Edexcel A Level Maths C4 · Core Mathematics 4 June 2019 question paper 6666

The binomial series expansion of

up to and including the term in x² is

where a and k are constants.

Find the value of a.

Find the value of k, giving your answer in its simplest form.

Hence find the numerical coefficient of x² in the series expansion of

Figure 1 shows a sketch of the curve with equation

The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis and the

The table below shows corresponding values of x and y for y = sin 2x sin x௘ ZLWK WKH YDOXHV of y given to 5 significant figures where appropriate.

Use the trapezium rule, with all the values of y in the table, to find an estimate for the area of R, giving your answer to 4 significant figures.

where a and b are constants to be found. (Solutions based entirely on graphical or numerical methods are not acceptable.)

The curve C has equation x²y³xxsin(௘y

The point P with coordinates (3, 2) lies on C. The tangent to C at P meets the y-axis at the point Q.

Find the y coordinate of Q, giving your answer in the form where a, b and c are integers to be found.

Figure 2 shows a sketch of part of the curve C with equation

The finite region R, shown shaded in Figure 2, is bounded by the line with equation x í WKH FXUYH C, the line with equation x = and the x-axis. The region R is rotated through 360° about the x-axis to form a solid of revolution with volume V. Use calculus to find the exact value of V, giving your answer in its simplest form. (Solutions based entirely on graphical or numerical methods are not acceptable.)

A container with a circular cross-section is shown in Figure 3. Initially the container is empty. At time t seconds after water begins to flow into the container, the height of water in the container is h cm. The height of water in the container satisfies the differential equation

where k is a constant. When h = 16, the height of water in the container is increasing at a rate of 0.6 cm s

Find the value of k.

Find the time taken to fill the container with water from empty to a height of 30 cm.

*LYHQ WKDW WKH ZDWHU IORZV LQWR WKH FRQWDLQHU DW D FRQVWDQW UDWH RI ³ s

find the volume of water in the container when h = 30 Give your answer in cm³ to 3 significant figures.

Find, in its simplest form,

Hence find

Use integration by parts to find

giving your answer in the form a + b ln 2, where a and b are constants to be found.

The curve C has parametric equations

where θ is a parameter.

Find an expression for in terms of . The curve C cuts the y-axis at the point A. The line l is the normal to C at the point A.

Show that an equation for l is

The line l intersects the curve C again at the point B.

are rational constants. (Solutions based entirely on graphical or numerical methods are not acceptable.)

With respect to a fixed origin O, the line l₁ is given by the equation

where is a scalar parameter. The point P lies on l₁ and has coordinates (2, c, d), where c and d are constants. The point A does not lie on l₁1 DQG KDV FRRUGLQDWHV í

Find the exact value of the distance PA.

The line l₂ passes through the point A and is parallel to the line l₁

Write down a vector equation for the line l₂ The acute angle between PA and l₂ is

The point B lies on the line l₂ such that PA = PB.

Find the coordinates of the point B.