The equations \(x^2-a x+b=0\) and \(x^2+b x-a=0\) have a common root, then

A complex number \(z\) among the following which satisfy \(z^3+27 i=0\) is

The circle possessing \(y\)-axis as its tangent at \((0,2)\) and passing through \((-1,0)\), also passes through

If \(1, \omega, \omega^2\) are the cube roots of unity then the value of \((x+y)^2+\left(x \omega+y \omega^2\right)^2+\left(x \omega^2+y \omega\right)^2\) is

The number of real values of \(m\) so that the equation \(x^2+(2 m+1) x+m=0\) has equal roots is

The value of ' \(k\) ' for which the equation \(x^2-4 x y-y^2+6 x+2 y+k=0\) represents a pair of straight lines is equal to ........

The lines \(y=2 x+\sqrt{76}\) and \(2 y+x=8\) touch the ellipse \(\frac{x^2}{16}+\frac{y^2}{12}=1\). If the point of intersection of these two lines lie on a circle. whose centre coincides with the centre of that ellipse, then the equation of that circle is

Which one of the following equation is correct for the reaction
\(\mathrm{N}_2(g) 3 \mathrm{H}_2(g) \longrightarrow 2 \mathrm{NH}_3(g) \text { ? }\)

If the planes \(2 x+3 y+4 z+7=0\) and \(4 x+k y+8 z+1\) \(=0\) are parallel, then the equation of the plane passing through is point \((\mathrm{k}, \mathrm{k}, \mathrm{k})\) and having the direction ratios of its normal as \((\mathrm{k}-1, \mathrm{k}, \mathrm{k}+1)\) is

Match the following
\(\begin{array}{ll}
\text{List - I} & \text{List - II} \\
\text{A) } \mathrm{Al}_2 \mathrm{O}_3+2 \mathrm{NaOH}+3 \mathrm{H}_2 \mathrm{O} \rightarrow 2 \mathrm{Na}\left[\mathrm{Al}(\mathrm{OH})_4\right] & \text{I) Roasting} \\
\text{B) } \mathrm{Ni}(\mathrm{CO})_4 \xrightarrow{230^{\circ} \mathrm{C}} \mathrm{Ni}+4 \mathrm{CO} & \text{II) Calcination} \\
\text{C) } \mathrm{Fe}_2 \mathrm{O}_3 \cdot 3 \mathrm{H}_2 \mathrm{O} \xrightarrow{\Delta} \mathrm{Fe}_2 \mathrm{O}_3+3 \mathrm{H}_2 \mathrm{O} & \text{II) Vapour phase refining} \\
\text{D) } 2 \mathrm{PbS}+3 \mathrm{O}_2 \rightarrow 2 \mathrm{PbO}+2 \mathrm{SO}_2 & \text{IV) Electrolysis} \\
& \text{V) Leaching}
\end{array}\)
The correct answer is

At $\mathrm{T}\left(\mathrm{K}\right)\mathrm{xg}$ of a non-volatile solid (molar mass $78 \mathrm{g} {\mathrm{mol}}^{-1}$ ) when added to $0.5 \mathrm{kg}$ water, lowered its freezing point by $1.0°\mathrm{C}$. What is $x$ (in $\mathrm{g}$) (${\mathrm{K}}_{\mathrm{f}}$ of water at $\mathrm{T}\left(\mathrm{K}\right)=1.86 {\mathrm{KKgmol}}^{-1}$)

Match the following
\(\begin{array}{|c|c|c|}
\hline & \text { List - I } & & \text { List - II } \\
\hline \text{b)} & \text { Vitamin } \mathrm{E} & \text { i) } & \text { Meat } \\
\hline \text{c)} & \text { Vitamin K } & \text { ii) } & \text { Sunflower oil } \\
\hline \text{d)} & \text { Vitamin } B_{12} & \text { iii) } & \text { Egg white } \\
\hline \text{e)} & \text { Vitamin } B_2 & \text { iv) } & \text { Exposure to sunlight } \\
\hline & & \text { v) } & \text { Green leafy vegetables } \\
\hline
\end{array}\)
The correct answer is

If \([\mathbf{a} \times \mathbf{b} \mathbf{b} \times \mathbf{c} \mathbf{c} \times \mathbf{a}]=\lambda[\mathbf{a} \mathbf{b} \mathbf{c}]^2\), then \(\lambda\) is equal to