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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 | 4 | 5 | 5 | 7 | 6 | 8 | 7 | 8 | 11 | 11 | 11 | 87 |

Score |

Question 1 Code: 9709/12/O/N/10/2, Topic: Trigonometry

Prove the identity$$\tan ^{2} x-\sin ^{2} x \equiv \tan ^{2} x \sin ^{2} x$$ $[4]$

Question 2 Code: 9709/12/O/N/18/2, Topic: Integration

Showing all necessary working, find $\displaystyle\int_{1}^{4}\left(\sqrt{x}+\frac{2}{\sqrt{x}}\right) \mathrm{d} x$. $[4]$Question 3 Code: 9709/11/O/N/18/3, Topic: Coordinate geometry

Two points $A$ and $B$ have coordinates $(3 a,-a)$ and $(-a, 2 a)$ respectively, where $a$ is a positive constant.

$\text{(i)}$ Find the equation of the line through the origin parallel to $A B$. $[2]$

$\text{(ii)}$ The length of the line $A B$ is $3 \frac{1}{3}$ units. Find the value of $a$. $[3]$

Question 4 Code: 9709/11/O/N/12/4, Topic: Series

$\text{(i)}$ Find the first 3 terms in the expansion of $\left(2 x-x^{2}\right)^{6}$ in ascending powers of $x$. $[3]$

$\text{(ii)}$ Hence find the coefficient of $x^{8}$ in the expansion of $(2+x)\left(2 x-x^{2}\right)^{6}$. $[2]$

Question 5 Code: 9709/11/O/N/19/5, Topic: Trigonometry

$\text{(i)}$ Given that $\displaystyle 4 \tan x+3 \cos x+\displaystyle\frac{1}{\cos x}=0$, show, without using a calculator, that $\sin x=-\frac{2}{3}.$ $[3]$

$\text{(ii)}$ Hence, showing all necessary working, solve the equation

$$ 4 \tan \left(2 x-20^{\circ}\right)+3 \cos \left(2 x-20^{\circ}\right)+\frac{1}{\cos \left(2 x-20^{\circ}\right)}=0 $$for $0^{\circ} \leqslant x \leqslant 180^{\circ}$. $[4]$

Question 6 Code: 9709/12/M/J/20/5, Topic: Functions

The function $\mathrm{f}$ is defined for $x \in \mathbb{R}$ by

$$ \text { f: } x \mapsto a-2 x $$where $a$ is a constant.

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

$\text{(b)}$ Given that $\mathrm{ff}(x)=\mathrm{f}^{-1}(x)$, find $x$ in terms of $a$. $[2]$

Question 7 Code: 9709/13/M/J/11/6, Topic: Series

$\text{(a)}$ A geometric progression has a third term of 20 and a sum to infinity which is three times the first term. Find the first term. $[4]$

$\text{(b)}$ An arithmetic progression is such that the eighth term is three times the third term. Show that the sum of the first eight terms is four times the sum of the first four terms. $[4]$

Question 8 Code: 9709/12/M/J/13/7, Topic: Coordinate geometry

The point $R$ is the reflection of the point $(-1,3)$ in the line $3 y+2 x=33$. Find by calculation the coordinates of $R$. $[7]$

Question 9 Code: 9709/11/O/N/10/8, Topic: Differentiation

The diagram shows a metal plate consisting of a rectangle with sides $x \mathrm{~cm}$ and $y \mathrm{~cm}$ and a quarter-circle of radius $x \mathrm{~cm}.$ The perimeter of the plate is $60 \mathrm{~cm}$.

$\text{(i)}$ Express $y$ in terms of $x$. $[2]$

$\text{(ii)}$ Show that the area of the plate, $A \mathrm{~cm}^{2}$, is given by $A=30 x-x^{2}$. $[2]$

Given that $x$ can vary,

$\text{(iii)}$ find the value of $x$ at which $A$ is stationary, $[2]$

$\text{(iv)}$ find this stationary value of $A$, and determine whether it is a maximum or a minimum value. $[2]$

Question 10 Code: 9709/11/M/J/10/9, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto 2 x^{2}-12 x+7$ for $x \in \mathbb{R}$.

$\text{(i)}$ Express $\mathrm{f}(x)$ in the form $a(x-b)^{2}-c$. $[3]$

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

$\text{(iii)}$ Find the set of values of $x$ for which $f(x)<21$. $[3]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 2 x+k$ for $x \in \mathbb{R}$.

$\text{(iv)}$ Find the value of the constant $k$ for which the equation $\operatorname{gf}(x)=0$ has two equal roots. $[4]$

Question 11 Code: 9709/13/M/J/11/9, Topic: Differentiation

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{2}{\sqrt{x}}-1$ and $P(9,5)$ is a point on the curve.

$\text{(i)}$ Find the equation of the curve. $[4]$

$\text{(ii)}$ Find the coordinates of the stationary point on the curve. $[3]$

$\text{(iii)}$ Find an expression for $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$ and determine the nature of the stationary point. $[2]$

$\text{(iv)}$ The normal to the curve at $P$ makes an angle of $\tan ^{-1} k$ with the positive $x$-axis. Find the value of $k$. $[2]$

Question 12 Code: 9709/11/M/J/19/11, Topic: Coordinate geometry, Integration

The diagram shows part of the curve $\displaystyle y=\frac{3}{\sqrt{(1+4 x)}}$ and a point $P(2,1)$ lying on the curve. The normal to the curve at $P$ intersects the $x$-axis at $Q$.

$\text{(i)}$ Show that the $x$-coordinate of $Q$ is $\frac{16}{9}$. $[5]$

$\text{(ii)}$ Find, showing all necessary working, the area of the shaded region. $[6]$