A block of mass \(0.18 \mathrm{~kg}\) is attached to a spring of force constant \(2 \mathrm{~N} / \mathrm{m}\). The coefficient of friction between the block and the floor is 0.1. Initially, the block is at rest and the spring is unstretched. An impulse is given to the block as shown in the figure. The block slides a distance of \(0.06 \mathrm{~m}\) and comes to rest for the first time. The initial velocity of the block in \(\mathrm{m} / \mathrm{s}\) is \(v=\frac{N}{10}\). Then, \(N\) is

An audio transmitter \((T)\) and a receiver \((R)\) are hung vertically from two identical massless strings of length 8 m with their pivots well separated along the \(X\) axis. They are pulled from the equilibrium position in opposite directions along the \(X\) axis by a small angular amplitude \(\theta_0=\cos ^{-1}(0.9)\) and released simultaneously. If the natural frequency of the transmitter is 660 Hz and the speed of sound in air is \(330 \mathrm{~m} / \mathrm{s}\), the maximum variation in the frequency (in Hz ) as measured by the receiver (Take the acceleration due to gravity \(g=10 \mathrm{~m} / \mathrm{s}^2\) ) is \(\qquad\)

A projectile is thrown from a point \(\mathrm{O}\) on the ground at an angle \(45^{\circ}\) from the vertical and with a speed \(5 \sqrt{2} \mathrm{~m} / \mathrm{s}\). The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, \(0.5 \mathrm{~s}\) after the splitting. The other part, \(t\) seconds after the splitting, falls to the ground at a distance \(x\) meters from the point O. The acceleration due to gravity \(g=10 \mathrm{~m} / \mathrm{s}^{2}\).
The value of $t$ is _________.

A double slit setup is shown in the figure. One of the slits is in medium $2$ of refractive index $n_2$. The other slit is at the interface of this medium with another medium $1$ of refractive index $n_1\left(\ne n_2\right)$. The line joining the slits is perpendicular to the interface and the distance between the slits is $d$. The slit widths are much smaller than $d$. A monochromatic parallel beam of light is incident on the slits from medium $1$. A detector is placed in medium $2$ at a large distance from the slits, and at an angle $\theta$ from the line joining them, so that $\theta$ equals the angle of refraction of the beam. Consider two approximately parallel rays from the slits received by the detector.
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Which of the following statement(s) is(are) correct?

In terms of potential difference $V$ , electic current $I$ , permittivity ${\epsilon }_0$ , permeability ${\mu }_0$ and speed of light $c$ , the dimensionally correct equation(s) is (are)

Sunlight of intensity $1.3 kW m^{-2}$ is incident normally on a thin convex lens of focal length $20 cm$ . Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light, in $kW m^{-2}$ , at a distance $22 cm$ from the lens on the other side is __________.

An infinitely long wire, located on the \(z\)-axis, carries a current \(I\) along the \(+z\)-direction and produces the magnetic field \(\vec{B}\). The magnitude of the line integral \(\int \vec{B} \cdot \overrightarrow{d l}\) along a straight line from the point \((-\sqrt{3} a, a, 0)\) to \((a, a, 0)\) is given by
[ \(\mu_0\) is the magnetic permeability of free space.]

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\(P\) and \(Q\) are isomers of dicarboxylic acid \(\mathrm{C}_{4} \mathrm{H}_{4} \mathrm{O}_{4}\). Both decolorize \(\mathrm{Br}_{2} / \mathrm{H}_{2} \mathrm{O}\). On heating, \(\boldsymbol{P}\) forms the cyclic anhydride.

Upon treatment with dilute alkaline \(\mathrm{KMnO}_{4}, \boldsymbol{P}\) as well as \(\boldsymbol{Q}\) could produce one or more than one from \(\boldsymbol{S}, \boldsymbol{T}\) and \(\boldsymbol{U}\).




Question:
Compounds formed from \(\boldsymbol{P}\) and \(\boldsymbol{Q}\) are, respectively

Let $\hat{i},\hat{j}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes. Let $\overrightarrow{a}=3\hat{i}+\hat{j}-\hat{k}$ $\overrightarrow{b}=\hat{i}+b_2\hat{j}+b_3\hat{k}, b_2,b_3\in \mathrm{ℝ}$ $\overrightarrow{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k},c_1,c_2,c_3\in \mathrm{ℝ}$ be three vectors such that $b_2{b}_3>0,\overrightarrow{a}·\overrightarrow{b}=0$ and $\left(\begin{array}{ccc}0& -c_3& c_2\\ c_3& 0& -c_1\\ -c_2& c_1& 0\end{array}\right)\left(\begin{array}{l}1\\ b_2\\ b_3\end{array}\right)=\left(\begin{array}{c}3-c_1\\ 1-c_2\\ -1-c_3\end{array}\right).$ Then, which of the following is/are TRUE?

For the following reaction scheme, percentage yields are given along the arrow:

\(\mathbf{x} g\) and \(\mathbf{y} g\) are mass of \(\mathbf{R}\) and \(\mathbf{U}\), respectively.
(Use: Molar mass (in \(\mathrm{g} \mathrm{mol}^{-1}\) ) of \(\mathrm{H}, \mathrm{C}\) and \(\mathrm{O}\) as 1,12 and 16 , respectively)
The value of $\mathrm{x}$ is ___.

The correct stability order of the following resonance structure is

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There are some deposits of nitrates and phosphates in earth's crust. Nitrates are more soluble in water. Nitrates are difficult to reduce under the laboratory conditions but microbes do it easily. Ammonia forms large number of complexes with transition metal ions. Hybridisation easily explains the ease of sigma donation capability of \(\mathrm{NH}_3\) and \(\mathrm{PH}_3\). Phosphine is a flammable gas and is prepared from white phosphorus.
Question:
Among the following, the correct statement is

For $x\in R$, the number of real roots of the equation $3x^2-4\left|x^2-1\right|+x-1=0$ is