The total number of local maxima and local minima of the function \(f(x)=\left\{\begin{array}{lc}(2+x)^3 ; & -3 < x \leq-1 \\ x^{\frac{2}{3}} ; & -1 < x < 2\end{array}\right.\) is

Consider the family of all circles whose centers lie on the straight line $\mathrm{y}=\mathrm{x}.$ If this family of circles is represented by the differential equation $\mathrm{P}\mathrm{y}"\mathrm{ }+\mathrm{ }\mathrm{Q}\mathrm{y}\mathrm{\prime }\mathrm{ }+\mathrm{ }1\mathrm{ }=\mathrm{ }0,\mathrm{ }$ where $\mathrm{P},\mathrm{ }\mathrm{Q}$ are functions of $\mathrm{x},\mathrm{ }\mathrm{y}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{y}\mathrm{\prime }$ $\left(\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{ }{\mathrm{y}}^{\mathrm{\prime }}=\mathrm{ }\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}},\mathrm{ }\mathrm{y}"\mathrm{ }=\mathrm{ }\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}\right)$ , then which of the following statements is (are) true ?

For any complex number $w=c+id$, let $\arg \left(w\right)\in (-\pi ,\pi ]$, where $i=\sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z=x+iy$ satisfying $\arg \left(\frac{z+\alpha }{z+\beta }\right)=\frac{\pi }{4}$, the ordered pair $(x, y)$ lies on the circle $x^2+y^2+5x-3y+4=0$
Then which of the following statements is(are) TRUE?

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A fair die is tossed repeatedly until a six is obtained. Let \(X\) denotes the number of tosses required.Question:
The conditional probability that \(X \geq 6\) given \(X>3\) equals

Three numbers are chosen at random, one after another with replacement, from the set \(S=\{1,2,3, \ldots, 100\}\). Let \(p_{1}\) be the probability that the maximum of chosen numbers is at least 81 and \(p_{2}\) be the probability that the minimum of chosen numbers is at most 40 .
The value of $\frac{625}{4}{p}_1$ is

Let \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be unit vectors such that \(\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}\). Which one of the following is correct?

Internal bisector of \(\angle A\) of \(\triangle A B C\) meets side \(B C\) at \(D\). A line drawn through \(D\) perpendicular to \(A D\) intersects the side \(A C\) at \(E\) and side \(A B\) at \(F\). If \(a, b\) and \(c\) represent sides of \(\triangle A B C\), then

Consider a system of three connected strings, \(S_1, S_2\) and \(S_3\) with uniform linear mass densities \(\mu \mathrm{kg} / \mathrm{m}\), \(4 \mu \mathrm{~kg} / \mathrm{m}\) and \(16 \mu \mathrm{~kg} / \mathrm{m}\), respectively, as shown in the figure. \(S_1\) and \(S_2\) are connected at the point \(P\), whereas \(S_2\) and \(S_3\) are connected at the point \(Q\), and the other end of \(S_3\) is connected to a wall. A wave generator O is connected to the free end of \(S_1\). The wave from the generator is represented by \(y=y_0\) \(\cos (\omega t-k x) \mathrm{cm}\), where \(y_0, \omega\) and \(k\) are constants of appropriate dimensions. Which of the following statements is/are correct:

Two metallic rings \(A\) and \(B\), identical in shape and size but having different resistivities \(\rho_A\) and \(\rho_B\), are kept on top of two identical solenoids as shown in the figure. When current \(I\) is switched on in both the solenoids in identical manner, the rings \(A\) and \(B\) jump to heights \(h_A\) and \(h_B\), respectively, with \(h_A>h_B\). The possible relation(s) between their resistivities and their masses \(m_A\) and \(m_B\) is(are)

The solubility of barium iodate in an aqueous solution prepared by mixing 200 mL of 0.010 M barium nitrate with 100 mL of 0.10 M sodium iodate is \(\boldsymbol{X} \times 10^{-6} \mathrm{~mol} \mathrm{dm}^{-3}\). The value of \(\boldsymbol{X}\) is \(\qquad\) .
Use: Solubility product constant \(\left(K_{\mathrm{sp}}\right)\) of barium iodate \(=1.58 \times 10^{-9}\)

List-$\mathrm{I}$ shows different radioactive decay processes and List-$\mathrm{II}$ provides possible emitted particles. Match each entry in List-$\mathrm{I}$ with an appropriate entry from List-$\mathrm{II}$, and choose the correct option.

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\mathrm{P}$	${{}^{238}}_{92}U{{\to }^{234}}_{91}Pa$	$1$	one $\alpha$ particle and one ${\beta }^{+}$ particle
$\mathrm{Q}$	${{}^{214}}_{82}Pb{{\to }^{210}}_{82}Pb$	$2$	three ${\beta }^{-}$ particles and one $\alpha$ particle
$\mathrm{R}$	${{}^{210}}_{81}Tl{{\to }^{206}}_{82}Pb$	$3$	two ${\beta }^{-}$ particles and one $\alpha$ particle
$\mathrm{S}$	${{}^{228}}_{91}Pa{{\to }^{224}}_{88}Ra$	$4$	one $\alpha$ particle and one ${\beta }^{-}$ particle
		$5$	one $\alpha$ particle and two ${\beta }^{+}$ particles

Column I contains a list of processes involving expansion of an ideal gas. Match this with Column II describing the thermodynamic change during this process. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

Suppose $\det \left[\begin{array}{cc}\sum _{k=0}^{n}k& \sum _{k=0}^n{}_{\mathrm{ }}{}^n{C}_k{k}^2\\ \sum _{k=0}^n{}_{\mathrm{ }}{}^n{C}_{k}k& \sum _{k=0}^n{}_{\mathrm{ }}{}^n{C}_k{3}^k\end{array}\right]=0,$ holds for some positive integer $n.$ Then $\sum _{k=0}^n\frac{{}_{ }{}^n{C}_k}{k+1}$ equals