1. Use the Trapezium Rule with five ordinates to find an approximate value for the integral

Transcription

1 1. Use the Trapezium Rule with five ordinates to find an approximate value for the integral Show your working and give your answer correct to three decimal places When When When When When Formula booklet page 7 { } { ( )} { } { }

2 2. Find all the values of in the range satisfying The angles and are the three angles of a triangle. Given that and that, find the values of and. Give each angle correct to the nearest degree. Let, EITHER OR ( ) ( ), , Angles cannot be negative Angles in a triangle add up to : Substituting into

3 3. The sum of the third and eighth terms of an arithmetic series is zero. The sum of the fifth, seventh and tenth terms of the series is 22. Find the first term and the common difference of the series. The first term of another arithmetic series is 9 and the common difference is 2. The sum of the first terms of this arithmetic series is 3 times the sum of the first terms of the series. Find the value of. The general formula for a term in an arithmetic series is. Formula booklet page 2 Using this formula, the third term is equal to eighth term is equal to. and the Fifth term Substituting into, First term,. Common difference,. when and. [ ] Substituting and into Formula booklet page 2 [ ] [ ] [ ] or

4 4. A geometric series has first term and common ratio. Prove that the sum of the first terms is given by The fourth term of a geometric series is and the seventh term is 4. (i) Find the common ratio of the series. (ii) Find the sum to infinity of the series. ( (i) The general term for in a geometric series is. Formula booklet page 2 (ii) The sum to infinity of a geometric series is. Substituting and into : ( )

5 5. The diagram shows a sketch of the triangle with The point is on such that (i) By applying the cosine rule in each of the triangles and, show that and find a similar expression for (ii) Noting that and are angles on a straight line, use the expressions derived in part (i) to write down an equation satisfied by. Hence show that. Find the area of triangle places.. Give your answer correct to two decimal (i) Recall that the cosine rule is. Using this rule to find, QED To find

6 (ii) Angles on a straight line add up to. ( ) ( ) Using ( ) ( ) ( ) ( ) This is k negative argument Let ( ) as cannot be negative. QED Recall that the area of a triangle is equal to. Applying this rule to triangle : From part (i), (from part (ii)) ( ) (2dp)

7 6. Find ( ). The diagram below shows a sketch of the curve with equation. The shaded region is bounded by the curve, the -axis and the lines. Find the area of the shaded region. ( ) ( ) ( ) Integrate between and. ( ) [ ] [ ] [ ] [ ]

8 7. Given that, show that. Solve the equation Show your working and give your answer correct to three decimal places. Solve the equation. Let Taking the th root of both sides, If, then As and, QED Taking log of both sides, ( ) ( ) (3dp)

9 8. The circle has centre and equation (i) Write down the coordinates of. (ii) The point has coordinates and lies on. Find the equation of the tangent to at. The line has equation. Show that and do not intersect. (i) Completing the square: (ii) The radius of the circle is perpendicular to the tangent at as shown. The gradient of the radius, i.e. the normal to the tangent, at point can be found by: Substituting, and into : Equation of a tangent

10 Substituting into (from part : ( ) If the line intersects the circle, then the above quadratic equation will have at least one solution. Looking at the discriminant, : As the discriminant is negative, there are no real solutions to this quadratic equation, so and do not intersect. QED If the discriminant is negative, then we would be taking the square root of a negative number in the quadratic formula, which is impossible. Looking at the graphs of and, it is also clear that they do not intersect.

11 9. The diagram shows two concentric circles with a common centre. The radius of the larger circle is 7cm and the radius of the smaller circle is 4cm. The points and lie on the larger circle and and cut the smaller circle at the points and respectively. The area of the shaded region is. Find the perimeter of. Let angle. Area of a sector Area of shaded region radians Length of cm Length of cm Length of Length of cm Perimeter of cm

12 10. The th term of a number sequence is denoted by The th term of the sequence satisfies for all positive integers. Given that, evaluate and, describe the behaviour of the sequence and hence, without carrying out any further calculation, write down the value of which means the pattern, will repeat indefinitely. When is a multiple of 3, When is 1 more than a multiple of 3, When is 2 more than a multiple of 3, For, is 2 more than a multiple of 3. where is a positive integer.,