### 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 $\frac { { l }_{ 1 }+{ l }_{ 2 }+{ l }_{ 3 } }{ 3 } ,\frac { { m }_{ 1 }+{ m }_{ 2 }+{ m }_{ 3 } }{ 3 } ,\frac { { n }_{ 1 }+{ n }_{ 2 }+{ n }_{ 3 } }{ 3 }$