CIE Oct 2021 9709 Pure Maths Paper 31 (pdf)
- Solve the equation 4|5x − 1| = 5x, giving your answers correct to 3 decimal places.
- (a) Express 5 sin x − 3 cos x in the form R sin(x − α), where R > 0 and 0 < α < 1/2π. Give the exact
value of R and give α correct to 2 decimal places
(b) Hence state the greatest and least possible values of (5 sin x − 3 cos x)2
- The curve with equation y = xe1−2x
has one stationary point.
(a) Find the coordinates of this point.
(b) Determine whether the stationary point is a maximum or a minimum.
- Using the substitution u = √x, find the exact value of
- (a) Show that the equation
cot 2θ + cot θ = 2
can be expressed as a quadratic equation in tan θ.
(b) Hence solve the equation cot 21 + cot θ = 2, for 0 < θ < π, giving your answers correct to 3 decimal places.
- When (a + bx)√(1 + 4x), where a and b are constants, is expanded in ascending powers of x, the
coefficients of x and x2 are 3 and −6 respectively.
Find the values of a and b.
- (a) Given that y = ln(ln x), show that
It is given that x = e when t = 2.
(b) Solve the differential equation obtaining an expression for x in terms of t, simplifying your
(c) Hence state what happens to the value of x as t tends to infinity.
- The constant a is such that
- Two lines l and m have equations r = 3i + 2j + 5k + s(4i − j + 3k) and r = i − j − 2k + t(−i + 2j + 2k)
(a) Show that l and m are perpendicular.
(b) Show that l and m intersect and state the position vector of the point of intersection.
- The complex number 1 + 2i is denoted by u. The polynomial 2x3 + ax2 + 4x + b, where a and b are
real constants, is denoted by p(x). It is given that u is a root of the equation p(x) = 0.
(a) Find the values of a and b.
(b) State a second complex root of this equation.
(c) Find the real factors of p(x).
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