Let $\overrightarrow{u},\overrightarrow{v}$ and $\overrightarrow{w}$ be vectors in three-dimensional space, where $\overrightarrow{u}$ and $\overrightarrow{v}$ are unit vectors which are not perpendicular to each other and $\overrightarrow{u}·\overrightarrow{w}=1, \overrightarrow{v}·\overrightarrow{w}=1, \overrightarrow{w}·\overrightarrow{w}=4$.
If the volume of the parallelopiped, whose adjacent sides are represented by the vectors $\overrightarrow{u},\overrightarrow{v}$ and $\overrightarrow{w}$, is $\sqrt{2}$, then the value of $|3\overrightarrow{u}+5\overrightarrow{v}|$ is

Let \(E\) and \(F\) be two independent events. The probability that exactly one of them occurs is \(11 / 25\) and the probability of none of them occurring is \(2 / 25\). If \(P(T)\) denotes the probability of occurrence of the event \(T\), then

For $a\in R, \left|a\right|>1,$ let $\underset{n\to \infty }{\lim }\left(\frac{1+\sqrt[3]{2}+\dots +\sqrt[3]{n}}{n^{7/3}\left(\frac{1}{{\left(an+1\right)}^2}+\frac{1}{{\left(an+2\right)}^2}+\dots +\frac{1}{{\left(an+n\right)}^2}\right)}\right)=54.$ Then the possible value(s) of $a$ is/are:

Let $P$ be the plane $\sqrt{3}x+2y+3z=16$ and let
$S=\left\{\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}: {\alpha }^2+{\beta }^2+{\gamma }^2=1\right.$ and the distance of $\left(\alpha , \beta , \gamma \right)$ from the plane $P$ is $\frac{7}{2}\}$.
Let $\overrightarrow{u}, \overrightarrow{v}$ and $\overrightarrow{w}$ be three distinct vectors in $S$ such that $\left|\overrightarrow{u}-\overrightarrow{v}\right|=\left|\overrightarrow{v}-\overrightarrow{w}\right|=\left|\overrightarrow{w}-\overrightarrow{u}\right|$. Let $V$ be the volume of the parallelepiped determined by vectors $\overrightarrow{u}, \overrightarrow{v}$ and $\overrightarrow{w}$. Then the value of $\frac{80}{\sqrt{3}}V$ is

Let $G$ be a circle of radius $R>0$. Let $G_1,G_2,\dots ,G_n$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_1,G_2,\dots ,G_n$ touches the circle $G$ externally. Also, for $i=1,2,\dots ,n-1$, the circle $G_i$ touches $G_{i+1}$ externally, and $G_n$ touches $G_1$ externally. Then, which of the following statements is/are TRUE?

A solution curve of differential equation $\left(x^2+xy+4x+2y+4\right)\frac{dy}{dx}-y^2=0, x>\mathrm{0 }$ passes through the point (1, 3). Then the solution curve -

Let ${\text{S}}_{\text{n}}=\underset{\text{k}=1}{\overset{4\text{n}}{\Sigma }}{\left(-1\right)}^{\frac{\text{k}\left(\text{k}+1\right)}{2}}{\text{k}}^2$. Then S_n can take value(s)

Oxidation states of the metal in the minerals haematite and magnetite, respectively, are

Consider the following transformations of a compound $\mathrm{P}$.

Choose the correct option(s).

The total number of carboxylic acid groups in the product P is

A linear octasaccharide (molar mass \(=1024 \mathrm{~g} \mathrm{~mol}^{-1}\) ) on complete hydrolysis produces three monosaccharides: ribose, 2-deoxyribose and glucose. The amount of 2-deoxyribose formed is \(58.26 \%(\mathrm{w} / \mathrm{w})\) of the total amount of the monosaccharides produced in the hydrolyzed products. The number of ribose unit(s) present in one molecule of octasaccharide is \(\qquad\)
Use : Molar mass \(\left(\right.\) in \(\left.\mathrm{g} \mathrm{mol}^{-1}\right)\) : ribose \(=150,2\)-deoxyribose \(=134\), glucose \(=180\); Atomic mass (in amu): \(\mathrm{H}=1, \mathrm{O}=16\)

Let $a_1,a_2,a_3,\dots$ be an arithmetic progression with $a_1=7$ and common difference $8$. Let $T_1,T_2,T_3,\dots$ be such that $T_1=3$ and $T_{n+1}-T_n=a_n$ fo $n\ge 1$. Then, which of the following is/are TRUE?

A double slit setup is shown in the figure. One of the slits is in medium $2$ of refractive index $n_2$. The other slit is at the interface of this medium with another medium $1$ of refractive index $n_1\left(\ne n_2\right)$. The line joining the slits is perpendicular to the interface and the distance between the slits is $d$. The slit widths are much smaller than $d$. A monochromatic parallel beam of light is incident on the slits from medium $1$. A detector is placed in medium $2$ at a large distance from the slits, and at an angle $\theta$ from the line joining them, so that $\theta$ equals the angle of refraction of the beam. Consider two approximately parallel rays from the slits received by the detector.
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Which of the following statement(s) is(are) correct?