A block is moving on an inclined plane making an angle \(45^{\circ}\) with the horizontal and the coefficient of friction is \(\mu\). The force required to just push it up the inclined plane is 3 times the force required to just prevent it from sliding down. If we define \(N=10 \mu\), then \(N\) is

Consider two identical galvanometers and two identical resistors with resistance R. If the internal resistance of the galvanometers $R_c<\frac{R}{2},$ which of the following statement(s) about any one of the galvanometers
is(are) true ?

At time \(t=0\), a battery of \(10 \mathrm{~V}\) is connected across points \(A\) and \(B\) in the given circuit. If the capacitors have no charge initially, at what time (in second) does the voltage across them become \(4 \mathrm{~V}\) ?
[Take \(: \ln 5=1.6, \ln 3=1.1]\)

A point source S is placed at the bottom of a transparent block of height 10 mm and refractive index 2.72. It is immersed in a lower refractive index liquid as shown in the figure. It is found that the light emerging from the block to the liquid forms a circular bright spot of diameter 11.54 mm on the top of the block. The refractive index of the liquid is

In the circuit shown in the figure, there are two parallel plate capacitors each of capacitance C. The switch S₁ is pressed first to fully charge the capacitor C₁ and the released. The switch S₂ is then pressed to charge the capacitor C₂. After some time, S₂ is released and then is S₃ pressed. After some time,

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A small block of mass \(M\) moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from \(60^{\circ}\) to \(30^{\circ}\) at point \(B\). The block is initially at rest at \(A\). Assume that collisions between the block and the incline are totally inelastic \(\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)\)

Question:
If collision between the block and the incline is completely elastic, then the vertical (upward) component of the velocity of the block at point \(B\), immediately after it strikes the second incline is

Three plane mirrors form an equilateral triangle with each side of length $L$. There is a small hole at a distance $l>0$ from one of the corners as shown in the figure. A ray of light is passed through the hole at an angle $\theta$ and can only come out through the same hole. The cross section of the mirror configuration and the ray of light lie on the same plane.

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

Match the following columns

The minimum value of the sum of real numbers \(a^{-5}, a^{-4}, 3 a^{-3}, 1, a^8\) and \(a^{10}\) with \(a>0\) is

The concentration of \(R\) in the reaction \(R \longrightarrow P\) was measured as a function of time and the following data is obtained :
\(\begin{array}{lllll}{[R] \text { (molar) }} & 1.0 & 0.75 & 0.40 & 0.10 \\ t \text { (min) } & 0.0 & 0.05 & 0.12 & 0.18\end{array}\)
The order of the reaction is

Consider a neutral conducting sphere. A positive point charge is placed outside the sphere. The net charge on the sphere is then

Let $M$ be a $3\times 3$ invertible matrix with real entries and let $I$ denote the $3\times 3$ identity matrix. $M^{-1}=adj\left(adjM\right)$, then which of the following statements is/are ALWAYS TRUE?

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Let \(M=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^{2}+y^{2} \leq r^{2}\right\}\),
where \(r>0 .\) Consider the geometric progression \(a_{n}=\frac{1}{2^{n-1}}, n=1,2,3, \ldots .\) Let \(S_{0}=0\) and, for \(n \geq 1\), let \(S_{n}\) denote the sum of the first \(n\) terms of this progression. For \(n \geq 1\), let \(C_{n}\) denote the circle with center \(\left(S_{n-1}, 0\right)\) and radius \(a_{n}\), and \(D_{n}\) denote the circle with center \(\left(S_{n-1}, S_{n-1}\right)\) and radius \(a_{n}\).

Question:
Consider \(M\) with \(r=\frac{\left(2^{199}-1\right) \sqrt{2}}{2^{198}} .\) The number of all those circles \(D_{n}\) that are inside \(M\) is