$\require{\cancel}$ $\require{\stix[upint]}$

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 |

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 $\displaystyle 18-\frac{42}{\mathrm{e}}$. $[5]$