2.1 Equations of straight lines questions for IB Maths AA

Find the equation of $L$, giving your answer in the form $y=mx$.

The line $L$ meets the unit circle, $x^2+y^2=1$, at the points A and B, where A is in the second quadrant.

Write down the coordinates of A.

Write down the coordinates of B.

The line $M$ passes through O and is perpendicular to $L$. The line $M$ meets the unit circle at the point C, which lies in the first quadrant.

Find the area of triangle ABC.

Write down the equation of $L$.

Find the $x$-coordinate of P.

Find the area of the region enclosed by $L$ and the graph of $f$.

Find the acute angle between the tangent to the graph of $f$ at O and the $x$-axis.

The line $y=x\tan \theta$, where $\theta$ is acute, divides the region enclosed by the graph of $f$ and the $x$-axis into two parts of equal area.

Find the value of $\theta$.

By using the identity $\cos 2\theta =2{\cos }^2\theta -1$, show that ${\cos }^2\left({\displaystyle \frac{\pi }{12}}\right)={\displaystyle \frac{2+\sqrt{3}}{4}}$.

Find the exact value of ${\sin }^2\left({\displaystyle \frac{\pi }{12}}\right)$.

Hence, show that $\tan \left({\displaystyle \frac{\pi }{12}}\right)=2-\sqrt{3}$.

Write down the exact value of $\tan \left({\displaystyle \frac{11\pi }{12}}\right)$.

Write down the exact value of $\tan \left({\displaystyle \frac{13\pi }{12}}\right)$.

The points on the unit circle corresponding to the angles ${\displaystyle \frac{\pi }{12}}$ and ${\displaystyle \frac{5\pi }{12}}$ are reflections of each other in the line $y=x$.

Use this fact to show that $\tan \left({\displaystyle \frac{5\pi }{12}}\right)=2+\sqrt{3}$.

The lines $y=x\tan \left({\displaystyle \frac{\pi }{12}}\right)$ and $y=x\tan \left({\displaystyle \frac{5\pi }{12}}\right)$ meet the line $x=2$ at the points C and D respectively.

Find the exact area of triangle OCD, where O is the origin.

By considering $\tan (\beta -\alpha )$, find the acute angle between $L_1$ and $L_2$. Give your answer in degrees.

The line $L_3$ passes through the point $(0,2)$ and makes an acute angle of $40\mathrm{°}$ with $L_1$.

Find the possible values of the gradient of $L_3$.

Find the Cartesian equation of $L$, giving your answer in the form $ax+by=c$, where $a,b,c\in \mathbb{Z}$.

The line $M$ has equation $y=3x-11$.

Write down a direction vector for $M$.

Find the coordinates of the point where $L$ and $M$ intersect, and the value of $\lambda$ at this point.

Find the exact value of the cosine of the acute angle between $L$ and $M$.

The regression line of $y$ on $x$ has equation $y=ax+b$. Find the value of $a$ and the value of $b$.

The regression line of $x$ on $y$ has equation $x=cy+d$. Find the value of $c$ and the value of $d$.

Write down the value of $r$, Pearson's product-moment correlation coefficient, for these data.

The two regression lines intersect at the point P. Find the coordinates of P.

Another athlete's best long jump is 7.45 m. Use an appropriate regression line to estimate his best 100 m time.

A different athlete's best 100 m time is 10.9 s. Use an appropriate regression line to estimate his best long jump.

A coach claims that, for these athletes, a 100 m time that is 0.2 s faster is associated with a longer long jump. Use the regression line of $x$ on $y$ to estimate the increase in the long-jump distance associated with a 100 m time that is 0.2 s faster.