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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 | 4 | 5 | 6 | 6 | 8 | 9 | 7 | 8 | 9 | 10 | 10 | 86 |

Score |

Question 1 Code: 9709/11/M/J/15/1, Topic: Trigonometry

Given that $\theta$ is an obtuse angle measured in radians and that $\sin \theta=k$, find, in terms of $k$, an expression for

$\text{(i)}$ $\cos \theta$, $[1]$

$\text{(ii)}$ $\tan \theta$, $[2]$

$\text{(iii)}$ $\sin (\theta+\pi)$. $[1]$

Question 2 Code: 9709/11/M/J/20/1, Topic: Series

The sum of the first nine terms of an arithmetic progression is 117. The sum of the next four terms is 91.

Find the first term and the common difference of the progression. $[4]$

Question 3 Code: 9709/13/M/J/10/3, Topic: Functions

The function $\mathrm{f}: x \mapsto a+b \cos x$ is defined for $0 \leqslant x \leqslant 2 \pi$. Given that $\mathrm{f}(0)=10$ and that $\mathrm{f}\left(\frac{2}{3} \pi\right)=1$, find

$\text{(i)}$ the values of $a$ and $b$, $[2]$

$\text{(ii)}$ the range of $\mathrm{f}$, $[1]$

$\text{(iii)}$ the exact value of $\mathrm{f}\left(\frac{5}{6} \pi\right)$. $[2]$

Question 4 Code: 9709/13/M/J/17/6, Topic: Quadratics

The line $3 y+x=25$ is a normal to the curve $y=x^{2}-5 x+k$. Find the value of the constant $k$. $[6]$

Question 5 Code: 9709/11/M/J/20/6, Topic: Functions

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

$$ \begin{aligned} &\mathrm{f}: x \mapsto \frac{1}{2} x-a \\ &\mathrm{~g}: x \mapsto 3 x+b \end{aligned} $$where $a$ and $b$ are constants.

$\text{(a)}$ Given that $\operatorname{gg}(2)=10$ and $\mathrm{f}^{-1}(2)=14$, find the values of $a$ and $b$. $[4]$

$\text{(b)}$ Using these values of $a$ and $b$, find an expression for $\operatorname{gf}(x)$ in the form $c x+d$, where $c$ and $d$ are constants. $[2]$

Question 6 Code: 9709/11/M/J/17/7, Topic: Integration, Quadratics, Differentiation

A curve for which $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=7-x^{2}-6 x$ passes through the point $(3,-10)$.

$\text{(i)}$ Find the equation of the curve. $[3]$

$\text{(ii)}$ Express $7-x^{2}-6 x$ in the form $a-(x+b)^{2}$, where $a$ and $b$ are constants. $[2]$

$\text{(iii)}$ Find the set of values of $x$ for which the gradient of the curve is positive. $[3]$

Question 7 Code: 9709/13/M/J/18/7, Topic: Trigonometry

$\text{(a)}$ $\quad\text{(i)}$ Express $\displaystyle \frac{\tan ^{2} \theta-1}{\tan ^{2} \theta+1}$ in the form $a \sin ^{2} \theta+b$, where $a$ and $b$ are constants to be found. $[3]$

$\qquad \text{(ii)}$ Hence, or otherwise, and showing all necessary working, solve the equation. $[2]$

$$ \frac{\tan ^{2} \theta-1}{\tan ^{2} \theta+1}=\frac{1}{4} $$$\qquad$ for $-90^{\circ} \leqslant \theta \leqslant 0^{\circ}$

$\text{(b)}$

The diagram shows the graphs of $y=\sin x$ and $y=2 \cos x$ for $-\pi \leqslant x \leqslant \pi$. The graphs intersect at the points $A$ and $B$.

$\text{(i)}$ Find the $x$-coordinate of $A$. $[2]$

$\text{(ii)}$ Find the $y$-coordinate of $B$. $[2]$

Question 8 Code: 9709/11/M/J/19/7, Topic: Vectors

The diagram shows a three-dimensional shape in which the base $O A B C$ and the upper surface $D E F G$ are identical horizontal squares. The parallelograms $O A E D$ and $C B F G$ both lie in vertical planes. The point $M$ is the mid-point of $A F$.

Unit vectors $\mathbf{i}$ and $\mathbf{j}$ are parallel to $O A$ and $O C$ respectively and the unit vector $\mathbf{k}$ is vertically upwards. The position vectors of $A$ and $D$ are given by $\overrightarrow{O A}=8 \mathbf{i}$ and $\overrightarrow{O D}=3 \mathbf{i}+10 \mathbf{k}$.

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

$\text{(ii)}$ Use a scalar product to find angle $G M A$ correct to the nearest degree. $[4]$

Question 9 Code: 9709/12/M/J/20/8, Topic: Integration, Coordinate geometry

The diagram shows part of the curve $\displaystyle y=\frac{6}{x}$. The points $(1,6)$ and $(3,2)$ lie on the curve. The shaded region is bounded by the curve and the lines $y=2$ and $x=1$.

$\text{(a)}$ Find the volume generated when the shaded region is rotated through $360^{\circ}$ about the $y$-axis. $[5]$

$\text{(b)}$ The tangent to the curve at a point $X$ is parallel to the line $y+2 x=0$. Show that $X$ lies on the line $y=2 x$. $[3]$

Question 10 Code: 9709/12/M/J/16/9, Topic: Series

A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that each day a greater amount of water is lost.

$\text{(i)}$ On the first day after filling, 10 litres of water are lost and this increases by 2 litres each day.

$\text{(a)}$ How many litres will be lost on the 30 th day after filling? $[2]$

$\text{(b)}$ The tank becomes empty during the $n$th day after filling. Find the value of $n$. $[3]$

$\text{(ii)}$ Assume instead that 10 litres of water are lost on the first day and that the amount of water lost increases by $10 \%$ on each succeeding day. Find what percentage of the original 2000 litres is left in the tank at the end of the 30 th day after filling. $[4]$

Question 11 Code: 9709/11/M/J/16/10, Topic: Vectors

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

$$ \overrightarrow{O A}=\left(\begin{array}{r} 2 \\ 1 \\ -2 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{r} 5 \\ -1 \\ k \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{r} 2 \\ 6 \\ -3 \end{array}\right) $$respectively, where $k$ is a constant.

$\text{(i)}$ Find the value of $k$ in the case where angle $A O B=90^{\circ}$. $[2]$

$\text{(ii)}$ Find the possible values of $k$ for which the lengths of $A B$ and $O C$ are equal. $[4]$

The point $D$ is such that $\overrightarrow{O D}$ is in the same direction as $\overrightarrow{O A}$ and has magnitude 9 units. The point $E$ is such that $\overrightarrow{O E}$ is in the same direction as $\overrightarrow{O C}$ and has magnitude 14 units.

$\text{(iii)}$ Find the magnitude of $\overrightarrow{D E}$ in the form $\sqrt{n}$ where $n$ is an integer. $[4]$

Question 12 Code: 9709/13/M/J/21/10, Topic: Coordinate geometry

Points $A(-2,3), B(3,0)$ and $C(6,5)$ lie on the circumference of a circle with centre $D$.

$\text{(a)}$ Show that angle $A B C=90^{\circ}$. $[2]$

$\text{(b)}$ Hence state the coordinates of $D$. $[1]$

$\text{(c)}$ Find an equation of the circle. $[2]$

The point $E$ lies on the circumference of the circle such that $BE$ is a diameter.

$\text{(d)}$ Find an equation of the tangent to the circle at $E$. $[5]$