An artist is designing a stained-glass panel. The boundary of the panel is modeled by the curve CCC with equation y=6x−x2y = 6x - x^2y=6x−x2, for x≥0x \ge 0x≥0. A decorative lead strip is placed along the line lll with equation y=kxy = kxy=kx, where kkk is a constant and 0<k<60 < k < 60<k<6.
The line lll and the curve CCC intersect at the origin OOO and at the point PPP.
Find, in terms of kkk, the coordinates of PPP.
The region R1R_1R1 is bounded by the curve CCC and the line lll. Show that the area of R1R_1R1 is
(6−k)36 \frac{(6 - k)^3}{6} 6(6−k)3The region R2R_2R2 is bounded by the curve CCC, the xxx-axis, and the line lll, such that R1R_1R1 and R2R_2R2 together comprise the total area under the curve for 0≤x≤60 \le x \le 60≤x≤6. Given that the area of R1R_1R1 is equal to the area of R2R_2R2, find the exact value of kkk.
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.