Area of the region $\{\left(x,y\right)\in R^2: y\ge \sqrt{\left|x+3\right|},$ $5y\le x+9\le 15\}$ is equal to

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Let \(V_r\) denotes the sum of the first \(r\) terms of an arithmetic progression \((A P)\) whose first term is \(r\) and the common difference is \((2 r-1)\). Let \(T_r=V_{r+1}-V_r-2\) and \(Q_r=T_{r+1}-T_r\) for \(r=1,2, \ldots\)
Question:
The sum \(V_1+V_2+\ldots+V_n\) is

The number of all possible values of \(\theta\), where \(0 < \theta < \pi\), for which the system of equations
\((y+z) \cos 3 \theta=(x y z) \sin 3 \theta \) \( x \sin 3 \theta=\frac{2 \cos 3 \theta}{y}+\frac{2 \sin 3 \theta}{z}\) and \((x y z) \sin 3 \theta=(y+2 z) \cos 3 \theta\) \(+y \sin 3 \theta\) have a solution \(\left(x_0, y_0, z_0\right)\) with \(\quad y_0 z_0 \neq 0\), is

Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z=x+iy \left(x, y\in \mathrm{ℝ}, i=\sqrt{-1}\right)$ of the equation $sz+t\stackrel{-}{z}+r=0$, where $\stackrel{-}{z}=x-iy$. Then, which of the following statement(s) is (are) TRUE?

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Let \(\psi_{1}:[0, \infty) \rightarrow \mathbb{R}, \psi_{2}:[0, \infty) \rightarrow \mathbb{R}, f:[0, \infty) \rightarrow \mathbb{R}\) and \(g:[0, \infty) \rightarrow \mathbb{R}\) be functions such that \(f(0)=g(0)=0\),

\(\psi_{1}(x)=e^{-x}+x, \quad x \geq 0\),

\(\psi_{2}(x)=x^{2}-2 x-2 e^{-x}+2, \quad x \geq 0\),

\(f(x)=\int_{-x}^{x}\left(|t|-t^{2}\right) e^{-t^{2}} d t, \quad x>0\)

and \(g(x)=\int_{0}^{x^{2}} \sqrt{t} e^{-t} d t, \quad x>0\)


Question:

Which of the following statements is TRUE ?

Let $\text{P}=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \text{α}\\ 3& -5& 0\end{array}\right]$ , where $\text{α}\in \text{R},$ Suppose $\text{Q}=\left[{\text{q}}_{\text{i}\text{j}}\right]$ is a matrix such that$PQ = kI$, where $\text{k}\in \text{R},\text{k}\ne 0$ and$I$ is the identity matrix of order 3. If ${\text{q}}_{23}=-\frac{\text{k}}{8}$ and $\text{det}\left(\text{Q}\right)=\frac{{\text{k}}^2}{2}$ then 

Let \(X=\left({ }^{10} C_1\right)^2+2\left({ }^{10} C_2\right)^2+3\left({ }^{10} C_3\right)^2+\ldots+\) \(10\left({ }^{10} C_{10}\right)^2\), where \({ }^{10} C_r, r \in\{1,2, \ldots, 10\}\) denote binomial coefficients. Then, the value of \(\frac{1}{1430} X\) is _____________.

For the reaction:
${\mathrm{I}}^{-}+\mathrm{C}\mathrm{l}{\mathrm{O}}_3^{-}+{\mathrm{H}}_2\mathrm{S}{\mathrm{O}}_4\to \mathrm{C}{\mathrm{l}}^{-}+\mathrm{H}\mathrm{S}{\mathrm{O}}_4^{-}+{\mathrm{I}}_2\mathrm{ }$
The correct statement(s) in the balanced equation is/are:

Let \(\alpha, \beta\) and \(\gamma\) be real numbers such that the system of linear equations
\[
\begin{array}{c}
x+2 y+3 z=\alpha \\
4 x+5 y+6 z=\beta \\
7 x+8 y+9 z=\gamma-1
\end{array}
\]
is consistent. Let \(|M|\) represent the determinant of the matrix
\[
M=\left[\begin{array}{ccc}
\alpha & 2 & \gamma \\
\beta & 1 & 0 \\
-1 & 0 & 1
\end{array}\right]
\]
Let \(P\) be the plane containing all those \((\alpha, \beta, \gamma)\) for which the above system of linear equations is consistent, and \(D\) be the square of the distance of the point \((0,1,0)\) from the plane \(P\).
The value of $\left|M\right|$ is

Four charges ${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_3$ and ${\mathrm{Q}}_4$ of same magnitude are fixed along the x axis at $\mathrm{x}=-2\mathrm{a},\mathrm{}-\mathrm{a},\mathrm{}+\mathrm{a}$ and $+2\mathrm{a}$ , respectively. A positive charge q is placed on the positive y axis at a distance $\mathrm{b}\mathrm{}>\mathrm{}0$. Four options of the signs of these charges are given in List I. The direction of the forces on the charge q is given in List II. Match List I with List II and select the correct answer using the code given below the lists.

	List I		List II
A.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_3\mathrm{}{\mathrm{Q}}_4$ all positive	$\mathrm{P}.$	$+\mathrm{x}$
B.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_2$ positive; ${\mathrm{Q}}_3,\mathrm{}{\mathrm{Q}}_4$ negative	Q.	$-\mathrm{x}$
C.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_4$ positive; ${\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_3$ negative	$\mathrm{R}.$	$+\mathrm{y}$
D.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_3$ positive; ${\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_4$ negative	$\mathrm{S}.$	$-\mathrm{y}$

A thin rod of mass $M$ and length $a$ is free to rotate in horizontal plane about a fixed vertical axis passing through point $O$. A thin circular disc of mass $M$ and of radius $\frac{a}{4}$ is pivoted on this rod with its centre at a distance $\frac{a}{4}$ from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary conserver finds the rod rotating with an angular velocity $\Omega$ and the disc rotating about its vertical axis with angular velocity $4\Omega$. The total angular momentum of the system about the point $O$ is $\left[\left(\frac{M{a}^2\Omega }{48}\right)n\right]$. The value of $n$ is _________ .

A disk of radius \(\frac{a}{4}\) having a uniformly distributed charge \(6 \mathrm{C}\) is placed in the \(x-y\) plane with its centre at \(\left(\frac{-a}{2}, 0,0\right)\). A rod of length a carrying a uniformly distributed charge \(8 \mathrm{C}\) is placed on the \(x\)-axis from \(x=\frac{a}{4}\) to \(x=\frac{5 a}{4}\). Two point charges \(-7 \mathrm{C}\) and \(3 \mathrm{C}\) are placed at \(\left(\frac{a}{4}, \frac{-a}{4}, 0\right)\) and \(\left(\frac{-3 a}{4}, \frac{3 a}{4}, 0\right)\), respectively. Consider a cubical surface formed by six surfaces \(x=\pm \frac{a}{2}, y=\pm \frac{a}{2}, z=\pm \frac{a}{2}\).
The electric flux through this cubical surface is

Let $a_1, a_2, a_3, .........$ be a sequence of positive integers in arithmetic progression with common difference $2$. Also, let $b_1, b_2, b_3, ........$ be a sequence of positive integers in geometric progression with common ratio $2$. If $a_1=b_1=c$, then the number of all possible values of $c$, for which the equality $2 \left(a_1+a_2+........+a_n\right)=b_1+b_2+.....+b_n$ holds for some positive integer $n$, is _______