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

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 | 4 | 5 | 5 | 5 | 6 | 8 | 8 | 11 | 12 | 15 | 10 | 94 |

Score |

Question 1 Code: 9709/11/M/J/19/1, Topic: Series

The term independent of $x$ in the expansion of $\displaystyle \left(2 x+\frac{k}{x}\right)^{6}$, where $k$ is a constant, is 540.

$\text{(i)}$ Find the value of $k$. $[3]$

$\text{(ii)}$ For this value of $k$, find the coefficient of $x^{2}$ in the expansion. $[2]$

Question 2 Code: 9709/13/M/J/16/2, Topic: Integration

The diagram shows part of the curve $\displaystyle y=\left(x^{3}+1\right)^{\frac{1}{2}}$ and the point $P(2,3)$ lying on the curve. Find, showing all necessary working, the volume obtained when the shaded region is rotated through $360^{\circ}$ about the $x$-axis. $[4]$

Question 3 Code: 9709/12/M/J/18/3, Topic: Series

A company producing salt from sea water changed to a new process. The amount of salt obtained each week increased by $2 \%$ of the amount obtained in the preceding week. It is given that in the first week after the change the company obtained $8000 \mathrm{~kg}$ of salt.

$\text{(i)}$ Find the amount of salt obtained in the 12 th week after the change. $[3]$

$\text{(ii)}$ Find the total amount of salt obtained in the first 12 weeks after the change. $[2]$

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: 9709/11/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 6 Code: 9709/13/M/J/17/4, Topic: Vectors

Relative to an origin $O$, the position vectors of points $A$ and $B$ are given by

$$ \overrightarrow{O A}=\left(\begin{array}{l} 5 \\ 1 \\ 3 \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{r} 5 \\ 4 \\ -3 \end{array}\right) $$The point $P$ lies on $A B$ and is such that $\overrightarrow{A P}=\frac{1}{3} \overrightarrow{A B}$.

$\text{(i)}$ Find the position vector of $P$. $[3]$

$\text{(ii)}$ Find the distance $O P$. $[1]$

$\text{(iii)}$ Determine whether $O P$ is perpendicular to $A B$. Justify your answer. $[2]$

Question 7 Code: 9709/13/M/J/18/8, Topic: Differentiation

$\text{(i)}$ The tangent to the curve $y=x^{3}-9 x^{2}+24 x-12$ at a point $A$ is parallel to the line $y=2-3 x$. Find the equation of the tangent at $A$. $[6]$

$\text{(ii)}$ The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=x^{3}-9 x^{2}+24 x-12$ for $x>k$, where $k$ is a constant. Find the smallest value of $k$ for $\mathrm{f}$ to be an increasing function. $[2]$

Question 8 Code: 9709/13/M/J/19/8, Topic: Integration

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=3 x^{2}+a x+b$. The curve has stationary points at $(-1,2)$ and $(3, k)$. Find the values of the constants $a, b$ and $k$. $[8]$

Question 9 Code: 9709/12/M/J/14/9, Topic: Differentiation, Integration

The diagram shows part of the curve $y=8-\sqrt{(} 4-x)$ and the tangent to the curve at $P(3,7)$.

$\text{(i)}$ Find expressions for $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$ and $\displaystyle\int y \mathrm{~d} x$. $[5]$

$\text{(ii)}$ Find the equation of the tangent to the curve at $P$ in the form $y=m x+c$. $[2]$

$\text{(iii)}$ Find, showing all necessary working, the area of the shaded region. $[4]$

Question 10 Code: 9709/12/M/J/12/10, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$$ \begin{aligned} &\mathrm{f}: x \mapsto 2 x+5 \quad \text { for } x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto \frac{8}{x-3} \quad \text { for } x \in \mathbb{R}, x \neq 3 \end{aligned} $$$\text{(i)}$ Obtain expressions, in terms of $x$, for $\mathrm{f}^{-1}(x)$ and $\mathrm{g}^{-1}(x)$, stating the value of $x$ for which $\mathrm{g}^{-1}(x)$ is not defined. $[4]$

$\text{(ii)}$ Sketch the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$ on the same diagram, making clear the relationship between the two graphs. $[3]$

$\text{(iii)}$ Given that the equation $f g(x)=5-k x$, where $k$ is a constant, has no solutions, find the set of possible values of $k$. $[5]$

Question 11 Code: 9709/12/M/J/14/10, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$$ \begin{aligned} &\mathrm{f}: x \mapsto 2 x-3, \quad x \in \mathbb{R}, \\ &\mathrm{g}: x \mapsto x^{2}+4 x, \quad x \in \mathbb{R}. \end{aligned} $$$\text{(i)}$ Solve the equation $\mathrm{ff}(x)=11$. $[2]$

$\text{(ii)}$ Find the range of $\mathrm{g}$. $[2]$

$\text{(iii)}$ Find the set of values of $x$ for which $\mathrm{g}(x)>12$. $[3]$

$\text{(iv)}$ Find the value of the constant $p$ for which the equation $\operatorname{gf}(x)=p$ has two equal roots. $[3]$

Function $\mathrm{h}$ is defined by $\mathrm{h}: x \mapsto x^{2}+4 x$ for $x \geqslant k$, and it is given that $\mathrm{h}$ has an inverse.

$\text{(v)}$ State the smallest possible value of $k$. $[1]$

$\text{(vi)}$ Find an expression for $\mathrm{h}^{-1}(x)$. $[4]$

Question 12 Code: 9709/13/M/J/12/11, Topic: Functions

The function $\mathrm{f}$ is such that $\mathrm{f}(x)=8-(x-2)^{2}$, for $x \in \mathbb{R}$.

$\text{(i)}$ Find the coordinates and the nature of the stationary point on the curve $y=\mathrm{f}(x)$. $[3]$

The function $\mathrm{g}$ is such that $\mathrm{g}(x)=8-(x-2)^{2}$, for $k \leqslant x \leqslant 4$, where $k$ is a constant.

$\text{(ii)}$ State the smallest value of $k$ for which $\mathrm{g}$ has an inverse. $[1]$

For this value of $k$,

$\text{(iii)}$ find an expression for $\mathrm{g}^{-1}(x)$, $[3]$

$\text{(iv)}$ sketch, on the same diagram, the graphs of $y=\mathrm{g}(x)$ and $y=\mathrm{g}^{-1}(x)$. $[3]$