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question_answer1) If the line \[\frac{x-3}{2}=\frac{y-4}{3}=\frac{z-5}{4}\]lies in the plane \[4x+4y-kz-d=0,\]then find the value of \[\left( k+d \right).\]
question_answer2) The distance of the point \[\left( -1,-5,-10 \right)\] from the point of intersection of line \[\frac{x-2}{2}=\frac{y+1}{4}=\frac{z-2}{12}\] and plane \[x-y+z=5\] is \['\lambda '\] then find value of \[\left[ \lambda /3 \right]\] (where \[[\,\cdot \,]\] represents greatest integer function)
question_answer3) The plane passing through the point \[(-2,-2,2)\] and containing the line joining the points \[\left( 1,\text{ }1,\text{ }1 \right)\] and \[\left( 1,\text{ -}1,\text{ }2 \right)\] makes intercepts on the co-ordinates axes, the sum of whose length is \[\lambda \] then find value of \[\lambda /2\].
question_answer4) The volume of the tetrahedron included between the plane \[3x+4y-5z\text{ - }60=0\] and the coordinate planes is \[\lambda \] then find value of \[\frac{\lambda }{100}\].
question_answer5) A plane is parallel to two lines whose direction ratios are \[\left( 1,0,-1 \right)\]and \[\left( \text{- }1,\text{ }1,\text{ }0 \right)\] it passes through the point \[\left( 1,\text{ }1,\text{ }1 \right),\]cuts the axis at A, B, C, then find the volume of the tetrahedron OABC.
question_answer6) If the reflection of the point \[P\left( 1,\text{ }0,\text{ }0 \right)\] in the line \[\frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+10}{8}\] is \[\left( \alpha ,\beta ,\gamma \right)\] then find value of \[-\left( \alpha +\beta +\gamma \right).\]
question_answer7) A line makes the same angle \[\theta \], with each of the x and z axes. If the angle \[\beta \], which it makes with y-axis is such that \[{{\sin }^{2}}\beta =3{{\sin }^{2}}\theta ,\] then find value of \[5{{\cos }^{2}}\theta .\]
question_answer8) If line joining \[A\text{ }\left( 3,\text{ }4,\text{ }5 \right)\] and \[B\text{ }\left( 4,3,9 \right)\] are parallel to plane \[\overrightarrow{r}.\left( \hat{i}+\lambda \hat{j}+\hat{k} \right)=5\] then find value of \[\lambda \].
question_answer9) A variable plane is at a distance 2 from origin O and meets co-ordinate axes at A, B and C. The centroid of tetrahedron OABC lies on \[\frac{1}{{{x}^{2}}}+\frac{1}{{{y}^{2}}}+\frac{1}{{{z}^{2}}}={{k}^{2}}\] then find value of k.
question_answer10) Find the distance of the point \[\left( 1,2,3 \right)\] from the line \[\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}.\]
question_answer11) If the sum of the squares of the distance of a point from the three coordinate axes is 36, if its distance from the origin is \[k\sqrt{2}\] then find k.
question_answer12) If the shortest distance between the line \[\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}\] and \[\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}\] is \[k\sqrt{30}\] then find k.
question_answer13) The image of point \[\left( 1,\text{ }2,-3 \right)\] in the plane \[3x-3y+10z-26=0\] is \[(\alpha ,\,\,\beta ,\,\,\gamma )\] then find \[\left( \alpha +\beta +\gamma \right).\]
question_answer14) Find algebraic sum of the intercepts made by the plane \[x+3y-4z+6=0\] on the axes.
question_answer15) If the straight line \[x=1+s,\,y=3-\lambda s,\,\] \[z=1+\lambda s\] and \[x=t/2,\,\,y=1+t,\,z=2-t,\] with parameters \['s'\] and \['t'\] respectively, are coplanar, then find the value of \[\lambda \].
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