n d fu y=1
qfr x=1. 5

Find the pafticular solution of the differential equation (x+y)dy+(x-y)dx=0, given
that y=1 when x=1.

3IetErT/oR

3{q6m HlflffiWT j} . , cot x = 4x cosec x (x * o) Or U6 frfr[S Ff, ElcT dBU, frrn pri I

t fu y=g qfr x=n/Z.
Find a particularsolution of thedifferential equation
f|.rcotx = 4xcosec x (x * 0),
T

J
given that y=6 when x=n/2.
)
2s. 3rTffiq FdRi t fr{T offitltiib sFilfd z=6ox+40y 6r qrdfto-{oT uei sfEroilftowT
f,rd dfrs- 5
3
By graphical method, minimise and maximise Z=6Ox+40y under the following
constraints: g
x+2y<LZ
2x+y<12 rl
4x+5y>20 e
XrY>0

).
428 (rcr) t7 I I P.T.O.

26. E6 qrfiTrko FHfilr b qrs A, B erqr 6 qqfr{ 3ffie( dt qerq3ffi{ A, 4o/o q{d
grfift Tilrftm ortTT t aefi 3trqtfi s 3f{ c F-rryr: 5olo oil{ 7olo Er{w rnqfi 3qrfu w}
dt ord q{ A e-iT Hrr{i $T 50o/o dFIIcrT t, t S-a Hrrq tnT 30o/o derl c graT srr€t 61 20olo
dIlrdr tt qfr ED s<ttr Hrcrft Jeqrka t fi St B 6RT sflTka H Hr+ # mkoar wr
ti5
A manufacturer has three machine operators A, B and C. The first operator A
produces 4a/o defective items, where as the other two operators B and C
produces 5o/o and 7o/o defective items respectively. A is on the job for 5Oo/o of
the time, B is on the job for30o/o of the time and C is on the job for2Oo/o of the
time. A defective item is produced, what is the probability that it was produced
bv B?

3Prqr/oR
qTdi b t6 + d +{ dl( ffi-ml q( mi (doubrets) 61 dqsT or mfr'-oar dm sna
frBur
Find the probability distribution of number of doublets in three throws of a pair
of dice.
*****

428(rcT) I8I

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