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1.9 Calculus

1.9 Calculus

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Question 516

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The cross-section of a high-tensile structural support beam is modeled by the curve defined by the equation

3y2−4xy+2x2+6x=20 3y^2 - 4xy + 2x^2 + 6x = 20 3y2−4xy+2x2+6x=20

where xxx and yyy are coordinates in decimetres. The boundary of the cross-section intersects the positive xxx-axis at the point RRR.

a.

State the coordinates of RRR.

[2]
b.

The curve has two stationary points, PPP and QQQ, where the tangent to the curve is horizontal.

Show that, at points PPP and QQQ,

ax2+bx+c=0 ax^2 + bx + c = 0 ax2+bx+c=0

where aaa, bbb and ccc are integers to be found.

[5]
c.

Find the xxx-coordinate of point QQQ, given that xQ>0x_Q > 0xQ​>0, giving your answer to 3 decimal places.

[2]
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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