AIEEE 2009

If the roots of the equation \(b{x^2} + cx + a = 0\) imaginary, then for all real values of \(x\), the expression \(3{b^2}{x^2} + 6bcx + 2{c^2}\) is :

The sum to infinite term of the series \(1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + {{14} \over {{3^4}}} + .....\) is

If $A, B$ and $C$ are three sets such that $A \cap B=A \cap C$ and $A \cup B=A \cup C$, then :

Statement - 1 : The variance of first n even natural numbers is \({{{n^2} - 1} \over 4}\)
Statement - 2 : The sum of first n natural numbers is \({{n\left( {n + 1} \right)} \over 2}\) and the sum of squares of first n natural numbers is $${...

If the mean deviation of number 1, 1 + d, 1 + 2d,........, 1 + 100d from their mean is 255, then the d is
equal to

The lines \(p\left( {{p^2} + 1} \right)x - y + q = 0\) and \(\left( {{p^2} + 1} \right){}^2x + \left( {{p^2} + 1} \right)y + 2q\) \(=0\) are perpendicular to a common line for :

The shortest distance between the line \(y - x = 1\) and the curve \(x = {y^2}\) is :

Let A and B denote the statements
A: \(\cos \alpha + \cos \beta + \cos \gamma = 0\)
B: \(\sin \alpha + \sin \beta + \sin \gamma = 0\)
If $$\cos \left( {\beta - \gamma } \right) + \cos \left( {\gamma - \alpha } \right) + \cos \left(...

If \(\overrightarrow u ,\overrightarrow v ,\overrightarrow w\) are non-coplanar vectors and \(p,q\) are real numbers, then the equality $$\left[ {3\overrightarrow u \,\,p\overrightarrow v \,\,p\overrightarrow w } \right] - \left[ {p\overri...

An inductor of inductance \(L=400\) \(mH\) and resistors of resistance \({R_1} = 2\Omega\) and \({R_2} = 2\Omega\) are connected to a battery of \(emf\) \(12\) \(V\) as shown in the figure. The internal resistance of the battery is neglig...

The transition from the state \(n=4\) to \(n=3\) in a hydrogen like atom result in ultra violet radiation. Infrared radiation will be obtained in the transition from :

The above is a plot of binding energy per nucleon \({E_b},\) against the nuclear mass \(M;A,B,C,D,E,F\) correspond to different nuclei. Consider four reactions :
$$\eqalign{
& \left( i \right)\,\,\,\,\,\,\,\,\,\,A + B \to C + \varepsil...

The surface of a metal is illuminated with the light of \(400\) \(nm.\) The kinetic energy of the ejected photoelectrons was found to be \(1.68\) \(eV.\) The work function of the metal is : \(\left( {hc = 1240eV.nm} \right)\)

The logic circuit shown below has the input waveforms \('A'\) and \('B'\) as shown. Pick out the correct output waveform.


Output is

A \(p\)-\(n\) junction \((D)\) shown in the figure can act as a rectifier. An alternating current source \((V)\) is connected in the circuit.

The current \((I)\) in the resistor \((R)\) can be shown by :

(This question contains Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements.)
Statement-1 : For a charged particle moving from point \(P\) to point \(Q\), the ne...

A charge \(Q\) is placed at each of the opposite corners of a square. A charge \(q\) is placed at each of the other two corners. If the net electrical force on \(Q\) is zero, then \(Q/q\) equals:

Let \(P\left( r \right) = {Q \over {\pi {R^4}}}r\) be the change density distribution for a solid sphere of radius \(R\) and total charge \(Q\). For a point \('p'\) inside the sphere at distance \({r_1}\) from the center of the sphere, the ...

Two points \(P\) and \(Q\) are maintained at the potentials of \(10\) \(V\) and \(-4\) \(V\), respectively. The work done in moving \(100\) electrons from \(P\) to \(Q\) is :

A transparent solid cylindrical rod has a refractive index of \({2 \over {\sqrt 3 }}.\) It is surrounded by air. A light ray is incident at the mid-point of one end of the rod as shown in the figure.

The incident angle \(\theta\) for whi...

In an optics experiment, with the position of the object fixed, a student varies the position of a convex lens and for each position, the screen is adjusted to get a clear image of the object. A graph between the object distance \(u\) and t...

The height at which the acceleration due to gravity becomes \({g \over 9}\) (where \(g=\) the acceleration due to gravity on the surface of the earth) in terms of \(R,\) the radius of the earth, is:

Two moles of helium gas are taken over the cycle \(ABCD\), as shown in the \(P\)-\(T\) diagram.

The net work done on the gas in the cycle \(ABCDA\) is:

A long metallic bar is carrying heat from one of its ends to the other end under steady-state. The variation of temperature \(\theta\) along the length \(x\) of the bar from its hot end is best described by which of the following figures?

Two moles of helium gas are taken over the cycle \(ABCD,\) as shown in the \(P\)-\(T\) diagram.

The work done on the gas in taking it from \(D\) to \(A\) is :

Two moles of helium gas are taken over the cycle \(ABCD,\) as shown in the \(P\)-\(T\) diagram.

Assuming the gas to be ideal the work done on the gas in taking it from \(A\) to \(B\) is :

One \(kg\) of a diatomic gas is at a pressure of \(8 \times {10^4}\,N/{m^2}.\) The density of the gas is \(4kg/{m^3}\). What is the energy of the gas due to its thermal motion ?

Statement - 1: The temperature dependence of resistance is usually given as \(R = {R_0}\left( {1 + \alpha \,\Delta t} \right).\) The resistance of wire changes from \(100\Omega\) to \(150\Omega\) when its temperature is increased from $$...

A current loop \(ABCD\) is held fixed on the plane of the paper as shown in the figure. The arcs \(BC\) (radius \(= b\)) and \(DA\) (radius \(=a\)) of the loop are joined by two straight wires \(AB\) and \(CD\). A steady current \(I\) is fl...

A current loop \(ABCD\) is held fixed on the plane of the paper as shown in the figure. The arcs \(BC\) (radius \(= b\)) and \(DA\) (radius \(=a\)) of the loop are joined by two straight wires \(AB\) and \(CD\). A steady current \(I\) is fl...

A particle has an initial velocity \(3\widehat i + 4\widehat j\) and an acceleration of \(0.4\widehat i + 0.3\widehat j\). Its speed after 10 s is:

Consider a rubber ball freely falling from a height \(h=4.9\) \(m\) onto a horizontal elastic plate. Assume that the duration of collision is negligible and the collision with the plate is totally elastic.
Then the velocity as a function ...

Two wires are made of the same material and have the same volume. However wire \(1\) has cross-sectional area \(A\) and wire \(2\) has cross-sectional area \(3A.\) If the length of wire \(1\) increases by \(\Delta x\) on applying force $$F,...

A thin uniform rod of length \(l\) and mass \(m\) is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is \(\omega\). Its center of mass rises to a maximum height of:

If \(x,\) \(v\) and \(a\) denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period \(T,\) then, which of the following does not change with time?

In an experiment the angles are required to be measured using an instrument, 29 divisions of the
main scale exactly coincide with the 30 divisions of the vernier scale. If the smallest division of the
main scale is half-a degree(=$$0.5^\cir...

A mixture of light, consisting of wavelength \(590\) \(nm\) and an unknown wavelength, illuminates Young's double slit and gives rise to two overlapping interference patterns on the screen. The central maximum of both lights coincide. Furth...

A motor cycle starts from rest and accelerates along a straight path at \(2m/{s^2}.\) At the starting point of the motor cycle there is a stationary electric siren. How far has the motor cycle gone when the driver hears the frequency of the...

Three sound waves of equal amplitudes have frequencies \(\left( {v - 1} \right),\,v,\,\left( {v + 1} \right).\) They superpose to give beats. The number of beats produced per second will be :