Further Pure 1: Further trigonometryEdexcel A-Level Further Maths: Topic test

1. 1
   The angle $\phi$ satisfies $\tan\frac{\phi}{2}=\frac34$.

   (a)

   Find the value of $\sin\phi$.

   [1 mark]

   • A$\frac34$
   • B$\frac35$
   • C$\frac{24}{25}$
   • D$\frac{7}{25}$

   (b)

   Find the value of $\cos\phi$.

   [1 mark]

   • A$\frac{7}{25}$
   • B$-\frac{7}{25}$
   • C$\frac45$
   • D$\frac{24}{25}$

   (c)

   Find the exact value of $\operatorname{cosec}\phi+\cot\phi$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A student solves $4\sin x-3\cos x=2$ for $0\le x<2\pi$ using the substitution $t=\tan\frac{x}{2}$.

   (a)

   Which quadratic equation in $t$ does the student obtain?

   [1 mark]

   • A$t^2+4t-5=0$
   • B$t^2+8t-5=0$
   • C$5t^2-8t-1=0$
   • D$t^2-8t-5=0$

   (b)

   Why must the student check $x=\pi$ separately?

   [1 mark]

   • ABecause $\sin\pi=0$ makes the equation undefined
   • BBecause $\cos\pi=-1$ gives a quadratic with no real roots
   • CBecause $x=\pi$ gives $t=0$, which cannot be divided by
   • DBecause $\tan\frac{\pi}{2}$ is undefined, so the substitution cannot find $x=\pi$ even if it is a solution

   (c)

   Solve the equation for $0\le x<2\pi$, giving your answers to 2 decimal places.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The definitions $\sec\theta=\frac{1}{\cos\theta}$ and $\tan\theta=\frac{\sin\theta}{\cos\theta}$ are used, with $t=\tan\frac{\theta}{2}$.

   (a)

   Show that $\sec\theta+\tan\theta=\frac{1+t}{1-t}$.

   [3 marks]

   (b)

   Hence solve $\sec\theta+\tan\theta=3$ for $0<\theta<\frac{\pi}{2}$, finding the exact value of $\tan\theta$ and the value of $\theta$ in radians to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In this question you must use the substitution $t=\tan\frac{x}{2}$, where $0\le x<2\pi$.

   (a)

   Solve $\sin x+7\cos x=5$, giving your answers to 3 significant figures.

   [6 marks]

   (b)

   Show that $5\cos x+12\sin x=k$, where $k$ is a constant, can be written as $(k+5)t^2-24t+(k-5)=0$, and hence find the set of values of $k$ for which the equation has a solution in the given interval.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The angle $\alpha$ is obtuse and $\sin\alpha=\frac45$.

   (a)

   Let $t=\tan\frac{\alpha}{2}$. Which equation does $t$ satisfy?

   [1 mark]

   • A$2t^2-5t+2=0$
   • B$4t^2-5t+4=0$
   • C$2t^2+5t+2=0$
   • D$2t^2-5t-2=0$

   (b)

   The equation for $t$ has roots $t=2$ and $t=\frac12$. Why is $t=\frac12$ rejected?

   [1 mark]

   • ABecause $\tan\frac{\alpha}{2}$ must be negative when $\alpha$ is obtuse
   • BBecause $t=\frac12$ does not satisfy $\sin\alpha=\frac45$
   • CBecause $\frac{\alpha}{2}$ lies between $45^\circ$ and $90^\circ$, so $\tan\frac{\alpha}{2}>1$
   • DBecause $\frac{\alpha}{2}$ must be greater than $90^\circ$

   (c)

   Find the exact value of $\tan\alpha$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Let $\mathrm{f}(\theta)=\operatorname{cosec}\theta-\cot\theta$ for $0<\theta<\pi$.

   (a)

   Which expression is equal to $\mathrm{f}(\theta)$, where $t=\tan\frac{\theta}{2}$?

   [1 mark]

   • A$\frac1t$
   • B$\frac{1+t^2}{t}$
   • C$\frac{2t}{1+t^2}$
   • D$t$

   (b)

   Which value of $\theta$ satisfies $\mathrm{f}(\theta)=\sqrt3$?

   [1 mark]

   • A$\frac{\pi}{3}$
   • B$\frac{2\pi}{3}$
   • C$\frac{4\pi}{3}$
   • D$\frac{\pi}{6}$

   (c)

   Find the range of $\mathrm{f}$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Let $t=\tan\frac{\theta}{2}$, where $\theta$ is such that the expressions below are defined.

   (a)

   Show that $1+\sin\theta+\cos\theta=\frac{2(1+t)}{1+t^2}$.

   [3 marks]

   (b)

   Show that $\dfrac{1+\sin\theta+\cos\theta}{1+\sin\theta-\cos\theta}=\cot\frac{\theta}{2}$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   This question is about the $t$-formulae, where $t=\tan\frac{x}{2}$.

   (a)

   Solve $2\sin x+\cos x+1=0$ for $0\le x<2\pi$, giving non-exact answers to 3 significant figures.

   [6 marks]

   (b)

   Prove that $\sin x=\frac{2t}{1+t^2}$ and $\cos x=\frac{1-t^2}{1+t^2}$.

   [6 marks]

   Total for question 8: 12 marks

End of questions