Let \(f: R \rightarrow R\) be a continuous function, which satisfies \(f(x)=\int_0^x f(t) d t\). Then, the value of \(f(\ln 5)\) is

Let the straight line \(\mathrm{y}=2 \mathrm{x}\) touch a circle with center \((0, \alpha), \alpha>0\), and radius \(\mathrm{r}\) at a point \(\mathrm{A}_1\). Let \(\mathrm{B}_1\) be the point on the circle such that the line segment \(A_1 B_1\) is a diameter of the circle. Let \(\alpha+r=5+\sqrt{5}\).
Match each entry in List-I to the correct entry in List-II.

The correct option is

Paragraph:
Consider the circle \(x^2+y^2=9\) and the parabola \(y^2=8 x\). They intersect at \(P\) and \(Q\) in the first and the fourth quadrants, respectively. Tangents to the circle at \(P\) and \(Q\) intersect the \(\mathrm{X}\)-axis at \(R\) and tangents to the parabola at \(P\) and \(Q\) intersect the \(\mathrm{X}\)-axis at \(S\).Question:
The ratio of the areas of the \(\triangle P Q S\) and \(\triangle P Q R\) is

For any real numbers $\alpha$ and $\beta$, let $y_{\alpha ,\beta }\left(x\right),x\in R$, be the solution of the differential equation $\frac{dy}{dx}+\alpha y=x{e}^{\beta x},y\left(1\right)=1$ Let $S=\left\{y_{\alpha ,\beta }\left(x\right) :\alpha ,\beta \in R\right\}$. Then which of the following functions belong(s) to the set $S$?

In a high school, a committee has to be formed from a group of $6$ boys
$M_1, M_2, M_3, M_4, M_5, M_6$ and $5$ girls $G_1, G_2, G_3, G_4, G_5$.
(i) Let ${\alpha }_1$ be the total number of ways in which the committee can be formed such that the committee has $5$ members, having exactly $3$ boys and $2$ girls.
(ii) Let ${\alpha }_2$ be the total number of ways in which the committee can be formed such that the committee has at least $2$ members, and having an equal number of boys and girls.
(iii) Let ${\alpha }_3$ be the total number of ways in which the committee can be formed such that the committee has $5$ members, at least $2$ of them being girls.
(iv) Let ${\alpha }_4$ be the total number of ways in which the committee can be formed such that the committee has $4$ members, having at least $2$ girls and such that both $M_1$ and $G_1$ are NOT in the committee together.

LIST-I	LIST-II
A. The value of ${\alpha }_1$ is	P. $136$
B. The value of ${\alpha }_2$ is	Q. $189$
C. The value of ${\alpha }_3$ is	R. $192$
D. The value of ${\alpha }_4$ is	S. $200$
	T. $381$
	U. $461$

The correct option is:

Let \(\overrightarrow{O P}=\frac{\alpha-1}{\alpha} \hat{i}+\hat{j}+\hat{k}, \overrightarrow{O Q}=\hat{i}+\frac{\beta-1}{\beta} \hat{j}+\hat{k}\) and \(\overrightarrow{O R}=\hat{i}+\hat{j}+\frac{1}{2} \hat{k}\) be three vectors, where \(\alpha, \beta \in \mathbb{R}-\{0\}\) and \(O\) denotes the origin. If \((\overrightarrow{O P} \times \overrightarrow{O Q}) \cdot \overrightarrow{O R}=0\) and the point \((\alpha, \beta, 2)\) lies on the plane \(3 x+3 y-z+l=0\), then the value of \(l\) is ____

Let $P$ be the plane $\sqrt{3}x+2y+3z=16$ and let
$S=\left\{\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}: {\alpha }^2+{\beta }^2+{\gamma }^2=1\right.$ and the distance of $\left(\alpha , \beta , \gamma \right)$ from the plane $P$ is $\frac{7}{2}\}$.
Let $\overrightarrow{u}, \overrightarrow{v}$ and $\overrightarrow{w}$ be three distinct vectors in $S$ such that $\left|\overrightarrow{u}-\overrightarrow{v}\right|=\left|\overrightarrow{v}-\overrightarrow{w}\right|=\left|\overrightarrow{w}-\overrightarrow{u}\right|$. Let $V$ be the volume of the parallelepiped determined by vectors $\overrightarrow{u}, \overrightarrow{v}$ and $\overrightarrow{w}$. Then the value of $\frac{80}{\sqrt{3}}V$ is

Let \(\alpha\) and \(\beta\) be the roots of \(x^2-6 x-2=0\) with \(\alpha>\beta\). If \(a_n=\alpha^n-\beta^n\) for \(n \geq 1\), then the value of \(\frac{a_{10}-2 a_8}{2 a_9}\) is

The centres of two circles \(C_1\) and \(C_2\) each of unit radius are at a distance of 6 units from each other. Let \(P\) be the mid-point of the line segment joining the centres of \(C_1\) and \(C_2\) and \(C\) be a circle touching circles \(C_1\) and \(C_2\) externally. If a common tangents to \(C_1\) and \(C\) passing through \(P\) is also a common tangent to \(C_2\) and \(C\), then the radius of the circle \(C\) is

Let the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by
\(f(x)=\frac{\sin x}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}+\frac{2}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}\).
Then the number of solutions of \(f(x)=0\) in \(\mathbb{R}\) is

Statement I A block of mass \(m\) starts moving on a rough horizontal surface with a velocity \(v\). It stops due to friction between the block and the surface after moving through a certain distance. The surface is now tilted to an angle of \(30^{\circ}\) with the horizontal and the same block is made to go up on the surface with the same initial velocity \(v\). The decrease in the mechanical energy in the second situation is smaller than that in the first situation.
Statement II The coefficient of friction between the block and the surface decreases with the increase in the angle of inclination.

In a photoemission experiment, the maximum kinetic energies of photoelectrons from metals $P,Q$ and $R$ are $E_P, E_Q$ and $E_R$, respectively, and they are related by $E_P=2E_Q=2E_R$. In this experiment, the same source of monochromatic light is used for metals $P$ and $Q$ while a different source of monochromatic light is used for the metal $R$. The work functions for metals $P,Q$ and $R$ are $4.0 \mathrm{eV}, 4.5 \mathrm{eV}$ and $5.5 \mathrm{eV}$, respectively. The energy of the incident photon used for metal $R$, in $eV$ is ___.

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An acylic hydrocarbon \(P\), having molecular formula \(\mathrm{C}_6 \mathrm{H}_{10}\), gave acetone as the only organic product through the following sequence of reactions, in which \(Q\) is an intermediate organic compounds.

Question:
The structure of compound P is