$\require{\cancel}$ $\require{\stix[upint]}$

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 | 5 | 5 | 5 | 7 | 9 | 5 | 8 | 8 | 9 | 75 |

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/19/2, Topic: Coordinate geometry

Two points $A$ and $B$ have coordinates $(1,3)$ and $(9,-1)$ respectively. The perpendicular bisector of $A B$ intersects the $y$-axis at the point $C$. Find the coordinates of $C$. $[5]$

Question 3 Code: 9709/11/M/J/11/3, Topic: Integration

$\text{(i)}$ Sketch the curve $y=(x-2)^{2}$. $[1]$

$\text{(ii)}$ The region enclosed by the curve, the $x$-axis and the $y$-axis is rotated through $360^{\circ}$ about the $x$-axis. Find the volume obtained, giving your answer in terms of $\pi$. $[4]$

Question 4 Code: 9709/12/M/J/11/3, Topic: Quadratics

The equation $x^{2}+p x+q=0$, where $p$ and $q$ are constants, has roots $-3$ and 5.

$\text{(i)}$ Find the values of $p$ and $q$. $[2]$

$\text{(ii)}$ Using these values of $p$ and $q$, find the value of the constant $r$ for which the equation $x^{2}+p x+q+r=0$ has equal roots. $[3]$

Question 5 Code: 9709/12/M/J/18/3, Topic: Series

A company producing salt from sea water changed to a new process. The amount of salt obtained each week increased by $2 \%$ of the amount obtained in the preceding week. It is given that in the first week after the change the company obtained $8000 \mathrm{~kg}$ of salt.

$\text{(i)}$ Find the amount of salt obtained in the 12 th week after the change. $[3]$

$\text{(ii)}$ Find the total amount of salt obtained in the first 12 weeks after the change. $[2]$

Question 6 Code: 9709/12/M/J/15/4, Topic: Differentiation

Variables $u, x$ and $y$ are such that $u=2 x(y-x)$ and $x+3 y=12$. Express $u$ in terms of $x$ and hence find the stationary value of $u$. $[5]$

Question 7 Code: 9709/11/M/J/17/4, Topic: Series

$\text{(a)}$ An arithmetic progression has a first term of 32 , a 5 th term of 22 and a last term of $-28$. Find the sum of all the terms in the progression. $[4]$

$\text{(b)}$ Each year a school allocates a sum of money for the library. The amount allocated each year increases by $2.5 \%$ of the amount allocated the previous year. In 2005 the school allocated $\$ 2000$. Find the total amount allocated in the years 2005 to 2014 inclusive. $[3]$

Question 8 Code: 9709/13/M/J/18/7, Topic: Trigonometry

$\text{(a)}$ $\quad\text{(i)}$ Express $\displaystyle \frac{\tan ^{2} \theta-1}{\tan ^{2} \theta+1}$ in the form $a \sin ^{2} \theta+b$, where $a$ and $b$ are constants to be found. $[3]$

$\qquad \text{(ii)}$ Hence, or otherwise, and showing all necessary working, solve the equation. $[2]$

$$ \frac{\tan ^{2} \theta-1}{\tan ^{2} \theta+1}=\frac{1}{4} $$$\qquad$ for $-90^{\circ} \leqslant \theta \leqslant 0^{\circ}$

$\text{(b)}$

The diagram shows the graphs of $y=\sin x$ and $y=2 \cos x$ for $-\pi \leqslant x \leqslant \pi$. The graphs intersect at the points $A$ and $B$.

$\text{(i)}$ Find the $x$-coordinate of $A$. $[2]$

$\text{(ii)}$ Find the $y$-coordinate of $B$. $[2]$

Question 9 Code: 9709/11/M/J/21/7, Topic: Trigonometry

$\text{(a)}$ Prove the identity $\displaystyle \frac{1-2 \sin ^{2} \theta}{1-\sin ^{2} \theta} \equiv 1-\tan ^{2} \theta$. $[2]$

$\text{(b)}$ Hence solve the equation $\displaystyle \frac{1-2 \sin ^{2} \theta}{1-\sin ^{2} \theta}=2 \tan ^{4} \theta$ for $0^{\circ} \leqslant \theta \leqslant 180^{\circ}$. $[3]$

Question 10 Code: 9709/13/M/J/14/8, Topic: Quadratics

$\text{(i)}$ Express $2 x^{2}-10 x+8$ in the form $a(x+b)^{2}+c$, where $a, b$ and $c$ are constants, and use your answer to state the minimum value of $2 x^{2}-10 x+8$. $[4]$

$\text{(ii)}$ Find the set of values of $k$ for which the equation $2 x^{2}-10 x+8=k x$ has no real roots. $[4]$

Question 11 Code: 9709/12/M/J/20/9, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are such that

$$ \begin{aligned} &\mathrm{f}(x)=2-3 \sin 2 x \text { for } 0 \leqslant x \leqslant \pi, \\ &\mathrm{g}(x)=-2 \mathrm{f}(x) \quad \text { for } 0 \leqslant x \leqslant \pi \end{aligned} $$$\text{(a)}$ State the ranges of $\mathrm{f}$ and $\mathrm{g}$. $[3]$

The diagram below shows the graph of $y=\mathrm{f}(x)$.

$\text{(b)}$ Sketch, on this diagram, the graph of $y=\mathrm{g}(x)$. $[2]$

The function $\mathrm{h}$ is such that

$$ \mathrm{h}(x)=\mathrm{g}(x+\pi) \quad \text { for }-\pi \leqslant x \leqslant 0 $$$\text{(c)}$ Describe fully a sequence of transformations that maps the curve $y=\mathrm{f}(x)$ on to $y=\mathrm{h}(x)$. $[3]$

Question 12 Code: 9709/13/M/J/13/11, Topic: Coordinate geometry, Integration

The diagram shows part of the curve $\displaystyle y=\frac{8}{\sqrt{x}}-x$ and points $A(1,7)$ and $B(4,0)$ which lie on the curve. The tangent to the curve at $B$ intersects the line $x=1$ at the point $C$.

$\text{(i)}$ Find the coordinates of $C$. $[4]$

$\text{(ii)}$ Find the area of the shaded region. $[5]$