Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The diagram shows the curve with equation y = ln(2x+1) / (x+3). The curve has a maximum point M. [Figure 6.1]
Prove that 2 sin θ cosec 2θ ≡ sec θ.
The diagram shows the curve with equation y = 8e⁻ˣ – e²ˣ. The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded reg...
The diagram shows the curve with equation y = (ln (2x+1))/(x+3). The curve has a maximum point M.
Expand (3+x)(1-2x)⁻¹ in ascending powers of x, up to and including the term in x², simplifying the coefficients.
Solve the equation ln(x-5) = 7-lnx. Give your answer correct to 2 decimal places.
The variables x and y satisfy the equation aʸ = bx, where a and b are constants. The graph of y against lnx is a straight line passing through the ...
The complex number u is given by u = -1-i√3.
The equation of a curve is y = e^(sinx) / cos²x for 0 ≤ x ≤ 2π. Find dy/dx and hence find the x-coordinates of the stationary points of the curve.
On a single Argand diagram sketch the loci given by the equations |z-3+2i|= 2 and |w-3+2i|=|w+3-4i| where z and w are complex numbers.
Use the substitution u = 1-sinx to find the exact value of integral from π/6 to (3/2)π of (sin(2x) / sqrt(1-sinx)) dx. Give your answer in the form...
The equations of two straight lines l₁ and l₂ are l₁: r = i-2j+3k + λ(2i-j+ak) and l₂: r = -i-j-k + μ(3i-2j-2k), where a is a constant. The lines l...
Given that 2x = tany, show that dy/dx = 2 / (1+4x²).
In a field there are 300 plants of a certain species, all of which can be infected by a particular disease. At time t after the first plant is infe...
Express (6x²-9x-16) / (2x²-5x-12) in partial fractions.
The variables x and y satisfy the equation a^(2y−1) = b^(x−y), where a and b are constants.
The equation of a curve is ye^(2x) + y²e^x = 6. Find the gradient of the curve at the point where y = 1.
The diagram shows the curve y = xe^(-ax), where a is a positive constant, and its maximum point M. [Figure 6.1]
The points A, B and C have position vectors OA =−2i+j+4k, OB = 5i+2j and OC = 8i+5j-3k, where O is the origin. The line l₁ passes through B and C.
The complex numbers z and ω are defined by z = 1-i and ω =-3+3√3i.
Solve the equation 8³⁻⁶ˣ = 4×5⁻²ˣ. Give your answer correct to 3 decimal places.
Find the exact coordinates of the stationary point of the curve y = e²ˣ sin2x for 0 ≤ x ≤ ½π.
The square roots of 24-7i can be expressed in the Cartesian form x+iy, where x and y are real and exact. By first forming a quartic equation in x o...
The variables x and y satisfy the equation ky = eᶜˣ, where k and c are constants. The graph of ln y against x is a straight line passing through th...
Express (6x² - 2x + 2) / ((x-1)(2x+1)) in partial fractions.
Let f(x) = 8x³ + 54x² - 17x-21.
A container in the shape of a cuboid has a square base of side x and a height of (10-x). It is given that x varies with time, t, where t > 0. The c...
The equations of two straight lines are r=i+j+2ak+λ(3i+4j+ak) and r=-3i-j+4k+μ(−i+2j+2k) where a is a constant.
Use the substitution 2x = tan θ to find the exact value of ∫(from 0 to 1/2) 12 / (1+4x²)² dx. Give your answer in the form a+bπ, where a and b are ...
A car starts from rest and accelerates at 2ms⁻² for 10s. It then travels at a constant speed for 30s. The car then uniformly decelerates to rest ov...
Two forces of magnitudes 20N and FN act at a point P in the directions shown in the diagram. [Figure 2.1]
A train of mass 180000kg ascends a straight hill of length 1.5km, inclined at an angle of 1.5° to the horizontal. As it ascends the hill, the total...
A car of mass 1700kg is pulling a trailer of mass 300kg along a straight horizontal road. The car and trailer are connected by a light inextensible...
A straight slope of length 60m is inclined at an angle of 12° to the horizontal. A bobsled starts at the top of the slope with a speed of 5ms⁻¹. Th...
A particle moves in a straight line, starting from a point O. The velocity of the particle at time t s after leaving O is v m s⁻¹. It is given that...
A particle P of mass 0.2kg is projected vertically upwards from horizontal ground with speed 25ms⁻¹.
A cyclist and bicycle have a total mass of 72kg. The cyclist rides along a horizontal road against a total resistance force of 28 N. Find the total...
A particle P moves in a straight line. At time ts after leaving a point O on the line, P has velocity vms⁻¹, where v = 44t-6t² - 36.
Four coplanar forces of magnitude PN, 10N, 16N and 2N act at a point in the directions shown in the diagram. It is given that the forces are in equ...
A car has mass 1400kg. When the speed of the car is vms⁻¹ the magnitude of the resistance to motion is kv²N where k is a constant.
A particle of mass 0.8kg lies on a rough plane which is inclined at an angle of 28° to the horizontal. The particle is kept in equilibrium by a for...
Three particles A, B and C of masses 5kg, 1kg and 2kg respectively lie at rest in that order on a straight smooth horizontal track XYZ. Initially A...