Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The diagram shows a sector AOB of the circle, centre O, radius 12cm, together with points C and D such that ABCD is a rectangle. The angle AOB is θ...
Do not use a calculator in this question. The polynomial p(x) is ax³ - 4x² + bx + 18. It is given that p(x) and p'(x) are both divisible by 2x-3.
The variables x and y are such that when lny is plotted against x, a straight line graph is obtained. This line passes through the points x = 4, ln...
Solve the equation |4x-3|=x.
Without using a calculator, find the integers a and b such that a/(√3+1) + b/(√3-1) = √3-3.
Solve the equation 2lgx - lg((x+10)/2) = 1.
The number of bacteria, N, present in a culture can be modelled by the equation N = 7000+2000e-0.05t, where t is measured in days. Find
The curve with equation y = x³ + 2x²-7x+2 passes through the point A (-2, 16). Find
3 cm P x cm R 8 S 14 cm [Figure 7]
The function f(x) is given by f(x) = (3x³-1)/(x³+1) for 0 ≤ x ≤ 3.
The line y = kx – 4, where k is a positive constant, passes through the point P (0, -4) and is a tangent to the curve x² + y² - 2y = 8 at the point...
The town of Cambley is 5 km east and p km north of Edwintown so that the position vector of Cambley from Edwintown is (5000 / 1000p) metres. Manjit...
Mr and Mrs Coldicott have 5 sons and 4 daughters. All 11 members of the family play tennis. Six members of the family enter a tennis competition wh...
The graph of y = a sin(bx) + c has an amplitude of 4, a period of π/3 and passes through the point (π/12, 2). Find the value of each of the constan...
Given that y = ln(3x² + 2) / (x² + 1), find the value of dy/dx when x = 2, giving your answer as a + b ln14, where a and b are fractions in their s...
When lgy is plotted against x, a straight line is obtained which passes through the points (0.6, 0.3) and (1.1, 0.2).
Functions f and g are defined, for x > 0, by f(x) = lnx, g(x) = 2x² + 3.
A polynomial p(x) is ax³ + 8x² + bx + 5, where a and b are integers. It is given that 2x - 1 is a factor of p(x) and that a remainder of -25 is obt...
The diagram shows a river which is 120m wide and is flowing at 4 ms⁻¹. Points A and B are on opposite sides of the river such that B is 50m downstr...
The diagram shows an isosceles triangle ABC, where AB = AC = 5cm. The arc BEC is part of the circle centre A and has length 6.2cm. The point D is t...
If z = 2 + √3 find the integers a and b such that az² + bz = 1 + √3.
Solve the equation 2x^1.5 + 6x^-0.5 / x^0.5+ 5x^-0.5 = x.
Solve the inequality |3x - 1| > 3 + x.
Solve the simultaneous equations log2(x + 4) = 2log2y, log2 (7y - x) = 4.
Naomi is going on holiday and intends to read 4 books during her time away. She selects these books from 5 mystery, 3 crime and 2 romance books. Fi...
The volume of a closed cylinder of base radius x cm and height h cm is 500 cm³.
The gradient of the normal to a curve at the point with coordinates (x, y) is given by √x / (1 - 3x).
The matrix A is (2 1 / 4 3).
Find d/dx (x ln x).
Without using a calculator, solve the equation 6c³ - 7c² + 1 = 0.
The diagram shows the curve y = 4 + 3x - x² intersecting the positive x-axis at the point A. The line y = mx + 8 is a tangent to the curve at the p...
Solve 1 + √2 sin(x+50°) = 0 for -180° ≤ x ≤ 180°.
Find the equation of the curve which has a gradient of 4 at the point (0, -3) and is such that d²y/dx² = 5 + e^(2x).
y = x³ ln(2x+1)
The 7th term in the expansion of (a+bx)¹² in ascending powers of x is 924x⁶. It is given that a and b are positive constants.
A = ( (a 3) / (4 a+4) )
The polynomial p(x) = ax³ + bx² + cx - 9 is divisible by x + 3. It is given that p'(0) = 36 and p''(0) = 86.
The diagram shows part of the curve y = a+4cosbx, where a and b are positive constants. The curve meets the y-axis at the point (0,6) and the x-axi...
The diagram shows the sector OPQ of a circle, centre O, radius r cm, where angle POQ = θ radians. The perimeter of the sector is 10cm. [Figure X.X]
The line y = 2x+5 intersects the curve y+xy = 5 at the points A and B. Find the coordinates of the point where the perpendicular bisector of the li...
Solve the inequality (x-3)(x+4) > x+13.