JEE Advanced 2014 · 25 May 2014 · Paper 1

Q1.At time $t=0$, terminal A in the circuit shown in the figure is connected to B by a key and an alternating current $I(t)=I_0\cos (\omega t)$, with $I_0=1$ A and $\omega =500\mathrm{rad}{\mathrm{s}}^{-1}$ starts flowing in it with the initial direction shown in the fig …

Q2.A light source, which emits two wavelengths ${\lambda }_1=400nm$ and ${\lambda }_2=600nm$, is used in a Young's double slit experiment. If recorded fringe widths for ${\lambda }_1$ and ${\lambda }_2$ are ${\beta }_1$ and ${\beta }_2$ and the number of fringes for them w …

Q3.One end of a taut string of length $3m$ along the $x$ axis is fixed at $x=0$. The speed of the waves in the string is $100m{s}^{-1}$. The other end of the string is vibrating in the $y$ direction so that stationary waves are set up in the string. The possible waveform( …

Q4.A parallel plate capacitor has a dielectric slab of dielectric constant $K$ between its plates that covers 1/3 of the area of its plates, as shown in the figure. The total capacitance of the capacitor is $C$ while that of the portion with dielectric in between is $C_1$. W …

Q5.Let $E_1(r),E_2(r)$ and $E_3(r)$ be the respective electric fields at a distance $r$ from a point charge $Q$, an infinitely long wire with constant linear charge density $\lambda$, and an infinite plane with uniform surface charge density $\sigma$. If …

Q6.A student is performing an experiment using a resonance column and a tuning fork of frequency $244s^{-1}$. He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas). If the minimum height at which resonance occurs …

Q7.Heater of an electric kettle is made of a wire of length $L$ and diameter $d$. It takes 4 minutes to raise the temperature of 0.5kg water by 40K. This heater is replaced by a new heater having two wires of the same material, each of length $L$ and diameter $2d$. The way t …

Q8.In the figure, a ladder of mass $m$ is shown leaning against a wall. It is in static equilibrium making an angle $\theta$ with the horizontal floor. The coefficient of friction between the wall and the ladder is ${\mu }_1$ and that between the floor and the ladder is …

Q9.A transparent thin film of uniform thickness and refractive index $n_1=1.4$ is coated on the convex spherical surface of radius Rat one end of a long solid glass cylinder of refractive index $n_2=1.5$, as shown in the figure. Rays of light parallel to the axis of the cyli …

Q10.Two ideal batteries of $emf{V}_1$ and $V_2$ and three resistances $R_1,R_2$ and $R_3$ are connected as shown in the figure. The current in resistance $R_2$ would be zero if

Q11.Airplanes A and B are flying with constant velocity in the same vertical plane at angles 30° and $60^{\circ }$ with respect to the horizontal respectively as shown in figure. The speed of A is $100\sqrt{3}m{s}^{-1}$. At time $t=0s$, an observer in A finds B at a distance o …

Q12.During Searle's experiment, zero of the Vernier scale lies between $3.20\times 10^{-2}m$ and $3.25\times 10^{-2}m$ of the main scale. The $20^{\text{th }}$ division of the Vernier scale exactly coincides with one of the main scale divisions. When an additional load of …

Q13.A uniform circular disc of mass $1.5kg$ and radius $0.5m$ is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude $F=0.5N$ are applied simultaneously along the three sides of an equilateral triangle XYZ with its vertices on the perimet …

Q14.Two parallel wires in the plane of the paper are distance $X_0$ apart. A point charge is moving with speed $u$ between the wires in the same plane at a distance $X_1$ from one of the wires. When the wires carry current of magnitude $I$ in the same direction, the radiu …

Q15.To find the distance $d$ over which a signal can be seen clearly in foggy conditions, a railways engineer uses dimensional analysis and assumes that the distance depends on the mass density $\rho$ of the fog, intensity (power/area) $S$ of the light from the signal and its f …

Q16.A galvanometer gives full scale deflection with $0.006A$ current. By connecting it to a 4990 $\Omega$ resistance, it can be converted into a voltmeter of range $0-30V$. If connected to a $\frac{2n}{249}\Omega$ resistance, it becomes an ammeter of range $0-1.5A$. Th …

Q17.Consider an elliptically shaped rail PQ in the vertical plane with $\mathrm{OP}=3m$ and $\mathrm{OQ}=4m$. A block of mass 1 kg is pulled along the rail from P to Q with a force of $18N$, which is always parallel to line PQ (see the figure given). Assuming no frictional …

Q18.A rocket is moving in a gravity free space with a constant acceleration of $2m{s}^{-2}$ along $+x$ direction (see figure). The length of a chamber inside the rocket is $4m$. A ball is thrown from the left end of the chamber in $+x$ direction with a speed of …

Q19.A horizontal circular platform of radius $0.5m$ and mass $0.45kg$ is free to rotate about its axis. Two massless spring toy-guns, each carrying a steel ball of mass 0.05 kg are attached to the platform at a distance $0.25m$ from the centre on its either sides along its …

Q20.A thermodynamic system is taken from an initial state $i$ with internal energy $U_i=100J$ to the final state $f$ along two different paths iaf and $ibf$, as schematically shown in the figure. The work done by the system along the paths $af,ib$ and $bf$ are …

Q1.The correct combination of names for isomeric alcohols with molecular formula ${\mathrm{C}}_4{\mathrm{H}}_{10}\mathrm{O}$ is/are

Q2.The reactivity of compound Z with different halogens under appropriate conditions is given below:

The observed pattern of electrophilic substitution can be explained by

Q3.In the reaction shown below, the major product(s) formed is/are

Q4.An ideal gas in a thermally insulated vessel at internal pressure $=P_1$, volume $=V_1$ and absolute temperature $=T_1$ expands irreversibly against zero external pressure, as shown in the diagram. The final internal pressure, volume and absolute temperature of the gas are …

Q5.Hydrogen bonding plays a central role in the following phenomena:

Q6.In a galvanic cell, the salt bridge

Q7.Upon heating with ${\mathrm{Cu}}_2\mathrm{S}$, the reagent(s) that give copper metal is/are

Q8.The correct statement(s) for orthoboric acid is/are

Q9.For the reaction:

The correct statement(s) in the balanced equation is/are:

Q10.The pair(s) of reagents that yield paramagnetic species is/are

Q11.Consider all possible isomeric ketones, including stereoisomers, of $\mathrm{MW}=100$. All these isomers are independently reacted with ${\mathrm{NaBH}}_4$ (NOTE: stereoisomers are also reacted separately). The total number of ketones that give a racemic product(s) is/are

Q12.A list of species having the formula ${\mathrm{XZ}}_4$ is given below. …

Q13.Among $\mathrm{PbS},\mathrm{CuS},\mathrm{HgS},\mathrm{MnS},{\mathrm{Ag}}_2\mathrm{S},\mathrm{NiS},\mathrm{CoS},{\mathrm{Bi}}_2{\mathrm{S}}_3$ and ${\mathrm{SnS}}_2$, the total number of BLACK coloured sulfides is

Q14.The total number(s) of stable conformers with non-zero dipole moment for the following compound is (are)

Q15.Consider the following list of reagents:

Acidified ${\mathrm{K}}_2{\mathrm{Cr}}_2{\mathrm{O}}_7$, alkaline ${\mathrm{KMnO}}_4,{\mathrm{CuSO}}_4,{\mathrm{H}}_2{\mathrm{O}}_2,{\mathrm{Cl}}_2,{\mathrm{O}}_3,{\mathrm{FeCl}}_3,{\mathrm{HNO}}_3$ and …

Q16.The total number of distinct naturally occurring amino acids obtained by complete acidic hydrolysis of the peptide shown below is

Q17.In an atom, the total number of electrons having quantum numbers $n=4,\left|m_l\right|=1$ and $m_s=-\frac{1}{2}$ is

Q18.If the value of Avogadro number is $6.023\times 10^{23}{\mathrm{mol}}^{-1}$ and the value of Boltzmann constant is $1.380\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$, then the number of significant digits in the calculated value of the universal gas constant is

Q19.A compound ${\mathbf{H}}_2\mathbf{X}$ with molar weight of $80g$ is dissolved in a solvent having density of $0.4\mathrm{g}{\mathrm{ml}}^{-1}$. Assuming no change in volume upon dissolution, the molality of a 3.2 molar solution is

Q20.${\mathbf{MX}}_2$ dissociates into ${\mathbf{M}}^{2+}$ and ${\mathbf{X}}^{-}$ ions in an aqueous solution, with a degree of dissociation $(\alpha )$ of 0.5. The ratio of the observed depression of freezing point of the aqueous solution to the value of the depression …

Q1.Let $M$ and $N$ be two $3\times 3$ matrices such that $MN=NM$. Further, if $M\ne N^2$ and $M^2=N^4$, then

Q2.For every pair of continuous functions $f,g:[0,1]\to \mathbb{R}$ such that

the correct statement(s) is(are) :

Q3.Let $f:(0,\infty )\to \mathbb{R}$ be given by

Then

Q4.Let $a\in \mathbb{R}$ and let $f:\mathbb{R}\to \mathbb{R}$ be given by

Then

Q5.Let $f:[a,b]\to [1,\infty )$ be a continuous function and let $g:\mathbb{R}\to \mathbb{R}$ be defined as …

Q6.Let $f:\left(-\frac{\pi }{2},\frac{\pi }{2}\right)\to \mathbb{R}$ be given by

Then

Q7.From a point $P(\lambda ,\lambda ,\lambda )$, perpendiculars $PQ$ and $PR$ are drawn respectively on the lines $y=x,z=1$ and $y=-x,z=-1$. If $P$ is such that $\angle QPR$ is a right angle, then the possible value(s) of $\lambda$ is(are)

Q8.Let $\vec{x},\vec{y}$ and $\vec{z}$ be three vectors each of magnitude $\sqrt{2}$ and the angle between each pair of them is $\frac{\pi }{3}$. If $\vec{a}$ is a nonzero vector perpendicular to $\vec{x}$ and $\vec{y}\times \vec{z}$ and $\vec{b}$ is a nonzero vecto …

Q9.A circle $S$ passes through the point $(0,1)$ and is orthogonal to the circles $(x-1{)}^2+y^2=16$ and $x^2+y^2=1$. Then

Q10.Let $M$ be a 2 × 2 symmetric matrix with integer entries. Then $M$ is invertible if

Q11.Let $a,b,c$ be positive integers such that $\frac{b}{a}$ is an integer. If $a,b,c$ are in geometric progression and the arithmetic mean of $a,b,c$ is $b+2$, then the value of

is

Q12.Let $n\ge 2$ be an integer. Take $n$ distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value …

Q13.Let $n_1<n_2<n_3<n_4<n_5$ be positive integers such that $n_1+n_2+n_3+n_4+n_5=20$. Then the number of such distinct arrangements $(n_1,n_2,n_3,n_4,n_5)$ is

Q14.Let $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ be respectively given by $f(x)=|x|+1$ and $g(x)=x^2+1$. Define $h:\mathbb{R}\to \mathbb{R}$ by …

Q15.The value of

is

Q16.The slope of the tangent to the curve ${\left(y-x^5\right)}^2=x{\left(1+x^2\right)}^2$ at the point $(1,3)$ is

Q17.The largest value of the nonnegative integer $a$ for which

is

Q18.Let $f:[0,4\pi ]\to [0,\pi ]$ be defined by $f(x)={\cos }^{-1}(\cos x)$. The number of points $x\in [0,4\pi ]$ satisfying the equation

Is

Q19.For a point $P$ in the plane, let $d_1(P)$ and $d_2(P)$ be the distances of the point $P$ from the lines $x-y=0$ and $x+y=0$ respectively. The area of the region $R$ consisting of all points $P$ lying in the first quadrant of the plane and satisfying …

Q20.Let $\vec{a},\vec{b}$, and $\vec{c}$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac{\pi }{3}$ . If $\vec{a}\times \vec{b}+\vec{b}\times \vec{c}=p\vec{a}+q\vec{b}+r\vec{c}$, where $p,q$ and $r$ are scalars, then the va …