JEE Advanced 2004 Question Paper with Solutions

If a → , b → , c → and d → are distinct vectors such that a → × c → = b → × d → and a → × b → = c → × d → . Prove that ( a → − d → ) . ( b → − c → ) ≠ 0 i . e . a → . b → + d → . c → ≠ d → . b → + a → . c →

In Searle's experiment, which is used to find Young's Modulus of elasticity, the diameter of experimental wire is D = 0.05 cm ( measured by a scale of least count 0.001 cm ) and length is L = 110 cm ( measured by a scale of least count 0.1 cm ). A weight of 50 N causes an extension of X = 0.125 cm (measured by a micrometer of least count 0.001 cm ). Find maximum possible error in the value of Young's modulus. Screw gauge and and meter scale are free from error.

A screw gauge having 100 equal divisions and a pitch of length 1 mm is used to measure the diameter of a wire of length 5.6 cm. The main scale reading is 1mm and 47 th circular division coincides with the main scale. Find the curved surface area of wire in cm 2 to appropriate significant figures. ( use π = 22 7 )

If $\overrightarrow a = \left( {\widehat i + \widehat j + \widehat k} \right),\overrightarrow a .\overrightarrow b = 1$ and $\overrightarrow a \times \overrightarrow b = \widehat j - \widehat k,$ then $\overrightarrow b$ is

The unit vector which is orthogonal to the vector $3\overrightarrow i + 2\overrightarrow j + 6\overrightarrow k$ and is coplanar with the vectors $\,2\widehat i + \widehat j + \widehat k$ and $\,\widehat i - \widehat j + \widehat k$ $\,\,\,$ is

(a) The Schrodinger wave equation for hydrogen atom is $ ψ = 1 4 2 π ( 1 a 0 ) 3 / 2 ( 2 − r 0 a 0 ) e − r 0 / a 0 $ Where a 0 is Bohr's radius. If the radial node in 2s be at r 0 , then find r 0 in terms of a 0 . (b) A baseball having mass 100 g moves with velocity 100 m/s. Determine the value of wavelength of baseball.

A ball of mass 100 g is moving with 100 ms -1 . Find it's wavelength.

Draw the structure of XeF 4 and OSF 4 according to VSEPR theory clearly indicating the state of hybridisation of the central atom and lone pair of electrons (if any) on the central atom.

Arrange the following : In the decreasing order of the O - O bond length present in them O 2 , KO 2 and O 2 [AsF 4 ]

A sodium salt on treatment with MgCl 3 gives white precipitate only on heating. The anion of the sodium salt is

Tangent is drawn to parabola ${y^2} - 2y - 4x + 5 = 0$ at a point $P$ which cuts the directrix at the point $Q$ . $A$ point $R$ is such that it divides $QP$ externally in the ratio $1/2:1$ . Find the locus of point $R$

Find the centre and radius of circle given by $\,\left| {{{z - \alpha } \over {z - \beta }}} \right| = k,k \ne 1\,$ where, ${\rm{z = x + iy, }}\alpha {\rm{ = }}\,{\alpha _1}{\rm{ + i}}{\alpha _2}{\rm{,}}\,\beta = {\beta _1}{\rm{ + i}}{\beta _2}{\rm{ }}$

If a , b , c are positive real numbers. Then prove that $ ( a + 1 ) 7 ( b + 1 ) 7 ( c + 1 ) 7 > 7 7 a 4 b 4 c 4 $

Find the equation of circle touching the line 2x + 3y + 1 = 0 at (1, -1) and cutting orthogonally the circle having line segment joining (0, 3) and (- 2, -1) as diameter.

Prove by permulation or otherwise ${{({n^2})!} \over {{{(n!)}^n}}}$ is an integer $(n \in {1^ + })$ .

${P_1}$ and ${P_2}$ are planes passing through origin. ${L_1}$ and ${L_2}$ are two line on ${P_1}$ and ${P_2}$ respectively such that their intersection is origin. Show that there exists points $A, B, C,$ whose permutation $A',B',C'$ can be chosen such that (i) $A$ is on ${L_1},$ $B$ on ${P_1}$ but not on ${L_1}$ and $C$ not on ${P_1}$ (ii) $A'$ is on ${L_2},$ $B'$ on ${P_2}$ but not on ${L_2}$ and $C'$ not on ${P_2}$

A curve $'C''$ passes through $(2,0)$ and the slope at $(x,y|)$ as $\,{{{{\left( {x + 1} \right)}^2} + \left( {y - 3} \right)} \over {x + 3}}$ . Find the equation of the curve. Find the area bounded by curve and $x$ -axis in fourth quadrant.

If $y\left( x \right) = \int\limits_{{x^2}/16}^{{x^2}} {{{\cos x\cos \sqrt \theta } \over {1 + {{\sin }^2}\sqrt \theta }}d\theta ,}$ then find ${{dy} \over {dx}}$ at $x = \pi$

A parallelopiped ′ S ′ has base points A , B , C and D and upper face points A ′ , B ′ , C ′ and D ′ . This parallelopiped is compressed by upper face A ′ B ′ C ′ D ′ to form a new parallelopiped ′ T ′ having upper face points A ″ , B ″ , C ″ and D ″ . Volume of parallelopiped T is 90 percent of the volume of parallelopiped S . Prove that the locus of ′ A ‴ , is a plane.

Using Rolle's theorem, prove that there is at least one root in $\left( {{{45}^{1/100}},46} \right)$ of the polynomial $P\left( x \right) = 51{x^{101}} - 2323{\left( x \right)^{100}} - 45x + 1035$ .

For the given reactions, A + B → Products, following data were obtained .tg {border-collapse:collapse;border-spacing:0;margin: 0 auto} .tg td{font-family:Arial, sans-serif;font-size:14px;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;} .tg th{font-family:Arial, sans-serif;font-size:14px;font-weight:normal;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;} .tg .tg-s6z2{text-align:center} .tg .tg-baqh{text-align:center;vertical-align:top} .tg .tg-s268{text-align:left} [A o ] [B o ] R o (mol L -1 s -1 ) 1 0.1 0.1 0.05 2 0.2 0.1 0.10 3 0.1 0.2 0.05 (a) Write the rate law expression (b) Find the rate constant

A compound AB has rock salt type structure. The formula weight of AB is 6.023 Y amu, and the closest A - B distance is Y 1/3 nm, where Y is an arbitrary number (a) Find the density of the lattice (b) If the density of lattice is found to be 20 kg m -3 , then predict the type of defect

1.22 g of benzoic acid is dissolved in 100 g of acetone and 100 g of benzene separately. Boiling point of the solution in acetone increases by 0.17 o C, while that in the benzene increases by 0.13 o C; K b for acetone and benzene is 1.7 K kg mol -1 and 2.6 K kg mol -1 . Find molecular weight of the benzoic acid in two cases and justify your answer.

The pair of compounds having metals in their highest oxidation state is