Geometry and trigonometry: 38 IB Maths AA SL practice questions

Consider the vectors $\mathbf{a} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}$.

(a) Find

(i) $2\mathbf{a} + \mathbf{b}$[2]

(ii) $|2\mathbf{a} + \mathbf{b}|$[2]

(b) Let $2\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}$, where $\mathbf{0}$ is the zero vector. Find $\mathbf{c}$.[2]

Let $\overrightarrow{OA} = \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$ and $\overrightarrow{AB} = \begin{pmatrix} 1 \\ 3 \\ 1 \end{pmatrix}$, where O is the origin. $L_1$ is the line that passes through A and B.

(a) Find a vector equation for $L_1$.[2]

(b) The vector $\begin{pmatrix} 2 \\ p \\ 0 \end{pmatrix}$ is perpendicular to $\overrightarrow{AB}$. Find the value of p.[3]

Let $\mathbf{v} = \begin{pmatrix} -10 \\ 2 \\ 1 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 3 \\ -4 \\ 0 \end{pmatrix}$. Find the angle between $\mathbf{v}$ and $\mathbf{w}$, giving your answer correct to one decimal place.[6]

Consider the points $A(-2, 20)$, $B(4, 6)$ and $C(-14, 12)$. The line $L$ passes through the point $A$ and is perpendicular to $[BC]$.

(a) Find the equation of $L$.[3]

The line $L$ passes through the point $(k, 2)$.

(b) Find the value of $k$.[2]

Consider the lines $L_1$ and $L_2$ with respective equations

$$L_1 : y = -\frac{2}{3}x + 9 \text{ and } L_2 : y = \frac{2}{5}x - \frac{19}{5}$$

(a) Find the point of intersection of $L_1$ and $L_2$.[2]

A third line, $L_3$, has gradient $-\frac{3}{4}$.

(b) Write down a direction vector for $L_3$.[1]

$L_3$ passes through the intersection of $L_1$ and $L_2$.

(c) Write down a vector equation for $L_3$.[2]

A line $L$ passes through $\text{A}(1, -1, 2)$ and is parallel to the line $\mathbf{r} = \begin{pmatrix} -2 \\ 1 \\ 5 \end{pmatrix} + s \begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix}$

(a) Write down a vector equation for $L$ in the form $\mathbf{r} = \mathbf{a} + t\mathbf{b}$.[2]

The line $L$ passes through point P when $t = 2$.

(b) Find

(i) $\overrightarrow{\text{OP}}$;[2]

(ii) $|\overrightarrow{\text{OP}}|$.[2]

Let $a$ be a constant, where $a > 1$.

(a) Show that $a^2 + \left(\frac{a^2-1}{2}\right)^2 = \left(\frac{a^2+1}{2}\right)^2$.[3]

Consider a right-angled triangle with sides of length $a$, $\left(\frac{a^2-1}{2}\right)$ and $\left(\frac{a^2+1}{2}\right)$.

(b) Find an expression for the area of the triangle in terms of $a$.[2]

Let $\overrightarrow{AB} = \begin{pmatrix} 4 \\ 1 \\ 2 \end{pmatrix}$.

(a) Find $|\overrightarrow{AB}|$.[2]

(b) Let $\overrightarrow{AC} = \begin{pmatrix} 3 \\ 0 \\ 0 \end{pmatrix}$. Find $\widehat{BAC}$.[4]

A line, $L_1$, has equation $\vec{r} = \begin{pmatrix} -3 \\ 9 \\ 10 \end{pmatrix} + s \begin{pmatrix} 6 \\ 0 \\ 2 \end{pmatrix}$. Point $P(15, 9, c)$ lies on $L_1$.

(a) Find $c$.[4]

A second line, $L_2$, is parallel to $L_1$ and passes through $(1, 2, 3)$.

(b) Write down a vector equation for $L_2$.[2]

The vectors $\mathbf{a} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} k+3 \\ k \end{pmatrix}$ are perpendicular to each other.

(a) Find the value of $k$.[4]

(b) Given that $\mathbf{c} = \mathbf{a} + 2\mathbf{b}$, find $\mathbf{c}$.[3]

Let $\mathbf{u} = 6\mathbf{i} + 3\mathbf{j} + 6\mathbf{k}$ and $\mathbf{v} = 2\mathbf{i} + 2\mathbf{j} + \mathbf{k}$.

(a) Find[5]

(i) $\mathbf{u} \cdot \mathbf{v}$;

(ii) $|\mathbf{u}|$;

(iii) $|\mathbf{v}|$.

(b) Find the angle between $\mathbf{u}$ and $\mathbf{v}$.[2]

A metal sphere has a radius 12.7 cm.

(a) Find the volume of the sphere expressing your answer in the form $a \times 10^k$, $1 \le a < 10$ and $k \in \mathbb{Z}$.[3]

The sphere is to be melted down and remoulded into the shape of a cone with a height of 14.8 cm.

(b) Find the radius of the base of the cone, correct to 2 significant figures.[3]

In triangle $ABC$, $AB = 6$ cm and $AC = 8$ cm. The area of the triangle is $16$ cm$^2$.

(a) Find the two possible values for $\hat{A}$.[4]

(b) Given that $\hat{A}$ is obtuse, find $BC$.[3]

Consider the triangle $ABC$, where $AB = 10$, $BC = 7$ and $\angle CAB = 30°$.

(a) Find the two possible values of $\angle ACB$.[4]

(b) Hence, find $\angle ABC$, given that it is acute.[2]

(a) Show that $\sin 2x + \cos 2x - 1 = 2\sin x(\cos x - \sin x)$.[2]

(b) Hence or otherwise, solve $\sin 2x + \cos 2x - 1 + \cos x - \sin x = 0$ for $0 < x < 2\pi$.[6]