# Solved papers for JEE Main & Advanced JEE Main Online Paper (Held On 23 April 2013)

### done JEE Main Online Paper (Held On 23 April 2013) Total Questions - 30

• 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)
$\log \left| 1+\sqrt{2+{{x}^{2}}} \right|+C$

B)
$-\log \left| 1+\sqrt{2-{{x}^{2}}} \right|+C$

C)
$-x\log \left| 1-\sqrt{2-{{x}^{2}}} \right|+C$

D)
$x\log \left| 1-\sqrt{2+{{x}^{2}}} \right|+C$

• 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

B)
$4/3$

C)
$1/2$

D)
$3/4$

• 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.

B)
Statement 1 is true; Statement 2 is true; Statement 2 is a correct explanation for Statement 1.

C)
Statement 1 is true; Statement 2 is false.

D)
Statement 1 is false; Statement 2 is true.

• 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)
$Q\wedge R\to P$

B)
$P\wedge Q\to R$

C)
$Q\to R$

D)
$Q\to P$

• 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

B)
32

C)
9

D)
41

• 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)
$\frac{5}{4}{{\operatorname{n}}^{2}}$

B)
$\frac{5}{2}{{\operatorname{n}}^{2}}$

C)
$\frac{25{{a}^{2}}}{4}$

D)
$5{{a}^{2}}$

• 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

B)
12100

C)
12100

D)
12300

• 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)
$0$

B)
$\pm 4$

C)
$\pm 2$

D)
$\pm 3$

• 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)
$\frac{20}{401+20\operatorname{n}}$

B)
$\frac{\operatorname{n}}{{{\operatorname{n}}^{2}}+20\operatorname{n}+1}$

C)
$\frac{20}{{{\operatorname{n}}^{2}}+20\operatorname{n}+1}$

D)
$\frac{\operatorname{n}}{401+20\operatorname{n}}$

• 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)
$\frac{3}{2}$

B)
$\frac{2}{3}$

C)
$-\frac{3}{2}$

D)
$-\frac{2}{3}$

• 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)
${{\alpha }^{2}}+3\alpha -4=0$

B)
${{\alpha }^{2}}-5\alpha +4=0$

C)
${{\alpha }^{2}}-7\alpha +6=0$

D)
${{\alpha }^{2}}+5\alpha -6=0$

• 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)
$21/64$

B)
$7/8$

C)
$7/32$

D)
$9/64$

• 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)
$\sqrt{\lambda }+\sqrt{\mu }=1$

B)
$\lambda \ne \mu$

C)
$\lambda +\mu =0$

D)
$\lambda =\mu$

• 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

B)
5

C)
4

D)
6

• 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

B)
7

C)
9

D)
6

• 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)

B)
[-1,1]

C)
[0, 1)

D)
(-1,0]

• 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.

B)
Statement 1 is true; Statement 2 is true; Statement 2 is not correct explanation for Statement 1.

C)
Statement 1 is false; Statement 2 is true.

D)
Statement 1 is true; Statement 2 is false.

• 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)
${{\cos }^{2}}x\cos (\sin x)$

B)
${{\sin }^{2}}x\cos (\cos x)$

C)
${{\sin }^{2}}x\sin (\cos x)$

D)
${{\cos }^{2}}x\sin (\sin x)$

• 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

B)
12 m

C)
13 m

D)
15 m

• 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)
­$\frac{4}{{{x}^{2}}}+\frac{2}{{{y}^{2}}}=1$

B)
$\frac{2}{{{x}^{2}}}-\frac{4}{{{y}^{2}}}=1$

C)
$\frac{2}{{{x}^{2}}}+\frac{4}{{{y}^{2}}}=1$

D)
$\frac{4}{{{x}^{2}}}-\frac{2}{{{y}^{2}}}=1$

• 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

B)
3

C)
2

D)
1

• 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)
$\pi$

B)
$\pi /2$

C)
$4\pi$

D)
$\pi /4$

• 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

B)
50

C)
60

D)
40

• 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)
$2\sqrt{2}$

B)
$2\sqrt{2}-2$

C)
$2\sqrt{2}+2$

D)
0

• 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.

B)
Statement 1 is true; Statement 2 is true; Statement 2 is not correct explanation for Statement 1.

C)
Statement 1 is true; Statement 2 is true; Statement 2 is a correct explanation for Statement 1.

D)
Statement 1 is false; Statement 2 is true.

• 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

B)
205

C)
215

D)
220

• 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

B)
10

C)
5

D)
14

• 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)
$-\frac{3}{2}$

B)
$\frac{3}{2}$

C)
$\frac{5}{2}$

D)
$-\frac{5}{2}$

• 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)

B)
(3,0)

C)
(0, 3)

D)
(2,0)

• 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

B)
symmetric and transitive.

C)
reflexive and symmetric.

D)
reflexive and transitive.