JEE Main 2010 · 11 Apr 2010 · Offline

Q1.A rectangular loop has a sliding connector PQ of length $\ell$ and resistance $\mathrm{R}\Omega$ 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 ${\mathrm{I}}_1,{\mathrm{I}}_2$ and …

Q2.Let C be the capacitance of a capacitor discharging through a resistor R. Suppose ${\mathrm{t}}_1$ 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. …

Q3.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.

Q4.Statement-1: When ultraviolet light is incident on a photocell, its stopping potential is ${\mathrm{V}}_0$ and the maximum kinetic energy of the photoelectrons is ${\mathrm{K}}_{\max }$. When the ultraviolet light is replaced by X rays, both ${\mathrm{V}}_0$ and …

Q5.A ball is made of a material of density $\rho$ where ${\rho }_{\text{oil}}<\rho <{\rho }_{\text{water}}$ with ${\rho }_{\text{oil}}$ and ${\rho }_{\text{water}}$ representing the densities of oil and water, respectively. The oil and water are immiscible. If the above ball is in equilibrium in …

Q6.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

Q7.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

Q8.In the circuit shown below, the key K is closed at $\mathrm{t}=0$. The current through the battery is

Q9.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

Q10.The binding energy per nucleon for the parent nucleus is ${\mathrm{E}}_1$ and that for the daughter nuclei is ${\mathrm{E}}_2$. Then

Q11.The speed of daughter nuclei is

Q12.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

Q13.A thin semi-circular ring of radius $r$ has a positive charge $q$ distributed uniformly over it. The net field $\overrightarrow{\mathrm{E}}$ at the centre O is

Q14.The combination of gates shown below yields

Q15.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 32 V the efficiency of the engine is

Q16.If a source of power 4 kW produces $10^{20}$ photons/second, the radiation belong to a part of the spectrum called

Q17.The respective number of significant figures for the numbers $23.023,0.0003$ and $2.1\times 10^{-3}$ are

Q18.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 c …

Q19.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 $r(r<R)$ f …

Q20.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 …

Q21.Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of $30^{\circ }$ with each other. When suspended in a liquid of density $0.8{\text{g cm}}^{-3}$, the angle remains the same. If density of the material of the sphere is …

Q22.Two conductors have the same resistance at $0^{\circ }\text{C}$ but their temperature coefficients of resistance are ${\alpha }_1$ and ${\alpha }_2$. The respective temperature coefficients of their series and parallel combinations are nearly

Q23.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 …

Q24.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?

Q25.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)

Q26.As the beam enters the medium, it will

Q27.The initial shape of the wave front of the beam is

Q28.The speed of light in the medium is

Q29.A small particle of mass $m$ is projected at an angle $\theta$ with the $x$-axis with an initial velocity ${\mathrm{v}}_0$ in the $\mathrm{x}-\mathrm{y}$ plane as shown in the figure. At a time $\mathrm{t}<\frac{{\mathrm{v}}_0\sin \theta }{\mathrm{g}}$, the angular momentum of t …

Q30.The equation of a wave on a string of linear mass density $0.04\mathrm{kg}{\mathrm{m}}^{-1}$ 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

Q1.The standard enthalpy of formation of ${\mathrm{NH}}_3$ is $-46.0\mathrm{kJ}{\mathrm{mol}}^{-1}$. If the enthalpy of formation of ${\mathrm{H}}_2$ from its atoms is $-436\mathrm{kJ}{\mathrm{mol}}^{-1}$ and that of ${\mathrm{N}}_2$ is …

Q2.The time for half life period of a certain reaction $\mathrm{A}\to$ products is 1 hour. When the initial concentration of the reactant ' $A$ ', is $2.0\mathrm{mol}{\mathrm{L}}^{-1}$, how much time does it take for its concentration to come from 0.50 to …

Q3.A solution containing 2.675 g of ${\mathrm{CoCl}}_3.6{\mathrm{NH}}_3$ (molar mass $=267.5\mathrm{g}{\mathrm{mol}}^{-1}$ ) is passed through a cation exchanger. The chloride ions obtained in solution were treated with excess of ${\mathrm{AgNO}}_3$ to give 4.78 g of AgCl …

Q4.Consider the reaction:
${\mathrm{Cl}}_2(\mathrm{aq})+{\mathrm{H}}_2\mathrm{S}(\mathrm{aq})\to \mathrm{S}(\mathrm{s})+2{\mathrm{H}}^{+}(\mathrm{aq})+2{\mathrm{Cl}}^{-}(\mathrm{aq})$
The rate equation for this reaction is rate …

Q5.If $10^{-4}{\mathrm{dm}}^3$ of water is introduced into a $1.0{\mathrm{dm}}^3$ flask to 300 K , how many moles of water are in the vapour phase when equilibrium is established ?
(Given : Vapour pressure of ${\mathrm{H}}_2\mathrm{O}$ at 300 K is …

Q6.One mole of a symmetrical alkene on ozonolysis gives two moles of an aldehyde having a molecular mass of 44 u . The alkene is

Q7.If sodium sulphate is considered to be completely dissociated into cations and anions in aqueous solution, the change in freezing point of water $(\Delta {\mathrm{T}}_{\mathrm{f}})$, when 0.01 mol of sodium sulphate is dissolved in 1 kg of water, is ( …

Q8.From amongst the following alcohols the one that would react fastest with conc. HCl and anhydrous ${\mathrm{ZnCl}}_2$, is

Q9.In the chemical reactions,

the compounds ' A ' and ' B ' respectively are

Q10.29.5 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 …

Q11.The energy required to break one mole of $\mathrm{Cl}-\mathrm{Cl}$ bonds in $\mathrm{Cl}{}_2$ is $242{\text{ kJ mol}}^{-1}$. The longest wavelength of light capable of breaking a single $\mathrm{Cl}-\mathrm{Cl}$ bond is
($c=3\times 10^8{\text{ ms}}^{-1}$ and $N_{\text{A}}=6.02\times 10^{23}{\text{ mol}}^{-1}$)

Q12.Ionisation energy of $\mathrm{He}{}^{+}$ is $19.6\times 10^{-18}{\text{ J atom}}^{-1}$. The energy of the first stationary state $(n=1)$ of $\mathrm{Li}{}^{2+}$ is

Q13.Consider the following bromides:

The correct order of $S_{N}1$ reactivity is

Q14.Which one of the following has an optical isomer?
(en = ethylenediamine)

Q15.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 25.0 g of heptane and 35 g of octane will be (molar mass …

Q16.The main product of the following reaction is $\mathrm{C}{}_6\mathrm{H}{}_5\mathrm{CH}{}_2\mathrm{CH}(\mathrm{OH})\mathrm{CH}(\mathrm{CH}{}_3){}_2\stackrel{\text{conc. }\mathrm{H}{}_2\mathrm{SO}{}_4}{\to }$ ?

Q17.Three reactions involving $\mathrm{H}{}_2\mathrm{PO}{}_4{}^{-}$ are given below:
(i) $\mathrm{H}{}_3\mathrm{PO}{}_4+\mathrm{H}{}_2\mathrm{O}\to \mathrm{H}{}_3\mathrm{O}{}^{+}+\mathrm{H}{}_2\mathrm{PO}{}_4{}^{-}$
(ii) $\mathrm{H}{}_2\mathrm{PO}{}_4{}^{-}+\mathrm{H}{}_2\mathrm{O}\to \mathrm{HPO}{}_4{}^{2-}+\mathrm{H}{}_3\mathrm{O}{}^{+}$
(iii) $\mathrm{H}{}_2\mathrm{PO}{}_4{}^{-}+\mathrm{OH}{}^{-}\to \mathrm{H}{}_3\mathrm{PO}{}_4+\mathrm{O}{}^{2-}$

In which of the above does $\mathrm{H}{}_2\mathrm{PO}{}_4{}^{-}$ act as an acid?

Q18.In aqueous solution the ionization constants for carbonic acid are
${\mathrm{K}}_1=4.2\times 10^{-7}$ and ${\mathrm{K}}_2=4.8\times 10^{-11}$
Select the correct statement for a saturated 0.034 M solution of the carbonic acid.

Q19.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

Q20.The correct order of increasing basicity of the given conjugate bases ($\mathrm{R}={\mathrm{CH}}_3$) is

Q21.The correct sequence which shows decreasing order of the ionic radii of the elements is

Q22.Solubility product of silver bromide is $5.0\times 10^{-13}$. The quantity of potassium bromide (molar mass taken as $120\mathrm{g}\mathrm{of}{\mathrm{mol}}^{-1}$) to be added to 1 litre of 0.05 M solution of silver nitrate to start the precipitation of AgBr is

Q23.The Gibbs energy for the decomposition of ${\mathrm{Al}}_2{\mathrm{O}}_3$ at $500^{\circ }\mathrm{C}$ is as follows: …

Q24.At $25^{\circ }\mathrm{C}$, the solubility product of $\mathrm{Mg}(\mathrm{OH}{)}_2$ is $1.0\times 10^{-11}$. At which pH, will ${\mathrm{Mg}}^{2+}$ ions start precipitating in the form of $\mathrm{Mg}(\mathrm{OH}{)}_2$ from a solution of $0.001\mathrm{M}$ ${\mathrm{Mg}}^{2+}$ ions?

Q25.Percentage of free space in cubic close packed structure and in body centred packed structure are respectively

Q26.Out of the following, the alkene that exhibits optical isomerism is

Q27.Biuret test is not given by

Q28.The correct order of $E_{{\mathrm{M}}^{2+}/\mathrm{M}}^0$ values with negative sign for the four successive elements $\mathrm{Cr}$, $\mathrm{Mn}$, $\mathrm{Fe}$ and Co is

Q29.The polymer containing strong intermolecular forces e.g. hydrogen bonding, is

Q30.For a particular reversible reaction at temperature $\mathrm{T},\Delta \mathrm{H}$ and $\Delta \mathrm{S}$ were found to be both +ve . If ${\mathrm{T}}_{\mathrm{e}}$ is the temperature at equilibrium, the reaction would be spontaneous when

Q1.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 $\tan 2\alpha =$

Q2.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 $P$ ?

Q3.Let $\vec{a}=\hat{j}-\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$. Then vector $\vec{b}$ satisfying $\vec{a}\times \vec{b}+\vec{c}=\overrightarrow{0}$ and $\vec{a}\cdot \vec{b}=3$ is

Q4.The equation of the tangent to the curve $y=x+\frac{4}{x^2}$, that is parallel to the $x$-axis, is

Q5.Solution of the differential equation $\cos xdy=y(\sin x-y)dx,0<x<\frac{\pi }{2}$ is

Q6.The area bounded by the curves $y=\cos x$ and $y=\sin x$ between the ordinates $x=0$ and $x=\frac{3\pi }{2}$ is

Q7.If two tangents drawn from a point $P$ to the parabola $y^2=4x$ are at right angles, then the locus of $P$ is

Q8.If the vectors $\vec{a}=\hat{i}-\hat{j}+2\hat{k},\vec{b}=2\hat{i}+4\hat{j}+\hat{k}$ and $\vec{c}=\lambda \hat{i}+\hat{j}+\mu \hat{k}$ are mutually orthogonal, then $(\lambda ,\mu )=$

Q9.Consider the following relations:
$R=\{(x,y)\mid x,y$ are real numbers and $x=wy$ for some rational number $w\}$; …

Q10.Let $f:R\to R$ be defined by $f(x)=\{\begin{array}{ll}k-2x,& \text{ if }x\le -1\\ 2x+3,& \text{ if }x>-1\end{array}$. If $f$ has a local minimum at $x=-1$, then a possible value of k is

Q11.The number of $3\times 3$ non-singular matrices, with four entries as 1 and all other entries as 0, is

Q12.Four numbers are chosen at random (without replacement) from the set $\{1,2,3,\dots ,20\}$.

Statement-1: The probability that the chosen numbers when arranged in some order will form an AP is $\frac{1}{85}$.
Statement-2: If the four chosen numbers from an AP, then the set …

Q13.Statement-1: The point A(3,1,6) is the mirror image of the point B(1,3,4) in the plane x-y+z=5.

Statement-2: The plane x-y+z=5 bisects the line segment joining A(3,1,6) and B(1,3,4).

Q14.Let $S_1={\sum }_{j=1}^{10}j(j-1{)}^{10}{C}_j$, $S_2={\sum }_{j=1}^{10}{j}^{10}{C}_j$ and $S_3={\sum }_{j=1}^{10}{j}^2{}^{10}{C}_j$.

Statement-1: $S_3=55\times 2^9$
Statement-2: $S_1=90\times 2^8$ and $S_2=10\times 2^8$.

Q15.Let A be a $2\times 2$ matrix with non-zero entries and let $A^2=I$, where I is $2\times 2$ identity matrix. Define
$\mathrm{Tr}(A)=$ sum of diagonal elements of A and $|A|=$ determinant of matrix A.

Statement-1: $\mathrm{Tr}(A)=0$
Statement-2: $|A|=1$

Q16.Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function defined by $f(x)=\frac{1}{e^x+2e^{-x}}$.

Statement-1: $f(c)=\frac{1}{3}$, for some $c\in \mathbb{R}$.
Statement-2: $0<f(x)\le \frac{1}{2\sqrt{2}}$, for all $x\in \mathbb{R}$

Q17.For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is

Q18.If $\alpha$ and $\beta$ are the roots of the equation $x^2-x+1=0$, then ${\alpha }^{2009}+{\beta }^{2009}=$

Q19.The number of complex numbers $z$ such that $|z-1|=|z+1|=|z-i|$ equals

Q20.A line AB in three-dimensional space makes angles $45^{\circ }$ and $120^{\circ }$ with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle $\theta$ with the positive z-axis, then $\theta$ equals

Q21.The line $L$ given by $\frac{x}{5}+\frac{y}{b}=1$ passes through the point $(13,32)$. The line $K$ is parallel to $L$ and has the equation $\frac{x}{c}+\frac{y}{3}=1$. Then the distance between $L$ and $K$ is

Q22.A person is to count 4500 currency notes. Let $a_n$ denote the number of notes he counts in the $n^{\text{th}}$ minute. If $a_1=a_2=\dots =a_{10}=150$ and $a_{10},a_{11},\dots$ are in A.P. with common difference $-2$, then the time taken by him to count all notes is

Q23.Let $f:\mathbb{R}\to \mathbb{R}$ be a positive increasing function with ${\lim }_{x\to \infty }\frac{f(3x)}{f(x)}=1$. Then ${\lim }_{x\to \infty }\frac{f(2x)}{f(x)}=$

Q24.Let $p(x)$ be a function defined on $\mathbb{R}$ such that $p^{\prime }(x)=p^{\prime }(1-x)$, for all $x\in [0,1]$, $p(0)=1$ and $p(1)=41$. Then ${\int }_0^{1}p(x)dx$ equals

Q25.Let $f:(-1,1)\to \mathbb{R}$ be a differentiable function with $f(0)=-1$ and $f^{\prime }(0)=1$. Let $g(x)=[f(2f(x)+2){]}^2$. Then $g^{\prime }(0)=$

Q26.There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is

Q27.Consider the system of linear equations:
$x_1+2x_2+x_3=3$
$2x_1+3x_2+x_3=3$
$3x_1+5x_2+2x_3=1$
The system has

Q28.An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colour is

Q29.For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is

Q30.The circle $x^2+y^2=4x+8y+5$ intersects the line $3x-4y=m$ at two distinct points if