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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 | 5 | 7 | 6 | 8 | 6 | 7 | 9 | 11 | 10 | 83 |

Score |

Question 1 Code: 9709/13/M/J/17/2, Topic: Series

The common ratio of a geometric progression is $r$. The first term of the progression is $\left(r^{2}-3 r+2\right)$ and the sum to infinity is $S$.

$\text{(i)}$ Show that $S=2-r$. $[2]$

$\text{(ii)}$ Find the set of possible values that $S$ can take. $[2]$

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: 9709/13/M/J/11/3, Topic: Coordinate geometry

The line $\displaystyle\frac{x}{a}+\frac{y}{b}=1$, where $a$ and $b$ are positive constants, meets the $x$-axis at $P$ and the $y$-axis at $Q$. Given that $P Q=\sqrt{(} 45)$ and that the gradient of the line $P Q$ is $-\frac{1}{2}$, find the values of $a$ and $b$. $[5]$

Question 4 Code: 9709/12/M/J/13/3, Topic: Quadratics

The straight line $y=m x+14$ is a tangent to the curve $\displaystyle y=\frac{12}{x}+2$ at the point $P$. Find the value of the constant $m$ and the coordinates of $P$. $[5]$

Question 5 Code: 9709/13/M/J/14/5, Topic: Differentiation, Quadratics

A function $\mathrm{f}$ is such that $\displaystyle\mathrm{f}(x)=\frac{15}{2 x+3}$ for $0 \leqslant x \leqslant 6$

$\text{(i)}$ Find an expression for $\mathrm{f}^{\prime}(x)$ and use your result to explain why $\mathrm{f}$ has an inverse. $[3]$

$\text{(ii)}$ Find an expression for $\mathrm{f}^{-1}(x)$, and state the domain and range of $\mathrm{f}^{-1}$. $[4]$

Question 6 Code: 9709/12/M/J/21/5, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=2 x^{2}+3$ for $x \geqslant 0$.

$\text{(a)}$ Find and simplify an expression for $\mathrm{ff}(x)$. $[2]$

$\text{(b)}$ Solve the equation $\mathrm{ff}(x)=34 x^{2}+19$. $[4]$

Question 7 Code: 9709/12/M/J/12/6, Topic: Circular measure

The diagram shows a metal plate made by removing a segment from a circle with centre $O$ and radius $8 \mathrm{~cm}.$ The line $A B$ is a chord of the circle and angle $A O B=2.4$ radians. Find

$\text{(i)}$ the length of $A B$, $[2]$

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

$\text{(iii)}$ the area of the plate. $[3]$

Question 8 Code: 9709/12/M/J/16/6, Topic: Circular measure

The diagram shows a circle with radius $r \mathrm{~cm}$ and centre $O$. The line $P T$ is the tangent to the circle at $P$ and angle $P O T=\alpha$ radians. The line $O T$ meets the circle at $Q.$

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

$\text{(ii)}$ In the case where $\alpha=\frac{1}{3} \pi$ and $r=10$, find the area of the shaded region correct to 2 significant figures. $[3]$

Question 9 Code: 9709/12/M/J/18/8, Topic: Coordinate geometry

Points $A$ and $B$ have coordinates $(h, h)$ and $(4 h+6,5 h)$ respectively. The equation of the perpendicular bisector of $A B$ is $3 x+2 y=k$. Find the values of the constants $\mathrm{h}$ and $k$. $[7]$

Question 10 Code: 9709/13/M/J/20/8, Topic: Series

The first term of a progression is $\sin ^{2} \theta$, where $0 < \theta < \frac{1}{2} \pi$. The second term of the progression is $\sin ^{2} \theta \cos ^{2} \theta$.

$\text{(a)}$ Given that the progression is geometric, find the sum to infinity. It is now given instead that the progression is arithmetic. $[3]$

$\text{(b)} \quad \text{(i)}$ Find the common difference of the progression in terms of $\sin \theta$. $[3]$

$ \quad \quad \text{(ii)}$ Find the sum of the first 16 terms when $\theta=\frac{1}{3} \pi$. $[3]$

Question 11 Code: 9709/12/M/J/16/11, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto 6 x-x^{2}-5$ for $x \in \mathbb{R}$.

$\text{(i)}$ Find the set of values of $x$ for which $\mathrm{f}(x) \leqslant 3$. $[3]$

$\text{(ii)}$ Given that the line $y=m x+c$ is a tangent to the curve $y=\mathrm{f}(x)$, show that $4 c=m^{2}-12 m+16$. $[3]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 6 x-x^{2}-5$ for $x \geqslant k$, where $k$ is a constant.

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

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

$\text{(v)}$ For this value of $k$, find an expression for $\mathrm{g}^{-1}(x)$. $[2]$

Question 12 Code: 9709/12/M/J/21/11, Topic: Differentiation, Integration

The gradient of a curve is given by $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=6(3 x-5)^{3}-k x^{2}$, where $k$ is a constant. The curve has a stationary point at $(2,-3.5)$.

$\text{(a)}$ Find the value of $k$. $[2]$

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

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

$\text{(d)}$ Determine the nature of the stationary point at $(2,-3.5)$. $[2]$