If \(\left|\begin{array}{lll}a & a^3 & a^4 \\ b & b^3 & b^4 \\ c & c^3 & c^4\end{array}\right|=k(a-b)(b-c)(c-a)\) then \(\mathrm{k}=\)

Three non-zero non-collinear vectors \(\hat{\mathbf{a}}, \mathbf{b}\) and \(\hat{\mathbf{c}}\) are such that \(\hat{\mathbf{a}}+3 \hat{\mathbf{b}}\) is collinear with \(\hat{\mathbf{c}}\), while \(\hat{\mathbf{c}}\) is \(3 \hat{\mathbf{b}}+2 \hat{\mathbf{c}}\) collinear with \(\hat{\mathbf{a}}\). Then \(\hat{\mathbf{a}}+3 \hat{\mathbf{b}}+2 \hat{\mathbf{c}}\) equals to

The distance between the two lines represented by \(8 x^2-24 x y+18 y^2-6 x+9 y-5=0\) is

One line of the pair of lines \(x^2+x y-2 y^2=0\) is perpendicular to one line of the pair of lines \(3 y^2-5 x y-2 x^2=0\). If the combined equation of the two lines other than those two perpendicular lines is \(a x^2+2 h x y+b y^2=0\), then \(a+2 h+b=\)

Let \(n\) be a positive integer. If the coefficients of \(2^{\text {nd }}, 3^{\text {rd }}\) and \(4^{\text {th }}\) terms in the expansion of \((1+x)^n\) are in A.P, then the value of \(n=\)

Let P be any point on the circle \(x^2+y^2=25\). Let L be the chord of contact of P with respect to the circle \(x^2+y^2=\) 9. The locus of the poles of the lines L with respect to the circle \(x^2+y^2=36\) is

If \(z=x+i y, x, y \in R\) and if the point \(P\) in the argand plane represents \(z\), then the locus of \(P\) satisfying the condition \(\arg \left(\frac{z-1}{z-3 i}\right)=\frac{\pi}{2}\), is

Names of units of some physical quantities are given in List-I and their dimensions formulae are given in List-Il. Match the correct pairs in the lists
\(\begin{array}{ll|ll}
\hline & \text{ List-I } & \text{ List-II } \\
\hline A. & \mathrm{Pa}-\mathrm{s} & (i) & {\left[\mathrm{L}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}\right]} \\
\hline B. & \mathrm{NmK}^{-1} & (ii) & {\left[\mathrm{MLT}^{-3} \mathrm{~K}^{-1}\right]} \\
\hline C. & \mathrm{J} \mathrm{kg}^{-1} \mathrm{~K}^{-1} & (iii) & {\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]} \\
\hline D. & \mathrm{Wm}^{-1} \mathrm{~K}^{-1} & (iv) & {\left[\mathrm{ML}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}\right]} \\
\hline
\end{array}\)
\(\begin{array}{llll}A & B & C & D\end{array}\)

\(\int_{\pi / 6}^{\pi / 3} \frac{1}{1+\sqrt{\cot x}} d x=\)

A particle is placed at rest inside a hollow hemisphere of radius \(R\). The coefficient of friction between the particle and the hemisphere is \(\mu=\frac{1}{\sqrt{3}}\). The maximum height upto which the particle can remain stationary is

An iron sphere having diameter \(D\) and mass \(M\) is immersed in hot water so that the temperature of the sphere increases by \(\delta T\). If \(\alpha\) is the coefficient of linear expansion of the iron then the change in the surface area of the sphere is

Which among the following is most common in alkyl halides?

If \(\tan \mathrm{h}^{-1}(x+i y)=\frac{1}{2} \tan \mathrm{h}^{-1}\left(\frac{2 x}{1+x^2+y^2}\right)+\frac{i}{2} \tan ^{-1}\left(\frac{2 y}{1-x^2-y^2}\right)\), where \(x, y \in \mathbf{R}\), then \(\tan \mathrm{h}^{-1}(i y)=\)