Skew Lines

Q. If $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-collinear vectors, the value of $x$ for which vectors $\overrightarrow{\alpha }=(x-2)\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{\beta }=(3+2x)\overrightarrow{a}-2\overrightarrow{b}$ are collinear.

1. $-\frac{1}{4}$
2. $\frac{1}{4}$
3. $1$
4. $-1$

Q. Skew lines are non coplanar lines.

1. True
2. False

Q. If the direction ratios of a line passing through two points P (1, 4, 5) and Q (2, 3, K) are (2, -2, 4) then find the value of K?
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Q. Match the statements in Column-I with the values in Column-II.
$$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(A)}& \text{A line from the origin meets the lines}& \text{(p)}& -4\\ & \frac{x-2}{1}=\frac{y-1}{-2}=\frac{z+1}{1}\text{and}& & \\ & \frac{x-\frac{8}{2}}{2}=\frac{y+3}{-1}=\frac{z-1}{1}\text{at}P\text{and}q& & \\ & \text{respectively. If length}PQ=d,\text{then}{d}^2\text{is}& & \\ \text{(B)}& \text{The values of}x\text{satisfying}& \text{(q)}& 0\\ & {\tan }^{-1}(x+3)-{\tan }^{-1}(x-3)={\sin }^{-1}\left(\frac{3}{5}\right)& & \\ & \text{are}& & \\ \text{(C)}& \text{Non-zero vectors}\overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}\text{satisfy}& \text{(r)}& 4\\ & \overrightarrow{a}.\overrightarrow{b}=0,(\overrightarrow{b}-\overrightarrow{a}).(\overrightarrow{b}+\overrightarrow{c})=0& & \\ & \text{and}2|\overrightarrow{b}+\overrightarrow{c}|=|\overrightarrow{b}-\overrightarrow{a}|& & \\ & \text{If}\overrightarrow{a}=\mu \overrightarrow{b}+4\overrightarrow{c},\text{then the possible values of}& & \\ & \mu \text{are}& & \\ \text{(D)}& \text{Let}f\text{be the function on}[-\pi ,\pi ]\text{given by}& \text{(s)}& 5\\ & f\left(0\right)=9\text{and}f\left(x\right)=\sin \left(\frac{9x}{2}\right)/\sin \left(\frac{x}{2}\right)& & \\ & \text{for}x\ne 0.\text{The value of}\frac{2}{\pi }\underset{-\pi }{\overset{\pi }{\int }}f\left(x\right)dx\text{is}& & \\ & & \text{(t)}& 6\end{array}$$

1. A-R; B-P; C-Q; D-S
2. A-S; B-Q; C-P; D-R
3. A-Q; B-P; R-R; D-T
4. A-T; B-P; C-Q; D-R

Q. If $4\hat{i}+7\hat{j}+8\hat{k},2\hat{i}+3\hat{j}+4\hat{k}$ and $2\hat{i}+5\hat{j}+7\hat{k}$ are the position vectors of the vertices $A,B$ and $C$ respectively of triangle $ABC$. The position vector of the point where the bisector of $\angle A$ meets $BC$

1. $\frac{1}{2}(4\hat{i}+8\hat{j}+11\hat{k})$
2. $\frac{1}{3}(6\hat{i}+11\hat{j}+15\hat{k})$
3. $\frac{1}{3}(6\hat{i}+13\hat{j}+18\hat{k})$
4. $\frac{1}{4}(8\hat{i}+14\hat{j}+9\hat{k})$

Q. If the lines $\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-k}{5}$ and $\frac{x+1}{-1}=\frac{y-2}{2}=\frac{z-5}{5}$ are coplanar, then the value of $k$ is

Q. For the given lines $\overrightarrow{r}=2\hat{i}+2\hat{j}-\hat{k}+\lambda (\hat{i}-2\hat{j}-\hat{k})$ and $\overrightarrow{r}=3\hat{i}-2\hat{j}+\hat{k}+\mu (-\hat{i}-\hat{j}+\hat{k}),$ which of the following is correct ?

1. both lines are parallel
2. both lines are intersecting
3. both lines are coincident
4. both lines are skew

Q. The vector equations of two lines $L_1$ and $L_2$ are respectivly
$\overrightarrow{r}=17\hat{i}-9\hat{j}+9\hat{k}+\lambda (3\hat{i}+\hat{j}+5\hat{k})$ and $\overrightarrow{r}=15\hat{i}-8\hat{j}-\hat{k}+\mu (4\hat{i}+3\hat{j})$
I $L_1$ and $L_2$ are skew lines
II $(11,-11,-1)$ is the point of intersection of $L_1$ and $L_2$
III $(-11,-11,1)$ is the point of intersection of $L_1$ and $L_2$
IV $co{s}^{-1}\left(3/\sqrt{35}\right)$ is the acute angle between $L_1$ and $L_2$
then, which of the following is true ?

1. II and IV
2. I and IV
3. IV only
4. III and IV

Q. The equation of line passing through point $A(2,-1,1)$ and parallel to vector $2\hat{i}+3\hat{j}-\hat{k}$ is

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

Q. The lines $\frac{x-a+d}{\alpha -\delta }=\frac{y-a}{\alpha }=\frac{z-a-d}{\alpha +\delta }$ and $\frac{x-b+c}{\beta -\gamma }=\frac{y-b}{\beta }=\frac{z-b-c}{\beta +\gamma }$ are:

1. Coplanar
2. Parallel
3. Skew
4. None of these

Q. Two lines $L_1:x=5,\frac{y}{3-\alpha }=\frac{z}{-2}$ and $L_2:x=\alpha ,\frac{y}{-1}=\frac{z}{2-\alpha }$ are coplanar. Then $\alpha$ can take value(s)

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

Q. Skew lines are non coplanar lines.

1. True
2. False

Q. Consider two lines $\frac{x+3}{-4}=\frac{y-6}{3}=\frac{z}{2}$and$\frac{x-2}{-4}=\frac{y+1}{1}=\frac{z-6}{1}$. Which of the following are correct?

1. The given lines are coplanar.
2. The shortest distance between the given lines is 9.
3. The given lines are non-coplanar.
4. $(\hat{i}-4\hat{j}+8\hat{k})$ is a vector perpendicular to both given lines.

Q. The value of $\lambda$ so that the points $P,Q,R,S$ on the sides $OA,OB,OC$ and $AB$ of a regular tetrahedron are co-planar. When $\frac{OP}{OA}=\frac{1}{3};\frac{OQ}{OB}=\frac{1}{2};\frac{OR}{OC}=\frac{1}{3}$ and $\frac{OS}{AB}=\lambda$ is

1. $\lambda =\frac{1}{2}$
2. $\lambda =-1$
3. $\lambda =0$
4. $\lambda =2$

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

1. $3(x-21)=3y-92=3z-32$
2. $\frac{x-\left(\frac{62}{3}\right)}{\frac{1}{3}}=\frac{y-31}{\frac{1}{3}}=\frac{z+\left(\frac{31}{3}\right)}{\frac{1}{3}}$
3. $\frac{x-21}{\frac{1}{3}}=\frac{y-\left(\frac{92}{3}\right)}{\frac{1}{3}}=\frac{z+\left(\frac{32}{3}\right)}{\frac{1}{3}}$
4. $\frac{x-2}{\frac{1}{3}}=\frac{y+3}{\frac{1}{3}}=\frac{z-1}{\frac{1}{3}}$

Q. Two lines $L_1:x=5,\frac{y}{3-\alpha }=\frac{z}{-2},L_2:x=\alpha ,\frac{y}{-1}=\frac{z}{2-\alpha }$ are coplanar. Then, $\alpha$ can take value $\left(s\right)$

1. $1,2,5$
2. $1,4,5$
3. $3,4,5$
4. $2,4,5$

Q. Assertion :Two perpendicular non-intersecting lines are not coplanar. Reason: Two skew lines are not coplanar.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. If the vectors $4\hat{i}-7\hat{j}-2\hat{k},\hat{i}+5\hat{j}-3\hat{k},3\hat{i}-\lambda \hat{j}+\hat{k},$ form a triangle, then the value of $\lambda$ is equal to

Q. If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect, then $k$ is equal to

1. $-1$
2. $\frac{2}{9}$
3. $\frac{9}{2}$
4. $0$

Q. If the two lines $\overrightarrow{r}=2\hat{i}-2\hat{k}+\lambda (\hat{j}-\hat{k})$ and $\overrightarrow{r}=\hat{j}+\hat{k}+\mu (\hat{i}+2\hat{j}-\alpha \hat{k})$ intersect at a point, then the value of $\alpha =$

Q. The equation of the line passing through the point (1, 2, –4) and perpendicular to the two lines $\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}$ and $\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}$,

1. $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6}$
   
2. $\frac{x-1}{-2}=\frac{y-2}{3}=\frac{z+4}{8}$
   
3. $\frac{x-1}{3}=\frac{y-2}{2}=\frac{z+4}{8}$
4. $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{4}$

Q. Find the value of $\lambda$ for which the four points $A,B,C,D$ with position vectors $-\hat{j}-\hat{k};4\hat{i}+5\hat{j}+\lambda \hat{k};3\hat{i}+9\hat{j}+4\hat{k}$ and $-4\hat{i}+4\hat{j}+4\hat{k}$ are coplanar.

Q. If two lines intersect each other, they should be in the same plane.

1. False
2. True

Q. The angle between a diagonal of cube and an angle of the cube intersecting the diagonal is

1. ${\cos }^{-1}\frac{1}{3}$
2. ${\cos }^{-1}\sqrt{\frac{2}{3}}$
3. ${\cos }^{-1}\sqrt{2}$
4. $Noneofthese$

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

1. $3(x-21)=3y-92=3z-32$
2. $\frac{x-\left(\frac{62}{3}\right)}{\frac{1}{3}}=\frac{y-31}{\frac{1}{3}}=\frac{z+\left(\frac{31}{3}\right)}{\frac{1}{3}}$
3. $\frac{x-21}{\frac{1}{3}}=\frac{y-\left(\frac{92}{3}\right)}{\frac{1}{3}}=\frac{z+\left(\frac{32}{3}\right)}{\frac{1}{3}}$
4. $\frac{x-2}{\frac{1}{3}}=\frac{y+3}{\frac{1}{3}}=\frac{z-1}{\frac{1}{3}}$

Q. Write the vector equations of the following line and henece find the shortest distance between them:$\frac{x+1}{2}=\frac{y+1}{-6}=\frac{z+1}{1}and\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$

Q. If the direction ratios of a line passing through two points P (1, 4, 5) and Q (2, 3, K) are (2, -2, 4) then find the value of K?
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Q. Skew lines are non-coplanar lines.

1. False
2. True

Q. Let point $\overrightarrow{OC}=x\hat{i}+y\hat{j}+z\hat{k}$ lies on line joining points $\overrightarrow{OA}=3\hat{i}-2\hat{j}+\hat{k}$ and $\overrightarrow{OB}=-2\hat{i}+\hat{j}-3\hat{k},$ then which of the following statement(s) is/are correct ?

1. If $x=y;\overrightarrow{OC}$ divides $\overrightarrow{AB}$ internally in ratio $5:3$
2. If $y=z;\overrightarrow{OC}$ divides $\overrightarrow{AB}$ internally in ratio $3:4$
3. If $z=x;\overrightarrow{OC}$ divides $\overrightarrow{AB}$ externally in ratio $2:1$
4. If $z=x;\overrightarrow{OC}$ divides $\overrightarrow{AB}$ internally in ratio $2:1$

Q. The value of $p$ for which the vectors $\overrightarrow{a}=\hat{i}-2\hat{j}-12\hat{k}$ and $\overrightarrow{b}=-3\hat{i}+6\hat{j}+p\hat{k}$ are parallel is