CBSE Questions for Class 11 Engineering Maths Introduction To Three Dimensional Geometry Quiz 8 - MCQExams.com

The point equidistant from the point $$O(0, 0, 0), A(a, 0, 0), B(0, b, 0)$$ and $$C(0, 0, c)$$ has the coordinates
  • $$(a, b, c)$$
  • $$(a/2, b/2, c/2)$$
  • $$(a/3, b/3, c/3)$$
  • $$(a/4, b/4, c/4)$$
If the distance between a point P and the point (1, 1, 1) on the line $$\frac{{x\, - \,1}}{3}\, = \,\frac{{y - \,1}}{4}\, = \,\frac{{z\, - 1}}{{12}}$$ is 13, then the coordinates of P are
  • (3, 4, 12)
  • $$\left( {\frac{3}{{13}},\,\frac{4}{{13}},\,\frac{{12}}{{13}}} \right)$$
  • (4, 5, 12)
  • (40, 53, 157)
$$A$$ point $$C$$ with position vector $$\frac{{3a + 4b - 5c}}{3}$$ (where a,b and c are non co-planar vectors) divides the line joining $$A$$ and $$B$$ in the ratio $$2:1$$. If the position vector of $$A$$ is $$a-2b+3c$$, then the position vector of $$B$$ is
  • $$2a+3b-4c$$
  • $$2a-3b+4c$$
  • $$2a+3b+4c$$
  • $$a+3b-4c$$
The xy-plane divides the line joining the points (-1, 3, 4) and (2,-5,6).
  • internally in the ratio $$2 : 3$$
  • externally in the ratio $$2 : 3$$
  • internally in the ratio $$3 : 2$$
  • externally in the ratio $$3 : 2$$
If  $$a,b$$ and $$c$$ are non coplanar vectors then $$-\bar {a}+4\bar {b}-3\bar {c},3\bar {a}+2\bar {b}-5\bar {c},-3\bar {a}+8\bar {b}-5\bar {c},-3\bar {a}+2\bar {b}+\bar {c}$$ are collinear.
  • True
  • False
Let $$O$$ be the origin and $$P$$ be the point at a distance $$3$$ units from origin. If d.x.s' of OP are 1, - 2, - 2, then coordinates of P is given by 
  • $$1, - 2, - 2$$
  • $$3, - 6, - 6$$
  • $$\dfrac{1}{3}, - \dfrac{2}{3}, - \dfrac{2}{3}$$
  • $$\dfrac{1}{9}, - \dfrac{2}{9}, - \dfrac{2}{9}$$
If $$A(2, 2, -3), B(5, 6, 9), C(2, 7, 9)$$ be the vertices of a triangle. The internal bisector of the angle $$A$$ meets $$BC$$ at the point $$D$$, then find the coordinates of $$D$$.
  • $$(\dfrac{13}{2},\dfrac{7}{2},9)$$
  • $$(\dfrac{7}{2},\dfrac{13}{2},9)$$
  • $$(\dfrac{9}{2},\dfrac{7}{2},9)$$
  • $$(\dfrac{13}{2},\dfrac{9}{2},9)$$
If the lines $$\frac{x - 0}{1} =\frac{y+1}{2}=\frac{z-1}{-1}$$ and $$\frac{x+1}{k}=\frac{y-3}{-2}=\frac{z-2}{1}$$ are at right angles, then the value of k is
  • $$5$$
  • $$0$$
  • $$3$$
  • $$-1$$
The vector $$\vec{AF}$$, is given by?
  • $$-\left|\dfrac{\vec{a}}{\vec{c}}\right|\vec{c}$$
  • $$\left|\dfrac{\vec{a}}{\vec{c}}\right|\vec{c}$$
  • $$\dfrac{2|\vec{a}|}{|\vec{c}|}\vec{c}$$
  • $$\dfrac{1}{3}\left|\dfrac{\vec{a}}{\vec{c}}\right|\vec{c}$$
The position vector of the vertices of a triangle $$ABC$$ are $$\hat { i } ,\hat { j } ,\hat { k } $$ then the position vector of its orthocentre is
  • $$\hat { i } +\hat { j } +\hat { k } $$
  • $$2(\hat { i } +\hat { j } +\hat { k } )$$
  • $$\cfrac{1}{3}(\hat { i } +\hat { j } +\hat { k } )$$
  • $$\cfrac{1}{\sqrt{3}}(\hat { i } +\hat { j } +\hat { k } )$$
The graph of the equation $$y^{2}+z^{2}=0$$ in three dimensional space is
  • x- axis
  • y- axis
  • z- axis
  • yz-plane
The area of triangle whose vertices are $$(1, 2, 3), (2, 5, -1)$$ and $$(-1, 1, 2)$$ is
  • $$150\ sq. units$$
  • $$145\ sq. units$$
  • $$\sqrt {155}/2\ sq. units$$
  • $$155/2\ sq. units$$
If the point $$(x, y)$$ is equidistant from the points $$(a + b, b - a)$$ and $$(a - b , a + b)$$, then  $$bx = ay$$.
  • True
  • False
A tangent to the curve $$y = f(x)$$ at $$p(x, y)$$ meets $$x - axis$$ at $$A$$ and $$y-axis$$ at $$B$$. If $$\overline {AP} : \overline {BP} = 1 : 3$$ and $$f(1) = 1$$ then the curve also passes through the point.
  • $$\left (\dfrac {1}{2}, 4\right )$$
  • $$\left (\dfrac {1}{3}, 24\right )$$
  • $$\left (2, \dfrac {1}{8}\right )$$
  • $$\left (3, \dfrac {1}{28}\right )$$
If the distance between a point $$P$$ and the point $$(1, 1, 1)$$ on the line $$\dfrac {x - 1}{3} = \dfrac {y - 1}{4} = \dfrac {z - 1}{12}$$ is $$13$$, then the coordinates of $$P$$ are
  • $$(3, 4, 12)$$
  • $$\left (\dfrac {3}{13}, \dfrac {4}{13}, \dfrac {12}{13}\right )$$
  • $$(4, 5, 13)$$
  • $$(40, 53, 157)$$
The points $$(10,7,0)$$, $$(6,6-1)$$ and $$(6,9,-4)$$ form a 
  • Right -angled triangle
  • Isosceles triangle
  • Both $$(1)$$ & $$(2)$$
  • Equilateral triangle
The ratio in which the line joining $$(3,4,-7)$$ and $$(4,2,1)$$ is dividing the xy-plane
  • $$3:4$$
  • $$2:1$$
  • $$7:1$$
  • $$4:3$$
The ratio in which the line joining points $$(2,4,5)$$ and $$(3,5,-4)$$ divide YZ -plane is 
  • $$-2:3$$
  • $$2:3$$
  • $$-3:2$$
  • $$3:2$$
The vertices of a triangle are $$)2, 3, 5), (-1, 3, 2), (3, 5, -2)$$, then the angles are 
  • $$30^o, 30^o, 30^o$$
  • $$\cos^{-1}\left(\dfrac{1}{\sqrt{5}}\right), 90^o, \cos^{-1} \left(\dfrac{\sqrt{5}}{\sqrt{3}}\right)$$
  • $$30^o, 60^o, 90^o$$
  • $$\cos^{-1}\left(\dfrac{1}{\sqrt{3}}\right), 90^o, \cos^{-1} \left(\dfrac{\sqrt{2}}{\sqrt{3}}\right)$$
The values of a for which $$(8, -7, a), (5, 2, 4)$$ and $$(6, -1, 2)$$ are collinear, is given by?
  • $$2$$
  • $$-2$$
  • $$-1$$
  • $$1$$
The shortest distance of the point $$(1,2,3)$$ from $${x}^{2}+{y}^{2}=0$$ is 
  • $$5$$
  • $$\sqrt{5}$$
  • $$2$$
  • $$\sqrt{14}$$
A triangle $$ABC$$ is placed so that the mid-points of the sides are on the $$x,y,z$$ axes. Lengths of the intercepts made by the plane containing the triangle on these axes are respectively $$\alpha, \beta, \gamma$$. Coordinates of the centroid of the triangle $$ABC$$ are
  • $$(-\alpha/ 3, \beta/ 3,\gamma/3)$$
  • $$(\alpha/3,-\beta/3,\gamma/3)$$
  • $$(\alpha/ 3, \beta/3, -\gamma/ 3)$$
  • $$(\alpha/3, \beta/ 3, \gamma/ 3)$$
The triangle with vertices $$A (1, 0, 1), B (2, - 1, 4) \ and \ C (3, - 4, - 1),$$ is right-angled.
  • True
  • False
The distance between two points $$(1,1)$$ and $$\left( {\dfrac{{2{t^2}}}{{1 + {t^2}}},\dfrac{{{{\left( {1 - t} \right)}^2}}}{{1 + {t^2}}}} \right)$$ is 
  • 4t
  • 3t
  • 1
  • none of these
The plane $$x = 0$$ divides the joinning of $$( - 2, 3, 4)$$ and $$(1, - 2, 3)$$ in the ratio :
  • $$2 : 1$$
  • $$1 : 2$$
  • $$3 : 2$$
  • $$ - 4 : 3$$
The points (-5,12), (-2,-3),(9,-10),(6,5) taken in order, form
  • Parallelogram
  • rectangle
  • rhombus
  • square
The vertices of a triangle are $$(2, 3, 5), (-1, 3, 2), (3, 5, -2)$$, then the angles are
  • $$30^{\circ}, 30^{\circ}, 120^{\circ}$$
  • $$\cos^{-1} \left (\dfrac {1}{\sqrt {5}}\right ), 90^{\circ}, \cos^{-1} \left (\dfrac {\sqrt {5}}{\sqrt {3}}\right )$$
  • $$30^{\circ}, 60^{\circ}, 90^{\circ}$$
  • $$\cos^{-1} \left (\dfrac {1}{\sqrt {3}}\right ), 90^{\circ}, \cos^{-1} \left (\sqrt{\dfrac { {2}}{ {3}}}\right )$$
The plane passing through the point $$\left(-2,-2,2\right)$$ and containing the line joining the points $$\left(1,1,1\right)$$ and $$\left(1,-1,2\right)$$ makes intercepts on the coordinates axes, the sum whose lengths is ?
  • $$3$$
  • $$4$$
  • $$6$$
  • $$12$$
The nearest point  from the origin is 
  • $$(2,3,-1)$$
  • $$(-3,2,1)$$
  • $$(2,2,2)$$
  • $$(1,2,-1)$$
The points $$(3,\ 2,\ 0),\ (5,\ 3,\ 2)$$ and $$(-9,\ 6,\ -3)$$, are the vertices of a triangle $$ABC.AD$$ is the internal bisector of $$\angle\ BAC$$ which meets $$BC$$ at $$D$$. Then the co-ordinates of $$D$$, are
  • $$\left[ {\dfrac{{17}}{{16}},\ \dfrac{{57}}{{16}},\ \dfrac{{19}}{8}} \right]$$
  • $$\left[ {\dfrac{{19}}{{8}},\ \dfrac{{57}}{{16}},\ \dfrac{{17}}{16}} \right]$$
  • $$\left[0,\ 0,\ {\dfrac{{17}}{{16}}}\right]$$
  • $$\left[{\dfrac{{17}}{{16}}},\ 0,\ 0\right]$$
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