For 3 x 3 matrices M and N, which of the following statement(s) is (are) not correct?

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Consider the function \(f:(-\infty, \infty) \rightarrow(-\infty, \infty)\) defined by \(f(x)=\frac{x^2-a x+1}{x^2+a x+1} ; 0 < a < 2\)Question:
Which of the following is true ?

Let \(X\) be a random variable, and let \(P(X=x)\) denote the probability that \(X\) takes the value \(x\). Suppose that the points \((x, P(X=x)), x=0,1,2,3,4\), lie on a fixed straight line in the \(x y\)-plane, and \(P(X=x)=0\) for all \(x \in \mathbb{R}-\{0,1,2,3,4\}\). If the mean of \(X\) is \(\frac{5}{2}\), and the variance of \(X\) is \(\alpha\), then the value of \(24 \alpha\) is ___

Let \(f\) and \(g\) be real valued functions defined on interval \((-1,1)\) such that \(g^{\prime \prime}(x)\) is continuous, \(g(0) \neq 0, g^{\prime}(0)=0, g^{\prime \prime}(0) \neq 0\) and \(f(x)=g(x) \sin x\).
Statement \(1 \lim _{x \rightarrow 0}[g(x) \cos x-g(0) \operatorname{cosec} x]=f^{\prime \prime}(0)\).
Statement \(2 f^{\prime}(0)=g(0)\).

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A fair die is tossed repeatedly until a six is obtained. Let \(X\) denotes the number of tosses required.Question:
The conditional probability that \(X \geq 6\) given \(X>3\) equals

Match the following.
 

	List – I		List – II
(A)	The number of polynomials f(x) with non – negative integer coefficients of degree $\le 2,$ satisfying f(0) =0 and $\underset{0}{\overset{1}{\int }}f\left(x\right)dx=1, is$	(P)	8
(B)	The number of point in the interval $\left[- \sqrt{13}, \sqrt{13}\right]$ at which $f\left(x\right)=\sin \left(x^2\right)+\cos \left(x^2\right)$ attains its maximum value, is	(Q)	2
(C)	$\underset{-2}{\overset{2}{\int }}\frac{3x^2}{\left(1+e^x\right)}dx$ equals	(R)	4
(D)	$\frac{\left(\underset{-\frac{1}{2}}{\overset{\frac{1}{2}}{\int }}\cos 2x .\log \left(\frac{1+x}{1-x}\right)dx\right)}{\left(\underset{0}{\overset{\frac{1}{2}}{\int }}\cos 2x .\log \left(\frac{1+x}{1-x}\right)dx\right)}$	(S)	0

Match the following

A parallel plate capacitor having plates of area $\mathrm{S}$ and plate separation $\mathrm{d}$ , has capacitance ${\mathrm{C}}_1$ in air. When two dielectrics of different relative permittivities ( ${\mathrm{\epsilon }}_1=2\mathrm{}$ and ${\mathrm{\epsilon }}_2=4$ ) are introduced between the two plates as shown in the figure, the capacitance becomes ${\mathrm{C}}_2$ . The ratio $\frac{{\mathrm{C}}_2}{{\mathrm{C}}_1}$ is

A solid sphere of radius \(R\) has moment of inertia \(J\) about its qeometrical axis. If it is melted into a disc of radius \(r\) and thickness \(t\). If its moment of inertia about the tangential axis (which is perpendicular to plane of the disc), is also equal to \(I\), then the value of \(r\) is equal to

Let $X$ be a set with exactly $5$ elements and $Y$ be a set with exactly $7$ elements. If $\alpha$ is the number of one-one functions from $X\to Y$ and $\beta$ is the number of onto functions from $Y\to X$, then the value of $\frac{1}{5!}\left(\beta -\alpha \right)$ is _____-.

A reversible cyclic process for an ideal gas is shown below. Here, $P,V,\mathrm{a}\mathrm{n}\mathrm{d}T$ are pressure, volume and temperature, respectively. The thermodynamic parameters $q,w,H\mathrm{a}\mathrm{n}\mathrm{d}U$ are heat, work, enthalpy and internal energy, respectively.

The correct option(s) is (are)

A student performed the experiment of determination of focal length of a concave mirror by \(u-v\) method using an optical bench of length \(1.5 \mathrm{~m}\). The focal length of the mirror used is 24 \(\mathrm{cm}\). The maximum error in the location of the image can be \(0.2 \mathrm{~cm}\).
The 5 sets of \((u, v)\) values recorded by the student (in \(\mathrm{cm})\) are : \((42,56),(48\), \(48),(60,40),(66,33),(78,39)\). The data set(s) that cannot come from experiment and is (are) incorrectly recorded, is(are)

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In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be ' \(r\) '.
Question:
The number of atoms in this HCP unit cell is