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The number of possible organobromine compounds, which can be obtained in the allylic bromination of 1 - butene with \(\mathrm{N}\) - bromosuccinimide is

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A double ordinate \(P Q\) of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) is such that \(\Delta O P Q\) is equilateral, \(O\) being the centre of the hyperbola. Then the eccentricity e satisfies the relation

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A chord \(A B\) is drawn from the point \(A(0,3)\) on the circle \(x^{2}+4 x+(y-3)^{2}=0\), and is extended to \(M\) such that \(\mathrm{AM}=2 \mathrm{AB} .\) The locus of \(\mathrm{M}\) is

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A simple pendulum of length \(\ell\) is displaced so that its taught string is horizontal and then released. A uniform bar pivoted at one end is simultaneously released from its horizontal position. If their motions are synchronous, what is the length of the bar?

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The point \(Q\) is the image of the point \(P(1,5)\) about the line \(y=x\) and \(R\) is the image of the point \(Q\) about the line \(y=-x\). The circumcenter of the \(\Delta P Q R\) is

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The normal to a curve at \(P(x, y)\) meets the \(x\)-axis at \(G\). If the distance of \(G\) from the origin is twice the abscissa of \(P\) then the curve is

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