Three Dimensional Geometry Test

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Three Dimensional Geometry Test - 60

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Question 1
1 / -0

The direction ratios of a line perpendicular to both the lines whose direction ratios are $$3,-2,4$$ and $$1,3,-2$$

A
$$2,-5,6$$

B
$$4,-10,12$$

C
$$-8,10,11$$

D
$$-8,10,-11$$

Solution
Question 2
1 / -0

The direction cosines of the line $$x=44z+3; y=2-2\sqrt{19}z$$ is...

A
$$\dfrac{44}{\sqrt{2013}};\dfrac{-2\sqrt{19}}{\sqrt{2013}};\dfrac{1}{\sqrt{2013}}$$

B
$$\dfrac{-44}{\sqrt{2013}};\dfrac{2\sqrt{19}}{\sqrt{2013}};\dfrac{1}{\sqrt{2013}}$$

C
$$\dfrac{44}{\sqrt{2013}};\dfrac{2\sqrt{19}}{\sqrt{2013}};\dfrac{1}{\sqrt{2013}}$$

D
$$\dfrac{-44}{\sqrt{2013}};\dfrac{-2\sqrt{19}}{\sqrt{2013}};\dfrac{-1}{\sqrt{2013}}$$

Solution
Question 3
1 / -0

The distance of the point $$3\hat {i}+5\hat {k}$$ from the line parallel to $$6\hat {i}+\hat {j} 2\hat {k}$$ and passing through the point $$8\hat {i}+3\hat {j}+\hat {k}$$ is

A
$$1$$

B
$$2$$

C
$$3$$

D
$$4$$

Solution
Question 4
1 / -0

Three points whose position vectors are $$x\bar{i}+y\bar{j}+z\bar{k}$$, $$\bar{i}+2\bar{j}$$ and $$-\bar{i}-\bar{j}$$ are collinear, then relation between $$x, y, z$$ is?

A
$$x-2y=1, z=0$$

B
$$z+y=1, z=0$$

C
$$x-y=1, z=0$$

D
None of these

Solution
Question 5
1 / -0

A line makes equal angles with the coordinate axis. The direction cosines of this line are

A
$$\left(\dfrac{1}{3},\dfrac{1}{3},\dfrac{1}{3}\right)$$

B
$$\left(\dfrac{1}{\sqrt{3}},\dfrac{1}{\sqrt{3}},\dfrac{1}{\sqrt{3}}\right)$$

C
$$\left(\dfrac{1}{\sqrt{3}},\dfrac{1}{3},\dfrac{1}{3}\right)$$

D
$$\left(\dfrac{1}{\sqrt{2}},\dfrac{1}{\sqrt{2}},\dfrac{1}{\sqrt{2}}\right)$$

Solution

Let the direction cosines of the line make an angle α with each of the coordinate axes.

$$∴l=cosα,m=cosα,n=cosα$$

$${ l }^{ 2 }+{ m }^{ 2 }+{ n }^{ 2 }=1$$

$${ cos }^{ 2 }α+{ cos }^{ 2 }α+{ cos }^{ 2 }α=1$$

$$3{ cos }^{ 2 }α=1$$

$${ cos }^{ 2 }α=\frac { 1 }{ 3 } $$

$$cosα=±\frac { 1 }{ \sqrt { 3 } } $$

Thus,the direction cosines of the line,which is equally inclined to the coordinate axes,are

$$±\frac { 1 }{ \sqrt { 3 } } ,±\frac { 1 }{ \sqrt { 3 } } ,and±\frac { 1 }{ \sqrt { 3 } } .$$
Question 6
1 / -0

A line passes through the point $$(6, -7, -1)$$ and $$(2, -3, 1)$$. Then the sum of the direction cosines of the line, if the line makes acute angle with positive direction of x-axis, is?

A
$$1/3$$

B
$$4/3$$

C
$$-1/3$$

D
$$2/3$$
Question 7
1 / -0

The direction Ratio's of normal of the plane through $$(1, 0, 0), (0, 1, 0)$$ which makes angle $$\pi/4$$ with plane $$x+y =3$$ are

A
$$1, \sqrt{2}, 1$$

B
$$1, \sqrt{2}, \sqrt{2}$$

C
$$ \sqrt{2}, 1, 1$$

D
$$1, 1, \sqrt{2}$$

Solution
Question 8
1 / -0

If a line makes angles $$\alpha, \beta, \gamma$$ with positive axes, then the range of $$\sin{\alpha}\sin{\beta}+\sin{\beta} \sin{\gamma} +\sin{\gamma} \sin {\alpha}$$ is

A
$$\left(\dfrac{-1}{2},1\right)$$

B
$$\left(\dfrac{1}{2},2\right)$$

C
$$(-1,2)$$

D
$$(-1,2]$$

Solution

If a line makes an angle $$\alpha , \beta , \gamma$$ with the axes, then
$$\cos^{2}\alpha +\cos^{2}\beta +\cos^{2}\gamma=1$$ $$[\because \cos \alpha,\cos \beta,\cos \gamma $$ are directional cosines$$ ]$$
$$3-(\sin^{2}\alpha +\sin^{2}\beta +\sin^{2}\gamma)=1$$
$$\Rightarrow \ \sin^{2}\alpha +\sin^{2}\beta +\sin^{2}\gamma=2$$

Also $$\sin^{2}\alpha +\sin^{2}\beta +\sin^{2}\gamma \ \ge \sin \alpha \sin \beta +\sin \beta \sin \gamma +\sin \gamma \sin \alpha$$

$$\Rightarrow \displaystyle \sum \sin \alpha \sin \beta \le 2$$ $$\dots(1)$$

Also $$(\displaystyle \sum \sin \alpha)^{2}=\displaystyle \sum \sin^{2}\alpha +2\displaystyle \sum \sin \alpha \sin \beta$$ $$[\because (a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca]$$

$$(\displaystyle \sum \sin \alpha)^{2} > 0$$

$$\displaystyle \sum \sin^{2}\alpha +2\displaystyle \sum \sin \alpha \sin \beta > 0$$

$$2+2\displaystyle \sum \sin \alpha \sin \beta > 0$$

$$2\displaystyle \sum \sin \alpha \sin \beta > -2$$

$$\displaystyle \sum \sin \alpha \sin \beta > \dfrac {-2}{2}$$

$$\displaystyle \sum \sin \alpha \sin \beta >-1$$ $$\dots(2)$$

From $$(1)\ and\ (2)$$

$$\displaystyle 2 \geq \sum \sin \alpha \sin \beta >-1$$

Hence the range of $$\displaystyle \sum \sin \alpha \sin \beta$$ is $$(-1, 2]$$
Question 9
1 / -0

The direction ratios of the normal to the plane through $$(1,0,0)$$ and $$(0,1,0)$$ which makes an angle of $$\dfrac{\pi }{4}$$ with the plane $$x + y = 3$$ are-

A
$$1,\sqrt {2},1$$

B
$$1,1,\sqrt {2}$$

C
$$1,1,2$$

D
$$\sqrt {2},1,1$$

Solution

Let the equation of the plane through $$\left(1,0,0\right)$$ and $$\left(0,1,0\right)$$ be
$$ax+by+cz+d=0$$
where $$\vec{n}=\left(a,b,c\right)$$ is unit normal vector to the plane.
$$\Rightarrow \sqrt{{a}^{2}+{b}^{2}+{c}^{2}}=1$$
$$\Rightarrow a+d=0$$ and $$b+d=0$$Subtracting these two we get $$a-b=0$$ ....$$(1)$$
Also since this plane makes angle $$\dfrac{\pi}{4}$$ with $$x+y=3$$
$$\Rightarrow \dfrac{1}{\sqrt{2}}=\dfrac{\left(a,b,c\right).\left(1,1,0\right)}{1.\sqrt{2}}$$
$$\Rightarrow a+b=1$$ ........$$(2)$$
Solving $$(1)$$ and $$(2)$$ we get
$$2a=1$$ or $$a=\dfrac{1}{2}$$
From $$(1)$$ we have $$a=b=\dfrac{1}{2}$$
$$\Rightarrow {c}^{2}=1-\dfrac{1}{4}-\dfrac{1}{4}=\dfrac{1}{2}$$
or $$c=\pm\dfrac{1}{\sqrt{2}}$$
$$\therefore$$ Direction ratios of normal is $$\left(a,b,c\right)=\left(\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{\sqrt{2}}\right)$$
$$=\left(1,1,\sqrt{2}\right)$$ by multiplying by $$2$$
Question 10
1 / -0

If $${l}_{1},{m}_{1},{n}_{1}$$ and $${l}_{2},{m}_{2},{n}_{2}$$ are the d.c.s of the two lines, then $${({l}_{1}{l}_{2}+{m}_{1}{m}_{2}+{n}_{1}{n}_{2})}^{2}+{({l}_{1}{m}_{2}-{l}_{2}{m}_{1})}^{2}+{({m}_{1}{n}_{2}-{m}_{2}{n}_{1})}^{2}+{({n}_{1}{l}_{2}-{n}_{2}{l}_{1})}^{2}=$$

A
$$3$$

B
$$2$$

C
$$1$$

D
$$4$$

Solution