Past Exam Questions

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

(b) Find the coordinates of the stationary point.

(c) Determine the nature of the stationary point.

A curve has equation \(y = \frac{1}{k}x^{\frac{1}{2}} + x^{-\frac{1}{2}} + \frac{1}{k^2}\) where \(x > 0\) and \(k\) is a positive constant.

It is given that when \(x = \frac{1}{4}\), the gradient of the curve is 3.

Find the value of \(k\).

The equation of a curve is \(y = 2x + 1 + \frac{1}{2x+1}\) for \(x > -\frac{1}{2}\).

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

(b) Find the coordinates of the stationary point and determine the nature of the stationary point.

A curve has equation \(y = 2x^{\frac{1}{2}} - 1\).

(a) Find the equation of the normal to the curve at the point \(A(4, 3)\), giving your answer in the form \(y = mx + c\).

A point is moving along the curve \(y = 2x^{\frac{1}{2}} - 1\) in such a way that at \(A\) the rate of increase of the \(x\)-coordinate is \(3 \text{ cm s}^{-1}\).

(b) Find the rate of increase of the \(y\)-coordinate at \(A\).

At \(A\) the moving point suddenly changes direction and speed, and moves down the normal in such a way that the rate of decrease of the \(y\)-coordinate is constant at \(5 \text{ cm s}^{-1}\).

(c) As the point moves down the normal, find the rate of change of its \(x\)-coordinate.

The point (4, 7) lies on the curve \(y = f(x)\) and it is given that \(f'(x) = 6x^{-\frac{1}{2}} - 4x^{-\frac{3}{2}}\).

A point moves along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.12 units per second.

Find the rate of increase of the y-coordinate when \(x = 4\).

Air is being pumped into a balloon in the shape of a sphere so that its volume is increasing at a constant rate of 50 cm³s^-1.

Find the rate at which the radius of the balloon is increasing when the radius is 10 cm.

A point P is moving along a curve in such a way that the x-coordinate of P is increasing at a constant rate of 2 units per minute. The equation of the curve is \(y = (5x - 1)^{1/2}\).

\((a) Find the rate at which the y-coordinate is increasing when x = 1. [4]\)

(b) Find the value of x when the y-coordinate is increasing at \(\frac{5}{8}\) units per minute. [3]

A weather balloon in the shape of a sphere is being inflated by a pump. The volume of the balloon is increasing at a constant rate of 600 cm³ per second. The balloon was empty at the start of pumping.

(a) Find the radius of the balloon after 30 seconds.

(b) Find the rate of increase of the radius after 30 seconds.

A curve has equation \(y = x^2 - 2x - 3\). A point is moving along the curve in such a way that at \(P\) the \(y\)-coordinate is increasing at 4 units per second and the \(x\)-coordinate is increasing at 6 units per second.

Find the \(x\)-coordinate of \(P\).

The dimensions of a cuboid are x cm, 2x cm and 4x cm, as shown in the diagram.

(i) Show that the surface area S cm² and the volume V cm³ are connected by the relation

\(S = 7V^{\frac{2}{3}}\).

(ii) When the volume of the cuboid is 1000 cm³ the surface area is increasing at 2 cm² s⁻¹. Find the rate of increase of the volume at this instant.

A curve is such that \(\frac{dy}{dx} = x^3 - \frac{4}{x^2}\). The point \(P(2, 9)\) lies on the curve.

A point moves on the curve in such a way that the \(x\)-coordinate is decreasing at a constant rate of 0.05 units per second. Find the rate of change of the \(y\)-coordinate when the point is at \(P\).

A curve has equation \(y = \frac{1}{2}(4x - 3)^{-1}\). The point \(A\) on the curve has coordinates \((1, \frac{1}{2})\).

(i) (a) Find and simplify the equation of the normal through \(A\). [5]

(b) Find the \(x\)-coordinate of the point where this normal meets the curve again. [3]

(ii) A point is moving along the curve in such a way that as it passes through \(A\) its \(x\)-coordinate is decreasing at the rate of 0.3 units per second. Find the rate of change of its \(y\)-coordinate at \(A\). [2]

A point is moving along the curve \(y = 2x + \frac{5}{x}\) in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.02 units per second. Find the rate of change of the \(y\)-coordinate when \(x = 1\).

Fig. 2 shows a cross-section of a bowl containing water. When the height of the water level is \(h\) cm, the volume, \(V\) cm\(^3\), of water is given by \(V = \pi \left( \frac{1}{2}h^2 + h \right)\). Water is poured into the bowl at a constant rate of 2 cm\(^3\) s\(^{-1}\). Find the rate, in cm s\(^{-1}\), at which the height of the water level is increasing when the height of the water level is 3 cm.

The diagram shows a cubical closed container made of a thin elastic material which is filled with water and frozen. During the freezing process the length, x cm, of each edge of the container increases at the constant rate of 0.01 cm per minute. The volume of the container at time t minutes is V cm³.

\(Find the rate of increase of V when x = 20.\)

A curve has equation \(y = 3 + \frac{12}{2-x}\).

(i) Find the equation of the tangent to the curve at the point where the curve crosses the x-axis. [5]

(ii) A point moves along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.04 units per second. Find the rate of change of the y-coordinate when \(x = 4\). [2]

The diagram shows a water container in the form of an inverted pyramid, which is such that when the height of the water level is h cm the surface of the water is a square of side \(\frac{1}{2}h\) cm.

(i) Express the volume of water in the container in terms of h.

[The volume of a pyramid having a base area A and vertical height h is \(\frac{1}{3}Ah\).]

Water is steadily dripping into the container at a constant rate of 20 cm³ per minute.

(ii) Find the rate, in cm per minute, at which the water level is rising when the height of the water level is 10 cm.

The equation of a curve is \(y = 2 + \frac{3}{2x - 1}\).

(i) Obtain an expression for \(\frac{dy}{dx}\).

(ii) Explain why the curve has no stationary points.

At the point \(P\) on the curve, \(x = 2\).

(iii) Show that the normal to the curve at \(P\) passes through the origin.

(iv) A point moves along the curve in such a way that its \(x\)-coordinate is decreasing at a constant rate of 0.06 units per second. Find the rate of change of the \(y\)-coordinate as the point passes through \(P\).

The point \(P(x, y)\) is moving along the curve \(y = x^2 - \frac{10}{3}x^{3/2} + 5x\) in such a way that the rate of change of \(y\) is constant. Find the values of \(x\) at the points at which the rate of change of \(x\) is equal to half the rate of change of \(y\).

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

A point \(P\) moves along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.3 units per second. Find the rate of change of the \(y\)-coordinate as \(P\) crosses the \(y\)-axis.