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If $\int \frac{x^4+1}{x^6+1}dx={\tan }^{-1}\left(f\right(x\left)\right)-\frac{2}{3}{\tan }^{-1}\left(g\right(x\left)\right)+c$, where $c$ is arbitrary constant, then the number of real root(s) of $f\left(x\right)=g\left(x\right)$ is

Why is your face upside down in the inward curve of the spoon?

In a horizontal pipeline of uniform cross-section the pressure falls by $8{\text{N/m}}^2$ between two points separated by $1\text{km}$. If oil of density $800{\text{kg/m}}^3$ flows through the pipe, find the change in $K.E$ per $\text{kg}$ of oil at these points.

In the circuit shown in the figure, reading of voltmeter is $V_1$ when only $S_1$ is closed, $V_2$ when only $S_2$ is closed and $V_3$ when both $S_1$ and $S_2$ are closed. Choose the correct relation between $V_1$, $V_2$ and $V_3$.

The velocity of a particle moving along a straight line increases according to the linear law v = $v_0$ + kx

where k is a constant . Then

1) the accelearation of the particle is k($v_0$ + kx)

2) the particle takes a time $\frac{1}{k}$ $lo{g}_e$ ($\frac{v_1}{v_0}$) to attain a velocity $v_1$

3) velocity varies linearly with displacement with slope of velocity displacement curve equal to k

4) data is insufficient to arrive at a conclusion.

Which of the following is/are necessary and sufficient condition(s) for pure rotation?

The centre of mass of a semi-circular ring lies $5\text{cm}$ above the centre of the semi-circle. What is the circumference of the ring?

1. 6.436

Two uniform identical rods each of mass $M$ and length $l$ are joined to form a cross as shown in figure. Find the moment of inertia of the cross about a bisector as shown dotted in the figure.

The figure shows the velocity (v) of a particle plotted against time (t).

Gauss's law should be invalid if

A charge particle projected with a velocity $v$ in a uniform magnetic field moves along a helical path of radius $3\text{cm}$ and pitch $6\pi \text{cm}$. Find the velocity of projection [Take, $B=0.3\text{T},q=0.02\text{C},m=3\times {10}^{-3}\text{kg}]$

A converging mirror $M_1$, a point source S and a diverging mirror $M_2$ are arranged as shown in figure. The source is placed at a distance of 30 cm from $M_1$. The focal length of each of the mirrors is 20 cm. Consider only the images formed by a maximum of two reflections. It is found that one image is formed on the source itself.

(a) Find the distance between the two mirrors.

(b) Find the location of the image formed by the single reflection from $M_2$.

A plane is inclined at an angle $\alpha ={30}^{\circ }$ with respect to the horizontal. A particle is projected with a speed $u=2\text{m/s}$, from the base of the plane, as shown in figure. The distance from the base, at which the particle hits the plane, is close to:

(Take $g=10{\text{m/s}}^2$)

Find the root mean square (rms) voltage for the given variation of voltage.

The largest wavelength in the ultraviolet region of the hydrogen spectrum is $122\text{nm}$. The smallest wavelength in the infrared region of the hydrogen spectrum (to the nearest integer) is

A body is executing simple harmonic motion with frequency $n,$ the frequency of its potential energy is

Calculate the acceleration of the block $B$ in the shown figure, assuming the surfaces and the pulleys $p_1$ and $p_2$ are smooth and the string is light.

The number of rotational degree of freedom for a diatomic gas is

Match List-I with List-II

List-I	List-II
(a) $\overrightarrow{C}-\overrightarrow{A}-\overrightarrow{B}=0$	(i)
(b) $\overrightarrow{A}-\overrightarrow{C}-\overrightarrow{B}=0$	(ii)
(c) $\overrightarrow{B}-\overrightarrow{A}-\overrightarrow{C}=0$	(iii)
(d) $\overrightarrow{A}+\overrightarrow{B}=\overrightarrow{C}$	(iv)

Curves in the graph shown give, as functions of radial distance $r$, the magnitude $B$ of the magnetic field inside and outside for long wires $\text{a}$, $\text{b}$, $\text{c}$ and $\text{d}$, carrying currents that are uniformly distributed across the cross-sections of the wires. The wires are far from one another. Which wire has the greatest radius ?

Select the correct statement(s)

Find the minimum force P required to avoid sliding of the block of weight W on the incline plane

An electric dipole consisting of two opposite charges of magnitude $2\times {10}^{-6}C$ each, separated by a distance of 3 cm is placed in an electric field of $2\times {10}^{5}N/C$. The maximum torque on the dipole will be

Pulley - massless, frictionless String - massless Wedge-frictionless In the figure find the acceleration of the blocks if the wedge is fixed.

In an $AC$ circuit, the current is given by $I=4\sin (100\pi t+{30}^{\circ })\text{A}$. The current becomes maximum for the first time $(\text{After}t=0)$ at $t$ equal to

The correct value of dip angle at a place is ${45}^{\circ }$. If the dip circle is rotated by ${45}^{\circ }$ out of the meridian, then the tangent of the angle of apparent dip at the place is

Two blocks of masses $m$ and $2m$ are placed on triangular wedge by means of a massless inextensible string over a frictionless fixed pulley. Surface of contact between the masses and wedge is frictionless and wedge makes ${45}^{\circ }$ with horizontal on both sides as shown in figure. The tensions in the string $\left(T\right)$ is

A capacitor of capacitance $25\mu \text{F}$ is charged to $300\text{V}$. It is then connected across a $10\text{mH}$ inductor. The resistance in the circuit is negligible. Find the magnetic and electric energy at $t=1.2\text{ms}$. $(\sin {137.6}^{\circ }=0.67\text{and}\cos {137.6}^{\circ }=-0.74)$