CIE Oct 2020 9709 Pure Maths Paper 1

CIE Oct 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 − 3 and the curve with equation y = 2x² + 5 do not meet.
2. The equation of a curve is such that. It is given that the curve passes through the point (2, 7). Find the equation of the curve.
3. Air is being pumped into a balloon in the shape of a sphere so that its volume is increasing at a constant rate of 50 cm³s⁻¹.
   Find the rate at which the radius of the balloon is increasing when the radius is 10 cm.
4. In the diagram, the lower curve has equation y = cos θ. The upper curve shows the result of applying a combination of transformations to y = cos θ.
   Find, in terms of a cosine function, the equation of the upper curve.
5. In the expansion of , the coefficients of x⁶ and x³ are equal.
   (a) Find the value of the non-zero constant a.
   (b) Find the coefficient of x⁶ in the expansion of

6. The equation of a curve is . Find the coordinates of the point on the curve at which the gradient is 4/3. 7 (a) Show that
   (b) Hence solve the equation
7. A geometric progression has first term a, common ratio r and sum to infinity S. A second geometric progression has first term a, common ratio R and sum to infinity 2S.
   (a) Show that r = 2R − 1.
   It is now given that the 3rd term of the first progression is equal to the 2nd term of the second progression.
   (b) Express S in terms of a.
8. The diagram shows a circle with centre A passing through the point B. A second circle has centre B and passes through A. The tangent at B to the first circle intersects the second circle at C and D.
   The coordinates of A are (−1, 4) and the coordinates of B are (3, 2).
   (a) Find the equation of the tangent CBD.
   (b) Find an equation of the circle with centre B
   (c) Find, by calculation, the x-coordinates of C and D.
9. The diagram shows a sector CAB which is part of a circle with centre C. A circle with centre O and radius r lies within the sector and touches it at D, E and F, where COD is a straight line and angle ACD is θ radians.
   (a) Find CD in terms of r and sin θ
   It is now given that r = 4 and θ = 1/6 π
   (b) Find the perimeter of sector CAB in terms of π
   (c) Find the area of the shaded region in terms of π and √3
10. The functions f and g are defined by
    f(x) = x2 + 3 for x > 0,
    g(x) = 2x + 1 for x > −1/2.
    (a) Find an expression for fg(x).
    (b) Find an expression for (fg)^-1(x) and state the domain of (fg)^-1.
    (c) Solve the equation fg(x) − 3 = gf(x).
11. The diagram shows a curve with equation y = 4x^1/2 − 2x for x ≥ 0, and a straight line with equation y = 3 − x. The curve crosses the x-axis at A(4, 0) and crosses the straight line at B and C.
    (a) Find, by calculation, the x-coordinates of B and C.
    (b) Show that B is a stationary point on the curve.
    (c) Find the area of the shaded region

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