JEE Advanced 2015 · 24 May 2015 · Paper 2

Q1.A large spherical mass $M$ is fixed at one position and two identical point masses $m$ are kept on a line passing through the centre of $M$ (see figure). The point masses are connected by a rigid massless rod of length $l$ and this assembly is free to move along the line connecti …

Q2.The energy of a system as a function of time $t$ is given as $E(t)=A^2\exp (-\alpha t)$, where $\alpha =0.2{\mathrm{s}}^{-1}$. The measurement of $A$ has an error of 1.25%. If the error in the measurement of time is 1.50%, the percentage error in the value of $E(t)$ at …

Q3.The densities of two solid spheres $A$ and $B$ of the same radii $R$ vary with radial distance $r$ as ${\rho }_A(r)=k\left(\frac{r}{R}\right)$ and ${\rho }_B(r)=k{\left(\frac{r}{R}\right)}^5$, respectively, where $k$ is a constant. The moments of inertia of the individual spheres a …

Q4.Four harmonic waves of equal frequencies and equal intensities $I_0$ have phase angles $0,\pi /3$, $2\pi /3$ and $\pi$. When they are superposed, the intensity of the resulting wave is $n{I}_0$. The value of $n$ is

Q5.For a radioactive material, its activity $A$ and rate of change of its activity $R$ are defined as $A=-\frac{dN}{dt}$ and $R=-\frac{dA}{dt}$, where $N(t)$ is the number of nuclei at time $t$. Two radioactive sources $P$ (mean life $\tau$) and $Q$ (mean life $2\tau$) have the …

Q6.A monochromatic beam of light is incident at 60° on one face of an equilateral prism of refractive index $n$ and emerges from the opposite face making an angle $\theta (n)$ with the normal (see the figure). For $n=\sqrt{3}$ the value of $\theta$ is $60^{\circ }$ and …

Q7.In the following circuit, the current through the resistor $R(=2\Omega )$ is $I$ Amperes. The value of $I$ is

Q8.An electron in an excited state of ${\mathrm{Li}}^{2+}$ ion has angular momentum $3h/2\pi$. The de Broglie wavelength of the electron in this state is $p\pi a_0$ (where $a_0$ is the Bohr radius). The value of $p$ is

Q9.Two spheres P and Q of equal radii have densities ${\rho }_1$ and ${\rho }_2$, respectively. The spheres are connected by a massless string and placed in liquids $L_1$ and $L_2$ of densities ${\sigma }_1$ and ${\sigma }_2$ and viscosities ${\eta }_1$ and ${\eta }_2$, respectively. They float in eq …

Q10.In terms of potential difference V, electric current I, permittivity ${\epsilon }_0$, permeability ${\mu }_0$ and speed of light c, the dimensionally correct equation(s) is(are)

Q11.Consider a uniform spherical charge distribution of radius $R_1$ centred at the origin O. In this distribution, a spherical cavity of radius $R_2$, centred at P with distance $OP=a=R_1-R_2$ (see figure) is made. If the electric field inside the cavity at position $\vec{r}$ …

Q12.In plotting stress versus strain curves for the materials P and Q, a student by mistake puts strain on the y-axis and stress on the x-axis as shown in the figure. Then the correct statement(s) is(are)

Q13.A spherical body of radius R consists of a fluid of constant density and is in equilibrium under its own gravity. If $P(r)$ is the pressure at $r$ $(r<R)$, then the correct option(s) is(are)

Q14.A parallel plate capacitor having plates of area S and plate separation d, has capacitance $C_1$ in air. When two dielectrics of different relative permittivities (${\epsilon }_1=2$ and ${\epsilon }_2=4$) are introduced between the two plates as shown in the figure, the capacitan …

Q15.An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the figure). Initially the gas is at temperature $T_1$, pressure $P_1$ and volume $V_1$ and the spring is in its relaxed state. The gas is then heated very slowly to temperature …

Q16.A fission reaction is given by ${}_{92}^{236}\mathrm{U}{\to }_{54}^{140}\mathrm{Xe}{+}_{38}^{94}\mathrm{Sr}+\mathrm{x}+\mathrm{y}$, where x and y are two particles. Considering ${}_{92}^{236}\mathrm{U}$ to be at rest, the kinetic energies of the products are denoted by …

Q17.PARAGRAPH 1

In a thin rectangular metallic strip a constant current I flows along the positive x-direction, as shown in the figure. The length, width and thickness of the strip are l, w and d, respectively.

A uniform magnetic field B is applied on the strip along the positive y …

Q18.PARAGRAPH 1

In a thin rectangular metallic strip a constant current I flows along the positive x-direction, as shown in the figure. The length, width and thickness of the strip are l, w and d, respectively.

A uniform magnetic field B is applied on the strip along the positive y …

Q19.PARAGRAPH 2

Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index $n_1$ surrounded by a medium of lower refractive index $n_2$. The light guidance in the structure takes place due to su …

Q20.PARAGRAPH 2

Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index $n_1$ surrounded by a medium of lower refractive index $n_2$. The light guidance in the structure takes place due to su …

Q1.The number of hydroxyl group(s) in Q is

Q2.Among the following, the number of reaction(s) that produce(s) benzaldehyde is

I.

II.

III.

IV.

Q3.In the complex acetylbromidodicarbonylbis(triethylphosphine)iron(II), the number of Fe-C bond(s) is

Q4.Among the complex ions, [Co(NH₂-CH₂-CH₂-NH₂)₂Cl₂]⁺, [CrCl₂(C₂O₄)₂]³⁻, [Fe(H₂O)₄(OH)₂]⁺, [Fe(NH₃)₂(CN)₄]⁻, [Co(NH₂-CH₂-CH₂-NH₂)₂(NH₃)Cl]²⁺ and [Co(NH₃)₄(H₂O)Cl]²⁺, the number of complex ion(s) that show(s) cis-trans isomerism is

Q5.Three moles of B₂H₆ are completely reacted with methanol. The number of moles of boron containing product formed is

Q6.The molar conductivity of a solution of a weak acid HX (0.01 M) is 10 times smaller than the molar conductivity of a solution of a weak acid HY (0.10 M). If λ⁰ₓ₋ ≈ λ⁰ᵧ₋, the difference in their pKₐ values, pKₐ(HX) - pKₐ(HY), is (consider degree of ionization of both acids to be < …

Q7.A closed vessel with rigid walls contains 1 mol of ²³⁸₉₂U and 1 mol of air at 298 K. Considering complete decay of ²³⁸₉₂U to ²⁰⁶₈₂Pb, the ratio of the final pressure to the initial pressure of the system at 298 K is

Q8.In dilute aqueous H₂SO₄, the complex diaquodioxalatoferrate(II) is oxidized by MnO₄⁻. For this reaction, the ratio of the rate of change of [H⁺] to the rate of change of [MnO₄⁻] is

Q9.In the following reactions, the product S is

Q11.In the following reactions, the major product W is

Q12.The correct statement(s) regarding, (i) HClO, (ii) HClO₂, (iii) HClO₃ and (iv) HClO₄, is(are)

Q13.The pair(s) of ions where BOTH the ions are precipitated upon passing H₂S gas in presence of dilute HCl, is(are)

Q14.Under hydrolytic conditions, the compounds used for preparation of linear polymer and for chain termination, respectively, are

Q15.When O₂ is adsorbed on a metallic surface, electron transfer occurs from the metal to O₂. The TRUE statement(s) regarding this adsorption is(are)

Q16.One mole of a monoatomic real gas satisfies the equation p(V-b) = RT where b is a constant. The relationship of interatomic potential V(r) and interatomic distance r for the gas is given by

Q19.PARAGRAPH 2

When 100 mL of 1.0 M HCl was mixed with 100 mL of 1.0 M NaOH in an insulated beaker at constant pressure, a temperature increase of 5.7 °C was measured for the beaker and its contents (Expt. 1). Because the enthalpy of neutralization of a strong acid with a strong ba …

Q20.PARAGRAPH 2

When 100 mL of 1.0 M HCl was mixed with 100 mL of 1.0 M NaOH in an insulated beaker at constant pressure, a temperature increase of 5.7 °C was measured for the beaker and its contents (Expt. 1). Because the enthalpy of neutralization of a strong acid with a strong ba …

Q1.For any integer $k$, let ${\alpha }_k=\cos \left(\frac{k\pi }{7}\right)+i\sin \left(\frac{k\pi }{7}\right)$, where $i=\sqrt{-1}$. The value of the expression …

Q2.Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6:11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

Q3.The coefficient of $x^9$ in the expansion of $(1+x)\left(1+x^2\right)\left(1+x^3\right)\dots \left(1+x^{100}\right)$ is

Q4.Suppose that the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$ are $\left(f_1,0\right)$ and $\left(f_2,0\right)$ where $f_1>0$ and $f_2<0$. Let $P_1$ and $P_2$ be two parabolas with a common vertex at (0, 0) and with foci at …

Q5.Let $m$ and $n$ be two positive integers greater than 1. If

then the value of $\frac{m}{n}$ is

Q6.If

where ${\tan }^{-1}x$ takes only principal values, then the value of $\left({\log }_e|1+\alpha |-\frac{3\pi }{4}\right)$ is

Q7.Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous odd function, which vanishes exactly at one point and $f(1)=\frac{1}{2}$. Suppose that $F(x)={\int }_{-1}^{x}f(t)dt$ for all $x\in [-1,2]$ and $G(x)={\int }_{-1}^{x}t|f(f(t))|dt$ for all $x\in [-1,2]$. If …

Q8.Suppose that $\vec{p},\vec{q}$ and $\vec{r}$ are three non-coplanar vectors in ${\mathbb{R}}^3$. Let the components of a vector $\vec{s}$ along $\vec{p},\vec{q}$ and $\vec{r}$ be 4, 3 and 5, respectively. If the components of this vector $\vec{s}$ along …

Q9.Let $S$ be the set of all non-zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2-x+\alpha =0$ has two distinct real roots $x_1$ and $x_2$ satisfying the inequality $\left|x_1-x_2\right|<1$. Which of the following intervals is(are) a subse …

Q10.If $\alpha =3{\sin }^{-1}\left(\frac{6}{11}\right)$ and $\beta =3{\cos }^{-1}\left(\frac{4}{9}\right)$, where the inverse trigonometric functions take only the principal values, then the correct option(s) is(are)

Q11.Let $E_1$ and $E_2$ be two ellipses whose centers are at the origin. The major axes of $E_1$ and $E_2$ lie along the $x$-axis and the $y$-axis, respectively. Let $S$ be the circle $x^2+(y-1{)}^2=2$. The straight line $x+y=3$ touches the curves …

Q12.Consider the hyperbola $H:x^2-y^2=1$ and a circle $S$ with center $N\left(x_2,0\right)$. Suppose that $H$ and $S$ touch each other at a point $P\left(x_1,y_1\right)$ with $x_1>1$ and $y_1>0$. The common tangent to $H$ and $S$ at $P$ intersec …

Q13.The option(s) with the values of $a$ and $L$ that satisfy the following equation is(are)

Q14.Let $f,g:[-1,2]\to \mathbb{R}$ be continuous functions which are twice differentiable on the interval $(-1,2)$. Let the values of $f$ and $g$ at the points $-1,0$ and 2 be as given in the following table:
| | $x=-1$ | $x=0$ | $x=2$ || --- | --- | --- | …

Q15.Let $f(x)=7{\tan }^{8}x+7{\tan }^{6}x-3{\tan }^{4}x-3{\tan }^{2}x$ for all $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the correct expression(s) is(are)

Q16.Let $f^{\prime }(x)=\frac{192x^3}{2+{\sin }^4\pi x}$ for all $x\in \mathbb{R}$ with $f\left(\frac{1}{2}\right)=0$. If $m\le {\int }_{1/2}^{1}f(x)dx\le M$, then the possible values of $m$ and $M$ are

Q17.PARAGRAPH 1
Let $n_1$ and $n_2$ be the number of red and black balls, respectively, in box I. Let $n_3$ and $n_4$ be the number of red and black balls, respectively, in box II.

One of the two boxes, box I and box II, was selected at random and a ball was drawn ra …

Q18.PARAGRAPH 1
Let $n_1$ and $n_2$ be the number of red and black balls, respectively, in box I. Let $n_3$ and $n_4$ be the number of red and black balls, respectively, in box II.

A ball is drawn at random from box I and transferred to box II. If the probability of …

Q19.PARAGRAPH 2
Let $F:\mathbb{R}\to \mathbb{R}$ be a thrice differentiable function. Suppose that $F(1)=0,F(3)=-4$ and $F^{\prime }(x)<0$ for all $x\in (1/2,3)$. Let $f(x)=xF(x)$ for all $x\in \mathbb{R}$.

The correct statement(s) is(are)

Q20.PARAGRAPH 2
Let $F:\mathbb{R}\to \mathbb{R}$ be a thrice differentiable function. Suppose that $F(1)=0,F(3)=-4$ and $F^{\prime }(x)<0$ for all $x\in (1/2,3)$. Let $f(x)=xF(x)$ for all $x\in \mathbb{R}$.

If …