AIEEE 2010 — 25 Apr Offline Set D (Previous Year Paper)

The standard enthalpy of formation of $NH_3$ is $-46.0$ kJ mol$^{-1}$. If the enthalpy of formation of $H_2$ from its atoms is $-436$ kJ mol$^{-1}$ and that of $N_2$ is $-712$ kJ mol$^{-1}$, the average bond enthalpy of N-H bond in $NH_3$ is

1. A.$-964$ kJ mol$^{-1}$
2. B.$+352$ kJ mol$^{-1}$✓
3. C.$+1056$ kJ mol$^{-1}$
4. D.$-1102$ kJ mol$^{-1}$

Correct answer: B

The time for half life period of a certain reaction $A \rightarrow$ products is 1 hour. When the initial concentration of the reactant 'A', is 2.0 mol L$^{-1}$, how much time does it take for its concentration to come from 0.50 to 0.25 mol L$^{-1}$ if it is a zero order reaction ?

1. A.4 h
2. B.0.5 h
3. C.0.25 h✓
4. D.1 h

Correct answer: C

A solution containing 2.675 g of $CoCl_3\cdot6NH_3$ (molar mass = 267.5 g mol$^{-1}$) is passed through a cation exchanger. The chloride ions obtained in solution were treated with excess of $AgNO_3$ to give 4.78 g of AgCl (molar mass = 143.5 g mol$^{-1}$). The formula of the complex is (At. Mass of Ag = 108 u)

1. A.$[Co(NH_3)_6]Cl_3$✓
2. B.$[CoCl_2(NH_3)_4]Cl$
3. C.$[CoCl_3(NH_3)_3]$
4. D.$[CoCl(NH_3)_5]Cl_2$

Correct answer: A

Consider the reaction : $Cl_2(aq) + H_2S(aq) \rightarrow S(s) + 2H^+(aq) + 2Cl^-(aq)$. The rate equation for this reaction is rate = $k[Cl_2][H_2S]$. Which of these mechanisms is/are consistent with this rate equation ? (A) $Cl_2 + H_2S \rightarrow H^+ + Cl^- + Cl^+ + HS^-$ (slow); $Cl^+ + HS^- \rightarrow H^+ + Cl^- + S$ (fast). (B) $H_2S \rightleftharpoons H^+ + HS^-$ (fast equilibrium); $Cl_2 + HS^- \rightarrow 2Cl^- + H^+ + S$ (slow)

1. A.B only
2. B.Both A and B
3. C.Neither A nor B
4. D.A only✓

Correct answer: D

If $10^{-4}$ dm$^3$ of water is introduced into a 1.0 dm$^3$ flask to 300 K, how many moles of water are in the vapour phase when equilibrium is established ? (Given : Vapour pressure of $H_2O$ at 300 K is 3170 Pa ; $R = 8.314$ J K$^{-1}$ mol$^{-1}$)

1. A.$5.56\times10^{-3}$ mol
2. B.$1.53\times10^{-2}$ mol
3. C.$4.46\times10^{-2}$ mol
4. D.$1.27\times10^{-3}$ mol✓

Correct answer: D

One mole of a symmetrical alkene on ozonolysis gives two moles of an aldehyde having a molecular mass of 44 u. The alkene is

1. A.propene
2. B.1-butene
3. C.2-butene✓
4. D.ethene

Correct answer: C

If sodium sulphate is considered to be completely dissociated into cations and anions in aqueous solution, the change in freezing point of water $(\Delta T_f)$, when 0.01 mol of sodium sulphate is dissolved in 1 kg of water, is ($K_f$ = 1.86 K kg mol$^{-1}$)

1. A.0.0372 K
2. B.0.0558 K✓
3. C.0.0744 K
4. D.0.0186 K

Correct answer: B

From amongst the following alcohols the one that would react fastest with conc. HCl and anhydrous $ZnCl_2$, is

1. A.2-Butanol
2. B.2-Methylpropan-2-ol✓
3. C.2-Methylpropanol
4. D.1-Butanol

Correct answer: B

In the chemical reactions, aniline $\xrightarrow{NaNO_2,\ HCl,\ 278\ K}$ A $\xrightarrow{HBF_4}$ B, the compounds 'A' and 'B' respectively are

1. A.nitrobenzene and fluorobenzene
2. B.phenol and benzene
3. C.benzene diazonium chloride and fluorobenzene✓
4. D.nitrobenzene and chlorobenzene

Correct answer: C

29.5 mg of an organic compound containing nitrogen was digested according to Kjeldahl's method and the evolved ammonia was absorbed in 20 mL of 0.1 M HCl solution. The excess of the acid required 15 mL of 0.1 M NaOH solution for complete neutralization. The percentage of nitrogen in the compound is

1. A.59.0
2. B.47.4
3. C.23.7✓
4. D.29.5

Correct answer: C

The energy required to break one mole of Cl-Cl bonds in $Cl_2$ is 242 kJ mol$^{-1}$. The longest wavelength of light capable of breaking a single Cl - Cl bond is ($c = 3\times10^8$ ms$^{-1}$ and $N_A = 6.02\times10^{23}$ mol$^{-1}$)

1. A.594 nm
2. B.640 nm
3. C.700 nm
4. D.494 nm✓

Correct answer: D

Ionisation energy of $He^+$ is $19.6\times10^{-18}$ J atom$^{-1}$. The energy of the first stationary state (n = 1) of $Li^{2+}$ is

1. A.$4.41\times10^{-16}$ J atom$^{-1}$
2. B.$-4.41\times10^{-17}$ J atom$^{-1}$✓
3. C.$-2.2\times10^{-15}$ J atom$^{-1}$
4. D.$8.82\times10^{-17}$ J atom$^{-1}$

Correct answer: B

Consider the following bromides : (A) 1-bromobutane ($CH_3CH_2CH_2CH_2Br$), (B) 3-bromobut-1-ene ($CH_2=CHCH(Br)CH_3$), (C) 2-bromobutane ($CH_3CH_2CH(Br)CH_3$). The correct order of $S_N1$ reactivity is

1. A.B > C > A✓
2. B.B > A > C
3. C.C > B > A
4. D.A > B > C

Correct answer: A

Which one of the following has an optical isomer ? (en = ethylenediamine)

1. A.$[Zn(en)(NH_3)_2]^{2+}$
2. B.$[Co(en)_3]^{3+}$✓
3. C.$[Co(H_2O)_4(en)]^{3+}$
4. D.$[Zn(en)_2]^{2+}$

Correct answer: B

On mixing, heptane and octane form an ideal solution. At 373 K, the vapour pressures of the two liquid components (heptane and octane) are 105 kPa and 45 kPa respectively. Vapour pressure of the solution obtained by mixing 25.0g of heptane and 35 g of octane will be (molar mass of heptane = 100 g mol$^{-1}$ and of octane = 114 g mol$^{-1}$).

1. A.72.0 kPa✓
2. B.36.1 kPa
3. C.96.2 kPa
4. D.144.5 kPa

Correct answer: A

The main product of the following reaction is $C_6H_5CH_2CH(OH)CH(CH_3)_2 \xrightarrow{conc.\ H_2SO_4}$ ?

1. A.trans-$C_6H_5CH=CHCH(CH_3)_2$✓
2. B.$C_6H_5CH_2CH=C(CH_3)_2$
3. C.cis-$C_6H_5CH=CHCH(CH_3)_2$
4. D.$C_6H_5CH_2CH_2C(CH_3)=CH_2$

Correct answer: A

Three reactions involving $H_2PO_4^-$ are given below : (i) $H_3PO_4 + H_2O \rightarrow H_3O^+ + H_2PO_4^-$ (ii) $H_2PO_4^- + H_2O \rightarrow HPO_4^{2-} + H_3O^+$ (iii) $H_2PO_4^- + OH^- \rightarrow H_3PO_4 + O^{2-}$. In which of the above does $H_2PO_4^-$ act as an acid ?

1. A.(ii) only✓
2. B.(i) and (ii)
3. C.(iii) only
4. D.(i) only

Correct answer: A

In aqueous solution the ionization constants for carbonic acid are $K_1 = 4.2\times10^{-7}$ and $K_2 = 4.8\times10^{-11}$. Select the correct statement for a saturated 0.034 M solution of the carbonic acid.

1. A.The concentration of $CO_3^{2-}$ is 0.034 M.
2. B.The concentration of $CO_3^{2-}$ is greater than that of $HCO_3^-$.
3. C.The concentration of $H^+$ and $HCO_3^-$ are approximately equal.✓
4. D.The concentration of $H^+$ is double that of $CO_3^{2-}$.

Correct answer: C

The edge length of a face centered cubic cell of an ionic substance is 508 pm. If the radius of the cation is 110 pm, the radius of the anion is

1. A.288 pm
2. B.398 pm
3. C.618 pm
4. D.144 pm✓

Correct answer: D

The correct order of increasing basicity of the given conjugate bases ($R = CH_3$) is

1. A.$RCOO^- < HC\equiv C^- < R^- < NH_2^-$
2. B.$R^- < HC\equiv C^- < RCOO^- < NH_2^-$
3. C.$RCOO^- < NH_2^- < HC\equiv C^- < R^-$
4. D.$RCOO^- < HC\equiv C^- < NH_2^- < R^-$✓

Correct answer: D

The correct sequence which shows decreasing order of the ionic radii of the elements is

1. A.$Al^{3+} > Mg^{2+} > Na^+ > F^- > O^{2-}$
2. B.$Na^+ > Mg^{2+} > Al^{3+} > O^{2-} > F^-$
3. C.$Na^+ > F^- > Mg^{2+} > O^{2-} > Al^{3+}$
4. D.$O^{2-} > F^- > Na^+ > Mg^{2+} > Al^{3+}$✓

Correct answer: D

Solubility product of silver bromide is $5.0\times10^{-13}$. The quantity of potassium bromide (molar mass taken as 120 g mol$^{-1}$) to be added to 1 litre of 0.05 M solution of silver nitrate to start the precipitation of AgBr is

1. A.$1.2\times10^{-10}$ g
2. B.$1.2\times10^{-9}$ g✓
3. C.$6.2\times10^{-5}$ g
4. D.$5.0\times10^{-8}$ g

Correct answer: B

The Gibbs energy for the decomposition of $Al_2O_3$ at $500^\circ$C is as follows : $\frac{2}{3}Al_2O_3 \rightarrow \frac{4}{3}Al + O_2$, $\Delta_rG = +966$ kJ mol$^{-1}$. The potential difference needed for electrolytic reduction of $Al_2O_3$ at $500^\circ$C is at least

1. A.4.5 V
2. B.3.0 V
3. C.2.5 V✓
4. D.5.0 V

Correct answer: C

At $25^\circ$C, the solubility product of $Mg(OH)_2$ is $1.0\times10^{-11}$. At which pH, will $Mg^{2+}$ ions start precipitating in the form of $Mg(OH)_2$ from a solution of 0.001 M $Mg^{2+}$ ions ?

1. A.9
2. B.10✓
3. C.11
4. D.8

Correct answer: B

Percentage of free space in cubic close packed structure and in body centred packed structure are respectively

1. A.30% and 26%
2. B.26% and 32%✓
3. C.32% and 48%
4. D.48% and 26%

Correct answer: B

Out of the following, the alkene that exhibits optical isomerism is

1. A.3-methyl-2-pentene
2. B.4-methyl-1-pentene
3. C.3-methyl-1-pentene✓
4. D.2-methyl-2-pentene

Correct answer: C

Biuret test is not given by

1. A.carbohydrates✓
2. B.polypeptides
3. C.urea
4. D.proteins

Correct answer: A

The correct order of $E^\circ_{M^{2+}/M}$ values with negative sign for the four successive elements Cr, Mn, Fe and Co is

1. A.Mn > Cr > Fe > Co✓
2. B.Cr > Fe > Mn > Co
3. C.Fe > Mn > Cr > Co
4. D.Cr > Mn > Fe > Co

Correct answer: A

The polymer containing strong intermolecular forces e.g. hydrogen bonding, is

1. A.teflon
2. B.nylon 6,6✓
3. C.polystyrene
4. D.natural rubber

Correct answer: B

For a particular reversible reaction at temperature T, $\Delta H$ and $\Delta S$ were found to be both +ve. If $T_e$ is the temperature at equilibrium, the reaction would be spontaneous when

1. A.$T_e > T$
2. B.$T > T_e$✓
3. C.$T_e$ is 5 times T
4. D.$T = T_e$

Correct answer: B

A rectangular loop has a sliding connector PQ of length $\ell$ and resistance R $\Omega$ and it is moving with a speed v as shown. The set-up is placed in a uniform magnetic field going into the plane of the paper. The three currents $I_1$, $I_2$ and I are

1. A.$I_1 = -I_2 = \frac{B\ell v}{R}$, $I = \frac{2B\ell v}{R}$
2. B.$I_1 = I_2 = \frac{B\ell v}{3R}$, $I = \frac{2B\ell v}{3R}$✓
3. C.$I_1 = I_2 = I = \frac{B\ell v}{R}$
4. D.$I_1 = I_2 = \frac{B\ell v}{6R}$, $I = \frac{B\ell v}{3R}$

Correct answer: B

Let C be the capacitance of a capacitor discharging through a resistor R. Suppose $t_1$ is the time taken for the energy stored in the capacitor to reduce to half its initial value and $t_2$ is the time taken for the charge to reduce to one-fourth its initial value. Then the ratio $t_1/t_2$ will be

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

Correct answer: C

Statement-1 : Two particles moving in the same direction do not lose all their energy in a completely inelastic collision. Statement-2 : Principle of conservation of momentum holds true for all kinds of collisions.

1. A.Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation of Statement-1.✓
2. B.Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation of Statement-1.
3. C.Statement-1 is false, Statement-2 is true.
4. D.Statement-1 is true, Statement-2 is false.

Correct answer: A

Statement-1 : When ultraviolet light is incident on a photocell, its stopping potential is $V_0$ and the maximum kinetic energy of the photoelectrons is $K_{max}$. When the ultraviolet light is replaced by X-rays, both $V_0$ and $K_{max}$ increase. Statement-2 : Photoelectrons are emitted with speeds ranging from zero to a maximum value because of the range of frequencies present in the incident light.

1. A.Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation of Statement-1.
2. B.Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation of Statement-1.
3. C.Statement-1 is false, Statement-2 is true.
4. D.Statement-1 is true, Statement-2 is false.✓

Correct answer: D

A ball is made of a material of density $\rho$ where $\rho_{oil} < \rho < \rho_{water}$ with $\rho_{oil}$ and $\rho_{water}$ representing the densities of oil and water, respectively. The oil and water are immiscible. If the above ball is in equilibrium in a mixture of this oil and water, which of the following pictures represents its equilibrium position ?

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

Correct answer: B

A particle is moving with velocity $\vec{v} = K(y\hat{i} + x\hat{j})$, where K is a constant. The general equation for its path is

1. A.$y = x^2 +$ constant
2. B.$y^2 = x +$ constant
3. C.$xy$ = constant
4. D.$y^2 = x^2 +$ constant✓

Correct answer: D

Two long parallel wires are at a distance 2d apart. They carry steady equal current flowing out of the plane of the paper as shown. The variation of the magnetic field along the line XX' is given by

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

Correct answer: A

In the circuit shown below, the key K is closed at t = 0. The current through the battery is

1. A.$\frac{VR_1R_2}{\sqrt{R_1^2+R_2^2}}$ at t = 0 and $\frac{V}{R_2}$ at $t = \infty$
2. B.$\frac{V}{R_2}$ at t = 0 and $\frac{V(R_1+R_2)}{R_1R_2}$ at $t = \infty$✓
3. C.$\frac{V}{R_2}$ at t = 0 and $\frac{VR_1R_2}{\sqrt{R_1^2+R_2^2}}$ at $t = \infty$
4. D.$\frac{V(R_1+R_2)}{R_1R_2}$ at t = 0 and $\frac{V}{R_2}$ at $t = \infty$

Correct answer: B

The figure shows the position - time (x - t) graph of one-dimensional motion of a body of mass 0.4 kg. The magnitude of each impulse is

1. A.0.4 Ns
2. B.0.8 Ns✓
3. C.1.6 Ns
4. D.0.2 Ns

Correct answer: B

A nucleus of mass $M + \Delta m$ is at rest and decays into two daughter nuclei of equal mass $\frac{M}{2}$ each. Speed of light is c. The binding energy per nucleon for the parent nucleus is $E_1$ and that for the daughter nuclei is $E_2$. Then

1. A.$E_2 = 2E_1$
2. B.$E_1 > E_2$
3. C.$E_2 > E_1$✓
4. D.$E_1 = 2E_2$

Correct answer: C

A nucleus of mass $M + \Delta m$ is at rest and decays into two daughter nuclei of equal mass $\frac{M}{2}$ each. Speed of light is c. The speed of daughter nuclei is

1. A.$c\frac{\Delta m}{M+\Delta m}$
2. B.$c\sqrt{\frac{2\Delta m}{M}}$✓
3. C.$c\sqrt{\frac{\Delta m}{M}}$
4. D.$c\sqrt{\frac{\Delta m}{M+\Delta m}}$

Correct answer: B

A radioactive nucleus (initial mass number A and atomic number Z) emits 3 $\alpha$-particles and 2 positrons. The ratio of number of neutrons to that of protons in the final nucleus will be

1. A.$\frac{A-Z-8}{Z-4}$
2. B.$\frac{A-Z-4}{Z-8}$✓
3. C.$\frac{A-Z-12}{Z-4}$
4. D.$\frac{A-Z-4}{Z-2}$

Correct answer: B

A thin semi-circular ring of radius r has a positive charge q distributed uniformly over it. The net field $\vec{E}$ at the centre O is

1. A.$\frac{q}{4\pi^2\epsilon_0r^2}\hat{j}$
2. B.$-\frac{q}{4\pi^2\epsilon_0r^2}\hat{j}$
3. C.$-\frac{q}{2\pi^2\epsilon_0r^2}\hat{j}$✓
4. D.$\frac{q}{2\pi^2\epsilon_0r^2}\hat{j}$

Correct answer: C

The combination of gates shown below yields

1. A.OR gate✓
2. B.NOT gate
3. C.XOR gate
4. D.NAND gate

Correct answer: A

A diatomic ideal gas is used in a Car engine as the working substance. If during the adiabatic expansion part of the cycle, volume of the gas increases from V to 32V the efficiency of the engine is

1. A.0.5
2. B.0.75✓
3. C.0.99
4. D.0.25

Correct answer: B

If a source of power 4 kW produces $10^{20}$ photons/second, the radiation belong to a part of the spectrum called

1. A.X-rays✓
2. B.ultraviolet rays
3. C.microwaves
4. D.$\gamma$-rays

Correct answer: A

The respective number of significant figures for the numbers 23.023, 0.0003 and $2.1\times10^{-3}$ are

1. A.5, 1, 2✓
2. B.5, 1, 5
3. C.5, 5, 2
4. D.4, 4, 2

Correct answer: A

In a series LCR circuit R = 200 $\Omega$ and the voltage and the frequency of the main supply is 220 V and 50 Hz respectively. On taking out the capacitance from the circuit the current lags behind the voltage by $30^\circ$. On taking out the inductor from the circuit the current leads the voltage by $30^\circ$. The power dissipated in the LCR circuit is

1. A.305 W
2. B.210 W
3. C.Zero W
4. D.242 W✓

Correct answer: D

Let there be a spherically symmetric charge distribution with charge density varying as $\rho(r) = \rho_0\left(\frac{5}{4} - \frac{r}{R}\right)$ upto r = R, and $\rho(r) = 0$ for r > R, where r is the distance from the origin. The electric field at a distance r (r < R) from the origin is given by

1. A.$\frac{4\pi\rho_0r}{3\epsilon_0}\left(\frac{5}{3}-\frac{r}{R}\right)$
2. B.$\frac{\rho_0r}{4\epsilon_0}\left(\frac{5}{3}-\frac{r}{R}\right)$✓
3. C.$\frac{4\rho_0r}{3\epsilon_0}\left(\frac{5}{4}-\frac{r}{R}\right)$
4. D.$\frac{\rho_0r}{3\epsilon_0}\left(\frac{5}{4}-\frac{r}{R}\right)$

Correct answer: B

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $U(x) = \frac{a}{x^{12}} - \frac{b}{x^6}$, where a and b are constants and x is the distance between the atoms. If the dissociation energy of the molecule is $D = [U(x = \infty) - U_{at\ equilibrium}]$, D is

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

Correct answer: C

Two conductors have the same resistance at $0^\circ$C but their temperature coefficients of resistance are $\alpha_1$ and $\alpha_2$. The respective temperature coefficients of their series and parallel combinations are nearly

1. A.$\frac{\alpha_1+\alpha_2}{2}$, $\alpha_1+\alpha_2$
2. B.$\alpha_1+\alpha_2$, $\frac{\alpha_1+\alpha_2}{2}$
3. C.$\alpha_1+\alpha_2$, $\frac{\alpha_1\alpha_2}{\alpha_1+\alpha_2}$
4. D.$\frac{\alpha_1+\alpha_2}{2}$, $\frac{\alpha_1+\alpha_2}{2}$✓

Correct answer: D

A point P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of 'P' is such that it sweeps out a length $s = t^3 + 5$, where s is in metres and t is in seconds. The radius of the path is 20 m. The acceleration of 'P' when t = 2 s is nearly

1. A.13 m/s$^2$
2. B.12 m/s$^2$
3. C.7.2 m/s$^2$
4. D.14 m/s$^2$✓

Correct answer: D

Two fixed frictionless inclined plane making an angle $30^\circ$ and $60^\circ$ with the vertical are shown in the figure. Two block A and B are placed on the two planes. What is the relative vertical acceleration of A with respect to B ?

1. A.4.9 ms$^{-2}$ in horizontal direction
2. B.9.8 ms$^{-2}$ in vertical direction
3. C.zero
4. D.4.9 ms$^{-2}$ in vertical direction✓

Correct answer: D

For a particle in uniform circular motion the acceleration $\vec{a}$ at a point $P(R, \theta)$ on the circle of radius R is (here $\theta$ is measured from the x-axis)

1. A.$-\frac{v^2}{R}\cos\theta\,\hat{i} + \frac{v^2}{R}\sin\theta\,\hat{j}$
2. B.$-\frac{v^2}{R}\sin\theta\,\hat{i} + \frac{v^2}{R}\cos\theta\,\hat{j}$
3. C.$-\frac{v^2}{R}\cos\theta\,\hat{i} - \frac{v^2}{R}\sin\theta\,\hat{j}$✓
4. D.$\frac{v^2}{R}\hat{i} + \frac{v^2}{R}\hat{j}$

Correct answer: C

An initially parallel cylindrical beam travels in a medium of refractive index $\mu(I) = \mu_0 + \mu_2I$, where $\mu_0$ and $\mu_2$ are positive constants and I is the intensity of the light beam. The intensity of the beam is decreasing with increasing radius. As the beam enters the medium, it will

1. A.diverge
2. B.converge✓
3. C.diverge near the axis and converge near the periphery
4. D.travel as a cylindrical beam

Correct answer: B

An initially parallel cylindrical beam travels in a medium of refractive index $\mu(I) = \mu_0 + \mu_2I$, where $\mu_0$ and $\mu_2$ are positive constants and I is the intensity of the light beam. The intensity of the beam is decreasing with increasing radius. The initial shape of the wave front of the beam is

1. A.convex
2. B.concave
3. C.convex near the axis and concave near the periphery
4. D.planar✓

Correct answer: D

An initially parallel cylindrical beam travels in a medium of refractive index $\mu(I) = \mu_0 + \mu_2I$, where $\mu_0$ and $\mu_2$ are positive constants and I is the intensity of the light beam. The intensity of the beam is decreasing with increasing radius. The speed of light in the medium is

1. A.minimum on the axis of the beam✓
2. B.the same everywhere in the beam
3. C.directly proportional to the intensity I
4. D.maximum on the axis of the beam

Correct answer: A

A small particle of mass m is projected at an angle $\theta$ with the x-axis with an initial velocity $v_0$ in the x-y plane as shown in the figure. At a time $t < \frac{v_0\sin\theta}{g}$, the angular momentum of the particle is (where $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors along x, y and z-axis respectively)

1. A.$-mgv_0t^2\cos\theta\,\hat{j}$
2. B.$mgv_0t\cos\theta\,\hat{k}$
3. C.$-\frac{1}{2}mgv_0t^2\cos\theta\,\hat{k}$✓
4. D.$\frac{1}{2}mgv_0t^2\cos\theta\,\hat{i}$

Correct answer: C

The equation of a wave on a string of linear mass density 0.04 kg m$^{-1}$ is given by $y = 0.02\,(m)\sin\left[2\pi\left(\frac{t}{0.04\,(s)} - \frac{x}{0.50\,(m)}\right)\right]$. The tension in the string is

1. A.4.0 N
2. B.12.5 N
3. C.0.5 N
4. D.6.25 N✓

Correct answer: D

Let $\cos(\alpha + \beta) = \frac{4}{5}$ and let $\sin(\alpha - \beta) = \frac{5}{13}$, where $0 \le \alpha, \beta \le \frac{\pi}{4}$, then $\tan2\alpha$ =

1. A.$\frac{56}{33}$✓
2. B.$\frac{19}{12}$
3. C.$\frac{20}{7}$
4. D.$\frac{25}{16}$

Correct answer: A

Let S be a non-empty subset of R. Consider the following statement: P: There is a rational number $x \in S$ such that $x > 0$. Which of the following statements is the negation of the statement P ?

1. A.There is no rational number $x \in S$ such that $x \le 0$
2. B.Every rational number $x \in S$ satisfies $x \le 0$✓
3. C.$x \in S$ and $x \le 0 \Rightarrow x$ is not rational
4. D.There is a rational number $x \in S$ such that $x \le 0$

Correct answer: B

Let $\vec{a} = \hat{j} - \hat{k}$ and $\vec{c} = \hat{i} - \hat{j} - \hat{k}$. Then vector $\vec{b}$ satisfying $\vec{a}\times\vec{b} + \vec{c} = \vec{0}$ and $\vec{a}\cdot\vec{b} = 3$ is

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