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

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 | 3 | 5 | 5 | 7 | 7 | 6 | 10 | 9 | 8 | 6 | 9 | 9 | 84 |

Score |

Question 1 Code: 9709/13/M/J/21/1, Topic: Integration

A curve with equation $y=\mathrm{f}(x)$ is such that $\displaystyle \mathrm{f}^{\prime}(x)=6 x^{2}-\frac{8}{x^{2}}.$ It is given that the curve passes through the point $(2,7)$.

Find $\mathrm{f}(x)$. $[3]$

Question 2 Code: 9709/13/M/J/11/2, Topic: Coordinate geometry

Find the set of values of $m$ for which the line $y=m x+4$ intersects the curve $y=3 x^{2}-4 x+7$ at two distinct points. $[5]$

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/15/4, Topic: Vectors

Relative to the origin $O$, the position vectors of points $A$ and $B$ are given by

$$ \overrightarrow{O A}=\left(\begin{array}{r} 3 \\ 0 \\ -4 \end{array}\right) \quad \text { and } \quad \overrightarrow{O B}=\left(\begin{array}{r} 6 \\ -3 \\ 2 \end{array}\right) $$$\text{(i)}$ Find the cosine of angle $A O B$. $[3]$

The position vector of $C$ is given by $\overrightarrow{O C}=\left(\begin{array}{c}k \\ -2 k \\ 2 k-3\end{array}\right)$

$\text{(ii)}$ Given that $A B$ and $O C$ have the same length, find the possible values of $k$. $[4]$

Question 5 Code: 9709/12/M/J/17/4, Topic: Circular measure

The diagram shows a circle with radius $r \mathrm{~cm}$ and centre $O$. Points $A$ and $B$ lie on the circle and $A B C D$ is a rectangle. Angle $A O B=2 \theta$ radians and $A D=r \mathrm{~cm}$.

$\text{(i)}$ Express the perimeter of the shaded region in terms of $r$ and $\theta$. $[3]$

$\text{(ii)}$ In the case where $r=5$ and $\theta=\frac{1}{6} \pi$, find the area of the shaded region. $[4]$

Question 6 Code: 9709/13/M/J/21/6, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are both defined for $x \in \mathbb{R}$ and are given by

$$ \begin{aligned} &\mathrm{f}(x)=x^{2}-2 x+5 \\ &\mathrm{~g}(x)=x^{2}+4 x+13 \end{aligned} $$$\text{(a)}$ By first expressing each of $\mathrm{f}(x)$ and $\mathrm{g}(x)$ in completed square form, express $\mathrm{g}(x)$ in the form $\mathrm{f}(x+p)+q$, where $p$ and $q$ are constants. $[4]$

$\text{(b)}$ Describe fully the transformation which transforms the graph of $y = f(x)$ to the graph of $y = g(x)$. $[2]$

Question 7 Code: 9709/11/M/J/13/8, Topic: Quadratics, Functions

$\text{(i)}$ Express $2 x^{2}-12 x+13$ in the form $a(x+b)^{2}+c$, where $a, b$ and $c$ are constants. $[3]$

$\text{(ii)}$ The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=2 x^{2}-12 x+13$ for $x \geqslant k$, where $k$ is a constant. It is given that $\mathrm{f}$ is a one-one function. State the smallest possible value of $k$. $[1]$

The value of $k$ is now given to be 7.

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

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

Question 8 Code: 9709/11/M/J/12/9, Topic: Coordinate geometry

The coordinates of $A$ are $(-3,2)$ and the coordinates of $C$ are $(5,6).$ The mid-point of $A C$ is $M$ and the perpendicular bisector of $A C$ cuts the $x$-axis at $B$.

$\text{(i)}$ Find the equation of $M B$ and the coordinates of $B$. $[5]$

$\text{(ii)}$ Show that $A B$ is perpendicular to $B C$. $[2]$

$\text{(iii)}$ Given that $A B C D$ is a square, find the coordinates of $D$ and the length of $A D$. $[2]$

Question 9 Code: 9709/13/M/J/16/9, Topic: Vectors

The position vectors of $A, B$ and $C$ relative to an origin $O$ are given by

$$ \overrightarrow{O A}=\left(\begin{array}{r} 2 \\ 3 \\ -4 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{l} 1 \\ 5 \\ p \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{l} 5 \\ 0 \\ 2 \end{array}\right) $$where $p$ is a constant.

$\text{(i)}$ Find the value of $p$ for which the lengths of $A B$ and $C B$ are equal. $[4]$

$\text{(ii)}$ For the case where $p=1$, use a scalar product to find angle $A B C$. $[4]$

Question 10 Code: 9709/12/M/J/21/9, Topic: Integration

The diagram shows part of the curve with equation $y^{2}=x-2$ and the lines $x=5$ and $y=1$. The shaded region enclosed by the curve and the lines is rotated through $360^{\circ}$ about the $x$-axis.

Find the volume obtained. $[6]$

Question 11 Code: 9709/13/M/J/21/9, Topic: Series

$\text{(a)}$ A geometric progression is such that the second term is equal to 24% of the sum to infinity.

Find the possible values of the common ratio. $[3]$

$\text{(b)}$ An arithmetic progression $P$ has first term $a$ and common difference $d$. An arithmetic progression $Q$ has first term $2(a+1)$ and common difference $(d+1)$. It is given that

$$ \frac{\text { 5th term of } P}{12\text{th term of } Q}=\frac{1}{3} \quad \text { and } \quad \frac{\text { Sum of first } 5 \text { terms of } P}{\text { Sum of first } 5 \text { terms of } Q}=\frac{2}{3} $$Find the value of $a$ and the value of $d$. $[6]$

Question 12 Code: 9709/12/M/J/20/10, Topic: Differentiation

The equation of a curve is $y=54 x-(2 x-7)^{3}$.

$\text{(a)}$ Find $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$ and $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$. $[4]$

$\text{(b)}$ Find the coordinates of each of the stationary points on the curve. $[3]$

$\text{(c)}$ Determine the nature of each of the stationary points. $[2]$