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

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 | 5 | 4 | 6 | 7 | 9 | 7 | 9 | 9 | 12 | 9 | 87 |

Score |

Question 1 Code: 9709/11/M/J/20/2, Topic: Series

The coefficient of $\displaystyle \frac{1}{x}$ in the expansion of $\displaystyle \left(k x+\frac{1}{x}\right)^{5}+\left(1-\frac{2}{x}\right)^{8}$ is 74.

Find the value of the positive constant $k$. $[5]$

Question 2 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 3 Code: 9709/12/M/J/19/3, Topic: Differentiation, Integration

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{3}-\frac{4}{x^{2}}.$ The point $P(2,9)$ lies on the curve.

$\text{(i)}$ A point moves on the curve in such a way that the $x$-coordinate is decreasing at a constant rate of $0.05$ units per second. Find the rate of change of the $y$-coordinate when the point is at $P$. $[2]$

$\text{(ii)}$ Find the equation of the curve. $[3]$

Question 4 Code: 9709/11/M/J/21/5, Topic: Series

The fifth, sixth and seventh terms of a geometric progression are $8 k,-12$ and $2 k$ respectively. Given that $k$ is negative, find the sum to infinity of the progression. $[4]$

Question 5 Code: 9709/13/M/J/21/5, Topic: Circular measure

The diagram shows a triangle $A B C$, in which angle $A B C=90^{\circ}$ and $A B=4 \mathrm{~cm}.$ The sector $A B D$ is part of a circle with centre $A$. The area of the sector is $10 \mathrm{~cm}^{2}$.

$\text{(a)}$ Find angle $B A D$ in radians. $[2]$

$\text{(b)}$ Find the perimeter of the shaded region. $[4]$

Question 6 Code: 9709/13/M/J/13/6, Topic: Differentiation

The non-zero variables $x, y$ and $u$ are such that $u=x^{2} y$. Given that $y+3 x=9$, find the stationary value of $u$ and determine whether this is a maximum or a minimum value. $[7]$

Question 7 Code: 9709/13/M/J/18/7, Topic: Trigonometry

$\text{(a)}$ $\quad\text{(i)}$ Express $\displaystyle \frac{\tan ^{2} \theta-1}{\tan ^{2} \theta+1}$ in the form $a \sin ^{2} \theta+b$, where $a$ and $b$ are constants to be found. $[3]$

$\qquad \text{(ii)}$ Hence, or otherwise, and showing all necessary working, solve the equation. $[2]$

$$ \frac{\tan ^{2} \theta-1}{\tan ^{2} \theta+1}=\frac{1}{4} $$$\qquad$ for $-90^{\circ} \leqslant \theta \leqslant 0^{\circ}$

$\text{(b)}$

The diagram shows the graphs of $y=\sin x$ and $y=2 \cos x$ for $-\pi \leqslant x \leqslant \pi$. The graphs intersect at the points $A$ and $B$.

$\text{(i)}$ Find the $x$-coordinate of $A$. $[2]$

$\text{(ii)}$ Find the $y$-coordinate of $B$. $[2]$

Question 8 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 9 Code: 9709/11/M/J/15/8, Topic: Functions

The function $\mathrm{f}: x \mapsto 5+3 \cos \left(\frac{1}{2} x\right)$ is defined for $0 \leqslant x \leqslant 2 \pi$.

$\text{(i)}$ Solve the equation $\mathrm{f}(x)=7$, giving your answer correct to 2 decimal places. $[3]$

$\text{(ii)}$ Sketch the graph of $y=\mathrm{f}(x)$. $[2]$

$\text{(iii)}$ Explain why $\mathrm{f}$ has an inverse. $[1]$

$\text{(iv)}$ Obtain an expression for $\mathrm{f}^{-1}(x)$. $[3]$

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

The equation of a curve is $y=\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$.

$\text{(i)}$ Find $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$. $[3]$

$\text{(ii)}$ Find the equation of the tangent to the curve at the point where the curve intersects the $y$-axis. $[3]$

$\text{(iii)}$ Find the set of values of $x$ for which $\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$ is an increasing function of $x$. $[3]$

Question 11 Code: 9709/13/M/J/11/10, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$$ \begin{aligned} &\mathrm{f}: x \mapsto 3 x-4, \quad x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto 2(x-1)^{3}+8, \quad x>1 \end{aligned} $$$\text{(i)}$ Evaluate $\mathrm{fg(2)}$. $[2]$

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

$\text{(iii)}$ Obtain an expression for $\mathrm{g}^{\prime}(x)$ and use your answer to explain why $\mathrm{g}$ has an inverse. $[3]$

$\text{(iv)}$ Express each of $\mathrm{f}^{-1}(x)$ and $\mathrm{g}^{-1}(x)$ in terms of $x$. $[4]$

Question 12 Code: 9709/13/M/J/18/10, Topic: Functions

The one-one function $\mathrm{f}$ is defined by $\mathrm{f}(x)=(x-2)^{2}+2$ for $x \geqslant c$, where $c$ is a constant.

$\text{(i)}$ State the smallest possible value of $c$. $[1]$

In parts $\text{(ii)}$ and $\text{(iii)}$ the value of $c$ is 4.

$\text{(ii)}$ Find an expression for $\mathrm{f}^{-1}(x)$ and state the domain of $\mathrm{f}^{-1}$. $[3]$

$\text{(iii)}$ Solve the equation $\mathrm{ff}(x)=51$, giving your answer in the form $a+\sqrt{b}$. $[5]$