Level 2 Further Maths - Post some hard questions (Includes unofficial practice paper)

hOW DO YOU DO THIS, PLS DO IT STEP BY STEP I GET TO (1+SIN0 + COS0)/(1 + COSSIN + SIN + COS) = 1
BTW YOU ARE MEANT TO ONLY WORK FROM ONE SIDE... NO MEETING IN THE MIDDLE APPROACHES PLS BECAUSE THAT'S IMPROPER FOR REASONS I DON'T REALLY UNDERSTAND, JUST IN CASE.
I SEE WHAT THE TRICKY PART IS BUT I DONT GET THE ANSWER.
ANYONE ASAP PLS

Here's my first one. There's a very tricky part in this question that you need to watch out for (most A Level students would miss it) but apart from that it's good practice for Level 2 FM.


Solve the equation


$\displaystyle \frac{\sin \theta}{\cos \theta + 1} + \frac{\cos \theta}{\sin \theta + 1} = 1$


for $0^o \leq \theta \leq 360^o$.

Is there a more elegant reason why 180 and 270 don't work from sin x . cos x = 0, besides the fact that when you happen to sub them back into the original equation the numbers don't add up?

How come they're a solution to sin x . cos x = 0 but not to the original if the former came from the latter?

Because you multiplied by (cos + 1)(sin + 1) you are introducing extra solutions which are not solutions to the original equation.

Here are some of the general things to show up each year:

Quadratic Nth Term
Simultaneous Equations Using Matrices
Representing Ratios (2 markers)
Algebraic Fractions
At least 1 or 2 3D Trig
Representing a Certain Siny or cos or tan value in terms of another sin or cos or tan value. (i:e if siny = 5/3 whats cosy)
Trigonometry Identities
Solving Trig Identities Questions

This is the hardest one I've posted so far and requires a very good undrstanding of transformation matrices. Don't worry if you can't do this one.


The transformation matrix $\displaystyle \mathbf{P}^3$ represents a clockwise rotation of $150^o$ about the origin.

$\displaystyle \mathbf{Q} = \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}$

The unit vector $\displaystyle \begin{pmatrix}0 \\ 1\end{pmatrix}$ is transformed by the matrix $\displaystyle \mathbf{Q}\mathbf{P}$ to give the vector $\displaystyle \begin{pmatrix}-\cos \theta \\ \cos \alpha \end{pmatrix}$.

Find the value of $\theta$ and $\alpha$.

YOUR a donut bruv

$675x^4-2505x^3+2407x^2-63x-162\equiv (ax-b)(cx+d)(ex-c)^2$

$a, b, c, d$ and $e$ are different positive integers. Find their values.

Does this include trial and error? I seem to be using a lot of trial and error/ logic and it makes me feel like im missing something. (I haven't expanded the brackets because I assume that is missing the point of the question)

Does this include trial and error? I seem to be using a lot of trial and error/ logic and it makes me feel like im missing something. (I haven't expanded the brackets because I assume that is missing the point of the question)

6 years later so i dont expect a reply..... by my calculator doesnt say error when i put 180,270 in. How will i know to exclude these in the exam?