Let $PQRS$ be a quadrilateral in a plane, where $QR=1,\angle PQR=\angle QRS=70°,\angle PQS=15°$ and $\angle PRS=40°$. If $\angle RPS=\theta °,PQ=\alpha$ and $PS=\beta$, then the interval(s) that contain(s) the value of $4\alpha \beta \sin \theta °$ is/are

The function $\mathit{\text{f}}\left(x\right)=2\left|x\right|+\left|x+2\right|-\left|\left|x+2\right|-2\left|x\right|\right|$ has a local minimum or a local maximum at $x=$

For any real number $x$, let $\left[x\right]$ denotes the largest integer less than or equal to $x$. If $I={\int }_0^{10}\left[\sqrt{\frac{10x}{x+1}}\right]dx$, then the value of $9I$ is ____.

For any complex number $w=c+id$, let $\arg \left(w\right)\in (-\pi ,\pi ]$, where $i=\sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z=x+iy$ satisfying $\arg \left(\frac{z+\alpha }{z+\beta }\right)=\frac{\pi }{4}$, the ordered pair $(x, y)$ lies on the circle $x^2+y^2+5x-3y+4=0$
Then which of the following statements is(are) TRUE?

Let $\mathrm{f}\left(\mathrm{x}\right)=\sin \left(\frac{\mathrm{\pi }}{6}\sin \left(\frac{\mathrm{\pi }}{2}\sin \mathrm{x}\right)\right)$, for all $\mathrm{x}\in R$ and $\mathrm{g}\left(\mathrm{x}\right)=\frac{\mathrm{\pi }}{2}\sin \mathrm{x}$, for all $\mathrm{x}\in R$. Let $\left(\mathrm{f}o\mathrm{g}\right)\mathrm{ }\left(\mathrm{x}\right)$ denote $\mathrm{f}\left(\mathrm{g}\left(\mathrm{x}\right)\right)$ and $\left(\mathrm{g}o\mathrm{f}\right)\mathrm{ }\left(\mathrm{x}\right)$ denote $\mathrm{g}\left(\mathrm{f}\left(\mathrm{x}\right)\right)$. Then which of the following is (are) true?

Let \(F(x)\) be an indefinite integral of \(\sin ^2 x\).
Statement I The function \(F(x)\) satisfies \(F(x+\pi)=F(x)\) for all real \(x\).
Statement II \(\sin ^2(x+\pi)=\sin ^2 x\) for all real \(x\).

Two lines ${\text{L}}_1\text{ : }\text{x}=5\text{,}\frac{\text{y}}{3-\alpha }=\frac{\text{z}}{-2}$ and ${\text{L}}_2\text{ : }\text{x}=\alpha \text{,}\frac{\text{y}}{-1}=\frac{\text{z}}{2-\alpha }$ are coplanar. Then $\alpha$ can take value(s)

In plotting stress versus strain curves for two materials \(P\) and \(Q\), a student by mistake puts strain on the \(y\)-axis and stress on the \(x\)-axis as shown in the figure. Then the correct statement(s) is(are)

Six point charges, each of the same magnitude \(q\), are arranged in different manners as shown in Column-II. In each case, a point \(M\) and a line \(P Q\) passing through \(M\) are shown. Let \(E\) be the electric field and \(V\) be the electric potential at \(M\) (potential at infinity is zero) due to the given charge distribution when it is at rest. Now, the whole system is set into rotation with a constant angular velocity about the line \(P Q\). Let \(B\) the magnetic field at \(M\) and \(\mu\) be the magnetic moment of the system in this condition. Assume each rotating charge to be equivalent to a steady current.

Consider a triangle $PQR$ having sides of lengths $p,q$ and $r$ opposite to the angles $P,Q$ and $R$, respectively. Then which of the following statements is (are) TRUE?

The correct statement about the following disaccharide is

For the resistance network shown in the figure, choose the correct option(s)

Let $a, b\mathbb{ }\in \mathbb{R}$ and $a^2+b^2\ne 0$ . Suppose $S=\left\{z\in \mathbb{C}\mathbb{ }:z=\frac{1}{a+ibt}, t\mathbb{ }\in \mathbb{R},\mathbb{ }t \ne 0\right\},$ where $i= \sqrt{-1}.$ If $z=x+iy$ and $z\in S,$ then $\left(x, y\right)$ lies on