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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 5 6 Total
Marks 7 6 8 7 7 8 43
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 6 questions Question 1 Code: 9709/31/M/J/13/5, Topic: Differentiation For each of the following curves, find the gradient at the point where the curve crosses the$y$-axis:$\text{(i)}\displaystyle y=\frac{1+x^{2}}{1+\mathrm{e}^{2 x}}$;$\text{(ii)}2 x^{3}+5 x y+y^{3}=8$.$$Question 2 Code: 9709/32/M/J/13/5, Topic: Differentiation The diagram shows the curve with equation $$x^{3}+x y^{2}+a y^{2}-3 a x^{2}=0,$$ where$a$is a positive constant. The maximum point on the curve is$M$. Find the$x$-coordinate of$M$in terms of$a$.$$Question 3 Code: 9709/33/M/J/13/5, Topic: Algebra The polynomial$8 x^{3}+a x^{2}+b x+3$, where$a$and$b$are constants, is denoted by$\mathrm{p}(x)$. It is given that$(2 x+1)$is a factor of$\mathrm{p}(x)$and that when$\mathrm{p}(x)$is divided by$(2 x-1)$the remainder is 1.$\text{(i)}$Find the values of$a$and$b$.$\text{(ii)}$When$a$and$b$have these values, find the remainder when$\mathrm{p}(x)$is divided by$2 x^{2}-1$.$$Question 4 Code: 9709/31/O/N/13/5, Topic: Trigonometry, Integration$\text{(i)}$Prove that$\cot \theta+\tan \theta \equiv 2 \operatorname{cosec} 2 \theta$.$\text{(ii)}$Hence show that$\displaystyle\int_{\frac{1}{6} \pi}^{\frac{1}{3} \pi} \operatorname{cosec} 2 \theta \mathrm{d} \theta=\frac{1}{2} \ln 3.$Question 5 Code: 9709/32/O/N/13/5, Topic: Trigonometry, Integration Question 6 Code: 9709/33/O/N/13/5, Topic: Integration, Numerical solutions of equations It is given that$\displaystyle\int_{0}^{p} 4 x \mathrm{e}^{-\frac{1}{2} x} \mathrm{~d} x=9$, where$p$is a positive constant.$\text{(i)}$Show that$\displaystyle p=2 \ln \left(\frac{8 p+16}{7}\right)$.$\text{(ii)}$Use an iterative process based on the equation in part$\text{(i)}$to find the value of$p$correct to 3 significant figures. Use a starting value of$3.5$and give the result of each iteration to 5 significant figures.$\$

Worked solutions: P1, P3 & P6 (S1)

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