1.16 Systems of linear equations questions for IB Maths AA

Determine whether $L_1$ and $L_2$ are coincident, parallel but distinct, intersecting or skew. Justify your answer.

Show that $L_1$ and $L_3$ intersect, and find the coordinates of the point of intersection.

Find $\left(\begin{array}{c}1\\ 1\\ -1\end{array}\right)\times \left(\begin{array}{c}2\\ -1\\ 1\end{array}\right)$.

Verify that the point $(2,-1,0)$ lies on ${\mathrm{\Pi }}_1$ and on ${\mathrm{\Pi }}_2$.

Hence write down a vector equation of $L$.

For $t\in \mathbb{R}$, the plane ${\mathrm{\Pi }}_t$ is defined by

Show that $L$ lies in ${\mathrm{\Pi }}_t$ for every value of $t$.

Find the value of $t$ for which ${\mathrm{\Pi }}_t$ passes through the origin, and find the Cartesian equation of this plane.

Show that there is no value of $t$ for which ${\mathrm{\Pi }}_t$ is perpendicular to ${\mathrm{\Pi }}_1$.

Show that ${\mathrm{\Pi }}_2$ is perpendicular to ${\mathrm{\Pi }}_1$, and explain why this does not contradict part (d)(i).

Find the Cartesian equation of the plane in the family ${\mathrm{\Pi }}_t$ that is parallel to the $z$-axis.

Let ${\theta }_t$ be the acute angle between ${\mathrm{\Pi }}_t$ and ${\mathrm{\Pi }}_1$.

Show that $\cos {\theta }_t={\displaystyle \frac{1}{\sqrt{2t^2+1}}}$.

Find the values of $t$ for which the angle between ${\mathrm{\Pi }}_t$ and ${\mathrm{\Pi }}_1$ is ${45}^{\circ }$.

Describe what happens to ${\theta }_t$ as $t\to \mathrm{\infty }$, and relate this to part (d)(ii).

The plane ${\mathrm{\Pi }}_3$ has equation $4x+y-z=c$, where $c\in \mathbb{R}$.

Find the value of $t$ for which ${\mathrm{\Pi }}_t$ is parallel to ${\mathrm{\Pi }}_3$, and hence find the value of $c$ for which ${\mathrm{\Pi }}_1$, ${\mathrm{\Pi }}_2$ and ${\mathrm{\Pi }}_3$ intersect in a line.

Describe the geometrical arrangement of ${\mathrm{\Pi }}_1$, ${\mathrm{\Pi }}_2$ and ${\mathrm{\Pi }}_3$ when $c\mathrm{\ne }7$, justifying your answer.

Find the value of $a$ and the value of $b$.

Find a vector equation of $L$.

Find the acute angle between the directions of $T_1$ and $T_2$.

Show that the tunnels do not meet.

At one point, the first tunnel passes directly above the second tunnel.

Hence find the vertical distance between the two tunnels at this point.

The engineers decide instead to connect the tunnels with the shortest possible straight shaft $[\mathrm{R}\mathrm{S}]$, where R lies on $T_1$ and S lies on $T_2$. The shaft is perpendicular to both tunnels.

Find the value of $\lambda$ at R and the value of $\mu$ at S.

Find the length of the shaft.

The graph of the quadratic function $y=a{x}^2+bx+c$ passes through the points $(-1,2)$, $(0,1)$ and $(1,4)$.

Find the value of $a$, the value of $b$ and the value of $c$.

Find ${\displaystyle {\int }_{-1}^1(a{x}^2+bx+c)\mathrm{d}x}$ for these values of $a$, $b$ and $c$.

Now consider the quadratic $y=a{x}^2+bx+c$ whose graph passes through the points $(-h,p)$, $(0,q)$ and $(h,r)$, where $h>0$ and $p,q,r\in \mathbb{R}$.

Show that, for all values of $p$, $q$ and $r$, this system has the unique solution

Hence show that ${\displaystyle {\int }_{-h}^h(a{x}^2+bx+c)\mathrm{d}x=\frac{h}{3}(p+4q+r)}$.

It can be shown that the result of part (d) holds for any three equally spaced points $(m-h,p)$, $(m,q)$ and $(m+h,r)$. The rule ${\displaystyle {\int }_{m-h}^{m+h}f(x)\mathrm{d}x\approx \frac{h}{3}(f(m-h)+4f(m)+f(m+h))}$ is used to approximate integrals.

Use this rule with $m=0.5$ and $h=0.5$ to find an approximate value for ${\displaystyle {\int }_0^1{\mathrm{e}}^{x^2}\mathrm{d}x}$, and find the percentage error of this approximation.

Show that the rule in part (d) gives the exact value of ${\displaystyle {\int }_{-h}^{h}f(x)\mathrm{d}x}$ for any cubic function $f(x)=k{x}^3+a{x}^2+bx+c$.

A rule of the form ${\displaystyle {\int }_{-1}^{1}f(x)\mathrm{d}x\approx \alpha f(-1)+\beta f(0)+\gamma f(1)}$ is required to give the exact value when $f(x)=1$, $f(x)=x$ and $f(x)=x^2$.

Write down a system of three equations in $\alpha$, $\beta$ and $\gamma$, and solve it. Show that your answer agrees with part (d).

A different rule uses unequally spaced points: ${\displaystyle {\int }_0^{3}f(x)\mathrm{d}x\approx Af(0)+Bf(1)+Cf(3)}$. It is required to give the exact value when $f(x)=1$, $f(x)=x$ and $f(x)=x^2$.

Find the value of $A$, the value of $B$ and the value of $C$.

Determine whether the rule in part (h) gives the exact value of ${\displaystyle {\int }_0^3{x}^3\mathrm{d}x}$.

Explain briefly the significance of the value of $A$ found in part (h).