Let \(a\) and \(b\) be two nonzero real numbers. If the coefficient of \(x^5\) in the expansion of \(\left(a x^2+\frac{70}{27 b x}\right)^4\) is equal to the coefficient of \(x^{-5}\) in the expansion of \(\left(a x-\frac{1}{b x^2}\right)^7\), then the value of \(2 b\) is

Let $\mathrm{\alpha }$ and $\mathrm{\beta }$ be the roots of ${\mathrm{x}}^2-\mathrm{x}-1=0,$ with $\mathrm{\alpha }>\mathrm{\beta }.$ For all positive integers $\mathrm{n},$ define
${\mathrm{a}}_{\mathrm{n}}=\frac{{\mathrm{\alpha }}^{\mathrm{n}}-{\mathrm{\beta }}^{\mathrm{n}}}{\mathrm{\alpha }-\mathrm{\beta }},\mathrm{n}\ge 1$
${\mathrm{b}}_1=1$ and ${\mathrm{b}}_{\mathrm{n}}={\mathrm{a}}_{\mathrm{n}-1}+{\mathrm{a}}_{\mathrm{n}+1},\mathrm{n}\ge 2.$
Then which of the following options is/are correct?

The line \(2 x+y=1\) is tangent to the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\). If this line passes through the point of intersection of the nearest directrix and the \(x\)-axis, then the eccentricity of the hyperbola is

Considering only the principal values of the inverse trigonometric functions, the value of $\frac{3}{2}{\cos }^{-1}\sqrt{\frac{2}{2+{\pi }^2}}+\frac{1}{4}{\sin }^{-1}\frac{2\sqrt{2}\pi }{2+{\pi }^2}+{\tan }^{-1}\frac{\sqrt{2}}{\pi }$ is _______. (If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places)  

Let \(E^c\) denotes the complement of an event \(E\). Let \(E, F, G\) be pairwise independent events with \(P(G)>0\) and \(P(E \cap F \cap G)=0\). Then, \(P\left(E^c \cap F^c \mid G\right)\) equals

Consider the set of eight vectors $\text{V}=\left\{\text{a}\widehat{i}+\text{b}\widehat{\text{j}}+\text{c}\widehat{\text{k}}:\; a,\; b,\; c \in \left\{-1,\; 1\right\}\right\}$. Three non-coplanar vectors can be chosen from $\text{V}$ in $2^{\text{p}}$ ways. Then $\text{p}$ is

Let P be the point on the parabola $y^2=4x$ which is at the shortest distance from the center S of the circle $x^2+y^2-4x-16y+64=0.$ Let Q be the point on the circle dividing the line segment SP internally. Then -

Two thin circular discs of mass m and 4m, having radii of a and 2a, respectively, are rigidly fixed by a massless, right rod of length $\mathcal{l}=\sqrt{24}a$ through their center. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $\omega$ . The angular momentum of the entire assembly about the point 'O' is $\overrightarrow{L}$ (see the figure). Which of the following statement(s) is(are) true?

A line $l$ passing through the origin is perpendicular to the lines
$l_1\text{ : }\left(3+\text{t}\right)\widehat{\text{i}}+\left(-1+2\text{t}\right)\widehat{\text{j}}+\left(4+2\text{t}\right)\widehat{\text{k}}\text{, }-\infty <\text{t}<\infty$
$l_2\text{ : }\left(3+2\text{s}\right)\widehat{\text{i}}+\left(3+2\text{s}\right)\widehat{\text{j}}+\left(2+\text{s}\right)\widehat{\text{k}}\text{, }-\infty <\text{s}<\infty$
Then, the coordinate(s) of the point(s) on $l_2$ at a distance of $\sqrt{17}$  from the point of intersection of $l$ and $l_1$ is (are)

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A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
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Question :
The centre of mass of the disk undergoes simple harmonic motion with angular frequency \(\omega\) equal to

The compound(s) which react(s) with ${\mathrm{NH}}_3$ to give boron nitride $\left(\mathrm{BN}\right)$ is(are)

Let $f:R\to R$ be a differentiable function with $f\left(0\right)=1$ and satisfying the equation $f\left(x+y\right)=f\left(x\right)f^{\text{'}}\left(y\right)+f^{\text{'}}\left(x\right)f\left(y\right)$ for all $x, y\in R$. Then, the value of ${\log }_e\left(f\left(4\right)\right)$ is 

Among the following, the number of reaction(s) that produce(s) benzaldehyde is