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Name of student | JAMES DONALD | Date | 2022-06-03 | ||

Adm. number | A22/0051/2006 | Year/grade | 13 | Stream | Blue |

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

Start time | 09:30hrs | Duration | 1hr 45mins | Stop time | 11:15hrs |

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

Marks | 5 | 5 | 6 | 7 | 6 | 6 | 6 | 7 | 8 | 8 | 9 | 11 | 84 |

Score |

Question 1 Code: 9709/12/M/J/14/2, Topic: Series

Find the coefficient of $x^{2}$ in the expansion of $\displaystyle\left(1+x^{2}\right)\left(\frac{x}{2}-\frac{4}{x}\right)^{6}$. $[5]$

Question 2 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 3 Code: 9709/12/M/J/12/4, Topic: Coordinate geometry

The point $A$ has coordinates $(-1,-5)$ and the point $B$ has coordinates $(7,1)$. The perpendicular bisector of $A B$ meets the $x$-axis at $C$ and the $y$-axis at $D$. Calculate the length of $C D$. $[6]$

Question 4 Code: 9709/11/M/J/17/4, Topic: Series

$\text{(a)}$ An arithmetic progression has a first term of 32 , a 5 th term of 22 and a last term of $-28$. Find the sum of all the terms in the progression. $[4]$

$\text{(b)}$ Each year a school allocates a sum of money for the library. The amount allocated each year increases by $2.5 \%$ of the amount allocated the previous year. In 2005 the school allocated $\$ 2000$. Find the total amount allocated in the years 2005 to 2014 inclusive. $[3]$

Question 5 Code: 9709/12/M/J/18/4, Topic: Functions

The function $\mathrm{f}$ is such that $\mathrm{f}(x)=a+b \cos x$ for $0 \leqslant x \leqslant 2 \pi$. It is given that $\mathrm{f}\left(\frac{1}{3} \pi\right)=5$ and $\mathrm{f}(\pi)=11$.

$\text{(i)}$ Find the values of the constants $a$ and $b$. $[3]$

$\text{(ii)}$ Find the set of values of $k$ for which the equation $\mathrm{f}(x)=k$ has no solution. $[3]$

Question 6 Code: 9709/12/M/J/11/5, Topic: Trigonometry

$\text{(i)}$ Prove the identity $\displaystyle\frac{\cos \theta}{\tan \theta(1-\sin \theta)} \equiv 1+\frac{1}{\sin \theta}$. $[3]$

$\text{(ii)}$ Hence solve the equation $\displaystyle\frac{\cos \theta}{\tan \theta(1-\sin \theta)}=4$, for $0^{\circ} \leqslant \theta \leqslant 360^{\circ}$. $[3]$

Question 7 Code: 9709/11/M/J/12/5, Topic: Coordinate geometry

The diagram shows the curve $y=7 \sqrt{x}$ and the line $y=6 x+k$, where $k$ is a constant. The curve and the line intersect at the points $A$ and $B$.

$\text{(i)}$ For the case where $k=2$, find the $x$-coordinates of $A$ and $B$. $[4]$

$\text{(ii)}$ Find the value of $k$ for which $y=6 x+k$ is a tangent to the curve $y=7 \sqrt{x}$. $[2]$

Question 8 Code: 9709/12/M/J/19/6, Topic: Trigonometry

The equation of a curve is $y=3 \cos 2 x$ and the equation of a line is $\displaystyle 2 y+\frac{3 x}{\pi}=5$.

$\text{(i)}$ State the smallest and largest values of $y$ for both the curve and the line for $0 \leqslant x \leqslant 2 \pi$. $[3]$

$\text{(ii)}$ Sketch, on the same diagram, the graphs of $y=3 \cos 2 x$ and $\displaystyle 2 y+\frac{3 x}{\pi}=5$ for $0 \leqslant x \leqslant 2 \pi$. $[3]$

$\text{(iii)}$ State the number of solutions of the equation $\displaystyle 6 \cos 2 x=5-\frac{3 x}{\pi}$ for $0 \leqslant x \leqslant 2 \pi$. $[1]$

Question 9 Code: 9709/13/M/J/17/8, Topic: Coordinate geometry

$A(-1,1)$ and $P(a, b)$ are two points, where $a$ and $b$ are constants. The gradient of $A P$ is 2.

$\text{(i)}$ Find an expression for $b$ in terms of $a$. $[2]$

$\text{(ii)}$ $B(10,-1)$ is a third point such that $A P=A B$. Calculate the coordinates of the possible positions of $P$. $[6]$

Question 10 Code: 9709/13/M/J/13/9, Topic: Series

$\text{(a)}$ In an arithmetic progression, the sum, $S_{n}$, of the first $n$ terms is given by $S_{n}=2 n^{2}+8 n$. Find the first term and the common difference of the progression. $[3]$

$\text{(b)}$ The first 2 terms of a geometric progression are 64 and 48 respectively. The first 3 terms of the geometric progression are also the $1 \mathrm{st}$ term, the 9 th term and the $n$th term respectively of an arithmetic progression. Find the value of $n$. $[5]$

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

The equation of a curve is $y=\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$.

$\text{(i)}$ Find $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$. $[3]$

$\text{(ii)}$ Find the equation of the tangent to the curve at the point where the curve intersects the $y$-axis. $[3]$

$\text{(iii)}$ Find the set of values of $x$ for which $\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$ is an increasing function of $x$. $[3]$

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

$\text{(a)}$ A circle is divided into 6 sectors in such a way that the angles of the sectors are in arithmetic progression. The angle of the largest sector is 4 times the angle of the smallest sector. Given that the radius of the circle is $5 \mathrm{~cm}$, find the perimeter of the smallest sector. $[6]$

$\text{(b)}$ The first, second and third terms of a geometric progression are $2 k+3, k+6$ and $k$, respectively. Given that all the terms of the geometric progression are positive, calculate

$\text{(i)}$ the value of the constant $k$, $[3]$

$\text{(ii)}$ the sum to infinity of the progression. $[2]$