1.05b Sine and cosine rules: including ambiguous case

\includegraphics{figure_1} Figure 1 shows the sector \(AOB\) of a circle, with centre \(O\) and radius 6.5 cm, and \(\angle AOB = 0.8\) radians.

1. Calculate, in cm\(^2\), the area of the sector \(AOB\). [2]
2. Show that the length of the chord \(AB\) is 5.06 cm, to 3 significant figures. [3]

The segment \(R\), shaded in Fig. 1, is enclosed by the arc \(AB\) and the straight line \(AB\).

1. Calculate, in cm, the perimeter of \(R\). [2]

\includegraphics{figure_2} Triangle \(ABC\) has \(AB = 9\) cm, \(BC = 10\) cm and \(CA = 5\) cm. A circle, centre \(A\) and radius 3 cm, intersects \(AB\) and \(AC\) at \(P\) and \(Q\) respectively, as shown in Fig. 2.

1. Show that, to 3 decimal places, \(\angle BAC = 1.504\) radians. [3]

Calculate,

1. the area, in cm\(^2\), of the sector \(APQ\), [2]
2. the area, in cm\(^2\), of the shaded region \(BPQC\), [3]
3. the perimeter, in cm, of the shaded region \(BPQC\). [4]

\includegraphics{figure_4} In the diagram, \(ABCD\) is a quadrilateral in which \(AD\) is parallel to \(BC\). It is given that \(AB = 9\), \(BC = 6\), \(CA = 5\) and \(CD = 15\).

1. Show that \(\cos BCA = -\frac{1}{3}\), and hence find the value of \(\sin BCA\). [4]
2. Find the angle \(ADC\) correct to the nearest \(0.1°\). [4]

In a triangle \(ABC\), \(AB = 5\sqrt{2}\) cm, \(BC = 8\) cm and angle \(B = 60°\).

1. Find the exact area of the triangle, giving your answer as simply as possible. [3]
2. Find the length of \(AC\), correct to 3 significant figures. [3]

\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]

\includegraphics{figure_7} Fig. 7 shows triangle ABC, with AB = 8.4 cm. D is a point on AC such that angle ADB = 79°, BD = 5.6 cm and CD = 7.8 cm. Calculate

1. angle BAD, [2]
2. the length BC. [3]

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_5} Fig. 5 shows triangle ABC, where angle ABC = \(72°\), AB = \(5.9\) cm and BC = \(8.5\) cm. Calculate the length of AC. [3]

\includegraphics{figure_4} Fig. 4 shows triangle ABC, where AB = 7.2 cm, AC = 5.6 cm and angle BAC = 68°. Calculate the size of angle ACB. [5]

\includegraphics{figure_1} Figure 1 shows triangle \(PQR\) in which \(PQ = x\), \(PR = 7 - x\), \(QR = x + 1\) and \(\angle PQR = 60°\). Using the cosine rule, find the value of \(x\). [4]

\includegraphics{figure_3} Figure 3 shows the circle \(C\) with equation $$x^2 + y^2 - 8x - 10y + 16 = 0.$$

1. Find the coordinates of the centre and the radius of \(C\). [3]

\(C\) crosses the \(y\)-axis at the points \(P\) and \(Q\).

1. Find the coordinates of \(P\) and \(Q\). [3]

The chord \(PQ\) subtends an angle of \(\theta\) at the centre of \(C\).

1. Using the cosine rule, show that \(\cos \theta = \frac{7}{25}\). [4]
2. Find the area of the shaded minor segment bounded by \(C\) and the chord \(PQ\). [4]

\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]

In Fig. 6, OAB is a thin bent rod, with OA = \(a\) metres, AB = \(b\) metres and angle OAB = 120°. The bent rod lies in a vertical plane. OA makes an angle \(\theta\) above the horizontal. The vertical height BD of B above O is \(h\) metres. The horizontal through A meets BD at C and the vertical through A meets OD at E. \includegraphics{figure_6}

1. Find angle BAC in terms of \(\theta\). Hence show that $$h = a\sin\theta + b\sin(\theta - 60°).$$ [3]
2. Hence show that \(h = (a + \frac{1}{2}b)\sin\theta - \frac{\sqrt{3}}{2}b\cos\theta\). [3]

The rod now rotates about O, so that \(\theta\) varies. You may assume that the formulae for \(h\) in parts (i) and (ii) remain valid.

1. Show that OB is horizontal when \(\tan\theta = \frac{\sqrt{3}b}{2a + b}\). [3]

In the case when \(a = 1\) and \(b = 2\), \(h = 2\sin\theta - \sqrt{3}\cos\theta\).

1. Express \(2\sin\theta - \sqrt{3}\cos\theta\) in the form \(R\sin(\theta - \alpha)\). Hence, for this case, write down the maximum value of \(h\) and the corresponding value of \(\theta\). [7]

A ship \(A\) has maximum speed 30 km h\(^{-1}\). At time \(t = 0\), \(A\) is 70 km due west of \(B\) which is moving at a constant speed of 36 km h\(^{-1}\) on a bearing of 300°. Ship \(A\) moves on a straight course at a constant speed and intercepts \(B\). The course of \(A\) makes an angle \(\theta\) with due north.

1. Show that \(-\arctan \frac{4}{3} \leq \theta \leq \arctan \frac{4}{3}\). [7]
2. Find the least time for \(A\) to intercept \(B\). [5]

\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]