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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 | 5 | 7 | 6 | 9 | 7 | 11 | 9 | 12 | 9 | 12 | 96 |

Score |

Question 1 Code: 9709/13/M/J/13/1, Topic: Integration

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\sqrt{\big(} 2 x+5\big)$ and $(2,5)$ is a point on the curve. Find the equation of the curve. $[4]$

Question 2 Code: 9709/12/M/J/14/1, Topic: Coordinate geometry

Find the coordinates of the point at which the perpendicular bisector of the line joining $(2,7)$ to $(10,3)$ meets the $x$-axis. $[5]$

Question 3 Code: 9709/12/M/J/15/3, Topic: Series

$\text{(i)}$ Find the coefficients of $x^{2}$ and $x^{3}$ in the expansion of $(2-x)^{6}$. $[3]$

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

Question 4 Code: 9709/13/M/J/19/5, Topic: Series

Two heavyweight boxers decide that they would be more successful if they competed in a lower weight class. For each boxer this would require a total weight loss of $13 \mathrm{~kg}$. At the end of week 1 they have each recorded a weight loss of $1 \mathrm{~kg}$ and they both find that in each of the following weeks their weight loss is slightly less than the week before.

Boxer $A$ 's weight loss in week 2 is $0.98 \mathrm{~kg}$. It is given that his weekly weight loss follows an arithmetic progression.

$\text{(i)}$ Write down an expression for his total weight loss after $x$ weeks. $[1]$

$\text{(ii)}$ He reaches his $13 \mathrm{~kg}$ target during week $n$. Use your answer to part $\text{(i)}$ to find the value of $n$. Boxer $B$ 's weight loss in week 2 is $0.92 \mathrm{~kg}$ and it is given that his weekly weight loss follows a geometric progression. $[2]$

$\text{(iii)}$ Calculate his total weight loss after 20 weeks and show that he can never reach his target. $[4]$

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/13/7, Topic: Quadratics, Differentiation, Coordinate geometry

A curve has equation $y=x^{2}-4 x+4$ and a line has equation $y=m x$, where $m$ is a constant.

$\text{(i)}$ For the case where $m=1$, the curve and the line intersect at the points $A$ and $B$. Find the coordinates of the mid-point of $A B$. $[4]$

$\text{(ii)}$ Find the non-zero value of $m$ for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve. $[5]$

Question 7 Code: 9709/11/M/J/16/8, Topic: Coordinate geometry

A curve has equation $\displaystyle y=3 x-\frac{4}{x}$ and passes through the points $A(1,-1)$ and $B(4,11)$. At each of the points $C$ and $D$ on the curve, the tangent is parallel to $A B$. Find the equation of the perpendicular bisector of $C D$. $[7]$

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: 9709/13/M/J/18/9, Topic: Vectors

The diagram shows a pyramid $O A B C D$ with a horizontal rectangular base $O A B C$. The sides $O A$ and $A B$ have lengths of 8 units and 6 units respectively. The point $E$ on $O B$ is such that $O E=2$ units. The point $D$ of the pyramid is 7 units vertically above $E$. Unit vectors $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$ are parallel to $O A$, $O C$ and $E D$ respectively.

$\text{(i)}$ Show that $\overrightarrow{O E}=1.6 \mathbf{i}+1.2 \mathbf{j}$. $[2]$

$\text{(ii)}$ Use a scalar product to find angle $B D O$. $[7]$

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

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

$$ \begin{aligned} &\mathrm{f}: x \mapsto 3 x-4, \quad x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto 2(x-1)^{3}+8, \quad x>1 \end{aligned} $$$\text{(i)}$ Evaluate $\mathrm{fg(2)}$. $[2]$

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

$\text{(iii)}$ Obtain an expression for $\mathrm{g}^{\prime}(x)$ and use your answer to explain why $\mathrm{g}$ has an inverse. $[3]$

$\text{(iv)}$ Express each of $\mathrm{f}^{-1}(x)$ and $\mathrm{g}^{-1}(x)$ in terms of $x$. $[4]$

Question 11 Code: 9709/12/M/J/20/10, Topic: Differentiation

The equation of a curve is $y=54 x-(2 x-7)^{3}$.

$\text{(a)}$ Find $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$ and $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$. $[4]$

$\text{(b)}$ Find the coordinates of each of the stationary points on the curve. $[3]$

$\text{(c)}$ Determine the nature of each of the stationary points. $[2]$

Question 12 Code: 9709/11/M/J/20/11, Topic: Coordinate geometry, Integration

The diagram shows part of the curve $\displaystyle y=\frac{8}{x+2}$ and the line $2 y+x=8$, intersecting at points $A$ and $B$. The point $C$ lies on the curve and the tangent to the curve at $C$ is parallel to $A B$.

$\text{(a)}$ Find, by calculation, the coordinates of $A, B$ and $C$. $[6]$

$\text{(b)}$ Find the volume generated when the shaded region, bounded by the curve and the line, is rotated through $360^{\circ}$ about the $x$-axis. $[6]$