JEE Advanced 2013 Question Paper with Solutions

One end of a horizontal thick copper wire of length 2L and radius 2R is welded to an end of another horizontal thin copper wire of length L and radius R. When the arrangement is stretched by applying forces at two ends, the ratio of the elongation in the thin wire to that in the thick wire is

match List $I$ with List $II$ and select the correct answer using the code given below the lists: $\,\,\,\,$ $\,\,\,\,$ $\,\,\,\,$ List $I$ (P.) $\,\,\,\,$ Volume of parallelopiped determined by vectors $\overrightarrow a ,\overrightarrow b$ and $\overrightarrow c$ is $2.$ Then the volume of the parallelepiped determined by vectors $2\left( {\overrightarrow a \times \overrightarrow b } \right),3\left( {\overrightarrow b \times \overrightarrow c } \right)$ and $\left( {\overrightarrow c \times \overrightarrow a } \right)$ is (Q.) $\,\,\,\,$ Volume of parallelopiped determined by vectors $\overrightarrow a ,\overrightarrow b$ and $\overrightarrow c$ is $5.$ Then the volume of the parallelepiped determined by vectors $3\left( {\overrightarrow a + \overrightarrow b } \right),\left( {\overrightarrow b + \overrightarrow c } \right)$ and $2\left( {\overrightarrow c + \overrightarrow a } \right)$ is (R.) $\,\,\,\,$ Area of a triangle with adjacent sides determined by vectors ${\overrightarrow a }$ and ${\overrightarrow b }$ is $20.$ Then the area of the triangle with adjacent sides determined by vectors $\left( {2\overrightarrow a + 3\overrightarrow b } \right)$ and $\left( {\overrightarrow a - \overrightarrow b } \right)$ is (S.) $\,\,\,\,$ Area of a parallelogram with adjacent sides determined by vectors ${\overrightarrow a }$ and ${\overrightarrow b }$ is $30.$ Then the area of the parallelogram with adjacent sides determined by vectors $\left( {\overrightarrow a + \overrightarrow b } \right)$ and ${\overrightarrow a }$ is $\,\,\,\,$ $\,\,\,\,$ $\,\,\,\,$ List $II$ (1.) $\,\,\,\,$ $100$ (2.) $\,\,\,\,$ $30$ (3.) $\,\,\,\,$ $24$ (4.) $\,\,\,\,$ $60$

A vertical line passing through the point $(h,0)$ intersects the ellipse ${{{x^2}} \over 4} + {{{y^2}} \over 3} = 1$ at the points $P$ and $Q$ . Let the tangents to the ellipse at $P$ and $Q$ meet at the point $R$ . If $\Delta \left( h \right)$ $=$ area of the triangle $PQR$ , ${{\Delta _1}}$ $= \mathop {\max }\limits_{1/2 \le h \le 1} \Delta \left( h \right)$ and ${{\Delta _2}}$ $= \mathop {\min }\limits_{1/2 \le h \le 1} \Delta \left( h \right)$ , then ${8 \over {\sqrt 5 }}{\Delta _1} - 8{\Delta _2} =$

Let $S = {S_1} \cap {S_2} \cap {S_3}$ , where ${S_1} = \left\{ {z \in C:\left| z \right| < 4} \right\},{S_2} = \left\{ {z \in C:{\mathop{\rm Im}\nolimits} \left[ {{{z - 1 + \sqrt 3 i} \over {1 - \sqrt 3 i}}} \right] > 0} \right\}$ and ${S_3} = \left\{ {z \in C:{\mathop{\rm Re}\nolimits} z > 0} \right\}\,$ . $\,\mathop {\min }\limits_{z \in S} \left| {1 - 3i - z} \right| =$

A line $L:y=mx+3$ meets $y$ -axis at R $(0, 3)$ and the arc of the parabola ${y^2} = 16x,$ $0 \le y \le 6$ at the point $F\left( {{x_0},{y_0}} \right)$ . The tangent to the parabola at $F\left( {{x_0},{y_0}} \right)$ intersects the $y$ -axis at $G\left( {0,{y_1}} \right)$ . The slope $m$ of the line $L$ is chosen such that the area of the triangle $EFG$ has a local maximum. Match List $I$ with List $II$ and select the correct answer using the code given below the lists: List $I$ P. $\,\,\,m =$ Q. $\,\,\,$ Maximum area of $\Delta EFG$ is R. $\,\,\,$ ${y_0} =$ S. $\,\,\,$ ${y_1} =$ List $II$ 1. $\,\,\,$ ${1 \over 2}$ 2. $\,\,\,$ $4$ 3. $\,\,\,$ $2$ 4. $\,\,\,$ $1$

In the Young's double-slit experiment using a monochromatic light of wavelength $\lambda$ , the path difference (in terms of an integer n) corresponding to any point having half the peak intensity is

Perpendiculars are drawn from points on the line $\frac{x+2}{2}=\frac{y+1}{-1}=\frac{z}{3}$ to the plane $x+y+$ $z=3$ . The foot of perpendiculars lie on the line

Of the three independent events ${E_1},{E_2}$ and ${E_3},$ the probability that only ${E_1}$ occurs is $\alpha ,$ only ${E_2}$ occurs is $\beta$ and only ${E_3}$ occurs is $\gamma .$ Let the probability $p$ that none of events ${E_1},{E_2}$ or ${E_3}$ occurs satisfy the equations $\left( {\alpha -2\beta } \right)p = \alpha \beta$ and $\left( {\beta - 3\gamma } \right)p = 2\beta \gamma .$ All the given probabilities are assumed to lie in the interval $(0, 1)$ . Then ${{\Pr obability\,\,of\,\,occurrence\,\,of\,\,{E_1}} \over {\Pr obability\,\,of\,\,occurrence\,\,of\,\,{E_3}}}$

Four persons independently solve a certain problem correctly with probabilities ${1 \over 2},{3 \over 4},{1 \over 4},{1 \over 8}.$ Then the probability that the problem is solved correctly by at least one of them is

A line $l$ passing through the origin is perpendicular to the lines $ l 1 : ( 3 + t ) i ^ + ( − 1 + 2 t ) j ^ + ( 4 + 2 t ) k ^ , − ∞ < t < ∞ $$ ${l_2}:\left( {3 + 2s} \right)\widehat i + \left( {3 + 2s} \right)\widehat j + \left( {2 + s} \right)\widehat k,\,\,\,\,\, - \infty < s < \infty$ $ Then, the coordinate(s) of the points(s) on ${l_2}$ at a distance of $\sqrt {17}$ from the point of intersection of $l$ and ${l_1}$ is (are)

A box ${B_1}$ contains $1$ white ball, $3$ red balls and $2$ black balls. Another box ${B_2}$ contains $2$ white balls, $3$ red balls and $4$ black balls. A third box ${B_3}$ contains $3$ white balls, $4$ red balls and $5$ black balls. If $2$ balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these $2$ balls are drawn from box ${B_2}$ is

An aqueous solution of X is added slowly to an aqueous solution of Y as shown in List – I. The variation in conductivity of these reactions in List – II. Match List – I with List – II and select the correct answer using the code given below the lists: List - I P. $\mathop {(C{}_2{H_5}){}_3N}\limits_X$ + $\mathop {C{H_3}COOH}\limits_Y$ Q. $\mathop {KI(0.1M)}\limits_X$ + $\mathop {AgN{O_3}(0.01M)}\limits_Y$ R. $\mathop {C{H_3}COOH}\limits_X$ + $\mathop {KOH}\limits_Y$ S. $\mathop {NaOH}\limits_X$ + $\mathop {HI}\limits_Y$ List - II 1. Conductivity decreases then increases 2. Conductivity decreases then does not change much 3. Conductivity increases then does not change much 4. Conductivity does not change much then increases