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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 |

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 $\displaystyle \left(x-\frac{2}{x}\right)^{6}$. $[3]$

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

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. $[2]$

$\text{(ii)}$ Given instead that the progression is arithmetic, find the number of terms in the progression if the sum of all the terms is 0. $[3]$

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 $S=2-r$. $[2]$

$\text{(ii)}$ Find the set of possible values that $S$ can take. $[2]$

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. $[4]$

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 $P$. $[4]$

$\text{(ii)}$ Find the angle that this tangent makes with the $x$-axis. $[2]$

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. $[4]$

$\text{(b)}$ 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 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. $[5]$

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 $\alpha$. $[3]$

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

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 $E F G$. $[7]$

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. $[3]$

$\text{(ii)}$ Find the perimeter of the shaded region. $[3]$

$\text{(iii)}$ Find the area of the shaded region. $[3]$

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 $c$. $[1]$

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 $\mathrm{f}^{-1}$. $[3]$

$\text{(iii)}$ Solve the equation $\mathrm{ff}(x)=51$, giving your answer in the form $a+\sqrt{b}$. $[5]$

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 $A B C=90^{\circ}$. $[2]$

$\text{(b)}$ Hence state the coordinates of $D$. $[1]$

$\text{(c)}$ Find an equation of the circle. $[2]$

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 $E$. $[5]$