AIEEE 2012

Three numbers are chosen at random without replacement from \(\left\{ {1,2,3,..8} \right\}.\) The probability that their minimum is \(3,\) given that their maximum is \(6,\) is :

In a \(\Delta PQR,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu}\) If \(3{\mkern 1mu} \sin {\mkern 1mu} P + 4{\mkern 1mu} \cos {\mkern 1mu} Q = 6\) and \(4\sin Q + 3\cos P = 1,\) then the angle R is equal to :

The equation \({e^{\sin x}} - {e^{ - \sin x}} - 4 = 0\) has:

Statement-1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) +.....+ (361 + 380 + 400) is 8000.
Statement-2: \(\sum\limits_{k = 1}^n {\left( {{k^3} - {{(k - 1)}^3}} \right)} = {n^3}\), for any natural number n.

Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can be formed such that Y \(\subseteq\) X, Z \(\subseteq\) X and Y \(\cap\) Z is empty, is :

Let x1, x2,........., xn be n observations, and let \(\overline x\) be their arithematic mean and \({\sigma ^2}\) be their variance.
Statement 1 : Variance of 2x1, 2x2,......., 2xn is 4\({\sigma ^2}\).
Statement 2 : : Arithmetic mean of 2x...

If the line \(2x + y = k\) passes through the point which divides the line segment joining the points \((1, 1)\) and \((2, 4)\) in the ratio \(3 : 2\), then \(k\) equals :

Let \(ABCD\) be a parallelogram such that \(\overrightarrow {AB} = \overrightarrow q ,\overrightarrow {AD} = \overrightarrow p\) and \(\angle BAD\) be an acute angle. If \(\overrightarrow r\) is the vector that coincide with the altitud...

Let \(\overrightarrow a\) and \(\overrightarrow b\) be two unit vectors. If the vectors \(\,\overrightarrow c = \widehat a + 2\widehat b\) and \(\overrightarrow d = 5\widehat a - 4\widehat b\) are perpendicular to each other, then the a...

Hydrogen atom is excited from ground state to another state with principal quantum number equal to \(4.\) Then the number of spectral lines in the emission spectra will be :

Assume that a neutron breaks into a proton and an electron. The energy released during this process is : (mass of neutron \(= 1.6725 \times {10^{ - 27}}kg,\) mass of proton \(= 1.6725 \times {10^{ - 27}}\,kg,\) mass of electron $$ = 9 \t...

A diatomic molecule is made of two masses \({m_1}\) and \({m_2}\) which are separated by a distance \(r.\) If we calculate its rotational energy by applying Bohr's rule of angular momentum quantization, its energy will be given by: (\(n\) i...

The figure shows an experimental plot for discharging of a capacitor in an R-C circuit. The time constant \(\tau\) of this circuit lies between

Two cars of masses m1 and m2 are moving in circles of radii r1 and r2, respectively. Their speeds are such that they make complete circles in the same time t. The ratio of their centripetal acceleration is

A radar has a power of \(1kW\) and is operating at a frequency of \(10\) \(GHz.\) It is located on a mountain top of height \(500\) \(m.\) The maximum distance upto which it can detect object located on the surface of the earth (Radius of e...

Two electric bulbs marked \(25W\) \(-\) \(220\) \(V\) and \(100W\) \(-\) \(220V\) are connected in series to a \(440\) \(V\) supply. Which of the bulbs will fuse?

This question has Statement 1 and Statement 2. Of the four choices given after the statements, choose the one that best describes the two statements.
Statement 1 : Davisson - Germer experiment established the wave nature of electrons. Stat...

A coil is suspended in a uniform magnetic field, with the plane of the coil parallel to the magnetic lines of force. When a current is passed through the coil it starts oscillating; It is very difficult to stop. But if an aluminium plate is...

An electromagnetic wave in vacuum has the electric and magnetic field \(\mathop E\limits^ \to\)and \(\mathop B\limits^ \to\), which are always perpendicular to each other. The direction of polarization is given by $$\mathop X\limits^ \t...

Truth table for system of four \(NAND\) gates as shown in figure is:

In a uniformly charged sphere of total charge \(Q\) and radius \(R,\) the electric field \(E\) is plotted as function of distance from the center. The graph which would correspond to the above will be:

This question has statement- \(1\) and statement- \(2.\) Of the four choices given after the statements, choose the one that best describe the two statements.
An insulating solid sphere of radius \(R\) has a uniformly positive charge densit...

An object \(2.4\) \(m\) in front of a lens forms a sharp image on a film \(12\) \(cm\) behind the lens. A glass plate \(1\) \(cm\) thick, of refractive index \(1.50\) is interposed between lens and film with its plane faces parallel to fil...

The mass of a spaceship is \(1000\) \(kg.\) It is to be launched from the earth's surface out into free space. The value of \(g\) and \(R\) (radius of earth ) are \(10\,m/{s^2}\) and \(6400\) \(km\) respectively. The required energy for thi...

A wooden wheel of radius \(R\) is made of two semicircular part (see figure). The two parts are held together by a ring made of a metal strip of cross sectional area \(S\) and length \(L.\) \(L\) is slightly less than \(2\pi R.\) To fit the...

Helium gas goes through a cycle \(ABCD\) (consisting of two isochoric and isobaric lines) as shown in figure efficiency of this cycle is nearly : (Assume the gas to be close to ideal gas)

A Carnot engine, whose efficiency is \(40\%\), takes in heat from a source maintained at a temperature of \(500\) \(K.\) It is desired to have an engine of efficiency \(60\% .\) Then, the intake temperature for the same exhaust (sink) temp...

A particle of mass \(m\) is at rest at the origin at time \(t=0.\) It is subjected to a force \(F\left( t \right) = {F_0}{e^{ - bt}}\) in the \(x\) direction. Its speed \(v(t)\) is depicted by which of the following curves?

Proton, deuteron and alpha particle of same kinetic energy are moving in circular trajectories in a constant magnetic field. The radii of proton, denuteron and alpha particle are respectively \({r_p},{r_d}\) and \({r_\alpha }\). Which one...

A charge \(Q\) is uniformly distributed over the surface of non-conducting disc of radius \(R.\) The disc rotates about an axis perpendicular to its plane and passing through its center with an angular velocity \(\omega .\) As a result of t...

A boy can throw a stone up to a maximum height of 10 m. The maximum horizontal distance that the boy
can throw the same stone up to will be

A thin liquid film formed between a U-shaped wire and a light slider supports a weight of \(1.5 \times {10^{ - 2}}\,\,N\) (see figure). The length of the slider is \(30\) \(cm\) and its weight negligible. The surface tension of the liquid ...

A liquid in a beaker has temperature \(\theta \left( t \right)\) at time \(t\) and \({\theta _0}\) is temperature of surroundings, then according to Newton's law of cooling the correct graph between $${\log _e}\left( {\theta - {\theta _0}}...

If a simple pendulum has significant amplitude (up to a factor of \(1/e\) of original ) only in the period between \(t = 0s\,\,to\,\,t = \tau \,s,\) then \(\tau \,\) may be called the average life of the pendulum When the spherical bob of t...

Resistance of a given wire is obtained by measuring the current flowing in it and the voltage difference
applied across it. If the percentage errors in the measurement of the current and the voltage difference
are 3% each, then error in the...

A spectrometer gives the following reading when used to measure the angle of a prism.
Main scale reading: 58.5 degree
Vernier scale reading : 09 divisions
Given that 1 division on main scale corresponds to 0.5 degree. Total divisions on the...

In Young's double slit experiment , one of the slit is wider than other, so that amplitude of the light from one slit is double of that from other slit. If \({{\rm I}_m}\) be the maximum intensity, the resultant intensity \({\rm I}\) when t...

A cylindrical tube, open at both ends, has a fundamental frequency, \(f,\) in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air-column is now :

This question has Statement \(1\) and Statement \(2.\) Of the four choices given after the Statements, choose the one that best describes the two Statements.
If two springs \({S_1}\) and \({S_2}\) of force constants \({k_1}\) and \({k_2}\)...