CIE Oct 2021 9709 Pure Maths Paper 31

CIE Oct 2021 9709 Pure Maths Paper 31 (pdf)

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1. Solve the equation 4|5^x − 1| = 5^x, giving your answers correct to 3 decimal places.
2. (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
3. The curve with equation y = xe^1−2x has one stationary point. (a) Find the coordinates of this point. (b) Determine whether the stationary point is a maximum or a minimum.
4. Using the substitution u = √x, find the exact value of
5. (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.
6. When (a + bx)√(1 + 4x), where a and b are constants, is expanded in ascending powers of x, the coefficients of x and x^{2 are 3 and −6 respectively. Find the values of a and b.}
7. (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 answer. (c) Hence state what happens to the value of x as t tends to infinity.
8. The constant a is such that
9. Two lines l and m have equations r = 3i + 2j + 5k + s(4i − j + 3k) and r = i − j − 2k + t(−i + 2j + 2k) respectively. (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.
10. 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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