Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
Find the coordinates of the points of intersection of the curve and the line with equations 2xy+5y2 = 24 and 2x+y+4 = 0.
The coefficient of x7 in the expansion of (px²+4/x)5 is 1280. Find the value of the constant p.
A point P is moving along the curve with equation y = ax^3/2 – 12x in such a way that the x-coordinate of P is increasing at a constant rate of 5 u...
The equation of a curve is y = 4 cos2x + 3 for 0 ≤ x ≤ 2π.
The diagram shows the curve with equation y = 9/(5x+4)^1/2 and the line y = 6-3x. The line and the curve intersect at the point P which has y-coord...
The diagram shows the circle with equation x²+y² – 14x+8y+36 = 0 and the line y=-2. The line intersects the circle at the points A and B. The centr...
The equation of a curve is such that d²y/dx² = 24/x³ . It is given that the curve has a stationary point at (-2, 19).
The first, second and third terms of an arithmetic progression are 4k, k² and 8k respectively, where k is a non-zero constant.
The diagram shows the curve with equation y = 6e^2x – e^3x. The shaded region is bounded by the axes and the curve.
The polynomial p(x) is defined by p(x) = ax³ + bx² - ax-24, where a and b are constants. It is given that (2x-3) is a factor of p(x) and that the r...
The parametric equations of a curve are x = (2t+1)/(3t+4), y = 2ln(3t+4), where t > -4/3.
Show that ∫ from 2 to 11 (8 / (4x+1)) dx = ln a, where a is an integer to be found.
Find the coordinates of the stationary points of the curve with equation y = 8x / (2x+3) - 6x+5.
The diagram shows parts of the curves with equations y = 4e⁻²ˣ and y = 1+0.5 sin3x. Point P is a point of intersection of the curves, and the shade...
The polynomial p(x) is defined by p(x) = ax⁴ + bx³ +13x²-35x+15, where a and b are constants. It is given that (2x−1) and (x-3) are factors of p(x).
A curve has equation (x²-3)lny+6x = 14.
Show that ∫₂¹¹ (8 / (4x+1)) dx = ln a, where a is an integer to be found.
Find the coordinates of the stationary points of the curve with equation y = 8x/(2x+3) - 6x + 5.
It is given that 2 lnp+ln(p-1)-½ln(q+1) = 3. Find q in terms of p.
Find the complex numbers z for which z+5i / z-5 is real and |z|= √17. Give your answers in the form z = x+iy, where x and y are real.
The parametric equations of a curve are x = etant, y = 3 tan²t. Find the equation of the tangent to the curve at the point (e, 3). Give your answer...
The polynomial 3x³ +pax²+7a²x+qa³ is denoted by f(x), where p, q and a are constants and a ≠ 0. When f(x) is divided by (x+2a) the remainder is -22...
It is given that z₁ = 3e^(i π/4), z₂ = (1/2)e^(i π/2) and w = 2e^(-i π/2).
With respect to the origin O, the points A and B have position vectors 2i+4k and 5i+j+6k respectively. The line l₁ passes through the points A and B.
The constant a is such that ∫(from 2 to a) 6x ln x dx = 4.
The diagram shows the curve y = cos x √sin 2x for 0 ≤ x ≤ π/2. The curve has a maximum point at M, where x = a. [Figure 11.1]
Solve the equation ex+2e-xex-3 = 4. Give your answer correct to 3 decimal places.
Solve the equation 3 cotx-4 cot2x = 3 for 0° ≤ x ≤ 180°.
The square roots of -1-4√5 i can be expressed in the Cartesian form x+iy, where x and y are real and exact. By first forming a quartic equation in ...
The variables x and θ satisfy the differential equation dx/dθ sin 2θ = (4x+3) cos 2θ, and x = 0 when θ = ¹⁄₁₂π. Solve the differential equation and...
With respect to the origin O, the points A, B and C have position vectors given by OA = [1, -4, 2] OB = [-2, 1, 3] OC = [2, 3, 5] (a) Find a vecto...
The diagram shows the graph of y = 5 sin 2x cos²x for 0 < x ≤ ½π and its maximum point M. [Figure X.X] (a) Find the exact x-coordinate of M. (b) By...
Solve the equation 2 ln (2x+3) - ln(2x + 5) = ln(3x).