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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 | 3 | 4 | 4 | 7 | 5 | 5 | 6 | 6 | 6 | 8 | 11 | 69 |

Score |

Question 1 Code: 9709/11/O/N/12/1, Topic: Series

The first term of an arithmetic progression is 61 and the second term is 57. The sum of the first $n$ terms is $n$. Find the value of the positive integer $n$. $[4]$

Question 2 Code: 9709/12/O/N/13/1, Topic: Trigonometry

Given that $\cos x=p$, where $x$ is an acute angle in degrees, find, in terms of $p$,

$\text{(i)}$ $\sin x$, $[1]$

$\text{(ii)}$ $\tan x$, $[1]$

$\text{(iii)}$ $\tan \left(90^{\circ}-x\right)$. $[1]$

Question 3 Code: 9709/13/O/N/11/2, Topic: Series

The first and second terms of a progression are 4 and 8 respectively. Find the sum of the first 10 terms given that the progression is

$\text{(i)}$ an arithmetic progression, $[2]$

$\text{(ii)}$ a geometric progression. $[2]$

Question 4 Code: 9709/11/O/N/16/2, Topic: Series

Find the term independent of $x$ in the expansion of $\displaystyle\left(2 x+\frac{1}{2 x^{3}}\right)^{8}$. $[4]$

Question 5 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 6 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 7 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 8 Code: 9709/11/M/J/21/3, Topic: Series

$\text{(a)}$ Find the first three terms in the expansion of $(3-2 x)^{5}$ in ascending powers of $x$. $[3]$

$\text{(b)}$ Hence find the coefficient of $x^{2}$ in the expansion of $(4+x)^{2}(3-2 x)^{5}$. $[3]$

Question 9 Code: 9709/12/O/N/10/5, Topic: Series

$\text{(a)}$ The first and second terms of an arithmetic progression are 161 and 154 respectively.

The sum of the first $m$ terms is zero. Find the value of $m$. $[3]$

$\text{(b)}$ A geometric progression, in which all the terms are positive, has common ratio $r$.

The sum of the first $n$ terms is less than $90 \%$ of the sum to infinity. Show that $r^{n}>0.1$ $[3]$

Question 10 Code: 9709/11/M/J/20/7, Topic: Trigonometry

$\text{(a)}$ Prove the identity $\displaystyle \frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta} \equiv \frac{2}{\cos \theta}$. $[3]$

$\text{(b)}$ Hence solve the equation $\displaystyle \frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}=\frac{3}{\sin \theta}$, for $0 \leqslant \theta \leqslant 2 \pi$. $[3]$

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

The function $\mathrm{f}$ is such that $\mathrm{f}(x)=3-4 \cos ^{k} x$, for $0 \leqslant x \leqslant \pi$, where $k$ is a constant.

$\text{(i)}$ In the case where $k=2$,

$\text{(a)}$ find the range of $\mathrm{f}$, $[2]$

$\text{(b)}$ find the exact solutions of the equation $\mathrm{f}(x)=1$. $[3]$

$\text{(ii)}$ In the case where $k=1$,

$\text{(a)}$ sketch the graph of $y=\mathrm{f}(x)$, $[2]$

$\text{(b)}$ state, with a reason, whether $\mathrm{f}$ has an inverse. $[1]$

Question 12 Code: 9709/11/O/N/16/11, Topic: Coordinate geometry, Differentiation

The point $P(3,5)$ lies on the curve $\displaystyle y=\frac{1}{x-1}-\frac{9}{x-5}$.

$\text{(i)}$ Find the $x$-coordinate of the point where the normal to the curve at $P$ intersects the $x$-axis. $[5]$

$\text{(ii)}$ Find the $x$-coordinate of each of the stationary points on the curve and determine the nature of each stationary point, justifying your answers. $[6]$