A curve passes through the point $\left(1,\frac{\pi }{6}\right)$. Let the slope of the curve at each point $\left(\text{x},\text{y}\right)$ be $\frac{\text{y}}{\text{x}}+\text{sec}\left(\frac{\text{y}}{\text{x}}\right),\; x > 0$. Then the equation of the curve is

If \(y(x)\) satisfies the differential equation \(y^{\prime}-y \tan x\) \(=2 x \sec x\) and \(y(0)=0\), then

Let \(\theta \in\left(0, \frac{\pi}{4}\right)\) and \(t_1=(\tan \theta)^{\tan \theta},\) \(t_2=(\tan \theta)^{\cot \theta}\) and \(t_4=(\cot \theta)^{\tan \theta}\), then

For a > b > c > 0, the distance between (1, 1) and the point of intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is less than $2\sqrt{2}$.Then

Let $w=\frac{\sqrt{3}+\text{i}}{2}$ and $\text{P}=\left\{w^{\text{n}}\text{: }\text{n}=\text{1, 2, 3,}\dots \right\}$.Further ${\text{H}}_1=\left\{\text{z}\in \text{C}\text{ : }\text{Re z}>\frac{1}{2}\right\}$ and ${\text{H}}_2=\left\{\text{z}\in \text{C}\text{ : }\text{Re z}<\frac{-1}{2}\right\}$, where C is the set of all complex numbers. If ${\text{z}}_1\in \text{P}\cap {\text{H}}_1\text{, }{\text{z}}_2\in \text{P}\cap {\text{H}}_2$ and O represents the origin, then $\angle {\text{z}}_1\text{O}{\text{z}}_2=$

The value of  $cot \left(\underset{\text{n}=1}{\overset{23}{\Sigma }}{\text{cot}}^{-1}\left(1+\underset{\text{k}=1}{\overset{\text{n}}{\Sigma }}2\text{k}\right)\right)$  is

Let $e$ denote the base of the natural logarithm. The value of the real number $a$ for which the right-hand limit $\underset{x\to 0^{+}}{\lim }\frac{{\left(1-x\right)}^{\frac{1}{\chi }}-e^{-1}}{x^a}$ is equal to a non-zero real number, is

A ray of light travelling in water is incident on its surface open to air. The angle of incidence is \(\theta\), which is less than the critical angle. Then there will be

Plotting $1/{\mathrm{\Lambda }}_{\mathrm{m}}$ against ${\mathrm{c\Lambda }}_{\mathrm{m}}$ for aqueous solutions of a monobasic weak acid $\left(\mathrm{HX}\right)$ resulted in a straight line with $\mathrm{y}-$axis intercept of $\mathrm{P}$ and slope of $\mathrm{S}$. The ratio $\mathrm{P}/\mathrm{S}$ is
$\left[{\mathrm{\Lambda }}_{\mathrm{m}}=\right.\text{ molar conductivity }$ $$\begin{gathered} {\mathrm{\Lambda }}_{\mathrm{m}}^{\mathrm{o}}= \mathrm{limiting} \mathrm{molar}conductivity \\ \mathrm{c}= \mathrm{molar}concentration \end{gathered}$$ $\left.{\mathrm{K}}_{\mathrm{a}}=\text{ dissociation constant of }\mathrm{HX}\right]$

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A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
Question :
The maximum value of \(v_0\) for which the disk will roll without slipping is

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A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity \(\omega\) is an example of a non-inertial frame of reference.

The relationship between the force \(\vec{F}_{\text {rot }}\) experienced by a particle of mass \(m\) moving on the rotating disc and the force \(\vec{F}_{\text {in }}\) experienced by the particle in an inertial frame of reference is

\(\vec{F}_{\mathrm{rot}}=\vec{F}_{\mathrm{in}}+2 m\left(\vec{v}_{\mathrm{rot}} \times \vec{\omega}\right)+m(\vec{\omega} \times \vec{r}) \times \vec{\omega}\),

where \(\vec{v}_{\text {rot }}\) is the velocity of the particle in the rotating frame of reference and \(\vec{r}\) is the position vector of the particle with respect to the centre of the disc.

Now consider a smooth slot along a diameter of a disc of radius \(R\) rotating counter-clockwise with a constant angular speed \(\omega\) about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the \(x\)-axis along the slot, the \(y\)-axis perpendicular to the slot and the \(z\)-axis along the rotation axis \((\vec{\omega}=\omega \hat{k}) .\) A small block of mass \(m\) is gently placed in the slot at \(\vec{r}=(R / 2) \hat{i}\) at \(t=0\) and is constrained to move only along the slot.

Question :
The net reaction of the disc on the block is

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Box \(1\) contains three cards bearing numbers \(1,2,3\); box \(2\) contains five cards bearing numbers \(1,2,3,4,5\); and box \(3\) contains seven cards bearing numbers \(1,2,3,4,5,6,7\). A card is drawn from each of the boxes. Let \(x_{i}\) be the number on the card drawn from the \(i^{t h}\) box, \(i=1,2,3\).


Question:

The probability that \(x_{1}+x_{2}+x_{3}\) is odd, is

The number of isomeric tetraenes (NOT containing sp-hybridized carbon atoms) that can be formed from the following reaction sequence is _____ .