AIEEE 2008

The value of \(cot\left( {\cos e{c^{ - 1}}{5 \over 3} + {{\tan }^{ - 1}}{2 \over 3}} \right)\) is :

Let \(f\left( x \right) = \left\{ {\matrix{ {\left( {x - 1} \right)\sin {1 \over {x - 1}}} & {if\,x \ne 1} \cr 0 & {if\,x = 1} \cr } } \right.\)
Then which one of the following is true?

Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”,
and r be the statement “x is a rational number iff y is a transcendental number”.
Statement –1: r is equivalent to either q or p.
State...

The statement \(p \to \left( {q \to p} \right)\) is equivalent to

Let \(a, b, c\) be any real numbers. Suppose that there are real numbers \(x, y, z\) not all zero such that \(x=cy+bz,\) \(y=az+cx,\) and \(z=bx+ay.\) Then \({a^2} + {b^2} + {c^2} + 2abc\) is equal to :

Let \(A\) be a square matrix all of whose entries are integers.
Then which one of the following is true?

Let \(A\) be \(a\,2 \times 2\) matrix with real entries. Let \(I\) be the \(2 \times 2\) identity matrix. Denote by tr\((A)\), the sum of diagonal entries of \(a\). Assume that \({a^2} = I.\)
Statement-1 : If \(A \ne I\) and \(A \ne - I\)...

A parabola has the origin as its focus and the line \(x=2\) as the directrix. Then the vertex of the parabola is at :

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?

In a shop there are five types of ice-cream available. A child buys six ice-cream.
Statement - 1: The number of different ways the child can buy the six ice-cream is \({}^{10}{C_5}\).
Statement - 2: The number of different ways the ch...

A die is thrown. Let \(A\) be the event that the number obtained is greater than \(3.\) Let \(B\) be the event that the number obtained is less than \(5.\) Then \(P\left( {A \cup B} \right)\) is :

It is given that the events \(A\) and \(B\) are such that
\(P\left( A \right) = {1 \over 4},P\left( {A|B} \right) = {1 \over 2}\) and \(P\left( {B|A} \right) = {2 \over 3}.\) Then \(P(B)\) is :

STATEMENT - 1 : For every natural number \(n \ge 2,\)
\(${1 \over {\sqrt 1 }} + {1 \over {\sqrt 2 }} + ........ + {1 \over {\sqrt n }} > \sqrt n .\)$
STATEMENT - 2 : For every natural number \(n \ge 2,\),
$$$\sqrt {n\left( {n + 1} \right...

The quadratic equations \({x^2} - 6x + a = 0\) and \({x^2} - cx + 6 = 0\) have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is

The first two terms of a geometric progression add up to 12. the sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is

Let R be the real line. Consider the following subsets of the plane \(R \times R\) :
\(S = \left\{ {(x,y):y = x + 1\,\,and\,\,0 < x < 2} \right\}\)
\(T = \left\{ {(x,y): x - y\,\,\,is\,\,an\,\,{\mathop{\rm int}} eger\,} \right\}\),
Which...

The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following
gives possible values of a and b?

The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has y-intercept -4. Then a possible value of k is :

The non-zero vectors are \({\overrightarrow a ,\overrightarrow b }\) and \({\overrightarrow c }\) are related by \({\overrightarrow a = 8\overrightarrow b }\) and \({\overrightarrow c = - 7\overrightarrow b \,\,.}\) Then the angle betwee...

The vector \(\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k\) lies in the plane of the vectors
\(\overrightarrow b = \widehat i + \widehat j\) and \(\overrightarrow c = \widehat j + \widehat k\) and bisects the ...

Suppose an electron is attracted towards the origin by a force \({k \over r}\) where \('k'\) is a constant and \('r'\) is the distance of the electron from the origin. By applying Bohr model to this system, the radius of the \({n^{th}}\) or...

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:
Energy is released when heavy nuclei undergo fission or light nucl...

A parallel plate capacitor with air between the plates has capacitance of \(9\) \(pF.\) The separation between its plates is \('d'.\) The space between the plates has dielectric constant \({k_1}\) \(=3\) and thickness \({d \over 3}\) while ...

A thin rod of length \('L'\) is lying along the \(x\)-axis with its ends at \(x=0\) and \(x=L\). Its linear density (mass/length) varies with \(x\) as \(k{\left( {{x \over L}} \right)^n},\) where \(n\) can be zero or any positive number. If...

A block of mass \(0.50\) \(kg\) is moving with a speed of \(2.00\) \(m{s^{ - 1}}\) on a smooth surface. It strike another mass of \(1.0\) \(kg\) and then they move together as a single body. The energy loss during the collision is :

A \(5V\) battery with internal resistance \(2\Omega\) and a \(2V\) battery with internal resistance \(1\Omega\) are connected to a \(10\Omega\) resistor as shown in the figure.

The current in the \(10\Omega\) resistor is

Consider a block of conducting material of resistivity \('\rho '\) shown in the figure. Current \('I'\) enters at \('A'\) and leaves from \('D'\). We apply superposition principle to find voltage \('\Delta V'\) developed between \('B'\) and...

Consider a block of conducting material of resistivity \('\rho '\) shown in the figure. Current \('I'\) enters at \('A'\) and leaves from \('D'\). We apply superposition principle to find voltage \('\Delta V'\) developed between \('B'\) and...

Shown in the figure below is a meter-bridge set up with null deflection in the galvanometer.

The value of the unknown resister \(R\) is

Wave property of electrons implies that they will show diffraction effects. Davisson and Germer demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is obtained by requiring that electro...

Wave property of electrons implies that they will show diffraction effects. Davisson and Germer demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is obtained by requiring that electro...

In an experiment, electrons are made to pass through a narrow slit of width \('d'\) comparable to their de Broglie wavelength. They are detected on a screen at a distance \('D'\) from the slit (see figure).

Which of the following graphs ca...

Two coaxial solenoids are made by winding thin insulated wire over a pipe of cross-sectional area \(A=\) \(10\,\,c{m^2}\) and length \(=20\) \(cm\) . If one of the solenoid has \(300\) turns and the other \(400\) turns, their mutual induct...

A working transistor with its three legs marked \(P,Q\) and \(R\) is tested using a multi-meter. No conduction is found between \(P\) and \(Q\). By connecting the common (negative) terminal of the multi-meter to \(R\) and the other (positiv...

In the circuit below, \(A\) and \(B\) represent two inputs and \(C\) represents the output.

The circuit represents

A thin spherical shell of radius \(R\) has charge \(Q\) spread uniformly over its surface. Which of the following graphs most closely represents the electric field \(E(r)\) produced by the shell in the range \(0 \le r < \infty ,\) where $$r...

A student measures the focal length of a convex lens by putting an object pin at a distance \('u'\) from the lens and measuring the distance \('v'\) of the image pin. The graph between \('u'\) and \('v'\) plotted by the student should look ...

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 mass \(M\) kept at the center of a cube of side ...

A planet in a distant solar system is \(10\) times more massive than the earth and its radius is \(10\) times smaller. Given that the escape velocity from the earth is \(11\,\,km\,{s^{ - 1}},\) the escape velocity from the surface of the p...

An insulated container of gas has two chambers separated by an insulating partition. One of the chambers has volume \({V_1}\) and contains ideal gas at pressure \({P_1}\) and temperature \({T_1}\). The other chamber has volume \({V_2}\) an...

The speed of sound in oxygen \(\left( {{O_2}} \right)\) at a certain temperature is \(460\,\,m{s^{ - 1}}.\) The speed of sound in helium \((He)\) at the same temperature will be (assume both gases to be ideal)

Relative permittivity and permeability of a material \({\varepsilon _r}\) and \({\mu _r},\) respectively. Which of the following values of these quantities are allowed for a diamagnetic material?

A horizontal overhead powerline is at height of \(4m\) from the ground and carries a current of \(100A\) from east to west. The magnetic field directly below it on the ground is
$$\left( {{\mu _0} = 4\pi \times {{10}^{ - 7}}\,\,Tm\,\,{A^{...

A body is at rest at \(x=0.\) At \(t=0,\) it starts moving in the positive \(x\)-direction with a constant acceleration. At the same instant another body passes through \(x=0\) moving in the positive \(x\) direction with a constant speed. T...

A jar is filled with two non-mixing liquids \(1\) and \(2\) having densities \({\rho _1}\) and \({\rho _2}\) respectively. A solid ball, made of a material of density \({\rho _3}\), is dropped in the jar. It comes to equilibrium in the pos...

A capillary tube (A) is dipped in water. Another identical tube (B) is dipped in a soap-water solution. Which of the following shows the relative nature of the liquid columns in the two tubes?

A spherical solid ball of volume \(V\) is made of a material of density \({\rho _1}\). It is falling through a liquid of density \({\rho _2}\left( {{\rho _2} < {\rho _1}} \right)\). Assume that the liquid applies a viscous force on the ball...

Consider a uniform square plate of side \(' a '\) and mass \('m'\). The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

Two full turns of the circular scale of a screw gauge cover a distance of 1 mm on its main scale. The
total number of divisions on the circular scale is 50. Further, it is found that the screw gauge has a
zero error of − 0.03 mm while measu...

A body of mass m = 3.513 kg is moving along the x-axis with a speed of 5.00 ms−1. The magnitude of its momentum is recorded as