QCEVault

Integration and applications of integration — Question 12

QCAA 2024, Paper 1 · 7 marks

Q12 · 2024 · Technology-freeSimple familiar7 marks

QUESTION 12 (7 marks)

Point A lies on a section of the ellipse 3x2+y2=103x^2+y^2=10 as shown. The coordinates of A are (2,y1)(\sqrt2,y_1).
Section of ellipse 3x^2+y^2=10 with point A in quadrant IV
a)
Determine the value of y1y_1.
[2 marks]
b)
Use implicit differentiation to determine an expression for dydx\frac{dy}{dx} in terms of xx and yy.
[2 marks]
c)
Use your results from Questions 12a) and 12b) to determine the gradient of the tangent to the curve at A.
[1 mark]
d)
Consider the region between the given section of the ellipse, the xx-axis and the lines x=0x=0 and x=2x=\sqrt2. Determine the volume of the solid of revolution formed by rotating this region about the xx-axis. Express your answer in simplest form.
[2 marks]
Question linkOriginal paper

Related questions

  1. Q1 · 2025 QCAA · Paper 1 · 1 mark
    Use the substitution u=x2u=x^2 to express ∫2xex2 dx\displaystyle\int 2xe^{x^2}\,dx in terms of uu.
    Integration and applications of integration
  2. Q15 · 2025 QCAA · Paper 1 · 5 marks
    Integration by parts and area.
    Integration and applications of integration
  3. Q10 · 2025 QCAA · Paper 2 · 1 mark
    Consider the solid of revolution formed by rotating a section of the curve y=sin⁡(x)y=\sin(x) around the xx-axis.
    Determine the volume of the solid of revolution.
    Integration and applications of integration
  4. Q18 · 2025 QCAA · Paper 2 · 7 marks
    Polar curves are defined by points that are a variable distance of rr units from the origin and dependent on the angle θ\theta (in radians) measured from the positive xx-axis. Consider the polar curve r=1+cos⁡(θ)r=1+\cos(\theta). A table of four polar coordinates on this curve is shown.
    | θ\theta | rr | | --- | --- | | 00 | 22 | | π/6\pi/6 | 1+3/21+\sqrt3/2 | |…
    Integration and applications of integration