For $a\in R, \left|a\right|>1,$ let $\underset{n\to \infty }{\lim }\left(\frac{1+\sqrt[3]{2}+\dots +\sqrt[3]{n}}{n^{7/3}\left(\frac{1}{{\left(an+1\right)}^2}+\frac{1}{{\left(an+2\right)}^2}+\dots +\frac{1}{{\left(an+n\right)}^2}\right)}\right)=54.$ Then the possible value(s) of $a$ is/are:

For \(r=0,1, \ldots, 10\), let \(A_r, B_r\) and \(C_r\) denote, respectively, the coefficient of \(x^r\) in the expansions of \((1+x)^{10}\), \((1+x)^{20}\) and \((1+x)^{30}\). Then \(\sum_{r=1}^{10} A_r\left(B_{10} B_r-C_{10} A_r\right)\) is equal to

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

Consider the system of equations
where \(\quad a, b, c, d \in\{0,1\}\)
Statement 1 The probability that the system of equations has a unique solution, is \(3 / 8\).
Statement 2 The probability that the system of equations has a solution, is 1 .

Let $\omega$ be a complex cube root of unity with $\omega \ne 1$ and $\text{P}=\left[{\text{p}}_{\text{ij}}\right]$ be a n x n matrix with ${\text{p}}_{\text{ij}}={\omega }^{\text{i}+\text{j}}$. Then ${\text{P}}^2\ne 0$, when n =

The positive integer value of \(n>3\) satisfying the equation
\(\frac{1}{\sin \left(\frac{\pi}{n}\right)}=\frac{1}{\sin \left(\frac{2 \pi}{n}\right)}+\frac{1}{\sin \left(\frac{3 \pi}{n}\right)} \text { is }\)

Let $A_1,A_2,A_3,\dots ,A_8$ be the vertices of a regular octagon that lie on a circle of radius $2$. Let $P$ be a point on the circle and let $P{A}_i$ denote the distance between the points $P$ and $A_i$ for $i=1,2,\dots ,8$. If $P$ varies over the circle, then the maximum value of the product $P{A}_1·P{A}_2\dots ·P{A}_8$, is

Using the expression $2d \sin \theta =\lambda$, one calculates the values of $d$ by measuring the corresponding angle $\text{θ}$ in the range $0$ to $90°$. The wavelength $\lambda$ is exactly known and the error in $\theta$ is constant for all the values of $\theta$. As $\text{θ}$ increases from $0°$,

Gravitational acceleration on the surface of a planet is \(\frac{\sqrt{6}}{11} \mathrm{~g}\), where \(g\) is the gravitational acceleration on the surface of the earth. The average mass density of the planet is \(\frac{2}{3}\) times that of the earth. If the escape speed on the surface of the earth is taken to be 11 \(\mathrm{kms}^{-1}\), the escape speed on the surface of the planet in \(\mathrm{kms}^{-1}\) will be

Let $f:R\to R$ be defined by $f\left(x\right)=\frac{x^2-3x-6}{x^2+2x+4}$ Then which of the following statements is(are) TRUE?

Let \(P\left(x_1, y_1\right)\) and \(Q\left(x_2, y_2\right)\) be two distinct points on the ellipse
\(\frac{x^2}{9}+\frac{y^2}{4}=1\)
such that \(y_1>0\), and \(y_2>0\). Let C denote the circle \(x^2+y^2=9\), and \(M\) be the point \((3,0)\).
Suppose the line \(x=x_1\) intersects \(C\) at \(R\), and the line \(x=x_2\) intersects \(C\) at \(S\), such that the \(y\)-coordinates of \(R\) and \(S\) are positive. Let \(\angle R O M=\frac{\pi}{6}\) and \(\angle S O M=\frac{\pi}{3}\), where \(O\) denotes the origin \((0,0)\). Let \(|X Y|\) denote the length of the line segment \(X Y\).
Then which of the following statements is (are) TRUE?

The CORRECT statement(s) for cubic close packed (ccp) three dimensional structure is (are)

In an $X$-ray tube, electrons emitted from a filament (cathode) carrying current $I$ hit a target (anode) at a distance $d$ from the cathode. The target is kept at a potential $V$ higher than the cathode resulting in emission of continuous and characteristic X-rays. If the filament current $I$ is decreased to $\frac{I}{2},$ the potential difference $V$ is increased to $2V,$ and the separation distance $d$ is reduced to $\frac{d}{2},$ then