Let $O$ be the center of the circle $x^2+y^2=r^2,$ where $r>\frac{\sqrt{5}}{2}.$ Suppose $PQ$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is  $2x+4y=5$. If the center of the circumcircle of the triangle $OPQ$ lies on the line $x+2y=4,$ then the value of $r$ is ______

Let \(z=\cos \theta+i \sin \theta\). Then, the value of \(\sum_{m=1}^{15} \operatorname{Im}\left(z^{2 m-1}\right)\) at \(\theta=2^{\circ}\) is

Let $f\text{ : }\left[\frac{1}{2},1\right]⟶R$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f^{\text{'}}\left(x\right)<2f\left(x\right)$ and $f\left(\frac{1}{2}\right)=1$. Then the value of $\underset{1/2}{\overset{1}{\int }}f\left(x\right)\mathit{\text{dx}}$ lies in the interval:

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Consider the polynomial \(f(x)=1+2 x+3 x^2+4 x^3\). Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t=|s|\).Question:
The function \(f^{\prime}(x)\) is

Let \(P=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix and let \(Q=\left[b_{i j}\right]\), where \(b_{i j}=2^{i+j} a_{i j}\) for \(1 \leq i, j \leq 3\). If the determinant of \(P\) is 2 , then the determinant of the matrix \(Q\) is

The slope of the tangent to the curve ${\left(y-x^5\right)}^2=\text{x}{\left(1+{\text{x}}^2\right)}^2$ at the point (1, 3) is ________.

Let \(S=\{a+b \sqrt{2}: a, b \in \mathbb{Z}\}, T_1=\)\(\left\{(-1+\sqrt{2})^n: n \in \mathbb{N}\right\}\), and \(T_2=\left\{(1+\sqrt{2})^n: n \in \mathbb{N}\right\}\). Then which of the following statements is (are) TRUE?

If the function \(f(x)=x^3+e^{\frac{x}{2}}\) and \(g(x)=f^{-1}(x)\), then the value of \(g^{\prime}(1)\) is

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Given that for each \(a \in(0,1)\),

\(\lim _{h \rightarrow 0^{+}} \int_{h}^{1-h} t^{-a}(1-t)^{a-1} d t\)

exists. Let this limit be \(g(a)\). In addition, it is given that the function \(g(a)\) is differentiable on \((0,1)\).


Question:

The value of \(g^{\prime}\left(\frac{1}{2}\right)\) is

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

A photoelectric material having work-function ${\varphi }_0$ is illuminated with a light of wavelength $\lambda \left(\lambda <\frac{hc}{{\varphi }_0}\right)$. The fastest photoelectron has a de Broglie wavelength ${\lambda }_d$. A change in wavelength of the incident light by $\mathrm{\Delta }\lambda$ results in a change $\mathrm{\Delta }{\lambda }_d$ in ${\lambda }_d$. The ratio $\frac{\mathrm{\Delta }{\lambda }_d}{\mathrm{\Delta }\lambda }$ is proportional to

The order of the oxidation state of the phosphorus atom in $H_{3}P{O}_2, H_{3}P{O}_4, H_{3}P{O}_3$ and $H_4{P}_2{O}_6$ is

A point charge $q$ of mass $m$ is suspended vertically by a string of length $l.$ A point dipole of dipole moment $\overrightarrow{p}$ is now brought towards $q$ from infinity so that the charge moves away. The final equilibrium position of the system including the direction of the dipole, the angles and distances is shown in the figure below. If the work done in bringing the dipole to this position is $N\times \left(mgh\right)$, where $g$ is the acceleration due to gravity, then the value of $N$ is ____________ (Note that for three coplanar forces keeping a point mass in equilibrium, $\frac{\mathrm{F}}{\sin \theta }$ is the same for all forces, where $F$ is any one of the forces and $\theta$ is the angle between the other two forces)