Mathematics - Three Dimensional Geometry

Question - 1

The points (0,-7,-10),(4,-9,-6) and (1,-6,-6) are the vertices of

  • A an equilateral triangle
  • B a scalene triangle
  • C a right angled isosceles triangle
  • D collinear

Question - 2

The ratio in which the xy-plane divides the join of (-2,4,7) and (3,-5,8) is

  • A 7:8
  • B -7:8
  • C -8:7
  • D NONE OF THESE

Question - 3

The coordinates of the points P and Q are respectively (2,5,-7) and (3,4,12). then direction cosines of PQ are

  • A \(\frac { 1 }{ 11\sqrt { 3 } } ,\frac { 1 }{ 11\sqrt { 3 } } ,\frac { -1 }{ 11\sqrt { 3 } } \)
  • B \(\frac { 1 }{ 11\sqrt { 3 } } ,\frac { -1 }{ 11\sqrt { 3 } } ,\frac { -1 }{ 11\sqrt { 3 } } \)
  • C \(\frac { 1 }{ 11\sqrt { 3 } } ,\frac { -1 }{ 11\sqrt { 3 } } ,\frac { -1 }{ 11\sqrt { 3 } } \)
  • D NONE OF THESE

Question - 4

If cos \(\alpha\), cos \(\beta\), cos\( \gamma \) are direction cosines of a line, then value of \({ sin }^{ 2 }\alpha +{ sin }^{ 2 }\beta +{ sin }^{ 2 }\gamma \) is

  • A 1
  • B 2
  • C 3
  • D NONE OF THESE

Question - 5

The three lines with direction cosines l1,m1,n1; l2,m2,n2 and l3,m3,n3 are coplanar, if

  • A \(\left| \begin{matrix} { l }_{ 1 } & { m }_{ 1 } & { n }_{ 1 } \\ { l }_{ 2 } & { m }_{ 2 } & { n }_{ 2 } \\ { l }_{ 3 } & { m }_{ 3 } & { n }_{ 3 } \end{matrix} \right| =0\)
  • B \(\left| \begin{matrix} { l }_{ 1 } & { m }_{ 1 } & { n }_{ 1 } \\ { l }_{ 2 } & { m }_{ 2 } & { n }_{ 2 } \\ { l }_{ 3 } & { m }_{ 3 } & { n }_{ 3 } \end{matrix} \right| =1\)
  • C \(\left| \begin{matrix} { l }_{ 1 } & { m }_{ 1 } & { n }_{ 1 } \\ { l }_{ 2 } & { m }_{ 2 } & { n }_{ 2 } \\ { l }_{ 3 } & { m }_{ 3 } & { n }_{ 3 } \end{matrix} \right| =2\)
  • D NONE OF THESE

Question - 6

If four points A,B,C,D have coordinates (3,4,5), (4,6,3), (-1,2,4) and (1,0,5) respectively then length of the projection of AB on CD is

  • A \(\frac { 3 }{ 4 }\)
  • B \(\frac { 4 }{ 3 }\)
  • C \(\frac { 2 }{ 3 }\)
  • D \(\frac { 1 }{ 3 }\)

Question - 7

The projection of a line on the coordinatesaxes are 2,3,6. The length of the line is

  • A 2
  • B 3
  • C 6
  • D 7

Question - 8

The coordinates of the foot of perpendicular drawn from the point (1,1,1) on the line joining the point (5,4,4) and (1,4,6), is

  • A (5,4,3)
  • B (3,4,5)
  • C (4,3,5)
  • D (4,5,3)

Question - 9

The area of the triangle with vertices (0,0,0), (x1,y1,z1) and (x2,y2,z2) is

  • A \(\frac { 1 }{ 2 } \sum { \left( \sqrt { { ({ x }_{ 1 }{ y }_{ 2 }+{ x }_{ 2 }{ y }_{ 1 }) }^{ 2 } } \right) } \)
  • B \(\frac { 1 }{ 2 } \sum { \left( \sqrt { { ({ y }_{ 1 }{ z }_{ 2 }-y_{ 2 }{ z }_{ 1 }) }^{ 2 } } \right) } \)
  • C \(\frac { 1 }{ 2 } \sum { \left( \sqrt { { ({ z }_{ 1 }{ x }_{ 2 }+z_{ 2 }x_{ 1 }) }^{ 2 } } \right) } \)
  • D NONE OF THESE

Question - 10

The direction cosines of a line equally inclined to the three mutually perpendicular lines having direction cosines as l1,m1,n1; l2,m2,n2; l3,m3,n3 are

  • A \(\frac { { l }_{ 1 }-{ l }_{ 2 }-{ l }_{ 3 } }{ \sqrt { 3 } } ,\frac { { m }_{ 1 }-{ m }_{ 2 }-{ m }_{ 3 } }{ \sqrt { 3 } } ,\frac { { n }_{ 1 }-{ n }_{ 2 }-{ n }_{ 3 } }{ \sqrt { 3 } } \)
  • B \(\frac { { l }_{ 1 }+{ l }_{ 2 }+{ l }_{ 3 } }{ \sqrt { 3 } } ,\frac { { m }_{ 1 }+{ m }_{ 2 }+{ m }_{ 3 } }{ \sqrt { 3 } } ,\frac { { n }_{ 1 }+{ n }_{ 2 }+{ n }_{ 3 } }{ \sqrt { 3 } } \)
  • C \({ l }_{ 1 }+{ l }_{ 2 }+{ l }_{ 3 },{ m }_{ 1 }+{ m }_{ 2 }+{ m }_{ 3 },{ n }_{ 1 }+{ n }_{ 2 }+{ n }_{ 3 }\)
  • D \(\frac { { l }_{ 1 }+{ l }_{ 2 }+{ l }_{ 3 } }{ 3 } ,\frac { { m }_{ 1 }+{ m }_{ 2 }+{ m }_{ 3 } }{ 3 } ,\frac { { n }_{ 1 }+{ n }_{ 2 }+{ n }_{ 3 } }{ 3 } \)
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