AS & A Level Mathematics (9709) — October–November 2023, Paper 13

A curve is such that its gradient at a point (x, y) is given by dy/dx = x − 3x^(−1/2). It is given that the curve passes through the point (4, 1). Find the equation of the curve.

The circle with equation (x − 3)² + (y − 5)² = 40 intersects the y-axis at points A and B. Find the y-coordinates of A and B, expressing your answers in terms of surds.

Show that the equation 5 cos θ − sin θ tan θ + 1 = 0 may be expressed in the form a cos²θ + b cos θ + c = 0, where a, b and c are constants to be found.

Expand the following in ascending powers of x up to and including the term in x². (1 + 2x)⁵

The first, second and third terms of a geometric progression are 2p + 6, 5p and 8p + 2 respectively. Find the possible values of the constant p.

A line has equation y = 6x − c and a curve has equation y = cx² + 2x − 3, where c is a constant. The line is a tangent to the curve at point P. Find the possible values of c and…

The function f is defined by f(x) = 1 + 3/(x − 2) for x 2. State the range of f.

The diagram shows part of the graph of y = sin(a(x + b)), where a and b are positive constants. State the value of a and one possible value of b.

A curve has equation y = 2x^(1/2) − 1. Find the equation of the normal to the curve at the point A (4, 3), giving your answer in the form y = mx + c.

The diagram shows points A, B and C lying on a circle with centre O and radius r. Angle AOB is 2.8 radians. The shaded region is bounded by two arcs. The upper arc is part of the…

The diagram shows part of the curve with equation y = x + 2/(2x − 1)². The lines x = 1 and x = 2 intersect the curve at P and Q respectively and R is the stationary point on the…