JEE Advanced 2005 Question Paper with Solutions

If the incident ray on a surface is along the unit vector $\widehat v\,\,,$ the reflected ray is along the unit vector $\widehat w\,\,$ and the normal is along unit vector $\widehat a\,\,$ outwards. Express $\widehat w\,\,$ in terms of $\widehat a\,\,$ and $\widehat v\,\,.$

The side of a cube is measured by vernier callipers (10 divisions of a vernier scale coincide with 9 divisions of main scale, where 1 division of main scale is 1 mm). The main scale reads 10 mm and first division of vernier scale coincides with the main scale. Mass of the cube is 2.736 g. Find the density of the cube in appropriate significant figures.

If $\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c$ are three non-zero, non-coplanar vectors and $\overrightarrow {{b_1}} = \overrightarrow b - {{\overrightarrow b .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a ,\overrightarrow {{b_2}} = \overrightarrow b + {{\overrightarrow b .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a ,$ $\overrightarrow {{c_1}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a + {{\overrightarrow b .\,\overrightarrow c } \over {{{\left| c \right|}^2}}}{\overrightarrow b _1},\,\,\overrightarrow {{c_2}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a - {{\overrightarrow b \,.\,\overrightarrow c } \over {{{\left| {{{\overrightarrow b }_1}} \right|}^2}}}{\overrightarrow b _1},$ $\overrightarrow {{c_3}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow c } \right|}^2}}}\overrightarrow a + {{\overrightarrow b .\,\overrightarrow c } \over {{{\left| c \right|}^2}}}{\overrightarrow b _1},\,\,\overrightarrow {{c_4}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow c } \right|}^2}}}\overrightarrow a - {{\overrightarrow b \,.\,\overrightarrow c } \over {{{\left| {{{\overrightarrow b }_1}} \right|}^2}}}{\overrightarrow b _1},$ then the set of orthogonal vectors is

Which of the following set have different dimensions?

A simple pendulum has time period T 1 . The point of suspension is now moved upward according to the relation y = Kt 2 , (K = 1 m/s 2 ) where y is the vertical displacement. The time period now become T 2 . The ratio of ${{T_1^2} \over {T_2^2}}$ is (g = 10 m/s 2 )

Find the velocity (ms -1 ) of electron in first Bohr's orbit of radius a 0 . Also find the de Broglie's wavelength (in m). Find the orbital angular momentum of 2p orbital of hydrogen atom in units of $h/2 \pi$ .

Which species has the maximum number of lone pair of electrons on the central atom?

The number of radial nodes of 3s and 2p orbitals are respectively

If one the vertices of the square circumscribing the circle $\left| {z - 1} \right| = \sqrt 2 \,is\,2 + \sqrt {3\,} \,i$ . Find the other vertices of the square.

The area of the triangle formed by intersection of a line parallel to $x$ -axis and passing through $P (h, k)$ with the lines $y = x$ and $x + y = 2$ is $4{h^2}$ . Find the locus of the point $P$ .

If total number of runs scored in n matches is $\left( {{{n + 1} \over 4}} \right)\,\,({2^{n + 1}} - n - 2)\,$ where $n > 1$ , and the runs scored in the ${k^{th}}$ match are given by k. $\,{2^{n + 1 - k}}$ , where $1 \le k \le n$ . Find n.

Tangents are drawn from any point on the hyperbola ${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$ to the circle ${x^2} + {y^2} = 9$ .Find the locus of mid-point of the chord of contact.

Circles with radii 3, 4 and 5 touch each other externally. It P is the point of intersection of tangents to these circles at their points of contact, find the distance of P from the points of contact.

If length of tangent at any point on the curve $y=f(x)$ intecepted between the point and the $x$ -axis is length $1.$ Find the equation of the curve.

If $\left| {f\left( {{x_1}} \right) - f\left( {{x_2}} \right)} \right| < {\left( {{x_1} - {x_2}} \right)^2},$ for all ${x_1},{x_2} \in R$ . Find the equation of tangent to the cuve $y = f\left( x \right)$ at the point $(1, 2)$ .

Evaluate $\,\int\limits_0^\pi {{e^{\left| {\cos x} \right|}}} \left( {2\sin \left( {{1 \over 2}\cos x} \right) + 3\cos \left( {{1 \over 2}\cos x} \right)} \right)\sin x\,\,dx$

A person goes to office either by car, scooter, bus or train, the probability of which being ${1 \over 7},{3 \over 7},{2 \over 7}$ and ${1 \over 7}$ respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is ${2 \over 9},{1 \over 9},{4 \over 9}$ and ${1 \over 9}$ respectively. Given that he reached office in time, then what is the probability that he travelled by a car.

Find the area bounded by the curves ${x^2} = y,{x^2} = - y$ and ${y^2} = 4x - 3.$

At constant temperature and volume, X decomposes as 2X(g) $\to$ 3Y(g) + 2Z(g); P x is the partial pressure of X. .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} Observation No. Time (in minute) P x (in mm of Hg) 1 0 800 2 100 400 3 200 200 (i) What is the order of the reaction to X? (ii) Find the rate constant (iii) Find the time for 75% completion of the reaction. (iv) Find the total pressure when pressure of X is 700 mm of Hg

(a). For the reaction Ag + (aq) + Cl - (aq) $\leftrightharpoons$ AgCl (s) Given: .tg {border-collapse:collapse;border-spacing:0;} .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} Species $\Delta G_f^o$ (kJ/mol) Ag + (aq) +77 Cl - (aq) -129 AgCl (s) -109 Write the cell representation of above reaction and calculate $E_{cell}^o$ at 298 K. Also find the solubility product if AgCl. (b) If 6.539 $\times$ 10 -2 g of metallic zinc is added to 100 ml saturated solution of AgCl. Find the value of ${\log _{10}}{{\left[ {Z{n^{2 + }}} \right]} \over {{{\left[ {A{g^ + }} \right]}^2}}}$ . How many moles of Ag will be precipitated in the above reaction. Given that Ag + + e - $\to$ Ag; E o = 0.80 V; Zn 2+ + 2e - $\to$ Zn; E o = -0.76 V; (It was given that atomic mass of Zn = 65.39)

The elevation in boiling point of a solution of 13.44 g of CuCl 2 in 1kg of water using the following information will be (Molecular weight of CuCl 2 = 134.4 and K b = 0.52 K molal -1 )