The number of solutions of the pair of equations \(2 \sin ^2 \theta-\cos 2 \theta=0\) and \(2 \cos ^2 \theta-3 \sin \theta=0\) in the interval \([0,2 \pi]\) 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

Paragraph:
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 \(r=\frac{1025}{513}\). Let \(k\) be the number of all those circles \(C_{n}\) that are inside \(M\). Let \(l\) be the maximum possible number of circles among these \(k\) circles such that no two circles intersect. Then

The total number of distinct $x\in R$ for which $\left|\begin{array}{ccc}x& x^2& 1+x^3\\ 2x& 4x^2& 1+8x^3\\ 3x& 9x^2& 1+27x^3\end{array}\right|=10$ is

Let \(S\) be the focus of the parabola \(y^{2}=8 \mathrm{x}\) and let \(P Q\) be the common chord of the circle \(x^{2}+y^{2}-2 x-4 y=0\) and the given parabola. The area of the triangle \(P Q S\) is

Let $f, g :\left[-1, 2\right]\to \mathbb{ }\mathbb{R}$   be continuous functions which are twice differentiable on the interval $\left(-1, 2\right)$. Let the values of $f$ and $g$ at the points $-1, 0$ and $2$ be as given in the following table:
 

	$x= -1$	$x = 0$	$x = 2$
$f\left(x\right)$	3	6	0
$g\left(x\right)$	0	1	-1

In each of the intervals $\left(-1, 0\right)$ and $\left(0, 2\right)$ the function $(f-3g)"$ never vanishes. Then the correct statement(s) is(are)

Column I		Column II
$\left(\mathrm{A}\right)$	In a triangle $∆XYZ$ let $a,b$ and $\mathrm{c}$ be the lengths of the angles $X,Y$ and $\mathrm{Z}$, respectively. If $2\left(a^2-b^2\right)=c^2$ and $\lambda =\frac{\sin \left(X-Y\right)}{\mathrm{sinZ}}$then possible values of $n$ for which $\cos \left(n\pi \lambda \right)=0$ is are	$\left(\mathrm{P}\right)$	$1$
$\left(\mathrm{B}\right)$	In a triangle $\mathrm{\Delta }XYZ,$ let $a,b$ and $c$ be the lengths of the sides opposite to the angles $X,Y$ and $Z$, respectively. If $1+\cos 2x-2\cos 2y$$=2\sin x\sin y$, then possible value(s) of $\frac{a}{b}$ is (are)	$\left(\mathrm{Q}\right)$	$2$
$\left(\mathrm{C}\right)$	In ${\mathbb{R}}^2,$ let $\sqrt{3}\mathrm{ }\hat{\mathrm{i}}+\hat{\mathrm{j}},\mathrm{ }\hat{\mathrm{i}}+\sqrt{3}\mathrm{ }\hat{\mathrm{j}}$ and $\beta \hat{\mathrm{i}}+\left(1-\beta \right)\mathrm{ }\hat{\mathrm{j}}$ be the position vectors of $X,Y$ and $Z$ with respect to the origin $O$, respectively. If the distance of $Z$ from the bisector of the acute angle of $\overrightarrow{OX}$ with $\overrightarrow{OY}$ is $\frac{3}{\sqrt{2}}$, then possible value (s) of $\left|\beta \right|$ is (are)	$\left(\mathrm{R}\right)$	$3$
$\left(\mathrm{D}\right)$	Suppose that $F\left(\alpha \right)$ denotes the area of the region bounded by $x=0,\mathrm{ }x=2, y^2=4x$ and$y=\left|ax-1\right|+$$\left|\alpha x-2\right|+\alpha x,$  where $\alpha \in \left\{0, 1\right\}.$ Then the value (s) of $F\left(\alpha \right)+\frac{8}{3} \sqrt{2},$ when $\alpha =0$ and $\alpha =1$, is (are)	$\left(\mathrm{S}\right)$	$5$
		$\left(\mathrm{T}\right)$	$6$

 

Among \(\mathrm{V}(\mathrm{CO})_6, \mathrm{Cr}(\mathrm{CO})_5, \mathrm{Cu}(\mathrm{CO})_3,\)\(\operatorname{Mn}(\mathrm{CO})_5, \mathrm{Fe}(\mathrm{CO})_5,\left[\mathrm{Co}(\mathrm{CO})_3\right]^{3-},\left[\mathrm{Cr}(\mathrm{CO})_4\right]^{4-}\), and \(\operatorname{Ir}(\mathrm{CO})_3\), the total number of species isoelectronic with \(\mathrm{Ni}(\mathrm{CO})_4\) is ________.
[Given, atomic number:\(\mathrm{V}=23, \mathrm{Cr}=24, \mathrm{Mn}=25,\) \(\mathrm{Fe}=26, \mathrm{Co}=27, \mathrm{Ni}=28, \mathrm{Cu}=29,\) \(\mathrm{Ir}=77\)]

Correct option(s) for the following sequence of reactions is(are)

The cyanide process of gold extraction involves leaching out gold from its ore with ${\mathrm{C}\mathrm{N}}^{-}$ in the presence of $\mathrm{Q}$ in water to form $\mathrm{R}$ . Subsequently, $\mathrm{R}$ is treated with $\mathrm{T}$ to obtain $\mathrm{A}\mathrm{u}$ and $\mathrm{Z}$ . Choose the correct option(s).

Which one of the following options represents the magnetic field $\overrightarrow{B}$at $O$ due to the current flowing in the given wire segments lying on the $xy$plane?

Paragraph:
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

Let $\overrightarrow{a}=2\widehat{i}+\widehat{j}-\widehat{k}$ and $\overrightarrow{b}=\widehat{i}+2\widehat{j}+\widehat{k}$ be two vectors. Consider a vector $\overrightarrow{c}=\alpha \overrightarrow{a}+\beta \overrightarrow{b}, \alpha , \beta \in R.$ If the projection of $\overrightarrow{c}$ on the vector $\left(\overrightarrow{a}+\overrightarrow{b}\right)$ is $3\sqrt{2},$ then the minimum value of $\left(\overrightarrow{c}-\left(\overrightarrow{a}\times \overrightarrow{b}\right)\right).\overrightarrow{c}$ equals