Past Exam Questions

The diagram shows the curve \(y = \sqrt{x^4 + 4x + 4}\).

(i) Find the equation of the tangent to the curve at the point \((0, 2)\).

(ii) Show that the \(x\)-coordinates of the points of intersection of the line \(y = x + 2\) and the curve are given by the equation \((x + 2)^2 = x^4 + 4x + 4\). Hence find these \(x\)-coordinates.

The diagram shows the curve \(y = (6x + 2)^{\frac{1}{3}}\) and the point \(A (1, 2)\) which lies on the curve. The tangent to the curve at \(A\) cuts the \(y\)-axis at \(B\) and the normal to the curve at \(A\) cuts the \(x\)-axis at \(C\).

(i) Find the equation of the tangent \(AB\) and the equation of the normal \(AC\). [5]

(ii) Find the distance \(BC\). [3]

(iii) Find the coordinates of the point of intersection, \(E\), of \(OA\) and \(BC\), and determine whether \(E\) is the mid-point of \(OA\). [4]

The curve \(y = \frac{10}{2x+1} - 2\) intersects the \(x\)-axis at \(A\). The tangent to the curve at \(A\) intersects the \(y\)-axis at \(C\).

(i) Show that the equation of \(AC\) is \(5y + 4x = 8\).

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

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

(i) Show that the equation of the normal to the curve at the point \((3, 6)\) is \(2y = x + 9\).

(ii) Given that the normal meets the coordinate axes at points \(A\) and \(B\), find the coordinates of the mid-point of \(AB\).

(iii) Find the coordinates of the point at which the normal meets the curve again.

The diagram shows part of the curve \(y = 2 - \frac{18}{2x+3}\), which crosses the x-axis at \(A\) and the y-axis at \(B\). The normal to the curve at \(A\) crosses the y-axis at \(C\).

(i) Show that the equation of the line \(AC\) is \(9x + 4y = 27\).

(ii) Find the length of \(BC\).

The equation of a curve is \(y = 5 - \frac{8}{x}\).

(i) Show that the equation of the normal to the curve at the point \(P(2, 1)\) is \(2y + x = 4\).

This normal meets the curve again at the point \(Q\).

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

(iii) Find the length of \(PQ\).

The equation of a curve is \(y = 2x + \frac{8}{x^2}\).

(i) Obtain expressions for \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\).

(ii) Find the coordinates of the stationary point on the curve and determine the nature of the stationary point.

(iii) Show that the normal to the curve at the point \((-2, -2)\) intersects the x-axis at the point \((-10, 0)\).

A curve has equation \(y = \frac{4}{\sqrt{x}}\).

The normal to the curve at the point \((4, 2)\) meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Find the length of \(PQ\), correct to 3 significant figures.

The point P lies on the line with equation \(y = mx + c\), where \(m\) and \(c\) are positive constants. A curve has equation \(y = -\frac{m}{x}\). There is a single point P on the curve such that the straight line is a tangent to the curve at P.

(a) Find the coordinates of P, giving the \(y\)-coordinate in terms of \(m\).

The normal to the curve at P intersects the curve again at the point Q.

(b) Find the coordinates of Q in terms of \(m\).

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

(ii) Find the coordinates of the stationary point on the curve. [3]

(iii) Find an expression for \(\frac{d^2y}{dx^2}\) and determine the nature of the stationary point. [2]

(iv) The normal to the curve at \(P\) makes an angle of \(\arctan k\) with the positive \(x\)-axis. Find the value of \(k\). [2]

A curve has equation \(y = \frac{4}{3x-4}\) and \(P(2, 2)\) is a point on the curve.

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

(ii) Find the angle that this tangent makes with the \(x\)-axis.

The diagram shows the line \(2y = x + 5\) and the curve \(y = x^2 - 4x + 7\), which intersect at the points \(A\) and \(B\). Find

(a) the \(x\)-coordinates of \(A\) and \(B\),

(b) the equation of the tangent to the curve at \(B\),

(c) the acute angle, in degrees correct to 1 decimal place, between this tangent and the line \(2y = x + 5\).

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

Find the equation of the normal to the curve at the point (4, 4), giving your answer in the form \(y = mx + c\).

The diagram shows the curve with equation \(y = 4x^{\frac{1}{2}}\).

(i) The straight line with equation \(y = x + 3\) intersects the curve at points \(A\) and \(B\). Find the length of \(AB\).

(ii) The tangent to the curve at a point \(T\) is parallel to \(AB\). Find the coordinates of \(T\).

(iii) Find the coordinates of the point of intersection of the normal to the curve at \(T\) with the line \(AB\).

The diagram shows the curve \(y = (x - 1)^{\frac{1}{2}}\) and points \(A(1, 0)\) and \(B(5, 2)\) lying on the curve.

(i) Find the equation of the line \(AB\), giving your answer in the form \(y = mx + c\).

(ii) Find, showing all necessary working, the equation of the tangent to the curve which is parallel to \(AB\).

(iii) Find the perpendicular distance between the line \(AB\) and the tangent parallel to \(AB\). Give your answer correct to 2 decimal places.

A curve has equation \(y = 2x^{\frac{3}{2}} - 3x - 4x^{\frac{1}{2}} + 4\). Find the equation of the tangent to the curve at the point (4, 0).

\(The line 3y + x = 25 is a normal to the curve y = x^2 - 5x + k. Find the value of the constant k.\)

The point A (2, 2) lies on the curve \(y = x^2 - 2x + 2\).

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

The normal to the curve at A intersects the curve again at B.

(ii) Find the coordinates of B.

The tangents at A and B intersect each other at C.

(iii) Find the coordinates of C.

The point \(P(3, 5)\) lies on the curve \(y = \frac{1}{x-1} - \frac{9}{x-5}\).

(i) Find the \(x\)-coordinate of the point where the normal to the curve at \(P\) intersects the \(x\)-axis.

(ii) Find the \(x\)-coordinate of each of the stationary points on the curve and determine the nature of each stationary point, justifying your answers.

The equation of a curve is \(y = f(x)\), where \(f(x) = (4x - 3)^{\frac{5}{3}} - \frac{20}{3}x\).

(a) Find the \(x\)-coordinates of the stationary points of the curve and determine their nature.

(b) State the set of values for which the function \(f\) is increasing.