Paragraph:

Let \(F_{1}\left(x_{1}, 0\right)\) and \(F_{2}\left(x_{2}, 0\right)\), for \(x_{1}<0\) and \(x_{2}>0\), be the foci of the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{8}=1 .\) Suppose a parabola having vertex at the origin and focus at \(F_{2}\) intersects the ellipse at point \(M\) in the first quadrant and at point \(N\) in the fourth quadrant.


Question:

If the tangents to the ellipse at \(M\) and \(N\) meet at \(R\) and the normal to the parabola at \(M\) meets the \(x\)-axis at \(Q\), then the ratio of area of the triangle \(M Q R\) to area of the quadrilateral \(M F_{1} N F_{2}\) is

If \(I_n=\int_{-\pi}^\pi \frac{\sin n x}{\left(1+\pi^x\right) \sin x} d x ; n=0,1,2\), \(\ldots\), then

The point \(P\) is the intersection of the straight line joining the points \(Q(2,3,5)\) and \(R(1,-1,4)\) with the plane \(5 x-4 y-\) \(z=1\). If \(S\) is the foot of the perpendicular drawn from the point \(T(2,1,4)\) to \(Q R\), then the length of the line segment \(P S\) is

A farmer $F_1$ has a land in the shape of a triangle with vertices at $P\left(\mathrm{0,\; 0}\right),Q\left(\mathrm{1,\; 1}\right)$ and $R\left(\mathrm{2,\; 0}\right).$ From this land, a neighbouring farmer $F_2$ takes away the region which lies between the side $PQ$ and a curve of the form $y=x^n, n>1$. If the area of the region taken away by the farmer $F_2$ is exactly $\mathrm{30\%}$ of the area of $∆PQR,$ then the value of $n$ is

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Let \(F: \mathbb{R} \rightarrow \mathbb{R}\) be a thrice differentiable function. Suppose that \(F(1)=0, F(3)=-4\) and \(F^{\prime}(x)<0\) for all \(x \in(1 / 2,3)\). Let \(f(x)=x F(x)\) for all \(x \in \mathbb{R}\).


Question:

The correct statement(s) is(are)

Let \(a_{n}\) denote the number of all \(n\)-digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0 . Let \(b_{n}=\) the number of such \(n\)-digit integers ending with digit 1 and \(c_{n}=\) the number of such \(n\)-digit integers ending with digit 0 .

Question:
The value of \(b_{6}\) is

In a $∆XYZ$ let $x,y,z$,  be the lengths of sides opposite to the angles, $X, Y, Z$, respectively and $2s=x+y+z.$ If $\frac{s-x}{4}=\frac{s-y}{3}=\frac{s-z}{2}$ and area of incircle of the triangle XYZ is $\frac{8\pi }{3},$ then

Let \(L_1\) be the line of intersection of the planes given by the equation- \(2 x+3 y+z=4 \text { and } x+2 y+z=5 .\)
Let \(L_2\) be the line passing through the point \(P(2,-1,3)\) and parallel to \(L_1\). Let \(M\) denote the plane given by the equation- \(2 x+y-2 z=6\)
Suppose that the line \(L_2\) meets the plane \(M\) at the point \(Q\). Let \(R\) be the foot of the perpendicular drawn from \(P\) to the plane \(M\).
Then which of the following statements is (are) TRUE?

In the List-I below, four different paths of a particle are given as functions of time. In these functions, $\alpha \mathrm{a}\mathrm{n}\mathrm{d} \beta$ are positive constants of appropriate dimensions and $\alpha \ne \beta$ . In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: $\overrightarrow{p}$ is the linear momentum, $\overrightarrow{L}$ is the angular momentum about the origin, K is the kinetic energy, U is the potential energy and E is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for that path.

	LIST -I		LIST-II
A)	$\overrightarrow{r}\left(t\right)=\alpha t\widehat{i}+\beta \widehat{j}$	P)	$\overrightarrow{p}$
B)	$\overrightarrow{r}\left(t\right)=\alpha \cos \omega t\widehat{i}+\beta \sin \omega t\widehat{j}$	Q)	$\overrightarrow{L}$
C)	$\overrightarrow{r}\left(t\right)=\alpha \cos \omega t\widehat{i}+\sin \omega t\widehat{j}$	R)	$K$
D)	$\overrightarrow{r}\left(t\right)=\alpha t\widehat{i}+\frac{\beta }{2}{t}^2\widehat{j}$	S)	$U$
		T)	$E$

If \(f(x)=\left\{\begin{array}{cc}-x-\frac{\pi}{2}, & x \leq-\frac{\pi}{2} \\ -\cos x, & -\frac{\pi}{2} < x \leq 0 \\ x-1, & 0 < x \leq 1 \\ \log x, & x>1\end{array}\right.\), then

The boiling point of water in a \(0.1\) molal silver nitrate solution (solution \(\mathbf{A}\) ) is \(\mathbf{x}^{\circ} \mathrm{C}\). To this solution \(\mathbf{A}\), an equal volume of \(0.1\) molal aqueous barium chloride solution is added to make a new solution B. The difference in the boiling points of water in the two solutions \(\mathbf{A}\) and \(\mathbf{B}\) is \(\mathbf{y} \times 10^{-2}{ }^{\circ} \mathrm{C}\).
(Assume: Densities of the solutions \(\mathbf{A}\) and \(\mathbf{B}\) are the same as that of water and the soluble salts dissociate completely.
Use: Molal elevation constant (Ebullioscopic Constant), \(K_{b}=0.5 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\); Boiling point of pure water as \(100^{\circ} \mathrm{C}\).)
The value of $\mathrm{x}$ is ___.

Let \(g(x)=\frac{(x-1)^n}{\log \cos ^m(x-1)} ; 0 < x < 2, m\) and \(n\) are integers, \(m \neq 0, n>0\) and let \(p\) be the left hand derivative of \(|x-1|\) at \(x=1\). If \(\lim _{x \rightarrow 1^{+}} g(x)=p\), then

Let $f:\left[0, 2\right]\to ℝ$ be the function defined by $f\left(x\right)=\left(3-\sin \left(2\pi x\right)\right)\sin \left(\pi x-\frac{\pi }{4}\right)-\sin \left(3\pi x+\frac{\pi }{4}\right)$. If $\alpha , \beta \in \left[0, 2\right]$ are such that $\left\{x\in \left[0, 2\right] :f\left(x\right)\ge 0\right\}=\left[\alpha , \beta \right]$, then the value of $\beta -\alpha$ is _____