A vector v in the first octant is inclined to the x-axis at 60∘, to the y-axis at 45 and to the z-axis at an acute angle. If a plane passing through the points (2,−1,1) and (a,b,c), is normal to v, then :
02Medium249×since 2002Q2458
If a plane passes through the points (−1,k,0),(2,k,−1),(1,1,2) and is parallel to the line 1x−1=22y+1=−1z+1, then the value of (k−1)(k−2)k2+1 is :
03Medium249×since 2002Q2459
The line l1 passes through the point (2, 6, 2) and is perpendicular to the plane 2x+y−2z=10. Then the shortest distance between the line l1 and the line 2x+1=−3y+4=2z is :
04Hard249×since 2002Q2460
The plane 2x−y+z=4 intersects the line segment joining the points A (a,−2,4) and B (2,b,−3) at the point C in the ratio 2 : 1 and the distance of the point C from the origin is 5. If ab<0 and P is the point (a−b,b,2b−a) then CP2 is equal to :
05Medium249×since 2002Q2461
If the lines 1x−1=2y−2=1z+3 and 2x−a=3y+2=1z−3 intersect at the point P, then the distance of the point P from the plane z=a is :
06Medium249×since 2002Q2462
If the foot of the perpendicular drawn from (1, 9, 7) to the line passing through the point (3, 2, 1) and parallel to the planes x+2y+z=0 and 3y−z=3 is (α,β,γ), then α+β+γ is equal to :
07Medium249×since 2002Q2463
Let the plane containing the line of intersection of the planes
P₁ : x+(λ+4)y+z=1 and
P₂ : 2x+y+z=2
pass through the points (0, 1, 0) and (1, 0, 1). Then the distance of
the point (2λ,λ,−λ) from the plane P₂ is :
08Medium249×since 2002Q2464
The distance of the point (−1,9,−16) from the plane
2x+3y−z=5 measured parallel to the line
3x+4=42−y=12z−3 is :
09Hard249×since 2002Q2465
The plane, passing through the points (0,−1,2) and (−1,2,1) and parallel to the line passing through (5,1,−7) and (1,−1,−1), also passes through the point :
10Hard249×since 2002Q2466
The distance of the point (−1,2,3) from the plane r⋅(i^−2j^+3k^)=10 parallel to the line of the shortest distance between the lines r=(i^−j^)+λ(2i^+k^) and r=(2i^−j^)+μ(i^−j^+k^) is :