2 Marks Questions

Question 12 Marks

What are the direction cosines of Z-axis?

Answer

The z-axis makes angles 90°, 90° and 0° with x, y and z axes, respectively.
Therefore, the direction cosines of x-axis are cos 90°, cos 90°, cos 0°, i.e. 0, 0, 1.

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Question 22 Marks

If a line has direction ratios proportional to 2, -1, -2, then what are its direction consines?

Answer

If a line has direction ratios proportional to 2, -1 and -2, then its direction cosines are
$\frac{2}{\sqrt{(2)^2+(-1)+(-2)^2}},\frac{-1}{\sqrt{(2)^2+(-1)+(-2)^2}},\frac{-2}{\sqrt{(2)^2+(-1)+(-2)^2}}$
$=\frac{2}{3},-\frac{1}{3},-\frac{2}{3}$
Thus, the direction cosines are $=\frac{2}{3},-\frac{1}{3},-\frac{2}{3}$.

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Question 32 Marks

Write the distance of the point P(x, y, z) from XOY plane.

Answer

The distance of point P(x, y, z) from the XOY plane is z.

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Question 42 Marks

Write the inclination a line with z-axis, if its direction ratios are proportional to 0, 1, -1.

Answer

We know that if a line has direction ratio (a, b, c), then the cosine of its angle with the z-axis is given by
$\cos\gamma=\frac{\text{c}}{\sqrt{\text{a}^2+\text{b}^2+\text{c}^2}}$
Suppose the inclination of the line with direction ratio (0, 1, -1) with z-axis is $\gamma$.
Now,
$\cos\gamma=\frac{1}{\sqrt{0+1+1}}$
$=-\frac{1}{\sqrt{2}}$
Implies that $\lambda=\frac{3\pi}{4}$
Hence, the inclination of the line with z-axis is $\frac{3\pi}{4}$.

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Question 52 Marks

Write the distance of the point (3, −5, 12) from X-axis?

Answer

The distance of a general point (x, y, z) from x-axis is $\sqrt{\text{y}^2+\text{z}^2}$.
$\therefore$ Distance of the point (3, -5, 12) from x-axis $=\sqrt{(-5)^2+12^2}$
$=\sqrt{169}$
= $13\text{ units}$.

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Question 62 Marks

What are the direction cosines of X-axis?

Answer

The x-axis makes angles 0°, 90° and 90° with x, y and z axes, respectively.
Therefore, the direction cosines of x-axis are cos 90°, cos 0°, cos 90°, i.e. 1, 0, 0.

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Question 72 Marks

Write direction cosines of a line parallel to z-axis.

Answer

A line parallel to z-axis, makes an angle of 90°, 90° and 0° with the x, y, and z axes, respectively.
Thus, the direction cosines are given by
$\text{l}=\cos90^\circ=0$
$\text{m}=\cos90^\circ=0$
$\text{n}=\cos0^\circ=1$
Therefore, direction cosines of a line parallel to the z-axis 0, 0, 1.

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Question 82 Marks

Find the distance of the point (2, 3, 4) from the x-axis.

Answer

A general point (x, y, z) is at a distance $\sqrt{\text{y}^2+\text{z}^2}$ of from the x-axis.
$\therefore$ Distance of the point (2, 3, 4) from x-axis
$=\sqrt{3^2+4^2}=\sqrt{25}$
$=5\text{ units}$.

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Question 92 Marks

Define direction cosines of a direction line.

Answer

The direction cosines of a direction line segment are the cosines of the direction angles of the line segment. Let two points $A(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ define the directed line segment AB.
The direction cosines of AB are given by cos
$\alpha=\frac{\text{x}_2-\text{x}_1}{\text{d}}$
$\cos\beta=\frac{\text{y}_2-\text{y}_1}{\text{d}}$
$\cos\gamma=\frac{\text{z}_2-\text{z}_1}{\text{d}}$
Here, d is the distance between A and B.

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Question 102 Marks

Write the ratio in which the line segment joining (a, b, c) and (-a, -b, -c) is divided by the xy-plane.

Answer

Suppose the line segment joining the points (a, b, c) and (-a, -c, -b) is divided by the XY-plane at a point R in the ratio $\lambda:1$.
Coordinates of R are
$\Big(\frac{\lambda(-\text{a})+1(\text{a})}{\lambda+1},\lambda(-\text{c})+\frac{1(\text{b})}{\lambda+1},\frac{\lambda(-\text{b})+1(\text{c})}{\lambda+1}\Big)$
Since R lies on XY-plane, Z-coordinate of R must be zero.
$\Rightarrow\frac{\lambda(-\text{b})+1(\text{c})}{\lambda+1}=0=\frac{\text{c}}{\text{b}}$
Thus, the required ratio is c/b : 1 or c : b.
Hence, the XY-plane divided the line in the ratio c : b.

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Question 112 Marks

If a line has direction ratios 2, -1, -2, determine its direction cosines.

Answer

Let the direction cosines of a line be l, m and n. Now,$\text{l}=\frac{2}{\sqrt{2^2+(-1)^2+(-2)^2}}=\frac{2}{3}$
$\text{m}=\frac{-1}{\sqrt{2^2+(-1)^2+(-2)^2}}=\frac{-1}{3}$ $\text{n}=\frac{-2}{\sqrt{2^2+(-1)^2+(-2)^2}}=\frac{-2}{3}$ $\therefore$ The direction consines of the line are $\frac{2}{3},\frac{-1}{3},\frac{-2}{3}.$

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Question 122 Marks

Write the distances of the point (7, -2, 3) from XY, YZ and XZ-planes.

Answer

The distance of a general point P (x, y, z) from XY-plane is z.
Thus, distance of (7, -2, 3) from XY-plane is 3.
Similarly, the distance of P (x, y, z) from YZ-plane is x.
Thus, distance of (7, -2, 3) from YZ-plane is 7.
The distance of P (x, y, z) from XZ-plane is y.
Thus, distance of (7, -2, 3) from XZ-plane is 2.

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Question 132 Marks

If line makes angle $\alpha,\beta$ and $\gamma$ with the coordinate axes, find the value of $\cos2\alpha+\cos2\beta+\cos2\gamma$.

Answer

It is given that the line makes angles $\alpha,\beta,\gamma$ with the coordinate axis.
$\therefore\text{l}=\cos\alpha,\text{m}=\cos\beta$ and $\text{n}=\cos\gamma$
$\Rightarrow\text{l}^2+\text{m}^2+\text{n}^2=1$
$\Rightarrow\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1\ ......(1)$
Now,
$\cos2\alpha+\cos2\beta+\cos2\gamma$
$=(2\cos^2\alpha-1)+(2\cos^2\beta-1)+(2\cos^2\gamma-1)$
$=2\big(\cos^2\alpha+\cos^2\beta+\cos^2\gamma\big)-3$
$=2(1)-3$
$=-1$

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Question 142 Marks

Write the ratio in which YZ-plane divides the segment joining P(-2, 5, 9) and Q(3, -2, 4).

Answer

Let the YZ-plane divides the line segment joining points P(-2, 5, 9) and Q(3, -2, 4) in the ratio k : 1.
Using the setion formula, the coordinates of the point of intersection are given by
$\Big(\frac{\text{k}(3)-2}{\text{k}+1},\frac{\text{k}(-2)+5}{\text{k}+1},\frac{\text{k}(4)+9}{\text{k}+1}\Big)$
On the YZ-plane, the X-coordinate of any point is zero.
$\frac{\text{k}(3)-2}{\text{k}+1}=0$
Implies that 3k - 2 = 0
Implies that $\text{k}=\frac{2}{3}$
Thus, the YZ-plane divides the line segment formed by joining the given points in the ratio 2 : 3 internally.

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Question 152 Marks

Answer the following quations in one word or one sentence or as per exact requirement of the question:
Write the distance of a point P(a, b, c) from x-axis.

Answer

We know that a general point (x, y, z) has distance $\sqrt{\text{y}^2+\text{z}^2}$
$\therefore$ Distance of a point P(x, y, z) from x-axis $=\sqrt{\text{b}^2+\text{c}^2}$.

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Question 162 Marks

Write the coordinates of the projection of point P(2, -3, 5) on Y-axis.

Answer

The coordinates of the projection of the point P(2, -3, 5) on Y-axis are (0, -3, 0) as both x and z coordinates of each point on the y-axis are equal to zero.

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Question 172 Marks

Write the coordinates of the projection of point P(x, y, z) on XOZ-plane.

Answer

The projection of the point P(x, y, z) on XOZ-plane is (x, 0, z) as Y-coordinates of any point on XOZ-plane are equal to zero.

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Question 182 Marks

If a line makes angles of 90°, 60° and 30° with the positive direction of x, y, and z-axis respectively, find its direction cosines.

Answer

Let l, m and n be the direction cosines of a line.$\text{l}=\cos90^{\circ}=0$
$\text{m}=\cos60^{\circ}=\frac{1}{2}$ $\text{n}=\cos30^{\circ}=\frac{\sqrt{3}}{2}$ $\therefore$ The direction consines of the line are $0,\frac{1}{2},\frac{\sqrt{3}}{2}.$

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Question 192 Marks

What are the direction cosines of Y-axis?

Answer

The y-axis makes angles 90°, 0° and 90° with x, y and z axes, respectively.
Therefore, the direction cosines of x-axis are cos 90°, cos 0°, cos 90°, i.e. 0, 1, 0.

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Question 202 Marks

If a line makes angles 90° and 60° respectively with the positive direction of x and y axes, find the angle which it makes with the positive direction of z-axis.

Answer

Let the direction cosines of the line be l, m and n.
We know that $l^2 + m^2 + n^2 = 1$
Let the line make angle $\theta$ with positive direction of the z-axis
$\alpha=90^\circ,\beta=60^\circ,\gamma=\theta$.
So, $\cos^290^\circ+\cos^260^\circ+\cos^2\theta=0$
$\Rightarrow0+\Big(\frac{1}{2}\Big)^2+\cos^2\theta=1$
$\Rightarrow\cos^2\theta=1-\frac{1}{4}$
$\Rightarrow\cos^2\theta=\frac{3}{4}$
$\Rightarrow\cos\theta=\pm\frac{\sqrt{3}}{4}$
$\Rightarrow\theta=30^\circ$or $150^\circ$

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