JEE Advanced 2006 · 09 Apr 2006

Q1.Given,


(I)

(II)

(III)

The time constants (in $\mu \mathrm{S}$) for the circuits I, II, III are respectively

Q2.Two blocks A and B of masses 2m and m, respectively, are connected by a massless and inextensible string. The whole system is suspended by a massless spring as shown in the figure. The magnitudes of acceleration of A and B, immediately after the string is cut, are respectively

Q4.A biconvex lens of focal length f forms a circular image of sun of radius r in focal plane. Then

Q6.Graph of position of image vs position of point object from a convex lens is shown. Then, focal length of the lens is

Q7.A massless rod is suspended by two identical strings AB and CD of equal length. A block of mass m is suspended from point O such that BO is equal to 'x'. Further, it is observed that the frequency of $1^{\text{st }}$ harmonic (fundamental frequency) in AB is equal to …

Q8.A system of binary stars of masses ${\mathrm{m}}_{\mathrm{A}}$ and ${\mathrm{m}}_{\mathrm{B}}$ are moving in circular orbits of radii ${\mathrm{r}}_{\mathrm{A}}$ and ${\mathrm{r}}_{\mathrm{B}}$ respectively. If ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{T}}_{\mathrm{B}}$ are the time periods …

Q9.A solid sphere of mass M, radius R and having moment of inertia about an axis passing through the centre of mass as I, is recast into a disc of thickness t, whose moment of inertia about an axis passing through its edge and perpendicular to its plane remains I. Then, radius of th …

Q10.A student performs an experiment for determination of $\mathrm{g}\left(=\frac{4{\pi }^2\ell }{{\mathrm{T}}^2}\right),\ell \approx 1\mathrm{m}$, and he commits an error of $\Delta \ell$. For T he takes the time of n oscillations with the stop watch of least count …

Q11.The circular divisions of shown screw gauge are 50. It moves 0.5 mm on main scale in one rotation. The diameter of the ball is

Q13.In the given diagram, a line of force of a particular force field is shown. Out of the following options, it can never represent

Q14.The electrostatic potential (${\varphi }_{\mathrm{r}}$) of a spherical symmetric system, kept at origin, is shown in the adjacent figure, and given as
${\varphi }_{\mathrm{r}}=\frac{\mathrm{q}}{4\pi {ϵ}_0\mathrm{r}}\left(\mathrm{r}\ge {\mathrm{R}}_0\right)$ …

Q15.A solid cylinder of mass m and radius r is rolling on a rough inclined plane of inclination $\theta$. The coefficient of friction between the cylinder and incline is $\mu$. Then

Q16.Function $\mathrm{x}=\mathrm{A}{\sin }^2\omega \mathrm{t}+\mathrm{B}{\cos }^2\omega \mathrm{t}+\mathrm{C}\sin \omega \mathrm{t}\cos \omega \mathrm{t}$ represents SHM

Q17.In a dark room with ambient temperature ${\mathrm{T}}_0$, a black body is kept at a temperature T. Keeping the temperature of the black body constant (at T), sunrays are allowed to fall on the black body through a hole in the roof of the dark room. Assuming that there is no cha …

Q18.The graph between $1/\lambda$ and stopping potential (V) of three metals having work functions ${\varphi }_1,{\varphi }_2$ and ${\varphi }_3$ in an experiment of photo-electric effect is plotted as shown in the figure. Which of the following statement(s) is/are correct? [Here …

Q19.An infinite current carrying wire passes through point O and in perpendicular to the plane containing a current carrying loop ABCD as shown in the figure. Choose the correct option (s).

Q21.The capacitor of capacitance C can be charged (with the help of a resistance R) by a voltage source V, by closing switch ${\mathrm{S}}_1$ while keeping switch ${\mathrm{S}}_2$ open. The capacitor can be connected in series with an inductor 'L' by closing switch …

Q22.The capacitor of capacitance C can be charged (with the help of a resistance R) by a voltage source V, by closing switch ${\mathrm{S}}_1$ while keeping switch ${\mathrm{S}}_2$ open. The capacitor can be connected in series with an inductor 'L' by closing switch …

Q23.The capacitor of capacitance C can be charged (with the help of a resistance R) by a voltage source V, by closing switch ${\mathrm{S}}_1$ while keeping switch ${\mathrm{S}}_2$ open. The capacitor can be connected in series with an inductor 'L' by closing switch …

Q33.There is a rectangular plate of mass M kg of dimensions $(a\times b)$. The plate is held in horizontal position by striking n small balls each of mass m per unit area per unit time. These are striking in the shaded half region of the plate. The balls are colliding elastically …

Q34.In an insulated vessel, 0.05 kg steam at 373 K and 0.45 kg of ice at 253 K are mixed. Then, find the final temperature of the mixture.

Given, $L_{\text{fusion}}=80$ cal/g$=336$ J/g, $L_{\text{vaporization}}=540$ cal/g$=2268$ J/g, $S_{\text{ice}}=2100$ J/kg K$=0.5$ ca …

Q36.A circular disc with a groove along its diameter is placed horizontally. A block of mass 1 kg is placed as shown. The co-efficient of friction between the block and all surfaces of groove in contact is $\mu =2/5$. The disc has an acceleration of $25$ m/s${}^2$. Find the acce …

Q37.Heat given to process is positive, match the following option of column I with the corresponding option of column II

Q38.Match the following Columns
| (C) $\beta$-decay | (R) Generally possible for nuclei with high …

Q39.Match the following Columns
| (C) Co …

Q40.A simple telescope used to view distant objects has eyepiece and objective lens of focal lengths $f_e$ and $f_0$, respectively. Then
| (B) Angular magnificatio …

Q1.$\mathrm{B}(\mathrm{OH}{)}_3+\mathrm{NaOH}\rightleftharpoons {\mathrm{NaBO}}_2+\mathrm{Na}[\mathrm{B}(\mathrm{OH}{)}_4]+{\mathrm{H}}_2\mathrm{O}$
How can this reaction is made to proceed in forward direction?

Q2.A solution when diluted with ${\mathrm{H}}_2\mathrm{O}$ and boiled, it gives a white precipitate. On addition of excess ${\mathrm{NH}}_4\mathrm{Cl}/{\mathrm{NH}}_4\mathrm{OH}$. the volume of precipitate decreases leaving behind a white gelatinous precipitate. Identify th …

Q3.When benzene sulfonic acid and p-nitrophenol are treated with ${\mathrm{NaHCO}}_3$, the gases released respectively are

Q4.A monatomic ideal gas undergoes a process in which the ratio of P to V at any instant is constant and equals to 1. What is the molar heat capacity of the gas?

Q7.The IUPAC name of ${\mathrm{C}}_6{\mathrm{H}}_5\mathrm{COCl}$ is

Q8.${\mathrm{Ag}}^{+}+{\mathrm{NH}}_3\rightleftharpoons [\mathrm{Ag}({\mathrm{NH}}_3{)}^{+}];{\mathrm{k}}_1=3.5\times 10^{-3}$
$[\mathrm{Ag}({\mathrm{NH}}_3){]}^{+}+{\mathrm{NH}}_3\rightleftharpoons [\mathrm{Ag}({\mathrm{NH}}_3{)}_2{]}^{+};{\mathrm{k}}_2=1.7\times 10^{-3}$
then th …

Q9.${\mathrm{CH}}_3{\mathrm{NH}}_2+{\mathrm{CHCl}}_3+\mathrm{KOH}\to$ Nitrogen containing compound $+\mathrm{KCl}+{\mathrm{H}}_2\mathrm{O}$. Nitrogen containing compound is

Q10.${\mathrm{CuSO}}_4$ decolourises on addition of KCN, the product is

Q11.The direct conversion of A to B is difficult, hence it is carried out by the following shown path:

Given $\Delta {\mathrm{S}}_{(\mathrm{A}\to \mathrm{C})}=50$ e.u.
$\Delta {\mathrm{S}}_{(\mathrm{C}\to \mathrm{D})}=30$ e.u. …

Q12.${\mathrm{N}}_2+3{\mathrm{H}}_2\rightleftharpoons 2{\mathrm{NH}}_3$
Which is correct statement if ${\mathrm{N}}_2$ is added at equilibrium condition?

Q13.If the bond length of CO bond in carbon monoxide is $1.128\mathring{A}$, then what is the value of CO bond length in Fe(CO)${}_5$?

Q14.The species present in solution when ${\mathrm{CO}}_2$ is dissolved in water are

Q15.Which of the following reactants on reaction with conc. NaOH followed by acidification gives the following lactone as the only product?

Q16.The major products P and Q are

Q17.The given graph represents the variation of Z (compressibility factor = $\frac{\text{PV}}{\text{nRT}}$) versus P, for three real gases A, B and C. Identify the only incorrect statement.

Q20.${\mathrm{MgSO}}_4$ on reaction with ${\mathrm{NH}}_4\mathrm{OH}$ and ${\mathrm{Na}}_2{\mathrm{HPO}}_4$ forms a white crystalline precipitate. What is its formula?

Q33.75.2 g of ${\mathrm{C}}_6{\mathrm{H}}_5\mathrm{OH}$ (phenol) is dissolved in a solvent of ${\mathrm{K}}_{\mathrm{f}}=14$. If the depression in freezing point is 7 K then find the % of phenol that dimerises.

Q34.For the reaction, $2\mathrm{CO}+{\mathrm{O}}_2⟶2{\mathrm{CO}}_2;\Delta \mathrm{H}=-560\mathrm{kJ}$. Two moles of CO and one mole of ${\mathrm{O}}_2$ are taken in a container of volume 1 L . They completely form two moles of ${\mathrm{CO}}_2$, the gases devia …

Q37.Match the extraction processes listed in Column I with metals listed in Column II:

Column I
(A) Self reduction
(B) Carbon reduction
(C) Complex formation and displacement by metal
(D) Decomposition of iodide

Column II
(P) Lead
(Q) Silver
(R) Copper
(S) Boron

Q38.Match the following:

Column I
(A) ${\mathrm{Bi}}^{3+}⟶(\mathrm{BiO}{)}^{+}$
(B) ${\left[{\mathrm{AlO}}_2\right]}^{-}⟶\mathrm{Al}(\mathrm{OH}{)}_3$
(C) ${\mathrm{SiO}}_4^{4-}⟶{\mathrm{Si}}_2{\mathrm{O}}_7^{6-}$
(D) …

Q40.Match the following:

Column I
(A) ${\mathrm{CH}}_3-\mathrm{CHBr}-{\mathrm{CD}}_3$ on treatment with alc. KOH gives ${\mathrm{CH}}_2=\mathrm{CH}-{\mathrm{CD}}_3$ as a major product.
(B) $\mathrm{Ph}-\mathrm{CHBr}-{\mathrm{CH}}_3$ reacts faster than …

Q1.For $x>0,{\lim }_{x\to 0}\left((\sin x{)}^{1/x}+(1/x{)}^{\sin x}\right)$ is

Q2.$\int \frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}dx$ is equal to

Q3.Given an isosceles triangle, whose one angle is 120° and radius of its incircle $=\sqrt{3}$. Then the area of the triangle in sq. units is

Q4.If $0<\theta <2\pi$, then the intervals of values of $\theta$ for which $2{\sin }^2\theta -5\sin \theta +2>0$, is

Q5.If $w=\alpha +i\beta$, where $\beta \ne 0$ and $z\ne 1$, satisfies the condition that $\left(\frac{w-\overline{w}z}{1-z}\right)$ is purely real, then the set of values of z is

Q6.Let a, b, c be the sides of a triangle. No two of them are equal and $\lambda \in \mathbb{R}$. If the roots of the equation $x^2+2(a+b+c)x+3\lambda (ab+bc+ca)=0$ are real, then

Q7.If $f^{\prime \prime }(x)=-f(x)$ and $g(x)=f^{\prime }(x)$ and $F(x)={\left(f\left(\frac{x}{2}\right)\right)}^2+{\left(g\left(\frac{x}{2}\right)\right)}^2$ and given that $F(5)=5$, then $F(10)$ is equal to

Q8.If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is $r^2{t}^4{s}^2$, then the number of ordered pair $(p,q)$ is

Q9.Let $\theta \in \left(0,\frac{\pi }{4}\right)$ and $t_1=(\tan \theta {)}^{\tan \theta },t_2=(\tan \theta {)}^{\cot \theta },t_3=(\cot \theta {)}^{\tan \theta }$ and $t_4=(\cot \theta {)}^{\cot \theta }$, then

Q10.The axis of a parabola is along the line $y=x$ and the distance of its vertex from origin is $\sqrt{2}$ and that from its focus is $2\sqrt{2}$. If vertex and focus both lie in the first quadrant, then the equation of the parabola is

Q11.A plane passes through (1, -2, 1) and is perpendicular to two planes $2x-2y+z=0$ and $x-y+2z=4$. The distance of the plane from the point (1, 2, 2) is

Q12.Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k},\vec{b}=\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$. A vector in the plane of $\vec{a}$ and $\vec{b}$ whose projection on $\vec{c}$ is $\frac{1}{\sqrt{3}}$, is

Q13.The equations of the common tangents to the parabola $y=x^2$ and $y=-(x-2{)}^2$ is/are

Q14.If $f(x)=\min \{1,x^2,x^3\}$, then

Q15.A tangent drawn to the curve $y=f(x)$ at $P(x,y)$ cuts the x-axis and y-axis at A and B respectively such that BP : AP $=3:1$, given that $f(1)=1$, then

Q16.If a hyperbola passes through the focus of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and its transverse and conjugate axes coincide with the major and minor axes of the ellipse, and the product of eccentricities is 1, then

Q17.Internal bisector of $\angle A$ of triangle ABC meets side BC at D. A line drawn through D perpendicular to AD intersects the side AC at E and the side AB at F. If a, b, c represent sides of $△ABC$ then

Q18.f(x) is cubic polynomial which has local maximum at $x=-1$. If $f(2)=18,f(1)=-1$ and $f^{\prime }(x)$ has local minima at $x=0$, then

Q19.Let $\vec{A}$ be vector parallel to line of intersection of planes $P_1$ and $P_2$ through origin. $P_1$ is parallel to the vectors $2\hat{j}+3\hat{k}$ and $4\hat{j}-3\hat{k}$ and $P_2$ is parallel to $\hat{j}-\hat{k}$ and $3\hat{i}+3\hat{j}$, then the angle between vecto …

Q20.$f(x)=\{\begin{array}{ll}{e}^x,& 0\le x\le 1\\ 2-e^{x-1},& 1<x\le 2\\ x-e,& 2<x\le 3\end{array}$ and $g(x)={\int }_0^{x}f(t)dt,x\in [1,3]$ then $g(x)$ has

Q21.There are n urns each containing $n+1$ balls such that the ith urn contains i white balls and $(n+1-i)$ red balls. Let $u_i$ be the event of selecting ith urn, $i=1,2,3\dots ,n$ and w denotes the event of getting a white ball.

If $P(u_i)\propto i$, where …

Q22.There are n urns each containing $n+1$ balls such that the ith urn contains i white balls and $(n+1-i)$ red balls. Let $u_i$ be the event of selecting ith urn, $i=1,2,3\dots ,n$ and w denotes the event of getting a white ball.

If $P(u_i)=c$, where c is a constant then …

Q23.There are n urns each containing $n+1$ balls such that the ith urn contains i white balls and $(n+1-i)$ red balls. Let $u_i$ be the event of selecting ith urn, $i=1,2,3\dots ,n$ and w denotes the event of getting a white ball.

If n is even and E denotes the event of choosing …

Q33.If roots of the equation $x^2-10cx-11d=0$ are a, b and those of $x^2-10ax-11b=0$ are c, d, then the value of $\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}$ is (a, b, c and d are distinct numbers)

Q34.The value of $5050\frac{{\int }_0^1{\left(1-x^{50}\right)}^{100}dx}{{\int }_0^1{\left(1-x^{50}\right)}^{101}dx}$ is

Q35.If ${\mathrm{a}}_{\mathrm{n}}=\frac{3}{4}-{\left(\frac{3}{4}\right)}^2+{\left(\frac{3}{4}\right)}^3+\cdots (-1{)}^{\mathrm{n}-1}{\left(\frac{3}{4}\right)}^{\mathrm{n}}$ and ${\mathrm{b}}_{\mathrm{n}}=1-{\mathrm{a}}_{\mathrm{n}}$, then find the minimum natural number ${\mathrm{n}}_0$ s …

Q36.If $\mathrm{f}(\mathrm{x})$ is a twice differentiable function such that $\mathrm{f}(\mathrm{a})=0,\mathrm{f}(\mathrm{b})=2,\mathrm{f}(\mathrm{c})=-1,\mathrm{f}(\mathrm{d})=2,\mathrm{f}(\mathrm{e})=0$, where $\mathrm{a}<\mathrm{b}<\mathrm{c}<\mathrm{d}<\mathrm{e}$, then …

Q37.Normals are drawn at points P, Q and R lying on the parabola $y^2=4x$ which intersect at (3, 0). Then
| …