For $x\in \mathrm{ℝ}$, let the function $y\left(x\right)$ be the solution of the differential equation $\frac{dy}{dx}+12y=\cos \left(\frac{\pi }{12}x\right),y\left(0\right)=0.$ Then, which of the following statements is/are TRUE?

An ellipse intersects the hyperbola \(2 x^2-2 y^2=1\) orthogonally. The eccentricity of the ellipse is reciprocal to that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and more over the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is

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Consider the polynomial \(f(x)=1+2 x+3 x^2+4 x^3\). Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t=|s|\).Question:
The area bounded by the curve \(y=f(x)\) and the lines \(x=0, y=0\) and \(x=t\), lies in the interval

For each positive integer $n$, let $y_n={\frac{1}{n}\left((n+1)(n+2)\mathrm{...}(n+n)\right)}^{\frac{1}{n}}$. If $\underset{n\to \infty }{\lim }{y}_n=L,$ then the value of $\left[L\right]$ (where $\left[x\right]$ is the greatest integer less than or equal to $x$) is ____

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is $6:11$ and the seventh terms lies in between $130$ and $140$, then the common difference of this A.P. is

If the line $x=\alpha$ divides the area of the region $R=\{\left(x,y\right)\in R^2:x^3\le y\le x, 0\le x\le 1\}$ into two equal parts, then

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p-amino-N, N-dimethylaniline is added to a strongly acidic solution of \(X\). The resulting solution is treated with a few drops of aqueous solution of \(Y\) to yield blue colouration due to the formation of methylene blue. Treatment of the aqueous solution of \(Y\) with the reagent potassium hexacyanoferrate(II) leads to the formation of an intense blue precipitate. The precipitate dissolves an excess addition of the reagent. Similarly, treatment of the solution of \(Y\) with the solution of potassium hexacyanoferrate(III) leads to a brown colouration due to the formation of \(Z\).
Question:
The compound \(Y\), is

The largest value of the non - negative integer a for which $\underset{x\to 1}{\lim }{\left\{\frac{-ax+\sin \left(x-1\right)+a}{x+\sin \left(x-1\right)-1}\right\}}^{\frac{1-x}{1- \sqrt{x}}}=\frac{1}{4}$ is _________

Let \(\left(x_0, y_0\right)\) be the solution of the following equations \((2 x)^{\log 2}=(3 y)^{\log 3}\), \(3^{\log x}=2^{\log y}\), then \(x_0\) is equal to

A decapeptide (molecular weight 796) on complete hydrolysis gives glycine (molecular weight 75), alanine and phenylalanine. Glycine contributes \(47.0 \%\) to the total weight of the hydrolysed products. The number of glycine units present in the decapeptide is

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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 empty space in this \(\mathrm{HCP}\) unit cell is

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Upon heating \(\mathrm{KClO}_{3}\) in the presence of catalytic amount of \(\mathrm{MnO}_{2}\), a gas \(\mathbf{W}\) is formed. Excess amount of \(\mathbf{W}\) reacts with white phosphorus to give \(\mathbf{X}\). The reaction of \(\mathbf{X}\) with pure \(\mathrm{HNO}_{3}\) gives \(\mathbf{Y}\) and \(\mathbf{Z}\).
Question:
\(\mathbf{W}\) and \(\mathbf{X}\) are, respectively