Introduction to Three Dimensional Geometry Test

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Introduction to Three Dimensional Geometry Test - 38

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Question 1
1 / -0

If O and O' are circumcenter and orthocenter of a $$\Delta ABC$$ where $$\overline{OA} + \overline{OB} + \overline{OC}$$ is $$\lambda \overline{OO'}$$ then the value of $$\lambda$$ is

A
$$1$$

B
$$2$$

C
$$3$$

D
$$4$$

Solution
Question 2
1 / -0

The distance between the orthocentre and circumcentre of the triangle formed by the points $$(1, 2, 3), (3, -1, 5), (4, 0, -3)$$ is

A
$$\dfrac {\sqrt {66}}{2}$$

B
$$\dfrac {\sqrt {17}}{2}$$

C
$$\dfrac {\sqrt {55}}{2}$$

D
$$\dfrac {\sqrt {37}}{2}$$

Solution
Question 3
1 / -0

If $$R$$ divides the line segment joining $$P(2,3,4)$$ and $$Q(4,5,6)$$ in the ratio $$-3:2$$, then the parameter which represent $$R$$ is

A
$$3$$

B
$$2$$

C
$$1$$

D
$$-1$$

Solution

$$x=\cfrac { -3\times 4+2\times 2 }{ -1 } \quad \quad y=\cfrac { -3\times 5+2\times 3 }{ -1 }$$
$$x=\cfrac { -12+4 }{ -1 } \quad \quad$$$$y =\cfrac { -15+6 }{ -1 } $$
$$x=8\quad \quad y=9$$
$$z=\cfrac { -3\times 6+2\times 4 }{ -1 } =\cfrac { -10 }{ -1 } $$
$$z=10$$
$$R=\left( 8,9,10 \right) $$
Question 4
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In the $$xy-plane$$, the length of the shortest from $$(0, 0)$$ to $$(12, 16)$$ that does not go inside the circle $$(x - 6)^{2} + (y + 8)^{2} = 25$$ is

A
$$10\sqrt {3}$$

B
$$10\sqrt {5}$$

C
$$10\sqrt {3} + \dfrac {5\pi}{3}$$

D
$$10 + 5\pi$$
Question 5
1 / -0

Consider at three dimensional figure represented by $$xy{ z }^{ 2 }=2$$, then its minimum distance from origin is

A
2

B
4

C
6

D
8

Solution
Question 6
1 / -0

Perpendicular distance from the origin to the line joining the points $$(a\cos{\theta},a\sin{\theta})(a\cos{\theta},a\sin{\theta})$$ is

A
$$2a\cos{(\theta-\phi)}$$

B
$$a\cos { \left( \cfrac { \theta -\phi }{ 2 } \right) } $$

C
$$4a\cos { \left( \cfrac { \theta -\phi }{ 2 } \right) } $$

D
$$a\cos { \left( \cfrac { \theta +\phi }{ 2 } \right) } $$
Question 7
1 / -0

Q, R, S are the points $$(-2, -1), (0, 3) (4, 0)$$ respectively. Then the coordinates of P such that PQRS is a parallelogram is ________________.

A
$$(2, -6)$$

B
$$(-6, 2)$$

C
$$(2, -4)$$

D
$$(-3, 2)$$
Question 8
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The point which is equidistant from the points $$(-1,1,3),(2,1,2),(0,5,6)$$ and $$(3,2,2)$$ is

A
$$(-1,3,4)$$

B
$$(3,1,4)$$

C
$$(1,3,4)$$

D
$$(4,1,3)$4

Solution
Question 9
1 / -0

Minimum distance between the curves
$$y^{2}=4x$$ & $$x^{2}+y^{2} -12x+31=0$$ is -

A
$$\sqrt {21}$$

B
$$\sqrt {26}-\sqrt {5}$$

C
$$\sqrt {20}-\sqrt {5}$$

D
$$\sqrt {21}-\sqrt {5}$$

Solution
Question 10
1 / -0

Directions For Questions

[passage-header]Let $$A(2,3,5),B(-1,3,2)$$ and $$C(\lambda ,5,\mu )$$ are the vertices of a triangle and its median through A meets side BC at D. AD is equally inclined with the axes. If E is the point on BC such that $$BE:EC=1:2.$$[/passage-header]

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Let $$A(2,3,5),B(-1,3,2)$$ and $$C(\lambda ,5,\mu )$$ are the vertices of a triangle and its median through A meets side BC at D. AD is equally inclined with the axes. If E is the point on BC such that $$BE:EC=1:2.$$
Project of BA on BC

A
$$\dfrac { 23 }{ \sqrt { 33 } } $$

B
$$\dfrac { \sqrt { 33 } }{ 24 } $$

C
$$\dfrac { 24 }{ \sqrt { 33 } } $$

D
$$None$$ $$of$$ $$these$$