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### HENRYTAIGO

#### Cambridge International AS and A Level

 Name of student HENRYTAIGO Date Adm. number Year/grade HenryTaigo Stream HenryTaigo Subject Pure Mathematics 3 (P3) Variant(s) P31, P32, P33 Start time Duration Stop time

Qtn No. 1 2 3 4 Total
Marks 5 7 5 5 22
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 4 questions Question 1 Code: 9709/32/M/J/19/3, Topic: Trigonometry Showing all necessary working, solve the equation$\cot 2 \theta=2 \tan \theta$for$0^{\circ} < \theta < 180^{\circ}$.$[5]$Question 2 Code: 9709/33/M/J/19/3, Topic: Trigonometry, Integration Let$\displaystyle\mathrm{f}(\theta)=\frac{1-\cos 2 \theta+\sin 2 \theta}{1+\cos 2 \theta+\sin 2 \theta}\text{(i)}$Show that$\mathrm{f}(\theta)=\tan \theta$.$[3]\text{(ii)}$Hence show that$\displaystyle\int_{\frac{1}{6} \pi}^{\frac{1}{4} \pi} \mathrm{f}(\theta) \, \mathrm{d} \theta=\frac{1}{2} \ln \frac{3}{2}$.$[4]$Question 3 Code: 9709/31/O/N/19/3, Topic: Differentiation The parametric equations of a curve are $$x=2 t+\sin 2 t, \quad y=\ln (1-\cos 2 t).$$ Show that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\operatorname{cosec} 2 t$.$[5]$Question 4 Code: 9709/32/O/N/19/3, Topic: Algebra The polynomial$x^{4}+3 x^{3}+a x+b$, where$a$and$b$are constants, is denoted by$\mathrm{p}(x).$When$\mathrm{p}(x)$is divided by$x^{2}+x-1$the remainder is$2 x+3$. Find the values of$a$and$b$.$[5]\$

Worked solutions: P1, P3 & P6 (S1)

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