1.05c Area of triangle: using 1/2 ab sin(C)

5 \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-06_761_460_258_840} The diagram shows a cord going around a pulley and a pin. The pulley is modelled as a circle with centre \(O\) and radius 5 cm . The thickness of the cord and the size of the pin \(P\) can be neglected. The pin is situated 13 cm vertically below \(O\). Points \(A\) and \(B\) are on the circumference of the circle such that \(A P\) and \(B P\) are tangents to the circle. The cord passes over the major arc \(A B\) of the circle and under the pin such that the cord is taut. Calculate the length of the cord.

12 \includegraphics[max width=\textwidth, alt={}, center]{5b8ddd32-c884-48a0-ad51-5582ef0d5128-16_598_609_264_769} The diagram shows a cross-section of seven cylindrical pipes, each of radius 20 cm , held together by a thin rope which is wrapped tightly around the pipes. The centres of the six outer pipes are \(A , B , C , D\), \(E\) and \(F\). Points \(P\) and \(Q\) are situated where straight sections of the rope meet the pipe with centre \(A\).

1. Show that angle \(P A Q = \frac { 1 } { 3 } \pi\) radians.
2. Find the length of the rope.
3. Find the area of the hexagon \(A B C D E F\), giving your answer in terms of \(\sqrt { 3 }\).
4. Find the area of the complete region enclosed by the rope.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 \includegraphics[max width=\textwidth, alt={}, center]{574b96b2-62f2-41b3-a178-8e68e16429ff-08_509_654_264_751} The diagram shows a sector \(A B C\) of a circle with centre \(A\) and radius \(r\). The line \(B D\) is perpendicular to \(A C\). Angle \(C A B\) is \(\theta\) radians.

1. Given that \(\theta = \frac { 1 } { 6 } \pi\), find the exact area of \(B C D\) in terms of \(r\).
2. Given instead that the length of \(B D\) is \(\frac { \sqrt { 3 } } { 2 } r\), find the exact perimeter of \(B C D\) in terms of \(r\). [4]

8 \includegraphics[max width=\textwidth, alt={}, center]{3bad1d9f-5b9e-4895-aa4e-3e6d9f6c072e-10_454_744_255_703} The diagram shows triangle \(A B C\) in which angle \(B\) is a right angle. The length of \(A B\) is 8 cm and the length of \(B C\) is 4 cm . The point \(D\) on \(A B\) is such that \(A D = 5 \mathrm {~cm}\). The sector \(D A C\) is part of a circle with centre \(D\).

1. Find the perimeter of the shaded region.
2. Find the area of the shaded region.

8 \includegraphics[max width=\textwidth, alt={}, center]{6bcc553c-4938-46ef-bba4-97391b4d58d4-10_348_700_262_721} In the diagram, \(A B C\) is an isosceles triangle with \(A B = B C = r \mathrm {~cm}\) and angle \(B A C = \theta\) radians. The point \(D\) lies on \(A C\) and \(A B D\) is a sector of a circle with centre \(A\).

1. Express the area of the shaded region in terms of \(r\) and \(\theta\).
2. In the case where \(r = 10\) and \(\theta = 0.6\), find the perimeter of the shaded region.

7 \includegraphics[max width=\textwidth, alt={}, center]{10b2ec29-adca-4313-ae24-bab8b2d9f8a4-08_556_751_255_696} In the diagram the lengths of \(A B\) and \(A C\) are both 15 cm . The point \(P\) is the foot of the perpendicular from \(C\) to \(A B\). The length \(C P = 9 \mathrm {~cm}\). An arc of a circle with centre \(B\) passes through \(C\) and meets \(A B\) at \(Q\).

1. Show that angle \(A B C = 1.25\) radians, correct to 3 significant figures.
2. Calculate the area of the shaded region which is bounded by the \(\operatorname { arc } C Q\) and the lines \(C P\) and \(P Q\).

6 \includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-2_389_995_1432_575} In the diagram, \(A B C\) is a triangle in which \(A B = 4 \mathrm {~cm} , B C = 6 \mathrm {~cm}\) and angle \(A B C = 150 ^ { \circ }\). The line \(C X\) is perpendicular to the line \(A B X\).

1. Find the exact length of \(B X\) and show that angle \(C A B = \tan ^ { - 1 } \left( \frac { 3 } { 4 + 3 \sqrt { } 3 } \right)\).
2. Show that the exact length of \(A C\) is \(\sqrt { } ( 52 + 24 \sqrt { } 3 ) \mathrm { cm }\).

4 \includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-2_358_618_1082_762} The diagram shows a sector of a circle with radius \(r \mathrm {~cm}\) and centre \(O\). The chord \(A B\) divides the sector into a triangle \(A O B\) and a segment \(A X B\). Angle \(A O B\) is \(\theta\) radians.

1. In the case where the areas of the triangle \(A O B\) and the segment \(A X B\) are equal, find the value of the constant \(p\) for which \(\theta = p \sin \theta\).
2. In the case where \(r = 8\) and \(\theta = 2.4\), find the perimeter of the segment \(A X B\).

3 \includegraphics[max width=\textwidth, alt={}, center]{0b047754-84f2-46ea-b441-7c68cef47641-2_485_623_790_760} The diagram shows part of a circle with centre \(O\) and radius 6 cm . The chord \(A B\) is such that angle \(A O B = 2.2\) radians. Calculate

1. the perimeter of the shaded region,
2. the ratio of the area of the shaded region to the area of the triangle \(A O B\), giving your answer in the form \(k : 1\).

7 The position vectors of points \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 6 \\ - 1 \\ 7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 2 \\ 4 \\ 7 \end{array} \right)$$

1. Show that angle \(B A C = \cos ^ { - 1 } \left( \frac { 1 } { 3 } \right)\).
2. Use the result in part (i) to find the exact value of the area of triangle \(A B C\).

6 \includegraphics[max width=\textwidth, alt={}, center]{8c358a10-a3e1-47b5-ae62-30ba6b76c167-3_655_1011_255_566} The diagram shows triangle \(A B C\) where \(A B = 5 \mathrm {~cm} , A C = 4 \mathrm {~cm}\) and \(B C = 3 \mathrm {~cm}\). Three circles with centres at \(A , B\) and \(C\) have radii \(3 \mathrm {~cm} , 2 \mathrm {~cm}\) and 1 cm respectively. The circles touch each other at points \(E , F\) and \(G\), lying on \(A B , A C\) and \(B C\) respectively. Find the area of the shaded region \(E F G\).

8 \includegraphics[max width=\textwidth, alt={}, center]{028c7979-6b24-42d0-9857-c616a169b2b2-14_590_691_260_726} In the diagram, \(O A X B\) is a sector of a circle with centre \(O\) and radius 10 cm . The length of the chord \(A B\) is 12 cm . The line \(O X\) passes through \(M\), the mid-point of \(A B\), and \(O X\) is perpendicular to \(A B\). The shaded region is bounded by the chord \(A B\) and by the arc of a circle with centre \(X\) and radius \(X A\).

1. Show that angle \(A X B\) is 2.498 radians, correct to 3 decimal places.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

9

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-4_433_476_264_872} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} In Fig. 1, \(O A B\) is a sector of a circle with centre \(O\) and radius \(r\). \(A X\) is the tangent at \(A\) to the arc \(A B\) and angle \(B A X = \alpha\).
   1. Show that angle \(A O B = 2 \alpha\).
   2. Find the area of the shaded segment in terms of \(r\) and \(\alpha\).
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-4_451_503_1162_861} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} In Fig. 2, \(A B C\) is an equilateral triangle of side 4 cm . The lines \(A X , B X\) and \(C X\) are tangents to the equal circular \(\operatorname { arcs } A B , B C\) and \(C A\). Use the results in part (a) to find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).
   [0pt] [6]

3 \includegraphics[max width=\textwidth, alt={}, center]{933cdfe1-27bb-450d-8b9a-b494916242cb-2_737_693_1484_726} In the diagram, \(A B E D\) is a trapezium with right angles at \(E\) and \(D\), and \(C E D\) is a straight line. The lengths of \(A B\) and \(B C\) are \(2 d\) and \(( 2 \sqrt { 3 } ) d\) respectively, and angles \(B A D\) and \(C B E\) are \(30 ^ { \circ }\) and \(60 ^ { \circ }\) respectively.

1. Find the length of \(C D\) in terms of \(d\).
2. Show that angle \(C A D = \tan ^ { - 1 } \left( \frac { 2 } { \sqrt { 3 } } \right)\).

5 \includegraphics[max width=\textwidth, alt={}, center]{9cdb00a6-1e86-4185-bb73-ed3ecab981ba-3_560_506_258_822} The diagram shows a metal plate \(O A B C\), consisting of a right-angled triangle \(O A B\) and a sector \(O B C\) of a circle with centre \(O\). Angle \(A O B = 0.6\) radians, \(O A = 6 \mathrm {~cm}\) and \(O A\) is perpendicular to \(O C\).

1. Show that the length of \(O B\) is 7.270 cm , correct to 3 decimal places.
2. Find the perimeter of the metal plate.
3. Find the area of the metal plate.

7. \begin{figure}[h] \end{figure} Not to scale Figure 3 shows the design for a structure used to support a roof. The structure consists of four wooden beams, \(A B , B D , B C\) and \(A D\). Given \(A B = 6.5 \mathrm {~m} , B C = B D = 4.7 \mathrm {~m}\) and angle \(B A C = 35 ^ { \circ }\)

1. find, to one decimal place, the size of angle \(A C B\),
2. find, to the nearest metre, the total length of wood required to make this structure.

5. \begin{figure}[h] \end{figure} Figure 3 shows the plan view of a viewing platform at a tourist site. The shape of the viewing platform consists of a sector \(A B C O A\) of a circle, centre \(O\), joined to a triangle \(A O D\). Given that

• \(O A = O C = 6 \mathrm {~m}\)
• \(A D = 14 \mathrm {~m}\)
• angle \(A D C = 0.43\) radians
• angle \(A O D\) is an obtuse angle
• \(O C D\) is a straight line
  find
  1. the size of angle \(A O D\), in radians, to 3 decimal places,
  2. the length of arc \(A B C\), in metres, to one decimal place,
  3. the total area of the viewing platform, in \(\mathrm { m } ^ { 2 }\), to one decimal place.

1. The triangle \(A B C\) is such that

• \(A B = 15 \mathrm {~cm}\)
• \(A C = 25 \mathrm {~cm}\)
• angle \(B A C = \theta ^ { \circ }\)
• area triangle \(A B C = 100 \mathrm {~cm} ^ { 2 }\)
  1. Find the value of \(\sin \theta ^ { \circ }\)

Given that \(\theta > 90\)

find the length of \(B C\), in cm , to 3 significant figures.

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable. Figure 1 shows the plan view of a flower bed.
The flowerbed is in the shape of a triangle \(A B C\) with

• \(A B = p\) metres
• \(A C = q\) metres
• \(B C = 2 \sqrt { 2 }\) metres
• angle \(B A C = 60 ^ { \circ }\)
  1. Show that

$$p ^ { 2 } + q ^ { 2 } - p q = 8$$ Given that side \(A C\) is 2 metres longer than side \(A B\), use algebra to find

1. the exact value of \(p\),
2. the exact value of \(q\). Using the answers to part (b),

calculate the exact area of the flower bed.

5. \begin{figure}[h] \end{figure} Figure 2 shows the plan view of a garden.
The shape of the garden \(A B C D E A\) consists of a triangle \(A B E\) and a right-angled triangle \(B C D\) joined to a sector \(B D E\) of a circle with radius 6 m and centre \(B\). The points \(A , B\) and \(C\) lie on a straight line with \(A B = 10.8 \mathrm {~m}\) Angle \(B C D = \frac { \pi } { 2 }\) radians, angle \(E B D = 1.3\) radians and \(A E = 12.2 \mathrm {~m}\)

1. Find the area of the sector \(B D E\), giving your answer in \(\mathrm { m } ^ { 2 }\)
2. Find the size of angle \(A B E\), giving your answer in radians to 2 decimal places.
3. Find the area of the garden, giving your answer in \(\mathrm { m } ^ { 2 }\) to 3 significant figures.

4. A parallelogram \(A B C D\) has area \(40 \mathrm {~cm} ^ { 2 }\) Given that \(A B\) has length \(10 \mathrm {~cm} , B C\) has length 6 cm and angle \(D A B\) is obtuse, find

1. the size of angle \(D A B\), in degrees, to 2 decimal places,
2. the length of diagonal \(B D\), in cm , to one decimal place.

3. \begin{figure}[h] \end{figure} Figure 1 shows the design for a badge.
The design consists of two congruent triangles, \(A O C\) and \(B O C\), joined to a sector \(A O B\) of a circle centre \(O\).

• Angle \(A O B = \alpha\)
• \(A O = O B = 3 \mathrm {~cm}\)
• \(O C = 5 \mathrm {~cm}\)

Given that the area of sector \(A O B\) is \(7.2 \mathrm {~cm} ^ { 2 }\)

1. show that \(\alpha = 1.6\) radians.
2. Hence find
   1. the area of the badge, giving your answer in \(\mathrm { cm } ^ { 2 }\) to 2 significant figures,
   2. the perimeter of the badge, giving your answer in cm to one decimal place.

      VIXV SIHIANI III IM IONOO

      VIAV SIHI NI JYHAM ION OO

      VI4V SIHI NI JLIYM ION OO

6. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 ( x + 1 ) ( x - 3 ) ^ { 2 }$$

1. Sketch a graph of \(C\). Show on your graph the coordinates of the points where \(C\) cuts or meets the coordinate axes.
2. Write \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\), where \(a , b , c\) and \(d\) are constants to be found.
3. Hence, find the equation of the tangent to \(C\) at the point where \(x = \frac { 1 } { 3 }\)

5. Figure 2 Diagram NOT accurately drawn Figure 2 shows the plan view of a frame for a flat roof.
The shape of the frame consists of triangle \(A B D\) joined to triangle \(B C D\).
Given that

• \(B D = x \mathrm {~m}\)
• \(C D = ( 1 + x ) \mathrm { m }\)
• \(B C = 5 \mathrm {~m}\)
• angle \(B C D = \theta ^ { \circ }\)
  1. show that \(\cos \theta ^ { \circ } = \frac { 13 + x } { 5 + 5 x }\)

Given also that