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question_answer1) Let f be an odd function defined on the set of real numbers such that for \[x\ge 0,\]\[f(x)=3sinx+4cosx.\] Then f(x) at \[x=-\frac{11\pi }{6}\]is equal to:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[\frac{3}{2}+2\sqrt{3}\]
done
clear
B)
\[-\frac{3}{2}+2\sqrt{3}\]
done
clear
C)
\[\frac{3}{2}-2\sqrt{3}\]
done
clear
D)
\[-\frac{3}{2}-2\sqrt{3}\]
done
clear
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question_answer2) If \[{{z}_{1}},{{z}_{2}}\]and \[{{z}_{3}},{{z}_{4}}\]are 2 pairs of complex conjugate numbers, then \[\arg \left( \frac{{{z}_{1}}}{{{z}_{4}}} \right)+\arg \left( \frac{{{z}_{2}}}{{{z}_{3}}} \right)\]equals:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
0
done
clear
B)
\[\frac{\pi }{2}\]
done
clear
C)
\[\frac{3\pi }{2}\]
done
clear
D)
\[\pi \]
done
clear
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question_answer3) If \[\alpha \] and \[\beta \] are roots of the equation, \[{{x}^{2}}-4\sqrt{2}kx+2{{e}^{4\ln k}}-1=0\] for some k, and \[{{\alpha }^{2}}+{{\beta }^{2}}=66,\]then \[{{\alpha }^{3}}+{{\beta }^{2}}\]is equal to:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[248\sqrt{2}\]
done
clear
B)
\[280\sqrt{2}\]
done
clear
C)
\[-32\sqrt{2}\]
done
clear
D)
\[-280\sqrt{2}\]
done
clear
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question_answer4) Let A be a \[3\times 3\]matrix such that\[A\left[ \begin{matrix} 1 & 2 & 3 \\ 0 & 2 & 3 \\ 0 & 1 & 1 \\ \end{matrix} \right]=\left[ \begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{matrix} \right]\]Then \[{{A}^{-1}}\]is:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[\left[ \begin{matrix} 3 & 1 & 2 \\ 3 & 0 & 2 \\ 1 & 0 & 1 \\ \end{matrix} \right]\]
done
clear
B)
\[\left[ \begin{matrix} 3 & 2 & 1 \\ 3 & 2 & 0 \\ 1 & 1 & 0 \\ \end{matrix} \right]\]
done
clear
C)
\[\left[ \begin{matrix} 0 & 1 & 3 \\ 0 & 2 & 3 \\ 1 & 1 & 0 \\ \end{matrix} \right]\]
done
clear
D)
\[\left[ \begin{matrix} 1 & 2 & 3 \\ 0 & 1 & 1 \\ 0 & 2 & 3 \\ \end{matrix} \right]\]
done
clear
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question_answer5) Let for i = 1, 2, 3, pi(x) be a polynomial of degree 2 in \[x,p{{'}_{i}}(x)\]and \[p'{{'}_{i}}(x)\]be the first and second order derivatives of \[{{p}_{i}}(x)\] respectively. Let,\[A(x)=\left[ \begin{matrix} {{p}_{1}}(x) & {{p}_{1}}'(x) & {{p}_{1}}''(x) \\ {{p}_{2}}(x) & {{p}_{2}}'(x) & {{p}_{2}}''(x) \\ {{p}_{3}}(x) & {{p}_{3}}'(x) & {{p}_{3}}''(x) \\ \end{matrix} \right]\]and \[B(x)={{[A(x)]}^{T}}A(x).\]Then determinant of B(x):
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
is a polynomial of degree 6 in x.
done
clear
B)
is a polynomial of degree 3 in x.
done
clear
C)
is a polynomial of degree 2 in x.
done
clear
D)
does not depend on x.
done
clear
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question_answer6) An eight digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
72 (7!)
done
clear
B)
18 (7!)
done
clear
C)
40 (7!)
done
clear
D)
36 (7!)
done
clear
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question_answer7) The coefficient of \[{{x}^{50}}\]in the binomial expansion of\[{{(1+x)}^{1000}}+x{{(1+x)}^{999}}+{{x}^{2}}\]\[{{(1+x)}^{998}}+....+{{x}^{1000}}\] is:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[\frac{\left( 1000 \right)!}{\left( 50 \right)!\left( 950 \right)!}\]
done
clear
B)
\[\frac{\left( 1000 \right)!}{\left( 49 \right)!\left( 951 \right)!}\]
done
clear
C)
\[\frac{\left( 1001 \right)!}{\left( 51 \right)!\left( 950 \right)!}\]
done
clear
D)
\[\frac{\left( 1001 \right)!}{\left( 50 \right)!\left( 951 \right)!}\]
done
clear
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question_answer8) In a geometric progression, if the ratio of the sum of first 5 terms to the sum of their reciprocals is 49, and the sum of the first and the third term is 35. Then the first term of this geometric progression is:
A)
7
done
clear
B)
21
done
clear
C)
28
done
clear
D)
42
done
clear
View Answer play_arrow
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question_answer9) The sum of the first 20 terms common between the series 3 + + 11 + 15 + ......... and 1 + 6 + 11 + 16 + ......, is
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
4000
done
clear
B)
4020
done
clear
C)
4200
done
clear
D)
4220
done
clear
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question_answer10) If\[\underset{x\to 2}{\mathop{\lim }}\,\frac{\tan \left( x-2 \right)\left\{ {{x}^{2}}+\left( k-2 \right)x-2k \right\}}{{{x}^{2}}-4x+4}=5,\]then k is equal to:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
0
done
clear
B)
1
done
clear
C)
2
done
clear
D)
3
done
clear
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question_answer11) Let f(x) = x|x|, g(x) = sin x and h(x) = (gof) (x). Then
A)
h(x) is not differentiable at x = 0.
done
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B)
h(x) is differentiable at x = 0, but h¢(x) is not continuous at x = 0
done
clear
C)
h?(x) is continuous at x = 0 but it is not differentiable at x = 0
done
clear
D)
h?(x) is differentiable at x = 0
done
clear
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question_answer12) For the curve \[y=3\sin \theta \cos \theta ,x={{e}^{\theta }}\sin \theta ,\]\[0\le \theta \le \pi ,\]the tangent is parallel to x-axis when \[\theta \]is:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[\frac{3\pi }{4}\]
done
clear
B)
\[\frac{\pi }{2}\]
done
clear
C)
\[\frac{\pi }{4}\]
done
clear
D)
\[\frac{\pi }{6}\]
done
clear
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question_answer13) Two ships A and B are sailing straight away from a fixed point O along routes such that \[\angle AOB\] is always \[120{}^\circ \]. At a certain instance, OA = 8 km, OB = 6 km and the ship A is sailing at the rate of 20 km/hr while the ship B sailing at the rate of 30 km/hr. Then the distance between A and B is changing at the rate (in km/ hr):
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[\frac{260}{\sqrt{37}}\]
done
clear
B)
\[\frac{260}{37}\]
done
clear
C)
\[\frac{80}{\sqrt{37}}\]
done
clear
D)
\[\frac{80}{37}\]
done
clear
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question_answer14) The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius \[=\sqrt{3}\] is:
A)
\[\frac{4}{3}\sqrt{3}\pi \]
done
clear
B)
\[\frac{8}{3}\sqrt{3}\pi \]
done
clear
C)
\[4\pi \]
done
clear
D)
\[2\pi \]
done
clear
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question_answer15) The integral\[\int_{{}}^{{}}{x{{\cos }^{-1}}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)dx(x>0)\]is equal to:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[-x+1(1+{{x}^{2}})ta{{n}^{-1}}x+c\]
done
clear
B)
\[x-(1+{{x}^{2}})co{{t}^{-1}}x+c\]
done
clear
C)
\[-x+(1+{{x}^{2}})co{{t}^{-1}}x+c\]
done
clear
D)
\[x-(1+{{x}^{2}})ta{{n}^{-1}}x+c\]
done
clear
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question_answer16) If for \[n\ge 1,{{P}_{n}}=\int\limits_{1}^{e}{{{\left( \log x \right)}^{n}}dx,}\]then\[{{P}_{10}}-90{{P}_{8}}\] then is equal to:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
-9
done
clear
B)
10e
done
clear
C)
-9 e
done
clear
D)
10
done
clear
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question_answer17) If the general solution of the differential equation\[y'=\frac{y}{x}+\Phi \left( \frac{x}{y} \right),\]for some function \[\Phi ,\] is given by \[\ln |cx|=x,\]where c is an arbitrary constant, then \[\Phi (2)\]is equal to:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
4
done
clear
B)
\[\frac{1}{4}\]
done
clear
C)
-4
done
clear
D)
\[-\frac{1}{4}\]
done
clear
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question_answer18) A stair-case of length l rests against a vertical wall and a floor of a room. Let P be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio 1 : 2. If the stair-case begins to slide on the floor, then the locus of P is:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
an ellipse of eccentricity\[\frac{1}{2}\]
done
clear
B)
an ellipse of eccentricity\[\frac{\sqrt{3}}{2}\]
done
clear
C)
a circle of radius\[\frac{1}{2}\]
done
clear
D)
a circle of radius\[\frac{\sqrt{3}}{2}l\]
done
clear
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question_answer19) The base of an equilateral triangle is along the line given by 3x + 4y = 9. If a vertex of the triangle is (1, 2), then the length of a side of the triangle is:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[\frac{2\sqrt{3}}{15}\]
done
clear
B)
\[\frac{4\sqrt{3}}{15}\]
done
clear
C)
\[\frac{4\sqrt{3}}{5}\]
done
clear
D)
\[\frac{2\sqrt{3}}{5}\]
done
clear
View Answer play_arrow
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question_answer20) The set of all real values of \[\lambda \]for which exactly two common tangents can be drawn to the circles \[{{x}^{2}}+{{y}^{2}}4x4y+6=0\]and \[{{x}^{2}}+{{y}^{2}}10x10y+\lambda =0\]is the interval:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
(12, 32)
done
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B)
(18, 42)
done
clear
C)
(12, 24)
done
clear
D)
(18, 48)
done
clear
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question_answer21) Let \[{{L}_{1}}\]be the length of the common chord of the curves \[{{x}^{2}}+{{y}^{2}}=9\] and\[{{y}^{2}}=8x,\] and \[{{L}_{2}}\] be the length of the latus rectum of \[{{y}^{2}}=8x,\] then:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[{{L}_{1}}>{{L}_{2}}\]
done
clear
B)
\[{{L}_{1}}={{L}_{2}}\]
done
clear
C)
\[{{L}_{1}}<{{L}_{2}}\]
done
clear
D)
\[\frac{{{L}_{1}}}{{{L}_{2}}}=\sqrt{2}\]
done
clear
View Answer play_arrow
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question_answer22) Let \[P(3sec\theta ,2tan\theta )\]and \[Q(3sec\phi ,2tan\phi )\] where \[\theta +\phi =\frac{\pi }{2},\]be two distinct points on the hyperbola \[\frac{{{x}^{2}}}{9}-\frac{{{y}^{2}}}{4}=1.\]Then the ordinate of the point of intersection of the normals at P and Q is:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[\frac{11}{3}\]
done
clear
B)
\[-\frac{11}{3}\]
done
clear
C)
\[\frac{13}{2}\]
done
clear
D)
\[-\frac{13}{2}\]
done
clear
View Answer play_arrow
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question_answer23) Let A (2, 3, 5), B (? 1, 3, 2) and \[C(\lambda ,5,\mu )\]be the vertices of a DABC. If the median through A is equally inclined to the coordinate axes, then:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[5\lambda -8\mu =0\]
done
clear
B)
\[8\lambda -5\mu =0\]
done
clear
C)
\[10\lambda -7\mu =0\]
done
clear
D)
\[7\lambda -10\mu =0\]
done
clear
View Answer play_arrow
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question_answer24) The plane containing the line \[\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}\]and parallel to the line\[\frac{x}{1}=\frac{y}{1}=\frac{z}{4}\]passes through the point:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[(1, - 2, 5)\]
done
clear
B)
\[(1, 0, 5)\]
done
clear
C)
\[(0, 3, -5)\]
done
clear
D)
\[(-1, -3, 0)\]
done
clear
View Answer play_arrow
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question_answer25) If \[\overset{\to }{\mathop{{{\left| c \right|}^{2}}}}\,=60\]and \[\overset{\to }{\mathop{c}}\,\times \left( \hat{i}+2\hat{j}+5\hat{k} \right)=\overset{\to }{\mathop{0}}\,,\]then a value of\[\overset{\to }{\mathop{c}}\,.\left( -7\hat{i}+2\hat{j}+3\hat{k} \right)\]is:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[4\sqrt{2}\]
done
clear
B)
12
done
clear
C)
24
done
clear
D)
122
done
clear
View Answer play_arrow
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question_answer26) A set S contains 7 elements. A non-empty subset A of S and an element x of S are chosen at random. Then the probability that \[x\in A\]is:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[\frac{1}{2}\]
done
clear
B)
\[\frac{64}{127}\]
done
clear
C)
\[\frac{63}{128}\]
done
clear
D)
\[\frac{31}{128}\]
done
clear
View Answer play_arrow
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question_answer27) If X has a binomial distribution, B(n, p) with parameters n and p such that P(X = 2) = P (X = 3), then E(X), the mean of variable X, is
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
2 ? p
done
clear
B)
3 ? p
done
clear
C)
\[\frac{p}{2}\]
done
clear
D)
\[\frac{p}{3}\]
done
clear
View Answer play_arrow
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question_answer28) If \[2\cos \theta +\sin \theta =1\left( \theta \ne \frac{\pi }{2} \right),\]then \[7\cos \theta +6\sin \theta \]is equal to:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[\frac{1}{2}\]
done
clear
B)
2
done
clear
C)
\[\frac{11}{2}\]
done
clear
D)
\[\frac{46}{5}\]
done
clear
View Answer play_arrow
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question_answer29) The angle of elevation of the top of a vertical tower from a point P on the horizontal ground was observed to be \[\alpha .\] After moving a distance 2 metres from P towards the foot of the tower, the angle of elevation changes to \[\beta .\] Then the height (in metres) of the tower is:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
\[\frac{2\sin \alpha \sin \beta }{\sin \left( \beta -\alpha \right)}\]
done
clear
B)
\[\frac{\sin \alpha \sin \beta }{\cos \left( \beta -\alpha \right)}\]
done
clear
C)
\[\frac{2\sin \left( \beta -\alpha \right)}{\sin \alpha \sin \beta }\]
done
clear
D)
\[\frac{\cos \left( \beta -\alpha \right)}{\sin \alpha \sin \beta }\]
done
clear
View Answer play_arrow
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question_answer30) The proposition \[\tilde{\ }\left( p\vee \tilde{\ }q \right)\vee \tilde{\ }\left( p\vee q \right)\]is logically equivalent to:
[JEE Main Online Paper ( Held On 11 Apirl 2014 )
A)
p
done
clear
B)
q
done
clear
C)
\[\tilde{\ }p\]
done
clear
D)
\[\tilde{\ }q\]
done
clear
View Answer play_arrow