IGCSE Mathematics - Additional (0606) — October–November 2023, Paper 21

Write 19 − 12x − 3x² in the form a(x + b)² + c where a, b and c are integers.

Solve the following simultaneous equations: 5x − 3 ln y = 2 and x + ln y = 1.

Find ∫ (4x + 5 − 1/(2x + 3)) dx.

In this question a and b are integers. Three terms in the expansion of (2 + ax)⁵(1 + bx) are 32 + 112x − 240x². Find the values of a and b.

In this question p and q are constants. The normal to the curve y = p/x² + 5x − 2, at the point where x = 1, has equation y = −x + q. Find the values of p and q.

Find the value of the constant a for which the line y = (2a + 1)x − 10 is a tangent to the curve y = ax² − 5x + 2.

A particle moves in a straight line. At time t seconds after passing through a fixed point O, its velocity, v m s⁻¹, is given by v = 10 sin 2t − 6 cos 2t. Find an expression for…

DO NOT USE A CALCULATOR IN THIS QUESTION. Solve the equation (2 − √10)x² + x + (2 + √10) = 0, giving your answers in the form a + b√10, where a and b are rational.

The functions f and g are defined as follows, for all real values of x: f(x) = 2x² − 1, g(x) = e^x + 1. Solve the equation fg(x) = 8.

In this question all lengths are in centimetres. The diagram shows a circle centre O with radius 6. The line AB is a tangent to the circle at the point B. The point C lies on the…

Show that 1/(sec x − cosec x) + 1/(sec x + cosec x) = 2 cos x/(1 − cot²x).