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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 | 6 | 5 | 7 | 7 | 8 | 8 | 11 | 6 | 10 | 10 | 11 | 95 |

Score |

Question 1 Code: 9709/12/M/J/12/4, Topic: Coordinate geometry

The point $A$ has coordinates $(-1,-5)$ and the point $B$ has coordinates $(7,1)$. The perpendicular bisector of $A B$ meets the $x$-axis at $C$ and the $y$-axis at $D$. Calculate the length of $C D$. $[6]$

Question 2 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 3 Code: 9709/12/M/J/15/5, Topic: Trigonometry

$\text{(i)}$ Prove the identity $\displaystyle\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta} \equiv \frac{\tan \theta-1}{\tan \theta+1}$. $[1]$

$\text{(ii)}$ Hence solve the equation $\displaystyle\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta}=\frac{\tan \theta}{6}$, for $0^{\circ} \leqslant \theta \leqslant 180^{\circ}$. $[4]$

Question 4 Code: 9709/13/M/J/12/6, Topic: Series

The first term of an arithmetic progression is 12 and the sum of the first 9 terms is 135.

$\text{(i)}$ Find the common difference of the progression. $[2]$

The first term, the ninth term and the $n$th term of this arithmetic progression are the first term, the second term and the third term respectively of a geometric progression.

$\text{(ii)}$ Find the common ratio of the geometric progression and the value of $n$. $[5]$

Question 5 Code: 9709/13/M/J/12/7, Topic: Coordinate geometry

The curve $\displaystyle y=\frac{10}{2 x+1}-2$ intersects the $x$-axis at $A$. The tangent to the curve at $A$ intersects the $y$-axis at $C$.

$\text{(i)}$ Show that the equation of $A C$ is $5 y+4 x=8$. $[5]$

$\text{(ii)}$ Find the distance $A C$. $[2]$

Question 6 Code: 9709/13/M/J/13/7, Topic: Coordinate geometry

The diagram shows three points $A(2,14), B(14,6)$ and $C(7,2).$ The point $X$ lies on $A B$, and $C X$ is perpendicular to $A B$. Find, by calculation,

$\text{(i)}$ the coordinates of $X$, $[6]$

$\text{(ii)}$ the ratio $A X: X B$. $[2]$

Question 7 Code: 9709/12/M/J/16/8, Topic: Coordinate geometry

Three points have coordinates $A(0,7), B(8,3)$ and $C(3 k, k).$ Find the value of the constant $k$ for which

$\text{(i)}$ $C$ lies on the line that passes through $A$ and $B$, $[4]$

$\text{(ii)}$ $C$ lies on the perpendicular bisector of $A B$. $[4]$

Question 8 Code: 9709/12/M/J/14/9, Topic: Differentiation, Integration

The diagram shows part of the curve $y=8-\sqrt{(} 4-x)$ and the tangent to the curve at $P(3,7)$.

$\text{(i)}$ Find expressions for $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$ and $\displaystyle\int y \mathrm{~d} x$. $[5]$

$\text{(ii)}$ Find the equation of the tangent to the curve at $P$ in the form $y=m x+c$. $[2]$

$\text{(iii)}$ Find, showing all necessary working, the area of the shaded region. $[4]$

Question 9 Code: 9709/12/M/J/21/9, Topic: Integration

The diagram shows part of the curve with equation $y^{2}=x-2$ and the lines $x=5$ and $y=1$. The shaded region enclosed by the curve and the lines is rotated through $360^{\circ}$ about the $x$-axis.

Find the volume obtained. $[6]$

Question 10 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 $E$. $[5]$

Question 11 Code: 9709/13/M/J/12/11, Topic: Functions

The function $\mathrm{f}$ is such that $\mathrm{f}(x)=8-(x-2)^{2}$, for $x \in \mathbb{R}$.

$\text{(i)}$ Find the coordinates and the nature of the stationary point on the curve $y=\mathrm{f}(x)$. $[3]$

The function $\mathrm{g}$ is such that $\mathrm{g}(x)=8-(x-2)^{2}$, for $k \leqslant x \leqslant 4$, where $k$ is a constant.

$\text{(ii)}$ State the smallest value of $k$ for which $\mathrm{g}$ has an inverse. $[1]$

For this value of $k$,

$\text{(iii)}$ find an expression for $\mathrm{g}^{-1}(x)$, $[3]$

$\text{(iv)}$ sketch, on the same diagram, the graphs of $y=\mathrm{g}(x)$ and $y=\mathrm{g}^{-1}(x)$. $[3]$

Question 12 Code: 9709/12/M/J/16/11, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto 6 x-x^{2}-5$ for $x \in \mathbb{R}$.

$\text{(i)}$ Find the set of values of $x$ for which $\mathrm{f}(x) \leqslant 3$. $[3]$

$\text{(ii)}$ Given that the line $y=m x+c$ is a tangent to the curve $y=\mathrm{f}(x)$, show that $4 c=m^{2}-12 m+16$. $[3]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 6 x-x^{2}-5$ for $x \geqslant k$, where $k$ is a constant.

$\text{(iii)}$ Express $6 x-x^{2}-5$ in the form $a-(x-b)^{2}$, where $a$ and $b$ are constants. $[2]$

$\text{(iv)}$ State the smallest value of $k$ for which $\mathrm{g}$ has an inverse. $[1]$

$\text{(v)}$ For this value of $k$, find an expression for $\mathrm{g}^{-1}(x)$. $[2]$