Past Exam Questions

The diagram shows part of the curve \(y = \frac{8}{x+2}\) and the line \(2y + x = 8\), intersecting at points \(A\) and \(B\). The point \(C\) lies on the curve and the tangent to the curve at \(C\) is parallel to \(AB\).

(a) Find, by calculation, the coordinates of \(A\), \(B\) and \(C\). [6]

(b) Find the volume generated when the shaded region, bounded by the curve and the line, is rotated through 360° about the \(x\)-axis. [6]

The diagram shows part of the curve with equation \(y = x^2 + 1\). The shaded region enclosed by the curve, the \(y\)-axis and the line \(y = 5\) is rotated through 360° about the \(y\)-axis.

Find the volume obtained.

A curve has equation \(y = f(x)\) and it is given that

\(f'(x) = \left( \frac{1}{2}x + k \right)^{-2} - (1 + k)^{-2}\),

where \(k\) is a constant. The curve has a minimum point at \(x = 2\).

(a) Find \(f''(x)\) in terms of \(k\) and \(x\), and hence find the set of possible values of \(k\).

It is now given that \(k = -3\) and the minimum point is at \((2, 3\frac{1}{2})\).

(b) Find \(f(x)\).

(c) Find the coordinates of the other stationary point and determine its nature.

A curve passes through (0, 11) and has an equation for which \(\frac{dy}{dx} = ax^2 + bx - 4\), where \(a\) and \(b\) are constants.

(i) Find the equation of the curve in terms of \(a\) and \(b\).

(ii) It is now given that the curve has a stationary point at (2, 3). Find the values of \(a\) and \(b\).

A curve has a stationary point at \((3, 9\frac{1}{2})\) and has an equation for which \(\frac{dy}{dx} = ax^2 + a^2 x\), where \(a\) is a non-zero constant.

1. Find the value of \(a\).
2. Find the equation of the curve.
3. Determine, showing all necessary working, the nature of the stationary point.

A curve is such that \(\frac{dy}{dx} = \sqrt{4x + 1}\) and \((2, 5)\) is a point on the curve.

(i) Find the equation of the curve. [4]

(ii) A point \(P\) moves along the curve in such a way that the \(y\)-coordinate is increasing at a constant rate of 0.06 units per second. Find the rate of change of the \(x\)-coordinate when \(P\) passes through \((2, 5)\). [2]

(iii) Show that \(\frac{d^2y}{dx^2} \times \frac{dy}{dx}\) is constant. [2]

A curve has equation \(y = f(x)\) and it is given that \(f'(x) = ax^2 + bx\), where \(a\) and \(b\) are positive constants.

(i) Find, in terms of \(a\) and \(b\), the non-zero value of \(x\) for which the curve has a stationary point and determine, showing all necessary working, the nature of the stationary point.

(ii) It is now given that the curve has a stationary point at \((-2, -3)\) and that the gradient of the curve at \(x = 1\) is 9. Find \(f(x)\).

The function \(f\) is defined for \(x \geq 0\). It is given that \(f\) has a minimum value when \(x = 2\) and that \(f''(x) = (4x + 1)^{-\frac{1}{2}}\).

(i) Find \(f'(x)\).

It is now given that \(f''(0), f'(0)\) and \(f(0)\) are the first three terms respectively of an arithmetic progression.

(ii) Find the value of \(f(0)\).

(iii) Find \(f(x)\), and hence find the minimum value of \(f\).

A curve for which \(\frac{dy}{dx} = 7 - x^2 - 6x\) passes through the point \((3, -10)\).

(i) Find the equation of the curve.

(ii) Express \(7 - x^2 - 6x\) in the form \(a - (x + b)^2\), where \(a\) and \(b\) are constants.

(iii) Find the set of values of \(x\) for which the gradient of the curve is positive.

A curve is such that \(\frac{dy}{dx} = \frac{2}{a}x^{-\frac{1}{2}} + ax^{-\frac{3}{2}}\), where \(a\) is a positive constant. The point \(A(a^2, 3)\) lies on the curve. Find, in terms of \(a\),

1. the equation of the tangent to the curve at \(A\), simplifying your answer,
2. the equation of the curve.

It is now given that \(B(16, 8)\) also lies on the curve.

3. Find the value of \(a\) and, using this value, find the distance \(AB\).

A curve has equation \(y = f(x)\) and it is given that \(f'(x) = 3x^{\frac{1}{2}} - 2x^{-\frac{1}{2}}\). The point \(A\) is the only point on the curve at which the gradient is \(-1\).

(i) Find the \(x\)-coordinate of \(A\).

(ii) Given that the curve also passes through the point \((4, 10)\), find the \(y\)-coordinate of \(A\), giving your answer as a fraction.

A curve is such that \(\frac{dy}{dx} = 6x^2 + \frac{k}{x^3}\) and passes through the point \(P(1, 9)\). The gradient of the curve at \(P\) is 2.

(i) Find the value of the constant \(k\).

(ii) Find the equation of the curve.

A curve has equation \(y = f(x)\), and it is given that \(f'(x) = 2x^2 - 7 - \frac{4}{x^2}\).

(a) Given that \(f(1) = -\frac{1}{3}\), find \(f(x)\).

(b) Find the coordinates of the stationary points on the curve.

(c) Find \(f''(x)\).

(d) Hence, or otherwise, determine the nature of each of the stationary points.

A curve is such that \(\frac{dy}{dx} = 2 - 8(3x + 4)^{-\frac{1}{2}}\).

The curve intersects the y-axis where \(y = \frac{4}{3}\).

Find the equation of the curve.

A curve passes through the point A (4, 6) and is such that \(\frac{dy}{dx} = 1 + 2x^{-\frac{1}{2}}\). A point P is moving along the curve in such a way that the x-coordinate of P is increasing at a constant rate of 3 units per minute.

(i) Find the rate at which the y-coordinate of P is increasing when P is at A.

(ii) Find the equation of the curve.

(iii) The tangent to the curve at A crosses the x-axis at B and the normal to the curve at A crosses the x-axis at C. Find the area of triangle ABC.

The curve \(y = f(x)\) has a stationary point at \((2, 10)\) and it is given that \(f''(x) = \frac{12}{x^3}\).

(i) Find \(f(x)\).

(ii) Find the coordinates of the other stationary point.

(iii) Find the nature of each of the stationary points.

A curve \(y = f(x)\) has a stationary point at \((3, 7)\) and is such that \(f''(x) = 36x^{-3}\).

(i) State, with a reason, whether this stationary point is a maximum or a minimum.

(ii) Find \(f'(x)\) and \(f(x)\).

A curve is such that \(\frac{d^2y}{dx^2} = \frac{24}{x^3} - 4\). The curve has a stationary point at \(P\) where \(x = 2\).

1. State, with a reason, the nature of this stationary point.
2. Find an expression for \(\frac{dy}{dx}\).
3. Given that the curve passes through the point \((1, 13)\), find the coordinates of the stationary point \(P\).

The function f is defined for x > 0 and is such that f'(x) = 2x - \(\frac{2}{x^2}\). The curve y = f(x) passes through the point P (2, 6).

1. Find the equation of the normal to the curve at P.
2. Find the equation of the curve.
3. Find the x-coordinate of the stationary point and state with a reason whether this point is a maximum or a minimum.