$\require{\cancel}$ $\require{\stix[upint]}$

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 | 4 | 6 | 7 | 8 | 8 | 7 | 7 | 10 | 11 | 11 | 10 | 94 |

Score |

Question 1 Code: 9709/13/M/J/11/1, Topic: Trigonometry

The coefficient of $x^{3}$ in the expansion of $(a+x)^{5}+(1-2 x)^{6}$, where $a$ is positive, is 90. Find the value of $a$. $[5]$

Question 2 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 3 Code: 9709/13/M/J/17/4, Topic: Vectors

Relative to an origin $O$, the position vectors of points $A$ and $B$ are given by

$$ \overrightarrow{O A}=\left(\begin{array}{l} 5 \\ 1 \\ 3 \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{r} 5 \\ 4 \\ -3 \end{array}\right) $$The point $P$ lies on $A B$ and is such that $\overrightarrow{A P}=\frac{1}{3} \overrightarrow{A B}$.

$\text{(i)}$ Find the position vector of $P$. $[3]$

$\text{(ii)}$ Find the distance $O P$. $[1]$

$\text{(iii)}$ Determine whether $O P$ is perpendicular to $A B$. Justify your answer. $[2]$

Question 4 Code: 9709/13/M/J/11/5, Topic: Vectors

In the diagram, $O A B C D E F G$ is a rectangular block in which $O A=O D=6 \mathrm{~cm}$ and $A B=12 \mathrm{~cm}$. The unit vectors i, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $\overrightarrow{O A}, \overrightarrow{O C}$ and $\overrightarrow{O D}$ respectively. The point $P$ is the mid-point of $D G, Q$ is the centre of the square face $C B F G$ and $R$ lies on $A B$ such that $A R=4 \mathrm{~cm}$.

$\text{(i)}$ Express each of the vectors $\overrightarrow{P Q}$ and $\overrightarrow{R Q}$ in terms of $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$. $[3]$

$\text{(ii)}$ Use a scalar product to find angle $R Q P$. $[4]$

Question 5 Code: 9709/13/M/J/10/6, Topic: Vectors

Relative to an origin $O$, the position vectors of the points $A, B$ and $C$ are given by

$$ \overrightarrow{O A}=\mathbf{i}-2 \mathbf{j}+4 \mathbf{k}, \quad \overrightarrow{O B}=3 \mathbf{i}+2 \mathbf{j}+8 \mathbf{k}, \quad \overrightarrow{O C}=-\mathbf{i}-2 \mathbf{j}+10 \mathbf{k}. $$$\text{(i)}$ Use a scalar product to find angle $A B C$. $[6]$

$\text{(ii)}$ Find the perimeter of triangle $A B C$, giving your answer correct to 2 decimal places. $[2]$

Question 6 Code: 9709/12/M/J/17/7, Topic: Series

$\text{(a)}$ The first two terms of an arithmetic progression are 16 and 24. Find the least number of terms of the progression which must be taken for their sum to exceed 20000. $[4]$

$\text{(b)}$ A geometric progression has a first term of 6 and a sum to infinity of 18. A new geometric progression is formed by squaring each of the terms of the original progression. Find the sum to infinity of the new progression. $[4]$

Question 7 Code: 9709/13/M/J/17/7, Topic: Circular measure

The diagram shows two circles with centres $A$ and $B$ having radii $8 \mathrm{~cm}$ and $10 \mathrm{~cm}$ respectively. The two circles intersect at $C$ and $D$ where $C A D$ is a straight line and $A B$ is perpendicular to $C D$.

$\text{(i)}$ Find angle $A B C$ in radians. $[1]$

$\text{(ii)}$ Find the area of the shaded region. $[6]$

Question 8 Code: 9709/11/M/J/11/8, Topic: Series

A television quiz show takes place every day. On day 1 the prize money is $\$ 1000$. If this is not won the prize money is increased for day 2. The prize money is increased in a similar way every day until it is won. The television company considered the following two different models for increasing the prize money.

Model 1: Increase the prize money by $\$ 1000$ each day.

Model 2: Increase the prize money by $10 \%$ each day.

On each day that the prize money is not won the television company makes a donation to charity. The amount donated is $5 \%$ of the value of the prize on that day. After 40 days the prize money has still not been won. Calculate the total amount donated to charity

$\text{(i)}$ if Model 1 is used, $[4]$

$\text{(ii)}$ if Model 2 is used. $[3]$

Question 9 Code: 9709/11/M/J/21/8, Topic: Circular measure

The diagram shows a symmetrical metal plate. The plate is made by removing two identical pieces from a circular disc with centre $C$. The boundary of the plate consists of two arcs $P S$ and $Q R$ of the original circle and two semicircles with $P Q$ and $R S$ as diameters. The radius of the circle with centre $C$ is $4 \mathrm{~cm}$, and $P Q=R S=4 \mathrm{~cm}$ also.

$\text{(a)}$ Show that angle $P C S=\frac{2}{3} \pi$ radians. $[2]$

$\text{(b)}$ Find the exact perimeter of the plate. $[3]$

$\text{(c)}$ Show that the area of the plate is $\left(\frac{20}{3}\pi + 8\sqrt{3}\right)$ cm$^{2}$. $[5]$

Question 10 Code: 9709/13/M/J/11/9, Topic: Differentiation

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{2}{\sqrt{x}}-1$ and $P(9,5)$ is a point on the curve.

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

$\text{(ii)}$ Find the coordinates of the stationary point on the curve. $[3]$

$\text{(iii)}$ Find an expression for $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$ and determine the nature of the stationary point. $[2]$

$\text{(iv)}$ The normal to the curve at $P$ makes an angle of $\tan ^{-1} k$ with the positive $x$-axis. Find the value of $k$. $[2]$

Question 11 Code: 9709/11/M/J/13/9, Topic: Differentiation, Integration

A curve has equation $y=\mathrm{f}(x)$ and is such that $\mathrm{f}^{\prime}(x)=3 x^{\frac{1}{2}}+3 x^{-\frac{1}{2}}-10$.

$\text{(i)}$ By using the substitution $u=x^{\frac{1}{2}}$, or otherwise, find the values of $x$ for which the curve $y=\mathrm{f}(x)$ has stationary points. $[4]$

$\text{(ii)}$ Find $\mathrm{f}^{\prime \prime}(x)$ and hence, or otherwise, determine the nature of each stationary point. $[3]$

$\text{(iii)}$ It is given that the curve $y=\mathrm{f}(x)$ passes through the point $(4,-7)$. Find $\mathrm{f}(x)$. $[4]$

Question 12 Code: 9709/13/M/J/15/11, Topic: Circular measure

In the diagram, $O A B$ is a sector of a circle with centre $O$ and radius $r$. The point $C$ on $O B$ is such that angle $A C O$ is a right angle. Angle $A O B$ is $\alpha$ radians and is such that $A C$ divides the sector into two regions of equal area.

$\text{(i)}$ Show that $\sin \alpha \cos \alpha=\frac{1}{2} \alpha$. $[4]$

It is given that the solution of the equation in part $\text{(i)}$ is $\alpha=0.9477$, correct to 4 decimal places.

$\text{(ii)}$ Find the ratio

perimeter of region $O A C$ : perimeter of region $A C B$,

giving your answer in the form $k: 1$, where $k$ is given correct to 1 decimal place. $[5]$

$\text{(iii)}$ Find angle $A O B$ in degrees. $[1]$