AIEEE 2007

If \(\,\left| {z + 4} \right|\,\, \le \,\,3\,\), then the maximum value of \(\left| {z + 1} \right|\) is :

The solution for \(x\) of the equation \(\int\limits_{\sqrt 2 }^x {{{dt} \over {t\sqrt {{t^2} - 1} }} = {\pi \over 2}}\) is

Let \(F\left( x \right) = f\left( x \right) + f\left( {{1 \over x}} \right),\) where \(f\left( x \right) = \int\limits_l^x {{{\log t} \over {1 + t}}dt,}\) Then \(F(e)\) equals

Let \(I = \int\limits_0^1 {{{\sin x} \over {\sqrt x }}dx}\) and \(J = \int\limits_0^1 {{{\cos x} \over {\sqrt x }}dx} .\) Then which one of the following is true?

The differential equation of all circles passing through the origin and having their centres on the \(x\)-axis is :

The largest interval lying in \(\left( { - {\pi \over 2},{\pi \over 2}} \right)\) for which the function
\(f\left( x \right) = {4^{ - {x^2}}} + {\cos ^{ - 1}}\left( {{x \over 2} - 1} \right)\)\(+ \log \left( {\cos x} \right)\),
is define...

A tower stands at the centre of a circular park. \(A\) and \(B\) are two points on the boundary of the park such that \(AB(=a)\) subtends an angle of \({60^ \circ }\) at the foot of the tower, and the angle of elevation of the top of the to...

For the Hyperbola \({{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1\) , which of the following remains constant when \(\alpha\) varies\(=\)?

The normal to a curve at \(P(x,y)\) meets the \(x\)-axis at \(G\). If the distance of \(G\) from the origin is twice the abscissa of \(P\), then the curve is a :

\(\int {{{dx} \over {\cos x + \sqrt 3 \sin x}}}\) equals

If sin-1\(\left( {{x \over 5}} \right)\) + cosec-1\(\left( {{5 \over 4}} \right)\) = \({\pi \over 2}\), then the value of x is :

Let \(f:R \to R\) be a function defined by
\(f(x) = \min \left\{ {x + 1,\left| x \right| + 1} \right\}\), then which of the following is true?

The function \(f:R/\left\{ 0 \right\} \to R\) given by
\(f\left( x \right) = {1 \over x} - {2 \over {{e^{2x}} - 1}}\)
can be made continuous at \(x\) = 0 by defining \(f\)(0) as

Let \(A = \left| {\matrix{ 5 & {5\alpha } & \alpha \cr 0 & \alpha & {5\alpha } \cr 0 & 0 & 5 \cr } } \right|.\) If \(\,\,\left| {{A^2}} \right| = 25,\) then \(\,\left| \alpha \right|\) equals

If \(D = \left| {\matrix{ 1 & 1 & 1 \cr 1 & {1 + x} & 1 \cr 1 & 1 & {1 + y} \cr } } \right|\) for \(x \ne 0,y \ne 0,\) then \(D\) is :

The equation of a tangent to the parabola \({y^2} = 8x\) is \(y=x+2\). The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is :

The set S = {1, 2, 3, ........., 12} is to be partitioned into three sets A, B, C of equal size. Thus \(A \cup B \cup C = S,\,A \cap B = B \cap C = A \cap C = \phi\). The number of ways to partition S is

A pair of fair dice is thrown independently three times. The probability of getting a score of exactly \(9\) twice is :

Two aeroplanes \({\rm I}\) and \({\rm I}\)\({\rm I}\) bomb a target in succession. The probabilities of \({\rm I}\) and \({\rm I}\)\({\rm I}\) scoring a hit correctly are \(0.3\) and \(0.2,\) respectively. The second plane will bomb only if...

If the difference between the roots of the equation \({x^2} + ax + 1 = 0\) is less than \(\sqrt 5 ,\) then the set of possible values of \(a\) is

The sum of series \({1 \over {2!}} - {1 \over {3!}} + {1 \over {4!}} - .......\) upto infinity is

In a geometric progression consisting of positive terms, each term equals the sum of the next two terns. Then the common ratio of its progression is equals

The average marks of boys in a class is 52 and that of girls is 42. The average marks of boys
and girls combined is 50. The percentage of boys in the class is

Let A \(\left( {h,k} \right)\), B\(\left( {1,1} \right)\) and C \((2, 1)\) be the vertices of a right angled triangle with AC as its hypotenuse. If the area of the triangle is \(1\) square unit, then the set of values which \('k'\) can take...

If one of the lines of \(m{y^2} + \left( {1 - {m^2}} \right)xy - m{x^2} = 0\) is a bisector of angle between the lines \(xy = 0,\) then \(m\) is :

Let \(P = \left( { - 1,0} \right),\,Q = \left( {0,0} \right)\) and \(R = \left( {3,3\sqrt 3 } \right)\) be three point. The equation of the bisector of the angle \(PQR\) is :

If \(\widehat u\) and \(\widehat v\) are unit vectors and \(\theta\) is the acute angle between them, then \(2\widehat u \times 3\widehat v\) is a unit vector for :

Let \(\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b = \widehat i - \widehat j + 2\widehat k\) and \(\overrightarrow c = x\widehat i + \left( {x - 2} \right)\widehat j - \widehat k\,\,.\) If the vectors $$\ov...

In an \(a.c.\) circuit the voltage applied is \(E = {E_0}\,\sin \,\omega t.\) The resulting current in the circuit is \(I = {I_0}\sin \left( {\omega t - {\pi \over 2}} \right).\) The power consumption in the circuit is given by

The half-life period of a ratio-active element \(X\) is same as the mean life time of another ratio-active element \(Y.\) Initially they have the same number of atoms. Then

If \({M_O}\) is the mass of an oxygen isotope \({}_8{O^{17}}\) , \({M_p}\) and \({M_N}\) are the masses of a proton and neutron respectively, the nuclear binding energy of the isotope is

Which of the following transitions in hydrogen atoms emit photons of highest frequency ?

In gamma ray emission from a nucleus

A battery is used to charge a parallel plate capacitor till the potential difference between the plates becomes equal to the electromotive force of the battery. The ratio of the energy stored in the capacitor and the work done by the batter...

A parallel plate condenser with a dielectric of dielectric constant \(K\) between the plates has a capacity \(C\) and is charged to a potential \(V\) volt. The dielectric slab is slowly removed from between the plates and then reinserted. T...

A circular disc of radius \(R\) is removed from a bigger circular disc of radius \(2R\) such that the circumferences of the discs coincide. The center of mass of the new disc is \(\alpha R\) form the center of the bigger disc. The value of ...

The resistance of a wire is \(5\) ohm at \({50^ \circ }C\) and \(6\) ohm at \({100^ \circ }C.\) The resistance of the wire at \({0^ \circ }C\) will be

Photon of frequency \(v\) has a momentum associated with it. If \(c\) is the velocity of light, the momentum is

An ideal coil of \(10H\) is connected in series with a resistance of \(5\Omega\) and a battery of \(5V\). \(2\) second after the connection is made, the current flowing in ampere in the circuit is

If in a \(p\)-\(n\) junction diode, a square input signal of \(10\) \(V\) is applied as shown

Then the output signal across \({R_L}\) will be

Carbon, silicon and germanium have four valence electrons each. At room temperature which one of the following statements is most appropriate ?

The potential at a point \(x\) (measured in \(\mu \,m\)) due to some charges situated on the \(x\)-axis is given by \(V\left( x \right) = 20/\left( {{x^2} - 4} \right)\) volt
The electric field \(E\) at \(x = 4\,\mu \,m\) is given by

An electric charge \({10^{ - 3}}\,\,\mu \,C\) is placed at the origin \((0,0)\) of \(X-Y\) co-ordinate system. Two points \(A\) and \(B\) are situated at \(\left( {\sqrt 2 ,\sqrt 2 } \right)\) and \(\left( {2,0} \right)\) respectively. The...

Charges are placed on the vertices of a square as shown. Let \(\overrightarrow E\) be the electric field and \(V\) the potential at the center. If the charges on \(A\) and \(B\) are interchanged with those on \(D\) and \(C\) respectively, ...

Two lenses of power \(-15\) \(D\) and \(+5\) \(D\) are in contact with each other. The focal length of the combination is

If \({g_E}\) and \({g_M}\) are the accelerations due to gravity on the surfaces of the earth and the moon respectively and if Millikan's oil drop experiment could be performed on the two surfaces, one will find the ratio
$${{electro\,\,ch...

If \({C_p}\) and \({C_v}\) denote the specific heats of nitrogen per unit mass at constant pressure and constant volume respectively, then

When a system is taken from state \(i\) to state \(f\) along the path iaf, it is found that \(Q=50\) cal and \(W=20\) \(cal\). Along the path \(ibf\) \(Q=36\) \(cal.\) \(W\) along the path \(ibf\) is

One end of a thermally insulated rod is kept at a temperature \({T_1}\) and the other at \({T_2}\). The rod is composed of two sections of length \({L_1}\) and \({L_2}\) and thermal conductivities \({K_1}\) and \({K_2}\) respectively. The t...

A Carnot engine, having an efficiency of \(\eta = 1/10\) as heat engine, is used as a refrigerator . If the work done on the system is \(10\) \(J\), the amount of energy absorbed from the reservoir at lower temperature is