Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The table shows values of the variables t and P. t: 1, 1.5, 2, 2.5 P: 4.39, 8.33, 15.8, 30.0
A particle moves in a straight line so that, t seconds after passing a fixed point O, its displacement, sm, from O is given by s = 1 + 3t - cos 5t.
Solve |5x + 3| = |1 - 3x|.
Without using a calculator, express ( (1 + √5) / (3 - √5) ) ^-2 in the form a + b√5, where a and b are integers.
Without using a calculator, factorise the expression 10x³ – 21x² + 4.
The point P lies on the curve y = 3x² – 7x + 11. The normal to the curve at P has equation 5y + x = k. Find the coordinates of P and the value of k.
Show that the roots of px² + (p - q)x - q = 0 are real for all real values of p and q.
Solutions to this question by accurate drawing will not be accepted. The points A and B are (-8, 8) and (4, 0) respectively.
A function f is defined, for x ≤ 3/2, by f(x) = 2x² - 6x + 5.
Solve the equation
The diagram shows part of the curve y = x³ + 4x² – 5x + 5 and the line y = 5. The curve and the line intersect at the points A, B and C. The points...
The function g is defined, for x > -1/2, by g(x) = 3/(2x + 1).
Solve the equations y − x = 4, x² + y² – 8x – 4y – 16 = 0.
Find the equation of the perpendicular bisector of the line joining the points (1, 3) and (4, −5). Give your answer in the form ax + by + c = 0, wh...
Diagrams A to D show four different graphs. In each case the whole graph is shown and the scales on the two axes are the same. [Figure A, B, C, D]
The population, P, of a certain bacterium t days after the start of an experiment is modelled by P = 800e^(kt), where k is a constant.
The diagram shows part of the graph of y = 16x + 27/x², which has a minimum at A. [Figure 11.1]
A curve is such that d²y/dx² = (2x - 5)⁻^(1/2). Given that the curve has a gradient of 6 at the point (9/2, 2/3), find the equation of the curve.
It is given that y = 1+tan3x.
Find the values of k for which the line y = 1-2kx does not meet the curve y = 9x² - (3k + 1)x + 5.
The variables x and y are such that when eʸ is plotted against x², a straight line graph passing through the points (5, 3) and (3, 1) is obtained. ...
A particle P moves so that its displacement, x metres from a fixed point O, at time t seconds, is given by x = ln (5t + 3).
Find the coordinates of the stationary point of the curve y = (x+2)/√(2x-1).
A population, B, of a particular bacterium, t hours after measurements began, is given by B = 1000e^(t/4).
Do not use a calculator in this question. All lengths in this question are in centimetres. [Figure showing a triangle ABC with angle B = 60 degrees...
The diagram shows the graph of the curve y= (e^(4x)+3)/8. The curve meets the y-axis at the point A. The normal to the curve at A meets the x-axis ...
A, B and C are subsets of the same universal set.
The variables x and y are such that y = ln(3x-1) for x > 1/3.
A 7-character password is to be selected from the 12 characters shown in the table. Each character may be used only once. Characters Upper-case let...
Do not use a calculator in this question. It is given that x + 4 is a factor of p(x) = 2x³ + 3x² + ax - 12. When p(x) is divided by x - 1 the remai...
The function f is defined by f(x) = 1/(2x-5) for x > 2.5.
In the diagram AOB and DOC are sectors of a circle centre O. The angle AOB is x radians. The length of the arc AB is 40 cm and the radius OB is 16 ...
An experiment was carried out recording values of y for certain values of x. The variables x and y are thought to be connected by the relationship ...
A particle moves in a straight line such that its displacement, s metres, from a fixed point O at time t seconds, is given by s = 4 + cos3t, where ...
In this question all lengths are in metres. A water container is in the shape of a triangular prism. The diagrams show the container and its cross-...