The number of real solutions of the equation ${\sin }^{-1}\left(\sum _{i=1}^{\infty }{x}^{i+1}-x\sum _{i=1}^{\infty }{\left(\frac{x}{2}\right)}^i\right)=\frac{\pi }{2}-{\cos }^{-1}\left(\sum _{i=1}^{\infty }{\left(-\frac{x}{2}\right)}^i-\sum _{i=1}^{\infty }{\left(-x\right)}^i\right)$ lying in the interval $\left(-\frac{1}{2},\frac{1}{2}\right)$ is____.
(Here, the inverse trigonometric functions ${\sin }^{-1}x and {\cos }^{-1}x$ assume values in $\left[-\frac{\pi }{2},\frac{\pi }{2}\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[0,\pi \right]$
respectively.)

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

A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then $\text{k}-20=$

If \(f^{\prime \prime}(x)=-f(x)\), where \(f(x)\) is a continuous double differentiable function and \(g(x)=f^{\prime}(x)\). If \(F(x)=\left(f\left(\frac{x}{2}\right)\right)^2+\left(g\left(\frac{x}{2}\right)\right)^2\) and \(F(5)=5\), then \(F(10)\) is

If \(e_1\) is the eccentricity of the ellipse \(\frac{x^2}{16}+\frac{y^2}{25}=1\) and \(e_2\) is the eccentricity of the hyperbola passing through the focii of the ellipse and \(e_1 e_2=1\), then equation of the hyperbola is

Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E:\frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P:y^2=12x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?

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Consider the function \(f:(-\infty, \infty) \rightarrow(-\infty, \infty)\) defined by \(f(x)=\frac{x^2-a x+1}{x^2+a x+1} ; 0 < a < 2\)Question:
Which of the following is true ?

Two equilateral-triangular prisms \(\mathrm{P}_1\) and \(\mathrm{P}_2\) are kept with their sides parallel to each other, in vacuum, as shown in the figure. A light ray enters prism \(P_1\) at an angle of incidence \(\theta\) such that the outgoing ray undergoes minimum deviation in prism \(P_2\). If the respective refractive indices of \(\mathrm{P}_1\) and \(\mathrm{P}_2\) are \(\sqrt{\frac{3}{2}}\) and \(\sqrt{3}\), then \(\theta=\sin ^{-1}\left[\sqrt{\frac{3}{2}} \sin \left(\frac{\pi}{\beta}\right)\right]\), where the value of \(\beta\) is _______ .

Let \(\vec{p}=2 \hat{i}+\hat{j}+3 \hat{k}\) and \(\vec{q}=\hat{i}-\hat{j}+\hat{k}\). If for some real numbers \(\alpha, \beta\), and \(\gamma\), we have \(15 \hat{i}+10 \hat{j}+6 \hat{k}=\alpha(2 \vec{p}+\vec{q})+\beta(\vec{p}-2 \vec{q})+\gamma(\vec{p} \times \vec{q})\) then the value of \(\gamma\) is

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A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity \(\omega\) is an example of a non-inertial frame of reference.

The relationship between the force \(\vec{F}_{\text {rot }}\) experienced by a particle of mass \(m\) moving on the rotating disc and the force \(\vec{F}_{\text {in }}\) experienced by the particle in an inertial frame of reference is

\(\vec{F}_{\mathrm{rot}}=\vec{F}_{\mathrm{in}}+2 m\left(\vec{v}_{\mathrm{rot}} \times \vec{\omega}\right)+m(\vec{\omega} \times \vec{r}) \times \vec{\omega}\),

where \(\vec{v}_{\text {rot }}\) is the velocity of the particle in the rotating frame of reference and \(\vec{r}\) is the position vector of the particle with respect to the centre of the disc.

Now consider a smooth slot along a diameter of a disc of radius \(R\) rotating counter-clockwise with a constant angular speed \(\omega\) about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the \(x\)-axis along the slot, the \(y\)-axis perpendicular to the slot and the \(z\)-axis along the rotation axis \((\vec{\omega}=\omega \hat{k}) .\) A small block of mass \(m\) is gently placed in the slot at \(\vec{r}=(R / 2) \hat{i}\) at \(t=0\) and is constrained to move only along the slot.

Question :
The net reaction of the disc on the block is

For the following electrochemical cell at $\text{289 K,}$
\(\operatorname{Pt}( s ) \mid H _2(g, 1\) bar \() \mid H ^{+}( aq , 1 M ) \| M ^{4+}(\) aq. \(), M ^{2+}(\) aq. \() \mid \operatorname{Pt}( s )\)
${\mathrm{E}}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}=0.092\mathrm{ }\mathrm{V}$ when $\frac{\left[{\mathrm{M}}^{2+}\left(\mathrm{a}\mathrm{q}.\right)\right]}{\left[{\mathrm{M}}^{4+}\left(\mathrm{a}\mathrm{q}.\right)\right]}={10}^{\mathrm{x}}$
Given: ${\mathrm{E}}_{{\mathrm{M}}^{4+}/{\mathrm{M}}^{2+}}^0=0.151\mathrm{ }\mathrm{V}\mathrm{ };2.303\frac{\mathrm{R}\mathrm{T}}{\mathrm{F}}=0.059$
The value of $\text{x}$ is -

An acidified solution of $0.05 \mathrm{M} {\mathrm{Zn}}^{2+}$ is saturated with $0.1 \mathrm{M} {\mathrm{H}}_2\mathrm{S}$. What is the minimum molar concentration $\left(\mathrm{M}\right)$ of ${\mathrm{H}}^{+}$ required to prevent the precipitation of $\mathrm{ZnS}$? Use ${\mathrm{K}}_{\mathrm{sp}}\left(\mathrm{ZnS}\right)=1.25\times {10}^{-22}$ and overall dissociation constant of ${\mathrm{H}}_2\mathrm{S}, {\mathrm{K}}_{\mathrm{NET}}={\mathrm{K}}_1{\mathrm{K}}_2=1\times {10}^{-21}$

Three randomly chosen non negative integers $x,y,z$ are found to satisfy the equation  $x+y+z=10$ . Then the probability that $z$ is even, is