Cambridge Past Paper Questions
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It is known that 1% of houses in a certain area have a wind turbine. A random sample of 400 houses in this area is chosen for a survey on domestic ...
The random variable X has the distribution B(8, 3/4). A random sample of 100 values of X is chosen, and the sample mean, X̅, is found.
The time, T minutes, for a certain daily bus journey is normally distributed. The bus company claims that the mean of T is 45. A passenger believes...
At an entertainment centre, the cost for using a particular video game is $0.40 per minute. The number of minutes for which people use the video ga...
The random variables W and X have the independent distributions Po(1.2) and Po(2.3) respectively.
A manufacturer of cell phones claims that 25% of students own a Pumpkin phone. Jeyeraj thinks that the proportion of students at his large college ...
X is a random variable with probability density function given by f(x) = { 1/π (1+cos πx) 0 ≤ x ≤ 1, { 0 otherwise.
One of a group of three students is to be chosen at random. The times, in minutes, taken by students to complete a test are normally distributed wi...
The height of a certain species of plant is denoted by Hcm. The heights of a random sample of 100 plants were measured, and the following results w...
The random variable X has the distribution Po(15). It is given that P(X = n) = P(X = n+1).
A biased spinner has four sides. Each side is of a different colour: yellow, red, green or black. The probability, p, that the spinner will land on...
The amount of time, in minutes, spent by a customer on one visit to a certain shop is modelled by the random variable X ~ N(μ, σ²). In the past, th...
Use suitable approximating distributions to answer the following.
The random variable X has probability density function given by f(x) = { kx² / a² for 0 ≤ x ≤ a, { 0 otherwise, where k and a are positive c...
Birgitte has a six-sided dice. She suspects that the dice is biased so that the probability, p, that it will show a six on one throw is less than 1...
At a certain shop, customers arrive independently and randomly at a constant average rate of 23.4 per hour.
The lengths of pencils made at a factory are normally distributed. The standard deviation of the lengths is σcm, and the mean is supposed to be 10c...
A machine dispenses coffee into cups. The volume, Vcm³, of coffee in a cup was measured for a random sample of 150 cups. The results were summarise...
Emma needs to choose one person at random from three people, P, Q and R. She plans to throw two fair coins and note the number, n, of heads. If n i...
In Urberia, the masses, in kilograms, of men have the distribution N(70.3, 5.9²). A certain footbridge in Urberia can take a maximum safe load of 1...
A random variable X has probability density function given by f(x) = { ax for 0 ≤ x ≤ b, { 0 otherwise, where a and b are constants.
In the past, one quarter of job applicants at a certain firm had first-class degrees. A change is made in the job description and a director of the...
The equation of a curve is such that dy/dx = 3/√x - x. Given that the curve passes through the point (4, 6), find the equation of the curve.
Find, in terms of the non-zero constant k, the first 4 terms in the expansion of (k + x)⁸ in ascending powers of x.
A progression has a second term of 96 and a fourth term of 54. Find the first term of the progression in each of the following cases:
The function f is defined by f : x → 5 – 3 sin 2x for 0 ≤ x ≤ π.
In the diagram, OABCDEFG is a cube in which each side has length 6. Unit vectors i, j and k are parallel to OA, OC and OD respectively. The point P...
A piece of wire of length 50 cm is bent to form the perimeter of a sector POQ of a circle. The radius of the circle is r cm and the angle POQ is θ ...
The function f is such that f(x) = 3/(2x + 5) for x ∈ R, x ≠ −2.5.
The diagram shows a rectangle ABCD. The point A is (0, -2) and C is (12, 14). The diagonal BD is parallel to the x-axis. [Figure X.X]
The length, x metres, of a Green Anaconda snake which is t years old is given approximately by the formula x = 0.7 √(2t – 1), where 1 ≤ t ≤ 10. Usi...
The diagram shows points A, C, B, P on the circumference of a circle with centre O and radius 3 cm. Angle AOC = angle BOC = 2.3 radians. [Figure 4.1]
A curve has equation y = kx² + 1 and a line has equation y = kx, where k is a non-zero constant.
The function f is defined by f(x) = x² – 4x +7 for x > 2.
The diagram shows part of the curve y = 2/(1-x) and the line y = 3x + 4. The curve and the line meet at points A and B. [Figure 8.1]
The diagram shows a pyramid OABCP in which the horizontal base OABC is a square of side 10 cm and the vertex P is 10cm vertically above O. The poin...
The diagram shows an open rectangular tank of height h metres covered with a lid. The base of the tank has sides of length x metres and 5/4x metres...
The diagram shows part of the curve y = 1/(3x + 1)⁴. The curve cuts the y-axis at A and the line x = 5 at B. [Figure 11.1]
The functions f and g are defined for x ∈ R by f: x → 3x + a, g: x → b - 2x, where a and b are constants. Given that ff(2) = 10 and g⁻¹(2) = 3, find
Relative to an origin O, the position vectors of points A and B are given by OĀ = 5i + j + 2k and OB = 2i + 7j + pk, where p is a constant.
The equation of a curve is y² + 2x = 13 and the equation of a line is 2y + x = k, where k is a constant.
The diagram shows a circle C₁ touching a circle C₂ at a point X. Circle C₁ has centre A and radius 6 cm, and circle C₂ has centre B and radius 10 c...
A curve is such that dy/dx = 5 - 8/x². The line 3y + x = 17 is the normal to the curve at the point P on the curve. Given that the x-coordinate of ...