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Name of student | Date | ||||

Adm. number | Year/grade | Stream | |||

Subject | Pure Mathematics 3 (P3) | Variant(s) | P31, P33 | ||

Start time | Duration | Stop time |

Qtn No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Marks | 3 | 6 | 5 | 5 | 6 | 8 | 8 | 9 | 10 | 10 | 10 | 13 | 93 |

Score |

Question 1 Code: 9709/33/O/N/10/1, Topic: Algebra

Expand $(1+2 x)^{-3}$ in ascending powers of $x$, up to and including the term in $x^{2}$, simplifying the coefficients. $[3]$

Question 2 Code: 9709/31/O/N/11/3, Topic: Algebra

The polynomial $x^{4}+3 x^{3}+a x+3$ is denoted by $\mathrm{p}(x)$. It is given that $\mathrm{p}(x)$ is divisible by $x^{2}-x+1$.

$\text{(i)}$ Find the value of $a$. $[4]$

$\text{(ii)}$ When $a$ has this value, find the real roots of the equation $\mathrm{p}(x)=0$. $[2]$

Question 3 Code: 9709/31/O/N/20/3, Topic: Differentiation

The parametric equations of a curve are

$$ x=3-\cos 2 \theta, \quad y=2 \theta+\sin 2 \theta $$for $0 < \theta < \frac{1}{2} \pi$.

Show that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\cot \theta$. $[5]$

Question 4 Code: 9709/33/O/N/13/4, Topic: Differentiation

A curve has equation $3 \mathrm{e}^{2 x} y+\mathrm{e}^{x} y^{3}=14$. Find the gradient of the curve at the point $(0,2)$. $[5]$

Question 5 Code: 9709/31/O/N/14/4, Topic: Differentiation

The parametric equations of a curve are

$$ x=\frac{1}{\cos ^{3} t}, \quad y=\tan ^{3} t, $$where $0 \leqslant t < \frac{1}{2} \pi$.

$\text{(i)}$ Show that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\sin t$. $[4]$

$\text{(ii)}$ Hence show that the equation of the tangent to the curve at the point with parameter $t$ is $y=x \sin t-\tan t$. $[3]$

Question 6 Code: 9709/33/O/N/14/5, Topic: Complex numbers

The complex numbers $w$ and $z$ are defined by $w=5+3 \mathrm{i}$ and $z=4+\mathrm{i}$.

$\text{(i)}$ Express $\displaystyle\frac{\mathrm{i} w}{z}$ in the form $x+\mathrm{i} y$, showing all your working and giving the exact values of $x$ and $y$. $[3]$

$\text{(ii)}$ Find $w z$ and hence, by considering arguments, show that $[4]$

$$ \tan ^{-1}\left(\frac{3}{5}\right)+\tan ^{-1}\left(\frac{1}{4}\right)=\frac{1}{4} \pi. $$Question 7 Code: 9709/33/O/N/15/5, Topic: Integration

Use the substitution $u=4-3 \cos x$ to find the exact value of $\displaystyle\int_{0}^{\frac{1}{2} \pi} \frac{9 \sin 2 x}{\sqrt{(} 4-3 \cos x)} \mathrm{d} x$. $[8]$

Question 8 Code: 9709/33/O/N/19/7, Topic: Vectors

The plane $m$ has equation $x+4 y-8 z=2$. The plane $n$ is parallel to $m$ and passes through the point $P$ with coordinates $(5,2,-2)$.

$\text{(i)}$ Find the equation of $n$, giving your answer in the form $a x+b y+c z=d$. $[2]$

$\text{(ii)}$ Calculate the perpendicular distance between $m$ and $n$. $[3]$

$\text{(iii)}$ The line $l$ lies in the plane $n$, passes through the point $P$ and is perpendicular to $O P$, where $O$ is the origin. Find a vector equation for $l$. $[4]$

Question 9 Code: 9709/33/O/N/12/8, Topic: Vectors

Two lines have equations

$$ \mathbf{r}=\left(\begin{array}{r} 5 \\ 1 \\ -4 \end{array}\right)+s\left(\begin{array}{r} 1 \\ -1 \\ 3 \end{array}\right) \quad \text { and } \quad \mathbf{r}=\left(\begin{array}{r} p \\ 4 \\ -2 \end{array}\right)+t\left(\begin{array}{r} 2 \\ 5 \\ -4 \end{array}\right) $$where $p$ is a constant. It is given that the lines intersect.

$\text{(i)}$ Find the value of $p$ and determine the coordinates of the point of intersection. $[5]$

$\text{(ii)}$ Find the equation of the plane containing the two lines, giving your answer in the form $a x+b y+c z=d$, where $a, b, c$ and $d$ are integers. $[5]$

Question 10 Code: 9709/31/O/N/18/9, Topic: Algebra, Integration

Let $\displaystyle\mathrm{f}(x)=\frac{6 x^{2}+8 x+9}{(2-x)(3+2 x)^{2}}$

$\text{(i)}$ Express $\mathrm{f}(x)$ in partial fractions. $[5]$

$\text{(ii)}$ Hence, showing all necessary working, show that $\displaystyle\int_{-1}^{0} \mathrm{f}(x) \, \mathrm{d} x=1+\frac{1}{2} \ln \left(\frac{3}{4}\right)$. $[5]$

Question 11 Code: 9709/33/O/N/20/9, Topic: Algebra

Let $\displaystyle\mathrm{f}(x)=\frac{8+5 x+12 x^{2}}{(1-x)(2+3 x)^{2}}$

$\text{(a)}$ Express $\mathrm{f}(x)$ in partial fractions. $[5]$

$\text{(b)}$ Hence obtain the expansion of $\mathrm{f}(x)$ in ascending powers of $x$, up to and including the term in $x^{2}$. $[5]$

Question 12 Code: 9709/33/O/N/10/10, Topic: Vectors

The polynomial $\mathrm{p}(z)$ is defined by

$$ \mathrm{p}(z)=z^{3}+m z^{2}+24 z+32 $$where $m$ is a constant. It is given that $(z+2)$ is a factor of $\mathrm{p}(z)$.

$\text{(i)}$ Find the value of $m$. $[2]$

$\text{(ii)}$ Hence, showing all your working, find

$\text{(a)}$ the three roots of the equation $\mathrm{p}(z)=0$, $[5]$

$\text{(b)}$ the six roots of the equation $\mathrm{p}\left(z^{2}\right)=0$. $[6]$