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Name of student | Date | ||||

Adm. number | Year/grade | Stream | |||

Subject | Pure Mathematics 1 (P1) | Variant(s) | P11 | ||

Start time | Duration | Stop time |

Qtn No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Marks | 4 | 6 | 5 | 7 | 7 | 8 | 7 | 8 | 7 | 9 | 10 | 14 | 92 |

Score |

Question 1 Code: 9709/11/M/J/12/1, Topic: Trigonometry

Solve the equation $\sin 2 x=2 \cos 2 x$, for $0^{\circ} \leqslant x \leqslant 180^{\circ}$. $[4]$

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: 9709/11/M/J/12/3, Topic: Circular measure

In the diagram, $A B C$ is an equilateral triangle of side $2 \mathrm{~cm}$. The mid-point of $B C$ is $Q.$ An arc of a circle with centre $A$ touches $B C$ at $Q$, and meets $A B$ at $P$ and $A C$ at $R.$ Find the total area of the shaded regions, giving your answer in terms of $\pi$ and $\sqrt{3}$. $[5]$

Question 4 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 5 Code: 9709/11/M/J/15/6, Topic: Coordinate geometry

The line with gradient $-2$ passing through the point $P(3 t, 2 t)$ intersects the $x$-axis at $A$ and the $y$-axis at $B$.

$\text{(i)}$ Find the area of triangle $A O B$ in terms of $t$. $[3]$

The line through $P$ perpendicular to $A B$ intersects the $x$-axis at $C$.

$\text{(ii)}$ Show that the mid-point of $P C$ lies on the line $y=x$. $[4]$

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/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 8 Code: 9709/11/M/J/12/8, Topic: Functions

The function $\mathrm{f}: x \mapsto x^{2}-4 x+k$ is defined for the domain $x \geqslant p$, where $k$ and $p$ are constants.

$\text{(i)}$ Express $\mathrm{f}(x)$ in the form $(x+a)^{2}+b+k$, where $a$ and $b$ are constants. $[2]$

$\text{(ii)}$ State the range of $\mathrm{f}$ in terms of $k$. $[1]$

$\text{(iii)}$ State the smallest value of $p$ for which $\mathrm{f}$ is one-one. $[1]$

$\text{(iv)}$ For the value of $p$ found in part $\text{(iii)}$, find an expression for $\mathrm{f}^{-1}(x)$ and state the domain of $\mathrm{f}^{-1}$, giving your answers in terms of $k$. $[4]$

Question 9 Code: 9709/11/M/J/19/9, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=2-3 \cos x$ for $0 \leqslant x \leqslant 2 \pi$.

$\text{(i)}$ State the range of $\mathrm{f}$. $[2]$

$\text{(ii)}$ Sketch the graph of $y=\mathrm{f}(x)$. The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=2-3 \cos x$ for $0 \leqslant x \leqslant p$, where $p$ is a constant. $[2]$

$\text{(iii)}$ State the largest value of $p$ for which $\mathrm{g}$ has an inverse. $[1]$

$\text{(iv)}$ For this value of $p$, find an expression for $\mathrm{g}^{-1}(x)$. $[3]$

Question 10 Code: 9709/11/M/J/14/10, Topic: Functions

The diagram shows the function $\mathrm{f}$ defined for $-1 \leqslant x \leqslant 4$, where

$$ \mathrm{f}(x)= \begin{cases}3 x-2 & \text { for }-1 \leqslant x \leqslant 1 \\ \frac{4}{5-x} & \text { for } 1 < x \leqslant 4\end{cases} $$$\text{(i)}$ State the range of $\mathrm{f}$. $[1]$

$\text{(ii)}$ Copy the diagram and on your copy sketch the graph of $y=\mathrm{f}^{-1}(x)$. $[2]$

$\text{(iii)}$ Obtain expressions to define the function $\mathrm{f}^{-1}$, giving also the set of values for which each expression is valid. $[6]$

Question 11 Code: 9709/11/M/J/14/11, Topic: Quadratics, Differentiation, Integration

A line has equation $y=2 x+c$ and a curve has equation $y=8-2 x-x^{2}$.

$\text{(i)}$ For the case where the line is a tangent to the curve, find the value of the constant $c$. $[3]$

$\text{(ii)}$ For the case where $c=11$, find the $x$-coordinates of the points of intersection of the line and the curve. Find also, by integration, the area of the region between the line and the curve. $[7]$

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 $x = 4$. $[4]$