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

Name of student | Date | ||||

Adm. number | Year/grade | Stream | |||

Subject | Pure Mathematics 1 (P1) | Variant(s) | P11, P12 | ||

Start time | Duration | Stop time |

Qtn No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Marks | 3 | 4 | 4 | 5 | 3 | 6 | 6 | 6 | 7 | 7 | 8 | 9 | 68 |

Score |

Question 1 Code: 9709/11/O/N/14/1, Topic: Series

In the expansion of $(2+a x)^{7}$, the coefficient of $x$ is equal to the coefficient of $x^{2}.$ Find the value of the non-zero constant $a$. $[3]$

Question 2 Code: 9709/11/O/N/17/1, Topic: Differentiation

A curve has equation $y=2 x^{\frac{3}{2}}-3 x-4 x^{\frac{1}{2}}+4$. Find the equation of the tangent to the curve at the point $(4,0)$. $[4]$

Question 3 Code: 9709/12/O/N/10/3, Topic: Differentiation

The length, $x$ metres, of a Green Anaconda snake which is $t$ years old is given approximately by the $0.4$ formula

$$x=0.7 \sqrt{(}2 t-1)$$where $1 \leqslant t \leqslant 10$. Using this formula, find

$\text{(i)}$ $\displaystyle\frac{\mathrm{d} x}{\mathrm{~d} t}$, $[2]$

$\text{(ii)}$ the rate of growth of a Green Anaconda snake which is 5 years old. $[2]$

Question 4 Code: 9709/12/O/N/12/3, Topic: Differentiation

The diagram shows a plan for a rectangular park $A B C D$, in which $A B=40 \mathrm{~m}$ and $A D=60 \mathrm{~m}$. Points $X$ and $Y$ lie on $B C$ and $C D$ respectively and $A X, X Y$ and $Y A$ are paths that surround a triangular playground. The length of $D Y$ is $x \mathrm{~m}$ and the length of $X C$ is $2 x \mathrm{~m}$.

$\text{(i)}$ Show that the area, $A \mathrm{~m}^{2}$, of the playground is given by $[2]$

$$ A=x^{2}-30 x+1200. $$$\text{(ii)}$ Given that $x$ can vary, find the minimum area of the playground. $[3]$

Question 5 Code: 9709/11/O/N/20/4, Topic: Trigonometry

In the diagram, the lower curve has equation $y=\cos \theta$. The upper curve shows the result of applying a combination of transformations to $y=\cos \theta$.

Find, in terms of a cosine function, the equation of the upper curve. $[3]$

Question 6 Code: 9709/11/O/N/10/5, Topic: Vectors

The diagram shows a pyramid $O A B C$ with a horizontal base $O A B$ where $O A=6 \mathrm{~cm}, O B=8 \mathrm{~cm}$ and angle $A O B=90^{\circ}$. The point $C$ is vertically above $O$ and $O C=10 \mathrm{~cm}$. Unit vectors $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$ are parallel to $O A, O B$ and $O C$ as shown.

Use a scalar product to find angle $A C B$. $[6]$

Question 7 Code: 9709/12/O/N/14/5, Topic: Trigonometry

$\text{(i)}$ Show that the equation $1+\sin x \tan x=5 \cos x$ can be expressed as $[3]$

$$ 6 \cos ^{2} x-\cos x-1=0 . $$$\text{(ii)}$ Hence solve the equation $1+\sin x \tan x=5 \cos x$ for $0^{\circ} \leqslant x \leqslant 180^{\circ}$. $[3]$

Question 8 Code: 9709/12/O/N/20/5, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$$ \begin{array}{ll} \mathrm{f}(x)=4 x-2, & \text { for } x \in \mathbb{R}, \\ \displaystyle \mathrm{g}(x)=\frac{4}{x+1}, & \text { for } x \in \mathbb{R}, x \neq-1. \end{array} $$$\text{(a)}$ Find the value of $\mathrm{fg}(7)$. $[1]$

$\text{(b)}$ Find the values of $x$ for which $\mathrm{f}^{-1}(x)=\mathrm{g}^{-1}(x)$. $[5]$

Question 9 Code: 9709/11/O/N/11/6, Topic: Series

$\text{(a)}$ The sixth term of an arithmetic progression is 23 and the sum of the first ten terms is 200. Find the seventh term. $[4]$

$\text{(b)}$ A geometric progression has first term 1 and common ratio $r$. A second geometric progression has first term 4 and common ratio $\frac{1}{4} r$. The two progressions have the same sum to infinity, $S$. Find the values of $r$ and $S$. $[3]$

Question 10 Code: 9709/12/O/N/20/7, Topic: Differentiation, Integration

The point $(4,7)$ lies on the curve $y=\mathrm{f}(x)$ and it is given that $\mathrm{f}^{\prime}(x)=6 x^{-\frac{1}{2}}-4 x^{-\frac{3}{2}}$.

$\text{(a)}$ A point moves along the curve in such a way that the $x$-coordinate is increasing at a constant rate of $0.12$ units per second. $[3]$

Find the rate of increase of the $y$-coordinate when $x=4$.

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

Question 11 Code: 9709/12/O/N/10/9, Topic: Vectors

The diagram shows a pyramid $O A B C P$ in which the horizontal base $O A B C$ is a square of side $10 \mathrm{~cm}$ and the vertex $P$ is $10 \mathrm{~cm}$ vertically above $O$. The points $D, E, F, G$ lie on $O P, A P, B P, C P$ respectively and $D E F G$ is a horizontal square of side $6 \mathrm{~cm}$. The height of $D E F G$ above the base is $a \mathrm{~cm}$. Unit vectors $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$ are parallel to $O A, O C$ and $O D$ respectively.

$\text{(i)}$ Show that $a=4$. $[2]$

$\text{(ii)}$ Express the vector $\overrightarrow{B G}$ in terms of $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$. $[2]$

$\text{(iii)}$ Use a scalar product to find angle $G B A$. $[4]$

Question 12 Code: 9709/11/O/N/12/9, Topic: Vectors

The position vectors of points $A$ and $B$ relative to an origin $O$ are $\mathbf{a}$ and $\mathbf{b}$ respectively. The position vectors of points $C$ and $D$ relative to $O$ are $3 \mathbf{a}$ and $2 \mathbf{b}$ respectively. It is given that

$$ \mathbf{a}=\left(\begin{array}{l} 2 \\ 1 \\ 2 \end{array}\right) \quad \text { and } \quad \mathbf{b}=\left(\begin{array}{l} 4 \\ 0 \\ 6 \end{array}\right) $$$\text{(i)}$ Find the unit vector in the direction of $\overrightarrow{C D}$. $[3]$

$\text{(ii)}$ The point $E$ is the mid-point of $C D$. Find angle $E O D$. $[6]$