Edexcel A Level Maths C2 · Core Mathematics 2 June 2014 question paper 6664

Figure 1 shows a sketch of part of the curve with equation y = ¥(x² + 1), x 0 The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis and the lines x = 1 and x = 2 The table below shows corresponding values for x and y for y = ¥(x² + 1).

Complete the table above, giving the missing value of y to 3 decimal places.

Use the trapezium rule, with all the values of y in the completed table, to find an approximate value for the area of R, giving your answer to 2 decimal places.

f(x) = 2x³ – 7x² + 4x + 4

Use the factor theorem to show that (x – 2) is a factor of f(x).

Factorise f(x) completely.

Find the first 3 terms, in ascending powers of x, of the binomial expansion of

giving each term in its simplest form.

Hence, or otherwise, find the first 3 terms, in ascending powers of x, of the expansion of

Use integration to find

The shape ABCDEA, as shown in Figure 2, consists of a right-angled triangle EAB and a triangle DBC joined to a sector BDE of a circle with radius 5 cm and centre B. The points A, B and C lie on a straight line with BC = 7.5 cm. Angle EAB = radians, angle EBD = 1.4 radians and CD = 6.1 cm.

Find, in cm, the area of the sector BDE.

Find the size of the angle DBC, giving your answer in radians to 3 decimal places.

Find, in cm², the area of the shape ABCDEA, giving your answer to 3 significant figures.

The first term of a geometric series is 20 and the common ratio is The sum to infinity of the series is S_’

Find the value of S_’

The sum to N terms of the series is S_N

Find, to 1 decimal place, the value of S₁₂

Find the smallest value of N, for which

Solve, for 0 ≤ θ < 360°, the equation

giving your answers to 1 decimal place. You must show each step of your working.

Solve, for – ≤ x , the equation 2tan x – 3sin x = 0 giving your answers to 2 decimal places where appropriate. [ Solutions based entirely on graphical or numerical methods are not acceptable. ]

Sketch the graph of

showing the coordinates of any points at which the graph crosses the axes.

Use algebra to solve the equation

giving your answers to 2 decimal places where appropriate.

Figure 3 shows a circle C with centre Q and radius 4 and the point T which lies on C. The tangent to C at the point T passes through the origin O and OT = 6 ¥ 5 Given that the coordinates of Q are (11, k), where k is a positive constant,

find the exact value of k,

find an equation for C.

A B A 4x cm B

Figure 4 shows a closed letter box ABFEHGCD, which is made to be attached to a wall of a house. The letter box is a right prism of length y cm as shown in Figure 4. The base ABFE of the prism is a rectangle. The total surface area of the six faces of the prism is S cm². The cross section ABCD of the letter box is a trapezium with edges of lengths DA = 9 x cm, AB = 4 x cm, BC = 6 x cm and CD = 5 x cm as shown in Figure 5. The angle DAB = 90° and the angle ABC = 90°. The volume of the letter box is 9600 cm³.

Show that

Hence show that the surface area of the letter box, S cm², is given by

Use calculus to find the minimum value of S.

Justify, by further differentiation, that the value of S you have found is a minimum.