Equation of a Plane : Vector Form

Q. Direction ratios of two lines are a, b, c and $\frac{1}{bc},\frac{1}{ca},\frac{1}{ab}$.The lines are

1. Mutually perpendicular

2. Parallel

3. Coincident

4. None of these

Q.

Consider a pyramid OPQRS located in the first octant $(x\ge 0,y\ge 0,z\ge 0)$ with O as origin, and OP and OR along the X-axis and the Y-axis, respectively. The base OPQR of the pyramid is a square with OP = 3. The point S is directly above the mid-point T of diagonal OQ such that TS = 3. Then.

1. the acute angle between OQ and OS is $\frac{\pi }{3}$

2. the equation of the plane containing the $\Delta OQS$ is x - y = 0.

3. the length of the perpendicular from P to the plane containing the $\Delta OQS$ is $\frac{3}{\sqrt{2}}$

4. the perpendicular distance from O to the straight line containing RS is $\sqrt{\frac{15}{2}}$

Q.

Let $f:S\to S$ where $S=\left(0,\infty \right)$ be a twice differentiable function such that $f(x+1)=xf\left(x\right)$. If $g:S\to R$ be defined as $g\left(x\right)={\log }_{e}f\left(x\right)$, then the value of $\left|g\left(5\right)-g\left(1\right)\right|$ is equal to:

1. $\frac{197}{144}$

2. $\frac{187}{144}$

3. $\frac{205}{144}$

4. $1$

Q. The plane passing through the point $(4,-1,2)$ and parallel to the lines $\frac{x+2}{3}=\frac{y-2}{-1}=\frac{z+1}{2}$ and $\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-4}{3}$ also passes through the point :

1. $(1,1,1)$
2. $(-1,-1,-1)$
3. $(-1,-1,1)$
4. $(1,1,-1)$

Q.

Solve the linear programming problem. Maximize$z=x+2y$, subject to constraints $x-y\le 10$, $2x+3y\le 20$ and $x\ge 0$ , $y\ge 0$.

1. $z=10$

2. $z=30$

3. $z=40$

4. $z=50$

5. None of the above

Q. If $\overrightarrow{p},\overrightarrow{q},\overrightarrow{r}$ is reciprocal system of vector triad $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c},$ then $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]\left[\overrightarrow{p}\overrightarrow{q}\overrightarrow{r}\right]=$

Q. In a $\Delta ABC$, $\angle A={30}^{\circ },H$ is the orthocentre and $D$ is the mid point of $\overrightarrow{BC}\cdot$ Segment $HD$ is produced to $T$ such that $HD=DT$. Then the ratio $\frac{AT}{BC}$ is equal to

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

Q. The vector equation of the plane passing through the origin and the line of intersection of the planes $\overrightarrow{r}\cdot \overrightarrow{a}=\lambda$ and $\overrightarrow{r}\cdot \overrightarrow{b}=\mu$ is

1. $\overrightarrow{r}\cdot (\lambda \overrightarrow{b}-\mu \overrightarrow{a})=0$
2. $\overrightarrow{r}\cdot (\lambda \overrightarrow{a}-\mu \overrightarrow{b})=0$
3. $\overrightarrow{r}\cdot (\lambda \overrightarrow{a}+\mu \overrightarrow{b})=0$
4. $\overrightarrow{r}\cdot (\lambda \overrightarrow{b}+\mu \overrightarrow{a})=0$

Q. A ray of light is sent along the line $x-2y-3=0$. Upon reaching the line $3x-2y-5=0$ the ray is reflected from it. The equation of the reflected ray is

1. $13x-y=14$
2. $17x-2y=19$
3. $29x-2y=31$
4. $x+y=11$

Q. The image of the point $(1,2,-1)$ with respect to the plane containing the line $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$ and the point $(0,7,-7),$ is

1. $(-\frac{1}{3},-\frac{7}{3},\frac{1}{3})$
2. $(-\frac{1}{3},\frac{2}{3},-\frac{7}{3})$
3. $(-\frac{1}{3},0,-\frac{7}{3})$
4. $(-\frac{1}{3},\frac{2}{3},\frac{7}{3})$

Q. Find the vector equation of the plane passing through the intersection of the planes and through the point (2, 1, 3)

Q.

The equations of the line passing through the point(1, 2, -4) and perpendicularto the two lines $\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}$ and $\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}$, will be

1. 
   

2. 
   

3. 
   

4. None of these

Q. The vector equation of the plane passing through the intersection of the planes

Q.

Which graph can be used to find the solution to the system of equations below?

$$\begin{gathered} 2x+y=-4 \\ -3y=2x+12 \end{gathered}$$.

1. 

2. 

3. 

4. 

Q. The equation of the plane passing through the intersection of the planes $x+2y+3z+4=0$ and $4x+3y+2z+1=0$ and the origin is

1. $3x+2y+z+1=0$
2. $3x+2y+z=0$
3. $2x+3y+z=0$
4. $x+y+z=0$

Q.

Find the equation of a line passing through the point $(2,3)$and parallel to the line $3x-4y+5=0$

1. $4x-3y+6=0$

2. $3x-4y+6=0$

3. $3x+3y-6=0$

4. None of these

Q.

Find the vector equation of the line joining the points $(2,1,3)$ and $(-4,3,-1)$.

Q. Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane

Q. Consider a plane $x+y-z=1$ and point $A(1,2,-3).$ A line $L$ has the equation $x=1+3r,y=2-r$ and $z=3+4r.$

The equation of the plane containing line $L$ and point $A$ has the equation

1. $x-3y+5=0$
2. $3x-y-1=0$
3. $x+3y-7=0$
4. $3x+y-5=0$

Q.

Find the position vector of point R which divides the line joining two points P(2a + b) and Q(a - 3b) externally in the ratio 1 : 2. Also, show that P is the middle point of the line segment RQ.

Q. Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector .

Q.

If $l_1,m_1,n_{1}and{l}_2,m_2,n_2$ are the direction cosines of two mutually perpendicular lines, show that the direction consines of the line perpendicular to both of these are $m_1{n}_2-m_2{n}_1,n_1{l}_2-n_2{l}_1,l_1{m}_2-l_2{m}_1$

Q.

The minimum value of $2{\log }_{10}\left(x\right)-{\log }_x(0.01)$, $x>1$ is

1. $2$

2. $4$

3. $6$

4. $8$

Q. Find the value of p, so that the lines
$l_1:\frac{1-x}{3}=\frac{7y-14}{p}=\frac{z-3}{2}and{l}_2:\frac{7-7x}{3p}=\frac{y-5}{1}=\frac{6-z}{5}$ are perpendicular to each other. Also find the equations of a line passing through a point $(3,2,-4)$ and parallel to line $l_1$.

Q. Let $\overrightarrow{a}=\hat{i}+2\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{c}=\hat{i}+\hat{j}-\hat{k}$.The vector in the plane of $\overrightarrow{a}$ and $\overrightarrow{b}$ , whose projection on C is

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

Q. Find the equation of plane through the points $(2,2,1)$ and $(9,3,6)$ and perpendicular to the plane $2x+6y+6z=9$

Q. Find the vector and Cartesian equation of the planes (a) that passes through the point (1, 0, −2) and the normal to the plane is . (b) that passes through the point (1, 4, 6) and the normal vector to the plane is .

Q.

Draw figures with the given vertices in a coordinate plane. Which figures are similar? Explain your reasoning.

Triangle $\mathrm{A}:(-4,4),(-2,4),(-2,0)$

Triangle $\mathrm{B}:(-2,2),(-1,2),(-1,0)$

Triangle $\mathrm{C}:(6,6),(3,6),(3,0)$

Q. Vector equation of the plane passing through a point having position vector $2\hat{i}+3\hat{j}-4\hat{k}$ and perpendicular to the vector $2\hat{i}-\hat{j}+2\hat{k}$ is

1. $\overrightarrow{r}\cdot (2\hat{i}-\hat{j}+2\hat{k})+7=0$
2. $\overrightarrow{r}\cdot (2\hat{i}-\hat{j}+2\hat{k})=7$
3. $\overrightarrow{r}\cdot (2\hat{i}-\hat{j}+2\hat{k})=9$
4. $\overrightarrow{r}\cdot (-3\hat{i}-2\hat{j}-3\hat{k})=0$

Q. Find the vector equation of line joining the points $(2,1,3)$ and $(-4,3,-1)$

1. $\overline{r}=2(1-3\lambda )\overline{i}-(1+2\lambda )\overline{j}-(3-4\lambda )\overline{k}$
2. $\overline{r}=2(1-3\lambda )\overline{i}-(1+2\lambda )\overline{j}+(3-4\lambda )\overline{k}$
3. $\overline{r}=2(1-3\lambda )\overline{i}+(1+2\lambda )\overline{j}+(3-4\lambda )\overline{k}$
4. $\overline{r}=2(1+3\lambda )\overline{i}+(1+2\lambda )\overline{j}+(3+4\lambda )\overline{k}$