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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 | 4 | 6 | 6 | 6 | 4 | 8 | 9 | 7 | 10 | 9 | 78 |

Score |

Question 1 Code: 9709/11/M/J/16/2, Topic: Trigonometry

Solve the equation $3 \sin ^{2} \theta=4 \cos \theta-1$ for $0^{\circ} \leqslant \theta \leqslant 360^{\circ}$. $[3]$

Question 2 Code: 9709/12/M/J/14/3, Topic: Trigonometry

The reflex angle $\theta$ is such that $\cos \theta=k$, where $0 < k < 1$.

$\text{(i)}$ Find an expression, in terms of $k$, for

$\text{(a)}$ $\sin \theta$, $[2]$

$\text{(b)}$ $\tan \theta$. $[1]$

$\text{(ii)}$ Explain why $\sin 2 \theta$ is negative for $0 < k < 1$. $[2]$

Question 3 Code: 9709/13/M/J/20/3, Topic: Coordinate geometry

In each of parts $\text{(a)}, \text{(b)}$ and $\text{(c)}$, the graph shown with solid lines has equation $y=\mathrm{f}(x)$. The graph shown with broken lines is a transformation of $y=\mathrm{f}(x)$.

$\text{(a)}$

State, in terms of $\mathrm{f}$, the equation of the graph shown with broken lines. $[1]$

$\text{(b)}$

State, in terms of f, the equation of the graph shown with broken lines. $[1]$

$\text{(c)}$

State, in terms of $\mathrm{f}$, the equation of the graph shown with broken lines. $[2]$

Question 4 Code: 9709/11/M/J/13/4, Topic: Series

The third term of a geometric progression is $-108$ and the sixth term is 32. Find

$\text{(i)}$ the common ratio, $[3]$

$\text{(ii)}$ the first term, $[1]$

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

Question 5 Code: 9709/11/M/J/12/5, Topic: Coordinate geometry

The diagram shows the curve $y=7 \sqrt{x}$ and the line $y=6 x+k$, where $k$ is a constant. The curve and the line intersect at the points $A$ and $B$.

$\text{(i)}$ For the case where $k=2$, find the $x$-coordinates of $A$ and $B$. $[4]$

$\text{(ii)}$ Find the value of $k$ for which $y=6 x+k$ is a tangent to the curve $y=7 \sqrt{x}$. $[2]$

Question 6 Code: 9709/12/M/J/18/6, Topic: Circular measure

The diagram shows points $A$ and $B$ on a circle with centre $O$ and radius $r$. The tangents to the circle at $A$ and $B$ meet at $T$. The shaded region is bounded by the minor $\operatorname{arc} A B$ and the lines $A T$ and $B T$. Angle $A O B$ is $2 \theta$ radians.

$\text{(i)}$ In the case where the area of the sector $A O B$ is the same as the area of the shaded region, show that $\tan \theta=2 \theta$. $[3]$

$\text{(ii)}$ In the case where $r=8 \mathrm{~cm}$ and the length of the minor $\operatorname{arc} A B$ is $19.2 \mathrm{~cm}$, find the area of the shaded region. $[3]$

Question 7 Code: 9709/12/M/J/21/6, Topic: Coordinate geometry

Points $A$ and $B$ have coordinates $(8,3)$ and $(p, q)$ respectively. The equation of the perpendicular bisector of $A B$ is $y=-2 x+4$.Find the values of $p$ and $q$. $[4]$

Question 8 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 9 Code: 9709/11/M/J/16/9, Topic: Series

$\text{(a)}$ The first term of a geometric progression in which all the terms are positive is 50. The third term is 32. Find the sum to infinity of the progression. $[3]$

$\text{(b)}$ The first three terms of an arithmetic progression are $2 \sin x, 3 \cos x$ and $(\sin x+2 \cos x)$ respectively, where $x$ is an acute angle.

$\text{(i)}$ Show that $\tan x=\frac{4}{3}$. $[3]$

$\text{(ii)}$ Find the sum of the first twenty terms of the progression. $[3]$

Question 10 Code: 9709/11/M/J/19/9, Topic: Functions

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=2-3 \cos x$ for $0 \leqslant x \leqslant 2 \pi$.

$\text{(i)}$ State the range of $\mathrm{f}$. $[2]$

$\text{(ii)}$ Sketch the graph of $y=\mathrm{f}(x)$. The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=2-3 \cos x$ for $0 \leqslant x \leqslant p$, where $p$ is a constant. $[2]$

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

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

Question 11 Code: 9709/13/M/J/19/9, Topic: Trigonometry

The function $\mathrm{f}: x \mapsto p \sin ^{2} 2 x+q$ is defined for $0 \leqslant x \leqslant \pi$, where $p$ and $q$ are positive constants. The diagram shows the graph of $y=\mathrm{f}(x)$.

$\text{(i)}$ In terms of $p$ and $q$, state the range of $\mathrm{f}$. $[2]$

$\text{(ii)}$ State the number of solutions of the following equations.

$\quad\text{(a)}$ $\mathrm{f}(x)=p+q$ $[1]$

$\quad\text{(b)}$ $\mathrm{f}(x)=q$ $[1]$

$\quad\text{(c)}$ $\displaystyle \mathrm{f}(x)=\frac{1}{2} p+q$ $[1]$

$\text{(iii)}$ For the case where $p=3$ and $q=2$, solve the equation $\mathrm{f}(x)=4$, showing all necessary working. $[5]$

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. $[5]$