Aieee Paper 2006

8/12/2019 Aieee Paper 2006

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1.

Ans.

Sol:

2.

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Sol:

MATHEMATICS

PART AABC is a triangle, right angled at A The resultant of the forces acting along AB, AC

with magnitudes and ~ respectively is the force along AD. where 0 is the

foot of the perpendicular from A onto BC. The magn1tude of the resultant 1s

{1 { : ~ ) ; { : g , {2) { : : ) . { ~ ~ )1 (4) _1_

{3

{ -A-8 -AC- AD '-{4) VMagnitude of resultant C

' · 1 , JAB · AC

I.As.J•I AcJ = AB·AC

BC BC 1

AB·AC AD·BC AD

Suppose a population A haspopulation B has 100

variances I i

( 1

(3)419

{1 {

L ..0

•

101. 102, , 200, a11d another, 250. If VA and V represent the

then VA is

{2) 914

{4)213

v

;s same m both the cases).

of the quadratic equation x' + px + q = 0 are tan30° a11d tan15°.

i I thevalueof2+q pis (2) 3(4) 1

Ans. (2)Sol: x +px+q=O

tan30°+\an15 p

tan 30°. tan 15° q

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tan45

' tan 30 - tan15

1 tan30 tan15''

=> P '1 q

_

. ___ ,H

=>q P '1 : .2 q P '3_

4 The value of the integral, j JX dx is

3 9 X t X

Ans.

Sol:

5

(1)112

(3) 2

Ans. (1)

'>I ' -'

Sol: 2sin' x + 5sin x

=>(sin x + 3) (2 sin x

=>sinx '2

'

(2) 312

(4) 1

6 a, b and C are any three vectors such that a· b 0,

1 c-a,(lJ,c). a b .. o b c o

(3 C)b-(b C)3-(3 C)b-(3 b)C

(a bJC- b Cja

3iiC

7. Let W der10te the words in the English dictio11ary. Define the relation R by :

8/12/2019 Aieee Paper 2006


Ans.

Sol:

8.

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Sol:

9.

R {(x, y) E W ~ W the words x andy have at least one letter in common}. Then R

(1) not reflexive. symmetric a11d tra11sitive

(2) reflexive, symmetric and not transitive(3) reftexive, symmetric and transitive

(4) reflexive, not symmetric and transitive

~ l ~ r l y {x, x) E R 1 x E W. So, R is reflexive.et {x, y) E R. then (y, x) ERas x andy have at least one letter in common. ,

symmetric.

But R is not tra11sitive for example 0et x ' DELHI, y DWARKA and z PARK

then {x. y) E Rand (y, z) E R but (x, Z) 1i R. ~If A and Bare square matrices of size n x n such that A' B' ( A ~ + B). then

wh1ch of the following Will be always true ? 0 +(1)A 'B

(2) AB BA(3) e1ther of A orB s a zero matnx

(2)

A B ' (A B)(A+B)

A BA B

=> AB BA

The value of I I Sin- '

~1)1 (2)1(3) 1 n {4) i

Ans. (4)

{ ..: ~ k r r 2kr. I '\£. 2krr ./.;,' .. 2kr.Sol: ..:.,. 1Sin-+< ..:.,.Sin-+li..,COS

1 . 1 11 , 11 11

10. ~ ' 1 m fm which bo1h ' oHheeq 'hood 2mP m' 1 0r an 2 but less than 4.1ie in the interval

<m<O {2)m>3

1< m< 3 {4 )1<m<4

Ans. {3)

Sol: Equation x 2mx + m' 1 ' 0{xm)'1=0

(x m+1)(x m 1)=0

X m 1.m+1

2<m 1andm+1<4

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11.

Ans.

Sol:

12.

m> 1 andm<3

1 <m<3_

A particle has two velocities of equal magnitude inclined to each other at an a11gle 0.

If one of them is halved, the angle between the other and the original resultant

velocity is bisected by the new resultant. The11 0 is

(1)90° (2)120°3) 45° 4) 60°

2)

0tan =

4 u • ~ c o s2

0 1 0 1. )> sm

4I

2sm

4cos0 =

2sm cos

4

2,0 ,3{1 3.0 . 0

Slll-=SIIl-= S i l l Slrl-

4 4 4 401 ()

51112 4= 4 => 4= 30 orH= 120°.

At a telephone enquiry system the comt.ec•

follow Poisson distribution with anintervals_ The probability that there i

time period is

1 / ).:_

leo ' ' ' d ic1g releva11t enquirycalls during 10-minute time

phone call during a 10-minute

,3, ; _ V

: , m

::x, a t • (X 1)

o

0 from P, 4s pnor to passmg through P If g ' 10 mls', then the he1ght above the

ml P from where the body began to fall1s

720 m {2) 900 m3) 320m {4) 680 m

Ans. 1)

8/12/2019 Aieee Paper 2006


Sol: We have h = and h + 400 = ~ g { t +4)2

_

Subtracting we get 400 = 8g + 4gt

t=8sec

14.

Ans.

Sol:

15.

1h = ~ 1 0 x 6 4 = 3 2 0 m2

.. Desired height= 320 + 400 = 720 m.

f f{sin x)dx is equal to

O(t)

400m

P(t+4)

{1) ' t :cosx)dx (2) ' t {sinx)dx J{3) ~ t{sinx)dx (4) f { c o s xn . , +

{4)CJ

I= J x f ( s m x ) d x ~ JtiT-x)f(smx)dx

' 'rrJf ls•nx)dx-1

2 · + '' ' ~I

%i '' '.

• '";{ >'

r r { (cosx)dx 0A straight lin hrou,ah the point A{3, 4) is such that its intercept between the axes is

bisected eq1'iation is

(1)x (2)3x 4y+7=0

(3)4x (4)3x+4y=25

16_ The two lines x = ay + b z = cy + d; and x = a'y +b. z = c'y + d' are perpendicular to

each other if

{1)aa'+cc'= 1

c3)-t-= 1a' c'

{2) aa' + cc' = 1

c4)-t-=1a' c'

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Ans.

Sol:

17.

Ans.

Sol:

18.

{1 {

x - b z - dEquation of lines y - --

• 0

x-b ' z -d '~a' c'

Lines are perpendicular=> aa + 1 + cc 0.

a'x' a'xThe locus of the vertices of the family of parabolas y -

3-

2(I) xy, 105

64

35(3) XY f f i

{1 {

a'x' a x

Parabola: y --.--ertex: (n, rll

35 a 35- - -x3---•

12 4 16

3 • 35 105- - - a - - .4a\ 16.' 64

Tho

a)(1 a)=O

3(2) XY 4

64(4 ) xy 105

•

A B C with position vectorsare the vertices of a right-angled

(2) 2 and 1

(4)2and 1

8/12/2019 Aieee Paper 2006


.19_ j [(x + :r)' + cos'(x + 3rr)Jctx is equal to

Ans.

Sol:

20.

Ans.

Sol:

21.

j

1).. :_32

(3) ~2

{3 {' . ' ,

I J ( x ~ R ) + c o s ( x + : h ) ] d xPutx+rr ' l

, 2 ' , . LI= J t + C o s t ] d t ~ 2 J os

2tdt

',. 2

2) ..':___ j _:_

32 2

(4) _:_ 14

J" ' T ( h c o s 2 t ) d t ~ ~ ~ o . CZ +

' . 3>' , , (lj.CJf X IS real, the ffi8XIffiUffi Value Of c C C • C C g i C ~ C ~ {(1)114 )4

(3)1 ~ 7 7{2{

+ 9 x ~ 1 7

;,:t1·, :;:,, 1 ) ' ' . \ ~0 2 0 xisreal

W=> {y l){y 41 _y,;41_

81(y-1)2 - 4 x 3 y - 1 ~ o 0

In an li dts nee between its foci is 6 and minor axis is 8. Then its

{1 {

6=:oae=3

'8 =>b=4

b' ' a'( I e )

16=a' a 'e 'a =16+9=25

8 '5

----

,

•(B) __ _

2

{D) _ 1

.J5

8/12/2019 Aieee Paper 2006


22.

Ans.

Sol:

23.

Ans.

Sol:

24.

Ans.

Sol:

tt 2\ t O jLe A" ' i JandB " ' i .a,bEN.Then

34 , o b .

(1) there cannot exist any B such that AB ' BA

(2) there exist more than one but finrte number of B's such that AB BA

(3) there exists exactly one B such that AS BA

(4) there exist infinitely ma11y B's such thetAS" BA

M-[' »3a 4 b ~• 0S A ~O

(1) ~11

{3) ~7

'] . ]~ ~ b 4b

Sx + 6 at the pomts (2 0) and (3. 0)

•'>"' 1.m,={2x 5)130 ,=1

b pl

a,ta , · - - ·a ,___ etermso an .. a,•a,+·· ·•a. ,

{2) ~2

(4).41

'

.. p"' q, t h en - equalsq· a ,

Ans. {4)

8/12/2019 Aieee Paper 2006


Sol:

26.

A11s.

Sol:

27.

E.[2a1+(p-1)d]

-.: [2a1+(q-1)d]

-a

1+:p-1']o

' 2 , p

q 1'·a,+:--10' 2 ,

p ' 2a1+(P-1)d~q ' 2a 1+(q 11d

o r ~ p-11. q-41.,_'

The set of points where f{x) = is differe11tiable is1t IX I

(1){ f.,0)c._;(Q,-r) (2){ f., 1)u{

{3) { x , :.:)

(3)

- -fixl- ,1-x

11+ x'

1,'f.) c{4) {0, x-) cz •

X•.O - f(x)- 1 :x)'' x, - i j fljcx: :O ~. -

: f {x) exist at everywhere

A tria11gular park 1s e11closed on s y a fence and on the th1rd stde by a

area enclosed by the park 1s

( 1 ) ~ x £ (2)

f{4) nx'3 i·· CZ

~ : . ~ .0 - ,a ts tnG-1 G--

' '

mber to be elected. There are 10 candidates and 4 are of be elected. If a votervotes for at least one ca11didate, the11the number of ways in which he ca11 vote is

(1)5040 {2)6210

(3)385 {4)1110

Ans. {3)Sol: ' c, + ' c, + '"c, + ' c,

= 10 + 45 + 120 + 210 ' 385

8/12/2019 Aieee Paper 2006


29.

Ans.

Sol:

30.

1If the expansion in powers of x of the function - - ~ O i i ; C > b C , ) is

a,+ a,x + a,x' + a,x' + .... then a, is

( b",,

(2) a" b'

b

'

b

''b-,.•

(4) b"'- a····{3) ab ' b '

{4 {

{1-ax)1(1-bx)

1- J -ax+a 'x '+ ... ( 1 + b x ~ b x +

: coefficient of x - b ' + ab" 1+ a'b" '+ b"'' a'1

b +a "-.c-"'-b '

For natural numbers m, n if 1

y1 + y) ' " 1 + 1

then {m, n) is

{1){20,45)

{3) {45, 35)

So.n m=10and =>m+n,80

: m=35.

•and a1 a2 10,

... + f{[a])}

... +f(a)}

(2)[a]f(a) {f(1)+f{2)+ +f{ a])}

(4)af{[a]) {f(1)+f(2)+ ... +f(a)}

k + h, where [a]" k and 0,;; h < 1

' 1 ' k _,' X j d x - J 1 1 ' ( x ' I : J x ~ J 2 1 ' ( x j d x - f ( k - 1 j d X ~ Jkf'(XjdX

' ' " '{1(2) 1(1)}+2{1(3) 1{2)}+3{1(4) f{3)}+ ......+{k 1)-{f{k) f(k 1 }

"1 (1 ) 1{2) 1{3) . f (k)+kf{k+h)

" [a] {a) {1{1) + 1{2) + 1 3) + .... + f( a])}

+ k{f(k +h) f{k)}

8/12/2019 Aieee Paper 2006


32. If the lines 3x 4y 7 0 and 2x 3y 5 0 are two diameters of a circle of area

49rr square units, the equation of the circle is

{1)x2 +y2 +2x 2y 47=0 (2)x2 +y2 +2x 2y 62=0

{3)x2

+y2

2x+2y 62 0 (4)x2

+y2

2x+2y 47 0

Ans. {4)

Sol: Point of intersection of 3x 4y 7 0 and 2x 3y 5 0 is {1centre of the circle and radius 7.

1 , which is the

33.

Ans.

Sol:

34.

Ans.

: Equation is x 1) + (y + 1)2

49 => x' + y' 2x + 2y 47 _

The differential equation whose solution is Ax + By' 1, where A and

constants is of

{1) second order and second degree{3) first order and first degree

{4{

A x ~ s / - 1 ... {1) 0 +0(A x ~ B y 0

... {2)J

0:; 0 {A-BY- ; ,+BI -1 -0 ... {3) f l j

x ,dx

From {2) and {3)

I ' 'x -Bydy_B:dy· ' -Bydy_Odx' \dxl dx

d2 'd L d~ •<-- »1-'1-<-'-odx ,dx , dx

Let C be the cm:le w 1 t h : : P e n rad1us 3 unrts The equation of the locus of

,,the m1d p01nts of the c e mcle C that subtend an angle of J at ts centre

' 3

(3) x'

{4

belongs to

1 ) o , ~(3) 3

•(B)x +y

2 1

9(D) X2 1 y ,

4

(2){3, x.)

(4) ( 3, il

8/12/2019 Aieee Paper 2006


Ans. {3)

Sol: .a<Oand a -- 0 - - > -<a 3

36. The image of the point ( 1, 3, 4) in the plene x 2y ' 0 is

Sol:

37.

Ans.

Sol:

1 ) l(3) 1

Ans. {3)

17

3

17

3

~ 413

19 1i.

(2){15,11,4)

(4) {8 4, 4)

(B) {4 .,ff3

(4){1•..fi)4

Sol: 3osx+s tnx - - -- '>1+stn2x-- -- '>Stn2x--- . sox is obtuse

d2tanx

~ t a n x-- 3 an' X+ 8 anx ~ 3 0

4

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39.

Ans.

Sol:

40.

Ans.

Sol:

-- tanx<O-4 - J7t n x ~

3

If a1

a,, , a, arein

H.P., then the expression a,a, + a,a, + ... +a, ,a,is

equal to{1)n(a, a,) (2){n 1)(a, a,)

(3)na,a, (4){n 1)a1a,

{ {

.,

Tha - a 2en a

1a

2~ .

4

a1a2 + a a +

· ~ da

-aa a - .

4

If x' -y; =(x + yf ' , then ~ ~ s1 l r

{say)

Aieee Paper 2006

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