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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 9 11 10 9 10 9 58
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 6 questions Question 1 Code: 9709/31/M/J/17/9, Topic: Differential equations$\text{(i)}$Express$\displaystyle\frac{1}{x(2 x+3)}$in partial fractions.$[2]\text{(ii)}$The variables$x$and$y$satisfy the differential equation $$x(2 x+3) \displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=y$$ and it is given that$y=1$when$x=1$. Solve the differential equation and calculate the value of$y$when$x=9$, giving your answer correct to 3 significant figures.$[7]$Question 2 Code: 9709/32/M/J/17/9, Topic: Vectors Relative to the origin$O$, the point$A$has position vector given by$\overrightarrow{O A}=\mathbf{i}+2 \mathbf{j}+4 \mathbf{k}$. The line$l$has equation$\mathbf{r}=9 \mathbf{i}-\mathbf{j}+8 \mathbf{k}+\mu(3 \mathbf{i}-\mathbf{j}+2 \mathbf{k})$.$\text{(i)}$Find the position vector of the foot of the perpendicular from$A$to$l$. Hence find the position vector of the reflection of$A$in$l$.$[5]\text{(ii)}$Find the equation of the plane through the origin which contains$l$. Give your answer in the form$a x+b y+c z=d$.$[3]\text{(iii)}$Find the exact value of the perpendicular distance of$A$from this plane.$[3]$Question 3 Code: 9709/33/M/J/17/9, Topic: Algebra, Integration Let$\displaystyle\mathrm{f}(x)=\frac{3 x^{2}-4}{x^{2}(3 x+2)}\text{(i)}$Express$\mathrm{f}(x)$in partial fractions.$[5]\text{(ii)}$Hence show that$\displaystyle\int_{1}^{2} \mathrm{f}(x) \mathrm{d} x=\ln \left(\frac{25}{8}\right)-1$.$[5]$Question 4 Code: 9709/31/O/N/17/9, Topic: Differentiation The diagram shows the curve$y=\left(1+x^{2}\right) \mathrm{e}^{-\frac{1}{2} x}$for$x \geqslant 0$. The shaded region$R$is enclosed by the curve, the$x$-axis and the lines$x=0$and$x=2$.$\text{(i)}$Find the exact values of the$x$-coordinates of the stationary points of the curve.$[4]\text{(ii)}$Show that the exact value of the area of$R$is$\displaystyle 18-\frac{42}{\mathrm{e}}$.$[5]$Question 5 Code: 9709/32/O/N/17/9, Topic: Integration, Numerical solutions of equations It is given that$\displaystyle\int_{1}^{a} x^{\frac{1}{2}} \ln x \mathrm{~d} x=2$, where$a>1.\text{(i)}$Show that$\displaystyle a^{\frac{3}{2}}=\frac{7+2 a^{\frac{3}{2}}}{3 \ln a}$.$[5]\text{(ii)}$Show by calculation that$a$lies between 2 and$4.[2]\text{(iii)}$Use the iterative formula $$a_{n+1}=\left(\frac{7+2 a_{n}^{\frac{3}{2}}}{3 \ln a_{n}}\right)^{\frac{2}{3}}$$ to determine$a$correct to 3 decimal places. Give the result of each iteration to 5 decimal places.$[3]$Question 6 Code: 9709/33/O/N/17/9, Topic: Differentiation The diagram shows the curve$y=\left(1+x^{2}\right) \mathrm{e}^{-\frac{1}{2} x}$for$x \geqslant 0$. The shaded region$R$is enclosed by the curve, the$x$-axis and the lines$x=0$and$x=2$.$\text{(i)}$Find the exact values of the$x$-coordinates of the stationary points of the curve.$[4]\text{(ii)}$Show that the exact value of the area of$R$is$\displaystyle 18-\frac{42}{\mathrm{e}}$.$[5]\$

Worked solutions: P1, P3 & P6 (S1)

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