JEE Advanced 2011 Question Paper with Solutions

Let $\overrightarrow a = \widehat i + \widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k$ and $\overrightarrow c = \widehat i - \widehat j - \widehat k$ be three vectors. A vector $\overrightarrow v$ in the plane of $\overrightarrow a$ and $\overrightarrow b ,$ whose projection on $\overrightarrow c$ is ${{1 \over {\sqrt 3 }}}$ , is given by

Match the statements given in Column - $I$ with the values given in Column- $II.$ $\,\,\,\,$ $\,\,\,\,$ $\,\,\,\,$ Column- $I$ (A) $\,\,\,\,$ If $\overrightarrow a = \widehat j + \sqrt 3 \widehat k,\overrightarrow b = - \widehat j + \sqrt 3 \widehat k$ and $\overrightarrow c = 2\sqrt 3 \widehat k$ form a triangle, then the internal angle of the triangle between $\overrightarrow a$ and $\overrightarrow b$ is (B) $\,\,\,\,$ If $\int\limits_a^b {\left( {f\left( x \right) - 3x} \right)dx = {a^2} - {b^2},}$ then the value of $f$ $\left( {{\pi \over 6}} \right)$ is (C) $\,\,\,\,$ The value of ${{{\pi ^2}} \over {\ell n3}}\int\limits_{7/6}^{5/6} {\sec \left( {\pi x} \right)dx}$ is (D) $\,\,\,\,$ The maximum value of $\left| {Arg\left( {{1 \over {1 - z}}} \right)} \right|$ for $\left| z \right| = 1,\,z \ne 1$ is given by $\,\,\,\,$ $\,\,\,\,$ $\,\,\,\,$ Column- $II$ (p) $\,\,\,\,$ ${{\pi \over 6}}$ (q) $\,\,\,\,$ ${{2\pi \over 3}}$ (r) $\,\,\,\,$ ${{\pi \over 3}}$ (s) $\,\,\,\,$ $\pi$ (t) $\,\,\,\,$ ${{\pi \over 2}}$

One mole of a monatomic gas is taken through a cycle ABCDA as shown in the PV diagram. Column II give the characteristics involved in the cycle. Match them with each of the processes given in Column I. Column I Column II (A) Process A $\to$ B (P) Internal energy decreases. (B) Process B $\to$ C (Q) Internal energy increase. (C) Process C $\to$ D (R) Heat is lost. (D) Process D $\to$ A (S) Heat is gained. (T) Work is done on the gas.

The density of a solid ball is to be determined in an experiment. The diameter of the ball is measured with a screw gauge, whose pitch is 0.5 mm and there are 50 divisions on the circular scale. The reading on the main scale is 2.5 mm and that on the circular scale is 20 divisions. If the measured mass of the ball has a relative error of 2 %, the relative percentage error in the density is

In 1 L saturated solution of AgCl [K sp (AgCl) = 1.6 $\times$ 10 -10 ], 0.1 mol of CuCl [K sp (CuCl) = 1.0 $\times$ 10 -6 ] is added. The resultant concentration of Ag + in the solution is 1.6 $\times$ 10 -x . The value of "x" is

Let $\left( {{x_0},{y_0}} \right)$ be the solution of the following equations ( 2 x ) ℓ n 2 = ( 3 y ) ℓ n 3 3 ℓ n x = 2 ℓ n y Then ${x_0}$ is

The positive integer value of $n\, > \,3$ satisfying the equation ${1 \over {\sin \left( {{\pi \over n}} \right)}} = {1 \over {\sin \left( {{{2\pi } \over n}} \right)}} + {1 \over {\sin \left( {{{3\pi } \over n}} \right)}}$ is

Let $f:(0,1) \to R$ be defined by $f(x) = {{b - x} \over {1 - bx}}$ , where b is a constant such that $0 < b < 1$ . Then

Let f(x) = x 2 and g(x) = sin x for all x $\in$ R. Then the set of all x satisfying $(f \circ g \circ g \circ f)(x) = (g \circ g \circ f)(x)$ , where $(f \circ g)(x) = f(g(x))$ , is

A long insulated copper wire is closely wound as a spiral of N turns. The spiral has inner radius a and outer radius b. The spiral lies in the xy-plane and a steady current I flows through the wire. The z-component of the magnetic field at the centre of the spiral is

If the point P(a, b, c), with reference to (E), lies on the plane 2x + y + z = 1, then the value of 7a + b + c is

Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\,{{11} \over {25}}$ and the probability of none of them occurring is $\,{{2} \over {25}}$ . If $P(T)$ denotes the probability of occurrence of the event $T,$ then