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

\includegraphics{figure_6} The diagram shows triangle \(ABC\), in which \(AB = 3\) cm, \(AC = 5\) cm and angle \(ABC = 2.1\) radians. Calculate

1. angle \(ACB\), giving your answer in radians, [2]
2. the area of the triangle. [3]

An arc of a circle with centre \(A\) and radius 3 cm is drawn, cutting \(AC\) at the point \(D\).

1. Calculate the perimeter and the area of the sector \(ABD\). [4]

Fig. 7 shows a sketch of a village green ABC which is bounded by three straight roads. AB = 92 m, BC = 75 m and AC = 105 m. \includegraphics{figure_7} Calculate the area of the village green. [5]

Fig. 10.1 shows Jean's back garden. This is a quadrilateral ABCD with dimensions as shown. \includegraphics{figure_10.1}

1. 1. Calculate AC and angle ACB. Hence calculate AD. [6]
   2. Calculate the area of the garden. [3]
2. The shape of the fence panels used in the garden is shown in Fig. 10.2. EH is the arc of a sector of a circle with centre at the midpoint, M, of side FG, and sector angle 1.1 radians, as shown. FG = 1.8 m. \includegraphics{figure_10.2} Calculate the area of one of these fence panels. [5]

\includegraphics{figure_6} The diagram shows triangle \(ABC\) in which \(AC = 8\) cm and \(\angle BAC = \angle BCA = 30°\).

1. Find the area of triangle \(ABC\) in the form \(k\sqrt{3}\). [4]

The point \(M\) is the mid-point of \(AC\) and the points \(N\) and \(O\) lie on \(AB\) and \(BC\) such that \(MN\) and \(MO\) are arcs of circles with centres \(A\) and \(C\) respectively.

1. Show that the area of the shaded region \(BNMO\) is \(\frac{8}{3}(2\sqrt{3} - \pi)\) cm\(^2\). [4]

Fig. 11.1 shows a village green which is bordered by 3 straight roads AB, BC and CA. The road AC runs due North and the measurements shown are in metres. \includegraphics{figure_1}

1. Calculate the bearing of B from C, giving your answer to the nearest 0.1°. [4]
2. Calculate the area of the village green. [2]

The road AB is replaced by a new road, as shown in Fig. 11.2. The village green is extended up to the new road. \includegraphics{figure_2} The new road is an arc of a circle with centre O and radius 130 m.

1. (A) Show that angle AOB is 1.63 radians, correct to 3 significant figures. [2] (B) Show that the area of land added to the village green is 5300 m² correct to 2 significant figures. [4]

\includegraphics{figure_3} For triangle ABC shown in Fig. 4, calculate

1. the length of BC, [3]
2. the area of triangle ABC. [2]

\emph{Arrowline Enterprises} is considering two possible logos: \includegraphics{figure_6}

1. Fig. 10.1 shows the first logo ABCD. It is symmetrical about AC. Find the length of AB and hence find the area of this logo. [4]
2. Fig. 10.2 shows a circle with centre O and radius 12.6 cm. ST and RT are tangents to the circle and angle SOR is 1.82 radians. The shaded region shows the second logo. Show that ST = 16.2 cm to 3 significant figures. Find the area and perimeter of this logo. [8]

\includegraphics{figure_2} Boat \(A\) is travelling with constant speed 7.9 m s\(^{-1}\) on a course with bearing 035°. Boat \(B\) is travelling with constant speed 10.5 m s\(^{-1}\) on a course with bearing 330°. At one instant, the boats are 1500 m apart with \(B\) on a bearing of 125° from \(A\) (see diagram).

1. Find the magnitude and the bearing of the velocity of \(B\) relative to \(A\). [5]
2. Find the shortest distance between \(A\) and \(B\) in the subsequent motion. [2]
3. Find the time taken from the instant when \(A\) and \(B\) are 1500 m apart to the instant when \(A\) and \(B\) are at the point of closest approach. [2]

Triangle \(ABC\), with \(BC = a\), \(AC = b\) and \(AB = c\) is inscribed in a circle. Given that \(AB\) is a diameter of the circle and that \(a^2\), \(b^2\) and \(c^2\) are three consecutive terms of an arithmetic progression (arithmetic series),

1. express \(b\) and \(c\) in terms of \(a\), [4]
2. verify that \(\cot A\), \(\cot B\) and \(\cot C\) are consecutive terms of an arithmetic progression. [3]

In an acute-angled triangle \(PQR\) the sides \(QR\), \(PR\) and \(PQ\) have lengths \(p\), \(q\) and \(r\) respectively.

1. Prove that $$\frac{p}{\sin P} = \frac{q}{\sin Q} = \frac{r}{\sin R}.$$ [3]

Given now that triangle \(PQR\) is such that \(p^2\), \(q^2\) and \(r^2\) are three consecutive terms of an arithmetic progression,

1. use the cosine rule to prove that $$\frac{2\cos Q}{q} = \frac{\cos P}{p} + \frac{\cos R}{r}.$$ [6]
2. Using the results given in parts \((c)\) and \((d)\), prove that \(\cot P\), \(\cot Q\) and \(\cot R\) are consecutive terms in an arithmetic progression. [3]

In this question you must show detailed reasoning. A disease that affects trees shows no visible evidence for the first few years after the tree is infected. A test has been developed to determine whether a particular tree has the disease. A positive result to the test suggests that the tree has the disease. However, the test is not 100% reliable, and a researcher uses the following model. • If the tree has the disease, the probability of a positive result is 0.95. • If the tree does not have the disease, the probability of a positive result is 0.1.

1. It is known that in a certain county, \(A\), 35% of the trees have the disease. A tree in county \(A\) is chosen at random and is tested. Given that the result is positive, determine the probability that this tree has the disease. [3]

A forestry company wants to determine what proportion of trees in another county, \(B\), have the disease. They choose a large random sample of trees in county \(B\). Each tree in the sample is tested and it is found that the result is positive for 43% of these trees.

1. By carrying out a calculation, determine an estimate of the proportion of trees in county \(B\) that have the disease. [4]

The diagram shows a ladder \(AB\), of length \(2a\) and mass \(m\), resting in equilibrium on a vertical wall of height \(h\). The ladder is inclined at an angle of \(30°\) to the horizontal. The end \(A\) is in contact with horizontal ground. An object of mass \(2m\) is placed on the ladder at a point \(C\) where \(AC = d\). The ladder is modelled as uniform, the ground is modelled as being rough, and the vertical wall is modelled as being smooth.

1. Show that the normal contact force between the ladder and the wall is \(\frac{mg(a + 2d)\sqrt{3}}{4h}\). [4]

It is given that the equilibrium is limiting and the coefficient of friction between the ladder and the ground is \(\frac{1}{3}\sqrt{3}\).

1. Show that \(h = k(a + 2d)\), where \(k\) is a constant to be determined. [7]
2. Hence find, in terms of \(a\), the greatest possible value of \(d\). [2]
3. State one improvement that could be made to the model. [1]

Triangle \(ABC\) has \(AB = 8.5\) cm, \(BC = 6.2\) cm and angle \(B = 35°\). Calculate the area of the triangle. [2]

A small ball \(B\) moves in the plane of a fixed horizontal axis \(Ox\), which lies on horizontal ground, and a fixed vertically upwards axis \(Oy\). \(B\) is projected from \(O\) with a velocity whose components along \(Ox\) and \(Oy\) are \(U \mathrm{m s}^{-1}\) and \(V \mathrm{m s}^{-1}\), respectively. The units of \(x\) and \(y\) are metres. \(B\) is modelled as a particle moving freely under gravity.

1. Show that the path of \(B\) has equation \(2U^2 y = 2UVx - gx^2\). [3]

During its motion, \(B\) just clears a vertical wall of height \(\frac{1}{3}a\) m at a horizontal distance \(a\) m from \(O\). \(B\) strikes the ground at a horizontal distance \(3a\) m beyond the wall.

1. Determine the angle of projection of \(B\). Give your answer in degrees correct to 3 significant figures. [5]
2. Given that the speed of projection of \(B\) is \(54.6 \mathrm{m s}^{-1}\), determine the value of \(a\). [2]
3. Hence find the maximum height of \(B\) above the ground during its motion. [3]
4. State one refinement of the model, other than including air resistance, that would make it more realistic. [1]

\(ABC\) is a right-angled triangle. \includegraphics{figure_6} \(D\) is the point on hypotenuse \(AC\) such that \(AD = AB\). The area of \(\triangle ABD\) is equal to half that of \(\triangle ABC\).

1. Show that \(\tan A = 2 \sin A\) [4 marks]

1. 1. Show that the equation given in part (a) has two solutions for \(0° \leq A \leq 90°\) [2 marks]
   2. State the solution which is appropriate in this context. [1 mark]

The diagram below shows a circle and four triangles. \(AB\) is a diameter of the circle. \(C\) is a point on the circumference of the circle. Triangles \(ABK\), \(BCL\) and \(CAM\) are equilateral. Prove that the area of triangle \(ABK\) is equal to the sum of the areas of triangle \(BCL\) and triangle \(CAM\). [5 marks]

A triangular field of grass, \(ABC\), has boundaries with lengths as follows: $$AB = 234 \text{ m} \qquad BC = 225 \text{ m} \qquad AC = 310 \text{ m}$$ The field is shown in the diagram below. \includegraphics{figure_7}

1. Find angle \(A\) [2 marks]
2. Farmers calculate the number of sheep they can keep in a field, by allowing one sheep for every \(1200 \text{ m}^2\) of grass. Find the maximum number of sheep which can be kept in the field \(ABC\) [3 marks]

A parallelogram has sides of length 6 cm and 4.5 cm. The larger interior angles of the parallelogram have size \(\alpha\) Given that the area of the parallelogram is 24 cm², find the exact value of \(\tan \alpha\) [4 marks]

A piece of wire of length 66 cm is bent to form the five sides of a pentagon. The pentagon consists of three sides of a rectangle and two sides of an equilateral triangle. The sides of the rectangle measure \(x\) cm and \(y\) cm and the sides of the triangle measure \(x\) cm, as shown in the diagram below. \includegraphics{figure_10}

1. 1. You are given that \(\sin 60° = \frac{\sqrt{3}}{2}\) Explain why the area of the triangle is \(\frac{\sqrt{3}}{4}x^2\) [1 mark]
   2. Show that the area enclosed by the wire, \(A\) cm\(^2\), can be expressed by the formula $$A = 33x - \frac{1}{4}(6 - \sqrt{3})x^2$$ [3 marks]
2. Use calculus to find the value of \(x\) for which the wire encloses the maximum area. Give your answer in the form \(p + q\sqrt{3}\), where \(p\) and \(q\) are integers. Fully justify your answer. [7 marks]

\includegraphics{figure_1} A triangular lawn is modelled by the triangle \(ABC\), shown in Figure 1. The length \(AB\) is to be \(30\text{m}\) long. Given that angle \(BAC = 70°\) and angle \(ABC = 60°\),

1. calculate the area of the lawn to \(3\) significant figures. [4]
2. Why is your answer unlikely to be accurate to the nearest square metre? [1]

In a triangle \(PQR\), \(PQ = 20\) cm, \(PR = 10\) cm and angle \(QPR = \theta\), where \(\theta\) is measured in degrees. The area of triangle \(PQR\) is 80 cm\(^2\).

1. Show that the two possible values of \(\cos \theta = \pm \frac{3}{5}\) [4]

Given that \(QR\) is the longest side of the triangle,

1. find the exact perimeter of the triangle \(PQR\), giving your answer as a simplified surd. [3]

The diagram below shows a triangle \(ABC\) with \(AC = 5\) cm, \(AB = x\) cm, \(BC = y\) cm and angle \(BAC = 120°\). The area of the triangle \(ABC\) is \(14\) cm\(^2\). \includegraphics{figure_14} Find the value of \(x\) and the value of \(y\). Give your answers correct to 2 decimal places. [6]