If \(|z|=1\) and \(z \neq \pm 1\), then all the values of \(\frac{z}{1-z^2}\) lie on

Let the function \(f:[1, \infty) \rightarrow \mathbb{R}\) be defined by
\(f(t)=\left\{\begin{array}{cc}(-1)^{n+1} 2, & \text { if } t=2 n-1, n \in \mathbb{N}, \\ \frac{(2 n+1-t)}{2} f(2 n-1)+\frac{(t-(2 n-1))}{2} f(2 n+1), & \text { if } 2 n-1 < t < 2 n+1, n \in \mathbb{N} .\end{array}\right.\)
Define \(g(x)=\int_1^x f(t) d t, x \in(1, \infty)\). Let \(\alpha\) denote the number of solutions of the equation \(g(x)=0\) in the interval \((1,8]\) and \(\beta=\lim _{x \rightarrow 1^+} \frac{g(x)}{x-1}\). Then the value of \(\alpha+\beta\) is equal to ________.

Let \(\int(x)=\left\{\begin{array}{r}x^{2}\left|\cos \frac{\pi}{x}\right|, \quad x \neq 0 \\ 0, \quad x=0\end{array}, x \in R\right.\) then \(\int\) is

Four persons independently solve a certain problem correctly with probabilities $\frac{1}{2},\frac{3}{4},\frac{1}{4},\frac{1}{8}$. Then the probability that the problem is solved correctly by at least one of them is

Consider the region \(R=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0\right.\) and \(\left.y^{2} \leq 4-x\right\}\). Let \(\mathcal{F}\) be the family of all circles that are contained in \(R\) and have centers on the \(x\)-axis. Let \(C\) be the circle that has largest radius among the circles in \(\mathcal{F}\). Let \((\alpha, \beta)\) be a point where the circle \(C\) meets the curve \(y^{2}=4-x\).
The radius of the circles $C$ is____.

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.	$60$

Three boys and two girls stand in a queue. The probability that the number of boys ahead of every girl is atleast one more than the number of girls ahead of her is

Match List I of the nuclear processes with List II containing parent nucleus and one of the end products of each process and then select the correct answer using the codes given below the lists :

List - i	List - II
A. Alpha decay	P. \({ }_8^{15} O \rightarrow{ }_7^{15} N+\ldots \ldots \ldots\).
B. \(\beta^{+}\)decay	Q. \({ }_{92}^{238} U \rightarrow{ }_{90}^{234} Th +\ldots \ldots \ldots\).
C. Fission	R. \({ }_{83}^{185} Bi \rightarrow{ }_{82}^{184} Pb+\ldots \ldots \ldots\).
D. Proton emission	S. \({ }_{94}^{239} Pu \rightarrow{ }_{57}^{140} La +\ldots \ldots \ldots\).

The correct statement(s) about the following sugars \(X\) and \(Y\) is (are)

Paragraph :
A wooden cylinder of diameter \(4 r\), height \(H\) and density \(\rho / 3\) is kept on a hole of diameter \(2 r\) of a tank, filled with liquid of density \(\rho\) as shown in the figure.

Question :
The block in the above question is maintained at the position by external means and the level of liquid is lowered. The height \(h_2\) when this external force reduces to zero is

A soft plastic bottle, filled with water of density \(1 \mathrm{gm} / \mathrm{cc}\), carries an inverted glass test-tube with some air (ideal gas) trapped as shown in the figure. The test-tube has a mass of \(5 \mathrm{gm}\), and it is made of a thick glass of density \(2.5 \mathrm{gm} / \mathrm{cc}\). Initially the bottle is sealed at atmospheric pressure \(p_{0}=10^{5} \mathrm{~Pa}\) so that the volume of the trapped air is \(v_{0}=3.3 \mathrm{cc}\). When the bottle is squeezed from outside at constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found that the test tube begins to sink at pressure \(p_{0}+\Delta p\) without changing its orientation. At this pressure, the volume of the trapped air is \(v_{0}-\Delta v\).
Let \(\Delta v=X\) cc and \(\Delta p=Y \times 10^{3} \mathrm{~Pa}\).

The value of $Y$ is _____.

Paragraph:
Let \(p\) be an odd prime number and \(T_p\) be the following set of \(2 \times 2\) matrices
\[
T_p=\left\{A=\left[\begin{array}{ll}
a & b \\
c & a
\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}
\]Question:
The number of \(A\) in \(T_p\) such that \(A\) is either symmetric or skew-symmetric or both, and det \((A)\) is divisible by \(p\) is

A uniform circular disc of mass 1.5 kg and radius 0.5 m is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude $\mathrm{F}\mathrm{}=\mathrm{}0.5\mathrm{}\mathrm{N}$ are applied simultaneously along the three sides of an equilateral triangle XYZ with its vertices on the perimeter of the disc (see figure). One second after applying the forces, the angular speed of the disc in rad ${\mathrm{s}}^{-1}$ is