Question Bank of 12 Class

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]


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]


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 ]


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| = .


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

.


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)+


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.


8/49

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


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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


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¿ .


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


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 .


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.


14/49

Q( Frove that the tan)ents to the %urve y= x3+6 at the oints / 1,#0 and

/1,


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.


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


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


18/49

INDE4INITE INTE9RALS

One /arks 2uestion

Q! ∫ s"c2

/


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


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


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.


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 !.


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


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 .


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!)


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


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

=.


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.


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.


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-


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.


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"!


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 .


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.


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 .

.


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.


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


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


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=


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.#,


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.


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.


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


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.


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.


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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


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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.


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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


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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

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