Cambridge Past Paper Questions

f(x) = x³ + px² + qx + 6 Given that (x + 1) is a factor of f(x) and that when f(x) is divided by (x - 2) the remainder is 30.

The circle C has equation x^2 + y^2 - 6x + 8y - 24 = 0. (a) Find the coordinates of the centre of C and the radius of C. (b) The line L has equati...

The curve C has equation y = 2e^x + 3. The line L has equation y = 7.

Prove that log_a(x) = (log_b(x)) / (log_b(a)), where a, b and x are positive and a ≠ 1, b ≠ 1.

The amount of a drug, D mg, in a patient's bloodstream t hours after it has been administered, is given by the formula D = 10e^(-0.2t).

Given that for all real values of x, 3x² + bx + 10 = A(x-2)² + C, where A, b and C are constants.

A rectangle has width x cm and length y cm. The perimeter of the rectangle is 20 cm.

Figure 1 shows a sketch of the curve C with equation y = 2^x. The curve C crosses the y-axis at the point A. The point B has x-coordinate 4 and li...

A curve has equation `y = 2x^3 - 6x^{4/3} - x^{-1}`. (a) Find `dy/dx`. (b) Find the exact value of `dy/dx` at `x=3`.

Solve the equation `2log_3(x+1) - log_3(x+11) = 2`.

The diagram shows a sketch of part of the curve with equation `y = f(x)`, where `f(x) = x(x^2 - 4x - 12)`. The curve crosses the x-axis at the orig...

(i) Find the exact value of `\sum_{r=1}^{10} (r^3 + 1)`. (ii) A sequence `u_1, u_2, u_3, ...` is defined by `u_n = 4n-1`. Find the value of `k` suc...

A geometric series has first term `a` and common ratio `r`. The second term of the series is `4` and the sum to infinity of the series is `18`. (a)...

The equation `2x^3 - x^2 + px + q = 0`, where `p` and `q` are constants, has roots `\alpha`, `\beta` and `\gamma`. Given that `\alpha = 2`, `\beta ...

(a) Sketch the graph of `y = |4x-3|`, showing the coordinates of the points where the graph meets the coordinate axes. (b) Solve the equation `|4x-...

(a) Find the first 3 terms, in ascending powers of `x`, of the binomial expansion of `(2 - 9x)^4`. Give each term in its simplest form. (b) Using t...

A curve has equation `y = x^3 - 3x^2 - 9x + 1`. (a) Find `dy/dx`. (b) Find the exact coordinates of the stationary points of the curve. (c) Determi...

(i) Prove that `tan \theta + cot \theta \equiv sec \theta cosec \theta`. (ii) Solve, for `0 \le x < 360^\circ`, the equation `2 tan x - 3 cot x - 1...

The curve `C` has equation `y = 2x^2 + kx + 8`, where `k` is a constant. The line `L` has equation `y = x + 4`. (a) Show that the x-coordinates of ...

(a) Show that `\frac{d}{dx} (\frac{\ln x}{x}) = \frac{1 - \ln x}{x^2}`. (b) Find the exact value of `\int_1^e \frac{1 - \ln x}{x^2} dx`.

A circle `C` has equation `x^2 + y^2 - 4x + 6y - 12 = 0`. (a) Find the coordinates of the centre of `C` and the radius of `C`. (b) The point `P(5, ...

(a) Show that `\sum_{r=1}^{n} (8r - 3) = n(4n + 1)`. (b) Hence, or otherwise, find `n` such that `\sum_{r=1}^{n} (8r - 3) = 1170`. (c) Find `\sum_{...

A curve has equation y = e^(-2x) sin x. (a) Show that dy/dx = e^(-2x) (cos x - 2sin x). (b) Find the x-coordinates of the stationary points of the ...

Solutions to this question by calculator technology are not acceptable. (a) Show that 2sin x tan x - 3 = 0 can be written as 2sin² x - 3cos x = 0.

Solve, for 0 ≤ x < 360°, the equation 2sin(x + 30°) = 1.

The curve C has equation y = (2x + 3)^3. (a) Find dy/dx. (b) Find the equation of the tangent to the curve C at the point where x = -1. Give your a...

Given that log₂ a = 3 and log₂ b = 5, find the value of log₂ (a²b).

A curve C has equation y = x³ - 6x² + 5x - 2.

A population of insects, $P$, is modelled by the differential equation $\frac{\mathrm{d}P}{\mathrm{d}t} = kP(1000-P)$, where $t$ is the time in day...

Use the substitution u = sec x to find the exact value of ∫(from 0 to π/3) (tan x)(sec^3 x) dx.

The curve $C$ has parametric equations $x = 3 \cos t$, $y = 2 \sin t$, $0 \le t < 2\pi$.

The value of a car, $V$, in pounds, can be modelled by the equation $V = A\mathrm{e}^{-kt}$, where $t$ is the age of the car in years, and $A$ and ...

Given that \theta is small and measured in radians, use the small angle approximations to show that \frac{\cos(3\theta) - 1}{\theta \sin(2\theta)} ...

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The curve C has equation x^2 + 2xy - 3y^2 + 16 = 0.

Figure 1 shows a sketch of the curve C with parametric equations x = 4 cos(2t), y = 3 sin t, 0 ≤ t ≤ π/2 [Figure 1]

Use the substitution u = x^2 to find ∫ (x / (√(1 - x^4))) dx

Relative to a fixed origin O, the points A and B have position vectors a = (i + 2j - 3k) and b = (5i - 3j - 2k) respectively.

The curve C has equation y = e^(2x) sin x.

A population of a species of animal is being studied. The population P is modelled by the differential equation dP/dt = kP(1 - P/1000) t ≥ 0 wher...

The curve C has equation y = x^2 - 4x + 5.

Given that y = 2^x.

A container is filled with liquid. The liquid is draining from the container. The rate at which the volume of liquid in the container is decreasing...

Figure 1 shows a sketch of the curve with equation $y = x^2 \ln x$, $x>0$. The curve has a minimum point $M$.

Relative to a fixed origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \\ -4 \end{pmatrix}$ and $\mathbf...

The amount of a drug in a patient's bloodstream, $D$ mg, $t$ hours after the drug has been administered, is modelled by the differential equation $...

The curve C has equation y = (2x + 1)e^(-2x). (b) Hence find the exact coordinates of the stationary point of C.

Show that ∫(from 1 to 4) (x + 4)/(x(x + 2)) dx = ln(k), where k is a rational number to be found.

A curve C has parametric equations x = 3cos(t) + 2, y = 4sin(t) - 1, for 0 ≤ t < 2π. (a) Show that the Cartesian equation of C is (x - 2)²/9 + (y ...