1 Marks Question

Question 11 Mark

Find the distance between (2, -1, 3) and (-2, 1, 3) pairs of points.

Answer

Let A(2, -1, 3) and B(-2, 1, 3) be two points. Then
$AB = \sqrt {{{( - 2 - 2)}^2} + {{[1 - ( - 1)]}^2} + {{(3 - 3)}^2}}$ [using distance formula]
$= \sqrt {{{( - 2 - 2)}^2} + {{(1 + 1)}^2} + {{(3 - 3)}^2}}$
$= \sqrt {16 + 4 + 0} = \sqrt {20} = 2\sqrt 5$ units

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Question 21 Mark

Find the distance between (-1, 3, -4) and (1, -3, 4) pairs of points.

Answer

Let A(-1, 3, -4) and B(1, -3, 4) be two points. Then
$AB = \sqrt {{{[1 - ( - 1)]}^2} + {{( - 3 - 3)}^2} + {{[4 - ( - 4)]}^2}}$ [using distance formula]
$ = \sqrt {4 + 36 + 64} = \sqrt {104} = 2\sqrt {26}$

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Question 31 Mark

Find the distance between (-3, 7, 2) and (2, 4, -1) pairs of points.

Answer

Let A(-3, 7, 2) and B(2, 4, -1) be two points. Then
$AB = \sqrt {{{[2 - ( - 3)]}^2} + {{(4 - 7)}^2} + {{( - 1 - 2)}^2}}$ [using distance formula]
$= \sqrt {{{(2 + 3)}^2} + {{(4 - 7)}^2} + {{( - 1 - 2)}^2}}$$= \sqrt {25 + 9 + 9} = \sqrt {43}$ units

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Question 41 Mark

Find the distance between (2, 3, 5) and (4, 3, 1) pairs of points.

Answer

Let A(2, 3, 5) and B(4, 3, 1) be two points. Then
$AB = \sqrt {{{(4 - 2)}^2} + {{(3 - 3)}^2} + {{(1 - 5)}^2}} $ [using distance formula]

$= \sqrt {4 + 0 + 16} = \sqrt {20} = 2\sqrt 5$ units

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Question 51 Mark

Name the octants in which the following points lie:
(1, 2, 3), (4, -2, 3), (4, -2, -5), (4, 2, -5), (-4, 2, -5), (-4, 2, 5), (-3, -1, 6), (-2, -4, -7)

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Question 61 Mark

A point is in the XZ-plane. What can you say about its y-coordinate?

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Question 71 Mark

A point is on the X-axis. What are its y and z-coordinates?

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Question 81 Mark

Find the coordinates of the point which divides the line segment joining the points (1, –2, 3) and (3, 4, –5) in the ratio 2 : 3 externally.

Answer

Let P (x, y, z) be the point which divides segment joining A (1, –2, 3) and B (3, 4, –5) externally in the ratio 2 : 3. Then
$x=\frac{2(3)+(-3)(1)}{2+(-3)}=-3, \quad y=\frac{2(4)+(-3)(-2)}{2+(-3)}=-14, z=\frac{2(-5)+(-3)(3)}{2+(-3)}=19$ [using section formula]
Therefore, the required point is (–3, –14, 19).

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Question 91 Mark

Find the coordinates of the point which divides the line segment joining the points (1, –2, 3) and (3, 4, –5) in the ratio 2 : 3 internally.

Answer

Let P (x, y, z) be the point which divides the line segment joining A(1, – 2, 3) and B (3, 4, –5) internally in the ratio 2 : 3. Therefore
$x=\frac{2(3)+3(1)}{2+3}=\frac{9}{5}, y=\frac{2(4)+3(-2)}{2+3}=\frac{2}{5}, z=\frac{2(-5)+3(3)}{2+3}=\frac{-1}{5}$ [using section formula]
Thus, the required point is $\left(\frac{9}{5}, \frac{2}{5}, \frac{-1}{5}\right)$

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Question 101 Mark

Find the octant in which the points (–3, 1, 2) and (–3, 1, – 2) lie.

Answer

The point (–3,1, 2) lies in the 2nd octant

and The point (–3, 1, – 2) lies in octant VI.

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