Shortest Distance between Two Skew Lines

Q. The shortest distance between the lines
$\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}$ and
$\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}$ is:

1. $2\sqrt{30}$
2. $\frac{7}{2}\sqrt{30}$
3. $3$
4. $3\sqrt{30}$

Q.

The image of the point $(3,5)$ in the line$x–y+1=0$, lies on :

1. ${(x–2)}^2+{(y–4)}^2=4$

2. ${(x–4)}^2+{(y+2)}^2=16$

3. ${(x–4)}^2+{(y–4)}^2=8$

4. ${(x–2)}^2+{(y–2)}^2=12$

Q. Let $\lambda$ be an integer. If the shortest distance between the lines $x-\lambda =2y-1=-2z$ and $x=y+2\lambda =z-\lambda$ is $\frac{\sqrt{7}}{2\sqrt{2}},$ then the value of $|\lambda |$ is

Q.

Find the shortest distance between the lines
$\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}and\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$

Q.

The equation of line whose midpoint is $(x_1,y_1)$ in between the axes is:

1. $\frac{x}{x_1}+\frac{y}{y_1}=2$

2. $\frac{x}{x_1}+\frac{y}{y_1}=\frac{1}{2}$

3. $\frac{x}{x_1}+\frac{y}{y_1}=1$

4. None of these

Q.

If $a_1,a_2,.....,a_n$are in AP, then $\frac{1}{a_1{a}_2}+\frac{1}{a_2{a}_3}+...\dots .+\frac{1}{a_{(n-1)}{a}_n}$ is

1. $\frac{(n-1)}{a_1{a}_n}$

2. $\frac{1}{a_1{a}_n}$

3. $\frac{(n+1)}{a_1{a}_n}$

4. $\frac{n}{a_1{a}_n}$

Q.

The points $A(5,-1,1),B(7,-4,7),C(1,-6,10)$ and $D(-1,-3,4)$ are vertices of a

1. Square

2. Rhombus

3. Rectangle

4. None of these

Q. Find the value of $\lambda ,$ so that the lines $\frac{1-x}{3}=\frac{7y-14}{\lambda }=\frac{z-3}{2}\text{and}\frac{7-7x}{3\lambda }=\frac{y-5}{1}=\frac{6-z}{5}$ are at right angles. Also, find whether the lines are intersecting or not.

Q.

Which of the following set of points are non-collinear?

1. $(1,–1,1),(–1,1,1),(0,0,1)$

2. $(1,2,3),(3,2,1),(2,2,2)$

3. $(–2,4,–3),(4,–3,–2),(–3,–2,4)$

4. $(2,0,–1),(3,2,–2),(5,6,–4)$

Q.

In the equation $y-y_1=m(x-x_1)$ if $m$ and $x_1$ are fixed and different lines are drawn from the different value of $y_1$ then

1. The lines will pass through a single point

2. There will be a set of parallel lines

3. There will be one line only

4. None of these

Q. A square ABCD of diagonal 2a is folded along the diagonal AC so that the planes DAC and BAC are at right angle. The shortest distance between DC and AB is

1. $\sqrt{2a}$
2. $\frac{2a}{\sqrt{3}}$
3. $\frac{2a}{\sqrt{5}}$
4. $\left(\frac{\sqrt{3}}{2}\right)a$

Q.

Draw a rough figure and label suitably in each of the following cases:

$\text{XY and PQ}$ Intersect at$\text{M.}$

Q. If angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ${30}^{\circ }$ and their magnitudes are respectively $\sqrt{3}$ and $4$ units, then the value of $\overrightarrow{a}\cdot \overrightarrow{b}$ is

1. $4\sqrt{3}$
2. $6$
3. $2\sqrt{3}$
4. $4$

Q.

A wooden cube with edge length ‘n’ (>2) units is painted red all over. By cutting parallel to faces, the cube is cut into $n^3$ smaller cubes each of unit edge length. If the number of smaller cubes with just one face painted Red is equal to the number of smaller cubes completely un painted, then n=

1. 2

2. 7

3. 8

4. 6

Q. The shortest distance between the lines $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$ and $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ is

1. $2\sqrt{3}$
2. $4\sqrt{3}$
3. $3\sqrt{6}$
4. $5\sqrt{6}$

Q. The unit vector bisecting $OY$ and $OZ$ is

1. $\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$
2. $\frac{\hat{j}+2\hat{k}}{\sqrt{5}}$
3. $\frac{2\hat{j}-\hat{k}}{\sqrt{5}}$
4. $\frac{\hat{j}+\hat{k}}{\sqrt{2}}$

Q.

Equation of the line of shortest distance between the lines $\frac{x}{2}=\frac{y}{-3}=\frac{z}{1}$ and $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$ is

1. $3x-62=3y-93=3z+31$

2. $\frac{x-21}{3}=\frac{y+\frac{92}{3}}{3}=\frac{z-\frac{32}{3}}{3}$

3. $3(x-21)=(3y-9z)=(3z-32)$

4. $x-\frac{62}{3}=y+31=z+\frac{31}{3}$

Q.

Suppose two straight lines $3x-2y=5and2x+ky+7=0$.Find the value of $k$ for which the given lines are parallel to each other.

Q. The sum and product of the slopes of the tangents to the hyperbola $\frac{x^2}{4}-\frac{y^2}{2}=1$ drawn from the point (3, –2) are

1. 
2. 
3. 
4. 

Q.

The distance between the parallel lines $y=x+a,y=x+b$ is:

1. $\frac{\left|\left(b-a\right)\right|}{\sqrt{2}}$

2. $\left|\left(a-b\right)\right|$

3. $a+b$

4. $\frac{\left|\left(b+a\right)\right|}{\sqrt{2}}$

Q. Let $\overline{PR}=3\hat{i}+\hat{j}-2\hat{k}$ and $\overline{SQ}=\hat{i}-3\hat{j}-4\hat{k}$ determine diagonals of a parallelogram PQRS and $\overline{PT}=\hat{i}+2\hat{j}$ be another vector. Then the volume of the parallelepiped determined by the vectors $\overline{PT},\overline{PQ}$ and $\overline{PS}$ is

1. 20
2. 30
3. 5
4. 10

Q. If two adjacent sides of a square are represented by the vectors $x\hat{i}+\hat{j}+4\hat{k}$ and $y\hat{i}+3\hat{j},$ then $\left(xy\right)=$

1. $3$
2. $-1$
3. $1$
4. $-3$

Q. If $x+y+z=0$ then what is the value of
$\frac{1}{x^2+y^2-z^2}+\frac{1}{y^2+z^2-x^2}+\frac{1}{z^2+x^2-y^2}$

1. $\frac{1}{x^2+y^2+z^2}$
2. $1$
3. $-1$
4. $0$

Q. The shortest distance $\left(\text{in units}\right)$ between the lines whose equations are $\overrightarrow{r}=3\hat{i}+5\hat{j}+7\hat{k}+\lambda (\hat{i}+2\hat{j}+\hat{k})$ and $\overrightarrow{r}=-\hat{i}-\hat{j}+\hat{k}+s(7\hat{i}-6\hat{j}+\hat{k})$ is:

1. $\frac{16}{5\sqrt{5}}$
2. $\frac{26}{5\sqrt{5}}$
3. $\frac{46}{5\sqrt{5}}$
4. $\frac{36}{5\sqrt{5}}$

Q. Find the shortest distance between the lines:
$\frac{x+1}{4}=\frac{y-3}{-6}=\frac{z+1}{1}$ and $\frac{x+3}{3}=\frac{y-5}{2}=\frac{z-7}{6}$

Q. Find the shortest distance between two lines : $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}$ is

1. $\frac{1}{6}$
2. $\frac{1}{\sqrt{6}}$
3. $\frac{1}{36}$
4. $\frac{1}{3}$

Q.

Find the shortest distance between the lines
$r=(\hat{i}+2\hat{j}+\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})andr=(2\hat{i}-\hat{j}-\hat{k})+\mu (2\hat{i}+\hat{j}+2\hat{k})$

Q. Let $f:R\to R$ be a continuous decreasing function. A point $x_0\in R$ is said to be a fixed point of $f$ if $f\left(x_0\right)=x_0.$
The number of fixed points of $f$ equals

1. $0$
2. $2$
3. Infinitely many
4. $1$

Q. Find the shortest distance between lines: $\frac{x-1}{1}=\frac{y-2}{3}=\frac{z-3}{2}$ and $\frac{x-4}{2}=\frac{y-5}{3}=\frac{z-6}{1}$

1. $\sqrt{6}$
2. $\sqrt{5}$
3. $\sqrt{3}$
4. $6$

Q.

Four points are graphed on the number line.

Part A: Choose the two points whose values have the greatest sum. Approximate this sum. Explain your reasoning.

Part B : Choose the two points whose values have the greatest difference. Approximate this difference. Explain your reasoning.

Part C: Choose the two points whose values have the greatest product. Approximate this product. Explain your reasoning.

Part D: Choose the two points whose values have the greatest quotient. Approximate this quotient. Explain your reasoning.