AHSEC HS 2nd Year Question Paper 2022 Maths

Total nunrber of pages - 16
32T MATH

2022

MATHENTATICS

Full Marks : 100
Pass Marks : 30

Time : Three hours

The figures in the margin inilicate full marks
lor the questions.

Q. No. 1 (a-j) mrries 1 nmrk ent:lr 1 x 10 : 10

Q. Nos.2-73 carry 4 nnrks caclr 4x-12 = 48

Q. Nos. 7tI-20 carry 6 nnrks ench 6x7 = 42

Total = 100

Contd.

1. Answer the following questions : 1x10:10

EE< AIfFK< ES< fiTI :

(a) Give an example of a column matrix which is also a row
matrix.

eA w dt-{ss< tsn<q fut fr6t {ft dt-q-sss e{ |

(b) "Diagonal elements of a skew-symmetric matrix are always
zero"
- Why
?

'frla-mfrs cfi-q-$ffi Fn"f cfrEr<t-< rqm W"
- ftrt
(c) Let f (x)= [x], where [x] is a greatest integer function and
s@) = x . Find the value of (f " s)t %)

<r<lqh ,f(r)=[r],{b [r] Ch trR6 q?rs saFI q1-6 s(x)=x.
(f " g)(- %)< ffi tsfrs{tr

(d) Differentiate sinx with respect to e'.

e'-i II(el(S sinx -{ q<FFdfst tsfrs<t I

2

(e) Write down the value ot l*lax.
-2

2

Jl
,l ar-< ffq fr<t I

2

32T MATH t2)

A Find the order of the differential equation

/ ,q \5
l-l
\dx )
. sin(y,,)= o.

l+41 . sin(y")= 0 q+qq rfr-nqdn or fifs Grr
\dx )

@) Find the principal value of "- '[#)

'- '(#) -<$rniiBfrs<rr

h) Fill in the blank :

{rft rE t< <'{ s
..
Ltm
1

,:r-ro -=
X

O What is the direction cosine of X-axis ?
X-qm fr{ti+ ft{la z

A) kt A and B be any two given sets. If f : A --> B is a onto
function, then hnd the range of /.

<<Iqh A ql$ Bfuqt $r x<& r <fr / : A+ B .4il qEq-s rdFI q{,
FN /< aR'5q ffis<l 1

32T MATH t3l

2. Deline an equivalence relation. check whether the following relation
R delined on the set of integers V, is an equivalence relation or not,
where R = \(a, bll o- b is an integer). 1+3=4

cN?Er-st {rii[.< cw't fr$ t z-\9- q(@Fq s{< cT{ R ctl m!{Jgl {TS
q{r{

de{{a$rsil,{bR=i(q b\l a- b ,sDisqsc(?[l]l
oR/ qqqt

Show that the function /:[i-+ni defined as /(x) =2x*3 is

invertible. Also find the inverse of ..f. 4

cr{srt c{ /: JIi'-+ iR-E csRq f (x)=zx-s qq-artt gffit f-<
sfr6;a6s Efrs< t

3. Show that 4

cq'{€<l c{

. ,3
sln -
,8 = cOS-,84
51785 -

oR/ qsqt

Solve the following equation : 4

vq< q'frc'q& rTl{Ft +lt s

2 tan'l (cos x) = tan-' (2cosec x)

32T MATH t4I

lz el l-r o-l
4. If a=l I and r=l l, thenlindthevalue ). andp
Lt 2) [o r]'

such that A2 + ).A*pI=9, where 0 is zero matrix of order 2.

4

o-l
u* o = ["-l2) ** r = [' 1] *, cs(s i qFF p -< rr{ Efre<t utrc
[r [o

A2 + ),A+ pI= o, {b o tqcR 2 qr< W dr-E+xl

oR / q?ril

Determine the value of a for which the system is consistent. 4

a -{ {l{ frfr mt rK <l-F T{(tlg q{ I

x*!tZ= L
2 x+3Y+22= 2

ctx + aA + 2az= 4

Contd.
32T MATH t5I

5. Find the value of k so that the following function
II-"'^---".
sinlOOx
if x+O
f\x)= 1 ee
I k , if x=o
is continuous at x = 0. 4

^,\ =il"''-'^oo', lfix+o
ffi^ ,f(x) ee
I t ,
{frx=o
T-ffi6t x = o RW qRfu {*, rers k < m fi'fu qtr

du
6. Find ;f if
-
2+2:4

du
sls\e<l
;; ffi{ -

0 sin2 x+ cos2y =1

(tu a= e

7. Prove that the greatest integer function defined by

f G) = k], O < x < 2 is not differentiable at x= l. 4

a:rtq Frl cq ,f (") =k], o < x<2-44Fil q(BK6 utFt o{?,ls pq+61 x= 1

fr-w w<-eqftr {sr
32T MATH 16I

oR/q?F{t

If ({fr) ee(x+l) =1, showthat(cq'{s<tcq) 4

d2u
J
(du\2
___=- |
-t
o*'- \ax) '

R Evaluate : 2+2=4

m fifr mt:
1)a*
(a) l( *r/, * 2., - x)
'\
(b)
I sin3 xcos2 xdx
oR/qqql

Evaluate : 4

ffifift<tc
r x*3
l---u
'l5-4x-x'
-

g. Find the equations of the tangent and normal to the curve
x2l3 + a2l3 =2 at (1, 1). 2+2=4

*'tf+ q+ qF{t.< cfr€q Bfrs<I
*rt, * Art" = 2 {utr{ ( 1, 1 ) fiTo t

32T MATH t7l Contrl.

oR/ qqqt

Find the local maxima and iocal minima, if any, of the function
f(x) = x"-6x2 + 9x + 15. 2+2=4

f (xl = *'- 6x2 + 9 x + 15 +E-{dr< qfft{ ttR* **t+ qfftr eRS $i tsfrs<t,
sft \ffP'r

10. A particle moves along the curve 6y- x3 +2. Find the point(s) on
the curve at which the y-coordinate is changing 8 times as fast as
the x-coordinate. 4

qDr +{tot 6a= *'+ 2 {@GI D6IIF{ qc< I T@tiK ffi frT (6{< ) Bfrs<l {b
x-Ql{l(s s?+ s eq m& c<rtis g-EFili$ "tR<frs qs t

oR/ qelqt

Show that the function /(r) = is neither strictly increasing
"o"Sx
nor decreasing on 1o, f l. 4

cq{sfl c{ /(x)=
"oss,
+f,drtl ( o, f ls qsis<<{-{{ Eflltq .{ble;rq{ I

5

11. Evaluate l@nt)a* as the limit of a sum. 4
0

5

c{plsan 5-$r rl{ forrt J("*t)dx-{ {14 fifi qt t

0

32T MATH t81

oR/ qq?l
4
Evaluate :
an fi"fi ql :
o lz
I Sln X
'0l-'- z dx
I +COS X

12. Show that the vector i + j + [ is equally inciined to the axes OX
4
OY arrd OZ.
qFF OZqs<E lEqwrK aqfr$q
cq-{s<l c{ i + j + t Faffir ox, oY
t

oR/q&t

State the triangle inequality for any two vectors and prove it'
1+3=4
frcorat kil Ffi< <lr<, fr-yu s{ffil frIl eqq mtt
13. Probability of solving a specilic problem independently by A and B
11u.rla resPectively' If both try to solve the probiem
ar" ) I
2+2:4
independently, hnd the probability that -

O the Probiem is soived

(it) exactly one of them solves the problem'

AqFI. B cr eBfirl{q{'fi {saslr<r$$qGFIqslkq @(n i *" i *
cnfllir c{l{-{< il(< Esrn EsEslc< cua or<, cE(g r-slkq fi'fu m {-F
-
(t) rrqldtc{{FI q{
(il csdrqffi ft+ qqr{ clcmil< c{{l4 qu<q{ |

L CONtd.
32T MATH 19 ]

oR/ qs?t
Let X denote the number of hours Rita studies during a randomly
selected school day. The probability that X can take the values x,
has the following form :

, 0.1 if x=0

P( X-xl- I
I kx, if x-lor2
I t(s-x), if x=3 or 4
0, otherwise

where k is an unknown constant.
(a) Find the value of k.
(b) What is the probability that Rita studies at least two hours,
exactly two hours and at most two hours ? 1+1+1+1=4

{I[fuqc< fi-6uq m ffu t-rs cflr{t{f qfi{ fr'-t< q<i-qi'ffit {'J s8t<
c$rl6l x c< lrql<l qh r x< TFr x RKF rstRsr$ firas <q\9- e-$f,.i Grl
?<qe

0.1, {fr x=0
kx, Ifr x: 1 n72
P(x=x) =
k(s-x), <ft x= 3 4t4
o, q;uafi
{b k qh \ss'ls g<rsr
(a) k-< rFr fr{x et r

(b ) ft -sft ovtra 1t sfi cfrs E sfi qt+ lr{w F sfi q<;T{ <qF c.etksrc{K
,

ftrta r

32T MATH [ 10 ]

14. Pind the minors and cofactors of the elements of the
de terminant 3+3=6

604
1 5 -7

lz -3 s
I

lu o 4
ficffiK cfiqr<F< wgtFtwt+ x<tft Bfrs<tt
l1 s -T

oR/ qqql

Find A-l by using elementary transformation, where - 6

dtfr+ <"ffs< Afua1 Acan oR a-t tsfr.g<l qt -

l2 o -1-l
o=l , 1o l

[.' .]

32T MATH [11] Contd.

15. Define homogeneous function of degree n. Solve the differential
equation 1+5=6

(*'+ *s) ay = (* +s, ) d*

n nEFr cffiks rffK ci@t firt r

(** ay = (* *s') a, w<oa qfr-fiqrtt< ml{i Efr\e<t r

"s)

oR/ qqqt

(t) Solve the differential equation :
|)

w+qq q'fi-qq-ca< 6"1qa ffis{t :

x!tclx Qx " 1)g . xe 2'

(ii) Form the differential equation of the family of circles touching
the X- axis at origin.

{qRao x-qs-s -"t f qt$< {fu{ qbt< \ra+q rfi-cqdr $q Ftr

16. Integrate :

q{ffi{ GI 3

c x-l
(a) tl-,l *, _t dx

(b) I xsinl xdx 2+4=6

32T MATH l12l

oR/w!41

(a) ll. 2cos x- 43sinxx
6cos x+ sin )o,
c x3+x+1 .
(b) )- *t, o* 2+4=6

17. For any three vectors a,6,a, prove that

t- \ : 6
dx(b+ i)= axb +axc'

ffi &ffit reE< a, r, a -< <tr< gIl6l sil c{
a"la+ e) = a"6 + d"xE 1

oR/qqqt

Threevectors[,Ianddsatisfytheconditiond+6+6=6.
Evaluate the quantity

p=a'6+6'c+e'd if lal=i'lrl=+ and lcl=z'
6

a,b qr+ c F?<&ffiRa+6+E=o u6fra+r<t
rFI fifr qr ffic< |al= ul6l= +
ql+ |al=2 s{I
t, = a.b + 6.a +6.4<

32T MATH t 13 I Contd'

18. Find the shortest distance between the lines

r =(i+ zj +i)+ 2(i - j+[ ) ana
r =(zi -j - [)* p Qi * j *zi). 6

;=(f + zj +i)+ t(i - j*[ ) wr+
r =(zi - j-[). p(zi * j+z[ ) r+.n T6w{atq<1arvrtrEEBecrr

oR/q*t

Find the equation of the plane passing through the point (-1, 3,2)
and perpendicular to each of the planes x+2A + 3z = 5 and
3x+39 +z = O. 6

(-1, 3,2) R-af< Cm Et+ x+2y + 3z = S qFF 3x+3y +z = O >FNat [cl{<l
e[s]-sGr d"{tEft< qfl {{E6FFF{ q.fi-<q tsfrsn
r

19. Minimizr Z=3x+SA
subject to x+3y > 3
x+a 22
x, a > 0 6

x+3A >3
x+U >2
x, g > o ffmaq ctcqffi z = 3x + sa -< q-ffir tlr Efrs<l r

32T MATH [ 14 ]

oR/q44t
Minimise and Maximise Z = 5x+lOg
subject to
x+ 29 3 72O
x+Y>60
x-29>O
x,a>o 6

y+29 < 12O
x+g>60
x-29>O
x,a>o clc4cfi z = $1 +10y-<cd6qR'{6-fln6ffie<ll

20. Of the students in a co11ege, it is known lhat 600/o reside in hostei
and, 4}yo are day schoiars (not residing in hostel)' Previous
year
results report that 30% of ail students who reside in hostel attain
A grade ind 2Oo/o of day schoiars attain A grade in their annual
eximination. At the end of the year, one student is chosen at
random from the coliege and he has an A grade, what is the
6
probability that the student is a hostlier "
qfi< q<t
,!1q RRkrrqr< 600/o c.qmRcE qt+ 40% fi qtqKlr\5 il?fics 1fr
qm {$rdl qf-Jr{
tts t qttFl <K< FffFd wflR <rqc<fixl etftmts qaTlcs
30% cn qK qqKlcs 4cfsl Sqq?FF{ 2Oo/o (A 6 6qs {mE I <q-{<
qw
w{RqiE{?F< {IFFqr< {6 cI ,{q{ {& A css 416l
qqq-{ WRlr{
qr<rfi m<< qetkq ftrn t
oR/ wqql

Find the mean number of heads in three tosses of a fair coin'
6
qDrfi{g {EI ffitq+R c"n<r Is crllFl rts] fift Ftr
--x-
32T MATH [15 ] Contd.

32r MATH [16 ] 50+