AIEEE 2010

Four numbers are chosen at random (without replacement) from the set \(\left\{ {1,2,3,....20} \right\}.\)
Statement - 1: The probability that the chosen numbers when arranged in some order will form an AP is \({1 \over {85}}.\)
Statement -...

For a regular polygon, let \(r\) and \(R\) be the radii of the inscribed and the circumscribed circles. A \(false\) statement among the following is :

If \(\alpha\) and \(\beta\) are the roots of the equation \({x^2} - x + 1 = 0,\) then \({\alpha ^{2009}} + {\beta ^{2009}} =\)

A person is to count 4500 currency notes. Let \({a_n}\) denote the number of notes he counts in the \({n^{th}}\) minute. If \({a_1}\) = \({a_2}\) = ....= \({a_{10}}\)= 150 and \({a_{10}}\), \({a_{11}}\),.... are in an AP with common differe...

Consider the following relations
$R=\{(x, y) \mid x, y$ are real numbers and $x=w y$ for some rational number $w\}$;
$S=\left\{\left(\frac{m}{n}, \frac{p}{q}\right) \mid m, n, p\right.$ and $q$ are integers such that $n, q \neq 0$ and $q ...

For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding
means are given to be 2 and 4, respectively. The variance of the combined data set is

The line \(L\) given by \({x \over 5} + {y \over b} = 1\) passes through the point \(\left( {13,32} \right)\). The line K is parrallel to \(L\) and has the equation \({x \over c} + {y \over 3} = 1.\) Then the distance between \(L\) and $$K$...

Let \(\cos \left( {\alpha + \beta } \right) = {4 \over 5}\) and \(\sin \,\,\,\left( {\alpha - \beta } \right) = {5 \over {13}},\) where \(0 \le \alpha ,\,\beta \le {\pi \over 4}.\)
Then \(tan\,2\alpha\) =

If the vectors \(\overrightarrow a = \widehat i - \widehat j + 2\widehat k,\,\,\,\,\,\overrightarrow b = 2\widehat i + 4\widehat j + \widehat k\,\,\,\) and \(\,\overrightarrow c = \lambda \widehat i + \widehat j + \mu \widehat k\) are m...

Let \(\overrightarrow a = \widehat j - \widehat k\) and \(\overrightarrow c = \widehat i - \widehat j - \widehat k.\) Then the vector \(\overrightarrow b\) satisfying $$\overrightarrow a \times \overrightarrow b + \overrightarrow c = ...

In the circuit shown below, the key \(K\) is closed at \(t=0.\) The current through the battery is

In a series \(LCR\) circuit \(R = 200\Omega\) and the voltage and the frequency of the main supply is \(220V\) and \(50\) \(Hz\) respectively. On taking out the capacitance from the circuit the current lags behind the voltage by $${30^ \ci...

A nucleus of mass \(M+\)\(\Delta m\) is at rest and decays into two daughter nuclei of equal mass \({M \over 2}\) each. Speed of light is \(c.\)
The binding energy per nucleon for the parent nucleus is \({E_1}\) and that for the daughter n...

A nucleus of mass \(M+\)\(\Delta m\) is at rest and decays into two daughter nuclei of equal mass \({M \over 2}\) each. Speed of light is \(c.\)
The speed of daughter nuclei is

A radioactive nucleus (initial mass number \(A\) and atomic number \(Z\) emits \(3\,\alpha\)- particles and \(2\) positrons. The ratio of number of neutrons to that of protons in the final nucleus will be

Let \(C\) be the capacitance of a capacitor discharging through a resistor \(R.\) Suppose \({t_1}\) is the time taken for the energy stored in the capacitor to reduce to half its initial value and \({t_2}\) is the time taken for the charge...

The figure shows the position\(-\)time \((x-t)\) graph of one-dimensional motion of body of mass \(0.4\) \(kg.\) The magnitude of each impulse is

Statement - 1 : Two particles moving in the same direction do not lose all their energy in a completely inelastic collision.
Statement - 2 : Principle of conservation of momentum holds true for all kinds of collisions.

A point \(P\) moves in counter-clockwise direction on a circular path as shown in the figure. The movement of \(P\) is such that it sweeps out a length \(s = {t^3} + 5,\) where \(s\) is in metres and \(t\) is in seconds. The radius of the p...

For a particle in uniform circular motion the acceleration \(\overrightarrow a\) at a point P(R, θ) on the circle of radius R is (here θ is measured from the x–axis)

Two conductors have the same resistance at \({0^ \circ }C\) but their temperature coefficients of resistance are \({\alpha _1}\) and \({\alpha _2}.\) The respective temperature coefficients of their series and parallel combinations are near...

If a source of power \(4kW\) produces \({10^{20}}\) photons/second, the radiation belongs to a part of the spectrum called

Statement - \(1\) : When ultraviolet light is incident on a photocell, its stopping potential is \({V_0}\) and the maximum kinetic energy of the photoelectrons is \({K_{\max }}\). When the ultraviolet light is replaced by \(X\)-rays, both...

A rectangular loop has a sliding connector \(PQ\) of length \(l\) and resistance \(R\) \(\Omega\) and it is moving with a speed \(v\) as shown. The set-up is placed in a uniform magnetic field going into the plane of the paper. The three c...

The combination of gates shown below yields

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of \({30^ \circ }\) with each other. When suspended in a liquid of density \(0.8g\) \(c{m^{ - 3}},\) the angle remains the same. If density o...

A thin semi-circular ring of radius \(r\) has a positive charges \(q\) distributed uniformly over it. The net field \(\overrightarrow E\) at the center \(O\) is

Let there be a spherically symmetric charge distribution with charge density varying as \(\rho \left( r \right) = {\rho _0}\left( {{5 \over 4} - {r \over R}} \right)\) upto \(r=R,\) and \(\rho \left( r \right) = 0\) for \(r>R,\) where \(r\)...

An initially parallel cylindrical beam travels in a medium of refractive index \(\mu \left( I \right) = {\mu _0} + {\mu _2}\,I,\) where \({\mu _0}\) and \({\mu _2}\) are positive constants and \(I\) is the intensity of the light beam. The i...

An initially parallel cylindrical beam travels in a medium of refractive index \(\mu \left( I \right) = {\mu _0} + {\mu _2}\,I,\) where \({\mu _0}\) and \({\mu _2}\) are positive constants and \(I\) is the intensity of the light beam. The i...

A diatomic ideal gas is used in a Carnot engine as the working substance. If during the adiabatic expansion part of the cycle the volume of the gas increases from \(V\) to \(32\) \(V\), the efficiency of the engine is

Two fixed frictionless inclined planes making an angle \({30^ \circ }\) and \({60^ \circ }\) with the vertical are shown in the figure. Two blocks \(A\) and \(B\) are placed on the two planes. What is the relative vertical acceleration of ...

Two long parallel wires are at a distance \(2d\) apart. They carry steady equal currents flowing out of the plane of the paper as shown. The variation of the magnetic field \(B\) along the line \(XX'\) is given by

A particle is moving with velocity \(\overrightarrow v = k\left( {y\widehat i + x\widehat j} \right)\), where K is a constant. The general equation for its path is

A ball is made of a material of density \(\rho\) where \({\rho _{oil}}\, < \rho < {\rho _{water}}\) with \({\rho _{oil}}\) and \({\rho _{water}}\) representing the densities of oil and water, respectively. The oil and water are immiscible...

A small particle of mass \(m\) is projected at an angle \(\theta\) with the \(x\)-axis with an initial velocity \({v_0}\) in the \(x\)-\(y\) plane as shown in the figure. At a time \(t < {{{v_0}\sin \theta } \over g},\) the angular moment...

The respective number of significant figures for the numbers 23.023, 0.0003 and 2.1 \(\times\) 10–3 are

An initially parallel cylindrical beam travels in a medium of refractive index \(\mu \left( I \right) = {\mu _0} + {\mu _2}\,I,\) where \({\mu _0}\) and \({\mu _2}\) are positive constants and \(I\) is the intensity of the light beam. The i...

The equation of a wave on a string of linear mass density \(0.04\,\,kg\,{m^{ - 1}}\) is given by
$$$y = 0.02\left( m \right)\,\sin \left[ {2\pi \left( {{t \over {0.04\left( s \right)}} - {x \over {0.50\left( m \right)}}} \right)} \right].$...

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by \(U\left( x \right) = {a \over {{x^{12}}}} - {b \over {{x^6}}},\) where \(a\) and \(b\) are constants and \(x\) is the distance b...