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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 | 2 | 3 | 5 | 5 | 6 | 5 | 7 | 7 | 8 | 7 | 10 | 10 | 75 |

Score |

Question 1 Code: 9709/11/M/J/14/1, Topic: Trigonometry

The diagram shows part of the graph of $y=a+b \sin x.$ State the values of the constants $a$ and $b$. $[2]$

Question 2 Code: 9709/11/M/J/16/1, Topic: Series

Find the term independent of $x$ in the expansion of $\displaystyle\left(x-\frac{3}{2 x}\right)^{6}$. $[3]$

Question 3 Code: 9709/13/M/J/19/1, Topic: Quadratics

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=x^{2}-4 x+8$ for $x \in \mathbb{R}$.

$\text{(i)}$ Express $x^{2}-4 x+8$ in the form $(x-a)^{2}+b$. $[2]$

$\text{(ii)}$ Hence find the set of values of $x$ for which $\mathrm{f}(x)<9$, giving your answer in exact form. $[3]$

Question 4 Code: 9709/13/M/J/13/2, Topic: Circular measure

The diagram shows a circle $C$ with centre $O$ and radius $3 \mathrm{~cm}$. The radii $O P$ and $O Q$ are extended to $S$ and $R$ respectively so that $O R S$ is a sector of a circle with centre $O$. Given that $P S=6 \mathrm{~cm}$ and that the area of the shaded region is equal to the area of circle $C$,

$\text{(i)}$ show that angle $P O Q=\frac{1}{4} \pi$ radians, $[3]$

$\text{(ii)}$ find the perimeter of the shaded region. $[2]$

Question 5 Code: 9709/11/M/J/10/3, Topic: Series

The ninth term of an arithmetic progression is 22 and the sum of the first 4 terms is $49$.

$\text{(i)}$ Find the first term of the progression and the common difference. $[4]$

The $n$th term of the progression is 46.

$\text{(ii)}$ Find the value of $n$. $[2]$

Question 6 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 7 Code: 9709/11/M/J/15/3, Topic: Series

$\text{(i)}$ Find the first three terms, in ascending powers of $x$, in the expansion of

$\text{(a)}$ $(1-x)^{6}$ $[2]$

$\text{(b)}$ $(1+2 x)^{6}$. $[2]$

$\text{(ii)}$ Hence find the coefficient of $x^{2}$ in the expansion of $[(1-x)(1+2 x)]^{6}$. $[3]$

Question 8 Code: 9709/12/M/J/13/6, Topic: Vectors

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

$$ \overrightarrow{O A}=\mathbf{i}-2 \mathbf{j}+2 \mathbf{k} \quad \text { and } \quad \overrightarrow{O B}=3 \mathbf{i}+p \mathbf{j}+q \mathbf{k} $$where $p$ and $q$ are constants.

$\text{(i)}$ State the values of $p$ and $q$ for which $\overrightarrow{O A}$ is parallel to $\overrightarrow{O B}$. $[2]$

$\text{(ii)}$ In the case where $q=2 p$, find the value of $p$ for which angle $B O A$ is $90^{\circ}$. $[2]$

$\text{(iii)}$ In the case where $p=1$ and $q=8$, find the unit vector in the direction of $\overrightarrow{A B}$. $[3]$

Question 9 Code: 9709/13/M/J/10/7, Topic: Circular measure

The diagram shows a metal plate $A B C D E F$ which has been made by removing the two shaded regions from a circle of radius 10 cm and centre $O.$ The parallel edges $A B$ and $E D$ are both of length 12 cm.

$\text{(i)}$ Show that angle $D O E$ is 1.287 radians, correct to 4 significant figures. $[2]$

$\text{(ii)}$ Find the perimeter of the metal plate. $[3]$

$\text{(iii)}$ Find the area of the metal plate. $[3]$

Question 10 Code: 9709/13/M/J/15/7, Topic: Coordinate geometry

The point $A$ has coordinates $(p, 1)$ and the point $B$ has coordinates $(9,3 p+1)$, where $p$ is a constant.

$\text{(i)}$ For the case where the distance $A B$ is 13 units, find the possible values of $p$. $[3]$

$\text{(ii)}$ For the case in which the line with equation $2 x+3 y=9$ is perpendicular to $A B$, find the value of $p$. $[4]$

Question 11 Code: 9709/11/M/J/15/9, Topic: Differentiation

The equation of a curve is $y=x^{3}+p x^{2}$, where $p$ is a positive constant.

$\text{(i)}$ Show that the origin is a stationary point on the curve and find the coordinates of the other stationary point in terms of $p$. $[4]$

$\text{(ii)}$ Find the nature of each of the stationary points. $[3]$

Another curve has equation $y=x^{3}+p x^{2}+p x$.

$\text{(iii)}$ Find the set of values of $p$ for which this curve has no stationary points. $[3]$

Question 12 Code: 9709/11/M/J/17/10, Topic: Differentiation, Integration

The diagram shows part of the curve $\displaystyle y=\frac{4}{5-3 x}$.

$\text{(i)}$ Find the equation of the normal to the curve at the point where $x=1$ in the form $y=m x+c$, where $m$ and $c$ are constants. $[5]$

The shaded region is bounded by the curve, the coordinate axes and the line $x=1$.

$\text{(ii)}$ Find, showing all necessary working, the volume obtained when this shaded region is rotated through $360^{\circ}$ about the $x$-axis. $[5]$