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### STEMRE INTERNATIONAL SCHOOL

#### Cambridge International AS and A Level

 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

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/14/2, Topic: Series Find the coefficient of$x^{2}$in the expansion of$\displaystyle\left(1+x^{2}\right)\left(\frac{x}{2}-\frac{4}{x}\right)^{6}$.$$Question 2 Code: 9709/12/M/J/18/3, Topic: Series A company producing salt from sea water changed to a new process. The amount of salt obtained each week increased by$2 \%$of the amount obtained in the preceding week. It is given that in the first week after the change the company obtained$8000 \mathrm{~kg}$of salt.$\text{(i)}$Find the amount of salt obtained in the 12 th week after the change.$\text{(ii)}$Find the total amount of salt obtained in the first 12 weeks after the change.$$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$C D$.$$Question 4 Code: 9709/11/M/J/17/4, Topic: Series$\text{(a)}$An arithmetic progression has a first term of 32 , a 5 th term of 22 and a last term of$-28$. Find the sum of all the terms in the progression.$\text{(b)}$Each year a school allocates a sum of money for the library. The amount allocated each year increases by$2.5 \%$of the amount allocated the previous year. In 2005 the school allocated$\$2000$. Find the total amount allocated in the years 2005 to 2014 inclusive. $$

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 $b$. $$

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

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

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

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

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 $B$. $$

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

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 $0 \leqslant x \leqslant 2 \pi$. $$

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

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

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 $a$. $$

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

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. $$

$\text{(b)}$ 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 $n$. $$

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 $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$. $$

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

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

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. $$

$\text{(b)}$ 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 $k$, $$

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

Worked solutions: P1, P3 & P6 (S1)

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