IIT-JEE 2005 Mains

In the following reaction

Identify X and Y.

Which of the following disaccharide will not reduce Tollen's reagent?

Predict whether the following molecules are iso-structural or not. Justify your answer.
(A) NMe\(_3\)
(B) N(SiMe\(_3\))\(_3\)

Given : that \({\mu _{obs}} = \sum {{\mu _i}\,{x_i}}\) where \({\mu _i}\) is the dipole moment of stable conformer and \({x_i}\) is the mole fraction of that conformer.
(A) Write stable conformer for Z-CH\(_2\)-CH\(_2\)-Z in Newman's proje...

Fill in the blanks:
(A) \(_{92}^{235}\)U + \(_{0}^{1}\)n \(\to\) \(_{52}^{137}\)A + \(_{40}^{97}\)B + ____________.
(B) \(_{34}^{82}\)Se \(\to\) 2 \({}_{ - 1}{e^0}\) + __________.

For the following reaction
2X(g) \(\to\) 3Y(g) + 2Z(g)
assuming ideal gas conditions, the data for change of partial pressure with time is as follows:
$$
\begin{array}{llll}
\hline \text { Time (in min) } & 0 & 100 & 200 \\
\hline \begin{ar...

In the following reaction, X is optically active.
$\underset{\mathbf{X}}{\mathrm{C}_5 \mathrm{H}_{13}} \mathrm{~N} \xrightarrow[\mathrm{~N}_2]{\mathrm{NaNO}_2 / \mathrm{HCl}} \mathbf{Y}($ Tertiary alcohol $)+$ Other products
Find X and Y. I...

In the following reaction sequence :

Identify A, B, C and D. Also write chemical equations for cons version of A to B and A to C.

\(F{e^{3 + }}\buildrel {SN{C^ - }(excess)} \over \longrightarrow\) blood red \(F{e^{3 + }}\buildrel {SN{C^ - }(excess)} \over \longrightarrow\) colourless (B):
(A) Identify compound A and B and write their IUPAC names.
(B) Determine the...

In the following reaction sequence, M is a transition metal.

Identify the metal M and MCl\(_4\). Explain the difference in colours of MCl\(_4\) and A.

Write the balanced chemical equation for developing a back and white photographic film. Also explain why the solution of sodium thiosulphate on acidification turns milky white.

(A) Calculate \(\Delta_r G^\circ\) of the following reaction
\(A{g^ + }(aq.) + C{l^ - }(aq.) \to AgCl(s)\)
Given :
\(\mathrm{\Delta_r G^\circ(AgCl)\quad-109~kJ/mole}\)
\(\mathrm{\Delta_r G^\circ(Cl^-)\quad-129~kJ/mole}\)
$$\mathrm{\Delta_r ...

Give reasons :

(A) Calculate the amount of calcium oxide required to react with 852 g of P\(_4\)O\(_{10}\).
(B) Write the structure of P\(_4\)O\(_{10}\).

Monomer A of a polymer on ozonolysis yields two moles of HCHO and one mole of CH\(_3\)COCHO.
(A) Deduce the structure of A.
(B) Write the structure of "all \(cis\)" forms of polymer of compound A.

An element crystallises in fcc lattice having edge length 400 pm. Calculate the maximum diameter of the atom which can be placed in interstitial site such that the structure is not distorted.

(A) Calculate velocity of electron in the first orbit of hydrogen atom (Given : \(r=a_0=0.529\) \(\mathop A\limits^o\)).
(B) Calculate the de Broglie's wavelength of the electron in first Bohr orbit.
(C) Calculate the orbital angular momen...

Nitrogen gas is absorbed on 20% surface sites. On heating N\(_2\) gas evolved from sites and was collected at 0.01 atm and 298 K in a container of volume 2.46 cm\(^3\). Find out the number of surface sites occupied per molecule of N, if the...

Find the equation of the plane containing the line \(2 x-y+z-3=0,3 x+y+z=5\) and at a distance of \(\frac{1}{\sqrt{6}}\) from the point \((2,1,-1)\).

If \(\left|f\left(x_{1}\right)-f\left(x_{2}\right)\right| \leq\left(x_{1}-x_{2}\right)^{2}\), for all \(x_{1}, x_{2} \in\) \(\mathbb{R}\). Find the equation of tangent to the curve \(y=f(x)\) at the point \((1,2)\).

If \(p(x)\) be a polynomial of degree 3 satisfying \(p(-1)=10, p(1)=-6\) and \(p(x)\) has maximum at \(x=-1\) and \(p'(x)\) has minima at \(x=1\). Find the distance between the local maximum and local minimum of the curve.

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

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

If $$\left[\begin{array}{lll}4 a^{2} & 4 a & 1 \\ 4 b^{2} & 4 b & 1 \\ 4 c^{2} & 4 c & 1\end{array}\right]\left[\begin{array}{c}f(-1) \\ f(1) \\ f(2)\end{array}\right]=\left[\begin{array}{c}3 a^{2}+3 a \\ 3 b^{2}+3 b \\ 3 c^{2}+3 c\end{arra...

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

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

Evatuate:
\(\int_\limits{0}^{\pi} e^{|\cos x|}\left[2 \sin \left(\frac{1}{2} \cos x\right)+3 \cos \left(\frac{1}{2} \cos x\right)\right] \sin x ~d x\)

If \(f(x)\) is a differentiable function and \(g(x)\) is a double differentiable function such that \(|f(x)| \leq 1\) and \(f'(x)=g(x)\), where,\(f^{2}(0)+g^{2}(0)=9\) then prove that there exists some \(c \in(-3,3)\) such that $$g(c) \circ...

Find the equation of the common tangent in the first quadrant to the circle \(x^{2}+y^{2}=16\) and the ellipse \(\frac{x^{2}}{25}+\frac{y^{2}}{4}=1\). Also find the length of the intercept of the tangent between the coordinate axes.

Find the range of value of \(t\) for which
\(2 \sin t=\frac{1-2 x+5 x^{2}}{3 x^{2}-2 x-1}, t \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)

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

If \(f(x-y)=f(x) \circ g(y)-f(y) \circ g(x)\) And \(g(x-y) =g(x) \circ g(y)+f(x) \circ f(y)\) for all \(x, y \in \mathrm{R}\). If right-hand derivative at \(x=0\) exists for \(f(x)\), find the derivative of \(g(x)\) at \(x=0\)

A person goes office either by car, scooter, bus or train, proability of which being \(\frac{1}{7}, \frac{3}{2}, \frac{2}{7}\) and \(\frac{1}{7}\), respectively. Probability that he reaches office late, if he takes car, scooter, bus or trai...

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

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

Incident ray is along the unit vector \(\hat{v}\) and the reflected ray is along the unit vector \(\widehat{w}\). The normal is along unit vector \(\hat{a}\) outwards. Express \(\hat{w}\), in terms of \(\hat{a}\) and \(\hat{v}\).

Highly energetic electrons are bombarded on a target of an element containing 30 neutrons. The ratio of radii of nucleus to that of Helium nucleus is \((14)^{\frac{1}{3}}\). Find
(A) Atomic number of the nucleus;
(B) the frequency of $$\mat...

In the given circuit, the switch S is closed at time \(t=0\). The charge Q on the capacitor at any instant t is given by \(Q(t)=Q_0(1-e^{-\alpha t})\). Find the value of Q\(_0\) and \(\alpha\) in terms of given parameters shown in the circu...

An unknown resistance X is to be
determined using resistances R\(_1\), R\(_2\) or R\(_3\). Their corresponding null points are A, B
and C. Find which of the above will give the
most accurate reading and why ?

The potential energy of a particle of mass m
is given by
$$\mathrm{U}(x)=\left\{\begin{array}{cc}\mathrm{E}_{0} & 0 \leq x \leq 1 \\ 0 & x>1\end{array}\right.$$
\(\lambda_{1}\) and \(\lambda_{2}\) are the de Broglie wavelengths of the part...

A long solenoid of radius a and number of turns per unit length \(n\) is enclosed by cylindrical shell of radius R, thickness \(d\) \((d < < R)\) and length L. A variable current \(\mathrm{I}=\mathrm{I}_{0} \sin \omega t\) flows through the...

A conducting liquid bubble of radius \(a\) and thickness \(t(t < < a)\) is charged to potential V. If the bubble collapses to a droplet, find the potential on the droplet.

What will be the minimum angle of incidence such that the total internal reflection occurs on both the surfaces?

Two identical prisms of refractive index \(\sqrt{3}\) are kept as shown in the figure. A light ray strikes the first prism at face AB. Find
(A) the angle of incidence so that the
emergent ray from the first prism has
minimum deviation;
(...

A cylinder of mass \(1 \mathrm{~kg}\) is given heat of \(20000 \mathrm{~J}\) at atmospheric pressure. If initially temperature of cylinder is \(20^{\circ} \mathrm{C}\), find
(A) The final temperature of the cylinder;
(B) The work done by th...

In a moving coil galvanometer, torque on the coil can be expressed as \(\tau=k i\), where \(i\) is current through the wire and \(k\) is constant. The rectangular coil of the galvanometer having numbers of turns \(\mathrm{N}\), area $$\math...

A U-tube is rotated about one of its limbs with an angular velocity \(\omega\). Find the difference in height \(\mathrm{H}\) of the liquid (density \(\rho\) ) level, where the diameter of the tube is \(d < <\mathrm{L}\).

A wooden log of mass M and length L is
hinged by a frictionless nail at O; a bullet
of mass m strikes with velocity \(v\) and sticks
to it. Find angular velocity of the system
immediately after the collision about O.

A cylinder of mass m and radius R rolls
down on an inclined plane of inclination \(\theta\).
Calculate the linear acceleration of axis of
cylinder.

Two identical ladders, each of mass M and
length L are resting on the rough horizontal
surface as shown in the figure. A block of
mass \(m\) hangs from P. If the system is in
equilibrium, find the magnitude and the
direction of frictio...