AIEEE 2006 Mock Test

$ABC$ is a triangle, right-angled at $A$. The resultant of the forces acting along $AB$ and $AC$ with magnitudes $\frac{1}{AB}$ and $\frac{1}{AC}$ respectively is a force along $AD$, where $D$ is the foot of the perpendicular from $A$ onto $BC$. The magnitude of the resultant is:

Suppose a population $A$ has $100$ observations $101,102,\dots ,200$, and another population $B$ has $100$ observations $151,152,\dots ,250$. If $V_A$ and $V_B$ represent the variances of the two populations, respectively, then $\frac{V_A}{V_B}$ is:

If the roots of the quadratic equation $x^2+px+q=0$ are $\tan 30^{\circ }$ and $\tan 15^{\circ }$, respectively, then the value of $2+q-p$ is:

The value of the integral ${\int }_3^6\frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}}dx$ is:

The number of values of $x$ in the interval $[0,3\pi ]$ satisfying the equation $2{\sin }^{2}x+5\sin x-3=0$ is:

If $(\vec{a}\times \vec{b})\times \vec{c}=\vec{a}\times (\vec{b}\times \vec{c})$, where $\vec{a},\vec{b}$ and $\vec{c}$ are any three vectors such that $\vec{a}\cdot \vec{b}\ne 0$ and $\vec{b}\cdot \vec{c}\ne 0$, then $\vec{a}$ and $\vec{c}$ are:

Let $W$ denote the words in the English dictionary. Define the relation $R$ by $R=\{(x,y)\in W\times W\mid \text{the words }x\text{ and }y\text{ have at least one letter in common}\}$. Then $R$ is:

If $A$ and $B$ are square matrices of size $n\times n$ such that $A^2-B^2=(A-B)(A+B)$, then which of the following will be always true?

The value of ${\sum }_{k=1}^{10}\left(\sin \frac{2k\pi }{11}+i\cos \frac{2k\pi }{11}\right)$ is:

All the values of $m$ for which both roots of the equation $x^2-2mx+m^2-1=0$ are greater than $-2$ but less than $4$, lie in the interval:

A particle has two velocities of equal magnitude $u$ inclined to each other at an angle $\theta$. If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then $\theta$ is:

At a telephone enquiry system, the number of phone calls regarding relevant enquiry follows a Poisson distribution with an average of $5$ phone calls during $10\text{-minute}$ time intervals. The probability that there is at most one phone call during a $10\text{-minute}$ time period is:

A body falling from rest under gravity passes a certain point $P$. It was at a distance of $400\text{m}$ from $P$, $4\text{s}$ prior to passing through $P$. If $g=10{\text{m/s}}^2$, then the height above the point $P$ from where the body began to fall is:

${\int }_0^{\pi }xf(\sin x)dx$ is equal to:

A straight line through the point $A(3,4)$ is such that its intercept between the axes is bisected at $A$. Its equation is:

The two lines $x=ay+b,z=cy+d$; and $x=a^{\prime }y+b^{\prime },z=c^{\prime }y+d^{\prime }$ are perpendicular to each other if:

The locus of the vertices of the family of parabolas $y=\frac{a^3{x}^2}{3}+\frac{a^{2}x}{2}-2a$ is:

The values of $a$ for which the points $A,B,C$ with position vectors $2\hat{i}-\hat{j}+\hat{k}$, $\hat{i}-3\hat{j}-5\hat{k}$ and $a\hat{i}-3\hat{j}+\hat{k}$ respectively are the vertices of a right-angled triangle with $\angle C=\pi /2$ are:

${\int }_{-\pi /2}^{\pi /2}\left[[x+\pi ]+\cos (x+3\pi )\right]dx$ is equal to (where $[\cdot ]$ denotes the greatest integer function):

If $x$ is real, the maximum value of $\frac{3x^2+9x+17}{3x^2+9x+7}$ is:

In an ellipse, the distance between its foci is $6$ and the minor axis is $8$. Then its eccentricity is:

Let $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$ and $B=\left(\begin{array}{cc}a& 0\\ 0& b\end{array}\right)$, $a,b\in \mathbb{N}$. Then:

The function $f(x)=\frac{x}{2}+\frac{2}{x}$ has a local minimum at:

Angle between the tangents to the curve $y=x^2-5x+6$ at the points $(2,0)$ and $(3,0)$ is:

Let $a_1,a_2,a_3,\dots$ be terms of an A.P. If $\frac{a_1+a_2+\cdots +a_p}{a_1+a_2+\cdots +a_q}=\frac{p^2}{q^2}$, $p\ne q$, then $\frac{a_6}{a_{21}}$ equals:

The set of points where $f(x)=\frac{x}{1+|x|}$ is differentiable is:

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $x$. The maximum area enclosed by the park is:

At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are $10$ candidates and $4$ are to be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote is:

If the expansion in powers of $x$ of the function $\frac{1}{(1-ax)(1-bx)}$ is $a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\dots$, then $a_n$ is:

For natural numbers $m,n$, if $(1-y{)}^m(1+y{)}^n=1+a_{1}y+a_2{y}^2+\dots$, and $a_1=a_2=10$, then $(m,n)$ is:

The value of ${\int }_1^a[x]f^{\prime }(x)dx,a>1$, where $[x]$ denotes the greatest integer not exceeding $x$, is:

If the lines $3x-4y-7=0$ and $2x-3y-5=0$ are two diameters of a circle of area $49\pi$ square units, the equation of the circle is:

The differential equation whose solution is $A{x}^2+B{y}^2=1$, where $A$ and $B$ are arbitrary constants, is of:

Let $C$ be the circle with centre $(0,0)$ and radius $3$ units. The equation of the locus of the mid points of the chords of the circle $C$ that subtend an angle of $2\pi /3$ at its centre is:

If $(a,a^2)$ falls inside the angle made by the lines $y=x/2,x>0$ and $y=3x,x>0$, then $a$ belongs to:

The image of the point $(-1,3,4)$ in the plane $x-2y=0$ is:

If $z^2+z+1=0$, where $z$ is a complex number, then the value of ${\left(z+\frac{1}{z}\right)}^2+{\left(z^2+\frac{1}{z^2}\right)}^2+\cdots +{\left(z^6+\frac{1}{z^6}\right)}^2$ is:

If $0<x<\pi$ and $\cos x+\sin x=1/2$, then $\tan x$ is:

If $a_1,a_2,\dots ,a_n$ are in H.P., then the expression $a_1{a}_2+a_2{a}_3+\cdots +a_{n-1}{a}_n$ is equal to:

If $x^m{y}^n=(x+y{)}^{m+n}$, then $\frac{dy}{dx}$ is:

A particle located at $x=0$ at time $t=0$, starts moving along the positive $x$ -direction with a velocity $v$ that varies as $v=\alpha \sqrt{x}$. The displacement of the particle varies with time as:

A mass of $M\text{kg}$ is suspended by a weightless string. The horizontal force $F$ that is required to displace it until the string makes an angle of $45^{\circ }$ with the initial vertical direction is:

A bomb of mass $16\text{kg}$ at rest explodes into two pieces of masses $4\text{kg}$ and $12\text{kg}$. The velocity of the $12\text{kg}$ mass is $4\text{m/s}$. The kinetic energy of the other mass is:

A particle of mass $100\text{g}$ is thrown vertically upwards with a speed of $5\text{m/s}$. The work done by the force of gravity during the time the particle goes up is:

A whistle producing sound waves of frequencies $9500\text{Hz}$ and above is approaching a stationary person with speed $v\text{m/s}$. The velocity of sound in air is $300\text{m/s}$. If the person can hear frequencies up to a maximum of $10000\text{Hz}$, the maximum value of $v$ up to which he can hear the whistle is:

An electric dipole is placed at an angle of $30^{\circ }$ to a non-uniform electric field. The dipole will experience:

A material $B$ has twice the specific resistance (resistivity) of $A$. A circular wire made of $B$ has twice the diameter of a wire made of $A$. For the two wires to have the same resistance, the ratio $L_A/L_B$ of their respective lengths must be:

Kirchhoff’s first law ($\sum i=0$) and second law ($\sum iR=\sum E$) are respectively based on the conservation of:

In a region, steady and uniform electric and magnetic fields are present. These two fields are parallel to each other. A charged particle is released from rest in this region. The path of the particle will be a:

Needles $N_1,N_2$ and $N_3$ are made of a ferromagnetic, a paramagnetic and a diamagnetic substance respectively. A magnet when brought close to them will:

Which of the following units denotes the dimensions $[M{L}^2/Q^2]$, where $Q$ denotes the electric charge?

A player caught a cricket ball of mass $150\text{g}$ moving at a rate of $20\text{m/s}$. If the catching process is completed in $0.1\text{s}$, the force of the blow exerted by the ball on the hand of the player is equal to:

A ball of mass $0.2\text{kg}$ is thrown vertically upwards by applying a force by hand. If the hand moves $0.2\text{m}$ while applying the force and the ball goes up to $2\text{m}$ height further, find the magnitude of the force. (Consider $g=10{\text{m/s}}^2$)

Consider a two particle system with particles having masses $m_1$ and $m_2$. If the first particle is pushed towards the centre of mass through a distance $d$, by what distance should the second particle be moved, so as to keep the centre of mass at the same position?

Starting from the origin, a body oscillates simple harmonically with a period of $2\text{s}$. After what time will its kinetic energy be $75\%$ of the total energy?

The maximum velocity of a particle executing simple harmonic motion with an amplitude $7\text{mm}$ is $4.4\text{m/s}$. The period of oscillation is:

A string is stretched between fixed points separated by $75\text{cm}$. It is observed to have resonant frequencies of $420\text{Hz}$ and $315\text{Hz}$. There are no other resonant frequencies between these two. Then, the lowest resonant frequency for this string is:

Assuming the sun to be a spherical body of radius $R$ at a temperature of $T\text{K}$, evaluate the total radiant power incident on Earth at a distance $r$ from the Sun. (where $r_0$ is the radius of the Earth and $\sigma$ is Stefan’s constant).

The refractive index of glass is $1.520$ for red light and $1.525$ for blue light. Let $D_1$ and $D_2$ be the angle of minimum deviation for red and blue light respectively in a thin prism of this glass. Then:

In a Wheatstone’s bridge, three resistances $P,Q$ and $R$ are connected in three arms and the fourth arm is formed by two resistances $S_1$ and $S_2$ connected in parallel. The condition for the bridge to be balanced is:

The current $I$ drawn from the $5\text{V}$ source in the following circuit is:

In a series resonant LCR circuit, the voltage across $R$ is $100\text{V}$ and $R=1\text{k}\Omega$ with $C=2\mu \text{F}$. The resonant frequency $\omega$ is $200\text{rad/s}$. At resonance, the voltage across $L$ is:

Two infinitely large insulating plates are uniformly charged such that the potential difference between them is $V_2-V_1=20\text{V}$. The plates are separated by $d=0.1\text{m}$. An electron is released from rest on the inner surface of plate 1. What is its speed when it hits plate 2? ($e=1.6\times 10^{-19}\text{C}$, $m_e=9.11\times 10^{-31}\text{kg}$)

The resistance of a bulb filament is $100\Omega$ at a temperature of $100^{\circ }\text{C}$. If its temperature coefficient of resistance be $0.005$ per ${}^{\circ }\text{C}$, its resistance will become $200\Omega$ at a temperature of:

In an AC generator, a coil with $N$ turns, all of the same area $A$ and total resistance $R$, rotates with frequency $\omega$ in a magnetic field $B$. The maximum value of emf generated in the coil is:

The flux linked with a coil at any instant $t$ is given by $\varphi =10t^2-50t+250$. The induced emf at $t=3\text{s}$ is:

A thermocouple is made from two metals, Antimony and Bismuth. If one junction of the couple is kept hot and the other is kept cold, then an electric current will:

The time taken by a photoelectron to come out after the photon strikes is approximately:

An alpha nucleus of energy $\frac{1}{2}m{v}^2$ bombards a heavy nuclear target of charge $Ze$. Then the distance of closest approach for the alpha nucleus will be proportional to:

The threshold frequency for a metallic surface corresponds to an energy of $6.2\text{eV}$, and the stopping potential for a radiation incident on this surface is $5\text{V}$. The incident radiation lies in the:

The energy spectrum of $\beta$ -particles [number $N(E)$ as a function of $\beta$ -energy $E$ ] emitted from a radioactive source is:

When ${}_3{\text{Li}}^7$ nuclei are bombarded by protons, and the resultant nuclei are ${}_4{\text{Be}}^8$, the emitted particles will be:

A solid which is transparent to visible light and whose conductivity increases with temperature is formed by:

If the ratio of the concentration of electrons to that of holes in a semiconductor is $7/5$ and the ratio of currents is $7/4$, then what is the ratio of their drift velocities?

In a common base mode of a transistor, the collector current is $5.488\text{mA}$ for an emitter current of $5.60\text{mA}$. The value of the base current amplification factor ($\beta$) will be:

The potential energy of a $1\text{kg}$ particle free to move along the $x$ -axis is given by $V(x)=\left(\frac{x^4}{4}-\frac{x^2}{2}\right)\text{J}$. The total mechanical energy of the particle is $2\text{J}$. Then, the maximum speed (in m/s) is:

A force of $-F\hat{k}$ acts on $O$, the origin of the coordinate system. The torque about the point $(1,-1,0)$ is:

A thin circular ring of mass $m$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two objects each of mass $M$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity ${\omega }^{\prime }$:

The terminal speed of a sphere of gold (density $=19.5{\text{kg/m}}^3$) is $0.2\text{m/s}$ in a viscous liquid (density $=1.5{\text{kg/m}}^3$). Find the terminal speed of a sphere of silver (density $=10.5{\text{kg/m}}^3$) of the same size in the same liquid.

Work of $146\text{kJ}$ is performed in order to compress one kilo mole of gas adiabatically and in this process the temperature of the gas increases by $7^{\circ }\text{C}$. The gas is: ($R=8.3{\text{J mol}}^{-1}{\text{K}}^{-1}$)

The rms value of the electric field of the light coming from the Sun is $720\text{N/C}$. The average total energy density of the electromagnetic wave is:

A coin is placed on a horizontal platform which undergoes vertical simple harmonic motion of angular frequency $\omega$. The amplitude of oscillation is gradually increased. The coin will leave contact with the platform for the first time:

An electric bulb is rated $220\text{V}-100\text{W}$. The power consumed by it when operated on $110\text{V}$ will be:

The anode voltage of a photocell is kept fixed. The wavelength $\lambda$ of the light falling on the cathode is gradually changed. The plate current $I$ of the photocell varies as follows:

The 'rad' is the correct unit used to report the measurement of:

If the binding energy per nucleon in ${}_3{\text{Li}}^7$ and ${}_2{\text{He}}^4$ nuclei are $5.60\text{MeV}$ and $7.06\text{MeV}$ respectively, then in the reaction ${}_1{\text{H}}^1+{}_3{\text{Li}}^7\to 2{}_2{\text{He}}^4$, the energy released is:

If the lattice constant of a semiconductor is decreased, then which of the following is correct?

In the following, which one of the diodes is reverse biased?

The circuit has two oppositely connected ideal diodes in parallel. What is the current flowing in the circuit?

A long solenoid has $200$ turns per cm and carries a current $i$. The magnetic field at its centre is $6.28\times 10^{-2}{\text{Wb/m}}^2$. Another long solenoid has $100$ turns per cm and it carries a current $i/3$. The value of the magnetic field at its centre is:

Four point masses, each of value $m$, are placed at the corners of a square $ABCD$ of side $\ell$. The moment of inertia through $A$ and parallel to $BD$ is:

A wire elongates by $\ell \text{mm}$ when a load $W$ is hanged from it. If the wire goes over a pulley and two weights $W$ each are hung at the two ends, the elongation of the wire will be (in mm):

Two rigid boxes containing different ideal gases are placed on a table. Box A contains one mole of nitrogen at temperature $T_0$, while Box B contains one mole of helium at temperature $(7/3)T_0$. The boxes are put into thermal contact. The final temperature $T_f$ is:

Two spherical conductors $A$ and $B$ of radii $1\text{mm}$ and $2\text{mm}$ are separated by a distance of $5\text{cm}$ and are uniformly charged. If the spheres are connected by a conducting wire, then in equilibrium, the ratio of the magnitude of the electric fields at the surface of spheres $A$ and $B$ is:

An inductor ($L=100\text{mH}$), a resistor ($R=100\Omega$) and a battery ($E=100\text{V}$) are connected in series. After a long time, the battery is disconnected after short-circuiting. The current in the circuit $1\text{ms}$ after the short-circuit is:

$\text{HBr}$ reacts with ${\text{CH}}_2=\text{CH}-{\text{OCH}}_3$ under anhydrous conditions at room temperature to give:

The IUPAC name of the compound shown below is:

The increasing order of the rate of $\text{HCN}$ addition to compounds A – D is:(A) $\text{HCHO}$ (B) ${\text{CH}}_3{\text{COCH}}_3$ (C) ${\text{PhCOCH}}_3$ (D) $\text{PhCOPh}$

How many moles of magnesium phosphate, ${\text{Mg}}_3({\text{PO}}_4{)}_2$ will contain $0.25\text{mole}$ of oxygen atoms?

According to Bohr’s theory, the angular momentum of an electron in $5\text{th}$ orbit is:

Which of the following molecules/ions does not contain unpaired electrons?

Total volume of atoms present in a face-centre cubic (FCC) unit cell of a metal is ($r$ is atomic radius):