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

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 | 5 | 5 | 4 | 4 | 6 | 3 | 5 | 8 | 9 | 8 | 8 | 9 | 74 |

Score |

Question 1 Code: 9709/13/M/J/10/1, Topic: Series

The first term of a geometric progression is 12 and the second term is $-$6. Find

$\text{(i)}$ the tenth term of the progression, $[3]$

$\text{(ii)}$ the sum to infinity. $[2]$

Question 2 Code: 9709/12/M/J/13/2, Topic: Series

Find the coefficient of $x^{2}$ in the expansion of

$\text{(i)}$ $\displaystyle\left(2 x-\frac{1}{2 x}\right)^{6}$, $[2]$

$\text{(ii)}$ $\displaystyle\left(1+x^{2}\right)\left(2 x-\frac{1}{2 x}\right)^{6}$. $[3]$

Question 3 Code: 9709/13/M/J/15/2, Topic: Integration

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=(2 x+1)^{\frac{1}{2}}$ and the point $(4,7)$ lies on the curve. Find the equation of the curve. $[4]$

Question 4 Code: 9709/11/M/J/18/2, Topic: Differentiation

A point is moving along the curve $\displaystyle y=2 x+\frac{5}{x}$ in such a way that the $x$-coordinate is increasing at a constant rate of $0.02$ units per second. Find the rate of change of the $y$-coordinate when $x=1$. $[4]$

Question 5 Code: 9709/12/M/J/16/4, Topic: Series

Find the term that is independent of $x$ in the expansion of

$\text{(i)}$ $\displaystyle\left(x-\frac{2}{x}\right)^{6}$, $[2]$

$\text{(ii)}$ $\displaystyle\left(2+\frac{3}{x^{2}}\right)\left(x-\frac{2}{x}\right)^{6}$. $[4]$

Question 6 Code: 9709/11/M/J/21/4, Topic: Trigonometry

The diagram shows part of the graph of $y=a \tan (x-b)+c$

Given that $0 < b < \pi$, state the values of the constants $a, b$ and $c$. $[3]$

Question 7 Code: 9709/12/M/J/21/4, Topic: Series

The coefficient of $x$ in the expansion of $\displaystyle \left(4 x+\frac{10}{x}\right)^{3}$ is $p$. The coefficient of $\displaystyle \frac{1}{x}$ in the expansion of $\displaystyle \left(2 x+\frac{k}{x^{2}}\right)^{5}$ is $q$.

Given that $p=6 q$, find the possible values of $k$. $[5]$

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

$\text{(i)}$ The tangent to the curve $y=x^{3}-9 x^{2}+24 x-12$ at a point $A$ is parallel to the line $y=2-3 x$. Find the equation of the tangent at $A$. $[6]$

$\text{(ii)}$ The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=x^{3}-9 x^{2}+24 x-12$ for $x>k$, where $k$ is a constant. Find the smallest value of $k$ for $\mathrm{f}$ to be an increasing function. $[2]$

Question 9 Code: 9709/11/M/J/11/9, Topic: Circular measure

In the diagram, $O A B$ is an isosceles triangle with $O A=O B$ and angle $A O B=2 \theta$ radians. Arc $P S T$ has centre $O$ and radius $r$, and the line $A S B$ is a tangent to the $\operatorname{arc} P S T$ at $S$.

$\text{(i)}$ Find the total area of the shaded regions in terms of $r$ and $\theta$. $[4]$

$\text{(ii)}$ In the case where $\theta=\frac{1}{3} \pi$ and $r=6$, find the total perimeter of the shaded regions, leaving your answer in terms of $\sqrt{3}$ and $\pi$. $[5]$

Question 10 Code: 9709/13/M/J/12/9, Topic: Integration, Differentiation

A curve is such that $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=-4 x.$ The curve has a maximum point at $(2,12)$.

$\text{(i)}$ Find the equation of the curve. $[6]$

A point $P$ moves along the curve in such a way that the $x$-coordinate is increasing at $0.05$ units per second.

$\text{(ii)}$ Find the rate at which the $y$-coordinate is changing when $x=3$, stating whether the $y$-coordinate is increasing or decreasing. $[2]$

Question 11 Code: 9709/13/M/J/14/9, Topic: Differentiation

The base of a cuboid has sides of length $x \mathrm{~cm}$ and $3 x \mathrm{~cm}$. The volume of the cuboid is $288 \mathrm{~cm}^{3}$.

$\text{(i)}$ Show that the total surface area of the cuboid, $A \mathrm{~cm}^{2}$, is given by $[3]$

$$ A=6 x^{2}+\frac{768}{x}. $$$\text{(ii)}$ Given that $x$ can vary, find the stationary value of $A$ and determine its nature. $[5]$

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. $[5]$