Let \(f(x)=2+\cos x\) for all real \(x\).
Statement I For each real \(t\), there exists a point \(c\) in \([t, t+\pi]\) such that \(f^{\prime}(c)=0\).
Statement II \(f(t)=f(t+2 \pi)\) for each real \(t\).

Let \(\mathbb{R}\) denote the set of all real numbers. Then the area of the region \(\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x>0, y>\frac{1}{x}, 5 x-4 y-1>0,4 x+4 y-17 <0\right\}\) 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 value of $\alpha$ is _____.

If \(\lim _{x \rightarrow 0}\left[1+x \log \left(1+b^2\right)\right]^{1 / x}=2 b \sin ^2 \theta b>0\) and \(\theta \in(-\pi, \pi]\), then the value of \(\theta\) is

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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 ?

For each positive integer $n$, let $y_n={\frac{1}{n}\left((n+1)(n+2)\mathrm{...}(n+n)\right)}^{\frac{1}{n}}$. If $\underset{n\to \infty }{\lim }{y}_n=L,$ then the value of $\left[L\right]$ (where $\left[x\right]$ is the greatest integer less than or equal to $x$) is ____

Let $f :R\to R$ and $g :R\to R$ be respectively given by $f\left(x\right)=\left|x\right|+1$ and $g\left(x\right)=x^2+1.$ Define $h :R\to R$ by $h\left(x\right)= \left\{\begin{array}{ccc}\max \left\{f\left(x\right), g\left(x\right)\right\}& if& x \le 0\\ \min \left\{f\left(x\right), g\left(x\right)\right\}& if& x>0\end{array}\right.$
Then number of points at which $h\left(x\right)$ is not differentiable is_________

A field line is shown in the figure. This field cannot represent

The tangent to the curve \(y=e^x\) drawn at the point \(\left(c, e^c\right)\) intersects the line joining the points \(\left(c-1, e^{c-1}\right)\) and \(\left(c+1, e^{c+1}\right)\)

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When a particle of mass \(m\) moves on the \(x\)-axis in a potential of the form \(V(x)=k x^2\), it performs simple harmonic motion.

The corresponding time period is proportional to \(\sqrt{\frac{m}{k}}\), as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of \(x=0\) in a way different from \(k x^2\) and its total energy is such that the particle does not escape to infinity. Consider a particle of mass \(\mathrm{m}\) moving on the \(x\)-axis. Its potential energy is \(V(x)=\alpha x^4(\alpha>0)\) for \(|x|\) near the origin and becomes a constant equal to \(V_0\) for \(|x| \geq X_0\) (see figure).
Question :
If the total energy of the particle is \(E\), it will perform periodic motion only if

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A circle \(C\) of radius 1 is inscribed in an equilateral \(\triangle P Q R\). The points of contact of \(C\) with the sides \(P Q, Q R, R P\) are \(D, E, F\) respectively. The line \(P Q\) is given by the equation
\(\sqrt{3} x+y-6=0\) and the point \(D\) is \(\left(\frac{3 \sqrt{3}}{2}, \frac{3}{2}\right)\). Further, it is given that the origin and the centre of \(C\) are on the same side of the line \(P Q\).Question:
Equations of the sides \(Q R, R P\) are

The total number of possible isomers for $\left[\mathrm{Pt}{\left({\mathrm{NH}}_3\right)}_4{\mathrm{Cl}}_2\right]{\mathrm{Br}}_2$ is___.

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity $X$ as follows: $\left[\mathrm{position}\right]=\left[X^{\alpha }\right];$ $\left[\mathrm{speed}\right]=\left[X^{\beta }\right]$; $\left[\mathrm{acceleration}\right]=\left[X^p\right]$;$\left[\mathrm{linear}\mathrm{momentum}\right]=\left[X^q\right]$; $\left[\mathrm{force}\right]=\left[X^r\right]$. Then