AIEEE 2011

The number of values of \(k\) for which the linear equations
\(4x + ky + 2z = 0,kx + 4y + z = 0\) and \(2x+2y+z=0\) possess a non-zero solution is :

These are 10 points in a plane, out of these 6 are collinear, if N is the number of triangles formed by joining these points. then:

Statement - 1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is emply is \({}^9{C_3}\).
Statement - 2: The number of ways of choosing any 3 places from 9 different places is \({}^9{C_3}\).

If \(C\) and \(D\) are two events such that \(C \subset D\) and \(P\left( D \right) \ne 0,\) then the correct statement among the following is :

Consider \(5\) independent Bernoulli's trials each with probability of success \(p.\) If the probability of at least one failure is greater than or equal to \({{31} \over 32},\) then \(p\) lies in the interval :

A man saves ₹ 200 in each of the first three months of his service. In each of the subsequent months his saving increases by ₹ 40 more than the saving of immediately previous month. His total saving from the start of service will be ₹ 110...

Let $R$ be the set of real numbers.
Statement I : $A=\{(x, y) \in R \times R: y-x$ is an integer $\}$ is an equivalence relation on $R$.
Statement II : $ B=\{(x, y) \in R \times R: x=\alpha y$ for some rational number $\alpha\}$ is an equiv...

If the mean deviation about the median of the numbers a, 2a,........., 50a is 50, then |a| equals

The lines \({L_1}:y - x = 0\) and \({L_2}:2x + y = 0\) intersect the line \({L_3}:y + 2 = 0\) at \(P\) and \(Q\) respectively. The bisector of the acute angle between \({L_1}\) and \({L_2}\) intersects \({L_3}\) at \(R\).
Statement-1: The ...

If \(A = {\sin ^2}x + {\cos ^4}x,\) then for all real \(x\):

The vectors \(\overrightarrow a\) and \(\overrightarrow b\) are not perpendicular and \(\overrightarrow c\) and \(\overrightarrow d\) are two vectors satisfying $$\overrightarrow b \times \overrightarrow c = \overrightarrow b \times ...

Let \(\overrightarrow a\), \(\overrightarrow b\), \(\overrightarrow c\) be three non-zero vectors which are pairwise non-collinear. If $\overrightarrow a+3 \overrightarrow b$ is collinear with $\overrightarrow c$ and $\overrightarrow b+2...

If \(\overrightarrow a = {1 \over {\sqrt {10} }}\left( {3\widehat i + \widehat k} \right)\) and \(\overrightarrow b = {1 \over 7}\left( {2\widehat i + 3\widehat j - 6\widehat k} \right),\) then the value
of $$\left( {2\overrightarrow a ...

A resistor \('R'\) and \(2\mu F\) capacitor in series is connected through a switch to \(200\) \(V\) direct supply. Across the capacitor is a neon bulb that lights up at \(120\) \(V.\) Calculate the value of \(R\) to make the bulb light up ...

A fully charged capacitor \(C\) with initial charge \({q_0}\) is connected to a coil of self inductance \(L\) at \(t=0.\) The time at which the energy is stored equally between the electric and the magnetic fields is :

Energy required for the electron excitation in \(L{i^{ + + }}\) from the first to the third Bohr orbit is :

The half life of a radioactive substance is \(20\) minutes. The approximate time interval \(\left( {{t_2} - {t_1}} \right)\) between the time \({{t_2}}\) when \({2 \over 3}\) of it had decayed and time \({{t_1}}\) when \({1 \over 3}\) of it...

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 -\(1\) : Sky wave signals are used for long distance radio communica...

If a wire is stretched to make it \(0.1\%\) longer, its resistance will:

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 - \(1\) : A metallic surface is irradiated by a monochromatic light...

A boat is moving due east in a region where the earth's magnetic fields is \(5.0 \times {10^{ - 5}}\) \(N{A^{ - 1}}\,{m^{ - 1}}\) due north and horizontal. The best carries a vertical aerial \(2\) \(m\) long. If the speed of the boat is $$1...

Two identical charged spheres suspended from a common point by two massless strings of length \(l\) are initially a distance \(d\left( {d < < 1} \right)\) apart because of their mutual repulsion. The charge begins to leak from both the sph...

The electrostatic potential inside a charged spherical ball is given by \(\phi = a{r^2} + b\) where \(r\) is the distance from the center and \(a,b\) are constants. Then the charge density inside the ball is:

A car is fitted with a convex side-view mirror of focal length \(20\) \(cm\). A second car \(2.8m\) behind the first car is overtaking the first car at a relative speed of \(15\) \(m/s\). The speed of the image of the second car as seen in ...

Let \(x\)-\(z\) plane be the boundary between two transparent media. Medium \(1\) in \(z \ge 0\) has a refractive index of \(\sqrt 2\) and medium \(2\) with \(z < 0\) has a refractive index of \(\sqrt 3 .\) A ray of light in medium \(1\)...

Two bodies of masses \(m\) and \(4\) \(m\) are placed at a distance \(r.\) The gravitational potential at a point on the line joining them where the gravitational field is zero is:

A thermally insulated vessel contains an ideal gas of molecular mass \(M\) and ratio of specific heats \(\gamma .\) It is moving with speed \(v\) and it's suddenly brought to rest. Assuming no heat is lost to the surroundings, Its temperatu...

A Carnot engine operating between temperatures \({{T_1}}\) and \({{T_2}}\) has efficiency \({1 \over 6}\). When \({T_2}\) is lowered by \(62\) \(K\) its efficiency increases to \({1 \over 3}\). Then \({T_1}\) and \({T_2}\) are, respectively...

Three perfect gases at absolute temperatures \({T_1},\,{T_2}\) and \({T_3}\) are mixed. The masses of molecules are \({m_1},{m_2}\) and \({m_3}\) and the number of molecules are \({n_1},\) \({n_2}\) and \({n_3}\) respectively. Assuming no l...

\(100g\) of water is heated from \({30^ \circ }C\) to \({50^ \circ }C\). Ignoring the slight expansion of the water, the change in its internal energy is (specific heat of water is \(4184\) \(J/kg/K\)):

A current \(I\) flows in an infinitely long wire with cross section in the form of a semi-circular ring of radius \(R.\) The magnitude of the magnetic induction along its axis is:

A water fountain on the ground sprinkles water all around it. If the speed of water coming out of the
fountain is v, the total area around the fountain that gets wet is :

An object, moving with a speed of 6.25 m/s, is decelerated at a rate given by :
\({{dv} \over {dt}} = - 2.5\sqrt v\) where v is the instantaneous speed. The time taken by the object, to come to rest, would be :

Water is flowing continuously from a tap having an internal diameter \(8 \times {10^{ - 3}}\,\,m.\) The water velocity as it leaves the tap is \(0.4\,\,m{s^{ - 1}}\) . The diameter of the water stream at a distance $$2 \times {10^{ - 1}}\,\...

Work done in increasing the size of a soap bubble from a radius of \(3\) \(cm\) to \(5\) \(cm\) is nearly (Surface tension of soap solution \(= 0.03N{m^{ - 1}},\)

A thin horizontal circular disc is rotating about a vertical axis passing through its center. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the jou...

A mass \(m\) hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass \(m\) and radius \(R.\) Assuming pulley to be a perfect uniform circular disc, the acceleration of the mass \(m,\) if the st...

A pulley of radius \(2\) \(m\) is rotated about its axis by a force \(F = \left( {20t - 5{t^2}} \right)\) newton (where \(t\) is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation i...

Two particles are executing simple harmonic motion of the same amplitude \(A\) and frequency \(\omega\) along the \(x\)-axis. Their mean position is separated by distance \({X_0}\left( {{X_0} > A} \right)\). If the maximum separation betwe...

A mass \(M,\) attached to a horizontal spring, executes \(S.H.M.\) with amplitude \({A_1}.\) When the mass \(M\) passes through its mean position then a smaller mass \(m\) is placed over it and both of them move together with amplitude $${A...

A screw gauge gives the following reading when used to measure the diameter of a wire.
Main scale reading : 0 mm
Circular scale reading : 52 divisions
Given that 1 mm on main scale corresponds to 100 divisions of the circular scale.
The dia...

This question has a paragraph followed by two statements, Statement \(-1\) and Statement \(-2\). Of the given four alternatives after the statements, choose the one that describes the statements.
A thin air film is formed by putting the c...

The transverse displacement \(y(x, t)\) of a wave on a string is given by \(y\left( {x,t} \right) = {e^{ - \left( {a{x^2} + b{t^2} + 2\sqrt {ab} \,xt} \right)}}.\) This represents \(a:\)