JEE Advanced 2012 Question Paper with Solutions

If $\overrightarrow a ,\overrightarrow b$ and $\overrightarrow c$ are unit vectors satisfying ${\left| {\overrightarrow a - \overrightarrow b } \right|^2} + {\left| {\overrightarrow b - \overrightarrow c } \right|^2} + {\left| {\overrightarrow c - \overrightarrow a } \right|^2} = 9,$ then $\left| {2\overrightarrow a + 5\overrightarrow b + 5\overrightarrow c } \right|$ is

In the determination of Young's modulus $\left( {Y = {{4MLg} \over {\pi l{d^2}}}} \right)$ by using Searle's method, a wire of length L = 2 m and diameter d = 0.5 mm is used. For a load M = 2.5 kg, an extension l = 0.25 mm in the length of the wire is observed. Quantities d and l are measured using a screw gauge and a micrometer, respectively. They have the same pitch of 0.5 mm. The number of divisions on their circular scale is 100. The contributions to the maximum probable error of the Y measurement

A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density and radius 2R, as shown in the figure. The moment of inertia of this lamina about axes passing through O and P is I O and I P respectively. Both these axes are perpendicular to the plane of the lamina. The ratio I P /I O to the nearest integer is ____________.

If $\overrightarrow a$ and $\overrightarrow b$ are vectors such that $\left| {\overrightarrow a + \overrightarrow b } \right| = \sqrt {29}$ and $\,\overrightarrow a \times \left( {2\widehat i + 3\widehat j + 4\widehat k} \right) = \left( {2\widehat i + 3\widehat j + 4\widehat k} \right) \times \widehat b,$ then a possible value of $\left( {\overrightarrow a + \overrightarrow b } \right).\left( { - 7\widehat i + 2\widehat j + 3\widehat k} \right)$ is

The figure shows a system consisting of (i) a ring of outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed $\omega$ and (ii) an inner disc of radius 2R rotating anti-clockwise with angular speed ${\omega \over 2}$ . The ring and disc are separated by frictionless ball bearings. The point P on the inner disc is at a distance R from the origin, where OP makes an angle of $30^\circ$ with the horizontal. Then with respect to the horizontal surface,

The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [ $\alpha_0$ is Bohr radius]

Let $\theta ,\,\varphi \, \in \,\left[ {0,2\pi } \right]$ be such that $2\cos \theta \left( {1 - \sin \,\varphi } \right) = {\sin ^2}\theta \,\,\left( {\tan {\theta \over 2} + \cot {\theta \over 2}} \right)\cos \varphi - 1,\,\tan \left( {2\pi - \theta } \right) > 0$ and $- 1 < \sin \theta \, < - {{\sqrt 3 } \over 2},$ then $\varphi$ cannot satisfy

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x - 5y = 20 to the circle ${x^2}\, + \,{y^2} = 9$ is

The value of $6 + {\log _{3/2}}\left( {{1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}...} } } } \right)$ is __________.

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

Let $\alpha$ (a) and $\beta$ (a) be the roots of the equation ( 1 + a 3 − 1 ) x 2 + ( 1 + a − 1 ) x + ( 1 + a 6 − 1 ) = 0 where $a > - 1$ . Then $\mathop {\lim }\limits_{a \to {0^ + }} \alpha (a)$ and $\mathop {\lim }\limits_{a \to {0^ + }} \beta (a)$ are

A cylinder cavity of diameter a exists inside a cylinder of diameter 2a as shown in the figure. Both the cylinder and the cavity are infinitely long. A uniform current density J flows along the length. If the magnitude of the magnetic field at the point P is given by ${N \over {12}}{\mu _0}aJ$ , then the value of N is ______________.

If $y(x)$ satisfies the differential equation $y' - y\,tan\,x = 2x\,secx$ and $y(0)=0,$ then

The point $P$ is the intersection of the straight line joining the points $Q(2, 3, 5)$ and $R(1, -1, 4)$ with the plane $5x-4y-z=1.$ If $S$ is the foot of the perpendicular drawn from the point $T(2, 1, 4)$ to $QR,$ then the length of the line segment $PS$ is

Let f ( x ) = { x 2 | cos ⁡ π x | , x ≠ 0 0 , x = 0 x $\in$ R, then f is

If $\mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + x + 1} \over {x + 1}} - ax - b} \right) = 4$ , then

The value of the integral $\int\limits_{ - \pi /2}^{\pi /2} {\left( {{x^2} + 1n{{\pi + x} \over {\pi - x}}} \right)\cos xdx}$ is

An organic compound undergoes first-order decomposition. The time taken for its decomposition to 1/8 and 1/10 of its initial concentration are t 1/8 and t 1/10 respectively. What is the value of $\left[ {{{{t_{1/8}}} \over {{t_{1/10}}}}} \right] \times 10$ ? ( ${\log _{10}}2 = 0.3$ )

The substituents R 1 and R 2 for nine peptides are listed in the table given below. How many of these peptides are positively charged at pH = 7.0 ? .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-baqh{text-align:center;vertical-align:top} Peptide ${R_1}$ ${R_2}$ I H H II H $C{H_3}$ III $C{H_2}COOH$ H IV $C{H_2}CON{H_2}$ ${(C{H_2})_4}N{H_2}$ V $C{H_2}CON{H_2}$ $C{H_2}CON{H_2}$ VI ${(C{H_2})_4}N{H_2}$ ${(C{H_2})_4}N{H_2}$ VII $C{H_2}COOH$ $C{H_2}CON{H_2}$ VIII $C{H_2}OH$ ${(C{H_2})_4}N{H_2}$ IX ${(C{H_2})_4}N{H_2}$ $C{H_3}$

The electrochemical cell shown below is a concentration cell. M | M 2+ (saturated solution of a sparingly soluble salt, MX 2 ) || M 2+ (0.001 mol dm –3 ) | M The emf of the cell depends on the difference in concentrations of M 2+ ions at the two electrodes. The emf of the cell at 298 K is 0.059 V. The solubility product (K sp ; mol 3 dm –9 ) of MX 2 at 298 K based on the information available for the given concentration cell is (take 2.303 $\times$ R $\times$ 298/F = 0.059 V)