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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 | 6 | 5 | 7 | 7 | 7 | 7 | 9 | 8 | 11 | 9 | 10 | 90 |

Score |

Question 1 Code: 9709/13/O/N/20/1, Topic: Quadratics

$\text{(a)}$ Express $x^{2}+6 x+5$ in the form $(x+a)^{2}+b$, where $a$ and $b$ are constants. $[2]$

$\text{(b)}$ The curve with equation $y=x^{2}$ is transformed to the curve with equation $y=x^{2}+6 x+5$. Describe fully the transformation(s) involved. $[2]$

Question 2 Code: 9709/12/O/N/17/2, Topic: Functions

A function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto 4-5 x$ for $x \in \mathbb{R}$.

$\text{(i)}$ Find an expression for $\mathrm{f}^{-1}(x)$ and find the point of intersection of the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$. $[3]$

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

Question 3 Code: 9709/11/O/N/19/4, Topic: Series

A runner who is training for a long-distance race plans to run increasing distances each day for 21 days. She will run $x \mathrm{~km}$ on day 1 , and on each subsequent day she will increase the distance by $10 \%$ of the previous day's distance. On day 21 she will run $20 \mathrm{~km}$.

$\text{(i)}$ Find the distance she must run on day 1 in order to achieve this. Give your answer in $\mathrm{km}$ correct to 1 decimal place. $[3]$

$\text{(ii)}$ Find the total distance she runs over the 21 days. $[2]$

Question 4 Code: 9709/12/O/N/17/5, Topic: Trigonometry

$\text{(i)}$ Show that the equation $\cos 2 x\left(\tan ^{2} 2 x+3\right)+3=0$ can be expressed as $[3]$

$$ 2 \cos ^{2} 2 x+3 \cos 2 x+1=0 $$$\text{(ii)}$ Hence solve the equation $\cos 2 x\left(\tan ^{2} 2 x+3\right)+3=0$ for $0^{\circ} \leqslant x \leqslant 180^{\circ}$. $[4]$

Question 5 Code: 9709/11/M/J/18/5, Topic: Coordinate geometry

The diagram shows a kite $O A B C$ in which $A C$ is the line of symmetry. The coordinates of $A$ and $C$ are $(0,4)$ and $(8,0)$ respectively and $O$ is the origin.

$\text{(i)}$ Find the equations of $A C$ and $O B$. $[4]$

$\text{(ii)}$ Find, by calculation, the coordinates of $B$. $[3]$

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

The diagram shows two circles with centres $A$ and $B$ having radii $8 \mathrm{~cm}$ and $10 \mathrm{~cm}$ respectively. The two circles intersect at $C$ and $D$ where $C A D$ is a straight line and $A B$ is perpendicular to $C D$.

$\text{(i)}$ Find angle $A B C$ in radians. $[1]$

$\text{(ii)}$ Find the area of the shaded region. $[6]$

Question 7 Code: 9709/12/M/J/18/7, Topic: Functions

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

$\text{(i)}$ Express $7-2 x^{2}-12 x$ in the form $a-2(x+b)^{2}$, where $a$ and $b$ are constants. $[2]$

$\text{(ii)}$ State the coordinates of the stationary point on the curve $\mathrm{y = f}(x)$. $[1]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 7-2 x^{2}-12 x$ for $x \geqslant k$.

$\text{(iii)}$ State the smallest value of $k$ for which $\mathrm{g}$ has an inverse. $[1]$

$\text{(iv)}$ For this value of $k$, find $\mathrm{g}^{-1}(x)$. $[3]$

Question 8 Code: 9709/11/M/J/18/8, Topic: Series

$\text{(a)}$ A geometric progression has a second term of 12 and a sum to infinity of 54. Find the possible values of the first term of the progression. $[4]$

$\text{(b)}$ The $n$th term of a progression is $p+q n$, where $p$ and $q$ are constants, and $S_{n}$ is the sum of the first $n$ terms.

$\text{(i)}$ Find an expression, in terms of $p, q$ and $n$, for $S_{n}$. $[3]$

$\text{(ii)}$ Given that $S_{4}=40$ and $S_{6}=72$, find the values of $p$ and $q$. $[2]$

Question 9 Code: 9709/13/O/N/20/8, Topic: Differentiation

The equation of a curve is $\displaystyle y=2 x+1+\frac{1}{2 x+1}$ for $x>-\frac{1}{2}$.

$\text{(a)}$ Find $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$ and $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$. $[3]$

$\text{(b)}$ Find the coordinates of the stationary point and determine the nature of the stationary point. $[5]$

Question 10 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 11 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. $[2]$

Question 12 Code: 9709/13/M/J/21/10, Topic: Coordinate geometry

Points $A(-2,3), B(3,0)$ and $C(6,5)$ lie on the circumference of a circle with centre $D$.

$\text{(a)}$ Show that angle $A B C=90^{\circ}$. $[2]$

$\text{(b)}$ Hence state the coordinates of $D$. $[1]$

$\text{(c)}$ Find an equation of the circle. $[2]$

The point $E$ lies on the circumference of the circle such that $BE$ is a diameter.

$\text{(d)}$ Find an equation of the tangent to the circle at $E$. $[5]$