Past Exam Questions

A curve has equation \(y = f(x)\) and is such that \(f'(x) = 3x^{\frac{1}{2}} + 3x^{-\frac{1}{2}} - 10\).

It is given that the curve \(y = f(x)\) passes through the point \((4, -7)\). Find \(f(x)\).

A curve is such that \(\frac{dy}{dx} = 2(3x + 4)^{\frac{3}{2}} - 6x - 8\).

It is now given that the stationary point on the curve has coordinates \((-1, 5)\). Find the equation of the curve.

A curve is such that \(\frac{dy}{dx} = -\frac{8}{x^3} - 1\) and the point (2, 4) lies on the curve. Find the equation of the curve.

A curve which passes through (0, 3) has equation \(y = f(x)\). It is given that \(f'(x) = 1 - \frac{2}{(x-1)^3}\).

(a) Find the equation of the curve.

The tangent to the curve at (0, 3) intersects the curve again at one other point, \(P\).

(b) Show that the \(x\)-coordinate of \(P\) satisfies the equation \((2x + 1)(x - 1)^2 - 1 = 0\).

(c) Verify that \(x = \frac{3}{2}\) satisfies this equation and hence find the \(y\)-coordinate of \(P\).

A curve is such that \(\frac{dy}{dx} = \frac{2}{\sqrt{x}} - 1\) and \(P(9, 5)\) is a point on the curve.

Find the equation of the curve.

A curve has equation \(y = f(x)\). It is given that \(f'(x) = 3x^2 + 2x - 5\).

Given that the curve passes through \((1, 3)\), find \(f(x)\).

The equation of a curve is such that \(\frac{dy}{dx} = \frac{3}{\sqrt{x}} - x\). Given that the curve passes through the point (4, 6), find the equation of the curve.

A curve is such that \(\frac{dy}{dx} = 2x^2 - 5\). Given that the point \((3, 8)\) lies on the curve, find the equation of the curve.

The equation of a curve is such that \(\frac{dy}{dx} = \frac{4}{(x-3)^3}\) for \(x > 3\). The curve passes through the point (4, 5).

Find the equation of the curve.

The equation of a curve is such that \(\frac{dy}{dx} = 6x^2 - 30x + 6a\), where \(a\) is a positive constant. The curve has a stationary point at \((a, -15)\).

(a) Find the value of \(a\).

(b) Determine the nature of this stationary point.

(c) Find the equation of the curve.

(d) Find the coordinates of any other stationary points on the curve.

At the point (4, -1) on a curve, the gradient of the curve is \(-\frac{3}{2}\). It is given that \(\frac{dy}{dx} = x^{-\frac{1}{2}} + k\), where \(k\) is a constant.

(a) Show that \(k = -2\).

(b) Find the equation of the curve.

(c) Find the coordinates of the stationary point.

(d) Determine the nature of the stationary point.

The curve \(y = f(x)\) is such that \(f'(x) = \frac{-3}{(x+2)^4}\).

(a) The tangent at a point on the curve where \(x = a\) has gradient \(-\frac{16}{27}\). Find the possible values of \(a\).

(b) Find \(f(x)\) given that the curve passes through the point \((-1, 5)\).

The equation of a curve is such that \(\frac{dy}{dx} = 3x^{\frac{1}{2}} - 3x^{-\frac{1}{2}}\). The curve passes through the point \((3, 5)\).

(a) Find the equation of the curve.

(b) Find the \(x\)-coordinate of the stationary point.

(c) State the set of values of \(x\) for which \(y\) increases as \(x\) increases.

The diagram shows part of the curve with equation \(y = x + \frac{2}{(2x-1)^2}\). The lines \(x = 1\) and \(x = 2\) intersect the curve at \(P\) and \(Q\) respectively and \(R\) is the stationary point on the curve.

(a) Verify that the \(x\)-coordinate of \(R\) is \(\frac{3}{2}\) and find the \(y\)-coordinate of \(R\). [4]

(b) Find the exact value of the area of the shaded region. [6]

The diagram shows the curve with equation \(y = (3x - 2)^{\frac{1}{2}}\) and the line \(y = \frac{1}{2}x + 1\). The curve and the line intersect at points \(A\) and \(B\).

(a) Find the coordinates of \(A\) and \(B\).

(b) Hence find the area of the region enclosed between the curve and the line.

The diagram shows the curves with equations \(y = x^{-\frac{1}{2}}\) and \(y = \frac{5}{2} - x^{-\frac{1}{2}}\). The curves intersect at the points \(A \left( \frac{1}{4}, 2 \right)\) and \(B \left( 4, \frac{1}{2} \right)\).

(a) Find the area of the region between the two curves.

(b) The normal to the curve \(y = x^{-\frac{1}{2}}\) at the point \((1, 1)\) intersects the y-axis at the point \((0, p)\).

Find the value of \(p\).

The diagram shows the line \(x = \frac{5}{2}\), part of the curve \(y = \frac{1}{2}x + \frac{7}{10} - \frac{1}{(x-2)^{\frac{1}{3}}}\) and the normal to the curve at the point \(A \left(3, \frac{6}{5}\right)\).

(a) Find the \(x\)-coordinate of the point where the normal to the curve meets the \(x\)-axis. [5]

(b) Find the area of the shaded region, giving your answer correct to 2 decimal places. [6]

The diagram shows part of the curve with equation \(y = x^{\frac{1}{2}} + k^2 x^{-\frac{1}{2}}\), where \(k\) is a positive constant.

(a) Find the coordinates of the minimum point of the curve, giving your answer in terms of \(k\).

The tangent at the point on the curve where \(x = 4k^2\) intersects the y-axis at \(P\).

(b) Find the y-coordinate of \(P\) in terms of \(k\).

The shaded region is bounded by the curve, the x-axis and the lines \(x = \frac{9}{4}k^2\) and \(x = 4k^2\).

(c) Find the area of the shaded region in terms of \(k\).

The equation of a curve is \(y = 2\sqrt{3x+4} - x\).

Find the exact area of the region bounded by the curve, the x-axis and the lines \(x = 0\) and \(x = 4\).

The diagram shows the curve with equation \(y = 9(x^{-\frac{1}{2}} - 4x^{-\frac{3}{2}})\). The curve crosses the x-axis at the point A.

(a) Find the x-coordinate of A.

(b) Find the equation of the tangent to the curve at A.

(c) Find the x-coordinate of the maximum point of the curve.

(d) Find the area of the region bounded by the curve, the x-axis and the line \(x = 9\).