JEE Main & Advanced Mathematics Three Dimensional Geometry Question Bank

done System of Co-ordinates, Direction cosines and direction ratios, Projection Total Questions - 75

• question_answer1)
  If the points (0, 1, 2), (2, ?1, 3) and (1, ?3, 1) are the vertices of a triangle, then the triangle is
  A)
             Right angled done clear
  B)
             Isosceles right angled done clear
  C)
             Equilateral done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer2)
  If the points (?1, 3, 2), (?4, 2, ?2) and \[(5,\,\,5,\,\,\lambda )\] are collinear, then \[\lambda \]=
  A)
             ? 10 done clear
  B)
             5 done clear
  C)
             ? 5 done clear
  D)
             10 done clear
  View Solution play_arrow

• question_answer3)
  The direction cosines of the normal to the plane \[2x+3y-6z=5\] are
  A)
             \[2,\,\,3,\,\,-6\] done clear
  B)
             \[\frac{2}{7},\,\frac{3}{7},\,-\frac{6}{7}\] done clear
  C)
             \[\frac{2}{5},\,\,\frac{3}{5},\,-\frac{6}{5}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer4)
  The point dividing the line joining the points \[(1,\,\,2,\,\,3)\]and \[(3,\,\,-5,\,\,6\,)\] in the ratio \[3:\,-5\] is
  A)
             \[\left( 2,\,\frac{-25}{2},\,\frac{3}{2} \right)\] done clear
  B)
             \[\left( -2,\,\frac{25}{2},\,\frac{-3}{2} \right)\] done clear
  C)
             \[\left( 2,\,\,\frac{25}{2},\,\frac{3}{2} \right)\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer5)
  From which of  the following the distance of the point \[(1,\,\,2,\,\,3)\]is \[\sqrt{10}\]
  A)
             Origin done clear
  B)
             x-axis done clear
  C)
             y-axis done clear
  D)
             z-axis done clear
  View Solution play_arrow

• question_answer6)
  If \[\alpha ,\,\,\beta ,\,\gamma \]be the angles which a line makes with the positive direction of co-ordinate axes, then             \[{{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma =\] [RPET 2000; AMU 2002; MP PET 1989, 98, 2000, 03, Pb. CET 2001]
  A)
             2 done clear
  B)
             1 done clear
  C)
             3 done clear
  D)
             0 done clear
  View Solution play_arrow

• question_answer7)
  If a,b,g  be the direction angles of a vector and \[\cos \alpha =\frac{14}{15}\], \[\cos \beta =\frac{1}{3}\] then \[\cos \gamma \]=
  A)
             \[\pm \frac{2}{15}\] done clear
  B)
             \[\frac{1}{5}\] done clear
  C)
             \[\pm \frac{1}{15}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer8)
  All the points on the x- axis have                 [MP PET 1988]
  A)
             \[x=0\] done clear
  B)
             \[y=0\] done clear
  C)
             \[x=0,y=0\] done clear
  D)
             \[y=0,z=0\] done clear
  View Solution play_arrow

• question_answer9)
  . Distance between the points (1, 3, 2) and (2, 1, 3) is [MP PET 1988]
  A)
             12 done clear
  B)
             \[\sqrt{12}\] done clear
  C)
             \[\sqrt{6}\] done clear
  D)
             6 done clear
  View Solution play_arrow

• question_answer10)
  The direction cosines of the line \[x=y=z\] are [MP PET 1989]
  A)
             \[\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\] done clear
  B)
             \[\frac{1}{3},\frac{1}{3},\frac{1}{3}\] done clear
  C)
             1, 1, 1 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer11)
  Distance of the point (1, 2, 3) from the co-ordinate axes are
  A)
             13, 10, 5 done clear
  B)
             \[\sqrt{13},\sqrt{10},\sqrt{5}\] done clear
  C)
             \[\sqrt{5},\sqrt{13},\sqrt{10}\] done clear
  D)
             \[\frac{1}{\sqrt{13}},\frac{1}{\sqrt{10}},\frac{1}{\sqrt{5}}\] done clear
  View Solution play_arrow

• question_answer12)
  If the centroid of triangle whose vertices are (a,1, 3), (? 2, b, ?5) and (4, 7, c) be the origin, then the values of a, b, c are
  A)
             ? 2, ?8, ?2 done clear
  B)
             2, 8, ?2 done clear
  C)
             ?2, ?8, 2 done clear
  D)
             7, ?1, 0 done clear
  View Solution play_arrow

• question_answer13)
  Which of the following set of points are non- collinear                          [MP PET 1990]
  A)
             (1, ?1, 1), (?1, 1, 1), (0, 0, 1) done clear
  B)
             (1, 2, 3), (3, 2, 1), (2, 2, 2) done clear
  C)
             (?2,4, ?3), (4, ?3, ?2), (?3, ?2, 4) done clear
  D)
             (2, 0, ?1), (3, 2, ?2), (5, 6, ?4) done clear
  View Solution play_arrow

• question_answer14)
  If a straight line in space is equally inclined to the co-ordinate axes, the cosine of its angle of inclination to any one of the axes is                                                                            [MP PET 1992]
  A)
             \[\frac{1}{3}\] done clear
  B)
             \[\frac{1}{2}\] done clear
  C)
             \[\frac{1}{\sqrt{3}}\] done clear
  D)
             \[\frac{1}{\sqrt{2}}\] done clear
  View Solution play_arrow

• question_answer15)
  If a line makes angles of \[{{30}^{o}}\]and \[{{45}^{o}}\]with x-axis and        y-axis, then the angle made by it with \[z-\]axis is
  A)
             \[{{45}^{o}}\] done clear
  B)
             \[{{60}^{o}}\] done clear
  C)
             \[{{120}^{o}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer16)
  Direction ratios of the normal to the plane passing through the points (0, 1, 1), (1, 1, 2) and (?1, 2, ? 2) are
  A)
             (1, 1, 1) done clear
  B)
             (2, 1, ?1) done clear
  C)
             (1, 2, ?1) done clear
  D)
             (1, ? 2, ? 1) done clear
  View Solution play_arrow

• question_answer17)
  If the length of a vector be 21 and direction ratios be 2, ? 3, 6 then its direction cosines are
  A)
             \[\frac{2}{21},\frac{-1}{7},\frac{2}{7}\] done clear
  B)
             \[\frac{2}{7},\frac{-3}{7},\frac{6}{7}\] done clear
  C)
             \[\frac{2}{7},\frac{3}{7},\frac{6}{7}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer18)
  If the co-ordinates of the points \[P,\,Q,R,\,S\] be (1, 2, 3),        (4, 5, 7), (? 4, 3, ? 6) and (2, 0, 2) respectively, then 
  A)
             \[PQ||RS\] done clear
  B)
             \[PQ\,\bot \,RS\] done clear
  C)
             \[PQ=RS\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer19)
  If the co-ordinates of  the points \[A,B,C,D\] be (2, 3, ?1),        (3, 5, ? 3), (1, 2, 3) and (3, 5, 7) respectively, then the projection of \[AB\] on \[CD\] is
  A)
             0 done clear
  B)
             1 done clear
  C)
             2 done clear
  D)
             \[\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer20)
  If the co-ordinates of the points  P and Q be (1, ?2, 1) and (2, 3, 4) and O be the origin, then
  A)
             \[OP=OQ\] done clear
  B)
             \[OP\,\bot \,OQ\] done clear
  C)
             \[OP||OQ\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer21)
  If the projections of a line on the co-ordinate axes be           2, ?1, 2, then the length of the lines is
  A)
             3 done clear
  B)
             4 done clear
  C)
             2 done clear
  D)
             \[\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer22)
  xy-plane divides the line joining the points (2, 4, 5) and        (? 4, 3, ? 2) in the ratio [MP PET 1988]
  A)
             3 : 5 done clear
  B)
             5 : 2 done clear
  C)
             1 : 3 done clear
  D)
             3 : 4 done clear
  View Solution play_arrow

• question_answer23)
  If \[A\,(1\,,\,\,2,\,\,-1)\] and \[B(-1,\,\,0,\,\,1)\] are given, then the              co-ordinates of P which divides \[AB\] externally in the ratio\[1:2\], are [MP PET 1989]
  A)
             \[\frac{1}{3}(1,\,4,-1)\] done clear
  B)
             (3, 4, ?3) done clear
  C)
             \[\frac{1}{3}(3,\,4,-3)\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer24)
  The point of intersection of the line joining the points (3, 4, 1) and (5, 1, 6) and the xy-plane is
  A)
             \[(13,\,23,\,0)\] done clear
  B)
             \[\left( \frac{13}{5},\frac{23}{5},0 \right)\] done clear
  C)
             (?13, 23, 0) done clear
  D)
             \[\left( -\frac{13}{5},\frac{23}{5},0 \right)\] done clear
  View Solution play_arrow

• question_answer25)
  If the co-ordinates of A and B be (1, 2, 3) and (7, 8, 7), then the projections of the line segment AB on the co-ordinate axes are
  A)
             6, 6, 4 done clear
  B)
             4, 6, 4 done clear
  C)
             3, 3, 2 done clear
  D)
             2, 3, 2 done clear
  View Solution play_arrow

• question_answer26)
  The co-ordinates of the point P are \[(x,y,z)\]and the direction cosines of the line OP when O is the origin, are \[l,\,m,\,n\]. If \[OP\], then
  A)
             \[l=x,\,m=y,\,n=z\] done clear
  B)
             \[l=xr,m=yr,n=zr\] done clear
  C)
             \[x=lr,\,y=mr,\,z=nr\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer27)
  A line makes angles \[\alpha ,\beta ,\gamma \] with the co-ordinate axes. If \[\alpha +\beta ={{90}^{o}}\],  then \[\gamma =\]
  A)
             0 done clear
  B)
             \[90{}^\circ \] done clear
  C)
             \[180{}^\circ \] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer28)
  Points (?2, 4, 7), (3, ?6, ?8) and (1, ?2, ?2) are            [AI CBSE 1982]
  A)
             Collinear                                     done clear
  B)
             Vertices of an equilateral triangle done clear
  C)
             Vertices of an isosceles triangle done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer29)
  If the points \[A(9,\,\,8,\,-10)\],\[B(3,\,\,2,\,\,-4)\]and \[C(5,\,\,4,\,-6)\]be collinear, then the point C divides the line \[AB\] in the ratio
  A)
             2 : 1 done clear
  B)
             3 : 1 done clear
  C)
             1 : 2 done clear
  D)
             ?1 : 2 done clear
  View Solution play_arrow

• question_answer30)
  The projections of a line on the co-ordinate axes are 4, 6, 12. The direction cosines of the line are
  A)
             \[\frac{2}{7},\frac{3}{7},\frac{6}{7}\] done clear
  B)
             2, 3, 6 done clear
  C)
             \[\frac{2}{11},\frac{3}{11},\frac{6}{11}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer31)
  If the sum of the squares of the distance of a point from the three co-ordinate axes be 36,then its distance from the origin is    
  A)
             6 done clear
  B)
             \[3\sqrt{2}\] done clear
  C)
             \[2\sqrt{3}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer32)
  The line joining the points \[(-2,\,\,1,\,-8)\] and \[(a,\,b,\,c)\] is parallel to the line whose direction ratios are 6, 2, 3. The values of \[a,b,c\] are                  
  A)
             4, 3, ?5 done clear
  B)
             1, 2, ?13/2 done clear
  C)
             10, 5, ?2 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer33)
  The direction ratios of the line joining the points (4, 3, ?5) and (?2, 1, ?8) are               [AI CBSE 1984; MP PET 1988]
  A)
             \[\frac{6}{7},\frac{2}{7},\frac{3}{7}\] done clear
  B)
             \[6,\,\,2,\,\,3\] done clear
  C)
             \[2,\,\ 4,\ -13\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer34)
  The co-ordinates of the point in which the line joining the points \[(3,\,\ 5,\ -7)\] and \[(-2,\,\ 1,\,\ 8)\] is intersected by the plane yz are given by                                                                      [MP PET 1993]
  A)
             \[\left( 0,\,\frac{13}{5},\,\,2 \right)\] done clear
  B)
             \[\left( 0,\,-\frac{13}{5},\,-2 \right)\] done clear
  C)
             \[\left( 0,-\frac{13}{5},\frac{2}{5} \right)\] done clear
  D)
             \[\left( 0,\frac{13}{5},\frac{2}{5} \right)\] done clear
  View Solution play_arrow

• question_answer35)
  The co-ordinates of a point which is equidistant from the points \[(0,\,0,\ 0),(a,\,0,\,0),(0,\,\,b,\,\,0)\] and \[(0,\,0,\,c)\] are given by                                                                     [MP PET 1993]
  A)
             \[\left( \frac{a}{2},\frac{b}{2},\frac{c}{2} \right)\] done clear
  B)
             \[\left( -\frac{a}{2},-\frac{b}{2},\frac{c}{2} \right)\] done clear
  C)
             \[\left( \frac{a}{2},\,\,-\frac{b}{2},\,-\frac{c}{2} \right)\] done clear
  D)
             \[\left( -\frac{a}{2}\,\,,\frac{b}{2},\,-\frac{c}{2} \right)\] done clear
  View Solution play_arrow

• question_answer36)
  The projection of the line segment joining the points (?1, 0, 3) and (2, 5, 1) on the line whose direction ratios are 6, 2, 3 is                                                                       [AI CBSE 1985]
  A)
             \[\frac{10}{7}\] done clear
  B)
             \[\frac{22}{7}\] done clear
  C)
             \[\frac{18}{7}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer37)
  Points (1, 1, 1), (?2, 4, 1), (?1, 5, 5) and (2, 2, 5) are the vertices of a
  A)
             Rectangle done clear
  B)
             Square done clear
  C)
             Parallelogram done clear
  D)
             Trapezium done clear
  View Solution play_arrow

• question_answer38)
  If \[{{l}_{1}},\,{{m}_{1}},\,{{n}_{1}}\] and \[{{l}_{2}},{{m}_{2}},{{n}_{2}}\] are the direction cosines of two perpendicular lines, then the direction cosine of the line which is perpendicular to both the lines, will be
  A)
             \[({{m}_{1}}{{n}_{2}}-{{m}_{2}}{{n}_{1}}),\,\,({{n}_{1}}{{l}_{2}}-{{n}_{2}}{{l}_{1}}),\,({{l}_{1}}{{m}_{2}}-{{l}_{2}}{{m}_{1}})\] done clear
  B)
             \[({{l}_{1}}{{l}_{2}}-{{m}_{1}}{{m}_{2}}),\,({{m}_{1}}{{m}_{2}}-{{n}_{1}}{{n}_{2}}),\,({{n}_{1}}{{n}_{2}}-{{l}_{1}}{{l}_{2}})\] done clear
  C)
             \[\frac{1}{\sqrt{l_{1}^{2}+m_{1}^{2}+n_{1}^{2}}},\frac{1}{\sqrt{l_{2}^{2}+m_{2}^{2}+n_{2}^{2}}},\frac{1}{\sqrt{3}}\] done clear
  D)
             \[\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\] done clear
  View Solution play_arrow

• question_answer39)
  If a line makes the angle \[\alpha ,\beta ,\gamma \] with three dimensional        co-ordinate axes respectively, then \[\cos 2\alpha +\cos 2\beta +\cos 2\gamma =\] [MP PET 1994, 95,99; RPET 2003; Kerala (Engg.) 2005]
  A)
             ? 2 done clear
  B)
             ? 1 done clear
  C)
             1 done clear
  D)
             2 done clear
  View Solution play_arrow

• question_answer40)
  Perpendicular distance of the point (3, 4, 5) from the y-axis, is [MP PET 1994, Pb. CET 2002]
  A)
             \[\sqrt{34}\] done clear
  B)
             \[\sqrt{41}\] done clear
  C)
             4 done clear
  D)
             5 done clear
  View Solution play_arrow

• question_answer41)
  The direction cosines of a line equally inclined to three mutually perpendicular lines having direction cosines as \[{{l}_{1}},{{m}_{1}},{{n}_{1}};{{l}_{2}},{{m}_{2}},{{n}_{2}}\] and \[{{l}_{3}},{{m}_{3}},{{n}_{3}}\] are
  A)
             \[{{l}_{1}}+{{l}_{2}}+{{l}_{3}},{{m}_{1}}+{{m}_{2}}+{{m}_{3}},{{n}_{1}}+{{n}_{2}}+{{n}_{3}}\] done clear
  B)
             \[\frac{{{l}_{1}}+{{l}_{2}}+{{l}_{3}}}{\sqrt{3}},\frac{{{m}_{1}}+{{m}_{2}}+{{m}_{3}}}{\sqrt{3}},\frac{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}}{\sqrt{3}}\] done clear
  C)
             \[\frac{{{l}_{1}}+{{l}_{2}}+{{l}_{3}}}{3},\frac{{{m}_{1}}+{{m}_{2}}+{{m}_{3}}}{3},\frac{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}}{3}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer42)
  A point \[(x,y,z)\] moves parallel to x-axis. Which of the three variable\[x,y,z\]remain fixed
  A)
             x done clear
  B)
             y and z done clear
  C)
             x and y done clear
  D)
             z and x done clear
  View Solution play_arrow

• question_answer43)
   If the direction cosines of a line are \[\left( \frac{1}{c},\frac{1}{c},\frac{1}{c} \right)\],  then  [JET 1989; CET 1993]
  A)
             \[c>0\] done clear
  B)
             \[c=\pm \sqrt{3}\] done clear
  C)
             \[0<c<1\] done clear
  D)
             \[c>2\] done clear
  View Solution play_arrow

• question_answer44)
  The plane \[XOZ\] divides the join of \[(1,\,-1,\,\,5)\] and (2, 3, 4) in the ratio \[\lambda :1\], then \[\lambda \] is [JET 1988]
  A)
             ? 3 done clear
  B)
             3 done clear
  C)
             \[-\frac{1}{3}\] done clear
  D)
             \[\frac{1}{3}\] done clear
  View Solution play_arrow

• question_answer45)
  The co-ordinates of a point P are (3, 12, 4) with respect to origin O, then the direction cosines of \[OP\]are [MP PET 1996]
  A)
             3, 12, 4 done clear
  B)
             \[\frac{1}{4},\frac{1}{3},\frac{1}{2}\] done clear
  C)
             \[\frac{3}{\sqrt{13}},\frac{1}{\sqrt{13}},\frac{2}{\sqrt{13}}\] done clear
  D)
             \[\frac{3}{13},\frac{12}{13},\frac{4}{13}\] done clear
  View Solution play_arrow

• question_answer46)
  The locus of a first degree equation in \[x,y,z\]is a
  A)
             Straight line done clear
  B)
             Sphere done clear
  C)
             Plane done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer47)
  The direction cosines of the normal to the plane \[x+2y-3z+4=0\] are       [MP PET  1996; Pb. CET  2000]
  A)
             \[-\frac{1}{\sqrt{14}},-\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\] done clear
  B)
             \[\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\] done clear
  C)
             \[-\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\] done clear
  D)
             \[\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},-\frac{3}{\sqrt{14}}\] done clear
  View Solution play_arrow

• question_answer48)
  The direction cosines of the line \[\frac{3x+1}{-3}=\frac{3y+2}{6}=\frac{z}{-1}\] are
  A)
             \[\left( \frac{1}{3},\frac{2}{3},\,0 \right)\] done clear
  B)
             \[\left( -1,\frac{2}{3},\,1 \right)\] done clear
  C)
             \[\left( -\frac{1}{2},\,\text{ }1,\,-\frac{1}{2} \right)\] done clear
  D)
             \[\left( -\frac{1}{\sqrt{6}},\frac{2}{\sqrt{6}},-\frac{1}{\sqrt{6}} \right)\] done clear
  View Solution play_arrow

• question_answer49)
  The co-ordinates of the point which divides the join of the points (2, ?1, 3) and (4, 3, 1) in the ratio 3 : 4 internally are given by [MP PET 1997]
  A)
             \[\frac{2}{7},\frac{20}{7},\frac{10}{7}\] done clear
  B)
             \[\frac{15}{7},\frac{20}{7},\frac{3}{7}\] done clear
  C)
             \[\frac{10}{7},\frac{15}{7},\frac{2}{7}\] done clear
  D)
             \[\frac{20}{7},\frac{5}{7},\frac{15}{7}\] done clear
  View Solution play_arrow

• question_answer50)
  If the direction ratios of a line are \[1,-3,\,2\], then the direction cosines of the line are            [MP PET 1997, Pb. CET 2002]
  A)
             \[\frac{1}{\sqrt{14}},\frac{-3}{\sqrt{14}},\frac{2}{\sqrt{14}}\] done clear
  B)
             \[\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\] done clear
  C)
             \[\frac{-1}{\sqrt{14}},\frac{3}{\sqrt{14}},\frac{-2}{\sqrt{14}}\] done clear
  D)
             \[\frac{-1}{\sqrt{14}},\frac{-2}{\sqrt{14}},\frac{-3}{\sqrt{14}}\] done clear
  View Solution play_arrow

• question_answer51)
  A line makes angles of \[45{}^\circ \]and\[60{}^\circ \] with the positive axes of X and Y respectively. The angle made by the same line with the positive axis of  Z, is                                                          [MP PET 1997]
  A)
             \[30{}^\circ \]or \[60{}^\circ \] done clear
  B)
             \[60{}^\circ \]or \[90{}^\circ \] done clear
  C)
             \[90{}^\circ \]or \[120{}^\circ \] done clear
  D)
             \[60{}^\circ \]or \[120{}^\circ \] done clear
  View Solution play_arrow

• question_answer52)
  The direction cosines of the normal to the plane \[3x+4y+12z=52\] will be [MP PET 1997]
  A)
             3, 4, 12 done clear
  B)
             ? 3, ? 4, ? 12 done clear
  C)
             \[\frac{3}{13},\frac{4}{13},\frac{12}{13}\] done clear
  D)
             \[\frac{3}{\sqrt{13}},\frac{4}{\sqrt{13}},\frac{12}{\sqrt{13}}\] done clear
  View Solution play_arrow

• question_answer53)
  The shortest distance of the point (a, b, c) from the x-axis is                     [MP PET 1999; DCE 1999]
  A)
             \[\sqrt{({{a}^{2}}+{{b}^{2}})}\] done clear
  B)
             \[\sqrt{({{b}^{2}}+{{c}^{2}})}\] done clear
  C)
             \[\sqrt{({{c}^{2}}+{{a}^{2}})}\] done clear
  D)
             \[\sqrt{({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}\] done clear
  View Solution play_arrow

• question_answer54)
  The direction ratios of the line \[x-y+z-5=\]\[0=x-3y-6\]  are [MP PET 1999; Pb. CET 2000]
  A)
             3, 1, ? 2 done clear
  B)
             2, ? 4, 1 done clear
  C)
             \[\frac{3}{\sqrt{14}},\frac{1}{\sqrt{14}},\frac{-2}{\sqrt{14}}\] done clear
  D)
             \[\frac{2}{\sqrt{41}},\frac{-4}{\sqrt{41}},\frac{1}{\sqrt{41}}\] done clear
  View Solution play_arrow

• question_answer55)
  If O is the origin and \[OP=3\]with direction ratios \[-1,\,2,-2\], then co-ordinates of P are [RPET 2000; DCE 2005]
  A)
             (1, 2, 2) done clear
  B)
             \[(-1,\,2,\,-2)\] done clear
  C)
             (?3, 6, ?9) done clear
  D)
             \[(-1/3\,,\,2/3,\,-2/3)\] done clear
  View Solution play_arrow

• question_answer56)
  If x co-ordinates of a point P of line joining the points \[Q(2,\,2,\,1)\] and \[R\,(5,\,2,-2)\]is 4, then the z-coordinates of P is                                                                     [RPET 2000]
  A)
             ? 2 done clear
  B)
             ?1 done clear
  C)
             1 done clear
  D)
             2 done clear
  View Solution play_arrow

• question_answer57)
  The points \[A(5,\,-1,\,\,1)\]; \[B\,(7,-4,\,7);\] \[C(1,\,-6,\,10)\] and \[D(-1,-3,\,4)\] are vertices of a [RPET 2000]
  A)
             Square done clear
  B)
             Rhombus done clear
  C)
             Rectangle done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer58)
  The direction cosines of the line joining the points (4, 3, ? 5) and (? 2, 1, ? 8) are [MP PET 2001]
  A)
             \[\left( \frac{6}{7},\frac{2}{7},\frac{3}{7} \right)\] done clear
  B)
             \[\left( \frac{2}{7},\frac{3}{7},\frac{6}{7} \right)\] done clear
  C)
             \[\left( \frac{6}{7},\frac{3}{7},\frac{2}{7} \right)\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer59)
  If a line lies in the octant \[OXYZ\] and it makes equal angles with the axes, then [MP PET 2001]
  A)
             \[l=m=n=\frac{1}{\sqrt{3}}\] done clear
  B)
             \[l=m=n=\pm \frac{1}{\sqrt{3}}\] done clear
  C)
             \[l=m=n=-\frac{1}{\sqrt{3}}\] done clear
  D)
             \[l=m=n=\pm \frac{1}{\sqrt{2}}\] done clear
  View Solution play_arrow

• question_answer60)
  The triangle formed by the points (0, 7, 10), (?1, 6, 6),            (? 4, 9, 6) is         [RPET 2001]
  A)
             Equilateral done clear
  B)
             Isosceles done clear
  C)
             Right angled done clear
  D)
             Right angled Isosceles done clear
  View Solution play_arrow

• question_answer61)
  If \[A(1,\,2,\,3),\,B(-1,-1,-1)\] be the points, then the distance AB is [MP PET 2001; Pb. CET 2001]
  A)
             \[\sqrt{5}\] done clear
  B)
             \[\sqrt{21}\] done clear
  C)
             \[\sqrt{29}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer62)
  If \[\alpha ,\beta ,\gamma \] be the angles which a line makes with the              co-ordinate axes, then [MP PET 2002; Orissa JEE 2002]
  A)
             \[{{\sin }^{2}}\alpha +{{\cos }^{2}}\beta +{{\sin }^{2}}\gamma =1\] done clear
  B)
             \[{{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma =1\] done clear
  C)
             \[{{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma =1\] done clear
  D)
             \[{{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\sin }^{2}}\gamma =1\] done clear
  View Solution play_arrow

• question_answer63)
  If \[P(3,\,4,\,5),\] \[Q(4,\,6,\,3),\] \[R(-1,\,2,\,4),\] \[S(1,\,0,\,5)\] then the  projection of RS on PQ is [Orissa JEE 2002; RPET 2002]
  A)
             ? 2/3 done clear
  B)
             ? 4/3 done clear
  C)
             ½ done clear
  D)
             2 done clear
  View Solution play_arrow

• question_answer64)
  If a line makes \[\alpha ,\beta ,\gamma \] with the positive direction of \[x,\ y\] and z-axis respectively. Then, \[{{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma \] is [Orissa JEE 2002]
  A)
             ½ done clear
  B)
             ?1/2 done clear
  C)
             ?1 done clear
  D)
             1 done clear
  View Solution play_arrow

• question_answer65)
  The projection of a line on a co-ordinate axes are 2, 3, 6. Then the length of the line is [Orissa JEE 2002]
  A)
             7 done clear
  B)
             5 done clear
  C)
             1 done clear
  D)
             11 done clear
  View Solution play_arrow

• question_answer66)
  A line which makes angle \[{{60}^{o}}\]with y-axis and z-axis, then the angle which it makes with x-axis is [RPET 2002; AMU 2005]
  A)
             \[45{}^\circ \] done clear
  B)
             \[60{}^\circ \] done clear
  C)
             \[75{}^\circ \] done clear
  D)
             \[30{}^\circ \] done clear
  View Solution play_arrow

• question_answer67)
  The points (5, ? 4, 2), (4, ?3, 1), (7, ? 6, 4) and (8, ?7, 5) are the vertices of [RPET 2002]
  A)
             A rectangle done clear
  B)
             A square done clear
  C)
             A parallelogram done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer68)
  If \[\left( \frac{1}{2},\frac{1}{3},n \right)\] are the direction cosines of a line, then the value of n is [Kerala (Engg.) 2002]
  A)
             \[\frac{\sqrt{23}}{6}\] done clear
  B)
             \[\frac{23}{6}\] done clear
  C)
             \[\frac{2}{3}\] done clear
  D)
             \[\frac{3}{2}\] done clear
  View Solution play_arrow

• question_answer69)
  The distance of the point  (4, 3, 5) from the y-axis is   [MP PET 2003]
  A)
             \[\sqrt{34}\] done clear
  B)
             5 done clear
  C)
             \[\sqrt{41}\] done clear
  D)
             \[\sqrt{15}\] done clear
  View Solution play_arrow

• question_answer70)
  If projection of any line on co-ordinate axis 3, 4, and 5, then its length is [Pb. CET 2000]
  A)
             12 done clear
  B)
             50 done clear
  C)
             \[5\sqrt{2}\] done clear
  D)
             \[3\sqrt{2}\] done clear
  View Solution play_arrow

• question_answer71)
  If a line makes angles \[\alpha ,\beta ,\gamma ,\delta \] with four diagonals of a cube, then the value of \[{{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +\] \[{{\sin }^{2}}\gamma +{{\sin }^{2}}\delta \] is                                [MP PET 2004]
  A)
             \[\frac{4}{3}\] done clear
  B)
             1 done clear
  C)
             \[\frac{8}{3}\] done clear
  D)
             \[\frac{7}{3}\] done clear
  View Solution play_arrow

• question_answer72)
  If \[\theta \] is the angle between the lines \[AB\] and \[CD\], then projection of line segment \[AB\]on line \[CD\], is [MP PET 1995]
  A)
             \[AB\sin \theta \] done clear
  B)
             \[AB\cos \theta \] done clear
  C)
             \[AB\tan \theta \] done clear
  D)
             \[CD\cos \theta \] done clear
  View Solution play_arrow

• question_answer73)
  The co-ordinates of points \[A,B,C,D\]are (a, 2, 1), (1, ?1, 1), (2, ? 3, 4) and (a+1, a+2, a+3) respectively. If \[AB=5\]and \[CD=6\], then \[a=\]
  A)
             2 done clear
  B)
             3 done clear
  C)
             ? 2 done clear
  D)
             ? 3 done clear
  View Solution play_arrow

• question_answer74)
  If the co-ordinates of the points \[A,B,C\]be \[(-1,\,3,\,2),\,\,(2,\,3,\,5)\] and (3, 5,?2) respectively, then \[\angle A=\]
  A)
             \[0{}^\circ \] done clear
  B)
             \[45{}^\circ \] done clear
  C)
             \[60{}^\circ \] done clear
  D)
             \[90{}^\circ \] done clear
  View Solution play_arrow

• question_answer75)
  The number of straight lines that are equally inclined to the three dimensional co-ordinate axes, is [MP PET 1994]
  A)
             2 done clear
  B)
             4 done clear
  C)
             6 done clear
  D)
             8 done clear
  View Solution play_arrow