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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 12 11 11 11 11 10 66
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 6 questions Question 1 Code: 9709/31/M/J/14/10, Topic: Differentiation The diagram shows the curve$y=10 \mathrm{e}^{-\frac{1}{2} x} \sin 4 x$for$x \geqslant 0$. The stationary points are labelled$T_{1}, T_{2}$,$T_{3}, \ldots$as shown.$\text{(i)}$Find the$x$-coordinates of$T_{1}$and$T_{2}$, giving each$x$-coordinate correct to 3 decimal places.$[6]\text{(ii)}$It is given that the$x$-coordinate of$T_{n}$is greater than 25. Find the least possible value of$n$.$[4]$Question 2 Code: 9709/32/M/J/14/10, Topic: Vectors Referred to the origin$O$, the points$A, B$and$C$have position vectors given by $$\overrightarrow{O A}=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}, \quad \overrightarrow{O B}=2 \mathbf{i}+4 \mathbf{j}+\mathbf{k} \quad \text { and } \quad \overrightarrow{O C}=3 \mathbf{i}+5 \mathbf{j}-3 \mathbf{k}$$$\text{(i)}$Find the exact value of the cosine of angle$B A C$.$[4]\text{(ii)}$Hence find the exact value of the area of triangle$A B C$.$[3]\text{(iii)}$Find the equation of the plane which is parallel to the$y$-axis and contains the line through$B$and$C$. Give your answer in the form$a x+b y+c z=d$.$[5]$Question 3 Code: 9709/33/M/J/14/10, Topic: Vectors The line$l$has equation$\mathbf{r}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}+\lambda(3 \mathbf{i}-2 \mathbf{j}+2 \mathbf{k})$and the plane$p$has equation$2 x+3 y-5 z=18\text{(i)}$Find the position vector of the point of intersection of$l$and$p$.$[3]\text{(ii)}$Find the acute angle between$l$and$p$.$[4]\text{(iii)}$A second plane$q$is perpendicular to the plane$p$and contains the line$l$. Find the equation of$q$, giving your answer in the form$a x+b y+c z=d$.$[5]$Question 4 Code: 9709/31/O/N/14/10, Topic: Vectors The line$l$has equation$\mathbf{r}=4 \mathbf{i}-9 \mathbf{j}+9 \mathbf{k}+\lambda(-2 \mathbf{i}+\mathbf{j}-2 \mathbf{k})$. The point$A$has position vector$3 \mathbf{i}+8 \mathbf{j}+5 \mathbf{k}$.$\text{(i)}$Show that the length of the perpendicular from$A$to$l$is$15.[5]\text{(ii)}$The line$l$lies in the plane with equation$a x+b y-3 z+1=0$, where$a$and$b$are constants. Find the values of$a$and$b$.$[5]$Question 5 Code: 9709/32/O/N/14/10, Topic: Vectors Question 6 Code: 9709/33/O/N/14/10, Topic: Integration By first using the substitution$u=\mathrm{e}^{x}$, show that$[10]\$

$$\displaystyle\int_{0}^{\ln 4} \frac{\mathrm{e}^{2 x}}{\mathrm{e}^{2 x}+3 \mathrm{e}^{x}+2} \mathrm{~d} x=\ln \left(\frac{8}{5}\right)$$

Worked solutions: P1, P3 & P6 (S1)

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