jee mains maths solutions year 2010 download doubtnut … · 2018. 10. 3. · 15 jee mains maths...
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JEEMAINSMATHSSOLUTIONS
YEAR2010
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QuesNo. Question
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
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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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121
223
13
112
132
52
x2 + y2 = 4x + 8y+ 5
3x4y = m 35 < m < 1515 < m < 65 35 < m < 85 85 < m < 35