1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta

3 A sector, \(P O Q\), of a circle centre \(O\) has radius 7 cm and angle 1.7 radians (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{41d9ff74-82de-4ac5-928f-f6ab008319d2-2_469_723_662_712}

1. Find the length of the line \(P Q\).
2. Hence find the perimeter of the shaded area.

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 9 cm . The angle \(A O B\) is \(100 ^ { \circ }\).

1. Express \(100 ^ { \circ }\) in radians, giving your answer in exact form.
2. Find the perimeter of the sector \(O A B\).
3. Find the area of the sector \(O A B\).

11 \includegraphics[max width=\textwidth, alt={}, center]{2f48a6ee-e8ce-47e4-a07f-2c55a6904e7d-3_661_953_767_596} The diagram shows a circle, centre \(O\), radius \(r\). The points \(R\) and \(S\) lie on the circumference of the circle, and the line \(R T\) is a tangent to the circle at \(R\). The angle \(R O S\) is \(\theta\) radians where \(0 < \theta < \frac { 1 } { 2 } \pi\).

1. Find expressions for the perimeter, \(P\), and the area, \(A\), of the shaded region in terms of \(r\) and \(\theta\).
2. Hence show that \(A \neq r P\).

5 \includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-3_570_736_292_667} The diagram shows a sector of a circle, \(O M N\). The angle \(M O N\) is \(2 x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.

1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and the perimeter, \(P\), of the sector.
2. Given that \(P = 20\), show that \(A = \frac { 100 x } { ( 1 + x ) ^ { 2 } }\).
3. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\), and hence find the value of \(x\) for which the area of the sector is a maximum.

5 \includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-3_577_743_287_662} The diagram shows a sector of a circle, \(O M N\). The angle \(M O N\) is \(2 x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.

1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and the perimeter, \(P\), of the sector.
2. Given that \(P = 20\), show that \(A = \frac { 100 x } { ( 1 + x ) ^ { 2 } }\).
3. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\), and hence find the value of \(x\) for which the area of the sector is a maximum.

6 \includegraphics[max width=\textwidth, alt={}, center]{f4b66aaa-16b9-4b15-b3f5-b9657fe98274-3_545_557_269_794} The diagram shows a sector \(A O B\) of a circle, centre \(O\) and radius \(r\). Angle \(A O B\) is \(\theta\) radians. The point \(C\) lies on \(O B\), and \(A C\) is perpendicular to \(O B\). The area of the triangle \(A O C\) is equal to the area of the segment bounded by the chord \(A B\) and the \(\operatorname { arc } A B\).

1. Show that \(\theta = \sin \theta ( 1 + \cos \theta )\). The equation \(\theta = \sin \theta ( 1 + \cos \theta )\) has only one positive root.
2. Use an iterative process based on this equation to find the value of the root correct to 3 significant figures. Use a starting value of 1 and show the result of each iteration. Use a change of sign to verify that the value you have found is correct to 3 significant figures.

5 \includegraphics[max width=\textwidth, alt={}, center]{48b63de9-f022-4881-a187-f08e3c7d9f1a-3_570_734_219_667} The diagram shows a sector of a circle, \(O M N\). The angle \(M O N\) is \(2 x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.

1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and the perimeter, \(P\), of the sector.
2. Given that \(P = 20\), show that \(A = \left( \frac { 10 } { 1 + x } \right) ^ { 2 }\).
3. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\), and hence find the value of \(x\) for which the area of the sector is a maximum.

5 \includegraphics[max width=\textwidth, alt={}, center]{8a0a6e46-99cf-4217-93ad-5ed6e9d7c4ef-3_565_730_219_669} The diagram shows a sector of a circle, \(O M N\). The angle \(M O N\) is \(2 x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.

1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and the perimeter, \(P\), of the sector.
2. Given that \(P = 20\), show that \(A = \left( \frac { 10 } { 1 + x } \right) ^ { 2 }\).
3. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\), and hence find the value of \(x\) for which the area of the sector is a maximum.

5 A circle \(S\) has centre at the point \(( 3,1 )\) and passes through the point \(( 0,5 )\).

1. Find the radius of \(S\) and hence write down its cartesian equation.
2. (a) Determine the two points on \(S\) where the \(y\)-coordinate is twice the \(x\)-coordinate.
   (b) Calculate the length of the minor arc joining these two points.

The diagram below shows a plan of the patio Eric wants to build. The walls \(O A\) and \(O C\) are perpendicular. The straight line \(A B\) is of length 4 m and is perpendicular to \(O A\). The shape \(O B C\) is a sector of a circle with centre \(O\) and radius OC.
The angle \(B O C\) is \(\frac { \pi } { 3 }\) radians. Calculate the area of the patio \(O A B C\). Give your answer correct to 2 decimal places. The sum to infinity of a geometric series with first term \(a\) and common ratio \(r\) is 120 . The sum to infinity of another geometric series with first term \(a\) and common ratio \(4 r ^ { 2 }\) is \(112 \frac { 1 } { 2 }\). Find the possible values of \(r\) and the corresponding values of \(a\). The function \(f ( x )\) is defined by $$f ( x ) = \frac { 6 x + 4 } { ( x - 1 ) ( x + 1 ) ( 2 x + 3 ) }$$ a) Express \(f ( x )\) in terms of partial fractions.
b) Find \(\int \frac { 3 x + 2 } { ( x - 1 ) ( x + 1 ) ( 2 x + 3 ) } \mathrm { d } x\), giving your answer in the form \(a \ln | g ( x ) |\), where \(a\) is a real number and \(g ( x )\) is a function of \(x\). Geraint opens a savings account. He deposits \(\pounds 10\) in the first month. In each subsequent month, the amount he deposits is 20 pence greater than the amount he deposited in the previous month.
a) Find the amount that Geraint deposits into the savings account in the 12th month.
b) Determine the number of months it takes for the total amount in the savings account to reach \(\pounds 954\).
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The diagram below shows a sketch of the curves \(y = x ^ { 2 }\) and \(y = 8 \sqrt { x }\). \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-3_508_869_2094_623} Find the area of the region bounded by the two curves.

\includegraphics{figure_6} The diagram shows a sector \(OAB\) of a circle with centre \(O\). Angle \(AOB = \theta\) radians and \(OP = AP = x\).

1. Show that the arc length \(AB\) is \(2x\theta \cos \theta\). [2]
2. Find the area of the shaded region \(APB\) in terms of \(x\) and \(\theta\). [4]

\includegraphics{figure_7} In the diagram, \(AOD\) and \(BC\) are two parallel straight lines. Arc \(AB\) is part of a circle with centre \(O\) and radius \(15\text{cm}\). Angle \(BOA = \theta\) radians. Arc \(CD\) is part of a circle with centre \(O\) and radius \(10\text{cm}\). Angle \(COD = \frac{1}{3}\pi\) radians.

1. Show that \(\theta = 0.7297\), correct to 4 decimal places. [1]
2. Find the perimeter and the area of the shape \(ABCD\). Give your answers correct to 3 significant figures. [7]

\includegraphics{figure_8} The diagram shows a symmetrical plate \(ABCDEF\). The line \(ABCD\) is straight and the length of \(BC\) is 2cm. Each of the two sectors \(ABF\) and \(DCE\) is of radius \(r\)cm and each of the angles \(ABF\) and \(DCE\) is equal to \(\frac{1}{4}\pi\) radians.

1. It is given that \(r = 0.4\)cm.
   1. Show that the length \(EF = 2.4\)cm. [2]
   2. Find the area of the plate. Give your answer correct to 3 significant figures. [4]
2. It is given instead that the perimeter of the plate is 6cm. Find the value of \(r\). Give your answer correct to 3 significant figures. [4]

\includegraphics{figure_4} The diagram shows the shape of a coin. The three arcs \(AB\), \(BC\) and \(CA\) are parts of circles with centres \(C\), \(A\) and \(B\) respectively. \(ABC\) is an equilateral triangle with sides of length 2 cm.

1. Find the perimeter of the coin. [2]
2. Find the area of the face \(ABC\) of the coin, giving the answer in terms of \(\pi\) and \(\sqrt{3}\). [4]

\includegraphics{figure_3} The diagram shows a sector of a circle, centre \(O\), where \(OB = OC = 15\) cm. The size of angle \(BOC\) is \(\frac{2}{5}\pi\) radians. Points \(A\) and \(D\) on the lines \(OB\) and \(OC\) respectively are joined by an arc \(AD\) of a circle with centre \(O\). The shaded region is bounded by the arcs \(AD\) and \(BC\) and by the straight lines \(AB\) and \(DC\). It is given that the area of the shaded region is \(\frac{90}{7}\pi\) cm\(^2\). Find the perimeter of the shaded region. Give your answer in terms of \(\pi\). [5]

\includegraphics{figure_6} The diagram shows a metal plate \(OABCDEF\) consisting of sectors of two circles, each with centre \(O\). The radii of sectors \(AOB\) and \(EOF\) are \(r\) cm and the radius of sector \(COD\) is \(2r\) cm. Angle \(AOB =\) angle \(EOF = \theta\) radians and angle \(COD = 2\theta\) radians. It is given that the perimeter of the plate is 14 cm and the area of the plate is 10 cm\(^2\). Given that \(r \geqslant \frac{3}{2}\) and \(\theta < \frac{3}{4}\), find the values of \(r\) and \(\theta\). [6]

\includegraphics{figure_7} The diagram shows a metal plate \(ABCDEF\) consisting of five parts. The parts \(BCD\) and \(DEF\) are semicircles. The part \(BAFO\) is a sector of a circle with centre \(O\) and radius 20 cm, and \(D\) lies on this circle. The parts \(OBD\) and \(ODF\) are triangles. Angles \(BOD\) and \(DOF\) are both \(\theta\) radians.

1. Given that \(\theta = 1.2\), find the area of the metal plate. Give your answer correct to 3 significant figures. [5]
2. Given instead that the area of each semicircle is \(50\pi \text{ cm}^2\), find the exact perimeter of the metal plate. [5]

\includegraphics{figure_6} The diagram shows a metal plate made by removing a segment from a circle with centre \(O\) and radius \(8\) cm. The line \(AB\) is a chord of the circle and angle \(AOB = 2.4\) radians. Find

1. the length of \(AB\), [2]
2. the perimeter of the plate, [3]
3. the area of the plate. [3]

\includegraphics{figure_8} In the diagram, \(AB\) is an arc of a circle with centre \(O\) and radius \(r\). The line \(XB\) is a tangent to the circle at \(B\) and \(A\) is the mid-point of \(OX\).

1. Show that angle \(AOB = \frac{1}{3}\pi\) radians. [2]

Express each of the following in terms of \(r\), \(\pi\) and \(\sqrt{3}\):

1. the perimeter of the shaded region, [3]
2. the area of the shaded region. [2]

A piece of wire of length 24 cm is bent to form the perimeter of a sector of a circle of radius \(r\) cm.

1. Show that the area of the sector, \(A\) cm\(^2\), is given by \(A = 12r - r^2\). [3]
2. Express \(A\) in the form \(a - (r - b)^2\), where \(a\) and \(b\) are constants. [2]
3. Given that \(r\) can vary, state the greatest value of \(A\) and find the corresponding angle of the sector. [2]

\includegraphics{figure_2} In the diagram, \(AYB\) is a semicircle with \(AB\) as diameter and \(OAXB\) is a sector of a circle with centre \(O\) and radius \(r\). Angle \(AOB = 2\theta\) radians. Find an expression, in terms of \(r\) and \(\theta\), for the area of the shaded region. [4]

\includegraphics{figure_7} The diagram shows two circles with centres \(A\) and \(B\) having radii 8 cm and 10 cm respectively. The two circles intersect at \(C\) and \(D\) where \(CAD\) is a straight line and \(AB\) is perpendicular to \(CD\).

1. Find angle \(ABC\) in radians. [1]
2. Find the area of the shaded region. [6]

\includegraphics{figure_3} The diagram shows triangle \(ABC\) which is right-angled at \(A\). Angle \(ABC = \frac{1}{4}\pi\) radians and \(AC = 8\) cm. The points \(D\) and \(E\) lie on \(BC\) and \(BA\) respectively. The sector \(ADE\) is part of a circle with centre \(A\) and is such that \(BDC\) is the tangent to the arc \(DE\) at \(D\).

1. Find the length of \(AD\). [3]
2. Find the area of the shaded region. [3]

\includegraphics{figure_3} In the diagram, \(CXD\) is a semicircle of radius \(7\) cm with centre \(A\) and diameter \(CD\). The straight line \(YAX\) is perpendicular to \(CD\), and the arc \(CYD\) is part of a circle with centre \(B\) and radius \(8\) cm. Find the total area of the region enclosed by the two arcs. [6]