Four combinations of two thin lenses are given in List I. The radius of curvature of all curved surfaces is r and the refractive index of all the lenses is 1.5. Match lens combinations in List I with their focal length in List II and select the correct answer using the code given below the lists.

	List I		List II
A.		$\mathrm{P}.$	$2\mathrm{r}$
B.		$\mathrm{Q}.$	$\frac{\mathrm{r}}{2}$
C.		$\mathrm{R}.$	$-\mathrm{r}$
D.		$\mathrm{S}.$	$\mathrm{r}$

A conducting loop in the shape of right angled isosceles triangle of height 10 cm is kept such that the ${90}^o$ vertex is very close to an infinitely long conducting wire (see the figure). The wire is electrically insulated from the loop. The hypotenuse of the triangle is parallel to the wire. The current in the triangular loop is in counterclockwise direction and increased at constant rate of $10A{s}^{-1}.$ Which of the following statement(s) is (are) true?

Answer the following by appropriately matching the lists based on the information given in the paragraph.
A musical instrument is made using four different metal strings, $1,\mathrm{ }2,\mathrm{ }3$ and $4$ with mass per unit length $\mathrm{\mu },2\mathrm{\mu },3\mathrm{\mu }$ and $4\mathrm{\mu }$ respectively. The instrument is played by vibrating the strings by varying the free length in between the range ${\mathrm{L}}_0$ and $2{\mathrm{L}}_0.$ It is found that in string- $1\left(\mathrm{\mu }\right)$ at free length ${\mathrm{L}}_0$ and tension ${\mathrm{T}}_0$ the fundamental mode frequency is ${\mathrm{f}}_0.$
List-I gives the above four strings while list-II lists the magnitude of some quantity.
 

	List I		List II
$\left(\mathrm{a}\right)$	String $-1\left(\mathrm{\mu }\right)$	$\left(\mathrm{p}\right)$	$1$
$\left(\mathrm{b}\right)$	String $-2\left(2\mathrm{\mu }\right)$	$\left(\mathrm{q}\right)$	$\frac{1}{2}$
$\left(\mathrm{c}\right)$	String $-3\left(3\mathrm{\mu }\right)$	$\left(\mathrm{r}\right)$	$\frac{1}{\sqrt{2}}$
$\left(\mathrm{d}\right)$	String $-4\left(4\mathrm{\mu }\right)$	$\left(\mathrm{s}\right)$	$\frac{1}{\sqrt{3}}$
		$\left(\mathrm{t}\right)$	$\frac{3}{16}$
		$\left(\mathrm{u}\right)$	$\frac{1}{16}$

If the tension in each string is ${\mathrm{T}}_0,$ the correct match for the highest fundamental frequency in ${\mathrm{f}}_0$ units will be,

An \(\alpha\)-particle and a proton are accelerated from rest by a potential difference of \(100 \mathrm{~V}\). After this, their de Broglie wavelengths are \(\lambda_\alpha\) and \(\lambda_p\) respectively. The ratio \(\frac{\lambda_p}{\lambda_\alpha}\), to the nearest integer, is

A particle of mass \(m\) and charge \(q\), moving with velocity \(v\) enters Region II normal to the boundary as shown in the figure. Region II has a uniform magnetic field \(B\) perpendicular to the plane of the paper. The length of the Region II is \(l\). Choose the correct choice(s)

The mass \(M\) shown in the figure oscillates in simple harmonic motion with amplitude \(A\). The amplitude of the point \(P\) is

Two point-like objects of masses $20\mathrm{gm}$ and $30\mathrm{gm}$ are fixed at the two ends of a rigid massless rod of length $10\mathrm{cm}$. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2\times {10}^{-8}\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$. The angular frequency of the oscillations in $n\times {10}^{-3}\mathrm{rad}{\mathrm{s}}^{-1}$. The value of $n$ is _____.

Let $\left[x\right]$ be the greatest integer less than or equals to $x$. Then, at which of the following point(s) the function $f\left(x\right)=x\cos \left(\pi \left(x+\left[x\right]\right)\right)$ is discontinuous?

Two lines ${\text{L}}_1\text{ : }\text{x}=5\text{,}\frac{\text{y}}{3-\alpha }=\frac{\text{z}}{-2}$ and ${\text{L}}_2\text{ : }\text{x}=\alpha \text{,}\frac{\text{y}}{-1}=\frac{\text{z}}{2-\alpha }$ are coplanar. Then $\alpha$ can take value(s)

Paragraph:

Let \(a, r, s, t\) be nonzero real numbers. Let \(P\left(a t^{2}, 2 a t\right), Q, R\left(a r^{2}, 2 a r\right)\) and \(S\left(a s^{2}, 2 a s\right)\) be distinct points on the parabola \(y^{2}=4 a x\). Suppose that \(P Q\) is the focal chord and lines \(Q R\) and \(P K\) are parallel, where \(K\) is the point \((2 a, 0)\).


Question:

The value of \(r\) is

Two non-reactive monoatomic ideal gases have their atomic masses in the ratio 2 : 3. The ratio of their partial pressures, when enclosed in a vessel kept at a constant temperature, is 4 : 3. The ratio of their densities is 

As per IUPAC nomenclature, the name of the complex \(\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{4}\left(\mathrm{NH}_{3}\right)_{2}\right] \mathrm{Cl}_{3}\) is :

A ball of mass \(0.2 \mathrm{~kg}\) rests on a vertical post of height \(5 \mathrm{~m}\). A bullet of mass \(0.01 \mathrm{~kg}\), travelling with a velocity \(v \mathrm{~m} / \mathrm{s}\) in a horizontal direction, hits the centre of the ball. After the collision, the ball and bullet travel independently. The ball hits the ground at a distance of \(20 \mathrm{~m}\) and the bullet at a distance of \(100 \mathrm{~m}\) from the foot of the post. The initial velocity \(v\) of the bullet is