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

Name of student | Date | ||||

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

Subject | Pure Mathematics 1 (P1) | Variant(s) | P11, P12, P13 | ||

Start time | Duration | Stop time |

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

Marks | 5 | 5 | 5 | 5 | 5 | 5 | 7 | 8 | 10 | 8 | 8 | 9 | 80 |

Score |

Question 1 Code: 9709/13/M/J/11/1, Topic: Trigonometry

The coefficient of $x^{3}$ in the expansion of $(a+x)^{5}+(1-2 x)^{6}$, where $a$ is positive, is 90. Find the value of $a$. $[5]$

Question 2 Code: 9709/12/M/J/18/1, Topic: Series

The coefficient of $x^{2}$ in the expansion of $\displaystyle \left(2+\frac{x}{2}\right)^{6}+(a+x)^{5}$ is 330 . Find the value of the constant $a$. $[5]$

Question 3 Code: 9709/11/M/J/12/3, Topic: Circular measure

In the diagram, $A B C$ is an equilateral triangle of side $2 \mathrm{~cm}$. The mid-point of $B C$ is $Q.$ An arc of a circle with centre $A$ touches $B C$ at $Q$, and meets $A B$ at $P$ and $A C$ at $R.$ Find the total area of the shaded regions, giving your answer in terms of $\pi$ and $\sqrt{3}$. $[5]$

Question 4 Code: 9709/13/M/J/18/3, Topic: Series

The common ratio of a geometric progression is $0.99$. Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures. $[5]$

Question 5 Code: 9709/12/M/J/19/3, Topic: Differentiation, Integration

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{3}-\frac{4}{x^{2}}.$ The point $P(2,9)$ lies on the curve.

$\text{(i)}$ A point moves on the curve in such a way that the $x$-coordinate is decreasing at a constant rate of $0.05$ units per second. Find the rate of change of the $y$-coordinate when the point is at $P$. $[2]$

$\text{(ii)}$ Find the equation of the curve. $[3]$

Question 6 Code: 9709/11/M/J/20/3, Topic: Series

Each year the selling price of a diamond necklace increases by $5 \%$ of the price the year before. The selling price of the necklace in the year 2000 was $\$ 36000$.

$\text{(a)}$ Write down an expression for the selling price of the necklace $n$ years later and hence find the selling price in 2008. $[3]$

$\text{(b)}$ The company that makes the necklace only sells one each year. Find the total amount of money obtained in the ten-year period starting in the year 2000. $[2]$

Question 7 Code: 9709/13/M/J/15/7, Topic: Coordinate geometry

The point $A$ has coordinates $(p, 1)$ and the point $B$ has coordinates $(9,3 p+1)$, where $p$ is a constant.

$\text{(i)}$ For the case where the distance $A B$ is 13 units, find the possible values of $p$. $[3]$

$\text{(ii)}$ For the case in which the line with equation $2 x+3 y=9$ is perpendicular to $A B$, find the value of $p$. $[4]$

Question 8 Code: 9709/12/M/J/10/9, Topic: Integration

The diagram shows the curve $y=(x-2)^{2}$ and the line $y+2 x=7$, which intersect at points $A$ and $B$. Find the area of the shaded region. $[8]$

Question 9 Code: 9709/11/M/J/17/9, Topic: Functions

The function $\mathrm{f}$ is defined by $\displaystyle\mathrm{f}: x \mapsto \frac{2}{3-2 x}$ for $x \in \mathbb{R}, x \neq \frac{3}{2}$.

$\text{(i)}$ Find an expression for $\mathrm{f}^{-1}(x)$. $[3]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 4 x+a$ for $x \in \mathbb{R}$, where $a$ is a constant.

$\text{(ii)}$ Find the value of $a$ for which $\operatorname{gf}(-1)=3$. $[3]$

$\text{(iii)}$ Find the possible values of $a$ given that the equation $\mathrm{f}^{-1}(x)=\mathrm{g}^{-1}(x)$ has two equal roots. $[4]$

Question 10 Code: 9709/12/M/J/19/9, Topic: Coordinate geometry

The curve $C_{1}$ has equation $y=x^{2}-4 x+7$. The curve $C_{2}$ has equation $y^{2}=4 x+k$, where $k$ is a constant. The tangent to $C_{1}$ at the point where $x=3$ is also the tangent to $C_{2}$ at the point $P$. Find the value of $k$ and the coordinates of $P$. $[8]$

Question 11 Code: 9709/12/M/J/20/9, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are such that

$$ \begin{aligned} &\mathrm{f}(x)=2-3 \sin 2 x \text { for } 0 \leqslant x \leqslant \pi, \\ &\mathrm{g}(x)=-2 \mathrm{f}(x) \quad \text { for } 0 \leqslant x \leqslant \pi \end{aligned} $$$\text{(a)}$ State the ranges of $\mathrm{f}$ and $\mathrm{g}$. $[3]$

The diagram below shows the graph of $y=\mathrm{f}(x)$.

$\text{(b)}$ Sketch, on this diagram, the graph of $y=\mathrm{g}(x)$. $[2]$

The function $\mathrm{h}$ is such that

$$ \mathrm{h}(x)=\mathrm{g}(x+\pi) \quad \text { for }-\pi \leqslant x \leqslant 0 $$$\text{(c)}$ Describe fully a sequence of transformations that maps the curve $y=\mathrm{f}(x)$ on to $y=\mathrm{h}(x)$. $[3]$

Question 12 Code: 9709/13/M/J/12/10, Topic: Coordinate geometry, Quadratics

The equation of a line is $2 y+x=k$, where $k$ is a constant, and the equation of a curve is $x y=6$.

$\text{(i)}$ In the case where $k=8$, the line intersects the curve at the points $A$ and $B$. Find the equation of the perpendicular bisector of the line $A B$. $[6]$

$\text{(ii)}$ Find the set of values of $k$ for which the line $2 y+x=k$ intersects the curve $x y=6$ at two distinct points. $[3]$