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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 | 4 | 3 | 4 | 5 | 7 | 7 | 6 | 9 | 8 | 8 | 9 | 13 | 83 |

Score |

Question 1 Code: 9709/11/M/J/10/1, Topic: Trigonometry

The acute angle $x$ radians is such that $\tan x=k$, where $k$ is a positive constant. Express, in terms of $k$,

$\text{(i)}$ $\tan (\pi-x)$, $[1]$

$\text{(ii)}$ $\tan \left(\frac{1}{2} \pi-x\right)$, $[1]$

$\text{(iii)}$ $\sin x$. $[2]$

Question 2 Code: 9709/13/M/J/18/1, Topic: Quadratics

Express $3 x^{2}-12 x+7$ in the form $a(x+b)^{2}+c$, where $a, b$ and $c$ are constants. $[3]$

Question 3 Code: 9709/11/M/J/14/3, Topic: Series

Find the term independent of $x$ in the expansion of $\displaystyle\left(4 x^{3}+\frac{1}{2 x}\right)^{8}$. $[4]$

Question 4 Code: 9709/13/M/J/16/3, Topic: Integration

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=6 x^{2}+\frac{k}{x^{3}}$ and passes through the point $P(1,9)$. The gradient of the curve at $P$ is $2.$

$\text{(i)}$ Find the value of the constant $k$. $[1]$

$\text{(ii)}$ Find the equation of the curve. $[4]$

Question 5 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 6 Code: 9709/13/M/J/16/5, Topic: Differentiation

A curve has equation $y=8 x+(2 x-1)^{-1}$. Find the values of $x$ at which the curve has a stationary point and determine the nature of each stationary point, justifying your answers. $[7]$

Question 7 Code: 9709/13/M/J/20/5, Topic: Circular measure

The diagram shows a cord going around a pulley and a pin. The pulley is modelled as a circle with centre $O$ and radius $5 \mathrm{~cm}$. The thickness of the cord and the size of the pin $P$ can be neglected. The pin is situated 13 cm vertically below $O$. Points $A$ and $B$ are on the circumference of the circle such that $A P$ and $B P$ are tangents to the circle. The cord passes over the major arc $A B$ of the circle and under the pin such that the cord is taut.

Calculate the length of the cord. $[6]$

Question 8 Code: 9709/11/M/J/13/7, Topic: Quadratics, Differentiation, Coordinate geometry

A curve has equation $y=x^{2}-4 x+4$ and a line has equation $y=m x$, where $m$ is a constant.

$\text{(i)}$ For the case where $m=1$, the curve and the line intersect at the points $A$ and $B$. Find the coordinates of the mid-point of $A B$. $[4]$

$\text{(ii)}$ Find the non-zero value of $m$ for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve. $[5]$

Question 9 Code: 9709/12/M/J/10/8, Topic: Differentiation

A solid rectangular block has a square base of side $x \mathrm{~cm}$. The height of the block is $h \mathrm{~cm}$ and the total surface area of the block is 96 cm$^{2}$.

$\text{(i)}$ Express $\mathrm{h}$ in terms of $x$ and show that the volume, $V \mathrm{~cm}^{3}$, of the block is given by $[3]$

$$ V=24 x-\frac{1}{2} x^{3} $$Given that $x$ can vary,

$\text{(ii)}$ find the stationary value of $V$, $[3]$

$\text{(iii)}$ determine whether this stationary value is a maximum or a minimum. $[2]$

Question 10 Code: 9709/12/M/J/13/10, Topic: Series

$\text{(a)}$ The first and last terms of an arithmetic progression are 12 and 48 respectively. The sum of the first four terms is 57. Find the number of terms in the progression. $[4]$

$\text{(b)}$ The third term of a geometric progression is four times the first term. The sum of the first six terms is $k$ times the first term. Find the possible values of $k$. $[4]$

Question 11 Code: 9709/11/M/J/19/10, Topic: Integration, Differentiation

A curve for which $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=2 x-5$ has a stationary point at $(3,6)$.

$\text{(i)}$ Find the equation of the curve. $[6]$

$\text{(ii)}$ Find the $x$-coordinate of the other stationary point on the curve. $[1]$

$\text{(iii)}$ Determine the nature of each of the stationary points. $[2]$

Question 12 Code: 9709/13/M/J/19/10, Topic: Coordinate geometry, Integration

The diagram shows part of the curve with equation $y=(3 x+4)^{\frac{1}{2}}$ and the tangent to the curve at the point $A$. The $x$-coordinate of $A$ is 4.

$\text{(i)}$ Find the equation of the tangent to the curve at $A$. $[5]$

$\text{(ii)}$ Find, showing all necessary working, the area of the shaded region. $[5]$

$\text{(iii)}$ A point is moving along the curve. At the point $P$ the $y$-coordinate is increasing at half the rate at which the $x$-coordinate is increasing. Find the $x$-coordinate of $P$. $[3]$