AS & A Level Mathematics (9709) — May–June 2023, Paper 13

The diagram shows the graph of y = f(x), which consists of the two straight lines AB and BC, where A is at (1, 3), B is at (2, 5) and C is at (4, 4). The lines A′B′ and B′C′ form…

The function f is defined for x ∈ ℝ by f(x) = x² − 6x + c, where c is a constant. It is given that f(x) 2 for all values of x. Find the set of possible values of c.

Give the complete expansion of (x + 2/x)⁵.

Show that the equation 3 tan²x − 3 sin²x − 4 = 0 may be expressed in the form a cos⁴x + b cos²x + c = 0, where a, b and c are constants to be found.

A circle has equation (x − 1)² + (y + 4)² = 40. A line with equation y = x − 9 intersects the circle at points A and B. Find the coordinates of the two points of intersection.

The diagram shows a sector OAB of a circle with centre O and radius r cm. Angle AOB = θ radians. It is given that the length of the arc AB is 9.6 cm and that the area of the…

The function f is defined by f(x) = 2 − 5/(x + 2) for x −2. State the range of f.

A progression has first term a and second term a²/(a + 2), where a is a positive constant. For the case where the progression is geometric and the sum to infinity is 264, find the…

A curve which passes through (0, 3) has equation y = f(x). It is given that f′(x) = 1 − 2/(x − 1)³. Find the equation of the curve.

The diagram shows the points A (1½, 5½) and B (7½, 3½) lying on the curve with equation y = 9x − (2x + 1)^(3/2). Find the coordinates of the maximum point of the curve.