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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 | 5 | 6 | 5 | 6 | 6 | 6 | 7 | 7 | 12 | 8 | 12 | 84 |

Score |

Question 1 Code: 9709/11/M/J/12/2, Topic: Series

Find the coefficient of $x^{6}$ in the expansion of $\displaystyle\left(2 x^{3}-\frac{1}{x^{2}}\right)^{7}$. $[4]$

Question 2 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 3 Code: 9709/13/M/J/21/3, Topic: Quadratics

A line with equation $y=m x-6$ is a tangent to the curve with equation $y=x^{2}-4 x+3$.

Find the possible values of the constant $m$, and the corresponding coordinates of the points at which the line touches the curve. $[6]$

Question 4 Code: 9709/11/M/J/14/4, Topic: Differentiation

A curve has equation $\displaystyle y=\frac{4}{(3 x+1)^{2}}$. Find the equation of the tangent to curve at the point where the line $x=-1$ intersects the curve. $[5]$

Question 5 Code: 9709/13/M/J/14/4, Topic: Trigonometry

$\text{(i)}$ Prove the identity $\displaystyle\frac{\tan x+1}{\sin x \tan x+\cos x} \equiv \sin x+\cos x$. $[3]$

$\text{(ii)}$ Hence solve the equation $\displaystyle\frac{\tan x+1}{\sin x \tan x+\cos x}=3 \sin x-2 \cos x$ for $0 \leqslant x \leqslant 2 \pi$. $[3]$

Question 6 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 7 Code: 9709/11/M/J/20/5, Topic: Quadratics

The equation of a line is $y=m x+c$, where $m$ and $c$ are constants, and the equation of a curve is $x y=16$

$\text{(a)}$ Given that the line is a tangent to the curve, express $m$ in terms of $c$. $[3]$

$\text{(b)}$ Given instead that $m=-4$, find the set of values of $c$ for which the line intersects the curve at two distinct points. $[3]$

Question 8 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 9 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 10 Code: 9709/12/M/J/12/10, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$$ \begin{aligned} &\mathrm{f}: x \mapsto 2 x+5 \quad \text { for } x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto \frac{8}{x-3} \quad \text { for } x \in \mathbb{R}, x \neq 3 \end{aligned} $$$\text{(i)}$ Obtain expressions, in terms of $x$, for $\mathrm{f}^{-1}(x)$ and $\mathrm{g}^{-1}(x)$, stating the value of $x$ for which $\mathrm{g}^{-1}(x)$ is not defined. $[4]$

$\text{(ii)}$ Sketch the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$ on the same diagram, making clear the relationship between the two graphs. $[3]$

$\text{(iii)}$ Given that the equation $f g(x)=5-k x$, where $k$ is a constant, has no solutions, find the set of possible values of $k$. $[5]$

Question 11 Code: 9709/12/M/J/13/10, Topic: Series

$\text{(a)}$ The first and last terms of an arithmetic progression are 12 and 48 respectively. The sum of the first four terms is 57. Find the number of terms in the progression. $[4]$

$\text{(b)}$ The third term of a geometric progression is four times the first term. The sum of the first six terms is $k$ times the first term. Find the possible values of $k$. $[4]$

Question 12 Code: 9709/12/M/J/16/10, Topic: Differentiation, Integration, Coordinate geometry

The diagram shows the part of the curve $\displaystyle y=\frac{8}{x}+2 x$ for $x>0$, and the minimum point $M$

$\text{(i)}$ Find expressions for $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}, \displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$ and $\displaystyle\int y^{2} \mathrm{~d} x$. $[5]$

$\text{(ii)}$ Find the coordinates of $M$ and determine the coordinates and nature of the stationary point on the part of the curve for which $x<0$. $[5]$

$\text{(iii)}$ Find the volume obtained when the region bounded by the curve, the $x$-axis and the lines $x=1$ and $x=2$ is rotated through $360^{\circ}$ about the $x$-axis. $[2]$