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question_answer1)
The direction cosines of a line segment \[AB\] are \[-2/\sqrt{17},\] \[3/\sqrt{17},\,\,-2/\sqrt{17}.\] If \[AB=\sqrt{17}\] and the co-ordinates of A are (3, -6, 10), then the co-ordinates of B are
A)
(1, -2, 4) done
clear
B)
(2, 5, 8) done
clear
C)
(-1, 3, -8) done
clear
D)
(1, -3, 8) done
clear
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question_answer2)
The projection of any line on co-ordinate axes be respectively 3, 4, 5 then its length is [MP PET 1995; RPET 2001]
A)
12 done
clear
B)
50 done
clear
C)
\[5\sqrt{2}\] done
clear
D)
None of these done
clear
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question_answer3)
If centroid of the tetrahedron \[OABC\], where \[A,B,C\]are given by (a, 2, 3),(1, b, 2) and (2, 1, c) respectively be (1, 2, -1), then distance of \[P(a,b,c)\] from origin is equal to
A)
\[\sqrt{107}\] done
clear
B)
\[\sqrt{14}\] done
clear
C)
\[\sqrt{107/14}\] done
clear
D)
None of these done
clear
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question_answer4)
If \[P\equiv (0,\,1,\,0),Q\equiv (0,\,0,\,1)\], then projection of \[PQ\] on the plane \[x+y+z=3\]is [EAMCET 2002]
A)
\[\sqrt{3}\] done
clear
B)
3 done
clear
C)
\[\sqrt{2}\] done
clear
D)
2 done
clear
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question_answer5)
The points \[A(4,\,5,\,1),B(0,-1,-1),C(3,\,9,\,4)\]and \[D(-4,\,4,\,4)\]are [Kurukshetra CEE 2002]
A)
Collinear done
clear
B)
Coplanar done
clear
C)
Non- coplanar done
clear
D)
Non- Collinear and non-coplanar done
clear
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question_answer6)
The angle between two diagonals of a cube will be [MP PET 1996, 2000; RPET 2000, 02; UPSEAT 2004]
A)
\[{{\sin }^{-1}}1/3\] done
clear
B)
\[{{\cos }^{-1}}1/3\] done
clear
C)
Variable done
clear
D)
None of these done
clear
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question_answer7)
The equations of the line passing through the point (1,2,-4) and perpendicular to the two lines \[\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}\] and \[\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}\], will be [AI CBSE 1983]
A)
\[\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6}\] done
clear
B)
\[\frac{x-1}{-2}=\frac{y-2}{3}=\frac{z+4}{8}\] done
clear
C)
\[\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6}\] done
clear
D)
None of these done
clear
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question_answer8)
If three mutually perpendicular lines have direction cosines \[({{l}_{1}},{{m}_{1}},{{n}_{1}}),({{l}_{2}},{{m}_{2}},{{n}_{2}})\]and \[({{l}_{3}},{{m}_{3}},{{n}_{3}})\], then the line having direction cosines \[{{l}_{1}}+{{l}_{2}}+{{l}_{3}}\], \[{{m}_{1}}+\,\,{{m}_{2}}+\,\,{{m}_{3}}\]and \[{{n}_{1}}+{{n}_{2}}+{{n}_{3}}\] make an angle of ..... with each other
A)
\[0{}^\circ \] done
clear
B)
\[30{}^\circ \] done
clear
C)
\[60{}^\circ \] done
clear
D)
\[90{}^\circ \] done
clear
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question_answer9)
The straight lines whose direction cosines are given by \[al+bm+cn=0,fmn+gnl+hlm=0\]are perpendicular, if
A)
\[\frac{f}{a}+\frac{g}{b}+\frac{h}{c}=0\] done
clear
B)
\[\sqrt{\frac{a}{f}}+\sqrt{\frac{b}{g}}+\sqrt{\frac{c}{h}}=0\] done
clear
C)
\[\sqrt{af}=\sqrt{bg}=\sqrt{ch}\] done
clear
D)
\[\sqrt{\frac{a}{f}}=\sqrt{\frac{b}{g}}=\sqrt{\frac{c}{h}}\] done
clear
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question_answer10)
If the straight lines \[x=1+s,\] \[y=-3-\lambda s,\] \[z=1+\lambda s\] and \[x=t/2,y=1+t,z=2-t\], with parameters s and \[t\] respectively, are co-planar, then \[\lambda \]equals [AIEEE 2004]
A)
0 done
clear
B)
-1 done
clear
C)
-1/2 done
clear
D)
-2 done
clear
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question_answer11)
The co-ordinates of the foot of perpendicular drawn from point \[P(1,\,0,\,3)\]to the join of points \[A(4,\,7,\,1)\]and \[B(3,\,5,\,3)\] is [RPET 2001]
A)
(5, 7, 1) done
clear
B)
\[\left( \frac{5}{3},\frac{7}{3},\frac{17}{3} \right)\] done
clear
C)
\[\left( \frac{2}{3},\frac{5}{3},\frac{7}{3} \right)\] done
clear
D)
\[\left( \frac{5}{3},\frac{2}{3},\frac{7}{3} \right)\] done
clear
View Solution play_arrow
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question_answer12)
If the lines \[\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}\] and \[\frac{x-3}{1}=\frac{y-k}{1}=\frac{z}{1}\] intersect, then k = [IIT Screening 2004]
A)
\[\frac{2}{9}\] done
clear
B)
\[\frac{9}{2}\] done
clear
C)
0 done
clear
D)
None of these done
clear
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question_answer13)
A square \[ABCD\] of diagonal 2a is folded along the diagonal \[AC\] so that the planes \[DAC\] and \[BAC\] are at right angle. The shortest distance between \[DC\] and \[AB\] is [Kurukshetra CEE 1998]
A)
\[\sqrt{2}a\] done
clear
B)
\[2a/\sqrt{3}\] done
clear
C)
\[2a/\sqrt{5}\] done
clear
D)
[(\sqrt{3}/2)a\] done
clear
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question_answer14)
A line with direction cosines proportional to 2,1, 2 meets each of the lines \[x=y+a=z\]and \[x+a=2y=2z\]. The co-ordinates of each of the points of intersection are given by [AIEEE 2004]
A)
\[(2a,\,\,a,\,3a),(2a,\,a,\,a)\] done
clear
B)
\[(3a,\,2a,\,3a),\ (a,\,a,\,a)\] done
clear
C)
\[(3a,\,2a,\,3a),(a,\,a,\,2a)\] done
clear
D)
\[(3a,\,3a,\,3a),(a,\,a,\,a)\] done
clear
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question_answer15)
The equation of the planes passing through the line of intersection of the planes \[3x-y-4z=0\] and \[x+3y+6=0\] whose distance from the origin is 1, are
A)
\[x-2y-2z-3=0\], \[2x+y-2z+3=0\] done
clear
B)
\[x-2y+2z-3=0\], \[2x+y+2z+3=0\] done
clear
C)
\[x+2y-2z-3=0\], \[2x-y-2z+3=0\] done
clear
D)
None of these done
clear
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question_answer16)
The co-ordinates of the points A and B are (2, 3, 4) (-2, 5,-4) respectively. If a point P moves so that \[P{{A}^{2}}-P{{B}^{2}}=k\] where k is a constant, then the locus of P is
A)
A line done
clear
B)
A plane done
clear
C)
A sphere done
clear
D)
None of these done
clear
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question_answer17)
The equation of the plane passing through the points \[(1,-3,-2)\] and perpendicular to planes \[x+2y+2z=5\] and \[3x+3y+2z=8\], is [AISSE 1987]
A)
\[2x-4y+3z-8=0\] done
clear
B)
\[2x-4y-3z+8=0\] done
clear
C)
\[2x+4y+3z+8=0\] done
clear
D)
None of these done
clear
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question_answer18)
A variable plane at a constant distance p from origin meets the co-ordinates axes in \[A,B,C\]. Through these points planes are drawn parallel to co-ordinate planes. Then locus of the point of intersection is
A)
\[\frac{1}{{{x}^{2}}}+\frac{1}{{{y}^{2}}}+\frac{1}{{{z}^{2}}}=\frac{1}{{{p}^{2}}}\] done
clear
B)
\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}={{p}^{2}}\] done
clear
C)
\[x+y+z=p\] done
clear
D)
\[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=p\] done
clear
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question_answer19)
P is a fixed point \[(a,\,a,\,a)\] on a line through the origin equally inclined to the axes, then any plane through P perpendicular to OP, makes intercepts on the axes, the sum of whose reciprocals is equal to
A)
a done
clear
B)
\[\frac{3}{2a}\] done
clear
C)
\[\frac{3a}{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer20)
The equation of the plane through the intersection of the planes \[x+2y+3z-4=0\], \[4x+3y+2z+1=0\] and passing through the origin will be [MP PET 1998]
A)
\[x+y+z=0\] done
clear
B)
\[17x+14y+11z=0\] done
clear
C)
\[7x+4y+z=0\] done
clear
D)
\[17x+14y+z=0\] done
clear
View Solution play_arrow
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question_answer21)
The d.r-s of normal to the plane through \[(1,\,0,\,0),\,\,(0,\,1,\,0)\] which makes an angle \[\frac{\pi }{4}\] with plane \[x+y=3\], are [AIEEE 2002]
A)
\[1,\sqrt{2},1\] done
clear
B)
1,1, \[\sqrt{2}\] done
clear
C)
1, 1, 2 done
clear
D)
\[\sqrt{2},\,1,\,1\] done
clear
View Solution play_arrow
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question_answer22)
Two systems of rectangular axes have the same origin. If a plane cuts them at distance a, b, c and a', b', c' from the origin, then [AIEEE 2003]
A)
\[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}+\frac{1}{a{{'}^{2}}}+\frac{1}{b{{'}^{2}}}+\frac{1}{c{{'}^{2}}}=0\] done
clear
B)
\[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}-\frac{1}{{{c}^{2}}}+\frac{1}{a{{'}^{2}}}+\frac{1}{b{{'}^{2}}}-\frac{1}{c{{'}^{2}}}=0\] done
clear
C)
\[\frac{1}{{{a}^{2}}}-\frac{1}{{{b}^{2}}}-\frac{1}{{{c}^{2}}}+\frac{1}{a{{'}^{2}}}-\frac{1}{b{{'}^{2}}}-\frac{1}{c{{'}^{2}}}=0\] done
clear
D)
\[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}-\frac{1}{a{{'}^{2}}}-\frac{1}{b{{'}^{2}}}-\frac{1}{c{{'}^{2}}}=0\] done
clear
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question_answer23)
If \[4x+4y-kz=0\] is the equation of the plane through the origin that contains the line \[\frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4},\]then k = [MP PET 1992]
A)
1 done
clear
B)
3 done
clear
C)
5 done
clear
D)
7 done
clear
View Solution play_arrow
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question_answer24)
The distance of the point (1, -2, 3) from the plane \[x-y+z=5\] measured parallel to the line \[\frac{x}{2}=\frac{y}{3}=\frac{z}{-6},\]is [AI CBSE 1984]
A)
1 done
clear
B)
6/7 done
clear
C)
7/6 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer25)
The distance of the point of intersection of the line \[\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2}\]and the plane \[x+y+z=17\] from the point (3, 4, 5) is given by
A)
3 done
clear
B)
3/2 done
clear
C)
\[\sqrt{3}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer26)
The lines \[\frac{x-a+d}{\alpha -\delta }=\frac{y-a}{\alpha }=\frac{z-a-d}{\alpha +\delta }\] and \[\frac{x-b+c}{\beta -\gamma }=\frac{y-b}{\beta }=\frac{z-b-c}{\beta +\gamma }\] are coplanar and then equation to the plane in which they lie, is
A)
\[x+y+z=0\] done
clear
B)
\[x-y+z=0\] done
clear
C)
\[x-2y+z=0\] done
clear
D)
\[x+y-2z=0\] done
clear
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question_answer27)
The line \[\frac{x-3}{2}=\frac{y-4}{3}=\frac{z-5}{4}\] lies in the plane \[4x+4y-kz-d=0\]. The values of k and d are
A)
4, 8 done
clear
B)
-5, -3 done
clear
C)
5, 3 done
clear
D)
- 4, - 8 done
clear
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question_answer28)
The value of k such that \[\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-k}{2}\] lies in the plane \[2x-4y+z=7\], is [IIT Screening 2003]
A)
7 done
clear
B)
- 7 done
clear
C)
No real value done
clear
D)
4 done
clear
View Solution play_arrow
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question_answer29)
The shortest distance from the plane \[12x+4y+3z=327\] to the sphere \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+\] \[4x-2y-6z=155\] is [AIEEE 2003]
A)
26 done
clear
B)
\[11\frac{4}{13}\] done
clear
C)
13 done
clear
D)
39 done
clear
View Solution play_arrow
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question_answer30)
The radius of the circle in which the sphere \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2x-2y-4z-19=0\] is cut by the plane \[x+2y+2z+7=0\]is [AIEEE 2003]
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
4 done
clear
View Solution play_arrow
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question_answer31)
The equation of motion of a rocket are: \[x=2t,\,y=-4t,\] \[\,z=4t\] where the time 't' is given in seconds, and the co-ordinates of a moving point in kilometers. What is the path of the rocket. At what distance will be the rocket be from the starting point 0(0, 0, 0) in 10 seconds
A)
Straight line, 60 km done
clear
B)
Straight line, 30 km done
clear
C)
Parabola, 60 km done
clear
D)
Ellipse, 60 km done
clear
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question_answer32)
The plane \[lx+my=0\] is rotated an angle \[\alpha \] about its line of intersection with the plane \[z=0\], then the equation to the plane in its new position is
A)
\[lx+my\pm z\sqrt{({{l}^{2}}+{{m}^{2}})}\tan \alpha =0\] done
clear
B)
\[lx-my\pm z\sqrt{({{l}^{2}}+{{m}^{2}})}\tan \alpha =0\] done
clear
C)
\[lx+my\pm z\sqrt{({{l}^{2}}+{{m}^{2}})}\cos \alpha =0\] done
clear
D)
\[lx-my\pm z\sqrt{({{l}^{2}}+{{m}^{2}})}\cos \alpha =0\] done
clear
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question_answer33)
The distance between two points P and Q is d and the length of their projections of PQ on the co-ordinate planes are \[{{d}_{1}},{{d}_{2}},{{d}_{3}}\]. Then \[d_{1}^{2}+d_{2}^{2}+d_{3}^{2}=k{{d}^{2}}\] where . k- is
A)
1 done
clear
B)
5 done
clear
C)
3 done
clear
D)
2 done
clear
View Solution play_arrow
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question_answer34)
If \[{{P}_{1}}\] and \[{{P}_{2}}\] are the lengths of the perpendiculars from the points (2,3,4) and (1,1,4) respectively from the plane \[3x-6y+2z+11=0\], then \[{{P}_{1}}\] and \[{{P}_{2}}\] are the roots of the equation
A)
\[{{P}^{2}}-23P+7=0\] done
clear
B)
\[7{{P}^{2}}-23P+16=0\] done
clear
C)
\[{{P}^{2}}-17P+16=0\] done
clear
D)
\[{{P}^{2}}-16P+7=0\] done
clear
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question_answer35)
The edge of a cube is of length -a- then the shortest distance between the diagonal of a cube and an edge skew to it is
A)
\[a\sqrt{2}\] done
clear
B)
a done
clear
C)
\[\sqrt{2}/a\] done
clear
D)
\[a/\sqrt{2}\] done
clear
View Solution play_arrow
