In a triangle $PQR, let \angle PQR={30}^o$ and the sides $PQ \mathrm{a}\mathrm{n}\mathrm{d} QR$ have lengths $10\sqrt{3} \mathrm{a}\mathrm{n}\mathrm{d} 10$ , respectively. Then, which of the following statement(s) is (are) TRUE?

Match List-I with List - II and select the correct answer using the code given below the lists 
 

	List-I		List-II
A.	Volume of parallelepiped determined by vectors  $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ is $2$. Then the volume of the parallelepiped determined by vectors $2\left(\overrightarrow{a}\times \overrightarrow{b}\right),3\left(\overrightarrow{b}\times \overrightarrow{c}\right)$ and $\left(\overrightarrow{c}\times \overrightarrow{a}\right)$ is	P.	$100$
B.	Volume of parallel piped determined by vectors  $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ is $5$. Then the volume of the parallelepiped determined by vectors $3\left(\overrightarrow{a}+\overrightarrow{b}\right),\left(\overrightarrow{b}+\overrightarrow{c}\right)$ and $2\left(\overrightarrow{c}+\overrightarrow{a}\right)$ is	Q.	$30$
C	Area of a triangle with adjacent sides determined by vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $20$.Then the area of the triangle with adjacent sides determined by vectors $\left(2\overrightarrow{a}+3\overrightarrow{b}\right)$ and $\left(\overrightarrow{a}-\overrightarrow{b}\right)$  is	R.	$24$
D	Area of a parallelogram with adjacent sides determined by vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $30$. Then the area of the parallelogram with adjacent sides determined by vectors $\left(\overrightarrow{a}+\overrightarrow{b}\right)$ and $\overrightarrow{a}$ is	S.	$60$

Paragraph:

Given that for each \(a \in(0,1)\),

\(\lim _{h \rightarrow 0^{+}} \int_{h}^{1-h} t^{-a}(1-t)^{a-1} d t\)

exists. Let this limit be \(g(a)\). In addition, it is given that the function \(g(a)\) is differentiable on \((0,1)\).


Question:

The value of \(g\left(\frac{1}{2}\right)\) is

If a chord, which is not a tangent, of the parabola $y^2=16x$ has the equation $2x+y=p$, and midpoint (h, k), then which of the following is (are) possible value(s) of $p,h$ and $k$?
 

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 $\alpha =\sum _{k=1}^{\infty }{\sin }^{2k}\left(\frac{\pi }{6}\right)$. Let $g:\left[0,1\right]\to \mathrm{ℝ}$ be the function defined by $g\left(x\right)=2^{ax}+2^{a\left(1-x\right)}$. Then, which of the following statements is/are TRUE?

Let $\mathrm{X}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{Y}$ be two arbitrary, $3\mathrm{ }\times 3$ , non - zero, skew - symmetric matrices and Z be an arbitrary $3\mathrm{ }\times 3$ , non - zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric ?

A mixture of two salts is used to prepare a solution

The correct option (s) for the salt mixture is(are)

Paragraph:
Chemical reactions involve interaction of atoms and molecules. A large number of atoms/molecules (approximately \(6.023 \times 10^{23}\) ) are present in a few grams of any chemical compound varying with their atomic/molecular masses. To handle such large numbers conveniently, the mole concept was introduced. This concept has implications in diverse areas such as analytical chemistry, biochemistry, electrochemistry and radiochemistry. The following example illustrates a typical case, involving chemical/electrochemical reaction, which requires a clear understanding of the mole concept.
A \(4.0\) molar aqueous solution of \(\mathrm{NaCl}\) is prepared and \(500 \mathrm{~mL}\) of this solution is electrolysed. This leads to the evolution of chlorine gas at one of electrodes (atomic mass: \(\mathrm{Na}=23, \mathrm{Hg}=200 ; 1\) faraday \(=96500\) coulombs).
Question:
The total charge (coulombs) required for complete electrolysis is

The correct statement(s) about of the following reaction sequence is(are)

Paragraph : In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $\left[E\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[B\right]$ stand for dimensions of electric and magnetic fields respectively, while $\left[{\epsilon }_0\right]$ and $\left[{\mu }_0\right]$ stand for dimensions of the permittivity and permeability of free space respectively. $\left[L\right]\mathrm{a}\mathrm{n}\mathrm{d} \left[T\right]$ are dimensions of length and time respectively. All the quantities are given in $SI$ units

Question : The relation between $\left[{ϵ}_0\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[{\mu }_0\right]$ is

Let $X$ be the set consisting of the first $2018$ terms of the arithmetic progression $\mathrm{1,6},11,\dots ,$ and $Y$ be the set consisting of the first $2018$ terms of the arithmetic progression $\mathrm{9,16,23},\dots$ . Then, the number of elements in the set $X\cup Y$ is___.

Let the circles $C_1:x^2+y^2=9$ and $C_2:{\left(x-3\right)}^2+{\left(y-4\right)}^2=16,$ intersect at the points $X$ and $Y.$ Suppose that another circle $C_3:{\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ satisfies the following conditions:
$\left(i\right)$ centre of $C_3$ is collinear with the centres of $C_1$ and $C_2$
$\left(ii\right) C_1$ and $C_2$ both lie inside $C_3,$ and
$\left(iii\right) C_3$ touches $C_1$ at $M$ and $C_2$ at $N.$
Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W,$ and let a common tangent of $C_1$ and $C_3$ be a tangent to the parabola $x^2=8\alpha y.$
There are some expression given in the List- $I$ whose values are given in List- $\mathrm{I}\mathrm{I}$ below:
 

	List- $\mathrm{I}$		List-  II
$\left(I\right)$	$2h+k$	$\left(P\right)$	$6$
$\left(II\right)$	$\frac{\mathrm{L}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }ZW}{\mathrm{L}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }XY}$	$\left(Q\right)$	$\sqrt{6}$
$\left(III\right)$	$\frac{\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}\mathrm{ }MZN}{\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}\mathrm{ }ZMW}$	$\left(R\right)$	$\frac{5}{4}$
$\left(IV\right)$	$\alpha$	$\left(S\right)$	$\frac{21}{5}$
		$\left(T\right)$	$2\sqrt{6}$
		$\left(U\right)$	$\frac{10}{3}$

Which of the following is the only INCORRECT combination?