Let \(P, Q, R\) and \(S\) be the points on the plane with position vectors \(-2 \hat{\mathbf{i}}-\hat{\mathbf{j}}, 4 \hat{\mathbf{i}}, 3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) an \(\mathrm{d}-3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}\) respectively. The quadrilateral \(P Q R S\) must be a

Let \(P\left(x_1, y_1\right)\) and \(Q\left(x_2, y_2\right)\) be two distinct points on the ellipse
\(\frac{x^2}{9}+\frac{y^2}{4}=1\)
such that \(y_1>0\), and \(y_2>0\). Let C denote the circle \(x^2+y^2=9\), and \(M\) be the point \((3,0)\).
Suppose the line \(x=x_1\) intersects \(C\) at \(R\), and the line \(x=x_2\) intersects \(C\) at \(S\), such that the \(y\)-coordinates of \(R\) and \(S\) are positive. Let \(\angle R O M=\frac{\pi}{6}\) and \(\angle S O M=\frac{\pi}{3}\), where \(O\) denotes the origin \((0,0)\). Let \(|X Y|\) denote the length of the line segment \(X Y\).
Then which of the following statements is (are) TRUE?

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors such that $\overrightarrow{a}.\overrightarrow{b}=0.$ For some $x,y\in \mathrm{R,}$ let $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}+\left(\overrightarrow{a}\times \overrightarrow{b}\right).$ If $\left|\overrightarrow{c}\right|=2$ and the vector $\overrightarrow{c}$ is inclined at the same angle $\alpha$ to both $\overrightarrow{a}$ and $\overrightarrow{b},$ then the value of $8{\cos }^2\alpha$ is

Of the three independent events ${\text{E}}_1$, ${\text{E}}_2$ and ${\text{E}}_3$, the probability that only ${\text{E}}_1$ occurs is $\alpha$, only ${\text{E}}_2$ occurs is $\beta$ and only  ${\text{E}}_3$ occurs is $\gamma$. Let the probability $p$ that none of events ${\text{E}}_1$, ${\text{E}}_2$ or ${\text{E}}_3$ occurs satisfy the equations $\left(\alpha -2\beta \right)p=\alpha \beta$ and $\left(\beta -3\gamma \right)p = 2\beta \gamma$. All the given probabilities are assumed to lie in the interval $\left(\text{0, 1}\right)$.
Then   $\frac{{\text{Probability of occurrence of E}}_1}{{\text{Probability of occurrence of E}}_3}=$ ​

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There are \(n\) urns each containing \((n+1)\) balls such that the ith urn contains \(i\) white balls and \((n+1-i)\) red balls. Let \(u_i\) be the event of selecting ith urn, \(i=1,2,3, \ldots, n\) and \(W\) denotes the event of getting a white balls.Question:
If \(n\) is even and \(E\) denotes the event of choosing even numbered urn \(\left(P\left(u_i\right)=\frac{1}{n}\right)\), then the value of \(P(W / E)\) is

If $\alpha = \underset{0}{\overset{1}{\int }}\left(e^{9x+3{\tan }^{-1}x}\right) \left(\frac{12+9x^2}{1+x^2}\right) dx$ where ${\tan }^{-1}x$ takes only principal values, then the value of $\left({\log }_e\left|1+\alpha \right|- \frac{3\pi }{4}\right)$ is

Two adjacent sides of a parallelogram \(A B C D\) are given by \(\overrightarrow{\mathbf{A B}}=2 \hat{\mathbf{i}}+10 \hat{\mathbf{j}}+11 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{A D}}=-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\). The side \(A D\) is rotated by an acute angle \(\alpha\) in the plane of the parallelogram so that \(A D\) becomes \(A D^{\prime}\). If \(A D^{\prime}\) makes a right angle with the side \(A B\), then the cosine of the angle \(\alpha\) is given by

Let \(\mathbb{N}\) denote the set of all natural numbers, and \(\mathbb{Z}\) denote the set of all integers. Consider the functions \(f: \mathbb{N} \rightarrow \mathbb{Z}\) and \(g: \mathbb{Z} \rightarrow \mathbb{N}\) defined by
\(f(n)= \begin{cases}(n+1) / 2 & \text { if } n \text { is odd } \\ (4-n) / 2 & \text { if } n \text { is even }\end{cases}\) And
\(g(n)=\left\{\begin{array}{cc}3+2 n & \text { if } n \geq 0 \\ -2 n & \text { if } n <0\end{array}\right.\)
Define \((g \circ f)(n)=g(f(n))\) for all \(n \in \mathbb{N}\), and \((f \circ g)(n)=f(g(n))\) for all \(n \in \mathbb{Z}\).
Then which of the following statements is (are) TRUE?

For any integer $k,$ let ${\alpha }_k=\cos \left(\frac{k\pi }{7}\right)+i\sin \left(\frac{k\pi }{7}\right), where i= \sqrt{-1}.$ The value of the expression $\frac{\sum _{k=1}^{12}\left|{\alpha }_{k+1}-{\alpha }_k\right|}{\sum _{k=1}^3\left|{\alpha }_{4k-1}-{\alpha }_{4k-2}\right|}$ is

The correct match of the group reagents in List-I for precipitating the metal ion given in List-II from solutions, is

List - I	List - II
(P) Passing \(H _2 S\) in the presence of \(NH _4 OH\)	(1) \(Cu ^{2+}\)
(Q) \(\left( NH _4\right)_2 CO _3\) in the presence of \(NH _4 OH\)	(2) \(Al ^{3+}\)
(R) \(NH _4 OH\) in the presence of \(NH _4 Cl\)	(3) \(Mn ^{2+}\)
(S) Passing \(H _2 S\) in the presence of dilute HCl	(4) \(Ba ^{2+}\)
	(5) \(Mg ^{2+}\)

A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency \(v_1\) and ejects the electron with a kinetic energy of 10 eV . The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneously emits a photon of frequency \(v_2\). The center of mass of the resulting positronium atom moves with a kinetic energy of 5 eV . It is given that positron has the same mass as that of electron and the positronium atom can be considered as a Bohr atom, in which the electron and the positron orbit around their center of mass. Considering no other energy loss during the whole process, the difference between the two photon energies (in eV ) is \(\qquad\)

(I) 1, 2-dihydroxy benzene
(II) 1, 3-dihydroxy benzene
(III) 1, 4-dihydroxy benzene
(IV) Hydroxy benzene
The increasing order of boiling points of above mentioned alcohols is

Let \(f, g\) and \(h\) be real-valued functions defined on the interval \([0,1]\) by \(f(x)=e^{x^2}+e^{-x^2}, \quad g(x)=x e^{x^2}+e^{-x^2}\) and \(h(x)=x^2 e^{x^2}+e^{-x^2}\). If \(a, b\) and \(c\) denote respectively, the absolute maximum of \(f, g\) and \(h\) on \([0,1]\), then