Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
In the diagram the lengths of AB and AC are both 15 cm. The point P is the foot of the perpendicular from C to AB. The length CP = 9cm. An arc of a...
The volume V m³ of a large circular mound of iron ore of radius r m is modelled by the equation V = (9/2)(r - 1/2)³ – 1 for r ≥ 2. Iron ore is adde...
The function f is defined by f(x) = x² + k/x + 2 for x > 0.
The diagram shows the line x = 5/2, part of the curve y = (1/2)x + 7/10 - 1/(x-2)³ and the normal to the curve at the point A (3, 5/2). [Figure X.X]
The diagram shows the circle with equation x² + y² – 6x + 4y – 27 = 0 and the tangent to the circle at the point P (5, 4). [Figure X.X]
Find the exact value of $\int_{-1}^{2} (4e^{2x} - 2e^{-x}) dx$.
The variables x and y satisfy the equation $a^y = kx$, where a and k are constants. The graph of y against In x is a straight line passing through ...
The curve with equation $y = xe^{2x} + 5e^{-x}$ has a minimum point M.
The diagram shows the curve with parametric equations $x = \ln(2t + 3)$, $y = \frac{2t-3}{2t+3}$. [Figure 5.1] The curve crosses the y-axis at the ...
The polynomials $f(x)$ and $g(x)$ are defined by $f(x) = 4x^3 + ax^2 + 8x + 15$ and $g(x) = x^2 + bx + 18$, where a and b are constants.
The polynomial p(x) is defined by p(x) = ax³ + bx – 10, where a and b are constants. It is given that (x + 2) is a factor of p(x) and that the rema...
The curve with equation y = 5x – 2 tan 2x has exactly one stationary point in the interval 0 ≤ x < ¼π. Find the coordinates of this stationary poin...
Given that ∫ (from a to a+14) (1/3x) dx = ln 2, find the value of the positive constant a.
A curve has equation x² + 4x cos 3y = 6. Find the exact value of the gradient of the normal to the curve at the point (√2, π/12).
Find the exact value of ∫₂₋₁(4e²ˣ - 2e⁻ˣ) dx.
The variables x and y satisfy the equation aʸ = kx, where a and k are constants. The graph of y against ln x is a straight line passing through the...
The curve with equation y = xe²ˣ + 5e⁻ˣ has a minimum point M.
The diagram shows the curve with parametric equations x = ln(2t + 3), y = (2t-3)/(2t+3). The curve crosses the y-axis at the point A and the x-axis...
The polynomials f(x) and g(x) are defined by f(x) = 4x³ + ax² + 8x + 15 and g(x) = x² + bx + 18, where a and b are constants.
The curve with equation y = xe¹⁻²ˣ has one stationary point.
The constant a is such that ∫ᵃ₁ (lnx / √x) dx = 6. [exp(x) is an alternative notation for eˣ.]
Two lines l and m have equations r = 3i + 2j + 5k + s(4i – j + 3k) and r = i − j − 2k + t(−i + 2j + 2k) respectively.
The complex number 1 + 2i is denoted by u. The polynomial 2x³ + ax² + 4x + b, where a and b are real constants, is denoted by p(x). It is given tha...
Find the value of x for which 3(2¹⁻ˣ) = 7ˣ. Give your answer in the form ln a / ln b, where a and b are integers.
Solve the inequality |3x – a| > 2|x + 2a|, where a is a positive constant.
Given the complex numbers u = a + ib and w = c + id, where a, b, c and d are real, prove that (u + w)* = u* + w*.
Express 4x² - 13x + 13 / (2x - 1)(x - 3) in partial fractions.
Using the expansions of sin(3x + 2x) and sin(3x - 2x), show that ½(sin 5x + sin x) = sin 3x cos 2x.
The variables x and y satisfy the differential equation e²ˣ dy/dx = 4xy², and it is given that y = 1 when x = 0. Solve the differential equation, o...
By first expanding (cos² θ + sin²θ)², show that cos⁴ θ + sin⁴θ = 1 − ½ sin² 2θ.
The equation of a curve is ye²ˣ – y²eˣ = 2.
With respect to the origin O, the position vectors of the points A and B are given by OA = (1, 2, -1) and OB = (0, 3, 1).
The equation of a curve is y = √tan x, for 0 ≤ x < ½π. [Figure X.X]
Find the quotient and remainder when 2x⁴ + 1 is divided by x² − x + 2.
Solve the equation 4ˣ⁻² = 4ˣ – 4², giving your answer correct to 3 decimal places.
Find the exact value of ∫ from π/2 to π of x sin(½x) dx.