JEE (Advanced) 2010 Paper-1
When a particle of mass m moves on the x-axis in a potential of the form $$V(x) = kx^2$$, it performs simple harmonic motion. The corresponding time period is proportional to $$\sqrt{\frac{m}{k}}$$, as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even whenits potential energy increases on both sides of x = O in a way different from $$kx^2$$ and its total energy is such that the particle does not escape to infinity. Consider a particle of mass m moving on the x-axis. Its potential energy is $$V(x) = \alpha x^4 (\alpha > 0)$$ for $$\mid x \mid$$ near the origin and becomes a constant equal to $$V_0$$ for $$\mid x \mid \geq X_0$$ (see figure).
For periodic motion of small amplitude A, the time period T of this particle is proportional to
Electrical resistance of certain materials, known as superconductors, changes abruptly from a nonzero value to zero as their temperature is lowered below a critical temperature $$T_c(0)$$. An interesting property of superconductors is that their critical temperature becomes smaller than $$T_c(0)$$ if they are placed in a magnetic field, i.e., the critical temperature $$T_c(B)$$ is a function of the magnetic field strength B. The dependence of $$T_c(B)$$ on B is shown in the figure.
In the graphs below, the resistance R of a super conductor is shown as a function of its temperature T for two different magnetic fields $$B_1$$ (solid line) and $$B_2$$ (dashed line). If $$B_2$$ is larger than $$B_1$$, which of the following graphs shows the correct variation of R with T in these fields ?
A superconductor has $$T_c (0) = 100 K$$. When a magnetic field of 7.5 Tesla is applied, its $$T_c$$ decreases to 75 K. For this material one candefinitely say that when
For the following questions answer them individually
The focal length of a thin biconvex lens is 20cm. When anobject is moved from a distance of 25cmin front of it to 50cm, the magnification of its image changes from $$m_{25}$$ to $$m_{50}$$. The ratio $$\frac{m_{25}}{m_{50}}$$ is
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An $$\alpha$$-particle and a proton are accelerated from rest by a potential difference of 100V. After this, their de Broglie wavelengths are $$\lambda_{\alpha}$$ and $$\lambda_{p}$$ respectively. The ratio $$\frac{\lambda_p}{\lambda_{\alpha}$$, to the nearest integer, is
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When two identical batteries of internal resistance 1 Ω each are connected in series across a resistor R, the rate of heat produced in R is $$J_1$$. When the samebatteries are connected in parallel across R, the rate is $$J_2$$. If $$J_1 = 2.25 J_2$$ then the value of R in Ω is
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Two spherical bodies A (radius 6 cm) and B (radius 18 cm) are at temperatures $$T_1$$ and $$T_2$$, respectively. The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm. Considering themto be black bodies, what will be the ratio of the rate of total energy radiated by A to that of B ?
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When two progressive waves $$y_1 = 4 \sin(2x - 6t)$$ and $$y_2 = 3 \sin\left(2x - 6t - \frac{\pi}{2}\right)$$ are superimposed, the amplitude of the resultant wave is
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A 0.1 kg mass is suspended from a wire of negligible mass. The length of the wire is 1m and its cross-sectional area is $$4.9 \times 10^{-7} m^2$$. If the mass is pulled a little in
the vertically downward direction and released, it performs simple harmonic motion of angular frequency 140 rad $$s^{-1}$$. If the Young’s modulus of the material of the wire is $$n \times 10^9 Nm^{-2}$$, the value of n is
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