CIE Mar 2020 9709 Pure Maths Paper 12

CIE Mar 2020 9709 Pure Maths Paper 12 (pdf)

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1. The function f is defined by f(x)
   Determine whether f is an increasing function, a decreasing function or neither.
2. The graph of y = f(x) is transformed to the graph Describe fully the two single transformations which have been combined to give the resulting transformation
3. The diagram shows part of the curve with equation y = x² + 1. The shaded region enclosed by the curve, the y-axis and the line y = 5 is rotated through 360° about the y-axis.
   Find the volume obtained
4. A curve has equation y = x² − 2x − 3. A point is moving along the curve in such a way that at P the y-coordinate is increasing at 4 units per second and the x-coordinate is increasing at 6 units per second.
   Find the x-coordinate of P
5. Solve the equation
6. The coefficient of 1/x in the expansion of
   (a) Find the possible values of the constant a
   (b) Hence find the coefficient of 1/x⁷in the expansion

7. The diagram shows a sector AOB which is part of a circle with centre O and radius 6 cm and with angle AOB = 0.8 radians. The point C on OB is such that AC is perpendicular to OB. The arc CD is part of a circle with centre O, where D lies on OA.
   Find the area of the shaded region.
8. A woman’s basic salary for her first year with a particular company is $30 000 and at the end of the year she also gets a bonus of $600.
   (a) For her first year, express her bonus as a percentage of her basic salary.
   At the end of each complete year, the woman’s basic salary will increase by 3% and her bonus will increase by $100.
   (b) Express the bonus she will be paid at the end of her 24th year as a percentage of the basic salary paid during that year
9. (a) Express 2x² + 12x + 11 in the form 2(x + a)² + b, where a and b are constants
   The function f is defined by f(x) = 2x² + 12x + 11 for x ≤ −4.
   (b) Find an expression for f^-1(x) and state the domain of f^-1
   The function g is defined by g(x) = 2x − 3 for x ≤ k.
   (c) For the case where k = −1, solve the equation fg(x) = 193
   (d) State the largest value of k possible for the composition fg to be defined
10. The gradient of a curve at the point (x, y) is given by x. The curve has a stationary point at (a, 14), where a is a positive constant.
    (a) Find the value of a.
    (b) Determine the nature of the stationary point.
    (c) Find the equation of the curve
11. (a) Solve the equation 3 tan²x − 5 tan x − 2 = 0 for 0° ≤ x ≤ 180°.
    (b) Find the set of values of k for which the equation 3 tan²x − 5 tan x + k = 0 has no solutions
    (c) For the equation 3 tan²x − 5 tan x + k = 0, state the value of k for which there are three solutions in the interval 0° ≤ x ≤ 180°, and find these solutions
12. A diameter of a circle C₁ has end-points at (−3, −5) and (7, 3).
    (a) Find an equation of the circle C₁
    The circle C₁ s translated by to give circle C₂, as shown in the diagram.
    (b) Find an equation of the circle C₂
    The two circles intersect at points R and S.
    (c) Show that the equation of the line RS is y = −2x + 13
    (d) Hence show that the x-coordinates of R and S satisfy the equation 5x² − 60x + 159 = 0

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