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

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 | 3 | 6 | 4 | 6 | 5 | 6 | 7 | 8 | 8 | 9 | 12 | 11 | 85 |

Score |

Question 1 Code: 9709/13/M/J/21/1, Topic: Integration

A curve with equation $y=\mathrm{f}(x)$ is such that $\displaystyle \mathrm{f}^{\prime}(x)=6 x^{2}-\frac{8}{x^{2}}.$ It is given that the curve passes through the point $(2,7)$.

Find $\mathrm{f}(x)$. $[3]$

Question 2 Code: 9709/11/M/J/17/2, Topic: Vectors

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

$$ \overrightarrow{O A}=\left(\begin{array}{r} 3 \\ -6 \\ p \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{r} 2 \\ -6 \\ -7 \end{array}\right) $$and angle $A O B=90^{\circ}$.

$\text{(i)}$ Find the value of $p$. $[2]$

The point $C$ is such that $\overrightarrow{O C}=\frac{2}{3} \overrightarrow{O A}$

$\text{(ii)}$ Find the unit vector in the direction of $\overrightarrow{B C}$. $[4]$

Question 3 Code: 9709/12/M/J/21/3, Topic: Coordinate geometry

The equation of a curve is $y=(x-3) \sqrt{x+1}+3$. The following points lie on the curve. Non-exact values are rounded to 4 decimal places.

$$ \begin{array}{lllll} A(2, k) & B(2.9,2.8025) & C(2.99,2.9800) & D(2.999,2.9980) & E(3,3) \end{array} $$$\text{(a)}$ Find $k$, giving your answer correct to 4 decimal places. $[1]$

$\text{(b)}$ Find the gradient of $A E$, giving your answer correct to 4 decimal places. $[1]$

The gradients of $B E, C E$ and $D E$, rounded to 4 decimal places, are 1.9748, $1.9975$ and $1.9997$ respectively.

$\text{(c)}$ State, giving a reason for your answer, what the values of the four gradients suggest about the gradient of the curve at the point $E$. $[2]$

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

The diagram shows a prism $A B C D P Q R S$ with a horizontal square base $A P S D$ with sides of length $6 \mathrm{~cm}$. The cross-section $A B C D$ is a trapezium and is such that the vertical edges $A B$ and $D C$ are of lengths $5 \mathrm{~cm}$ and $2 \mathrm{~cm}$ respectively. Unit vectors $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$ are parallel to $A D, A P$ and $A B$ respectively.

$\text{(i)}$ Express each of the vectors $\overrightarrow{C P}$ and $\overrightarrow{C Q}$ in terms of $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$. $[2]$

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

Question 5 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 6 Code: 9709/12/M/J/20/5, Topic: Functions

The function $\mathrm{f}$ is defined for $x \in \mathbb{R}$ by

$$ \text { f: } x \mapsto a-2 x $$where $a$ is a constant.

$\text{(a)}$ Express $\mathrm{ff}(x)$ and $\mathrm{f}^{-1}(x)$ in terms of $a$ and $x$. $[4]$

$\text{(b)}$ Given that $\mathrm{ff}(x)=\mathrm{f}^{-1}(x)$, find $x$ in terms of $a$. $[2]$

Question 7 Code: 9709/12/M/J/19/7, Topic: Functions

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

$$ \begin{aligned} \mathrm{f}: x & \mapsto 3 x-2, \quad x \in \mathbb{R} \\ \mathrm{g}: x & \mapsto \frac{2 x+3}{x-1}, \quad x \in \mathbb{R}, x \neq 1 \end{aligned} $$$\text{(i)}$ Obtain expressions for $\mathrm{f}^{-1}(x)$ and $\mathrm{g}^{-1}(x)$, stating the value of $x$ for which $\mathrm{g}^{-1}(x)$ is not defined. $[4]$

$\text{(ii)}$ Solve the equation $\mathrm{fg}(x)=\frac{7}{3}$. $[3]$

Question 8 Code: 9709/13/M/J/17/8, Topic: Coordinate geometry

$A(-1,1)$ and $P(a, b)$ are two points, where $a$ and $b$ are constants. The gradient of $A P$ is 2.

$\text{(i)}$ Find an expression for $b$ in terms of $a$. $[2]$

$\text{(ii)}$ $B(10,-1)$ is a third point such that $A P=A B$. Calculate the coordinates of the possible positions of $P$. $[6]$

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

The diagram shows the curve $y=(x-2)^{2}$ and the line $y+2 x=7$, which intersect at points $A$ and $B$. Find the area of the shaded region. $[8]$

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

The equation of a curve is $y=\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$.

$\text{(i)}$ Find $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$. $[3]$

$\text{(ii)}$ Find the equation of the tangent to the curve at the point where the curve intersects the $y$-axis. $[3]$

$\text{(iii)}$ Find the set of values of $x$ for which $\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$ is an increasing function of $x$. $[3]$

Question 11 Code: 9709/11/M/J/13/10, Topic: Coordinate geometry, Integration

The diagram shows part of the curve $y=(x-2)^{4}$ and the point $A(1,1)$ on the curve. The tangent at $A$ cuts the $x$-axis at $B$ and the normal at $A$ cuts the $y$-axis at $C$.

$\text{(i)}$ Find the coordinates of $B$ and $C$. $[6]$

$\text{(ii)}$ Find the distance $A C$, giving your answer in the form $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are integers. $[2]$

$\text{(iii)}$ Find the area of the shaded region. $[4]$

Question 12 Code: 9709/11/M/J/11/11, Topic: Functions, Coordinate geometry

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined for $x \in \mathbb{R}$ by

$$ \begin{aligned} &\mathrm{f}: x \mapsto 2 x+1 \\ &\mathrm{~g}: x \mapsto x^{2}-2 \end{aligned} $$$\text{(i)}$ Find and simplify expressions for $\mathrm{fg}(x)$ and $\operatorname{gf}(x)$. $[2]$

$\text{(ii)}$ Hence find the value of $a$ for which $\mathrm{fg}(a)=\operatorname{gf}(a)$. $[3]$

$\text{(iii)}$ Find the value of $b(b \neq a)$ for which $\mathrm{g}(b)=b$. $[2]$

$\text{(iv)}$ Find and simplify an expression for $\mathrm{f}^{-1} \mathrm{~g}(x)$. $[2]$

The function $\mathrm{h}$ is defined by

$$ \mathrm{h}: x \mapsto x^{2}-2, \quad \text { for } x \leqslant 0 $$$\text{(v)}$ Find an expression for $\mathrm{h}^{-1}(x)$. $[2]$