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question_answer1) The integral \[\int{\frac{x\operatorname{d}x}{2-{{x}^{2}}+\sqrt{2-{{x}^{2}}}}}\]equals :
JEE Main Online Paper ( Held On 23 April 2013 )
A)
\[\log \left| 1+\sqrt{2+{{x}^{2}}} \right|+C\]
done
clear
B)
\[-\log \left| 1+\sqrt{2-{{x}^{2}}} \right|+C\]
done
clear
C)
\[-x\log \left| 1-\sqrt{2-{{x}^{2}}} \right|+C\]
done
clear
D)
\[x\log \left| 1-\sqrt{2+{{x}^{2}}} \right|+C\]
done
clear
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question_answer2) If the curves\[\frac{{{x}^{2}}}{\alpha }+\frac{{{y}^{2}}}{4}=1\] and \[{{y}^{3}}=16x\] intersect at right angles, then a value of \[\alpha \]is:
JEE Main Online Paper ( Held On 23 April 2013 )
A)
2
done
clear
B)
\[4/3\]
done
clear
C)
\[1/2\]
done
clear
D)
\[3/4\]
done
clear
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question_answer3) Statement 1: The system of linear equations \[x+(\sin \alpha )y+(\cos \alpha )z=0\] \[x+(\cos \alpha )y+(\sin \alpha )z=0\] \[x-(\sin \alpha )y-(\cos \alpha )z=0\] has non-trivial solution of only one value of a lying in the interval \[0,\frac{\pi }{2}.\] Statement 2 : The equation in \[\alpha \] \[\left| \begin{matrix} \cos \alpha & \sin \alpha & \cos \alpha \\ \sin \alpha & \cos \alpha & \sin \alpha \\ \cos \alpha & -\sin \alpha & -\cos \alpha \\ \end{matrix} \right|=0\] has only one solution lying in the interval \[\left( 0,\frac{\pi }{2}. \right)\]
JEE Main Online Paper ( Held On 23 April 2013 )
A)
Statement 1 is true; Statement 2 is true; Statement 2 is not a correct explanation for Statement 1.
done
clear
B)
Statement 1 is true; Statement 2 is true; Statement 2 is a correct explanation for Statement 1.
done
clear
C)
Statement 1 is true; Statement 2 is false.
done
clear
D)
Statement 1 is false; Statement 2 is true.
done
clear
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question_answer4) For integers m and n both greater than 1, consider the following three statement: P: m divides n Q : m divides \[{{\text{n}}^{2}}\] R: m is prime, then
JEE Main Online Paper ( Held On 23 April 2013 )
A)
\[Q\wedge R\to P\]
done
clear
B)
\[P\wedge Q\to R\]
done
clear
C)
\[Q\to R\]
done
clear
D)
\[Q\to P\]
done
clear
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question_answer5) The sum of the rational terms in the binomial expansion of \[{{\left( {{2}^{\frac{1}{2}}}+{{3}^{\frac{1}{5}}} \right)}^{10}}\]is
JEE Main Online Paper ( Held On 23 April 2013 )
A)
25
done
clear
B)
32
done
clear
C)
9
done
clear
D)
41
done
clear
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question_answer6) If the extremities of the base of an isosceles triangle are the points (2a, 0) and (0, a) and the equation of one of the sides is \[x=2a,\] then the area of the square units, is:
JEE Main Online Paper ( Held On 23 April 2013 )
A)
\[\frac{5}{4}{{\operatorname{n}}^{2}}\]
done
clear
B)
\[\frac{5}{2}{{\operatorname{n}}^{2}}\]
done
clear
C)
\[\frac{25{{a}^{2}}}{4}\]
done
clear
D)
\[5{{a}^{2}}\]
done
clear
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question_answer7) The sum of the series: \[{{(2)}^{2}}+2{{(4)}^{2}}+3{{(6)}^{2}}+.....\operatorname{upto}10\operatorname{trems}\operatorname{is}:\]
JEE Main Online Paper ( Held On 23 April 2013 )
A)
11300
done
clear
B)
12100
done
clear
C)
12100
done
clear
D)
12300
done
clear
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question_answer8) If the circle \[{{x}^{2}}+{{y}^{2}}-6x-8y+(25-{{a}^{2}})=0\]touches the axis of \[x,\] then a equals.
JEE Main Online Paper ( Held On 23 April 2013 )
A)
\[0\]
done
clear
B)
\[\pm 4\]
done
clear
C)
\[\pm 2\]
done
clear
D)
\[\pm 3\]
done
clear
View Answer play_arrow
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question_answer9) If\[\operatorname{S}={{\tan }^{-1}}\left( \frac{1}{{{\operatorname{n}}^{2}}+\operatorname{n}+1} \right)+\] \[{{\tan }^{-1}}\left( \frac{1}{{{\operatorname{n}}^{2}}+3\operatorname{n}+3} \right)+.......\] \[+{{\tan }^{-1}}\left( \frac{1}{1+(\operatorname{n}+19)(\operatorname{n}+20)} \right),\]then tan S is equal to :
JEE Main Online Paper ( Held On 23 April 2013 )
A)
\[\frac{20}{401+20\operatorname{n}}\]
done
clear
B)
\[\frac{\operatorname{n}}{{{\operatorname{n}}^{2}}+20\operatorname{n}+1}\]
done
clear
C)
\[\frac{20}{{{\operatorname{n}}^{2}}+20\operatorname{n}+1}\]
done
clear
D)
\[\frac{\operatorname{n}}{401+20\operatorname{n}}\]
done
clear
View Answer play_arrow
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question_answer10) If \[\overset{\to }{\mathop{a}}\,\] and \[\overset{\to }{\mathop{b}}\,\] are non-collinear vectors, then the value of \[\alpha \]for which the vectors \[\overset{\to }{\mathop{\operatorname{u}}}\,=(\alpha -2)\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,\]and \[\overset{\to }{\mathop{\operatorname{v}}}\,=(2+3\alpha )\overset{\to }{\mathop{a}}\,-\overset{\to }{\mathop{3b}}\,\] are collinear is:
JEE Main Online Paper ( Held On 23 April 2013 )
A)
\[\frac{3}{2}\]
done
clear
B)
\[\frac{2}{3}\]
done
clear
C)
\[-\frac{3}{2}\]
done
clear
D)
\[-\frac{2}{3}\]
done
clear
View Answer play_arrow
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question_answer11) The least integral value \[a\] of \[x\]such that\[\frac{x-5}{{{x}^{2}}+5x-14}>0\], satisfies:
JEE Main Online Paper ( Held On 23 April 2013 )
A)
\[{{\alpha }^{2}}+3\alpha -4=0\]
done
clear
B)
\[{{\alpha }^{2}}-5\alpha +4=0\]
done
clear
C)
\[{{\alpha }^{2}}-7\alpha +6=0\]
done
clear
D)
\[{{\alpha }^{2}}+5\alpha -6=0\]
done
clear
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question_answer12) A, B,C, try to hit a target simultaneously but independently. Their respective probabilities of hitting the targets are \[\frac{3}{4},\frac{1}{2},\frac{5}{8}.\] The probability that the target is hit by A or B but not by C is :
JEE Main Online Paper ( Held On 23 April 2013 )
A)
\[21/64\]
done
clear
B)
\[7/8\]
done
clear
C)
\[7/32\]
done
clear
D)
\[9/64\]
done
clear
View Answer play_arrow
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question_answer13) If two lines \[{{\operatorname{L}}_{1}}\]and \[{{\operatorname{L}}_{2}}\] in space, are defined by \[{{\operatorname{L}}_{2}}=\{x=\sqrt{\lambda }y+(\sqrt{\lambda }-1)\}\] \[z=\left( \sqrt{\lambda }-1 \right)y+\sqrt{\lambda }\}\]and \[{{L}_{2}}=\{x=\sqrt{\mu }y+(1-\sqrt{\mu }),\] \[z=(1-\sqrt{\mu })y+\sqrt{\mu }\},\] then\[{{L}_{1}}\] is perpendicular to \[{{L}_{2}},\] for all non-negative reals\[\lambda \] and \[\mu \] such that:
JEE Main Online Paper ( Held On 23 April 2013 )
A)
\[\sqrt{\lambda }+\sqrt{\mu }=1\]
done
clear
B)
\[\lambda \ne \mu \]
done
clear
C)
\[\lambda +\mu =0\]
done
clear
D)
\[\lambda =\mu \]
done
clear
View Answer play_arrow
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question_answer14) The number of solutions of the equation\[\sin 2x-2\cos x+4\sin x=4\]in the interval\[[0,5\pi ]\]is:
JEE Main Online Paper ( Held On 23 April 2013 )
A)
3
done
clear
B)
5
done
clear
C)
4
done
clear
D)
6
done
clear
View Answer play_arrow
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question_answer15) If the projections of a line segment of the \[x,y\]and z-axes in 3-dimensional space are 2, 3 and 6 respectively, the length of the line segment is:
JEE Main Online Paper ( Held On 23 April 2013 )
A)
12
done
clear
B)
7
done
clear
C)
9
done
clear
D)
6
done
clear
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question_answer16) Let a =Im\[\left( \frac{1+{{z}^{2}}}{2iz} \right),\]where z is any non- zero complex ?complex number. The set \[\operatorname{A}=\{a:\left| z \right|1\operatorname{and}z\ne \pm 1\}\] is equal to :
JEE Main Online Paper ( Held On 23 April 2013 )
A)
(-1,1)
done
clear
B)
[-1,1]
done
clear
C)
[0, 1)
done
clear
D)
(-1,0]
done
clear
View Answer play_arrow
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question_answer17) Let \[{{\theta }_{1}}\]be the angle between two lines \[2x+3y+{{\operatorname{c}}_{1}}=0\]and \[-x+5y+{{c}_{2}}=0,\]and\[{{\theta }_{2}}\] be the angle between two lines \[2x+3y+{{c}_{1}}=0\], and\[-x+5y+{{c}_{3}}=0,\]where, \[{{c}_{1}},{{c}_{2}},\,\,{{c}_{3}}\]are any real numbers : Statement 1: If \[{{c}_{2}}\] and \[{{c}_{3}}\] are proportional, then \[{{\theta }_{1}}={{\theta }_{2}}\] Statement 2: \[{{\theta }_{1}}={{\theta }_{2}}\] for all \[{{\operatorname{c}}_{2}}\] and \[{{\operatorname{c}}_{3}}\]
JEE Main Online Paper ( Held On 23 April 2013 )
A)
Statement 1 is true; Statement 2 is true; Statement 2 is a correct explanation for Statement 1.
done
clear
B)
Statement 1 is true; Statement 2 is true; Statement 2 is not correct explanation for Statement 1.
done
clear
C)
Statement 1 is false; Statement 2 is true.
done
clear
D)
Statement 1 is true; Statement 2 is false.
done
clear
View Answer play_arrow
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question_answer18) If \[f(x)=\sin (\sin x)\]and \[f''(x)+\tan xf(x)+g(x)=0,\]then g(\[x\]) is :
JEE Main Online Paper ( Held On 23 April 2013 )
A)
\[{{\cos }^{2}}x\cos (\sin x)\]
done
clear
B)
\[{{\sin }^{2}}x\cos (\cos x)\]
done
clear
C)
\[{{\sin }^{2}}x\sin (\cos x)\]
done
clear
D)
\[{{\cos }^{2}}x\sin (\sin x)\]
done
clear
View Answer play_arrow
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question_answer19) If\[{{a}_{1}},{{a}_{2}},{{a}_{.3.......}}{{a}_{n}}......\] are in A.P. such that \[{{a}_{4}}-{{a}_{7}}+{{a}_{10}}=\operatorname{m},\] then sum of first 13 terms of this A.P., is :
JEE Main Online Paper ( Held On 23 April 2013 )
A)
10 m
done
clear
B)
12 m
done
clear
C)
13 m
done
clear
D)
15 m
done
clear
View Answer play_arrow
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question_answer20) A tangent to the hyperbola \[\frac{{{x}^{2}}}{4}-\frac{{{y}^{2}}}{2}=1\] meets \[x-\]axis at P and \[y-\]axis at \[Q.\]Lines PR and QR are drawn such that OPRQ is a rectangle (where O is the origin). The R lies on:
JEE Main Online Paper ( Held On 23 April 2013 )
A)
\[\frac{4}{{{x}^{2}}}+\frac{2}{{{y}^{2}}}=1\]
done
clear
B)
\[\frac{2}{{{x}^{2}}}-\frac{4}{{{y}^{2}}}=1\]
done
clear
C)
\[\frac{2}{{{x}^{2}}}+\frac{4}{{{y}^{2}}}=1\]
done
clear
D)
\[\frac{4}{{{x}^{2}}}-\frac{2}{{{y}^{2}}}=1\]
done
clear
View Answer play_arrow
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question_answer21) Let\[f\] be a composite function of \[x\] defined by \[f(u)=\frac{1}{{{\operatorname{u}}^{2}}+\operatorname{u}-2},\operatorname{u}(x)=\frac{1}{x-1}.\] Then the number of points \[x\] where f of is discontinuous is:
JEE Main Online Paper ( Held On 23 April 2013 )
A)
4
done
clear
B)
3
done
clear
C)
2
done
clear
D)
1
done
clear
View Answer play_arrow
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question_answer22) The value of \[\int\limits_{-\pi /2}^{\pi /2}{\frac{{{\sin }^{2}}x}{1+{{2}^{x}}}\operatorname{d}}x\]is :
JEE Main Online Paper ( Held On 23 April 2013 )
A)
\[\pi \]
done
clear
B)
\[\pi /2\]
done
clear
C)
\[4\pi \]
done
clear
D)
\[\pi /4\]
done
clear
View Answer play_arrow
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question_answer23) The cost of running a bus from A to B, is \[\operatorname{Rs}.\left( \operatorname{av}\frac{\operatorname{b}}{\operatorname{v}} \right),\] where v km/h is the cost speed of the bus. When the bus travels at 30 km /h, the cost comes out to be Rs. 75 while at 40 km/h, it is Rs. 65 Then the most economical speed (in Km/h)of the bus is :
JEE Main Online Paper ( Held On 23 April 2013 )
A)
45
done
clear
B)
50
done
clear
C)
60
done
clear
D)
40
done
clear
View Answer play_arrow
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question_answer24) The area under the curve\[y=\left| \cos x-\sin x \right|,\]\[0\le x\le \frac{\pi }{2},\]and above \[x-\operatorname{axis}\] is:
JEE Main Online Paper ( Held On 23 April 2013 )
A)
\[2\sqrt{2}\]
done
clear
B)
\[2\sqrt{2}-2\]
done
clear
C)
\[2\sqrt{2}+2\]
done
clear
D)
0
done
clear
View Answer play_arrow
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question_answer25) Let A, other than I or-I, a\[2\times 2\] areal matrix such that\[{{\operatorname{A}}^{2}}=I,\]I being the unit matrix. Let Tr be the sum of diagonal elements of A. Statement 1 :\[\operatorname{T}\operatorname{r}(\operatorname{A})=0\] Statement 1 :\[\det (\operatorname{A})=-1\]
JEE Main Online Paper ( Held On 23 April 2013 )
A)
Statement 1 is true; Statement 2 is false.
done
clear
B)
Statement 1 is true; Statement 2 is true; Statement 2 is not correct explanation for Statement 1.
done
clear
C)
Statement 1 is true; Statement 2 is true; Statement 2 is a correct explanation for Statement 1.
done
clear
D)
Statement 1 is false; Statement 2 is true.
done
clear
View Answer play_arrow
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question_answer26) On the sides AB,BC, CA of \[\Delta \operatorname{ABC},\] 3, 4, 5 distinct points (excluding vertices A, B, C) are respectively chosen. The number of triangles that can be constructed using these chosen points as vertices are:
JEE Main Online Paper ( Held On 23 April 2013 )
A)
210
done
clear
B)
205
done
clear
C)
215
done
clear
D)
220
done
clear
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question_answer27) If the median and the range of four numbers \[\{x,y,2x+y,x-y\},\] where \[0<y<x<2y\] are 10 and 28 respectively, then the mean of the numbers is :
JEE Main Online Paper ( Held On 23 April 2013 )
A)
18
done
clear
B)
10
done
clear
C)
5
done
clear
D)
14
done
clear
View Answer play_arrow
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question_answer28) If curve passes through the point \[\left( 2,\frac{7}{2} \right)\] and has slope \[\left( 1-\frac{1}{{{x}^{2}}} \right)\] at any point \[(x,y)\] on it, then the ordinate of the point on the curve whose abscissa is -2 is:
JEE Main Online Paper ( Held On 23 April 2013 )
A)
\[-\frac{3}{2}\]
done
clear
B)
\[\frac{3}{2}\]
done
clear
C)
\[\frac{5}{2}\]
done
clear
D)
\[-\frac{5}{2}\]
done
clear
View Answer play_arrow
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question_answer29) The point of intersection of the parabola \[{{y}^{2}}=4x\] at the ends of its latus rectum is
JEE Main Online Paper ( Held On 23 April 2013 )
A)
(0,2)
done
clear
B)
(3,0)
done
clear
C)
(0, 3)
done
clear
D)
(2,0)
done
clear
View Answer play_arrow
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question_answer30) Let \[\operatorname{R}=\{(x,y):x,y\in \operatorname{N}\] and \[{{x}^{2}}-4xy\] \[+3{{y}^{2}}=0\},\] where N is the set of all natural numbers. Then the relation R is :
JEE Main Online Paper ( Held On 23 April 2013 )
A)
reflexive but neither symmetric nor transitive
done
clear
B)
symmetric and transitive.
done
clear
C)
reflexive and symmetric.
done
clear
D)
reflexive and transitive.
done
clear
View Answer play_arrow
