Edexcel C4 June 2011 exam answers review

Find the values of the constants A, B and C.

(4)

Find the first three non-zero terms of the binomial expansion of f (x) in ascending powers of x. Give each coefficient as a simplified fraction.

(6)

A hollow hemispherical bowl is shown in Figure 1. Water is flowing into the bowl. When the depth of the water is h m, the volume V m3 is given by

1. Find, in terms of π, dV/dh when h = 0.1(6)

Water flows into the bowl at a rate of π/800m³s^-1

2. Find the rate of change of h, in m s^–1, when h = 0.1(6)

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Figure 2 shows a sketch of the curve with equation y = x³ ln (x2 + 2), x≥0. The finite region R, shown shaded in Figure 2, is bounded by the curve, the x-axis and the line x =√2.

The table below shows corresponding values of x and y for y = x³ln (x² + 2).

x	0	√2/4	√2/2	3√2/4	√2
y	0		0.3240		3.9210

1. Complete the table above giving the missing values of y to 4 decimal places.
2. Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 2 decimal places.(2)
3. Use the substitution u = x² + 2 to show that the area of R is
   ½\( \int_{4}^{2} \)(u – 2)lnu du

   (3)

4. Hence, or otherwise, find the exact area of R. (6)

Find the gradient of the curve with equation

at the point on the curve where x = 2. Give your answer as an exact value. (7)

With respect to a fixed origin O, the lines l1 and l2 are given by the equations

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where λ and µ are scalar parameters.

1. Show that l¹ and l² meet and find the position vector of their point of intersection A.(6)
2. Find, to the nearest 0.1°, the acute angle between l₁ and l₁.(3)

The point B has position vector [IMAGE]

3. Show that B lies on l¹(1)
4. Find the shortest distance from B to the line 2 l , giving your answer to 3 significant figures.(4)

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Figure 3 shows part of the curve C with parametric equations

The point P lies on C and has coordinates (√2, ½√3)

1. Find the value of θ at the point P.(2)

The line l is a normal to C at P. The normal cuts the x-axis at the point Q.

2. how that Q has coordinates (k√3, 0), giving the value of the constant k.(6)

The finite shaded region S shown in Figure 3 is bounded by the curve C, the line x = √3 and the x-axis. This shaded region is rotated through 2π radians about the x-axis to form a solid of revolution.

3. Find the volume of the solid of revolution, giving your answer in the form pπ√3 + qπ², where p and q are constants.(7)

1. Find \( \int \)(4y+3)^½dy(2)
2. Given that y =1.5 at x = – 2, solve the differential equation
   dy/dx = √(4y+3)/x²

   giving your answer in the form y = f (x).(6)