If \(1, \omega, \omega^2\) are the cube roots of unity and \(1, \alpha, \alpha^2, \alpha^3\) are the fourth roots of unity in usual notation then \(\alpha+\alpha \omega-\alpha^3 \omega^2=\)

An ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)\) is inscribed in a rectangle of dimensions \(2 a\) and \(2 b\) respectively, If the angle between the diagonals of the rectangle is \(\tan ^{-1}(4 \sqrt{3})\), then the eccentricity of that ellipse is

\(\bar{i}+\bar{j}+\bar{k}, a_1 \bar{i}+b_1 \bar{j}+c_1 \bar{k}, a_2 \bar{i}+b_2 \bar{j}+c_2 \bar{k}, a_3 \bar{i}+b_3 \bar{j}+c_3 \bar{k}\) are the position vectors of the points \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}\) respectively. \(\frac{2}{3}(\bar{i}+\bar{j}+\bar{k})\) is the position vector of the centroid of the triangular face BCD of the tetrahedron ABCD. If \(\alpha \bar{i}+\beta \bar{j}+\gamma \bar{k}\) is the position vector of the centroid of the tetrahedron, then \(2 \alpha+\beta+\gamma=\)

In the triangle with vertices at \(A(6,3), B(-6,3)\) and \(C(-6,-3)\), the median through \(A\) meets \(B C\) at \(P\), the line \(A C\) meets the \(x\)-axis at \(Q\), while \(R\) and \(S\) respectively denote the orthocentre and centroid of the triangle. Then the correct matching of the coordinates of points in List-I to List-II is \(\begin{array}{llll} & \text { List-I } & & \text { List-II } \ \text { (i) } & P & \text { (A) } & (0,0) \ \text { (ii) } & Q & \text { (B) } & (6,0) \ \text { (iii) } & R & \text { (C) } & (-2,1) \ \text { (iv) } & S & \text { (D) }(-6,0) \ & & \text { (E) }(-6,-3) \ & & \text { (F) }(-6,3)\end{array}\) (i) (ii) (iii) (iv)

If the direction cosines of two lines are such that \(l+m+n=0, l^2+m^2-n^2=0\), then the angle between them is :

If \(\cos ^{-1}\left(\frac{1}{2}\right)=\cot \left(\cos ^{-1} x\right)\), then the value of \(x\) is

If the line $x+y+1=0$ intersects the circle $x^2+y^2+x+3y=0$ at two points $A$ and $B,$ then the centre of the circle which passes through the points $A, B$ and the point of intersection of the tangents drawn at $A$ and $B$ to the given circle is

A solution of concentration \(C g\) equiv/L has a specific resistance \(R\). The equivalent conductance of the solution is

A particle \(A\) moves along the line, \(y=30 \mathrm{~m}\) with a constant velocity, \(\mathbf{v}\) parallel to \(x\)-axis. At the moment particle \(A\) passes the \(y\)-axis, a particle \(B\) starts from the origin with zero initial speed and a constant acceleration, \(\mathbf{a}=0.40 \mathrm{~m} / \mathrm{sec}^2\). The angle between \(\mathbf{a}\) and \(y\)-axis is \(60^{\circ}\). If the particles \(A\) and \(B\) collide after sometimes, then the value of \(|\mathbf{v}|\) will be

The domain and range of \(f(x)=\frac{1}{\sqrt{|x|-x^2}}\) are A and B respectively. Then \(\mathrm{A} \cup \mathrm{B}=\)

When the engine is switched off a vehicle of mass \(M\) is moving on a rough horizontal road with momentum \(p\). If the coefficient of friction between the road and tyres of the vehicle is \(\mu_k\), the distance travelled by the vehicle before it comes to rest is

\(\begin{aligned}
& 4 \mathrm{Ag}(\mathrm{s})+8 \mathrm{CN}^{-}(\mathrm{aq})+2 \mathrm{H}_2 \mathrm{O}(\mathrm{aq})+\mathrm{O}_2(\mathrm{~g}) \longrightarrow \longrightarrow \\
& 4\left[\mathrm{Ag}(\mathrm{CN})_2\right]^{-}(\mathrm{aq})+4^{-} \mathrm{OH}(\mathrm{aq})
\end{aligned}\)
The above reaction represents the process of concentration of ore in the extraction of silver. This process is

Match the following lists

The correct answer is \(A \quad B \quad C \quad D\)