Three Dimensional Geometry Test

Result

Three Dimensional Geometry Test - 63

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Question 1
1 / -0

$${ L }_{ 1 }$$ and $${ L }_{ 2 }$$ are two lines whose vector equations are $${ L }_{ 1 }:\vec { r } =\lambda \left[ \left( cos\theta +\sqrt { 3 } \right) \hat { i } +\left( \sqrt { 2 } sin\theta \right) \hat { j } +\left( cos\theta -\sqrt { 3 } \right) \hat { k } \right] $$
$${ L }_{ 2 }:\vec { r } =\mu \left( a\hat { i } +b\hat { j } +c\hat { k } \right) ,$$ where$$\lambda$$ and $$\mu $$ are scalars and$$\alpha $$ is the acute angle between $${ L }_{ 1 }$$and$${ L }_{ 2 }$$ If the angle$$ '\alpha '$$ is independent of $$\theta $$ then the value of $$\alpha $$ is

A
$$\dfrac { \pi }{ 6 } $$

B
$$\dfrac { \pi }{ 4 } $$

C
$$\dfrac { \pi }{ 3 } $$

D
$$\dfrac { \pi }{ 2 } $$

Solution
Question 2
1 / -0

Direction ratio of two lines are $$l_{1}, m_{1},n_{1}$$ and $$l_{2},m_{2},n_{2}$$ then direction ratios of the line perpendicular to both the lines are

A
$$l_{1}-l_{2}, m_{1}-m_{2}, n_{1}-n_{2}$$

B
$$l_{1}+l_{2}, m_{1}+m_{2}, n_{1}+n_{2}$$

C
$$m_{1}n_{2}-n_{1}m_{2}, n_{1}l_{2}-n_{2}l_{1}, l_{1}m_{2}-m_{1}l_{2}$$

D
$$m_{1}n_{2}-n_{1}m_{2}, n_{1}l_{2}-n_{1}l_{1}, l_{1}m_{2}-m_{1}l_{2}$$

Solution

Let $${u}_{1}=\left({l}_{1},{m}_{1},{n}_{1}\right)$$ be a unit vector along one line and $${u}_{2}=\left({l}_{2},{m}_{2},{n}_{2}\right)$$ be a unit vector along other line.
As $${u}_{1}\times {u}_{2}$$ is a unit vector perpendicular to both $${u}_{1}$$ and $${u}_{2}$$
$${u}_{1}\times {u}_{2}=\left|\begin{matrix}\hat{i} & \hat{j} & \hat{k} \\ {l}_{1} & {m}_{1} & {n}_{1} \\ {l}_{2} & {m}_{2} & {n}_{2}\end{matrix}\right|$$
$$=\hat{i}\left({m}_{1}{n}_{2}-{n}_{1}{m}_{2}\right)-\hat{j}\left({n}_{2}{l}_{1}-{n}_{1}{l}_{2}\right)+\hat{k}\left({l}_{1}{m}_{2}-{m}_{1}{l}_{2}\right)$$
$$=\hat{i}\left({m}_{1}{n}_{2}-{n}_{1}{m}_{2}\right)+\hat{j}\left({n}_{1}{l}_{2}-{n}_{2}{l}_{1}\right)+k\left({l}_{1}{m}_{2}-{m}_{1}{l}_{2}\right)$$
So, the required direction cosines are:
$${m}_{1}{n}_{2}-{n}_{1}{m}_{2}$$,$${n}_{1}{l}_{2}-{n}_{2}{l}_{1}$$ and $${l}_{1}{m}_{2}-{m}_{1}{l}_{2}$$
Question 3
1 / -0

The plane passing through $$(1, 1, 1), (1, -1, 1)$$ and $$(-7, -3, -5)$$ is parallel to

A
$$X-axis$$.

B
$$Y-axis$$.

C
$$Z-axis$$.

D
None of these

Solution
Question 4
1 / -0

Each group from the alternatives represents lengths of sides of a triangleStare which group does not represent a right-angled triangle.

A
$$ ( 8,40,41 ) $$

B
$$ ( 20,25,30 ) $$

C
$$ ( 8,15,17 ) $$

D
$$ ( 6,8,10 ) $$
Question 5
1 / -0

The lines $$\vec{r}=i-j+\lambda(2i+k)$$ and $$\vec{r}=(2i-j)+\mu(i+j-k)$$ intersect for

A
$$\lambda=1, \mu =1$$

B
$$\lambda=2, \mu =1$$

C
All values of $$\lambda$$ and $$\mu$$

D
No value of $$\lambda$$ and $$\mu$$
Question 6
1 / -0

there are 20 points in the plane on three of which are collinear. the number of straight lines by joining them is

A
190

B
200

C
40

D
500
Question 7
1 / -0

What is the area of the triangle with vertices $$(0,2,2),\,(2,0,-1)$$ and $$(3,4,0)$$ ?

A
$$\frac{{15}}{2}sq$$ unit

B
$$15sq$$ unit

C
$$\frac{{7}}{2}sq$$ unit

D
$$7sq$$ unit

Solution
Question 8
1 / -0

In the three points with position vectors (1, a. b) : (a, b, 3) are collinear in space, then the value of a + b is

A
3

B
4

C
5

D
none

Solution
Question 9
1 / -0

The number of straight lines that can be drawn through any two points out of $$10$$ points, of which $$7$$ are collinear.

A
$$25$$

B
$$30$$

C
$$35$$

D
$$45$$

Solution
Question 10
1 / -0

If the vectors $$2\hat{i} + 3\hat{j} , ~5\hat{i} + 6\hat{j} ,$$ and $$ 8\hat{i} +\lambda{\hat{j}}$$ have their initial points at $$(1 , 1)$$, then the value of $$\lambda$$ so that the vectors terminate on one straight line is

A
$$0$$

B
$$3$$

C
$$6$$

D
$$9$$