Given $$P\left(x\right)=x^4+a{x}^3+b{x}^2+cx+d$$ such that $$x=0$$ is the only real root of $$P^{\prime }\left(x\right)=0.$$ If $$P\left(-1\right)<P\left(1\right),$$ then in the interval $$\left[-1,1\right]:$$

The area of the region bounded by the parabola $${\left(y-2\right)}^2=x-1,$$ the tangent of the parabola at the point $$(2,3)$$ and the $$x$$-axis is :

Statement-1 : $$\sim \left(p\leftrightarrow \sim q\right)$$ is equivalent to $$p\leftrightarrow q$$. Statement-2 : $$\sim \left(p\leftrightarrow \sim q\right)$$ is a tautology.

In an experiment the angles are required to be measured using an instrument, 29 divisions of the main scale exactly coincide with the 30 divisions of the vernier scale. If the smallest division of the main scale is half-a degree(=$$0.5^{\circ }$$), then the least count of the instrument is:

In an optics experiment, with the position of the object fixed, a student varies the position of a convex lens and for each position, the screen is adjusted to get a clear image of the object. A graph between the object distance $$u$$ and the image distance $$v,$$ from the lens, is plotted using the same scale for the two axes. A straight line passing through the origin and making an angle of $$45^{\circ }$$ with the $$x$$-axis meets the experimental curve at $$P.$$ The coordinates of $$P$$ will be :

A transparent solid cylindrical rod has a refractive index of {2 \over {\sqrt 3 }}. It is surrounded by air. A light ray is incident at the mid-point of one end of the rod as shown in the figure. The incident angle $$\theta$$ for which the light ray grazes along the wall of the rod is :

Two wires are made of the same material and have the same volume. However wire $$1$$ has cross-sectional area $$A$$ and wire $$2$$ has cross-sectional area $$3A.$$ If the length of wire $$1$$ increases by $$\Delta x$$ on applying force $$F,$$ how much force is needed to stretch wire $$2$$ by the same amount?

A $$p$$-$$n$$ junction $$(D)$$ shown in the figure can act as a rectifier. An alternating current source $$(V)$$ is connected in the circuit. The current $$(I)$$ in the resistor $$(R)$$ can be shown by :

The logic circuit shown below has the input waveforms $${}^{\prime }{A}^{\prime }$$ and $${}^{\prime }{B}^{\prime }$$ as shown. Pick out the correct output waveform. Output is

If $$x,$$ $$v$$ and $$a$$ denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period $$T,$$ then, which of the following does not change with time?

An inductor of inductance $$L=400$$ $$mH$$ and resistors of resistance $$R_1=2\Omega$$ and $$R_2=2\Omega$$ are connected to a battery of $$emf$$ $$12$$ $$V$$ as shown in the figure. The internal resistance of the battery is negligible. The switch $$S$$ is closed at $$t=0.$$ The potential drop across $$L$$ as a function of time is :

The transition from the state $$n=4$$ to $$n=3$$ in a hydrogen like atom result in ultra violet radiation. Infrared radiation will be obtained in the transition from :

The above is a plot of binding energy per nucleon $$E_b,$$ against the nuclear mass $$M;A,B,C,D,E,F$$ correspond to different nuclei. Consider four reactions : \eqalign{ & \left( i \right)\,\,\,\,\,\,\,\,\,\,A + B \to C + \varepsilon \,\,\,\,\,\,\,\,\,\,\left( {ii} \right)\,\,\,\,\,\,\,\,\,\,C \to A + B + \varepsilon \,\,\,\,\,\,\,\,\,\, \cr & \left( {iii} \right)\,\,\,\,\,\,D + E \to F + \varepsilon \,\,\,\,\,\,\,\,\,\,\left( {iv} \right)\,\,\,\,\,\,\,\,\,F \to D + E + \varepsilon ,\,\,\,\,\,\,\,\,\,\, \cr} where $$\epsilon$$ is the energy released? In which reactions is $$\epsilon$$ positive?

Two moles of helium gas are taken over the cycle $$ABCD,$$ as shown in the $$P$$-$$T$$ diagram. Assuming the gas to be ideal the work done on the gas in taking it from $$A$$ to $$B$$ is :

A long metallic bar is carrying heat from one of its ends to the other end under steady-state. The variation of temperature $$\theta$$ along the length $$x$$ of the bar from its hot end is best described by which of the following figures?

Two moles of helium gas are taken over the cycle $$ABCD,$$ as shown in the $$P$$-$$T$$ diagram. The work done on the gas in taking it from $$D$$ to $$A$$ is :

Two moles of helium gas are taken over the cycle $$ABCD$$, as shown in the $$P$$-$$T$$ diagram. The net work done on the gas in the cycle $$ABCDA$$ is:

One $$kg$$ of a diatomic gas is at a pressure of $$8\times 10^{4}N/m^2.$$ The density of the gas is $$4kg/m^3$$. What is the energy of the gas due to its thermal motion ?

Statement - 1: The temperature dependence of resistance is usually given as $$R=R_0\left(1+\alpha \Delta t\right).$$ The resistance of wire changes from $$100\Omega$$ to $$150\Omega$$ when its temperature is increased from $$27^{\circ }C$$ to $$227^{\circ }C$$. This implies that $$\alpha =2.5\times 10^{-3}/C.$$ Statement - 2: $$R=R_0\left(1+\alpha \Delta t\right)$$ is valid only when the change in the temperature $$\Delta T$$ is small and $$\Delta T=\left(R-R_0\right)<<R_0.$$

A charge $$Q$$ is placed at each of the opposite corners of a square. A charge $$q$$ is placed at each of the other two corners. If the net electrical force on $$Q$$ is zero, then $$Q/q$$ equals: