DP Mathematics: Analysis and Approaches Questionbank

Given that $L$ meets ${\Pi }_1$ at the point $\text{P}$, find the coordinates of $\text{P}$.

Find the shortest distance from the point $\text{O}(0, 0, 0)$ to ${\Pi }_1$.

Find the equation of ${\Pi }_2$, giving your answer in the form $\mathit{r}\mathbf{.}\mathit{n}=d$.

Determine the acute angle between ${\Pi }_1$ and ${\Pi }_2$.

Find the vectors $\overrightarrow{\text{AB}}$ and $\overrightarrow{\text{AC}}$.

Use a vector method to show that $\text{B}\hat{\text{A}}\text{C}=60°$.

Show that the Cartesian equation of the plane $\Pi$ that contains the triangle $\text{ABC}$ is $-x+y+z=-2$.

Find a vector equation of the line $L$.

Hence determine the minimum distance, $d_{\text{min}}$, from $\text{D}$ to $\Pi$.

Find the volume of right-pyramid $\text{ABCD}$.

Find a Cartesian equation of the plane ${\text{Π}}_3$ which is perpendicular to ${\text{Π}}_1$ and ${\text{Π}}_2$ and passes through the origin $(0, 0, 0)$.

Find the coordinates of the point where ${\text{Π}}_1$, ${\text{Π}}_2$ and ${\text{Π}}_3$ intersect.

Find the vector $\overrightarrow{\text{AB}}$ and the vector $\overrightarrow{\text{AC}}$.

Hence find the equation of ${\Pi }_1$, expressing your answer in the form $ax+by+cz=d$, where $a, b, c, d\in \mathrm{\mathbb{Z}}$.

Show that at the point $\text{P}, \lambda =\frac{3}{4}$.

Find the reflection of the point $\text{B}$ in the plane ${\Pi }_3$.

The line $L$ is the intersection of ${\Pi }_1$ and ${\Pi }_2$. Verify that the vector equation of $L$ can be written as $\mathit{r}=\left(\begin{array}{c}0\\ -2\\ 0\end{array}\right)+\lambda \left(\begin{array}{c}1\\ 1\\ -1\end{array}\right)$.

Hence find the coordinates of $\text{P}$.

Hence find the vector equation of the line formed when $L$ is reflected in the plane ${\Pi }_3$.

Verify that the point $\text{P}(1, -2, 0)$ lies on both $\prod _1$ and $\prod _2$.

Find a vector equation of $L$, the line of intersection of $\prod _1$ and $\prod _2$.

Show that the three planes do not intersect.

Find the distance between $L$ and $\prod _3$.

Find the Cartesian equation of the plane ${\mathrm{\Pi }}_1$, passing through the points A , B and D.

Find the angle between the faces ABD and BCD.

Find the Cartesian equation of ${\mathrm{\Pi }}_3$.

Show that P is the midpoint of AD.

Find the area of the triangle OPQ.

Explain why ABCD is a parallelogram.

Using vector algebra, show that $\overrightarrow{\text{AD}}=\overrightarrow{\text{BC}}$.

Show that p = 1, q = 1 and r = 4.

Find the area of the parallelogram ABCD.

Find the vector equation of the straight line passing through M and normal to the plane $\mathrm{\Pi }$ containing ABCD.

Find the Cartesian equation of $\mathrm{\Pi }$.

Find the coordinates of X, Y and Z.

Find YZ.

Find the Cartesian equation of the plane containing P, Q and R.

Given that П₁ and П₂ meet in a line $L$, verify that the vector equation of $L$ can be given by r $=\left(\begin{array}{c}\frac{5}{4}\\ 0\\ -\frac{7}{4}\end{array}\right)+\lambda \left(\begin{array}{c}\frac{1}{2}\\ 1\\ -\frac{5}{2}\end{array}\right)$.

Given that П₃ is parallel to the line $L$, show that $a+2b-5c=0$.

Show that $5a-7c=4$.

Given that П₃ is equally inclined to both П₁ and П₂, determine two distinct possible Cartesian equations for П₃.

Find, in terms of $b$, a Cartesian equation of the plane Π containing this triangle.

Find, in terms of $b$, the equation of the line L which passes through M and is perpendicular to the plane П.

Show that L does not intersect the $y$-axis for any negative value of $b$.

Find the coordinates of the point of intersection of the planes defined by the equations $x+y+z=3,x-y+z=5$ and $x+y+2z=6$.

Find a vector equation of the line L passing through the points A and B.

Find the coordinates of the point of intersection of the line L with the plane Π.

find the value of $a$;

determine the coordinates of the point of intersection P.

Find the two possible coordinates of $\text{V}$.

Comment on the positions of $\text{V}$ in relation to the plane $\text{ABC}$.

At $\text{X}$, find the value of $p$ and the value of $\theta$.

Find the equation of the horizontal asymptote of the graph.

Three planes have equations:

$2x-y+z=5$

$x+3y-z=4$     , where $a\text{, }b\in \mathbb{R}$.

$3x-5y+az=b$

Find the set of values of $a$ and $b$ such that the three planes have no points of intersection.

A straight line, $L_{\theta }$, has vector equation r $=\left(\begin{array}{c}5\\ 0\\ 0\end{array}\right)+\lambda \left(\begin{array}{c}5\\ \text{sin}\theta \\ \text{cos}\theta \end{array}\right)\text{, }\lambda \text{, }\theta \in \mathbb{R}$.

The plane ${\Pi }_p$, has equation $x=p\text{, }p\in \mathbb{R}$.

Show that the angle between $L_{\theta }$ and ${\Pi }_p$ is independent of both $\theta$ and $p$.

Find the acute angle between the planes with equations $x+y+z=3$ and $2x-z=2$.