JEE Advanced 2010 Question Paper with Solutions

A student uses a simple pendulum of exactly 1m length to determine g, the acceleration due to gravity. He uses a stop watch with the least count of 1 sec for this and records 40 seconds for 20 oscillations. For this observation, which of the following statement(s) is (are) true?

Gravitational acceleration on the surface of a planet is ${{\sqrt 6 } \over {11}}g$ , where $g$ is the gravitational acceleration on the surface of the earth. The average mass density of the planet is ${2 \over 3}$ times that of the earth. If the escape speed on the surface of the earth is taken to be 11 kms -1 , the escape speed on the surface of the planet in kms -1 will be

Two adjacent sides of a parallelogram $ABCD$ are given by $\overrightarrow {AB} = 2\widehat i + 10\widehat j + 11\widehat k$ and $\,\overrightarrow {AD} = -\widehat i + 2\widehat j + 2\widehat k$ The side $AD$ is rotated by an acute angle $\alpha$ in the plane of the parallelogram so that $AD$ becomes $AD'.$ If $AD'$ makes a right angle with the side $AB,$ then the cosine of the angle $\alpha$ is given by

Silver (atomic weight = 108 g mol -1 ) has a density of 10.5 g.cm -3 . The number of silver atoms on a surface of area 10 -12 m 2 can be expressed in scientific notation as y $\times$ 10 x . The value of x is?

Let ${S_k}$ = 1, 2,....., 100, denote the sum of the infinite geometric series whose first term is $\,{{k - 1} \over {k\,!}}$ and the common ratio is ${1 \over k}$ . Then the value of ${{{{100}^2}} \over {100!}}\,\, + \,\,\sum\limits_{k = 1}^{100} {\left| {({k^2} - 3k + 1)\,\,{S_k}} \right|\,\,}$ is

The circle ${x^2} + {y^2} - 8x = 0$ and hyperbola ${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$ intersect at the points $A$ and $B$ . Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

The number of all possible values of $\theta$ where $0 < \theta < \pi ,$ for which the system of equations $ ( y + z ) cos ⁡ 3 θ = ( x y z ) sin ⁡ 3 θ $$ $x\sin 3\theta = {{2\cos 3\theta } \over y} + {{2\sin 3\theta } \over z}$ $$ $\left( {xyz} \right){\mkern 1mu} \sin 3\theta = \left( {y + 2z} \right){\mkern 1mu} \cos 3\theta + y{\mkern 1mu} sin3\theta$ $ have a solution $\left( {{x_0},{y_0},{z_0}} \right)$ with ${y_0}{z_0}{\mkern 1mu} \ne {\mkern 1mu} 0,$ is

Let $z_1$ and $z_2$ be two distinct complex numbers let $z=(1-t) z_1+t z_2$ for some real number t with $0 < t < 1$ . If $\operatorname{Arg}(w)$ denotes the principal argument of a nonzero complex number $w$ , then :

For $r = 0,\,1,....,$ let ${A_r},\,{B_r}$ and ${C_r}$ denote, respectively, the coefficient of ${X^r}$ in the expansions of ${\left( {1 + x} \right)^{10}},$ ${\left( {1 + x} \right)^{20}}$ and ${\left( {1 + x} \right)^{30}}.$ Then $\sum\limits_{r = 1}^{10} {{A_r}\left( {{B_{10}}{B_r} - {C_{10}}{A_r}} \right)}$ is equal to

Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, 100, 60 and 40 W bulbs have filament resistances R 100 , R 60 and R 40 respectively, the relation between these resistances is

A ray OP of monochromatic light is incident on the face AB of prism ABCD near vertex B at an incident angle of 60 $^\circ$ (see figure). If the refractive index of the material of the prism is $\sqrt3$ , which of the following is(are) correct?

Equation of the plane containing the straight line ${x \over 2} = {y \over 3} = {z \over 4}$ and perpendicular to the plane containing the straight lines ${x \over 3} = {y \over 4} = {z \over 2}$ and ${x \over 4} = {y \over 2} = {z \over 3}$ is

Match the statement in Column- $I$ with the values in Column- $II$ $\,\,\,\,$ $\,\,\,\,$ $\,\,\,\,$ Column- $I$ (A) $\,\,\,\,$ A line from the origin meets the lines $\,{{x - 2} \over 1} = {{y - 1} \over { - 2}} = {{z + 1} \over 1}$ and ${{x - {8 \over 3}} \over 2} = {{y + 3} \over { - 1}} = {{z - 1} \over 1}$ at $P$ and $Q$ respectively. If length $PQ=d,$ then ${d^2}$ is (B) $\,\,\,\,$ The values of $x$ satisfying ${\tan ^{ - 1}}\left( {x + 3} \right) - {\tan ^{ - 1}}\left( {x - 3} \right) = {\sin ^{ - 1}}\left( {{3 \over 5}} \right)$ are (C) $\,\,\,\,$ Non-zero vectors $\overrightarrow a ,\overrightarrow b$ and $\overrightarrow c \,\,$ satisfy $\overrightarrow a \,.\,\overrightarrow b \, = 0.$ $\left( {\overrightarrow b - \overrightarrow a } \right).\left( {\overrightarrow b + \overrightarrow c } \right) = 0$ and $2\left| {\overrightarrow b + \overrightarrow c } \right| = \left| {\overrightarrow b - \overrightarrow a } \right|.$ If $\overrightarrow a = \mu \overrightarrow b + 4\overrightarrow c \,\,,$ then the possible values of $\mu$ are (D) $\,\,\,\,$ Let $f$ be the function on $\left[ { - \pi ,\pi } \right]$ given by $f(0)=9$ and $f\left( x \right) = \sin \left( {{{9x} \over 2}} \right)/\sin \left( {{x \over 2}} \right)$ for $x \ne 0$ The value of ${2 \over \pi }\int_{ - \pi }^\pi {f\left( x \right)dx}$ is $\,\,\,\,$ $\,\,\,\,$ $\,\,\,\,$ Column- $II$ (p) $\,\,\,\,$ $-4$ (q) $\,\,\,\,$ $0$ (r) $\,\,\,\,$ $4$ (s) $\,\,\,\,$ $5$ (t) $\,\,\,\,$ $6$

The concentration of potassium ions inside a biological cell is at least twenty times higher than the outside. The resulting potential difference across the cell is important in several processes such as transmission of nerve impulses and maintaining the ion balance. A simple model for such a concentration cell involving a metal M is : M(s) | M + (aq ; 0.05 molar) || M + (aq ; 1 molar) | M(s) For the above electrolytic cell the magnitude of the cell potential | E cell | = 70 mV. For the above cell :