A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $100$, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are:

Two adjacent sides of a parallelogram \(A B C D\) are given by \(\overrightarrow{\mathbf{A B}}=2 \hat{\mathbf{i}}+10 \hat{\mathbf{j}}+11 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{A D}}=-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\). The side \(A D\) is rotated by an acute angle \(\alpha\) in the plane of the parallelogram so that \(A D\) becomes \(A D^{\prime}\). If \(A D^{\prime}\) makes a right angle with the side \(A B\), then the cosine of the angle \(\alpha\) is given by

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Let \(A_1, G_1, H_1\) denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For \(n \geq 2\), let \(A_{n-1}\) and \(H_{n-1}\) has arithmetic, geometric and harmonic means as \(A_n, G_n, H_n\), respectively.
Question:
Which of the following statements is correct?

Let \(k \in \mathbb{R}\). If \(\lim _{x \rightarrow 0^+}(\sin (\sin k x)+\cos x+x)^{\frac{2}{x}}=e^6\), then the value of \(k\) is

A line with positive direction cosines passes through the point \(P(2,-1,2)\) and makes equal angles with the coordinate axes. The line meets the plane \(2 x+y+z=9\) at point \(Q\). The length of the line segment \(P Q\) equals

A number is chosen at random from the set $\left\{1,2,3,\dots ,2000\right\}$. Let $p$ be the probability that the chosen number is a multiple of 3 or $7$. Then the value of $500p$ is ____.

Axis of a parabola is \(y=x\) and vertex and focus are at a distance \(\sqrt{2}\) and \(2 \sqrt{2}\) respectively, from the origin. Then, equation of the parabola is

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A small block of mass \(1 \mathrm{~kg}\) is released from rest at the top of a rough track. The track is a circular arc of radius \(40 \mathrm{~m}\). The block slides along the track without toppling and a frictional force acts on it in the direction opposite to the instantaneous velocity. The work done in overcoming the friction up to the point \(Q\), as shown in the figure below, is \(150 \mathrm{~J}\). (Take the acceleration due to gravity, \(g=10 \mathrm{~m} \mathrm{~s}^{-2}\) ).





Question:

The speed of the block when it reaches the point \(Q\) is

In a radioactive decay process, the activity is defined as $A=-\frac{dN}{dt}$, where $N\left(t\right)$ is the number of radioactive nuclei at time $\left(t\right)$. Two radioactive sources, $S_1$ and $S_2$ have same activity at time $t=0$. At a later time, the activities of $S_1$ and $S_2$ are $A_1$ and $A_2$, respectively. When $S_1$ and $S_2$ have just completed their $3^{\mathrm{rd}}$ and $7^{\mathrm{th}}$ half-lives, respectively, the ratio $\frac{A_1}{A_2}$ is _____.

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Consider a simple \(R C\) circuit as shown in Figure \(1\).

Process 1: In the circuit the switch \(S\) is closed at \(t=0\) and the capacitor is fully charged to voltage \(V_{0}\) (i.e., charging continues for time \(T>>R C\) ). In the process some dissipation \(\left(E_{D}\right)\) occurs across the resistance \(R\). The amount of energy finally stored in the fully charged capacitor is \(E_{C}\).

Process 2: In a different process the voltage is first set to \(\frac{V_{0}}{3}\) and maintained for a charging time \(T>>R C\). Then the voltage is raised to \(\frac{2 V_{0}}{3}\) without discharging the capacitor and again maintained for a time \(T>>R C\). The process is repeated one more time by raising the voltage to \(V_{0}\) and the capacitor is charged to the same final voltage \(V_{0}\) as in Process 1.

These two processes are depicted in Figure \(2 .\)







Question:

In Process 1 , the energy stored in the capacitor \(E_{C}\) and heat dissipated across resistance \(E_{D}\) are related by:

A block of mass \(5 \mathrm{~kg}\) moves along the \(x\)-direction subject to the force \(F=(-20 x+10) \mathrm{N}\), with the value of \(x\) in metre. At time \(t=0 \mathrm{~s}\), it is at rest at position \(x=1 \mathrm{~m}\). The position and momentum of the block at \(t=(\pi / 4) \mathrm{s}\) are

In a bimolecular reaction, the steric factor $\mathrm{P}$ was experimentally determined to be $4.5$. The correct option(s) among the following is(are)

Let \(A B C D\) be a quadrilateral with area 18 , with side \(A B\) parallel to the side \(C D\) and \(A B=2 C D\). Let \(A D\) be perpendicular to \(A B\) and \(C D\). If a circle is drawn inside the quadrilateral \(A B C D\) touching all the sides, then its radius is