Past Exam Questions

The diagram shows part of the curve \(y = \frac{2}{(3 - 2x)^2} - x\) and its minimum point \(M\), which lies on the \(x\)-axis.

(a) Find expressions for \(\frac{dy}{dx}\), \(\frac{d^2y}{dx^2}\) and \(\int y \, dx\).

(b) Find, by calculation, the \(x\)-coordinate of \(M\).

(c) Find the area of the shaded region bounded by the curve and the coordinate axes.

The diagram shows a curve with equation \(y = 4x^{\frac{1}{2}} - 2x\) for \(x \geq 0\), and a straight line with equation \(y = 3 - x\). The curve crosses the x-axis at \(A(4, 0)\) and crosses the straight line at \(B\) and \(C\).

(a) Find, by calculation, the x-coordinates of \(B\) and \(C\).

(b) Show that \(B\) is a stationary point on the curve.

(c) Find the area of the shaded region.

The diagram shows part of the curve with equation \(y = x^3 - 2bx^2 + b^2x\) and the line \(OA\), where \(A\) is the maximum point on the curve. The \(x\)-coordinate of \(A\) is \(a\) and the curve has a minimum point at \((b, 0)\), where \(a\) and \(b\) are positive constants.

(a) Show that \(b = 3a\).

(b) Show that the area of the shaded region between the line and the curve is \(ka^4\), where \(k\) is a fraction to be found.

The diagram shows part of the curve \(y = 1 - \frac{4}{(2x+1)^2}\). The curve intersects the x-axis at \(A\). The normal to the curve at \(A\) intersects the y-axis at \(B\).

(i) Obtain expressions for \(\frac{dy}{dx}\) and \(\int y \, dx\).

(ii) Find the coordinates of \(B\).

(iii) Find, showing all necessary working, the area of the shaded region.

The diagram shows curves with equations \(y = 2x^{\frac{1}{2}} + 13x^{-\frac{1}{2}}\) and \(y = 3x^{-\frac{1}{2}} + 12\). The curves intersect at points \(A\) and \(B\).

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

(b) Hence find the area of the shaded region.

The diagram shows part of the curve with equation \(y = (3x + 4)^{\frac{1}{2}}\) and the tangent to the curve at the point A. The \(x\)-coordinate of A is 4.

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

(ii) Find, showing all necessary working, the area of the shaded region.

(iii) A point is moving along the curve. At the point P the \(y\)-coordinate is increasing at half the rate at which the \(x\)-coordinate is increasing. Find the \(x\)-coordinate of P.

The diagram shows part of the curve \(y = \sqrt{4x+1} + \frac{9}{\sqrt{4x+1}}\) and the minimum point \(M\).

(i) Find expressions for \(\frac{dy}{dx}\) and \(\int y \, dx\).

(ii) Find the coordinates of \(M\).

The shaded region is bounded by the curve, the \(y\)-axis and the line through \(M\) parallel to the \(x\)-axis.

(iii) Find, showing all necessary working, the area of the shaded region.

The diagram shows part of the curve \(y = \frac{3}{\sqrt{1 + 4x}}\) and a point \(P(2, 1)\) lying on the curve. The normal to the curve at \(P\) intersects the \(x\)-axis at \(Q\).

(i) Show that the \(x\)-coordinate of \(Q\) is \(\frac{16}{9}\).

(ii) Find, showing all necessary working, the area of the shaded region.

The diagram shows part of the curve \(y = 3\sqrt{4x + 1} - 2x\). The curve crosses the y-axis at \(A\) and the stationary point on the curve is \(M\).

(i) Obtain expressions for \(\frac{dy}{dx}\) and \(\int y \, dx\).

(ii) Find the coordinates of \(M\).

(iii) Find, showing all necessary working, the area of the shaded region.

The diagram shows part of the curve with equation \(y = k(x^3 - 7x^2 + 12x)\) for some constant \(k\). The curve intersects the line \(y = x\) at the origin \(O\) and at the point \(A (2, 2)\).

1. Find the value of \(k\).
2. Verify that the curve meets the line \(y = x\) again when \(x = 5\).
3. Find, showing all necessary working, the area of the shaded region.

The curve with equation \(y = x^3 - 2x^2 + 5x\) passes through the origin.

Showing all necessary working, find the area of the region enclosed by the curve, the \(x\)-axis and the line \(x = 6\).

The diagram shows part of the curve \(y = 1 - 2x - (1 - 2x)^3\) intersecting the x-axis at the origin \(O\) and at \(A \left( \frac{1}{2}, 0 \right)\). The line \(AB\) intersects the y-axis at \(B\) and has equation \(y = 1 - 2x\).

(i) Show that \(AB\) is the tangent to the curve at \(A\).

(ii) Show that the area of the shaded region can be expressed as \(\int_0^{\frac{1}{2}} (1 - 2x)^3 \, dx\).

(iii) Hence, showing all necessary working, find the area of the shaded region.

The diagram shows parts of the graphs of \(y = 3 - 2x\) and \(y = 4 - \frac{3}{\sqrt{x}}\) intersecting at points \(A\) and \(B\).

(i) Find by calculation the \(x\)-coordinates of \(A\) and \(B\).

(ii) Find, showing all necessary working, the area of the shaded region.

The diagram shows part of the curve \(y = \sqrt{5x - 1}\) and the normal to the curve at the point \(P(2, 3)\). This normal meets the x-axis at \(Q\).

(i) Find the equation of the normal at \(P\).

(ii) Find, showing all necessary working, the area of the shaded region.

The diagram shows the curve \(y = f(x)\) defined for \(x > 0\). The curve has a minimum point at \(A\) and crosses the \(x\)-axis at \(B\) and \(C\). It is given that \(\frac{dy}{dx} = 2x - \frac{2}{x^3}\) and that the curve passes through the point \(\left(4, \frac{189}{16}\right)\).

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

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

(iii) Find the \(x\)-coordinates of \(B\) and \(C\).

(iv) Find, showing all necessary working, the area of the shaded region.

The diagram shows the curves with equations \(y = 2(2x - 3)^4\) and \(y = (2x - 3)^2 + 1\) meeting at points \(A\) and \(B\).

(a) By using the substitution \(u = 2x - 3\), find, by calculation, the coordinates of \(A\) and \(B\). [4]

(b) Find the exact area of the shaded region. [5]

The diagram shows parts of the curves \(y = (2x - 1)^2\) and \(y^2 = 1 - 2x\), intersecting at points \(A\) and \(B\).

(i) State the coordinates of \(A\).

(ii) Find, showing all necessary working, the area of the shaded region.

The diagram shows part of the curve \(y = \frac{1}{16}(3x-1)^2\), which touches the \(x\)-axis at the point \(P\). The point \(Q (3, 4)\) lies on the curve and the tangent to the curve at \(Q\) crosses the \(x\)-axis at \(R\).

(i) State the \(x\)-coordinate of \(P\).

Showing all necessary working, find by calculation

(ii) the \(x\)-coordinate of \(R\),

(iii) the area of the shaded region \(PQR\).

The points \(A\left(-\frac{1}{2}, 3\right)\) and \(B\left(1, 2\frac{1}{4}\right)\) lie on the curve \(y = 2x + (x+1)^{-2}\), as shown in the diagram.

(ii) Find the distance \(AB\).

(iii) Find, showing all necessary working, the area of the shaded region.

Points A (2, 9) and B (3, 0) lie on the curve \(y = 9 + 6x - 3x^2\), as shown in the diagram. The tangent at A intersects the x-axis at C. Showing all necessary working,

(i) find the equation of the tangent AC and hence find the x-coordinate of C,

(ii) find the area of the shaded region ABC.