Let $\mathrm{\omega }\ne 1$ be a cube root of unity. Then the minimum of the set $\{{\left|\mathrm{a}+\mathrm{b}\mathrm{\omega }+\mathrm{c}{\mathrm{\omega }}^2\right|}^2:\mathrm{a},\mathrm{b},\mathrm{c}$ distinct non$–$zero integers $\}$ equals ________

An ellipse intersects the hyperbola \(2 x^2-2 y^2=1\) orthogonally. The eccentricity of the ellipse is reciprocal to that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then

In a triangle $PQR$ , let $\overrightarrow{a}=\overrightarrow{QR}, \overrightarrow{b}=\overrightarrow{RP}$ and $\overrightarrow{c}=\overrightarrow{PQ}$ . If $\left|\overrightarrow{a}\right|=3, \left|\overrightarrow{b}\right|=4$ and $\frac{\overrightarrow{a}.\left(\overrightarrow{c}-\overrightarrow{b}\right)}{\overrightarrow{c}.\left(\overrightarrow{a}-\overrightarrow{b}\right)}=\frac{\left|\overrightarrow{a}\right|}{\left|\overrightarrow{a}\right|+\left|\overrightarrow{b}\right|}$, then the value of ${\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2$ is _______

If \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) and \(\mathbf{d}\) are the unit vectors such that \((\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=1\) and \(\mathbf{a} \cdot \mathbf{c}=\frac{1}{2}\), then

If \(z\) is any complex number satisfying \(|z-3-2 i| \leq 2\), then the maximum value of \(|2 z-6+5 i|\) is

Three lines are given by $\overrightarrow{\mathrm{r}}=\mathrm{\lambda }\hat{\mathrm{i}}, \lambda \in \mathrm{R}$
$\overrightarrow{\mathrm{r}}=\mathrm{\mu }\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}\right),\mathrm{\mu }\in \mathrm{R}$ and
$\overrightarrow{\mathrm{r}}=\mathrm{\nu }\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}+\widehat{\mathrm{k}}\right),\mathrm{\nu }\in \mathrm{R}.$
Let the lines cut the plane $\mathrm{x}+\mathrm{y}+\mathrm{z}=1$ at the points $\mathrm{A},\mathrm{ }\mathrm{B}$ and $\mathrm{C}$ respectively. If the area of the triangle $\mathrm{A}\mathrm{B}\mathrm{C}$ is $\mathrm{\Delta }$ then the value of ${\left(6\mathrm{\Delta }\right)}^2$ equals_________

Three numbers are chosen at random, one after another with replacement, from the set \(S=\{1,2,3, \ldots, 100\}\). Let \(p_{1}\) be the probability that the maximum of chosen numbers is at least 81 and \(p_{2}\) be the probability that the minimum of chosen numbers is at most 40 .
The value of $\frac{625}{4}{p}_1$ is

For any complex number $w=c+id$, let $\arg \left(w\right)\in (-\pi ,\pi ]$, where $i=\sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z=x+iy$ satisfying $\arg \left(\frac{z+\alpha }{z+\beta }\right)=\frac{\pi }{4}$, the ordered pair $(x, y)$ lies on the circle $x^2+y^2+5x-3y+4=0$
Then which of the following statements is(are) TRUE?

When \(20 \mathrm{~g}\) of naphthoic acid \(\left(\mathrm{C}_{11} \mathrm{H}_8 \mathrm{O}_2\right)\) is dissolved in \(50 \mathrm{~g}\) of benzene \(\left(K_f=1.72 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right)\), a freezing point depression of \(2 \mathrm{~K}\) is observed. The van't Hoff factor \((i)\) is

Let \(\omega\) be a complex cube root of unity with \(\omega \neq 1\). A fair die is thrown three times. If \(r_1, r_2\) and \(r_3\) are the numbers obtained on the die, then the probability that \(\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0\) is

In the figure, the inner (shaded) region $A$ represents a sphere of radius $r_A=1$, within which the electrostatic charge density varies with the radial distance $r$ from the center as ${\rho }_A=kr$, where $k$ is positive. In the spherical shell $B$ of outer radius $r_B$, the electrostatic charge density varies as ${\rho }_B=\frac{2k}{r}$. Assume that dimensions are taken care of. All physical quantities are in their SI units.

Which of the following statement(s) is/(are) correct?

Two parallel chords of a circle of radius 2 are at a distance \(\sqrt{3}+1\) apart. If the chords subtend at the centre, angles of \(\frac{\pi}{k}\) and \(\frac{2 \pi}{k}\), where \(k>0\), then the value of \([k]\) is
[Note : \([k]\) denotes the largest integer less than or equal to \(k\) ]

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When a particle of mass \(m\) moves on the \(x\)-axis in a potential of the form \(V(x)=k x^2\), 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 when its potential energy increases on both sides of \(x=0\) in a way different from \(k x^2\) and its total energy is such that the particle does not escape to infinity. Consider a particle of mass \(\mathrm{m}\) moving on the \(x\)-axis. Its potential energy is \(V(x)=\alpha x^4(\alpha>0)\) for \(|x|\) near the origin and becomes a constant equal to \(V_0\) for \(|x| \geq X_0\) (see figure).
Question :
The acceleration of this particle for \(|x|>X_0\) is