Upload
shantanusingh
View
225
Download
0
Embed Size (px)
Citation preview
8/9/2019 Question Bank of 12 Class
1/49
DELHI PUBLIC SCHOOL
VARANASI
MATHEMATICS - XII
MATRIX
One Mar k Questions
Q ! If [ x+3 y y7− x 4 ] = [4 −10 4 ] , nd the values of x and y.
Q" If matrix A= [1 2 3] write AA’, where A’ is the transose of matrix A.
Q# If [1 23 4] [3 12 5] = [7 11k 23 ] , then write the value of !.
Q$ If A = [cosα −sinα sinα cosα ] , then for what value of α is A an identity
matrix"
Q% If a matrix has # elements, write all ossi$le orders it %an have.
Q& If [2 35 7][ 1 −3−2 4 ] = [−4 6−9 x] , write the value of x.
Q' &imlify %os θ [ cosθ sinθ−sinθ cosθ] 'sin θ[ sinθ −cosθcosθ sinθ ] .
Q( (ind the value of x' y from the followin) e*uation+
2 [ x 57 y−3]+[3 −41 2 ] = [ 7 615 14]
8/9/2019 Question Bank of 12 Class
2/49
Q) If A = [ 3 4
−1 20 1] and -= [−1 2 11 2 3] , then nd AT -
Q!* If A is a s*uare matrix su%h that A2 =A, then write the value of /I' A0 2
3 A.
$+& Marks Questions
Q ! If A = [ cos θ τ sin θτ sin θ cos θ ] , then rove $y rin%ile of mathemati%al Indu%tionthat
An
= [ cos nθ τ sin nθτ sin nθ cosnθ ] .
Q " et A= [3 2 54 1 30 6 7] . xress A as a sum of two matri%es su%h that one issymmetri% and the other is skew symmetri%.
Q # If A= [1 2 2
2 1 2
2 2 1] , verify that A2 A#I =4.Q $ usin) elementary row oeration nd the inverse of the followin) matrix+
[2 51 3]
8/9/2019 Question Bank of 12 Class
3/49
Q%xress the followin) matrix as the sum of a symmetri% and s!ew
symmetri%, and verify your result+
[ 3 −2 −43 −2 −5−1 1 2 ]
Q& (or the followin) matri%es A and -, verify that /A-0’ =-’A’.
A= [ 1−43 ] , -= /1 2 10Q' 5sin) elementary transformations, nd the inverse of the matrix+
[
−1 1 21 2 3
3 1 1
]
Q( If A−1
= [ 3 −1 1−15 6 −5
5 −2 2 ]andB=[ 1 2 −2−1 3 0
0 −2 1 ] , nd /A-0 1.Q) &how that the elements on the main dia)onal of a s!ew symmetri%
matrix are all 6eros.
Q!* ,or the matrix A, show that A' AT
is a symmetri% matrix.
7eterminants
SET- I.
One /ark Questions.
Q! 8rite the value of the determinant [ 2 3 45 6 86 x 9 x 12 x ]
8/9/2019 Question Bank of 12 Class
4/49
Q" 8hat is value of the determinant |0 2 02 3 44 5 6
| "Q# (ind the minor of the element of se%ond row and third %olumn (a23) in
the followin) determinant+
|2 −3 56 0 41 5 7
| Q$ If A is a s*uare matrix of order 3 and |3 A| =9 | A| , then write the
value of 9.
Q% (or what value of x, the matrix [5− x x +12 4 ] is sin)ular"
Q& 8rite A−1
for A = [2 51 3] .
Q' valuate+ |cos15° sin15 °sin 75° cos75°|.
Q( If | x x1 x| = |3 41 2| , write the ositive value of x.
Q) If ∆
=
|5 3 8
2 o 11 2 3
|,
write the minor of the element
a23.
Q!* et A $e a s*uare matrix of order 3 × 3. 8rite the value of |2 A| ,
where | A| = .
8/9/2019 Question Bank of 12 Class
5/49
$+& Mark Question
Q! 5sin) roerties of determinant, rove that+
|a−b−c 2 a 2 a2 b b−c−a 2 b2 c 2 c c−a−b| = /a' $' %0 3.
Q" 5sin) matri%es, solve the followin) system of linear e*uation+
x' y' 6 =: 2xy'6 =1: 2x'y36 =;
Q$ 5sin) roerties of determinants, rove that+
|1 1 1
a b c
a3
b3
c3| = /a$0 /$%0 /%a0 /a' $' %0
Q% &how that the matrix A= [ 3 1−1 2] satises the e*uation A2 #A ' = π , show that |sin ( A +B+C ) sinB cosC
−sinB 0 tan Acos ( A+B) −tanA 0 | = 4.
Q' (or the matrix A =
[
3 2
1 1
], nd the num$ers a and $ su%h that A
2
'
α A '$I =4, hen%e nd A−1
.
8/9/2019 Question Bank of 12 Class
6/49
Q( If A = [2 −3 53 2 −41 1 −2 ] 0 nd A−1 5sin) A−1 solve the followin)
system of linear e*uation+
2x3y'#6 =11, 3x'2y6=#, x'y26 =3.
Q) 5sin) roerties of determinants, rove the followin)+
|1+a
2−b2 2 ab −2b2 ab 1−a2+b2 2 a2 b −2a 1−a2−b2
| = /1'a2'$203.
Q!* | x x2
1+ px3
y y2
1+ py3
z z2
1+ pz3| = /1' pxyz¿ ( x− y ) ( y− z ) ( z− x ) , where p is any s%alar.
Q!! If A =
[
2 −3 53 2 −4
1 1 −2
], nd A
−1 . 5sin) A−1
solve the followin)
system of e*uation+
2 x −3 y+5 z=16 ;3 x+2 y−4 z=−4 ; x + y−2 z=−3
Q !" &olve for x, y, 6
2
x '
3
y '
10
z = :
4
x
−6
y '
5
z =1:
6
x '
9
y
20
z = 2
13. 5sin) roerties of determinants rove the followin)+
8/9/2019 Question Bank of 12 Class
7/49
| a
2bc ca+c2
a2+ab b2 caab b
2+bc c2 | = a2 b2 c2
CONTINUIT1
One /ark Questions
Q! xamine the %ontinuity of the fun%tion f ( x)= x2+5 at x=−1.
Q" xamine the %ontinuity of the fun%tionf ( x)=
1
x+3, , x ϵ R .
Q# ?ive an examle of fun%tion whi%h %ontinuous at x =1, $ut not
di@erentia$le at x = 1.
8/9/2019 Question Bank of 12 Class
8/49
Q$ If fun%tion, f ( x)=
2 x+3sin x3 x+2sin x for x
o , t!"nfind f ( x)
Q% &tate the oints of dis%ontinuity for the fun%tion
x
∫( x)= [¿,∈−3
8/9/2019 Question Bank of 12 Class
9/49
Q$ (ind the value of ! su%h that the fun%tion $ f $ dened $y
f ( x )={k cos x
π −2 x , x
π
2
3 , x=π
2
%ontinuous at =π
2 .
Q% (ind the value of ! so that the fun%tion f , dened $y
f ( x)={kx +1 ,i f x# π cosx,if x>π is %ontinuous at x=π .
Q& (or what value of % , is the fun%tion
f ( x )={ %( x
2
−2 x) ,if x# 04 x+1 ,ifx>0 >ontinuous at x=4"
Q' If the fun%tionf ( x)={
3 ax +b, if x>111 ,ifx=1
5 ax−2 b ,if x< 1
Is %ontinuous at x=1, nd the value of a and $.
Q( 7etermine the values of a, $ and % for whi%h the fun%tion
8/9/2019 Question Bank of 12 Class
10/49
f ( x)=
{
sin (a+1 ) x+sin x x
, x0
&ay b" contin'o's at x=0.
DI44ERENTIATION
One /ark 2uestions
Q! Ifif x=sin θ , y=−tanθ, find
dy
dx .
Q" 7i@erentiate, cos−1 √ x , with rese%t to x.
Q# 7i@erentiate, "& tan−1 x ,
with rese%t to x.
Q$ 7i@erentiate, sin { log ( x2−1 ) } , with rese%t to x.
Q% ow will you ro%eed to nd derivative of (sin x¿ x (
Q& 7i@erentiate, x , )it! *"sp"ct
cos ¿" x¿ .
8/9/2019 Question Bank of 12 Class
11/49
Q' 7i@erentiate the followin) w.r.t.x+ y =
xsin ¿
¿log ¿
5¿
.
$ Mark 2uestions
Q! If x y
= " x− y
, rove that
1++ox ¿2 .
¿dy
dx=
+ox¿
Q" If y=tan−1[ √ + x
2+√ 1− x2
√ 1+ x2−√ 1− x2 ] . &how thatdy
dx=
− x
√ 1− x4 . .
Q# If x+ y ¿ p+-
x p
y-=¿ , rove that
(i ) dy
dx=
y
x∧ (ii )
d2 y
dx2 = 4.
Q$ If
3+ x1+ x
¿2+3 x
f ( x )=¿ , nd f’/40.
Q% If = "ax
sin bx ,t!"n p*o"t!at x2 d
2 y
dx2 −2 a
dy
dx+(a2+b2 ) y=0.
Q& If x√ 1+ y ' y √ 1+ x = 4 for
1+ x ¿2.
¿
−1
8/9/2019 Question Bank of 12 Class
12/49
Q' If y=
sin−1
x
√ 1− x2 1− x2
,s!o)t!at ¿ 0 d
2 y
dx2 / 3 x
dy
dx / y=0.
Q( If x =
t
t −tcost
sin ¿ ,t!"nfind d
2 y
dx2
¿cost +t sin ¿∧ y=a ¿
a¿
.
Q) If x= ¿a (θ−sinθ ) , y=a (1+cosθ ) , find d
2 y
dx2 .
Q!* If sin y = x sin /a'y0, rove thatdy
dx ¿
sin2(a+ y )sina
.
Q!! If y = / x
2+1 ¿2 d
2 y
dx2 + 2 x ( x2+1)
dy
dx=2.
tan−1
x ¿2 , p*o"t!at ¿
Q!" If x = √ asin−1t
, y = √ acos−1 t , show that
dy
dx =− y x .
Q!# If x= a /%os t' t sin t0 and y =a /sin t Bt %ot t0, 4 ¿ t <
π
2 , ndd
2 x
dt 2 ,
d2 y
dt 2 .
,∧d2 ydx
2 .
8/9/2019 Question Bank of 12 Class
13/49
Q!$ If y = "acos−1 x
,−1# x# , s!o) t!at (1− x2 ) d2 y
dx2 .− x
dy
dx−a2 y=0.
Q!% If x16
y9
= / x2
'y0, rove thatdy
dx = y
x
A556i7ations o, Deri8ati8es
One /arks 2uestion
Q! (ind the rate of %han)e of volume of the %one of %onstant hei)ht, withrese%t to radius of the $ase.
Q" (ind the an)le C, whi%h in%reases twi%e as its sine.
Q# he total %ost >/x0, asso%iated with the rodu%tion of x units, of an item
is )iven $y >/x0 =4.42 x3
' x2+1000. (ind the mar)inal %ost, when #
items are rodu%ed.
Q$ (or what value of a, the fun%tion f ( x )=a ( x+sinx )+a , is in%reasin) on D.
Q% &how the, the fun%tion f ( x)=log ( cosx ) is de%reasin), in [0, π 2 ] .
Q& &how that f ( x )=( x−1)" x
+1 is an in%reasin) fun%tion, for x E4.
Q' (ind the sloe of the normal to the %urve x=
1
t , y =2t at t = 2.
8/9/2019 Question Bank of 12 Class
14/49
Q( Frove that the tan)ents to the %urve y= x3+6 at the oints / 1,#0 and
/1,
8/9/2019 Question Bank of 12 Class
15/49
[Kisi$le area A at hei)ht h is )iven $y A=2 π*
2!
*+! ].
Q$Awater tan! has a shae of an inverted ri)ht %ir%ular %one with its axis
verti%al and vertex lowermost. Its semiverti%al an)le istan
−1 (0.5 ) . 8ater is
oured into it at a %onstant rate of # %u$i% metre er minute. (ind the rate atwhi%h the level of the water is risin) at the instant when the deth of waterin the tan! is 14 m.
Q% f ( x )=2 x3−15 x2+36 x+17 Q&
f ( x )= x3+ 1
x3 , x 0
Q' (ind the intervals in whi%h the fun%tion f i"nby f ( x)=sin x+cos x , 0 # x 2 π ,
is stri%tly in%reasin) or stri%tly de%reasin).
Q( (ind the values of x for whi%h x ( x−2)¿2
f ( x )=¿ is an in%reasin) fun%tion. Also
nd the oints on the %urve, where the tan)ent is arallel to x Baxis.
Q) Frove that y ¿
4sin θ
2+cos θ−θ
is an in%reasin) fun%tion of L [4 ,
π
2 ¿ .
Q!*&how that y=log (1+ x )−
2 x
2+ x , x>−1
is in%reasin) fun%tion of x, throu)h
out its domain.
Q!! (ind the e*uations of the tan)ent and normal to the %urve x1, y 1
16 x2+9 y2=144 at ¿ 0, where
x1 =2 and y1 ¿0. also, nd the oints of
interse%tion where $oth tan)ent and normal %ut the xaxis.
Q!"(ind a oint on the ara$ola x−3 ¿2
f ( x )=¿ , where the tan)ent is arallel to
the %hord Moinin) the oints, /3.40 and /, 10.
Q!# &how that the area of the trian)le formed $y the tan)ent and the
normal at the oint /a, a0 on the %urve y2
/2a Bx0 = x3
and the line x=2a,
is5 a
2
4 s- . units.
8/9/2019 Question Bank of 12 Class
16/49
Q!$ Frove that the %urves y2=4 ax∧ xy=c2 %ut and ri)ht an)les if c
4
=
32 a4
.
Q!% (ind the oints on the %urve y= x3
N 3 x2
' 2x at whi%h tan)ent to the
%urve is %urve is arallel to the line yN2x'3 =4.
Q!&(ind e*uation of the tan)ent to the %urve x = sin 3t, y = %os 2t, at t =π
4 .
Q!'(ind the oints on the %urve y= x3
at whi%h the sloe of the tan)ent is
e*ual to y%oordinate of the oint.
Q!( (ind the e*uation of the tan)ent to the %urve y = √ 3 x−2 whi%h isarallel to the line x2y'#=4.
Q!) (ind the e*uation of the tan)ent and normal to the %urve x=1N
cosθ , y=θ−sin θatθ=π
4 .
Q"* (ind the oint on the %urve y= x3−11 x+5 at whi%h tan)ent has
e*uation y =x B 11.
Q"! An oen $ox, with a s*uare $ase, is to $e made out of a )iven *uantity
of metal sheet of area c2
. &how that maximum volume of the $ox isc
3
6 √ 3
.
Q"" A ri)ht %ir%ular %ylinder is ins%ri$ed in a )iven %one. &how that the%urved surfa%e area of %ylinder is maximum when diameter of %ylinder ise*ual to radius of $ase of %one.
Q"# A window is in the form of a re%tan)le a$ove whi%h there is a
semi%ir%le. If the erimeter of the window is p
%m. &how that the windowwill allow the maximum ossi$le li)ht only when the radius of the semi%ir%le
is p
π +4 %m.
Q"$ An oen tan! with a s*uare $ase and verti%al side is to $e %onstru%tedfrom a metal sheet so as to hold a )iven *uantity of water. &how that the
8/9/2019 Question Bank of 12 Class
17/49
%ost of the material will $e the least when the deth of the tan! is half of itswidth.
Q"% Of all the re%tan)les ea%h of whi%h has erimeter 4 metres, nd onewhi%h has maximum area. (ind the area also.
Q"& &how that the volume of the )reatest %ylinder whi%h %an $e ins%ri$ed
in a %one of hei)ht ! and semiverti%al an)le 34G is4
81 π!
3
.
Q"' &how that volume of )reatest %ylinder whi%h %an $e ins%ri$ed in a
)iven ellise x
2
a2+
y2
b2=1 .
Q"( &how that the semi verti%al an)le of a ri)ht %ir%ular %one of )iven total
surfa%e area and maximum volume is sin−1 1
3 .
Q")A oint on the hyotenuse of a ri)htan)led trian)le is at distan%es aand $ from the side .&how that the len)th of the hyotenuse is at least /
a2 /3+b2/3¿2 /3 .
Q#*&how that the hei)ht of the %ylinder of maximum volume that %an $e
ins%ri$ed in a shere of radius R is
2 R
√ 3.
also nd the maximum volume.
Q#! If the len)th of three side of a trae6ium other than the $ase is e*ualto 14 %m ea%h, then nd the maximum area of the trae6ium.
Q#" &how that of all the re%tan)les of )iven area, the s*uare has thesmallest erimeter.
Q## A window has the shae of a re%tan)le surmounted $y an e*uilateraltrian)le. If the erimeter of the window is 12 m, nd the dimensions of there%tan)le that will rodu%e the lar)est area of the window.
Q#$ &how that the hei)ht of altitude of a ri)ht %ir%ular %one of maximum
volume that
8/9/2019 Question Bank of 12 Class
18/49
INDE4INITE INTE9RALS
One /arks 2uestion
Q! ∫ s"c2
/
8/9/2019 Question Bank of 12 Class
19/49
Q# ∫ dx
√ 1− x2 . Q$ valuate+
∫ 21+cos2 x dx
Q% ?iven ∫" x
/tan x'10 se% x dx = " x∫( x) ' %. 8rite ∫ ( x ) satisfyin)
the a$ove.
Q& ∫2−3sin x
cos2 x
dx . Q' ∫
x
√ x+2 dx.
Q(
x1+log ¿
.¿ x cos
2 ¿dx¿
∫ ¿
Q)
tan
¿¿−1 x ¿2
¿¿¿
∫ ¿
dx.
Q!* If ∫( x−1 x2 )" x dx=f ( x ) " x +c , t!"n)*it" t!" a+'" of f ( x ) .
$+& Marks Questions
Q ∫ x
2+1
x4+ x2+1
dx . Q" 0 "
2 x+1"
2 x1−1dx
8/9/2019 Question Bank of 12 Class
20/49
Q#
log x¿2
¿¿ x
16+¿√ ¿¿∫¿
Q$
x x
2−cos¿¿¿
1−cos ¿¿
¿sin x¿
∫ ¿
Q% ∫sin ( x−a)sin ( x+a)
dx Q& ∫√ tan x dx .
Q' ∫{ 1
+ox− 1
(+ox ¿2 } d x Q( ∫ x
x4− x2+1
dx
Q) ∫√ 2 ax− x2
dx.
Q!* ∫ 1
a2
sin2 x +b2cos2 x
dx Q!! ∫ x
2+4 x
4+16dx
Q!" ∫ x
4dx
( x−1 )( x2+1) Q!#∫" x( sin 4 x−41 / cos 4 x )dx
Q!$ ∫ 1− x2
x (1−2 x ) dx Q!%
x+2 ¿2
¿¿
3 x−1
¿∫ ¿
Q!& ∫ x2
tan−1
x dx Q!' ∫√ 1−√ x1+√ x dx
8/9/2019 Question Bank of 12 Class
21/49
Q!( ∫ sin x+cos x
√ 9+16sin2 xdx
Q!) ∫√ a− xa+ x dx.
Q"* ∫" x
/se% x 'se% x tan x0 dx Q"!
ax+b ¿ ]f ¿
f $ (ax+b)¿
∫¿n dx.
Q"" ∫ dx
√ ( x −α )( 2 − x) , 2>α Q"# ∫
x2
x4+ x2−2 dx
Q"$ ∫sin6 x+cos6 xsin
2 x cos
2 x dx Q"%
∫ √ x√ a3+ x3 dx
Q"&
1− x ¿2
¿¿
(2− x )" x
¿∫¿
. Q"' ∫ x
4+1 x
2+1 dx.
Q"( ∫ x cot−1
x dx . Q") ∫tan θ+ tan2θ
1+ yan3θdθ .
Q#*
x sin x+cos x ¿2
¿¿
x2
¿
∫ ¿
dx Q#! ∫" x
/se% x' se%
x tan x0 dx.
8/9/2019 Question Bank of 12 Class
22/49
Q#" ∫ 1
sin ( x− p ) cos( x−-) dx. Q## ∫√ cot x dx.
Q#$
x
sin¿¿¿¿
x+log ¿1+cot x
¿∫ ¿
dx. Q#% ∫"
x+ x"−1
" x+ x" dx.
DE4INITE INTE9RALS
! Marks Questions3-
Q! ∫0
1dx
1+ x2 . Q"∫
/ π / 2
π /2
sin5 x dx .
Q# ∫2
31
x dx Q$ If
3 x2dx¿
∫0
a
¿ =J,
write the value of a’.
Q% If
3 x2+2 x+k ¿
∫0
1
¿ 0 dx =4, (ind the value of !.
8/9/2019 Question Bank of 12 Class
23/49
Q&
3sin x−4 sin3
¿
∫0
π /3
¿ x0dx. Q' ∫
π / 2
π / 2
log|2 / sin x2+sin x| dx.
Q( ∫ / 1
1 x
3
1+ x2 dx. Q) ow will you re%ede
¿ "a+'at"∫−1
1
| x| dx"
Q!* 8hi%h roerty will you use to evaluate, ∫3
8
√ 11− x√ x +√ 11− x dx.
$+& Marks Questions3-
Q! ∫−5
0
f ( x ) dx,)!"*"f ( x )=| x|+|1+ x|+| x+5| Q" ∫π /6
π /3dx
1+√ tan x
Q# ∫0
1
log( 1− x x )dx .∨∫01
log( 1 x −1)dx . Q$
1− x2¿3/2
¿¿
(sin−1 x)¿
∫0
1/√ 2
¿
dx.
Q% ∫1
4
[| x−1|+| x−2|+| x−4|] dx Q& ∫−1
3/ 2
| x sin π x| dx
8/9/2019 Question Bank of 12 Class
24/49
Q' ∫0
3/ 2
| x cosπx| dx. Q(
∫0
x /2 x−sin x1+cos x dx.
Q) ∫0
π / 4sin x+cos x9+16sin2 x dx Q!*
∫−1
2
| x3− x| dx.
Q!! Frove that+ ∫0
π / 2
(√ tanx+√ cot x) dx =P
Q!" Frove that+ ∫0
π / 4log (1+tanθ ) d θ=
π 8 lo) 2.
Q!# Frove that+ ∫0
a
f ( x ) dx=∫0
a
f ( a− x ) dx . usin) it, evaluate+ ∫0
2
x √ 2− x dx.
Q!$ Frove that+ ∫0
a
f ( x ) dx=∫0
a
f ( a− x ) dx and hen%e, rove that and hen%e,
rove that ∫0
π /2sin x
sin x+cos x dx =π
4 .
8/9/2019 Question Bank of 12 Class
25/49
Q!% Frove that+ ∫0
a
sin−1√ xa+ x dx = a2 /PN20.
Q!& Frove that, ∫0
π /4
(√
tanx+√
cot x )dx= √
2.
π
2 Q!' ∫0
1
log (1+2 x )dx
Q!(
1−cos ¿5/2
¿¿
√ 1+cos x¿
∫π /3
π /2
¿
dx Q!)
8/9/2019 Question Bank of 12 Class
26/49
a¿
¿2cos2 x+b2sin2¿2
¿¿
dx
¿∫
0
π /2
¿
Q"*
x
sin x+cos¿dx¿
log ¿
∫−π /4
π /4
¿
Q"!
∫0
π / 2tan x
1+&2 tan2 x dx
Q"" ∫0
π x sin x
1+cos2 xdx . Q"#
∫0
1
cot−1 (1− x+ x2 ) dx.
Q"$ ∫0
π x tan x
s"c x+ tan x dx Q"%
∫π / 6
π /3sin x+cos x√ sin2 x dx
Q"&
1− x ¿n
∫0
1
x2
¿ dx. Q"' ∫
0
π /2 x sin x cos x
sin4
x+cos4
x dx
8/9/2019 Question Bank of 12 Class
27/49
Q"( ∫1
3dx
x2 ( x+1 ) . Q ")
∫0
π / 2
x cot x dx.
Q#* ∫2
3
√ x√ x +√ 5− x dx. Q#!
∫π / 6
π /31
1+√ tan x d x.
A556i7ations o, t:e inte;ra6s $+& /arks Questions+
Q ! (ind the area of smaller re)ion $ounded $y the ellise x
2
a2 <
y2
b2 =
1 and the strai)ht line x
a < y
b = 1
Q" (ind the area of the re)ion $ounded $y the %urve y = √ 1− x2
, line y
=x and the ositive x axis.
Q# 5sin) inte)ration, nd the area of the re)ion in the rst *uadrant
en%losed $y theaxis, the line x = √ 3 y and the %ir%le x2
' y2
=.
8/9/2019 Question Bank of 12 Class
28/49
Q$ (ind the area of the re)ion en%losed $etween the two %ir%les+
x−1¿2+3 2
x2+ y2=1,¿ =1.
Q% 5sin) inte)ration, nd the area of the %ir%le x2
' y2
=1H whi%h is
exterior to the ara$ola y2=Hx.
Q& 5sin) inte)ration, nd the area of the re)ion $ounded $y the ara$ola
y2 =x and the %ir%le x2 'y2 =;.
Q' Frove that the %urves y2 =x and x2 =y divide the area of the s*uare
$ounded $y x=4, x =, y=, and y=4 into three e*ual arts.
Q( 5sin) inte)ration, nd the area of the re)ion +Q /x, y0+ ;x2'y2 R 3H and
3x 'y S HT
Q) 5sin) inte)ration, nd the area of the followin) re)ion+
{( x , y ) :| x−1|# y #√ 5− x2 }
Q!* 5sin) inte)ration, nd the area of the re)ion+
{( x , y ): x2
9 +
y2
4 #1 #
x
3+
y
2 } Q!! 5sin) inte)ration, nd the area of the re)ion+
Q/x, y0+ | x +2|# y#√ 20− x2
T.
Q!" nds the area of the re)ion Q/x, y0+ x2
' y2
# 4, x+ y 4 2 }.
Q!# 5sin) the method of inte)ration, nd the area of the UA->,
%oordinates of whose are A/2 verti%es,40, -/,#0and >/H,30.Q!$ 7etermine the area en%losed $etween the %urve y =x B x2 and the
xaxis.
8/9/2019 Question Bank of 12 Class
29/49
Q!% 7raw a rou)h s!et%h of the re)ion Q/x, y0+ y2
R Hax, x2
' y2
R
1H a2
T. Also, nd the area of the re)ion s!et%hed, usin) method of
inte)ration.
Q!& 5sin) the method of inte)ration, nd the area $ounded $y the %urve
| x| ' | y| =1.
Q!' (ind the area of the re)ion $ounded $y the ara$ola y=x2 and y =
| x| .
Q!( &!et%h the )rah of y ¿| x+3| and evaluate ∫−6
0
| x+3| dx. 8ith 5sin)
inte)ration "
Q!) 5sin) inte)ration, nd the area of the re)ion )iven $elow+
Q/x, y0+4 R y R x2 '1, 4R y R x'1, 4R x R 2T.
Q"* (ind the area en%losed $y the %urve > =3 %os t, y=2sin t.
8/9/2019 Question Bank of 12 Class
30/49
Di?erentia6 E2uation
One Marks Question3-
Q! 8rite the order and de)ree of the di@erential e*uation /d
2 y
dx2 ¿3−5
dy
dx
'H =4.
Q" 8rite the order and de)ree of the di@erential e*uation xN %os (dy
dx
)=4.
Q# (ind the order and de)ree of the di@erential e*uation y= x '
√ 1+ p2. . 8here =dy
dx .
Q$ ow will you ro%eed to solve the di@erential e*uationdy
dy =
1'x'y'x y"
Q% 8rite the order and de)ree of the di@erential e*uationdy
dx ' sin
( dydx ) =4.
$+& Marks Questions3-
8/9/2019 Question Bank of 12 Class
31/49
Q!dy
dx+ y cot x=2 x+ x2 cot x , i"n t!at y (0 )=0.
Q" 2 x2
dy
dx−2 xy+ y2=0; y (" )=" Q# ( y
2− x2 ) dy
=3xy dx
Q$cos
2 x
dy
dx+ y tan x . Q%
( y +3 x2) dydx
= x .
Q&
dy
dx−
y
x '%ose% ( y
x
) =4: y=4 when x=1.
Q'dy
dx+ y cot x=4 x cos"c x , ( x 0 ) , i"nt!at y=0 )!"nx=
π
2 .
Q(( x2+1 ) dy
dx+2 xy=√ x2+4 .
Q) " x
tan y dx ' /1 " x
0 s"c2 y dy=0.
Q!* x d y '/y' x3
0 d x =4
Q!!dy
dx+2 y tan x=sin x, i"nt!at y=0, )!"n x=
π
3
Q!"dy
dx=1+ x2+ y2+ x2 y2,
)iven that y = 1 when x =4.
Q!# (ind the arti%ular solution of the di@erential e*uation+ x / x2
N10
dy
dx 1; y=0)!"n x=2.
8/9/2019 Question Bank of 12 Class
32/49
Q!$ &olve the followin) di@erential e*uation+ /1' x2
0 dy '2xy dx =%ot xdx:
xV4.
Q!% (ind the arti%ular solution of the followin) di@erential e*uation+
" x
√ 1− y2
dx ' y
x dy =4, x =o, y =1.
Q!& (orm the di@erential e*uation reresentin) the family of %urves y=A %os
/x'-0, where A and - are %onstants.
Q!' (orm the di@erential e*uation of the family of %ir%les tou%hin) the y axis at
ori)in.
Q!( (orm the di@erential e*uation reresentin) the family of %urves )iven $y
/x−a ¿2+2 y2 = a2, where a is an ar$itrary %onstant.
Q!) &how that the followin) di@erential e*uation is homo)eneous, and then solve
it+
Q"* (orm the di@erential e*uation of the family of ara$olas havin) vertex at the
ori)in and axis alon) ositive y Baxis.
Q"!
8/9/2019 Question Bank of 12 Class
33/49
Ve7tors
One Mark Questions
Q! 8rite the dire%tion %osines of a line e*ually in%lined to the three %oordinate
axes.
Q" 8hat is the %osine of the an)le whi%h the ve%tor √ 2 î ' ̂5 ' k̂
ma!e with yaxis"
Q# If ⃗ a and b are two ve%tor su%h that |⃗a .⃗ b| = |⃗a ×⃗ b| , then what is the
an)le $etween ⃗a and b "
Q$ Ke%tors ⃗a and b are su%h that |⃗a| = √ 3, |⃗b| =2
3 and
(⃗a ×⃗b ) is a unit ve%tor. 8rite the an)le $etween ⃗a and b .
Q% (or that value of $ a $ the ve%tors 2 î−3 ̂5 '
k̂ and a î 'H ̂5
−8 k̂ are %olliner"
Q& 8rite the osition ve%tor of the midoint of the ve%tor Moinin) the
oints /2, 3, 0 and W /, 1, 20.
Q' 8rite the value of ( ̂i× ̂5 ) . k̂ '
î . ̂5
Q( 8rite the value the area of the arallelo)ram determined $y the
ve%tors 2 î and 3
̂5 .
Q) ?iven ⃗AB=3 î− ̂5−5 k̂
and %oordinates of the terminal oint are /4, 1,30. (ind the %oordinates of the initial oint.
Q!* If ⃗a is a unit ve%tor and (⃗ x+⃗a ) . (⃗ x−⃗a )=15, (ind |⃗ x| .
Q!! If ⃗ a .b =4, then what you %an say a$out ve%tors ⃗ a∧b .
8/9/2019 Question Bank of 12 Class
34/49
Q!" (ind X’ when the roMe%tion of ⃗a =X î '
̂5 ' k̂ on b̂ =2
î 'H ̂5 '3
k̂ is units.
Q!# (or what value of X are the ve%tors⃗a
= 2
î 'X
^ 5 '
^k and
b = î 2
̂5 '3 k̂ erendi%ular to ea%h other"
$ /ark Questions
Q ! If ⃗a .b and ⃗c are three mutually erendi%ular ve%tors of e*ual
ma)nitude, rove that the an)le whi%h (⃗a+⃗b +⃗c ) ma!es with any of the
ve%tors ⃗a ,b or ⃗c is cos−1
( 1√ 3 ) .
Q" If the verti%es A,-,>of a UA-> have osition ve%tor /1,2,30,/1,4,40
/4,1,20 rese%tively, what is the ma)nitude of YA->.
Q# (ind the roMe%tion of b < ⃗ c on ⃗ a , where ⃗a = 2 î N2
̂5 ' k̂ ,
b̂ = î '2
̂5 N2 k̂ and ⃗c 2
î− ̂5 ' k̂ .
Q$ &how that the area of the arallelo)ram havin) dia)onals (3 î+ ̂5+2 k̂ )
and ( ̂i−3̂ 5+4 k̂ ) is5√ 3 s*. units.
8/9/2019 Question Bank of 12 Class
35/49
Q% 7ene the s%alar and ve%tor rodu%t of two ve%tors ⃗a and b . If for
three non6ero ve%tors ⃗ a ,b and ⃗c ;⃗ a .b = ⃗ a .⃗c and ⃗a ×b=⃗a×⃗ c , then
show that b=⃗c .
Q& (ind the osition ve%tor of a oint D whi%h divides the line Moinin) two
oints F and W whose osition ve%tors are ( 2⃗ a+⃗b ) and (⃗a−3⃗ b )
rese%tively, externally, in the ratio1+2. Also, show that is the midoint of
the line se)ment DW.
Q' et ⃗a=î+¿ ̂5 '2
k̂ ,⃗ b=3 î N 2 ̂5 '<
k̂ and ⃗c=2 î− ̂5+4 k̂ . (ind a
ve%tor d whi%h is erendi%ular to $oth ⃗a and b and ⃗ c . d =1J.
Q( (ind a unit ve%tor erendi%ular to ea%h of the ve%tors ⃗a+b and ⃗ a−b
, where ⃗a=3 î ' 2 ̂5 ' 2
k̂ and⃗b=î+2 ̂5−2 k̂ .
Q) If two ve%tors ⃗a and b are su%h that |⃗a|=2,|b|=1∧⃗a .⃗b=1, then nd
the value of ( 3⃗ a−5⃗ b ) . (2⃗ a+7⃗ b ) .
Q!* If
⃗a b⃗c
are three ve%tors su%h that
|⃗a|=5|⃗b|12∧|⃗c|=13 and
⃗a+⃗b +⃗c=⃗0 , nd the value of ⃗a .⃗ b .+⃗b .⃗ c+⃗c .⃗ a .
Q!! et ⃗a=î+4 ̂5+2 k̂ ,⃗ b=3 î−2 ̂5+7 k̂ and ⃗c=2 î− ̂5+4 k̂ . (ind a ve%tor ⃗ p
whi%h is erendi%ular to $oth ⃗ a∧b∧⃗ p .⃗ c=18.
Q!" 8rite the value of the area of the arallelo)ram determined $y the
ve%tors 2 î∧3 ̂5 .
Q!# If ⃗α =3 î+4 ̂5+5 k̂ and ⃗2=2 î+ ̂5−4 k̂ , then exress in the form
2= 21+ 22 where 2 is arallel to ⃗α and 2 is erendi%ular to⃗α .
.
8/9/2019 Question Bank of 12 Class
36/49
T:ree-Di/ensiona6 9eo/etr@
One Mark Questions
Q! If e*uation of the line A- is x−3
1 =
y+2−2
= z−5
4 , nd the dire%tion ratios
of line arallel to A-.
Q" 8rite the ve%tor e*uation of the followin) line+ x−5
3 =
y+47
=6− z
2
Q # 8rite the osition ve%tor of the midoint of the ve%tor Moinin) theoints /2,3,0and W/,1, 20.
Q$ 8rite the inter%et %ut o@ $y the lane 2 x + y − z=5 onxaxis.
Q% 8rite the dire%tion %osines of a line arallel to 6 axis.
$+& Marks Questions
Q! (ind the e*uation of the lane assin) throu)h the oints /4, 1, 10,/,#, 10 and /3,;,0.
Q" (ind the shortest distan%e $etween the followin) lines+
x−31
= y −5−2
=
z−71
∧ x+1
7 =
y +1−6
= z+1
1 .
Q# (ind the e*uation of the lane assin) throu)h the oint /1,1, 20 and
erendi%ular to ea%h of the followin) lanes+ 2 x+3 y−3 z=2∧5 x−4 y+ z=6.
Q$(rom the oint /1, 2, 0, a erendi%ular is drawn on the lane2 x + y −2 z+3=0. (ind the e*uation, the len)th and the %oordinates of the
foot of the erendi%ular.
8/9/2019 Question Bank of 12 Class
37/49
Q% (ind the shortest distan%e $etween the lines ⃗*=î+2 ̂5+3 k̂ + % /
2 î+3 ̂5+4 k̂ ¿ and
⃗*=2 î+4 ̂5+5 k̂ + 6 (3 î+4 ̂5+5 k̂ ) .
Q& (ind the e*uation of the line assin) throu)h the oints F /, H, 20 and
the oint of interse%tion of the line x−1
3 =
y
2=
z+17 and lane x' y B 6 = J.
Q'(ind the distan%e of the oint /N2, 3N0 from the line x +2
3 =
2 y +34
=3 z+4
5 measured arallel to the lane x'12y B 36'1 =4.
Q( (ind the value of X, so that the line 1− x3 = 7 y−142 % = 5 z−1011 and
7−7 x3 %
= y−5
1 =
6− z5 are erendi%ular to ea%h other.
Q) (ind the e*uation of the lane assin) throu)h the oint /1, 3, 20 anderendi%ular to ea%h of the followin) lanes x'2y'36=# and 3x'3y'6=4.
Q!* &how that the lines x +3−3
= y−1
1 =
z−55
; x +1−1
= y −2
2 =
z−55 are
%olanar. Also nd the e*uation of the lane %ontainin) the line.
Q!! 8rite the ve%tor e*uation of the followin) line and hen%e determine the
distan%e $etween them+ x−1
2 =
y−23
= z+4
6 ;
x−34
= y−3
6 =
z+512
Q!" (ind the e*uation of the lane assin) throu)h the oint F /1, 1, 10 and
%ontainin) the line ⃗*=(−3 î+ ̂5+5 k̂ )+ % (3 î− ̂5−5 k̂ ) . Also, show that the lane
%ontains the line ⃗*=(−î+2̂ 5+5 k̂ )+ 6 ( ̂i−2 ̂5−5 k̂ ) .
Q!# (ind the shortest distan%e $etween the followin) air of lines andhen%e write whether the lives are interse%tin) or not+
x−12 =
y+13 = z ;
x+15 =
y−21
; z=2
8/9/2019 Question Bank of 12 Class
38/49
Q!$ (ind the an)le $etween the followin) air of lines+
− x+2−2
= y−1
7 =
z+3−3
∧ x+2
−1 =
2 y−84 =
z−54
And %he%! whether the lines are arallel or erendi%ular.
Q!% (ind the e*uation of the lane whi%h %ontains the line of interse%tion of
the lanes ⃗* . ( î+2 ̂5+3 k̂ )−4=0,⃗* (2 î+ ̂5+k̂ )+5=0 and whi%h is erendi%ular to
the lane
⃗* (5 î+3 ̂5+6 k̂ )+8=0.
Q!& (ind the distan%e of the oint /N1, #N140 from the oint of interse%tionof the line
⃗* (2 î− ̂5+2 k̂ )+ % (3 î+4 ̂5+2 k̂ ) and he lane ⃗* .( î− ̂5+k̂ )=5.
Q!' (ind the e*uation of the lane assin) throu)h the line of interse%tionof the lanes
⃗* . ( î− ̂5+k̂ )=1∧. ⃗*=( 2 î+3 ̂5− k̂ )+4=0 and arallel to x axis.
Q!( (ind the oint on the line x +23 = y +12 = z−32 at a distan%e of # units
from the oint F /1, 3, 30.
Q!) (ind the shortest distan%e $etween two lines whose ve%tor e*uations
are ⃗*=(1−t )î ' /t −2 0 ̂5 ' /32t0
k̂ and ⃗* ¿ (s+1 )i+(2 s−1) ̂5
/ (2 s+1 ) k̂ .
Q"* (ind the e*uation of a line assin) throu)h the oint F /2,1, 30 and
erendi%ular to the lines. ⃗*=( î+ ̂5+ k̂ )+ % ( 2 î−2 ̂5+ k̂ )∧⃗*=(2 î− ̂5−3 k̂ )+ 6 ( ̂i+2 ̂5+2 k̂ ) .
Q"! (ind the %oordinates of the oint where the line throu)h the oints A /3,, 10 and -/#,1,H0 %rosses the Z lane
8/9/2019 Question Bank of 12 Class
39/49
Q"" If the lines x−1−3
= y−2−2 k
=
z−32
∧ x−1
k =
y−21
= z−3
5 are erendi%ular,
nd the value of ! and hen%e nd the e*uation of lane %ontainin) these
lines.Q"" (ind the %oordinates of the oint where the line throu)h the oints /3,,#0 and /23, 10 %rosses the lane 2x 'y ' 6=
8/9/2019 Question Bank of 12 Class
40/49
Linear Pro;ra//in;
$+& Marks Questions SET-!.
Q! &olve the followin) linear ro)rammin) ro$lem )rahi%ally+
\aximi6e =H4x'1#y
&u$Me%t to %onstraints
Z ' y R #4
3x ' y R ;4
Z, y S 4
Q" A dealer wishes to ur%hase a num$er of fans and sewin) ma%hines. ehas only Ds.#,
8/9/2019 Question Bank of 12 Class
41/49
on a $at is Ds. 24 and Ds.14 rese%tively, nd the num$er of tennis ra%!etsand %ri%!et $ats that the fa%tory must manufa%ture to earn the maximumrot. \a!e it as an .F.F. and solve it )rahi%ally.
Q' A mer%hant lans to sell two tyes of ersonal %omuter B a des!tomodel and a orta$le model that will %ost Ds. 2#,444 and Ds. 4,444rese%tively. e estimates that the total monthly demand of %omuters willnot ex%eed 2#4 units. 7etermine the num$er of units of ea%h tye of %omuters whi%h the mer%hant should sto%! to )et maximum rot if hedoes not want to invest more than Ds.
8/9/2019 Question Bank of 12 Class
42/49
availa$le er day. ow many $elt of ea%h tye should the %omany rodu%eso as to maximi6e the rot"
Q$ A manufa%turer of atent medi%ines is rearin) a rodu%tion lan onmedi%ines A and -. here are su`%ient raw \aterials availa$le to ma!e24,444$ottles of a and 4,444 $ottles of -, $ut there are only #,444 $ottlesinto whi%h either of the medi%ines %an $e ut. (urther, it ta!e 3 hours toreare enou)h material to ll 1444 $ottles of A, it ta!es 1 hour to reareenou)h material to ll 1444 $ottles of - and there are HH hours availa$le forthis oeration. he rot is Ds. J er $ottle for A and Ds. < er $ottle for -.ow should the manufa%turer s%hedule his rodu%tion in order to maximi6ehis rot"
Proai6it@
$+& MARS QUESTIONS SET- I.
Q! A man is !nown to sea! truth 3 out of times. e throws a die andreorts that it is a six. (ind the ro$a$ility that is a%tually a six.
Q" he ro$a$ility that a student enterin) a university will )raduate is 4..(ind the ro$a$ility that out of 3students of the university+
/I0 none will )raduate /II0 only one will )raduate /III0 all will)raduate.
Q# A %oin is $iased so that the head is 3 times as li!ely to o%%ur as a tail. If the %oin is tossed twi%e, nd the ro$a$ility distri$ution for the num$er of
tails. Q$ A and - toss a %oin alternately till one of them tosses a head and winsthe )ame. If A starts the )ame, nd their rese%tive ro$a$ilities of winnin).
Q% In a %lass, havin) H4^ $oys, #^of the $oys and 14^ of the )irls have anI.W. of more than 1#4. A student is sele%ted at random and found to have anI. W of more than 1#4. (ind the ro$a$ility that the sele%ted student is a $oy.
Q& -a) A %ontains H red and # $lue $alls and another $a) - %ontains # redand J $lue $alls. A $all is drawn from $a) A without seein) its %olour and it isut into the $a) -. hen a $all is drawn from $a) - at random. (ind the
ro$a$ility that the $all drawn is $lue in %olour.
Q' here are 2444 s%ooter drivers, 444 %ar drivers and H444 tru%! driversall insured. he ro$a$ilities of an a%%ident involvin) a s%ooter, a %ar a tru%!are 4.41, 4.43, 4.1# rese%tively. One of the insured drivers meets with ana%%ident. 8hat is the ro$a$ility that he is s%ooter driver"
Q(A air of di%e is thrown times. If )ettin) a dou$let is %onsidered asu%%ess, nd the ro$a$ility distri$ution of num$er of su%%esses.
8/9/2019 Question Bank of 12 Class
43/49
Q) 12 %ards num$ered 1 to 12, are la%ed in $ox, mixed u thorou)hly andthen a %ard is drawn at random from the $ox. If it !nown that the num$er onthe drawn %ard is more than 3, nd the ro$a$ility that it is an even num$er.
Q!* hree $a)s %ontain $alls as shown in the ta$le $elow+
Ba; Nu/er o, :itea66s
Nu/er o, B6a7k a66s
Nu/er o, Rea66s
I 1 2 3II 2 1 1III 3 2
A $a) is %hosen at random and two $alls are drawn from it. hey haen to$e white and red. 8hat is the ro$a$ility that they %ame from the III $a)"
Q!! wo )rous are %ometin) for the ositions on -oard of 7ire%tors of a%ororation. he ro$a$ilities that the rst and the se%ond )rous will winare 4.H and 4. rese%tively. (urther, if the rst )rou wins, the ro$a$ility of
introdu%in) a new rodu%t is 4.< and %orresondin) ro$a$ility is 4.3 if se%ond )rou wins. (ind the ro$a$ility that the rodu%t introdu%ed was $yse%ond )rou.
Q!" Fro$a$ilities of solvin) a se%i% ro$lem indeendently $y A and - are1
2 and1
3 rese%tively. If $oth try to solve the ro$lem indeendently,
nd the ro$a$ility that /i0 the ro$lem is solved /ii0 exa%tly one of themsolves the ro$lem.
Q!# &uose that #^of men and o.2#^of women have )rey hair. A )reyhaired erson is sele%ted at random. 8hat is the ro$a$ility of this erson$ein) male" Assume that there are e*ual num$er of males and females.
Q!$ A fa%tory has two ma%hines A and -. Fast re%ord shows that ma%hine Arodu%ed H4^ of items of outut and ma%hine - rodu%ed 4^ of the items.(urther 2^ of the items rodu%ed $y ma%hine A and 1^ rodu%ed $yma%hine - were defe%tive. All the items are ut into one sto%!ile and thenone item is %hosen at random from this and is found to $e defe%tive. 8hat isthe ro$a$ility that it was rodu%ed $y ma%hine -"
Q!% here is three %oins. One is a two headed %oin /havin) heads on $oth
fa%es0, another is a $iased %oin that %omes u heads
8/9/2019 Question Bank of 12 Class
44/49
a $all is drawn from -a) II. he $all so drawn is found to $e red in %olour. (indthe ro$a$ility that the transferred $alls were $oth $la%!.
Q!' In a %ertain %olle)e, ^ of $oys and 1^of )irls are taller than 1.
8/9/2019 Question Bank of 12 Class
45/49
$a) %ontainin) /i0 maximum num$er of $la%! $alls /ii0 maximum num$er of white $alls.
Q"' wo %ards are drawn su%%essively with rela%ement from a a%! of #2%ards. (ind the ro$a$ility distri$ution of the num$er of a%es. (ind its meanand standard deviation.
Q"( In an examination, an examinee either )uesses or %oies or !nows theanswer of multile %hoi%e *uestions with four %hoi%es. he ro$a$ility that he
ma!es a )uess is1
3 and ro$a$ility that he %oies the answer is1
6 .
he ro$a$ility that his answer is %orre%t, )iven that he %oied it, is1
8 . (ind
the ro$a$ility that he !new the answer to the *uestion, )iven that he%orre%tly answered it.
Q") he ro$a$ility that a $ul$ rodu%ed $y a fa%tory will fuse after 1#4days of use is 4.4#. (ind the ro$a$ility that out of # su%h $ul$s+
/i0 gone /ii0 not more than on /iii0 more than one will fuse after 1#4days of us/iv0At least one
Q#* In a hurdle ra%e, a layer has to %ross 14 hurdles. he ro$a$ility that
he will %lear ea%h hurdle is5
6 . 8hat is the ro$a$ility that he will !no%!
down fewer than 2 hurdles"Q#!If on an avera)e 1 shi in every 14 sin!s: nd the %han%e that out of #shis at least will arrive safely.
Q#" he items rodu%ed $y a %omany %ontain 14^ defe%tive items. &howthat the ro$a$ility of )ettin) 2 defe%tive items in a samle of J items is
28 × 96
108 .
Q## A sea!s truth in H4^ of the %ases and - in ;4^ of the %ases. In what
er%enta)e of %ases they are li!ely to %ontradi%t ea%h other in statin) thesame fa%t.
Q#$ (our ersons are %hosen at random from a )rou %onsistin) of 3 men, 2women and 3 %hildren. (ind the ro$a$ility that out of %hosen ersons,exa%tly 2 are %hildren.
8/9/2019 Question Bank of 12 Class
46/49
Q#% A letter is !nown to have %ome either from AA gA?AD or from>A>5A. On the enveloe Must two %onse%utive letters A are visi$le. 8hatis the ro$a$ility that the letters %ame from AA gA?AD"
Q#& A %ommittee of students is sele%ted at random from a )rou%onsistin) of J $oys and )irls. ?iven that there is at least one )irl in the%ommittee, %al%ulate the ro$a$ility that there are exa%tly 2 )irls in the%ommittee.
Q#' A $a) %ontains # red mar$les and 3 $la%! mar$les. hree mar$les aredrawn one $y one without rela%ement. 8hat is the ro$a$ility that at leastone of the three mar$les drawn $e $la%! if the rst mar$le is red"
Q#( A and - throw a air of di%e alternately. A wins the )ame if he )ets atotal of H and - wins if she )ets a total of
8/9/2019 Question Bank of 12 Class
47/49
Q! et set A $e the set of eole of di@erent a)e )rous asso%iated with>AgIg&&’ drive, et $e a $inary oeration of set A dened $y a $ =older of /a, $0, a, $j A. Is %ommutative, asso%iative in A" Are you also aart of set A"
Matri7es3-
Q" wo s%hools A and - de%ided to award ri6es to their students for threevalues onesty /x0, Fun%tuality /y0 and O$edien%e /60. &%hool A de%ided toaward a total of Ds. 1#,444 for the three values to , 3 and 2 studentrese%tively, while s%hool - dei%ed to award Ds.1;,444 for the three valuesto #, and 3student rese%tively. If all the three ri6es to)ether amount toDs. #,444, then/i0Deresent the a$ove situation $y matrix e*uation and from liner e*uation
usin) matrix multili%ation./ii0 8hi%h value you refer to $e rewarded most and why"Q# A store in a mall has three do6en shirts with &AK gKIDOg\g’rinted, two do6en shirts &AK I?D’ rinted and ve do6en shirts with
?DO8 FAg&’ rinted. he %ost of ea%h shirt is Ds. #;#, Ds.H14 and Ds.
8/9/2019 Question Bank of 12 Class
48/49
Q% wo si)n $oards, one %ir%ular and one s*uare to $e made usin) a wire of
len)th 4 m and %uttin) it into two ie%es. he si)n $oards are to dei%t -
Og& and - F5g>5A and these are $e dislayed near the main )ate
of the s%hool. 8hat should $e the len)ths of the two ie%es, so that the
%om$ined area of the s*uare and the %ir%le is minimum" 7o you thin! these
values are imortant in life"
Q& A %ylindri%al $ox is to made, whi%h is oen at the to and has a )iven
surfa%e area. &ouvenirs of di@erent life values are to $e stored in the $ox, so
we would li!e to have maximum volume of the $ox. 8hat should $e the
dimensions of the %ylindri%al $ox" game some of the values whi%h are
imortant to ea%h erson.
Linear Pro;ra//in;3-
Q' A retired erson has Ds.an you name some avenues"
Q( A %omany manufa%tures two tyes of sti%!ers A+ &AK gKIDOg\g
and -+ - >O5DO5&. ye A re*uires # minutes ea%h for %uttin) and 14
minutes ea%h for assem$lin). ye - re*uires J minutes ea%h for %uttin) and
J minutes ea%h for assem$lin). here are 3 hours and 24 minutes availa$le
for %uttin) and hours availa$le for assem$lin) in a day. e earns a rot of
Ds. #4 on ea%h tye A and Ds.H4 on ea%h tye -. ow many sti%!ers of ea%h
tye should the %omany manufa%ture in a day to maximi6e rot" ?ive your
views a$out &AK gKIDOg\g and - >O5DO5&.
Proai6it@3-Q) In answerin) a *uestion on a multile %hoi%e *uestions test with four
%hoi%es er *uestion. A student !nows the answer, )uesses or %oies the
answer. If k $e the ro$a$ility that he !nows the answer, $e the
ro$a$ility that he )uesses it and that he %oies it. Assumin) that a
student who %oies the answer will $e %orre%t with the ro$a$ility 3. 8hat
is ro$a$ility that the student !nows the answer )iven that he answered it
%orre%tly"
Q!* An insuran%e %omany insured ,444 %y%list, J,444 s%ooter drivers and
12,444%ar drivers. he ro$a$ility of an a%%ident involvin) a %y%list, s%ooter
8/9/2019 Question Bank of 12 Class
49/49
driver and a %ar driver are o.o2, o.oH and o.34 rese%tively. One of the
insured erson meets with an a%%ident. 8hat is the ro$a$ility that he is a
s%ooter driver" 8hi%h mode of transort would you su))est to a student and
why"
Q!! A sea!s truth in H4^ of the %ases and - in ;4^ of the %ases. In what
er%enta)e of %ases are they li!ely to %ontradi%t ea%h other in statin) the
same fa%t" 8hi%h values A is la%!in) and should imrove uon"
Q!" here is a )rou of 144 eole who are atrioti% out of whi%h