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### MATHEMATICS 9709

#### Cambridge International AS and A Level

 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 4 4 6 7 5 6 7 9 9 10 77
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/13/M/J/10/2, Topic: Series$\text{(i)}$Find the first three terms, in descending powers of$x$, in the expansion of$\displaystyle \left(x-\frac{2}{x}\right)^{6}$.$\text{(ii)}$Find the coefficient of$x^{4}$in the expansion of$\displaystyle \left(1+x^{2}\right)\left(x-\frac{2}{x}\right)^{6}$.$$Question 2 Code: 9709/13/M/J/14/2, Topic: Series The first term in a progression is 36 and the second term is 32.$\text{(i)}$Given that the progression is geometric, find the sum to infinity.$\text{(ii)}$Given instead that the progression is arithmetic, find the number of terms in the progression if the sum of all the terms is 0.$$Question 3 Code: 9709/13/M/J/17/2, Topic: Series The common ratio of a geometric progression is$r$. The first term of the progression is$\left(r^{2}-3 r+2\right)$and the sum to infinity is$S$.$\text{(i)}$Show that$S=2-r$.$\text{(ii)}$Find the set of possible values that$S$can take.$$Question 4 Code: 9709/13/M/J/20/2, Topic: Integration The equation of a curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=3 x^{\frac{1}{2}}-3 x^{-\frac{1}{2}}.$It is given that the point$(4,7)$lies on the curve. Find the equation of the curve.$$Question 5 Code: 9709/12/M/J/11/4, Topic: Differentiation, Coordinate geometry A curve has equation$\displaystyle y=\frac{4}{3 x-4}$and$P(2,2)$is a point on the curve.$\text{(i)}$Find the equation of the tangent to the curve at$P$.$\text{(ii)}$Find the angle that this tangent makes with the$x$-axis.$$Question 6 Code: 9709/11/M/J/17/4, Topic: Series$\text{(a)}$An arithmetic progression has a first term of 32 , a 5 th term of 22 and a last term of$-28$. Find the sum of all the terms in the progression.$\text{(b)}$Each year a school allocates a sum of money for the library. The amount allocated each year increases by$2.5 \%$of the amount allocated the previous year. In 2005 the school allocated$\$2000$. Find the total amount allocated in the years 2005 to 2014 inclusive. $$

Question 7 Code: 9709/12/M/J/20/4, Topic: Series

The $n$th term of an arithmetic progression is $\frac{1}{2}(3 n-15)$.

Find the value of $n$ for which the sum of the first $n$ terms is 84. $$

Question 8 Code: 9709/12/M/J/16/6, Topic: Circular measure The diagram shows a circle with radius $r \mathrm{~cm}$ and centre $O$. The line $P T$ is the tangent to the circle at $P$ and angle $P O T=\alpha$ radians. The line $O T$ meets the circle at $Q.$

$\text{(i)}$ Express the perimeter of the shaded region $P Q T$ in terms of $r$ and $\alpha$. $$

$\text{(ii)}$ In the case where $\alpha=\frac{1}{3} \pi$ and $r=10$, find the area of the shaded region correct to 2 significant figures. $$

Question 9 Code: 9709/13/M/J/16/6, Topic: Circular measure The diagram shows triangle $A B C$ where $A B=5 \mathrm{~cm}, A C=4 \mathrm{~cm}$ and $B C=3 \mathrm{~cm}$. Three circles with centres at $A, B$ and $C$ have radii $3 \mathrm{~cm}, 2 \mathrm{~cm}$ and $1 \mathrm{~cm}$ respectively. The circles touch each other at points $E, F$ and $G$, lying on $A B, A C$ and $B C$ respectively. Find the area of the shaded region $E F G$. $$

Question 10 Code: 9709/11/M/J/17/8, Topic: Circular measure In the diagram, $O A X B$ is a sector of a circle with centre $O$ and radius $10 \mathrm{~cm}.$ The length of the chord $A B$ is $12 \mathrm{~cm}$. The line $O X$ passes through $M$, the mid-point of $A B$, and $O X$ is perpendicular to $A B$. The shaded region is bounded by the chord $A B$ and by the arc of a circle with centre $X$ and radius $X A$.

$\text{(i)}$ Show that angle $A X B$ is $2.498$ radians, correct to 3 decimal places. $$

$\text{(ii)}$ Find the perimeter of the shaded region. $$

$\text{(iii)}$ Find the area of the shaded region. $$

Question 11 Code: 9709/13/M/J/18/10, Topic: Functions

The one-one function $\mathrm{f}$ is defined by $\mathrm{f}(x)=(x-2)^{2}+2$ for $x \geqslant c$, where $c$ is a constant.

$\text{(i)}$ State the smallest possible value of $c$. $$

In parts $\text{(ii)}$ and $\text{(iii)}$ the value of $c$ is 4.

$\text{(ii)}$ Find an expression for $\mathrm{f}^{-1}(x)$ and state the domain of $\mathrm{f}^{-1}$. $$

$\text{(iii)}$ Solve the equation $\mathrm{ff}(x)=51$, giving your answer in the form $a+\sqrt{b}$. $$

Question 12 Code: 9709/13/M/J/21/10, Topic: Coordinate geometry

Points $A(-2,3), B(3,0)$ and $C(6,5)$ lie on the circumference of a circle with centre $D$.

$\text{(a)}$ Show that angle $A B C=90^{\circ}$. $$

$\text{(b)}$ Hence state the coordinates of $D$. $$

$\text{(c)}$ Find an equation of the circle. $$

The point $E$ lies on the circumference of the circle such that $BE$ is a diameter.

$\text{(d)}$ Find an equation of the tangent to the circle at $E$. $$

Worked solutions: P1, P3 & P6 (S1)

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