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Consider an expanding sphere of instantaneous radius R whose total mass remains constant. The expansion is such that the instantaneous density $$\rho$$ remains uniform throughout the volume. The rate of fractional change in density $$(\frac{1}{\rho}\frac{d\rho}{dt})$$ is constant. The velocity $$\nu$$ of any point on the surface of the expanding sphere is proportional to
Consider regular polygons with number of sides n=3,4,5.....as shown in the figure. The center of mass of all the polygons is at height h from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is $$\triangle$$. Then $$\triangle$$ depends on n and has
A photoelectric material having work-function $$\phi_{0}$$ is illuminated with light of wavelength $$\lambda (\lambda < \frac{hc}{\phi_{0}})$$. The fastest photoelectron has a de Broglie wavelength $$\lambda_{d}$$. A change in wavelength of the incident light by $$\triangle \lambda$$ results in a change $$\triangle \lambda_{d}$$ in $$\lambda_{d}$$. Then the ratio $$\frac{\triangle\lambda_{d}}{\triangle\lambda}$$ is proportional to
A symmetric star shaped conducting wire loop is carrying a steady state current I as shown in the figure. The distance between the diametrically opposite vertices of the star is $$4a$$. The magnitude of the magnetic field at the center of the loop is
Three vectors $$\overrightarrow{P},\overrightarrow{Q}$$ and $$\overrightarrow{R}$$ are shown in the figure. Let be any point on the vector $$\overrightarrow{R}$$. The distance between the points $$P$$ and $$S$$ is $$b\mid\overrightarrow{R}\mid$$. The general relation among vectors $$\overrightarrow{P},\overrightarrow{Q}$$ and $$\overrightarrow{S}$$ is
A rocket is launched normal to the surface of the Earth, away from the Sun, along the line joining the Sun and the Earth. The Sun is $$3\times10^{5}$$ times heavier than the Earth and is at a distance $$2.5\times10^{4}$$ times larger than the radius of the Earth. The escape velocity from Earth’s gravitational field is
$$\nu_{e}=11.2 kms^{-1}$$. The minimum initial velocity $$\nu_{s}$$ required for the rocket to be able to leave the Sun-Earth system is closest to
(Ignore the rotation and revolution of the Earth and the presence of any other planet)
A person measures the depth of a well by measuring the time interval between dropping a stone and receiving the sound of impact with the bottom of the well. The error in his measurement of time is $$\delta T=0.01$$ seconds and he measures the depth of the well to be $$L=20$$ meters. Take the acceleration due to gravity $$g = 10 ms^{-2}$$ and the velocity of sound is $$300 ms^{-1}$$. Then the fractional error in the measurement, $$\frac{\delta L}{L}$$, is closest to
A uniform magnetic field B exists in the region between $$x=0$$ and $$x=\frac{3R}{2}$$ (region 2 in the figure) pointing normally into the plane of the paper. A particle with charge $$+Q$$ and momentum $$p$$ directed along x-axis enters region 2 from region 1 at point $$P_{1}(Y=-R)$$.Which of the following option(s) is/are correct?
The instantaneous voltages at three terminals marked X,Y and Z are given by $$V_{X}= V_{0} \sin \omega t$$, $$V_{Y}= V_{0} \sin (\omega t+\frac{2\pi}{3})$$
and $$V_{Z}= V_{0} \sin (\omega t+\frac{4\pi}{3})$$. An ideal voltmeter is configured to read rms value of the potential difference between its terminals. It is connected between points X and Y and then between Y and Z The reading(s) of the voltmeter will be
A point charge $$+Q$$ is placed just outside an imaginary hemispherical surface of radius R as shown in the figure. Which of the following statements is/are correct?
Two coherent monochromatic point sources $$S_{1}$$ and $$S_{2}$$ of wavelength $$\lambda=600$$ nm are placed symmetrically on either side of the center of the circle as shown. The sources are separated by a distance $$d=1.8$$ mm. This arrangement produces interference fringes visible as alternate bright and dark spots on the circumference of the circle. The angular separation between two consecutive bright spots is $$\triangle\theta$$. Which of the following options is/are correct?
A source of constant voltage V is connected to a resistance R and two ideal inductors $$L_{1}$$ and $$L_{2}$$ through a switch as shown. There is no mutual inductance between the two inductors. The switch S is initially open. At = 0, the switch is closed and current begins to flow. Which of the following options is/are correct?
A rigid uniform bar AB of length L is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is $$\theta$$ Which of the following statements about its motion is/are correct?
A wheel of radius R and mass M is placed at the bottom of a fixed step of height R as shown in the figure. A constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque $$\tau$$ about an axis normal to the plane of the paper passing through the point Q. Which of the following options is/are correct?
Consider a simple 𝑅𝐶 circuit as shown in Figure 1.
Process 1: In the circuit the switch 𝑆 is closed at t = 0 and the capacitor is fully charged to voltage $$V_0$$ (i.e., charging continues for time $$T \gg RC$$). In the process some dissipation $$(E_D)$$ occurs across the resistance 𝑅. The amount of energy finally stored in the fully charged capacitor is $$E_c$$.
Process 2: In a different process the voltage is first set to $$\frac{V_0}{3}$$ and maintained for a charging time $$T \gg RC$$. Then the voltage is raised to $$\frac{2 V_0}{3}$$ without discharging the capacitor and again maintained for a time $$T \gg RC$$. The process is repeated one more time by raising the voltage
to $$V_0$$ and the capacitor is charged to the same final voltage $$V_0$$ as in Process 1.
These two processes are depicted in Figure 2.
In Process 1, the energy stored in the capacitor $$E_{C}$$ and heat dissipated across resistance $$E_{D}$$ are related by:
In Process 2, total energy dissipated across the resistance $$E_{D}$$ is:
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One twirls a circular ring (of mass M and radius R) near the tip of one’s finger as shown in Figure 1. In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is r. The finger rotates with an angular velocity $$\omega_{0}$$. The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is $$\mu$$ and the acceleration due to gravity is g
The total kinetic energy of the ring is
The minimum value of $$\mu$$ below which the ring will drop down is
Pure water freezes at 273 K and 1 bar. The addition of 34.5 g of ethanol to 500 g of water changes the freezing point of the solution. Use the freezing point depression constant of water as 2 K kg $$mol^{-1}$$ . The figures shown below represent plots of vapour pressure (V.P.) versus temperature (T). [molecular weight of ethanol is 46 g $$mol^{-1}$$] Among the following, the option representing change in the freezing point is
For the following cell, $$Zn(s)\mid ZnSO_{4}(aq)\parallel CuSO_{4}(aq)\mid Cu(s)$$ when the concentration of $$Zn^{2+}$$ is 10 times the concentration of $$Cu^{2+}$$, the expression for $$\triangle G$$ ((in J $$mol^{-1}$$) is [F is Faraday constant; R is gas constant; T is temperature; $$E^{0}(cell)=1.1V$$]
The standard state Gibbs free energies of formation of C(graphite) and C(diamond) at T = 298 K are
$$\triangle_{f}G^{0}[C(graphite)]=0 kJ\ mol^{-1}$$
$$\triangle_{f}G^{0}[C(diamond)]=2.9 kJ\ mol^{-1}$$
The standard state means that the pressure should be 1 bar, and substance should be pure at a given temperature. The conversion of graphite [C(graphite)] to diamond [C(diamond)] reduces its volume by $$2\times10^{-6}m^{3}mol^{-1}$$ If C(graphite) is converted to C(diamond) isothermally at T = 298 K, the pressure at which C(graphite) is in equilibrium with
[Useful information: 1 J = 1 kg $$m^{2}s^{-2}$$;1 Pa=1kg $$m^{-1}s^{-2}$$; $$1 bar = 10^{5} Pa$$] C(diamond), is
Which of the following combination will produce $$H_{2}$$ gas?
The order of the oxidation state of the phosphorus atom in $$H_{3}PO_{2},H_{3}PO_{4},H_{3}PO_{3}$$ and $$H_{4}P_{2}O_{6}$$ is
The major product of the following reaction is
The order of basicity among the following compounds is
The correct statement(s) about surface properties is(are)
For a reaction taking place in a container in equilibrium with its surroundings, the effect of temperature on its equilibrium constant K in terms of change in entropy is described by
In a bimolecular reaction, the steric factor P was experimentally determined to be 4.5. The correct option(s) among the following is(are)
For the following compounds, the correct statement(s) with respect to nucleophilic substitution reactions is(are)
Among the following, the correct statement(s) is(are)
The option(s) with only amphoteric oxides is(are)
Compounds P and R upon ozonolysis produce Q and S, respectively. The molecular formula of Q and S is $$C_{8}H_{8}O$$. Q undergoes Cannizzaro reaction but not haloform reaction, whereas S undergoes haloform reaction but not Cannizzaro reaction.
The option(s) with suitable combination of P and R, respectively, is(are)
Upon heating $$KClO_{3}$$ in the presence of catalytic amount of $$MnO_{2}$$ a gas W is formed. Excess amount of W reacts with white phosphorus to give X. The reaction of X with pure $$HNO_{3}$$ gives Y and Z.
W and X are, respectively
Y and Z are, respectively
The reaction of compound P with $$CH_{3}MgBr$$ (excess) in $$(C_{2}H_{5})_{2}O$$ followed by addition of $$H_{2}O$$ gives Q. The compound Q on treatment with $$H_{2}SO_{4}$$ at $$0^{0}_{C}$$ gives R. The reaction of R with $$CH_{3}COCl$$ in the presence of anhydrous $$AlCl_{3}$$ in $$CH_{2}Cl_{2}$$ followed by treatment with $$H_{2}O$$ produces compound S. [Et in compound P is ethyl group]
The product S is
The reactions, Q to R and R to S, are
The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes $$2x+y-2z=5$$ and $$3x-6y-2z=7$$, is
Let O be the origin and let PQR be an arbitrary triangle. The point S is such that $$\overrightarrow{OP}.\overrightarrow{OQ}+\overrightarrow{OR}.\overrightarrow{OS}=\overrightarrow{OR}.\overrightarrow{OP}+\overrightarrow{OQ}.\overrightarrow{OS}=\overrightarrow{OQ}.\overrightarrow{OR}+\overrightarrow{OP}.\overrightarrow{OS}$$ Then the triangle PQR has S as its
If $$y=y(x)$$ ) satisfies the differential equation $$8\sqrt{x}(\sqrt{9+\sqrt{x}})dy=(\sqrt{4+\sqrt{9+\sqrt{x}}})^{-1}dx$$, x>0 and $$Y(0)=\sqrt{7}$$, then y(256)=
If $$f:R\rightarrow R$$is a twice differentiable function such that $$f''(x)>0$$ for all $$x\epsilon R$$, and $$f(\frac{1}{2})=\frac{1}{2},f(1)=1$$, then
How many $$3\times3$$ matrices M with entries from {0,1,2} are there, for which the sum of the diagonal entries of $$M^{T}M$$ is 5?
Lat S={1,2,3,...,9}. For = 1, 2, … ,5, let $$N_{k}$$ be the number of subsets of S, each containing five elements out of which exactly k are odd, Then $$N_{1}+N_{2}+N_{3}+N_{4}+N_{5}=$$
Three randomly chosen nonnegative integers x,y and z are found to satisfy the equation $$x+y+z=10$$. Then the probability that z is even, is
If $$g(x) = \int_{\sin x}^{\sin(2x)} \sin^{-1}(t)dt$$ then
Let $$\alpha$$ and $$\beta$$ be nonzero real numbers such that $$2(\cos\beta-\cos\alpha)+\cos\alpha\cos\beta=1$$. Then which of the following is/are true?
If $$f:R\rightarrow R$$ is a differentiable function such that $$f'(x)>2f(x)$$ for all $$x\epsilon R$$, and $$f(0)=1$$, then
Let $$f(x)=\frac{1-x(1+\mid1-x\mid)}{\mid1-x\mid}Cos(\frac{1}{1-x})$$ for $$x\neq1$$. Then
If f(x) =
, then
If the line $$x=\alpha$$ divides the area of region $$R={(x,y)\epsilon R^{2}:x^{3}\leq y\leq x},0\leq x\leq1$$ into two equal parts, then
If $$1=\sum^{98}_{k=1}\int_{k}^{k+1} \frac{k+1}{x(x+1)}dx$$, then
$$\mid\overrightarrow{OX}\times\overrightarrow{OY}\mid=$$
If the triangle PQRvaries, then the minimum value of $$\cos(P+Q)+\cos(Q+R)+\cos(R+P)$$ is
Let p,q be integers and let $$\alpha,\beta$$ be the roots of the equation, $$X^{2}-x-1=0$$, Where $$\alpha\neq\beta$$. For n=0,1,2..., Let $$a_{n}=p\alpha^{n}+q\beta^{n}$$.
FACT: If a and b are rational numbers and $$a+b\sqrt{5}=0$$, then $$a=0=b$$.
$$a_{12}=$$
If $$a_{4}=28$$, then $$p+2q=$$
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