The minimum value of the sum of real numbers \(a^{-5}, a^{-4}, 3 a^{-3}, 1, a^8\) and \(a^{10}\) with \(a>0\) is

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Let \(a\), b and \(c\) be three real numbers satisfying \(\left[\begin{array}{lll}a & b & c\end{array}\right]\left[\begin{array}{lll}1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7\end{array}\right]=\left[\begin{array}{lll}0 & 0 & 0\end{array}\right]\)Question:
Let \(b=6\), with \(a\) and \(c\) satisfying Eq. (E). If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(a x^2+b x+c=0\), then \(\sum_{n=0}^{\infty}\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)^n\) is

Let the function \(g:(-\infty, \infty) \rightarrow\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) be given by \(g(u)=2 \tan ^{-1}\left(e^u\right)-\frac{\pi}{2}\). Then, \(g\) is

Let $X=\left\{\left(x, y\right)\in \mathrm{\mathbb{Z}}\times \mathrm{\mathbb{Z}} : \frac{x^2}{8}+\frac{y^2}{20}<1 \mathrm{and} y^2<5\mathrm{x}\right\}$. Three distinct point $P, Q$ and $R$ are randomly chosen from $X$. Then the probability that $P, Q$ and $R$ form a triangle whose area is a positive integer, is

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

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Let \(A_1, G_1, H_1\) denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For \(n \geq 2\), let \(A_{n-1}\) and \(H_{n-1}\) has arithmetic, geometric and harmonic means as \(A_n, G_n, H_n\), respectively.
Question:
Which of the following statements is correct?

Tangents are drawn to the hyperbola \(\frac{x^{2}}{9}-\frac{y^{2}}{4}=1\), parallel to the straight line \(2 x-y=1\). The points of contact of the tangents on the hyperbola are

A block of weight $100\mathrm{}\mathrm{N}$ is suspended by copper and steel wires of same cross-sectional area $0.5\mathrm{}\mathrm{c}{\mathrm{m}}^2$ and, length $\sqrt{3}\mathrm{}\mathrm{m}$ and $1\mathrm{}\mathrm{m},$ respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are ${30}^{\mathrm{o}}$ and ${60}^{\mathrm{o}},$ respectively. If elongation in copper wire is $\left(\mathrm{\Delta }{\mathrm{l}}_{\mathrm{C}}\right)$ and elongation in steel wire is $\left(\mathrm{\Delta }{\mathrm{l}}_{\mathrm{S}}\right),$ then the ratio $\frac{\mathrm{\Delta }{\mathrm{l}}_{\mathrm{C}}}{\mathrm{\Delta }{\mathrm{l}}_{\mathrm{S}}}$ is _____.
[Young's modulus for copper and steel are $1\times {10}^{11}\mathrm{N}/{\mathrm{m}}^2$ and $2\times {10}^{11}\mathrm{N}/{\mathrm{m}}^2$ respectively]

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:

Two capacitors with capacitance values $C_1=\left(2000\pm 10\right) \mathrm{pF}$  and $C_2=\left(3000\pm 15\right) \mathrm{pF}$ are connected in series. The voltage applied across this combination is $V=\left(5.00\pm 0.02\right) \mathrm{V}$. The percentage error in the calculation of the energy stored in this combination of capacitors is __________.

In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is $10\pm 0.1 \mathrm{cm}$ and the distance of its real image from the lens is $20\pm 0.2 \mathrm{cm}$. The error in the determination of focal length of the lens is $n\%$. The value of $n$ is _____.

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The coordination number of \(\mathrm{Ni}^{2+}\) is 4 .
\(\mathrm{NiCl}_2+\mathrm{KCN}\) (excess) \(\rightarrow A\) (cyano complex)
\(\mathrm{NiCl}_2+\) conc. \(\mathrm{HCl}\) (excess) \(\rightarrow B\) (chloro complex)
Question:
The IUPAC name of \(A\) and \(B\) are

Two non-conducting spheres of radii $R_1$ and $R_2$ and carrying uniform volume charge densities $+\rho$ and $-\rho$, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region,