CIE May 2020 9709 Pure Maths Paper 1

CIE May 2020 9709 Pure Maths Paper 1 (pdf)

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1. Find the set of values of m for which the line with equation y = mx + 1 and the curve with equation
   y = 3x² + 2x + 4 intersect at two distinct points.
2. The equation of a curve is such that . It is given that the point (4, 7) lies on the curve. Find the equation of the curve
3. In each of parts (a), (b) and (c), the graph shown with solid lines has equation y = f(x). The graph shown with broken lines is a transformation of y = f(x).
   (a) State, in terms of f, the equation of the graph shown with broken line
   (b) State, in terms of f, the equation of the graph shown with broken line
   (c) State, in terms of f, the equation of the graph shown with broken line
4. (a) Expand (1 + a)⁵ in ascending powers of a up to and including the term in a³
   (b) Hence expand [1 + (x + x²)]⁵ in ascending powers of x up to and including the term in x³, simplifying your answer.
5. The diagram shows a cord going around a pulley and a pin. The pulley is modelled as a circle with centre O and radius 5 cm. The thickness of the cord and the size of the pin P can be neglected. The pin is situated 13 cm vertically below O. Points A and B are on the circumference of the circle such that AP and BP are tangents to the circle. The cord passes over the major arc AB of the circle and under the pin such that the cord is taut.
   Calculate the length of the cord.
6. A point P is moving along a curve in such a way that the x-coordinate of P is increasing at a constant rate of 2 units per minute. The equation of the curve is y = (5x − 1)^1/2
   (a) Find the rate at which the y-coordinate is increasing when x = 1.
   (b) Find the value of x when the y-coordinate is increasing at 5/8 units per minute.

7. (a) Show that (b) Hence solve the equation
8. The first term of a progression is sin² θ, where 0 < θ < 1/2 π. The second term of the progression is sin² θ cos² θ. (a) Given that the progression is geometric, find the sum to infinity It is now given instead that the progression is arithmetic. (b) (i) Find the common difference of the progression in terms of sin θ (ii) Find the sum of the first 16 terms when θ = 1/3 π
9. The functions f and g are defined by (a) Express f(x) in the form x − a² + b It is given that f is a one-one function. (b) State the smallest possible value c It is now given that c = 5. (c) Find an expression for f^-1(x) and state the domain of f^-1 (d) Find an expression for gf(x) and state the range of gf
10. (a) The coordinates of two points A and B are (−7, 3) and (5, 11) respectively. Show that the equation of the perpendicular bisector of AB is 3x + 2y = 11 (b) A circle passes through A and B and its centre lies on the line 12x − 5y = 70. Find an equation of the circle.
11. The diagram shows part of the curve with equation y = x³ − 2bx² + b²x and the line OA, where A is the maximum point on the curve. The x-coordinate of A is a and the curve has a minimum point at (b, 0), where a and b are positive constants. (a) Show that b = 3a. (b) Show that the area of the shaded region between the line and the curve is ka⁴, where k is a fraction to be found

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