AIEEE 2006 — 30 Apr Offline (Previous Year Paper)

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

1. A.$\frac{AB^2+AC^2}{(AB)^2(AC)^2}$
2. B.$\frac{(AB)(AC)}{AB+AC}$
3. C.$\frac{1}{AB}+\frac{1}{AC}$
4. D.$\frac{1}{AD}$✓

Correct answer: D

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

1. A.1✓
2. B.9/4
3. C.4/9
4. D.2/3

Correct answer: A

If the roots of the quadratic equation $x^2 + px + q = 0$ are $\tan30^\circ$ and $\tan15^\circ$, respectively then the value of $2 + q - p$ is

1. A.2
2. B.3✓
3. C.0
4. D.1

Correct answer: B

The value of the integral, $\int_3^6\frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}}dx$ is

1. A.1/2
2. B.3/2✓
3. C.2
4. D.1

Correct answer: B

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

1. A.4✓
2. B.6
3. C.1
4. D.2

Correct answer: A

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 $\vec{a}\cdot\vec{b} \ne 0$, $\vec{b}\cdot\vec{c} \ne 0$, then $\vec{a}$ and $\vec{c}$ are

1. A.inclined at an angle of $\pi/3$ between them
2. B.inclined at an angle of $\pi/6$ between them
3. C.perpendicular
4. D.parallel✓

Correct answer: D

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

1. A.not reflexive, symmetric and transitive
2. B.reflexive, symmetric and not transitive✓
3. C.reflexive, symmetric and transitive
4. D.reflexive, not symmetric and transitive

Correct answer: B

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 ?

1. A.A = B
2. B.AB = BA✓
3. C.either of A or B is a zero matrix
4. D.either of A or B is an identity matrix

Correct answer: B

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

1. A.i
2. B.1
3. C.-1
4. D.-i✓

Correct answer: D

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

1. A.-2 < m < 0
2. B.m > 3
3. C.-1 < m < 3✓
4. D.1 < m < 4

Correct answer: C

A particle has two velocities of equal magnitude 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

1. A.$90^\circ$
2. B.$120^\circ$✓
3. C.$45^\circ$
4. D.$60^\circ$

Correct answer: B

At a telephone enquiry system the number of phone cells regarding relevant enquiry follow Poisson distribution with an average of 5 phone calls during 10-minute time intervals. The probability that there is at the most one phone call during a 10-minute time period is

1. A.$\frac{6}{5^e}$
2. B.$\frac{5}{6}$
3. C.$\frac{6}{55}$
4. D.$\frac{6}{e^5}$✓

Correct answer: D

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

1. A.720 m✓
2. B.900 m
3. C.320 m
4. D.680 m

Correct answer: A

$\int_0^\pi xf(\sin x)\,dx$ is equal to

1. A.$\pi\int_0^\pi f(\cos x)dx$
2. B.$\pi\int_0^\pi f(\sin x)dx$
3. C.$\frac{\pi}{2}\int_0^{\pi/2}f(\sin x)dx$
4. D.$\pi\int_0^{\pi/2}f(\cos x)dx$✓

Correct answer: D

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

1. A.x + y = 7
2. B.3x - 4y + 7 = 0
3. C.4x + 3y = 24✓
4. D.3x + 4y = 25

Correct answer: C

The two lines x = ay + b, z = cy + d; and x = a'y + b', z = c'y + d' are perpendicular to each other if

1. A.aa' + cc' = -1✓
2. B.aa' + cc' = 1
3. C.$\frac{a}{a'}+\frac{c}{c'}=-1$
4. D.$\frac{a}{a'}+\frac{c}{c'}=1$

Correct answer: A

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

1. A.$xy = \frac{105}{64}$✓
2. B.$xy = \frac{3}{4}$
3. C.$xy = \frac{35}{16}$
4. D.$xy = \frac{64}{105}$

Correct answer: A

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 $C = \frac{\pi}{2}$ are

1. A.2 and 1✓
2. B.-2 and -1
3. C.-2 and 1
4. D.2 and -1

Correct answer: A

$\int_{-3\pi/2}^{-\pi/2}\left[(x+\pi)^3 + \cos^2(x+3\pi)\right]dx$ is equal to

1. A.$\frac{\pi^4}{32}$
2. B.$\frac{\pi^4}{32}+\frac{\pi}{2}$
3. C.$\frac{\pi}{2}$✓
4. D.$\frac{\pi}{4}-1$

Correct answer: C

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

1. A.1/4
2. B.41✓
3. C.1
4. D.17/7

Correct answer: B

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

1. A.$\frac{3}{5}$✓
2. B.$\frac{1}{2}$
3. C.$\frac{4}{5}$
4. D.$\frac{1}{\sqrt{5}}$

Correct answer: A

Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$, $a, b \in N$. Then

1. A.there cannot exist any B such that AB = BA
2. B.there exist more than one but finite number of B's such that AB = BA
3. C.there exists exactly one B such that AB = BA
4. D.there exist infinitely many B's such that AB = BA✓

Correct answer: D

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

1. A.x = 2✓
2. B.x = -2
3. C.x = 0
4. D.x = 1

Correct answer: A

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

1. A.$\frac{\pi}{2}$
2. B.$\frac{\pi}{2}$ (duplicate as printed)✓
3. C.$\frac{\pi}{6}$
4. D.$\frac{\pi}{4}$

Correct answer: B

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

1. A.$\frac{41}{11}$
2. B.$\frac{7}{2}$
3. C.$\frac{2}{7}$
4. D.$\frac{11}{41}$✓

Correct answer: D

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

1. A.$(-\infty, 0)\cup(0, \infty)$
2. B.$(-\infty, -1)\cup(-1, \infty)$
3. C.$(-\infty, \infty)$✓
4. D.$(0, \infty)$

Correct answer: C

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

1. A.$\frac{3}{2}x^2$
2. B.$\sqrt{\frac{x^3}{8}}$
3. C.$\frac{1}{2}x^2$✓
4. D.$\pi x^2$

Correct answer: C

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 of be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote is

1. A.5040
2. B.6210
3. C.385✓
4. D.1110

Correct answer: C

If the expansion in powers of x of the function $\frac{1}{(1-ax)(1-bx)}$ is $a_0 + a_1x + a_2x^2 + a_3x^3 + \dots$, then $a_n$ is

1. A.$\frac{b^n-a^n}{b-a}$
2. B.$\frac{a^n-b^n}{b-a}$
3. C.$\frac{a^{n+1}-b^{n+1}}{b-a}$
4. D.$\frac{b^{n+1}-a^{n+1}}{b-a}$✓

Correct answer: D

For natural numbers m, n if $(1 - y)^m(1 + y)^n = 1 + a_1y + a_2y^2 + \dots$, and $a_1 = a_2 = 10$, then (m, n) is

1. A.(20, 45)
2. B.(35, 20)
3. C.(45, 35)
4. D.(35, 45)✓

Correct answer: D

The value of $\int_1^a[x]f'(x)dx$, a > 1, where [x] denotes the greatest integer not exceeding x is

1. A.$af(a) - \{f(1) + f(2) + \dots + f([a])\}$
2. B.$[a]f(a) - \{f(1) + f(2) + \dots + f([a])\}$✓
3. C.$[a]f([a]) - \{f(1) + f(2) + \dots + f(a)\}$
4. D.$af([a]) - \{f(1) + f(2) + \dots + f(a)\}$

Correct answer: B

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

1. A.$x^2 + y^2 + 2x - 2y - 47 = 0$
2. B.$x^2 + y^2 + 2x - 2y - 62 = 0$
3. C.$x^2 + y^2 - 2x + 2y - 62 = 0$
4. D.$x^2 + y^2 - 2x + 2y - 47 = 0$✓

Correct answer: D

The differential equation whose solution is $Ax^2 + By^2 = 1$, where A and B are arbitrary constants is of

1. A.second order and second degree
2. B.first order and second degree
3. C.first order and first degree
4. D.second order and first degree✓

Correct answer: D

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 $\frac{2\pi}{3}$ at its centre is

1. A.$x^2 + y^2 = \frac{3}{2}$
2. B.$x^2 + y^2 = 1$
3. C.$x^2 + y^2 = \frac{27}{4}$
4. D.$x^2 + y^2 = \frac{9}{4}$✓

Correct answer: D

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

1. A.$\left(0, \frac{1}{2}\right)$
2. B.$(3, \infty)$
3. C.$\left(\frac{1}{2}, 3\right)$✓
4. D.$\left(-3, -\frac{1}{2}\right)$

Correct answer: C

If $z^2 + z + 1 = 0$, where z is a complex number, then the value of $\left(z+\frac{1}{z}\right)^2 + \left(z^2+\frac{1}{z^2}\right)^2 + \left(z^3+\frac{1}{z^3}\right)^2 + \dots + \left(z^6+\frac{1}{z^6}\right)^2$ is

1. A.18
2. B.54
3. C.6
4. D.12✓

Correct answer: D

If $0 < x < \pi$ and $\cos x + \sin x = \frac{1}{2}$, then $\tan x$ is

1. A.$\frac{1-\sqrt{7}}{4}$
2. B.$\frac{4-\sqrt{7}}{3}$
3. C.$-\frac{4+\sqrt{7}}{3}$✓
4. D.$\frac{1+\sqrt{7}}{4}$

Correct answer: C

If $a_1, a_2, \dots, a_n$ are in H.P., then the expression $a_1a_2 + a_2a_3 + \dots + a_{n-1}a_n$ is equal to

1. A.$n(a_1 - a_n)$
2. B.$(n-1)(a_1 - a_n)$
3. C.$na_1a_n$
4. D.$(n-1)a_1a_n$✓

Correct answer: D

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

1. A.$\frac{y}{x}$✓
2. B.$\frac{x+y}{xy}$
3. C.xy
4. D.$\frac{x}{y}$

Correct answer: A

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

1. A.$t^3$
2. B.$t^2$✓
3. C.t
4. D.$t^{1/2}$

Correct answer: B

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

1. A.$Mg(\sqrt{2}-1)$✓
2. B.$Mg(\sqrt{2}+1)$
3. C.$Mg\sqrt{2}$
4. D.$\frac{Mg}{\sqrt{2}}$

Correct answer: A

A bomb of mass 16 kg at rest explodes into two pieces of masses of 4 kg and 12 kg. The velocity of the 12 kg mass is 4 ms$^{-1}$. The kinetic energy of the other mass is

1. A.96 J
2. B.144 J
3. C.288 J✓
4. D.192 J

Correct answer: C

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

1. A.0.5 J
2. B.-0.5 J
3. C.-1.25 J✓
4. D.1.25 J

Correct answer: C

A whistle producing sound waves of frequencies 9500 Hz and above is approaching a stationary person with speed v ms$^{-1}$. The velocity of sound in air is 300 ms$^{-1}$. If the person can hear frequencies upto a maximum of 10,000 Hz, the maximum value of v upto which he can hear the whistle is

1. A.30 ms$^{-1}$
2. B.$15\sqrt{2}$ ms$^{-1}$
3. C.$15/\sqrt{2}$ ms$^{-1}$
4. D.15 ms$^{-1}$✓

Correct answer: D

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

1. A.a torque only
2. B.a translational force only in the direction of the field
3. C.a translational force only in a direction normal to the direction of the field
4. D.a torque as well as a translational force✓

Correct answer: D

The Kirchhoff's first law ($\sum i = 0$) and second law ($\sum iR = \sum E$), where the symbols have their usual meanings, are respectively based on

1. A.conservation of charge, conservation of energy✓
2. B.conservation of charge, conservation of momentum
3. C.conservation of energy, conservation of charge
4. D.conservation of momentum, conservation of charge

Correct answer: A

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

1. A.circle
2. B.helix
3. C.straight line✓
4. D.ellipse

Correct answer: C

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

1. A.attract all three of them
2. B.attract $N_1$ and $N_2$ strongly but repel $N_3$
3. C.attract $N_1$ strongly, $N_2$ weakly and repel $N_3$ weakly✓
4. D.attract $N_1$ strongly, but repel $N_2$ and $N_3$ weakly

Correct answer: C

Which of the following units denotes the dimensions $ML^2/Q^2$, where Q denotes the electric charge?

1. A.Weber (Wb)
2. B.Wb/m$^2$
3. C.Henry (H)✓
4. D.H/m$^2$

Correct answer: C

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

1. A.300 N
2. B.150 N
3. C.3 N
4. D.30 N✓

Correct answer: D

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?

1. A.d
2. B.$\frac{m_2}{m_1}d$
3. C.$\frac{m_1}{m_1+m_2}d$
4. D.$\frac{m_1}{m_2}d$✓

Correct answer: D

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

1. A.$\frac{1}{12}$ s
2. B.$\frac{1}{6}$ s✓
3. C.$\frac{1}{4}$ s
4. D.$\frac{1}{3}$ s

Correct answer: B

The maximum velocity of a particle, executing simple harmonic motion with an amplitude 7 mm, is 4.4 m/s. The period of oscillation is

1. A.100 s
2. B.0.01 s✓
3. C.10 s
4. D.0.1 s

Correct answer: B

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

1. A.10.5 Hz
2. B.105 Hz✓
3. C.1.05 Hz
4. D.1050 Hz

Correct answer: B

Assuming the sun to be a spherical body of radius R at a temperature of T 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)

1. A.$\frac{R^2\sigma T^4}{r^2}$
2. B.$\frac{4\pi r_0^2R^2\sigma T^4}{r^2}$
3. C.$\frac{\pi r_0^2R^2\sigma T^4}{r^2}$✓
4. D.$\frac{r_0^2R^2\sigma T^4}{4\pi r^2}$

Correct answer: C

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

1. A.$D_1 > D_2$
2. B.$D_1 < D_2$✓
3. C.$D_1 = D_2$
4. D.$D_1$ can be less than or greater than depending upon the angle of prism

Correct answer: B

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

1. A.$\frac{P}{Q} = \frac{R}{S_1+S_2}$
2. B.$\frac{P}{Q} = \frac{2R}{S_1+S_2}$
3. C.$\frac{P}{Q} = \frac{R(S_1+S_2)}{S_1S_2}$✓
4. D.$\frac{P}{Q} = \frac{R(S_1+S_2)}{2S_1S_2}$

Correct answer: C

The current I drawn from the 5 volt source will be

1. A.0.17 A
2. B.0.33 A
3. C.0.5 A✓
4. D.0.67 A

Correct answer: C

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

1. A.$4\times10^{-3}$ V
2. B.$2.5\times10^{-2}$ V
3. C.40 V
4. D.250 V✓

Correct answer: D

Two insulating plates are both uniformly charged in such a way that the potential difference between them is $V_2 - V_1 = 20$ V. (i.e. plate 2 is at a higher potential). The plates are separated by d = 0.1 m and can be treated as infinitely large. An electron is released from rest on the inner surface of plate 1. What is its speed when it hits plate 2? ($e = 1.6\times10^{-19}$ C, $m_e = 9.11\times10^{-31}$ kg)

1. A.$32\times10^{-19}$ m/s
2. B.$2.65\times10^6$ m/s✓
3. C.$7.02\times10^{12}$ m/s
4. D.$1.87\times10^6$ m/s

Correct answer: B

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

1. A.$200^\circ$C
2. B.$300^\circ$C✓
3. C.$400^\circ$C
4. D.$500^\circ$C

Correct answer: B

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

1. A.N.A.B.$\omega$✓
2. B.N.A.B.R.$\omega$
3. C.N.A.B
4. D.N.A.B.R

Correct answer: A

The flux linked with a coil at any instant 't' is given by $\phi = 10t^2 - 50t + 250$. The induced emf at t = 3 s is

1. A.190 V
2. B.-190 V
3. C.-10 V✓
4. D.10 V

Correct answer: C

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

1. A.flow from Antimony to Bismuth at the cold junction✓
2. B.flow from Antimony to Bismuth at the hot junction
3. C.flow from Bismuth to Antimony at the cold junction
4. D.not flow through the thermocouple

Correct answer: A

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

1. A.$10^{-1}$ s
2. B.$10^{-4}$ s
3. C.$10^{-10}$ s✓
4. D.$10^{-16}$ s

Correct answer: C

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

1. A.$\frac{1}{Ze}$
2. B.$v^2$
3. C.$\frac{1}{m}$✓
4. D.$\frac{1}{v^4}$

Correct answer: C

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

1. A.X-ray region
2. B.ultra-violet region✓
3. C.infra-red region
4. D.visible region

Correct answer: B

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

1. A.Graph (1)
2. B.Graph (2)
3. C.Graph (3)
4. D.Graph (4)✓

Correct answer: D

When $_3Li^7$ nuclei are bombarded by protons, and the resultant nuclei are $_4Be^8$, the emitted particles will be

1. A.neutrons
2. B.alpha particles
3. C.beta particles
4. D.gamma photons✓

Correct answer: D

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

1. A.Metallic binding
2. B.Ionic binding
3. C.Covalent binding✓
4. D.Van der Waals binding

Correct answer: C

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

1. A.$\frac{4}{7}$
2. B.$\frac{5}{8}$
3. C.$\frac{4}{5}$
4. D.$\frac{5}{4}$✓

Correct answer: D

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

1. A.48
2. B.49✓
3. C.50
4. D.51

Correct answer: B

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

1. A.2
2. B.$3/\sqrt{2}$✓
3. C.$\sqrt{2}$
4. D.$1/\sqrt{2}$

Correct answer: B

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

1. A.$-F(\hat{i}-\hat{j})$
2. B.$F(\hat{i}-\hat{j})$
3. C.$-F(\hat{i}+\hat{j})$✓
4. D.$F(\hat{i}+\hat{j})$

Correct answer: C

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 $\omega'$ =

1. A.$\frac{\omega m}{(m+2M)}$✓
2. B.$\frac{\omega(m+2M)}{m}$
3. C.$\frac{\omega(m-2M)}{(m+2M)}$
4. D.$\frac{\omega m}{(m+M)}$

Correct answer: A

If the terminal speed of a sphere of gold (density = 19.5 kg/m$^3$) is 0.2 m/s in a viscous liquid (density = 1.5 kg/m$^3$), find the terminal speed of a sphere of silver (density = 10.5 kg/m$^3$) of the same size in the same liquid.

1. A.0.2 m/s
2. B.0.4 m/s
3. C.0.133 m/s
4. D.0.1 m/s✓

Correct answer: D

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

1. A.monoatomic
2. B.diatomic✓
3. C.triatomic
4. D.a mixture of monoatomic and diatomic

Correct answer: B

The rms value of the electric field of the light coming from the Sun is 720 N/C. The average total energy density of the electromagnetic wave is

1. A.$3.3\times10^{-3}$ J/m$^3$
2. B.$4.58\times10^{-6}$ J/m$^3$✓
3. C.$6.37\times10^{-9}$ J/m$^3$
4. D.$81.35\times10^{-12}$ J/m$^3$

Correct answer: B

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

1. A.at the highest position of the platform
2. B.at the mean position of the platform
3. C.for an amplitude of $\frac{g}{\omega^2}$✓
4. D.for an amplitude of $\frac{g^2}{\omega^2}$

Correct answer: C

An electric bulb is rated 220 volt - 100 watt. The power consumed by it when operated on 110 volt will be

1. A.50 watt
2. B.75 watt
3. C.40 watt
4. D.25 watt✓