Edexcel A Level Maths C1 · Core Mathematics 1 June 2008 question paper 6663

Factorise completely

Figure 1 shows a sketch of the curve with equation y = f(x). The curve passes through the point (0, 7) and has a minimum point at (7, 0). On separate diagrams, sketch the curve with equation

y = f(x) + 3,

y = f(2 x).

On each diagram, show clearly the coordinates of the minimum point and the coordinates of the point at which the curve crosses the y-axis.

f(x) = 3x + x³, x > 0.

Differentiate to find f (x).

Given that f (x) = 15,

find the value of x.

A sequence x₁, x₂, x₃, … is defined by

where a is a constant.

Find an expression for x₂ in terms of a.

Show that x₃ = a² – 3a – 3. Given that x₃ = 7,

find the possible values of a.

The curve C has equation y=y = and the line l has equation y = 2x + 5.

On the axes below, sketch the graphs of C and l, indicating clearly the coordinates of any intersections with the axes.

Find the coordinates of the points of intersection of C and l.

Sue is training for a marathon. Her training includes a run every Saturday starting with a run of 5 km on the first Saturday. Each Saturday she increases the length of her run from the previous Saturday by 2 km.

Show that on the 4th Saturday of training she runs 11 km.

Find an expression, in terms of n, for the length of her training run on the n th Saturday.

Show that the total distance she runs on Saturdays in n weeks of training is n (n + 4) km.

On the n th Saturday Sue runs 43 km.

Find the value of n.

Find the total distance, in km, Sue runs on Saturdays in n weeks of training.

Given that the equation 2qx² + qx – 1 = 0, where q is a constant, has no real roots,

Hence find the set of possible values of q.

The curve C has equation y = kx³ – x² + x – 5, where k is a constant.

The point A with x-coordinate – lies on C. The tangent to C at A is parallel to the line with equation 2 y – 7 x + 1 = 0. Find

the value of k,

the value of the y-coordinate of A.

The points Q (1, 3) and R (7, 0) lie on the line l₁, as shown in Figure 2. The length of QR is a5.

Find the value of a.

The line l₂ is perpendicular to l₁, passes through Q and crosses the y-axis at the point P, as shown in Figure 2. Find

the coordinates of P,

The gradient of a curve C is given by = x 0.

The point (3, 20) lies on C.

Find an equation for the curve C in the form y = f(x).