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1.9 Calculus

1.9 Calculus

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Question 573

The figure shows

  • the curve CCC with equation y=2x−x2y = 2x - x^2y=2x−x2
  • the line lll with equation y=mxy = mxy=mx, where mmm is a constant and 0<m<20 < m < 20<m<2

The line and the curve intersect at the origin OOO and at the point PPP.

Coordinate diagram showing the parabola C, the line l through O and P, and hatched regions R1 and R2.

a.

Find, in terms of mmm, the coordinates of PPP.

[2]
b.

The region R1R_1R1​, shown shaded in the figure, is bounded by CCC and lll.

Show that the area of R1R_1R1​ is

(2−m)36 \frac{(2 - m)^3}{6} 6(2−m)3​
[3]
c.

The region R2R_2R2​, also shown shaded in the figure, is bounded by CCC, the xxx-axis and lll. Given that the area of R1R_1R1​ is equal to the area of R2R_2R2​,

find the exact value of mmm.

[3]
Markscheme

1.9 Calculus Questions

  1. A Level
  2. /Maths
  3. /1.9 Calculus

825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.

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