Let a, b, c be positive integers such that $\frac{b}{a}$ is an integer. If a, b, c are in geometric progression and the arithmetic mean of a, b, c is b + 2, then the value of $\frac{a^2+a-14}{a+1}$ is ________

Let $x\in R$ and let $P=\left[\begin{array}{ccc}1& 1& 1\\ 0& 2& 2\\ 0& 0& 3\end{array}\right], Q=\left[\begin{array}{ccc}2& x& x\\ 0& 4& 0\\ x& x& 6\end{array}\right]$ and $R=PQ{P}^{-1}.$
Then which of the following options is/are correct?

Let $\mathrm{g}\mathrm{ }:\mathbb{R}\to \mathbb{R}$ be a differentiable function with $\mathrm{g}\left(0\right)=0,\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }{\mathrm{g}}^{\mathrm{\prime }}\left(0\right)=0$ and ${\mathrm{g}}^{\mathrm{\prime }}\left(1\right)\mathrm{ }\ne 0.\mathrm{ }$ Let $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{ }\left\{\begin{array}{cc}\frac{\mathrm{x}}{\left|\mathrm{x}\right|}\mathrm{ }\mathrm{g}\left(\mathrm{x}\right),& \mathrm{x}\mathrm{ }\ne 0\\ 0,& \mathrm{x}=0\end{array}\right.$ and $\mathrm{h}\left(\mathrm{x}\right)={\mathrm{e}}^{\left|\mathrm{x}\right|}$ for all $\mathrm{x}\in \mathbb{R}.$ Let $\left(\mathrm{foh}\right)\mathrm{ }\left(\mathrm{x}\right)$ denote $\mathrm{f}\left(\mathrm{h}\left(\mathrm{x}\right)\right)\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\left(\mathrm{hof}\right)\mathrm{ }\left(\mathrm{x}\right)\mathrm{ }$denote $\mathrm{h}\left(\mathrm{f}\left(\mathrm{x}\right)\right)$ .Then which of the following 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

Paragraph:
Let \(p, q\) be integers and let \(\alpha, \beta\) be the roots of the equation, \(x^{2}-x-1=0\), where \(\alpha \neq \beta\). For \(n=0,1,2, \ldots\), let \(a_{n}=p \alpha^{n}+q \beta^{n}\)
FACT: If \(a\) and \(b\) are rational numbers and \(a+b \sqrt{5}=0\), then \(a=0=b\).

Question:
\(a_{12}=\)

For $x\in R$, the number of real roots of the equation $3x^2-4\left|x^2-1\right|+x-1=0$ is

Three randomly chosen non negative integers $x,y,z$ are found to satisfy the equation  $x+y+z=10$ . Then the probability that $z$ is even, is
 

An audio transmitter \((T)\) and a receiver \((R)\) are hung vertically from two identical massless strings of length 8 m with their pivots well separated along the \(X\) axis. They are pulled from the equilibrium position in opposite directions along the \(X\) axis by a small angular amplitude \(\theta_0=\cos ^{-1}(0.9)\) and released simultaneously. If the natural frequency of the transmitter is 660 Hz and the speed of sound in air is \(330 \mathrm{~m} / \mathrm{s}\), the maximum variation in the frequency (in Hz ) as measured by the receiver (Take the acceleration due to gravity \(g=10 \mathrm{~m} / \mathrm{s}^2\) ) is \(\qquad\)

The differential equation \(\frac{d y}{d x}=\frac{\sqrt{1-y^2}}{y}\) determines a family of circles with

Column I gives a list of possible set of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column II. Match the set of parameters given in Column I with the graphs given in Column II. Indicate your answer by darkening the appropriate bubble of the \(4 \times 4\) matrix given in the ORS.

A series \(R\) - \(C\) circuit is connected to \(A C\) voltage source. Consider two cases; \((A)\) when \(C\) is without a dielectric medium and \((B)\) when \(C\) is filled with dielectric of constant 4. The current \(I_R\) through the resistor and voltage \(V_C\) across the capacitor are compared in the two cases. Which of the following is/are true?

Two satellites $P$ and $Q$ are moving in different circular orbits around the Earth(radius $R$). The heights of $P$ and $Q$ from the Earth surface are $h_P$ and $h_Q$, respectively, where $h_P=\frac{R}{3}$. The accelerations of $P$ and $Q$ due to Earth’s gravity are $g_P$ and $g_Q$, respectively. If $\frac{g_P}{g_Q}=\frac{36}{25}$, what is the value of $h_Q$?

A positive, singly ionized atom of mass number \(A_{\mathrm{M}}\) is accelerated from rest by the voltage \(192 \mathrm{~V}\). Thereafter, it enters a rectangular region of width \(w\) with magnetic field \(\vec{B}_0=0.1 \hat{k}\) Tesla, as shown in the figure. The ion finally hits a detector at the distance \(x\) below its starting trajectory.
[Given: Mass of neutron/proton \(=(5 / 3) \times 10^{-27} \mathrm{~kg}\), charge of the electron \(=1.6 \times 10^{-19} \mathrm{C}\).]

Which of the following option(s) is(are) correct?