121231 Let the line \(\frac{x-1}{\lambda}=\frac{y-2}{1}=\frac{z-3}{2}\) and \(\frac{x+26}{-2}=\frac{y+18}{3}=\frac{z+28}{\lambda}\) be coplanar and \(P\) be the plane containing these two lines. Then which of the following points does NOT lies on P ?

121232 The shortest distance between the lines \(\frac{x+7}{-6}=\frac{y-6}{7}=z\) and \(\frac{7-x}{2}=y-2=z-6\) is

121236 If the vectors \(\alpha=\hat{\mathbf{i}}+\mathbf{a} \hat{\mathbf{j}}+\mathbf{a}^2 \hat{\mathbf{k}}, \boldsymbol{\beta}=\hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}+\mathbf{b}^2 \hat{\mathbf{k}}\) and \(\gamma=\hat{\mathbf{i}}+\mathbf{c} \hat{\mathbf{j}}+\mathbf{c}^2 \hat{\mathbf{k}}\) are three non- coplanar vectors and \(\left|\begin{array}{lll}a & a^2& 1+a^3 \\ b& b^2 &1+b^3 \\ c &c^2 &1+c^2\end{array}\right|=0\), then the value of abc is

121238 The value of ' \(x\) ' for which the points \((1,2,1)\). \((0,1,3),(1,0,1)\) and \((2,0, x)\) are coplanar is

121212 Lines \(\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-K}\) and \(\frac{x-1}{K}=\frac{y-4}{2}=\) \(\frac{\mathrm{z}-5}{1}\) are coplanar if

121231 Let the line \(\frac{x-1}{\lambda}=\frac{y-2}{1}=\frac{z-3}{2}\) and \(\frac{x+26}{-2}=\frac{y+18}{3}=\frac{z+28}{\lambda}\) be coplanar and \(P\) be the plane containing these two lines. Then which of the following points does NOT lies on P ?

121232 The shortest distance between the lines \(\frac{x+7}{-6}=\frac{y-6}{7}=z\) and \(\frac{7-x}{2}=y-2=z-6\) is

121236 If the vectors \(\alpha=\hat{\mathbf{i}}+\mathbf{a} \hat{\mathbf{j}}+\mathbf{a}^2 \hat{\mathbf{k}}, \boldsymbol{\beta}=\hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}+\mathbf{b}^2 \hat{\mathbf{k}}\) and \(\gamma=\hat{\mathbf{i}}+\mathbf{c} \hat{\mathbf{j}}+\mathbf{c}^2 \hat{\mathbf{k}}\) are three non- coplanar vectors and \(\left|\begin{array}{lll}a & a^2& 1+a^3 \\ b& b^2 &1+b^3 \\ c &c^2 &1+c^2\end{array}\right|=0\), then the value of abc is

121238 The value of ' \(x\) ' for which the points \((1,2,1)\). \((0,1,3),(1,0,1)\) and \((2,0, x)\) are coplanar is

121212 Lines \(\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-K}\) and \(\frac{x-1}{K}=\frac{y-4}{2}=\) \(\frac{\mathrm{z}-5}{1}\) are coplanar if

121231 Let the line \(\frac{x-1}{\lambda}=\frac{y-2}{1}=\frac{z-3}{2}\) and \(\frac{x+26}{-2}=\frac{y+18}{3}=\frac{z+28}{\lambda}\) be coplanar and \(P\) be the plane containing these two lines. Then which of the following points does NOT lies on P ?

121232 The shortest distance between the lines \(\frac{x+7}{-6}=\frac{y-6}{7}=z\) and \(\frac{7-x}{2}=y-2=z-6\) is

121236 If the vectors \(\alpha=\hat{\mathbf{i}}+\mathbf{a} \hat{\mathbf{j}}+\mathbf{a}^2 \hat{\mathbf{k}}, \boldsymbol{\beta}=\hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}+\mathbf{b}^2 \hat{\mathbf{k}}\) and \(\gamma=\hat{\mathbf{i}}+\mathbf{c} \hat{\mathbf{j}}+\mathbf{c}^2 \hat{\mathbf{k}}\) are three non- coplanar vectors and \(\left|\begin{array}{lll}a & a^2& 1+a^3 \\ b& b^2 &1+b^3 \\ c &c^2 &1+c^2\end{array}\right|=0\), then the value of abc is

121238 The value of ' \(x\) ' for which the points \((1,2,1)\). \((0,1,3),(1,0,1)\) and \((2,0, x)\) are coplanar is

121212 Lines \(\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-K}\) and \(\frac{x-1}{K}=\frac{y-4}{2}=\) \(\frac{\mathrm{z}-5}{1}\) are coplanar if

121231 Let the line \(\frac{x-1}{\lambda}=\frac{y-2}{1}=\frac{z-3}{2}\) and \(\frac{x+26}{-2}=\frac{y+18}{3}=\frac{z+28}{\lambda}\) be coplanar and \(P\) be the plane containing these two lines. Then which of the following points does NOT lies on P ?

121232 The shortest distance between the lines \(\frac{x+7}{-6}=\frac{y-6}{7}=z\) and \(\frac{7-x}{2}=y-2=z-6\) is

121236 If the vectors \(\alpha=\hat{\mathbf{i}}+\mathbf{a} \hat{\mathbf{j}}+\mathbf{a}^2 \hat{\mathbf{k}}, \boldsymbol{\beta}=\hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}+\mathbf{b}^2 \hat{\mathbf{k}}\) and \(\gamma=\hat{\mathbf{i}}+\mathbf{c} \hat{\mathbf{j}}+\mathbf{c}^2 \hat{\mathbf{k}}\) are three non- coplanar vectors and \(\left|\begin{array}{lll}a & a^2& 1+a^3 \\ b& b^2 &1+b^3 \\ c &c^2 &1+c^2\end{array}\right|=0\), then the value of abc is

121238 The value of ' \(x\) ' for which the points \((1,2,1)\). \((0,1,3),(1,0,1)\) and \((2,0, x)\) are coplanar is

121212 Lines \(\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-K}\) and \(\frac{x-1}{K}=\frac{y-4}{2}=\) \(\frac{\mathrm{z}-5}{1}\) are coplanar if

121231 Let the line \(\frac{x-1}{\lambda}=\frac{y-2}{1}=\frac{z-3}{2}\) and \(\frac{x+26}{-2}=\frac{y+18}{3}=\frac{z+28}{\lambda}\) be coplanar and \(P\) be the plane containing these two lines. Then which of the following points does NOT lies on P ?

121232 The shortest distance between the lines \(\frac{x+7}{-6}=\frac{y-6}{7}=z\) and \(\frac{7-x}{2}=y-2=z-6\) is

121236 If the vectors \(\alpha=\hat{\mathbf{i}}+\mathbf{a} \hat{\mathbf{j}}+\mathbf{a}^2 \hat{\mathbf{k}}, \boldsymbol{\beta}=\hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}+\mathbf{b}^2 \hat{\mathbf{k}}\) and \(\gamma=\hat{\mathbf{i}}+\mathbf{c} \hat{\mathbf{j}}+\mathbf{c}^2 \hat{\mathbf{k}}\) are three non- coplanar vectors and \(\left|\begin{array}{lll}a & a^2& 1+a^3 \\ b& b^2 &1+b^3 \\ c &c^2 &1+c^2\end{array}\right|=0\), then the value of abc is

121238 The value of ' \(x\) ' for which the points \((1,2,1)\). \((0,1,3),(1,0,1)\) and \((2,0, x)\) are coplanar is

121212 Lines \(\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-K}\) and \(\frac{x-1}{K}=\frac{y-4}{2}=\) \(\frac{\mathrm{z}-5}{1}\) are coplanar if