jee mains maths solutions year 2010 download doubtnut … · 2018. 10. 3. · 15 jee mains maths...

JEEMAINSMATHSSOLUTIONS

YEAR2010

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1

JEEMAINSMATHSSOLUTIONS-2010

Let and let where , then

(1) (2) (3) (4)

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2

JEEMAINSMATHSSOLUTIONS-2010

LetSbeanon-empty subset ofR.Consider the following statement:P:There is arationalnumber such that .Which of the following statements is thenegationofthestatementP?Thereisnorationalnumber suchthat (9)Everyrationalnumber satisfies (18) and (27)isnotrationalThereisarationalnumber suchthat (36)

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3

JEEMAINSMATHSSOLUTIONS-2010

Let and . Then vector satisfying

and is (1) (2) (3)(4)

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cos(α + β) =45

s ∈ (αβ) =5

130 ≤ α, β ≤

π

4tan2α =

5633

1912

207

2516

x ∈ S x > 0x ∈ S x ≤ 0

x ∈ S x ≤ 0 x ∈ S x ≤ 0 ⇒ xx ∈ S x ≤ 0

→a = j − k

→c = i − j − k

→b

→a ×

→b + →

c =→0 →

a

.→b = 3 2

→i −

→j + 2

→k i − j − 2k

i + j − 2k − i + j − 2k

4

JEEMAINSMATHSSOLUTIONS-2010

Theequationofthetangenttothecurve ,thatisparalleltothex-axis,is

(1) (2) (3) (4)

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5

JEEMAINSMATHSSOLUTIONS-2010

Solutionofthedifferentialequation

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6

JEEMAINSMATHSSOLUTIONS-2010

Theareaboundedbythecurves

between theordinates is (1) (2) (3)

(4)

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7

JEEMAINSMATHSSOLUTIONS-2010

IftwotangentsdrawnfromapointPtotheparabola areatrightangles,thenthelocusofPis

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y = x +4

x2

y = 1 y = 2 y = 3 y = 0

cos xdy = y(sinx

− y)dx, 0 < x <π

2

y = cosxandy= sinx

x = 0andx =3π2

4√2 + 2 4√21 4√2 + 1

4√22

y2 = 4x

8

JEEMAINSMATHSSOLUTIONS-2010

Ifthevectors

aremutuallyorthogonal,then (1) (2) (3) (4)

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9

JEEMAINSMATHSSOLUTIONS-2010

Consider the following relations:R= {(x, y) | x, y are real numbers and x =wy forsomerationalnumberw};

.Then(1)neitherRnorSisanequivalencerelation(2)SisanequivalencerelationbutRisnotanequivalencerelation(3)RandSbothareequivalencerelations(4)RisanequivalencerelationbutSisnotanequivalencerelation

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10

JEEMAINSMATHSSOLUTIONS-2010

Let bedefinedby

.Iffhasalocalminimumat ,thenapossiblevalueofkis(1)0(2) (3)

→a = i − j + 2

→k ,

→b = 2 i + 4j

+ kand→c = λi + j

+ mk

(λ, μ) = (2, 3) (2, 3) (3, 2) (3, 2)

S

= {( ,

)m,n,pandqareintegerssuchthatn,q ≠ 0andqm=pn}

m

np

q

f : R → Rf(x) = {k − 2x,

if x ≤ − 12x+ 3, fx ≻ 1}

x = 1 −12

−1

(4)1

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11

JEEMAINSMATHSSOLUTIONS-2010

Thenumberof3´3non-singularmatrices,withfourentriesas1andallotherentriesas0,is(1)5(2)6(3)atleast7(4)lessthan4

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12

JEEMAINSMATHSSOLUTIONS-2010

Fournumbersarechosenatrandom(withoutreplacement)fromtheset{1,2,3,.....,20}. Statement-1: The probability that the chosen numberswhen arranged in some

orderwill formanAPIs .Statement-2: If the fourchosennumbers fromanAP,

thenthesetofallpossiblevaluesofcommondifferenceis{1,2,3,4,5}.

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13

JEEMAINSMATHSSOLUTIONS-2010

Statement-1:ThepointA(3,1,6)isthemirrorimageofthepointB(1,3,4)intheplane.Statement-2:Theplanex bisectsthelinesegmentjoining

A(3,1,6)andB(1,3,4).(1)Statement-1istrue,Statement-2istrue;Statement-2isnot the correct explanation for Statement-1 (2) Statement-1 is true, Statement-2 isfalse(3)Statement-1isfalse,Statement-2istrue(4)Statement-1istrue,Statement-2istrue;Statement-2isthecorrectexplanationforStatement-1

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JEEMAINSMATHSSOLUTIONS-2010

2−1

185

xy + z = 5 xy + z = 5

14

Let

Statement-1: Statement-2:

. (1) Statement-1 is true, Statement-2 is true; Statement-2 is not the correctexplanation for Statement-1 (2) Statement-1 is true, Statement-2 is false (3)Statement-1isfalse,Statement-2istrue(4)Statement-1istrue,Statement-2istrue;Statement-2isthecorrectexplanationforStatement-1

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15

JEEMAINSMATHSSOLUTIONS-2010

Let A be a matrix with non-zero entries and let , where I isidentitymatrix.DefineTr(A)=sumofdiagonalelementsofAand|A|=determinantofmatrix A. Statement-1: Statement-2: (1) Statement-1 is true,Statement-2 is true; Statement-2 is not the correct explanation for Statement-1 (2)Statement-1istrue,Statement-2isfalse(3)Statement-1isfalse,Statement-2istrue(4)Statement-1istrue,Statement-2istrue;Statement-2isthecorrectexplanationforStatement-1

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16

JEEMAINSMATHSSOLUTIONS-2010

Let f:RRbeacontinuous functiondefinedby .Statement-1:

forsome .Statement-2: forall . (1)

Statement-1istrue,Statement-2istrue;Statement-2isnotthecorrectexplanationfor

S1 =10

∑j= 1

j(j − 1)10Cj , S2

=10

∑j = 1

j10Ci(andS)3

=10

∑j = 1

j210Cj.

S3 = 55 × 29

S1 = 90 × 28andS2

= 10 × 28

2 × 2 A2 = I 2 × 2

T r(A) = 0 |A| = 1

f(x) =1

ex + 2e−x

f(c) = ,13

c ∈ R 0 < f(x) ≤ ,1

2√2x ∈ R

Statement-1 (2) Statement-1 is true, Statement-2 is false (3) Statement-1 is false,Statement-2 is true (4) Statement-1 is true, Statement-2 is true; Statement-2 is thecorrectexplanationforStatement-1

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17

JEEMAINSMATHSSOLUTIONS-2010

Foraregularpolygon,letrandRbetheradiioftheinscribedandthecircumscribedcircles. A false statement among the following is There is a regular polygon with

(17) There is a regular polygon with (30) There is a regular

polygonwith (47)Thereisaregularpolygonwith (60)

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18

JEEMAINSMATHSSOLUTIONS-2010

If and aretherootsoftheequation ,then (1)(2)1(3)2(4)

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19

JEEMAINSMATHSSOLUTIONS-2010

Thenumberofcomplexnumberszsuchthat equals(1)1(2)2(3) (4)0

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20

JEEMAINSMATHSSOLUTIONS-2010

AlineABinthree-dimensionalspacemakesangles and withthepositivex-axis and the positive y-axis respectively. If AB makes an acute angle q with thepositivez-axis,thenqequals

=r

R

1

√2=

r

R

23

=r

R

√32

=r

R

12

α β x2-x + 1 = 0 α2009 + β2009 = 12

|z1| = |z + 1| = |zi|∞

45o 120o

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21

JEEMAINSMATHSSOLUTIONS-2010

The line L given by passes through the point (13, 32). The line K is

paralleltoLandhastheequation ThenthedistancebetweenLandKis

(1) (2) (3) (4)

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22

JEEMAINSMATHSSOLUTIONS-2010

A person is to count 4500 currency notes. Let an denote the number of notes hecountsinthenthminute.If

and are inA.P.withcommondifference2, then the time takenbyhim to count all notes is (1) 34 minutes (2) 125 minutes (3) 135 minutes (4) 24minutes

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23

JEEMAINSMATHSSOLUTIONS-2010

Let be a positive increasing function with . Then

(1) (2) (3)3(4)1

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JEEMAINSMATHSSOLUTIONS-2010

+ = 1x

5y

b+ = 1

x

c

y

3

√1717

√15

23

√17

23

√15

a1 = a2 = . . . . . .= a10 = 150

a10, a11, . . . . . .

f : R → R limx→ ∞

= 1f(3x)

f(x)

limx→ ∞

=f(2x)

f(x)23

32

( ) = (1 )

24Let p(x) be a function defined on R such that , for all

and .Then equals(1)21(2)41(3)42

(4)

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25

JEEMAINSMATHSSOLUTIONS-2010

Let beadifferentiablefunctionwith

.Let

.Then (1) (2)0(3) (4)4

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26

JEEMAINSMATHSSOLUTIONS-2010

Therearetwourns.UrnAhas3distinctredballsandurnBhas9distinctblueballs.Fromeachurn twoballsare takenoutat randomand then transferred to theother.The number of ways in which this can be done is (1) 36 (2)66(3)108(4)3

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27

JEEMAINSMATHSSOLUTIONS-2010

Considerthesystemoflinearequations:

Thesystemhas(1)exactly3solutions(2)auniquesolution(3)nosolution(4)infinitenumberofsolutions

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p' (x) = p' (1x)

x ∈ [0, 1], p(0) = 1 p(1) = 41 ∫ 1

0p(x)dx

√41

f : (1, 1) → Rf(0) = − 1 and

f ' (0) = 1

g(x)

= [f(2f(x) + 2)]2

g' (0) = 4 2

x1 + 2x2 + x3 = 3 2x1 + 3x2 + x3 = 33x1 + 5x2 + 2x3

= 1

28

JEEMAINSMATHSSOLUTIONS-2010

Anurncontainsnineballsofwhich threeare red, fourareblueand twoaregreen.Three balls are drawn at randomwithout replacement from the urn. The probability

thatthethreeballshavedifferentcolouris(1) (2) (3) (4)

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29

JEEMAINSMATHSSOLUTIONS-2010

For two data sets, each of size 5, the variances are given to be 4 and 5 and thecorresponding means are given to be 2 and 4, respectively. The variance of the

combineddatasetis(1) (2)2(3) (4)

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30

JEEMAINSMATHSSOLUTIONS-2010

Thecircle

intersects the line at two distinct points if (1) (2)(3) (4)

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27

121

223

13

112

132

52

x2 + y2 = 4x + 8y+ 5

3x4y = m 35 < m < 1515 < m < 65 35 < m < 85 85 < m < 35