In a triangle ABC, AB = 8cm, BC = a, AC = b and \(\angle BAC\) = 30°

a) Show that \(b^2-8\sqrt{3}b+64-a^2=0\)

b) Hence find the possible values of a (in cm) for which the triangle has two possible solutions.

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a) Show that \(\large \tan 2x \cot 2x\equiv \frac{2}{1-\tan^2x}\)

b) Hence, solve \(\large \tan 2x \cot 2x=3\) , for \(\large -\frac{\pi}{2}

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\(\overrightarrow { AB }\) and \(\overrightarrow { AC }\) are two vectors such that \(\overrightarrow { AB } =\left( \begin{matrix} 3 \\ -1 \\ 2 \end{matrix} \right) \) and \(\overrightarrow { AC } =\left( \begin{matrix} 2 \\ 0 \\ 1 \end{matrix} \right) \)

Find \(\hat { BAC } \) to the nearest degree.

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The angle between the line \({ L }_{ 1 }\) and \({ L }_{ 2 }\) is \(\frac{\pi}{2}\).

\( { L }_{ 1 }:\quad \frac { x+2 }{ 3 } =2y+1=\frac { 5-z }{ 2 } \)

\( { L }_{ 2 }: \quad x =\frac { y-2}{ 3} =kz \)

Find k.

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Find the angle between the planes \({ \Pi }_{ 1 }\) and \({ \Pi }_{ 2 }\) to the nearest degree.

\({ \Pi }_{ 1 }: 2x-3y+z=0\)

\({ \Pi }_{ 2 }: x+2y+5z=-4\)

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Find the value of x for which the vectors \(\left( \begin{matrix} sinx \\ \sqrt{3} \\ 0 \end{matrix} \right) \) and \(\left( \begin{matrix} 4cosx \\-1\\ 2 \end{matrix} \right) \)are perpendicular, \(0\le x\le \frac { \pi }{ 2 } \).

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OABC is a parallelogram.

\(\overrightarrow { OA } =\textbf{a}\) \(\overrightarrow { OB } =\textbf{b}\) \(\overrightarrow { OC } =\textbf{a}+\textbf{b}\)

Given that \((\textbf{ a }+\textbf{ b })\cdot (\textbf{ a }-\textbf{ b })=0\) what can you conclude

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ACB is a right-angled triangle

\(\overrightarrow { CB } = \textbf {a }\) \(\overrightarrow { AC } = \textbf {b }\)

a) Write \(\overrightarrow { AB } \) in terms of a and b

b) Find \(\textbf{ a }\cdot \textbf{ b }\)

c) Show that \({ \left| \textbf { a+b } \right| }^{ 2 }={ \left| \textbf { a } \right| }^{ 2 }+{ \left| \textbf{ b } \right| }^{ 2 }\) and hence prove Pythagoras' Theorem.

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A line L passes through the points A(0,2,-4) and B(3,-3,2)

Point C also lies on the line L. Find the coordinates of C given that \(\left| \overrightarrow { AC } \right| =\left| \overrightarrow { AB } \right| \)

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Two boats A and B, move so that a time t hours, their positions, in kilometres, are given by

​\(\textbf{r}_{A}=\left( \begin{matrix} -2 \\ -12 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -4 \end{matrix} \right) \)​

\(\textbf{r}_{B}=\left( \begin{matrix} 11 \\ -11 \end{matrix} \right) +t\left( \begin{matrix} -2 \\ 3 \end{matrix} \right) \)

1. Find the position where the two boats cross.
2. Show that the boats do not collide.

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The line L₁ has equation ​\(\textbf{r}=\left( \begin{matrix} 2\\-1 \\ 3 \end{matrix} \right) +t\left( \begin{matrix} -1 \\ -2\\k \end{matrix} \right) \)​

The line L₂ has equation \(\textbf{r}=\left( \begin{matrix} 2\\1 \\ -3 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -1\\-1 \end{matrix} \right) \)

1. The point A(3,1,-1) lies on the line L₁. Show that k = 4.
2. Show that the lines and L₁ are L₂ perpendicular.
3. Show that the lines and L₁ do not L₂ intersect.
4. The point B lies on the line The point C has coordinates (2,1,-3). ABC forms an isosceles triangles with AC=BC. Find the coordinates of B.

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a = 3i + 2j + k and b = 2i + 3j + 2k

Find a unit vector that is perpendicular to a and b

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The area of a parallelogram formed by two adjacent vectors a and b is 7 square units.

a = \(\left( \begin{matrix} -3 \\ 4 \\ k \end{matrix} \right) \) b = \(\left( \begin{matrix} 3 \\ -2 \\ -2 \end{matrix} \right) \)

Find k

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Given that \(\textbf{u}\times \textbf{v} = \textbf{u}\times \textbf{w}\) show that \(\textbf{v}- \textbf{w} \) is parallel to \(\textbf{u}\)

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a) For any two vectors v and w prove Lagrange's Identity

\({ \left| \textbf{v}\times \textbf{w} \right| }^{ 2 }+{ \left( \textbf{v}\cdot \textbf{w} \right) }^{ 2 }={ \left| \textbf{v} \right| }^{ 2 }{ \left| \textbf{w} \right| }^{ 2 }\)

b) Hence, find \(\textbf{v}\cdot \textbf{w}\) if

\({ \left| \textbf{v} \right| }=3\)

\({ \left| \textbf{w} \right| }=4\)

\(\textbf{v}\times \textbf{w} =\left( \begin{matrix} -1 \\ 2 \\ 3 \end{matrix} \right) \)

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The points A, B and C are given by the position vectors a, b and c.

If A, B and C are collinear, show that

\(\textbf{b}\times \textbf{c}=\textbf{a}\times (\textbf{c}-\textbf{b})\)

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a and b are vectors

Show that \(|\textbf{a}×\textbf{b}|^{ 2 }+|\textbf{a} ∙\textbf{b}|^{ 2 }=(\textbf{a}\textbf{b})^{ 2 }\)

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The coordinates of points A, B and C are given as (5,4,1) , (5,1,-2) and (1,-1,2) respectively.

a) Find the equation of the plane that passes through A, B and C

b) Find the equation of the plane that is perpendicular to AB and passes through C

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A plane has vector equation \(\textbf{ r }=\left( \begin{matrix} 1 \\ 2 \\ 0 \end{matrix} \right) +\mu \left( \begin{matrix} -2 \\ 0 \\ 5 \end{matrix} \right) +\lambda \left( \begin{matrix} 0 \\ -4 \\ 5 \end{matrix} \right) \)

a) Find the Cartesian equation of the plane

b) The plane meets the x, y and z axes at A, B and C respectively. OABC forms a pyramid. Find the volume of the pyramid.

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Find the Cartesian equation of the plane that is perpendicular to the plane 2x - y + z = 8 and contains the points A(4,2,-3) and B(6,1,-1).

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\(\ \ \ x + \ \ y +\ \ z =8\\ \ \ ax − \ y \ \ \quad \ \ \ =3\\ −x+3y+4z=b\)

1. There is no unique solution solution to the system of equations. Find the value of a.
2. Given that the system can be solved, find the value of b.

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The three planes \({ \Pi }_{ 1 }\) , \({ \Pi }_{ 2 }\) and \({ \Pi }_{ 3 }\) meet at a line

\({ \Pi }_{ 1 }:\quad 2x+\ y+3z=a\\ { \Pi }_{ 2 }:\quad \ x−2y+2z=−9 \\ { \Pi }_{ 3 }:\quad 3x+4y+4z=−1\)

1. Find a
2. Find the equation of the straight line in the form \(\textbf {r}=\textbf {a}+\lambda\textbf {b}\) where the components of b are integers

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Find the value of k which makes the following system of equations inconsistent:

\(x +2y+kz=−1\\ 2x+ \ y− \ z=3\\ kx−2y+ z=1\)

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Consider the plane \(x−2y+4z=−15\) and the line

\(x=3+kλ\\ y=−2+λ\\ z=(2k+6)−2λ\)

The line and the plane are perpendicular. Find

1. The value of k
2. The coordinates of the point of intersection of the line and the plane.

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\(Π_{ 1 }\)and \(Π_{ 2 }\) are planes such that

\(Π_{ 1 }:2x−y−2z=0\)

and

\(Π_{ 2 }:−2x+3y+3z=4 \)

L is the intersection of planes \(Π_{ 1 }\) and \(Π_{ 2 }\)

1. Find the equation of the line

A third plane \(Π_{ 3 }\) is defined by the equation \(kx+(k−1)y−z=5\)

2. Find the value of k such that the line L does not intersect with \(Π_{ 3 }\)

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The point A (3, 1, –2) is on the line L, which is perpendicular to the plane \(2x−3y−z+9=0\).

1. Find the Cartesian equation of the line L.
2. Find the point R which is the intersection of the line L and the plane.
3. The point A is reflected in the plane. Find the coordinates of the image of A.

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