A person moved from \(A\) to \(B\) on a circular path as shown in figure. If the distance travelled by him is \(60 \,m\), then the magnitude of displacement would be\(.....\,m\) (Given \(\left.\cos 135^{\circ}=-0.7\right)\)

Assume that the displacement\((s)\) of air is proportional to the pressure difference \((\Delta p)\) created by a sound wave. Displacement\((s)\) further depends on the speed of sound \((v),\) density of air \((\rho)\) and the frequency \((f)\) If \(\Delta p \approx 10\, Pa , v  \approx 300\, m / s , p  \approx 1\, kg / m ^{3}\) and \(f \approx  1000 \,Hz\), then \(s\) will be the order of (take multiplicative constant to be \(1\) )

A motor operating on 100 V draws a current of 1 A. If the efficiency of the motor is \(91.6 \%\), then the loss of power in units of \(\mathrm{cal} / \mathrm{s}\) is

From a solid sphere of mass \(M\) and radius \(R\), a spherical portion of radius \(R/2\) is removed, as shown in the figure. Taking gravitational potential \(V = 0\) at \(r = \infty\), the potential at the centre of the cavity thus formed is 
( \(G =\) gravitational constant)

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : The density of the copper \(\left({ }_{29}^{64} \mathrm{Cu}\right)\) nucleus is greater than that of the carbon \(\left({ }_6^{12} \mathrm{C}\right)\) nucleus.
Reason (R): The nucleus of mass number A has a radius proportional to \(\mathrm{A}^{1 / 3}\).
In the light of the above statements, choose the most appropriate answer from the options given below :

A wire of length \(1\,m\) moving with velocity \(8\,m / s\) at right angles to a magnetic field of \(2\,T\). The magnitude of induced emf, between the ends of wire will be \(............\,V\)

A particle of mass \(M\) originally at rest is subjected to a force whose direction is constant but magnitude varies with time according to the relation \(F=F_{0}\left(1-\left(\frac{t-T}{T}\right)^{2}\right)\) Where \(F_{0}\) and \(T\) are constants. The force acts only for the time internal \(2 T\). The velocity \(v\) of the particle after time \(2 {T}\) is -

Let \(\lambda \in R , \vec{a}=\lambda \hat{i}+2 \hat{j}-3 \hat{k}, \vec{b}=\hat{i}-\lambda \hat{j}+2 \hat{k}\) If \(((\vec{a}+\vec{b}) \times(\vec{a} \times \vec{b})) \times(\vec{a}-\vec{b})=8 \hat{i}-40 \hat{j}-24 \hat{k}\), then \(|\lambda(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})|^2\) is equal to

Two identical small bar magnets each of dipole moment \(3\sqrt{5}\) J/T are placed at a center to center separation of \(10\) cm, with their axes perpendicular to each other as shown in figure. The value of magnetic field at the point P midway between the magnets is \(\alpha \times 10^{-3}\) T. The value of \(\alpha\) is ______.
(\(\mu_0=4\pi \times 10^{-7}\) Tm/A)

Let \(\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}\) and \(\vec{b}=\hat{i}+\hat{j}-\hat{k}\). If \(\vec{c}\) is a vector such that \(\vec{a} \cdot \vec{c}=11, \vec{b} \cdot(\vec{a} \times \vec{c})=27\) and \(\vec{b} \cdot \vec{c}=-\sqrt{3}|\vec{b}|\), then \(|\vec{a} \times \vec{c}|^2\) is equal to

A radiation is emitted by \(1000\, W\) bulb and it generates an electric field and magnetic field at \(P\), placed at a distance of \(2\, m\). The efficiency of the bulb is \(1.25 \%\). The value of peak electric field at \(P\) is \(x \times 10^{-1} \,V / m\). Value of \(x\) is. (Rounded-off to the nearest integer) [Take \(\varepsilon_{0}=8.85 \times 10^{-12} C ^{2} N ^{-1} m ^{-2}, c =3 \times 10^{8}\) \(ms ^{-1}\) ]

Let \(\vec{a}, \vec{b}, \vec{c}\) be three vectors mutually perpendicular to each other and have same magnitude. If a vector \(\overrightarrow{\mathrm{r}}\) satisfies. \(\overrightarrow{\mathrm{a}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{a}}\}+\overrightarrow{\mathrm{b}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{c}}) \times \overrightarrow{\mathrm{b}}\}+\overrightarrow{\mathrm{c}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{a}}) \times \overrightarrow{\mathrm{c}}\}=\overrightarrow{0}\) then \(\overrightarrow{\mathrm{r}}\) is equal to:

The area (in square units) of the region enclosed by the ellipse \(x^2+3 y^2=18\) in the first quadrant below the line \(y=x\) is :