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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 | 6 | 7 | 8 | 6 | 8 | 8 | 7 | 11 | 11 | 86 |

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/19/2, Topic: Quadratics

The line $4 y=x+c$, where $c$ is a constant, is a tangent to the curve $y^{2}=x+3$ at the point $P$ on the curve.

$\text{(i)}$ Find the value of $c$. $[3]$

$\text{(ii)}$ Find the coordinates of $P$. $[2]$

Question 3 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 4 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 5 Code: 9709/11/M/J/15/5, Topic: Quadratics

A piece of wire of length $24 \mathrm{~cm}$ is bent to form the perimeter of a sector of a circle of radius $r \mathrm{~cm}$.

$\text{(i)}$ Show that the area of the sector, $A \mathrm{~cm}^{2}$, is given by $A=12 r-r^{2}$. $[3]$

$\text{(ii)}$ Express $A$ in the form $a-(r-b)^{2}$, where $a$ and $b$ are constants. $[2]$

$\text{(iii)}$ Given that $r$ can vary, state the greatest value of $A$ and find the corresponding angle of the sector. $[2]$

Question 6 Code: 9709/11/M/J/17/6, Topic: Differentiation

The horizontal base of a solid prism is an equilateral triangle of side $x \mathrm{~cm}$. The sides of the prism are vertical. The height of the prism is $h \mathrm{~cm}$ and the volume of the prism is $2000 \mathrm{~cm}^{3}$.

$\text{(i)}$ Express $\mathrm{h}$ in terms of $x$ and show that the total surface area of the prism, $A \mathrm{~cm}^{2}$, is given by $[3]$

$$ A=\frac{\sqrt{3}}{2} x^{2}+\frac{24000}{\sqrt{3}} x^{-1} $$$\text{(ii)}$ Given that $x$ can vary, find the value of $x$ for which $A$ has a stationary value. $[3]$

$\text{(iii)}$ Determine, showing all necessary working, the nature of this stationary value. $[2]$

Question 7 Code: 9709/13/M/J/17/6, Topic: Quadratics

The line $3 y+x=25$ is a normal to the curve $y=x^{2}-5 x+k$. Find the value of the constant $k$. $[6]$

Question 8 Code: 9709/11/M/J/15/7, Topic: Series

$\text{(a)}$ The third and fourth terms of a geometric progression are $\frac{1}{3}$ and $\frac{2}{9}$ respectively. Find the sum to infinity of the progression. $[4]$

$\text{(b)}$ A circle is divided into 5 sectors in such a way that the angles of the sectors are in arithmetic progression. Given that the angle of the largest sector is 4 times the angle of the smallest sector, find the angle of the largest sector. $[4]$

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

A solid rectangular block has a square base of side $x \mathrm{~cm}$. The height of the block is $h \mathrm{~cm}$ and the total surface area of the block is 96 cm$^{2}$.

$\text{(i)}$ Express $\mathrm{h}$ in terms of $x$ and show that the volume, $V \mathrm{~cm}^{3}$, of the block is given by $[3]$

$$ V=24 x-\frac{1}{2} x^{3} $$Given that $x$ can vary,

$\text{(ii)}$ find the stationary value of $V$, $[3]$

$\text{(iii)}$ determine whether this stationary value is a maximum or a minimum. $[2]$

Question 10 Code: 9709/13/M/J/12/8, Topic: Circular measure

In the diagram, $A B$ is an arc of a circle with centre $O$ and radius $r$. The line $X B$ is a tangent to the circle at $B$ and $A$ is the mid-point of $O X$.

$\text{(i)}$ Show that angle $A O B=\frac{1}{3} \pi$ radians. $[2]$

Express each of the following in terms of $r, \pi$ and $\sqrt{3}$ :

$\text{(ii)}$ the perimeter of the shaded region, $[3]$

$\text{(iii)}$ the area of the shaded region. $[2]$

Question 11 Code: 9709/12/M/J/17/10, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=3 \tan \left(\frac{1}{2} x\right)-2$, for $-\frac{1}{2} \pi \leqslant x \leqslant \frac{1}{2} \pi$.

$\text{(i)}$ Solve the equation $\mathrm{f}(x)+4=0$, giving your answer correct to 1 decimal place. $[3]$

$\text{(ii)}$ Find an expression for $\mathrm{f}^{-1}(x)$ and find the domain of $\mathrm{f}^{-1}$. $[5]$

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

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. $[6]$