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1EDEXCEL C4 June 2014 Q7
Figure 4 shows a sketch of part of the curve C with parametric equations x = 3tan(theta), y = 4cos^2(theta), 0 <= theta < pi/2. The point P lies on C and has coordinates (3, 2). The line l is the normal to C at P. The normal cuts the x-axis at the point Q. (a) Find the x coordinate of the point Q. (6) The finite region S, shown shaded in Figure 4, is bounded by the curve C, the x-axis, the y-axis and the line l. This shaded region is rotated 2pi radians about the x-axis to form a solid of revolution. (b) Find the exact value of the volume of the solid of revolution, giving your answer in the form p*pi + q*pi^2, where p and q are rational numbers to be determined. [You may use the formula V = (1/3)pi*r^2*h for the volume of a cone.] (9)
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2EDEXCEL C4 June 2013 Q4
A curve C has parametric equations x = 2sin(t), y = 1 - cos(2t), -pi/2 <= t <= pi/2. (a) Find dy/dx at the point where t = pi/6. (4) (b) Find a cartesian equation for C in the form y = f(x), -k <= x <= k, stating the value of the constant k. (3) (c) Write down the range of f(x). (2)
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3EDEXCEL C4 June 2012 Q6
Figure 2 shows a sketch of the curve C with parametric equations x = sqrt(3)sin(2t), y = 4cos^2(t), 0 <= t <= pi. (a) Show that dy/dx = k*sqrt(3)tan(2t), where k is a constant to be determined. (5) (b) Find an equation of the tangent to C at the point where t = pi/3. Give your answer in the form y = ax + b, where a and b are constants. (4) (c) Find a cartesian equation of C. (3)
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4EDEXCEL C4 January 2011 Q6
The curve C has parametric equations x = ln(t), y = t^2 - 2, t > 0. Find (a) an equation of the normal to C at the point where t = 3, (6) (b) a cartesian equation of C. (3) The finite area R, shown in Figure 1, is bounded by C, the x-axis, the line x = ln 2 and the line x = ln 4. The area R is rotated through 360 degrees about the x-axis. (c) Use calculus to find the exact volume of the solid generated. (6)
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5EDEXCEL C4 June 2010 Q4
A curve C has parametric equations x = sin^2(t), y = 2tan(t), 0 <= t < pi/2. (a) Find dy/dx in terms of t. (4) The tangent to C at the point where t = pi/3 cuts the x-axis at the point P. (b) Find the x-coordinate of P. (6)
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6EDEXCEL C4 June 2009 Q5
Figure 2 shows a sketch of the curve with parametric equations x = 2cos(2t), y = 6sin(t), 0 <= t <= pi/2. (a) Find the gradient of the curve at the point where t = pi/3. (4) (b) Find a cartesian equation of the curve in the form y = f(x), -k <= x <= k, stating the value of the constant k. (4) (c) Write down the range of f(x). (2)
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7EDEXCEL C4 January 2009 Q7
The curve C shown in Figure 3 has parametric equations x = t^3 - 8t, y = t^2, where t is a parameter. Given that the point A has parameter t = -1, (a) find the coordinates of A. (1) The line l is the tangent to C at A. (b) Show that an equation for l is 2x - 5y - 9 = 0. (5) The line l also intersects the curve at the point B. (c) Find the coordinates of B. (6)
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8EDEXCEL C4 June 2008 Q8
Figure 3 shows the curve C with parametric equations x = 8cos(t), y = 4sin(2t), 0 <= t <= pi/2. The point P lies on C and has coordinates (4, 2sqrt(3)). (a) Find the value of t at the point P. (2) The line l is a normal to C at P. (b) Show that an equation for l is y = -x*sqrt(3) + 6*sqrt(3). (6) The finite region R is enclosed by the curve C, the x-axis and the line x = 4, as shown shaded in Figure 3. (c) Show that the area of R is given by the integral from pi/3 to pi/2 of 64 sin^2(t)cos(t) dt. (4) (d) Use this integral to find the area of R, giving your answer in the form a + b*sqrt(3), where a and b are constants to be determined. (4)
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9Edexcel A Level June 2022 Paper 2 Q16
Figure 6 shows a sketch of the curve C with parametric equations x = 2tan(t) + 1, y = 2sec^2(t) + 3, -pi/4 <= t <= pi/3. The line l is the normal to C at the point P where t = pi/4. (a) Using parametric differentiation, show that an equation for l is y = -(1/2)x + 17/2. (5) (b) Show that all points on C satisfy the equation y = (1/2)(x - 1)^2 + 5. (2) The straight line with equation y = -(1/2)x + k, where k is a constant, intersects C at two distinct points. (c) Find the range of possible values for k. (5)
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10Edexcel A Level June 2023 Paper 2 Q9
The curve C has parametric equations x = t^2 + 6t - 16, y = 6ln(t + 3), t > -3. (a) Show that a Cartesian equation for C is y = A ln(x + B), x > -B, where A and B are integers to be found. (3) The curve C cuts the y-axis at the point P. (b) Show that the equation of the tangent to C at P can be written in the form ax + by = c ln 5, where a, b and c are integers to be found. (4)
