Area of the region enclosed by the curves \(3 x^2-y^2-2 x y\) \(+4 x+1=0\) and \(3 x^2-y^2-2 x y+6 x+2 y=0\) is

Real part of \(\frac{(\cos a+i \sin a)^6}{(\sin b+i \cos b)^8}\) is

The domain of defined of the function \(f(x)=\sqrt{\frac{1-|x|}{2-|x|}}\) is

If \(f: R \rightarrow R\) is defined by \(f(x)=\left[\frac{x}{5}\right]\) for \(x \in R\), where \([y]\) denotes the greatest integer not exceeding \(y\), then \(\{f(x):|x| < 71\}\) is equal to

The range of the real valued function \(f(x)=\sin ^{-1}\left(\frac{1+x^2}{2 x}\right)+\cos ^{-1}\left(\frac{2 x}{1+x^2}\right)\) is

A true statement among the following identities is

If \(\alpha, \beta, \gamma\) are the roots of \(x^3+p x^2+q x+r=0\), then the value of \(\left(1+\alpha^2\right)\left(1+\beta^2\right)\left(1+\gamma^2\right)\) is

Match the items of List - I with those of the entires of List - II
\(List - I\)
\(\begin{aligned} & \text { (I) } \sin ^2 5^{\circ}+\sin ^2 10^{\circ}+ \\ & \sin ^2 15^{\circ}+\ldots+\sin ^2 90^{\circ}=\end{aligned}\)
\(\begin{aligned} & \text { (II) } \tan ^2 5^{\circ} \cdot \tan ^2 10^{\circ} \text {. } \\ & \tan ^2 15^{\circ} \ldots \tan ^2 85^{\circ}= \\ & \end{aligned}\)
\(\begin{aligned} & \text { (III) } \cos ^2 5^{\circ}+\cos ^2 10^{\circ} \\ & +\cos ^2 15^{\circ}+\ldots+\cos ^2 180^{\circ}=\end{aligned}\)
\(\begin{gathered}\text { (IV) } \cot 5^{\circ}+\cot 10^{\circ}+\cot 15^{\circ} \\ +\ldots .+\cot 175^{\circ}=\end{gathered}\)
\(List - II\)
(A) \(0\)
(B) \(\frac{19}{2}\)
(C) \(18\)
(D) \(1\)
(E) \(-1\)

Find the condition for the line \(a x+b y+c=0\) to be a normal to an ellipse \(\frac{x^2}{4}+\frac{y^2}{36}=1\)

If \(f^{\prime}(x)=\sqrt{2 x^2-1}\) and \(y=f\left(x^3\right)\), then find the value of \(\frac{d y}{d x}\) at \(x=1\)

The rate of reaction of \(t\)-butyl bromide and . \(\mathrm{NaOH}\) in water depends on the concentration of

If a focal chord of the ellipse \(\frac{x^2}{25}+\frac{y^2}{16}=1\) meets its minor axis at the point \((0,3)\), then the perpendicular distance from the centre of the ellipse to this focal chord is

In \(\triangle \mathrm{ABC}\), if \(\cos \mathrm{A} \cdot \cos \mathrm{B} \cdot \cos \mathrm{C}=\frac{1}{5}\), then \(\tan \mathrm{A} \tan \mathrm{B}+\) \(\tan \mathrm{B} \tan \mathrm{C}+\tan \mathrm{C} \tan \mathrm{A}=\)