Two parallel chords of a circle of radius 2 are at a distance \(\sqrt{3}+1\) apart. If the chords subtend at the centre, angles of \(\frac{\pi}{k}\) and \(\frac{2 \pi}{k}\), where \(k>0\), then the value of \([k]\) is
[Note : \([k]\) denotes the largest integer less than or equal to \(k\) ]

The product of all positive real values of $x$ satisfying the equation $x^{\left(16{\left({\log }_{5}x\right)}^3-68{\log }_{5}x\right)}=5^{-16}$ is

Let \(P(3,2,6)\) be a point in space and \(Q\) be a point on the line \(\mathbf{r}=(\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})+\mu(-3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+5 \hat{\mathbf{k}})\).
Then, the value of \(\mu\) for which the vector \(\mathbf{P Q}\) is parallel to the plane \(x-4 y+3 z=1\) is

If \(n(X)={ }^m C_6\), then the value of \(m\) is

\[
\text { Let } M \text { be a } 3 \times 3 \text { matrix satisfying }
\]

\[
M\left[\begin{array}{l}
0 \\
1 \\
0
\end{array}\right]=\left[\begin{array}{c}
-1 \\
2 \\
3
\end{array}\right], M\left[\begin{array}{c}
1 \\
-1 \\
0
\end{array}\right]=\left[\begin{array}{c}
1 \\
1 \\
-1
\end{array}\right] \text { and } M\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right]=\left[\begin{array}{c}
0 \\
0 \\
12
\end{array}\right]
\]
Then, the sum of the diagonal entries of \(M\) is

Let $F:R\to R$ be a function. We say that $f$ has:
 PROPERTY $1$  if $\underset{h\to 0}{\lim }\frac{f\left(h\right)-f\left(0\right)}{\sqrt{\left|h\right|}}$ exists and is finite, and 
PROPERTY $2$  if $\underset{h\to 0}{\lim }\frac{f\left(h\right)-f\left(0\right)}{h^2}$ exists and is finite.
Then which of the following options is/are correct?

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 balls, having linear momenta \(\overrightarrow{\mathbf{p}}_1=p \hat{\mathbf{i}}\) and \(\overrightarrow{\mathbf{p}}_2=-p \hat{\mathbf{i}}\), undergo a collision in free space. There is no external force acting on the balls. Let \(\overrightarrow{\mathbf{p}}_1^{\prime}\) and \(\overrightarrow{\mathbf{p}}_2^{\prime}\) be their final momenta. The following option(s) is/are NOT ALLOWED for any non-zero value of \(p, a_1, a_2, b_1, b_2, c_1\) and \(c_2\)

A block $M$ hangs vertically at the bottom end of a uniform rope of constant mass per unit length. The top end of the rope is attached to a fixed rigid support at O. A transverse wave pulse (Pulse 1) of wavelength ${\lambda }_0$ is produced at point O on the rope. The pulse takes time $T_{OA}$ to reach point A. If the wave pulse of wavelength ${\lambda }_0$ is produced at point A (Pulse 2) without disturbing the position of M it takes time $T_{OA}$ to reach point O. Which of the following options is/are correct?

A person blows into open-end of a long pipe. As a result, a high pressure pulse of air travels down the pipe.

When this pulse reaches the other end of the pipe,

Let \(\frac{\pi}{2} < x < \pi\) be such that \(\cot x=\frac{-5}{\sqrt{11}}\). Then \(\left(\sin \frac{11 x}{2}\right)(\sin 6 x-\cos 6 x)+\left(\cos \frac{11 x}{2}\right)(\sin 6 x+\cos 6 x)\) is equal to

Paragraph:

The figure shows a circular loop of radius \(a\) with two long parallel wires (numbered \(1\) and \(2\) ) all in the plane of the paper. The distance of each wire from the centre of the loop is \(d\). The loop and the wires are carrying the same current \(I\). The current in the loop is in the counterclockwise direction if seen from above.



Question:

Consider \(d \gg a\), and the loop is rotated about its diameter parallel to the wires by \(30^{\circ}\) from the position shown in the figure. If the currents in the wires are in the opposite directions, the torque on the loop at its new position will be (assume that the net field due to the wires is constant over the loop)

Sulfide ores are common for the metals