# Solved papers for JEE Main & Advanced JEE Main Paper (Held On 22 April 2013)

### done JEE Main Paper (Held On 22 April 2013) Total Questions - 30

• question_answer1) The number of ways in which an examiner can assign 30 marks to 8 question, giving not less than 2 marks to any question, is:     JEE Main  Online Paper (Held On 22 April 2013)

A)
$^{30}{{C}_{7}}$

B)
$^{21}{{C}_{8}}$

C)
$^{21}{{C}_{7}}$

D)
$^{30}{{C}_{8}}$

• question_answer2) If the system of linear equations                 ${{x}_{1}}+2{{x}_{2}}+3{{x}_{3}}=6$                 ${{x}_{1}}+3{{x}_{2}}+5{{x}_{3}}=9$                 $2{{x}_{1}}+5{{x}_{2}}+a{{x}_{3}}=b$                 is consistent and has infinite number of solutions, then:     JEE Main  Online Paper (Held On 22 April 2013)

A)
$a=8,b$ can be any real number

B)
$b=15,$a cab be any real number

C)
$a=R-\{8\}$ and $\operatorname{b}\in \operatorname{R}-[15]$

D)
$\operatorname{a}=8,b=15$

• question_answer3) Given sum of the first n terms of an A.P. is $2n+3{{n}^{2}}$. Anther A.P. is formed with the same first term and double of the common difference, the sum of n terms of the new A.P. is :     JEE Main  Online Paper (Held On 22 April 2013)

A)
$n+4{{n}^{2}}$

B)
$6{{n}^{2}}-n$

C)
${{n}^{2}}+4n$

D)
$3n+2{{n}^{2}}$

• question_answer4) Statement 1: The function ${{x}^{2}}({{\operatorname{e}}^{x}}+{{\operatorname{e}}^{-x}})$is increasing for all $x>0.$                 Statement 2: The functions ${{x}^{2}}{{e}^{x}}$ and ${{x}^{2}}{{e}^{-x}}$ are increasing for all $x>0$ and the sum of two increasing functions in any interval (a, b) is an increasing function in (a, b).     JEE Main  Online Paper (Held On 22 April 2013)

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

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

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

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

• question_answer5) Mean of 5 observations is 7. If four of these observations are 6, 7, 8, 10 and one is missing then      the variance of all the five observations is :     JEE Main  Online Paper (Held On 22 April 2013)

A)
4

B)
6

C)
8

D)
2

• question_answer6) The area of the region (in sq. units), in the first quadrant, bounded by the parabola $y=9{{x}^{2}}$ and the lines $x=0,$ $y=1$ and $y=4$ is:     JEE Main  Online Paper (Held On 22 April 2013)

A)
$7/9$

B)
$14/3$

C)
$7/3$

D)
$14/9$

• question_answer7) If the $x-$ intercept of some line L is double as that of the line, $3x+4y=12$ and the $y-$intercept of L is half as that of the same line. Then the slope of L is :     jEE Main  Online Paper (Held On 22 April 2013)

A)
-3

B)
-3/8

C)
-3/2

D)
-3/16

• question_answer8) The sum $\frac{3}{{{1}^{2}}}+\frac{5}{{{1}^{2}}+{{2}^{2}}}+\frac{7}{{{1}^{2}}+{{2}^{2}}+{{3}^{2}}}+...$ upto 11 ?terms is:     JEE Main  Online Paper (Held On 22 April 2013)

A)
$\frac{7}{2}$

B)
$\frac{11}{4}$

C)
$\frac{11}{2}$

D)
$\frac{60}{11}$

• question_answer9) The integral $\int\limits_{7\pi /4}^{7\pi /3}{\sqrt{{{\tan }^{2}}}x\operatorname{d}x}$ is equal to :     JEE Main  Online Paper (Held On 22 April 2013)

A)
$\log 2\sqrt{2}$

B)
$\log 2$

C)
$2\log 2$

D)
$\log \sqrt{2}$

• question_answer10) Let $\operatorname{R}=\{(3,3),(5,5),(9,9)(12,12),$ $(5,12),(3,9),(3,12)(3,5),\}$ be a relation on the set A = {3, 5, 9, 12} . Then, R is :     JEE Main  Online Paper (Held On 22 April 2013)

A)
reflexive, symmetric but not transitive.

B)
symmetric, transitive but not reflexive

C)
an equivalence relation.

D)
reflexive, transitive but not symmetric

• question_answer11) If a complex number z satisfies the equation $z+\sqrt{2}\left| z+1 \right|+i=0,\operatorname{then}\left| z \right|$ is equal to :     JEE Main  Online Paper (Held On 22 April 2013)

A)
2

B)
$\sqrt{3}$

C)
$\sqrt{5}$

D)
$1$

• question_answer12) If the 7th  term in binomial expansion of ${{\left( \frac{3}{^{3}\sqrt{84}}+\sqrt{3}\operatorname{In}x \right)}^{9}},x>0,$ equal to  729, then $x$cab be :                                   JEE Main  Online Paper (Held On 22 April 2013)

A)
${{e}^{2}}$

B)
$e$

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

D)
$2e$

• question_answer13) Statement 1: The line $x-2y=2$ meets the parabola, ${{y}^{2}}+2x=0$ only at the point $(-2,-2):$                 Statement 2: The line $y=mx-\frac{1}{2m}(\operatorname{m}\#0)$is tangent to the parabola, ${{y}^{2}}=-2x$ at the point $\left( -\frac{1}{2{{\operatorname{m}}^{2}}},\frac{1}{\operatorname{m}} \right).$     JEE Main  Online Paper (Held On 22 April 2013)

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

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

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

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

• question_answer14) If a circle C passing though (4, 0) touches the circle ${{x}^{2}}+{{y}^{2}}+4x-6y-12=0$ externally at point (1, -1), then the radius of the circle C is:     JEE Main  Online Paper (Held On 22 April 2013)

A)
5

B)
$2\sqrt{5}$

C)
4

D)
$\sqrt{57}$

• question_answer15) Let Q be the foot of perpendicular from the origin to the plane $4x-3y+z+13=0$ and R be point (-1, 1, -6) on the plane Then length QR is :     JEE Main  Online Paper (Held On 22 April 2013)

A)
$\sqrt{14}$

B)
$\sqrt{\frac{19}{2}}$

C)
$3\sqrt{\frac{7}{2}}$

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

• question_answer16) Given two independent events, if the probability that exactly one of them occurs is $\frac{26}{49}$ and the probability that nine of them occurs is $\frac{15}{49},$ then the probability of more probable of the two events is :     JEE Main  Online Paper (Held On 22 April 2013)

A)
$4/7$

B)
$6/7$

C)
$3/7$

D)
$5/7$

• question_answer17) The statement $p\to (q\to p)$ is equivalent to:     JEE Main  Online Paper (Held On 22 April 2013)

A)
$\operatorname{p}\to \operatorname{q}$

B)
$\operatorname{p}\left( \operatorname{p}\,\vee \operatorname{q} \right)$

C)
$\operatorname{p}\to \left( \operatorname{p}\to \operatorname{q} \right)$

D)
$\operatorname{p}\to \left( \operatorname{p}\wedge \operatorname{q} \right)$

• question_answer18) The maximum area of a right angled triangle with hypotenuse h is :     JEE Main  Online Paper (Held On 22 April 2013)

A)
$\frac{{{\operatorname{h}}^{2}}}{2}\sqrt{2}$

B)
$\frac{{{\operatorname{h}}^{2}}}{2}$

C)
$\frac{{{\operatorname{h}}^{2}}}{\sqrt{2}}$

D)
$\frac{{{\operatorname{h}}^{2}}}{4}$

• question_answer19) If $\int{\frac{{{x}^{2}}-x+1}{{{x}^{2}}+1}e{{\cot }^{-1}}}x\operatorname{d}x=\operatorname{A}(x){{\operatorname{e}}^{{{\cot }^{-1}}}}x+C,$ then A$\left( x \right)$ is equal to :     JEE Main  Online Paper (Held On 22 April 2013)

A)
$-x$

B)
$x$

C)
$\sqrt{1-x}$

D)
$\sqrt{1+x}$

• question_answer20) If two vertices of an equilateral triangle are $\operatorname{A}(-a,0)$ and $B(a,\,0),\,a>0$, the third vertex C lies above $x-$axis then the equation of the circumcircle of   $\Delta \operatorname{ABC}$       is :       JEE Main  Online Paper (Held On 22 April 2013)

A)
$3{{x}^{2}}+3{{y}^{2}}-2\sqrt{3}ay=3{{a}^{2}}$

B)
$3{{x}^{2}}+3{{y}^{2}}-2ay=3{{a}^{2}}$

C)
${{x}^{2}}+{{y}^{2}}-2ay={{a}^{2}}$

D)
${{x}^{2}}+{{y}^{2}}-\sqrt{3}ay={{a}^{2}}$

• question_answer21) The acute angle between two lines such that the direction cosines I, m, n of each of them satisfy the equations I + m + n = 0 and ${{\operatorname{I}}^{2}}+{{\operatorname{m}}^{2}}-{{\operatorname{n}}^{2}}=0$ is:     JEE Main  Online Paper (Held On 22 April 2013)

A)
${{15}^{0}}$

B)
${{30}^{0}}$

C)
${{60}^{0}}$

D)
${{45}^{0}}$

• question_answer22) Consider the differential equation $\frac{dy}{dx}=\frac{{{y}^{3}}}{2(x{{y}^{2}}-{{x}^{2}})}:$ Statement 1: The substitution $z={{y}^{2}}$ transforms the above equation into a first order homogenous differential equation. Statement 2: The solution of this differential equation is ${{y}^{2}}e\frac{-{{y}^{2}}}{x}=C.$

A)
Both statements are false.

B)
Statement 1 is rue and statement 2 is false.

C)
Statement 1 is false. Statement 2 is true

D)
Both statement s are true.

• question_answer23) The number of solution of the equation, ${{\sin }^{-1}}$ $x=2$ ${{\tan }^{-1}}$ $x$ (in principal values is :)     JEE Main  Online Paper (Held On 22 April 2013)

A)
1

B)
4

C)
2

D)
3

• question_answer24) For $>0,\operatorname{t}\in \left( 0,\frac{\pi }{2} \right),$ let $x=\sqrt{{{a}^{\sin -1}}t}$  and $y=\sqrt{a{{\cos }^{-1}}\operatorname{t}}.$ Then , $1+{{\left( \frac{\operatorname{dy}}{dx} \right)}^{2}}$equals:     JEE Main  Online Paper (Held On 22 April 2013)

A)
$\frac{{{x}^{2}}}{{{y}^{2}}}$

B)
$\frac{{{y}^{2}}}{{{x}^{2}}}$

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

D)
$\frac{{{x}^{2}}+{{y}^{2}}}{{{x}^{2}}}$

• question_answer25) If p, q, r are 3 real numbers satisfying the matrix equation, $[p\,q\,r]\left[ \begin{matrix} 3 & 4 & 1 \\ 3 & 2 & 3 \\ 2 & 0 & 2 \\ \end{matrix} \right]=[3\,\,0\,\,1]$ then $2\operatorname{p}+\operatorname{q}-\operatorname{r}$ equals:     JEE Main  Online Paper (Held On 22 April 2013)

A)
-3

B)
-1

C)
4

D)
2

• question_answer26) If $\hat{a},\hat{b}$ and $\hat{c}$ are unit vectors satisfying $\hat{a}-\sqrt{3}$ $\hat{b}+\hat{c}=\overset{\to }{\mathop{0}}\,$ then the angle between the vectors $\hat{a}$ and $\hat{c}$ is :     JEE Main  Online Paper (Held On 22 April 2013)

A)
$\frac{\pi }{4}$

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

C)
$\frac{\pi }{6}$

D)
$\frac{\pi }{2}$

• question_answer27) Let the equations of two ellipses be                 ${{\operatorname{E}}_{1}}:\frac{{{x}^{2}}}{3}+\frac{{{y}^{2}}}{2}=1$ and                 ${{\operatorname{E}}_{2}}:\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{{{\operatorname{b}}^{2}}}=1$. If the product of their eccentricities is $\frac{1}{2},$ then the length of the minor axis of ellipse ${{\operatorname{E}}_{2}}$ is:     JEE Main  Online Paper (Held On 22 April 2013)

A)
8

B)
9

C)
4

D)
2

• question_answer28) If $\alpha$ and $\beta$ are roots of the equation ${{x}^{2}}+\operatorname{p}x+\frac{3\operatorname{p}}{4}=0,$ such that $\left| \alpha -\beta \right|$=$\sqrt{10},$then p belongs to the set:     JEE Main  Online Paper (Held On 22 April 2013)

A)
{2, -5}

B)
{-3, 2}

C)
{-2, 5}

D)
{3, -5}

• question_answer29) Statement 1: The number of common solution of the trigonometric equations $2{{\sin }^{2}}\theta -\cos 2\theta =0$ and 2${{\cos }^{2}}\theta -3$ $\sin \theta =0$in the interval [0, 2$\pi$] is two :                 Statement 2: The number of solutions of the equation, $2{{\cos }^{2}}\theta -3$$\sin \theta =0$ in the interval $\left[ 0,\pi \right]$ is two     JEE Main  Online Paper (Held On 22 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 a correct explanation for Statement 1.

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

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

• question_answer30) Let $f(x)=-1+\left| x-2 \right|,$and g$\left( x \right)=1-\left| x \right|;$ then the set of all points where $fog$ us discontinuous is :

A)
{0, 2}

B)
{0, 1, 2}

C)
{0}

D)
an empty set