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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 | 6 | 7 | 6 | 7 | 7 | 8 | 7 | 7 | 5 | 9 | 9 | 14 | 92 |

Score |

Question 1 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 2 Code: 9709/13/M/J/10/5, Topic: Differentiation, Integration

The equation of a curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6}{\sqrt{(} 3 x-2)}.$ Given that the curve passes through the point $P(2,11)$, find

$\text{(i)}$ the equation of the normal to the curve at $P$, $[3]$

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

Question 3 Code: 9709/11/M/J/12/5, Topic: Coordinate geometry

The diagram shows the curve $y=7 \sqrt{x}$ and the line $y=6 x+k$, where $k$ is a constant. The curve and the line intersect at the points $A$ and $B$.

$\text{(i)}$ For the case where $k=2$, find the $x$-coordinates of $A$ and $B$. $[4]$

$\text{(ii)}$ Find the value of $k$ for which $y=6 x+k$ is a tangent to the curve $y=7 \sqrt{x}$. $[2]$

Question 4 Code: 9709/13/M/J/19/5, Topic: Series

Two heavyweight boxers decide that they would be more successful if they competed in a lower weight class. For each boxer this would require a total weight loss of $13 \mathrm{~kg}$. At the end of week 1 they have each recorded a weight loss of $1 \mathrm{~kg}$ and they both find that in each of the following weeks their weight loss is slightly less than the week before.

Boxer $A$ 's weight loss in week 2 is $0.98 \mathrm{~kg}$. It is given that his weekly weight loss follows an arithmetic progression.

$\text{(i)}$ Write down an expression for his total weight loss after $x$ weeks. $[1]$

$\text{(ii)}$ He reaches his $13 \mathrm{~kg}$ target during week $n$. Use your answer to part $\text{(i)}$ to find the value of $n$. Boxer $B$ 's weight loss in week 2 is $0.92 \mathrm{~kg}$ and it is given that his weekly weight loss follows a geometric progression. $[2]$

$\text{(iii)}$ Calculate his total weight loss after 20 weeks and show that he can never reach his target. $[4]$

Question 5 Code: 9709/11/M/J/12/6, Topic: Vectors

Two vectors $\mathbf{u}$ and $\mathbf{v}$ are such that $\mathbf{u}=\left(\begin{array}{c}p^{2} \\ -2 \\ 6\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}2 \\ p-1 \\ 2 p+1\end{array}\right)$, where $p$ is a constant.

$\text{(i)}$ Find the values of $p$ for which $\mathbf{u}$ is perpendicular to $\mathbf{v}$. $[3]$

$\text{(ii)}$ For the case where $p=1$, find the angle between the directions of $\mathbf{u}$ and $\mathbf{v}$. $[4]$

Question 6 Code: 9709/11/M/J/10/7, Topic: Coordinate geometry

The diagram shows part of the curve $\displaystyle y=2-\frac{18}{2 x+3}$, which crosses the $x$-axis at $A$ and the $y$-axis at $B$. The normal to the curve at $A$ crosses the $y$-axis at $C$.

$\text{(i)}$ Show that the equation of the line $A C$ is $9 x+4 y=27$. $[6]$

$\text{(ii)}$ Find the length of $B C$. $[2]$

Question 7 Code: 9709/11/M/J/12/7, Topic: Series

$\text{(a)}$ The first two terms of an arithmetic progression are 1 and $\cos ^{2} x$ respectively. Show that the sum of the first ten terms can be expressed in the form $a-b \sin ^{2} x$, where $a$ and $b$ are constants to be found. $[3]$

$\text{(b)}$ The first two terms of a geometric progression are 1 and $\frac{1}{3} \tan ^{2} \theta$ respectively, where $0< \theta <\frac{1}{2} \pi$.

$\text{(i)}$ Find the set of values of $\theta$ for which the progression is convergent. $[2]$

$\text{(ii)}$ Find the exact value of the sum to infinity when $\theta=\frac{1}{6} \pi$. $[2]$

Question 8 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 9 Code: 9709/12/M/J/21/7, Topic: Coordinate geometry

The point $A$ has coordinates $(1,5)$ and the line $l$ has gradient $-\frac{2}{3}$ and passes through $A$. A circle has centre $(5,11)$ and radius $\sqrt{52}$.

$\text{(a)}$ Show that $l$ is the tangent to the circle at $A$. $[2]$

$\text{(b)}$ Find the equation of the other circle of radius $\sqrt{52}$ for which l is also the tangent at $A$. $[3]$

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

The diagram shows part of the curve $\displaystyle y=\frac{8}{\sqrt{x}}-x$ and points $A(1,7)$ and $B(4,0)$ which lie on the curve. The tangent to the curve at $B$ intersects the line $x=1$ at the point $C$.

$\text{(i)}$ Find the coordinates of $C$. $[4]$

$\text{(ii)}$ Find the area of the shaded region. $[5]$

Question 12 Code: 9709/11/M/J/21/11, Topic: Differentiation, Integration

The equation of a curve is $y=2 \sqrt{3 x+4}-x$

$\text{(a)}$ Find the equation of the normal to the curve at the point $(4,4)$, giving your answer in the form $y=m x+c$ $[5]$

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

$\text{(c)}$ Determine the nature of the stationary point. $[2]$

$\text{(d)}$ Find the exact area of the region bounded by the curve, the $x$-axis and the lines $x = 0$ and $x = 4$. $[4]$