Cambridge Past Paper Questions

Without using a calculator, find the exact value of ∫₂⁰ 4e⁻ˣ(e³ˣ + 1) dx.

The diagram shows the curve with equation y = 5 ln x / (2x + 1) and has a maximum point M. The curve crosses the x-axis at the point P. [Figure 4]

The parametric equations of a curve are x = 2 cos 2θ + 3 sin θ, y = 3 cos θ for 0 < θ < ½π.

The cubic polynomial f(x) is defined by f(x) = x³ + ax² + 14x + a + 1, where a is a constant. It is given that (x + 2) is a factor of f(x).

Solve the inequality |3x-2| < |x + 5|.

A curve has equation y = 3 ln(2x + 9) – 2 lnx.

A curve has equation y³ sin 2x + 4y = 8. Find the equation of the tangent to the curve at the point where it crosses the y-axis.

It is given that ∫ (1 + e^(x/2))² dx from 0 to a = 10, where a is a positive constant.

A curve has equation y³ sin 2x + 4y = 8.

It is given that ∫₀ᵃ (1 + e½ˣ)² dx = 10, where a is a positive constant.

Let I = ∫(from 1/4 to 3/4) √((x)/(1-x)) dx.

In a certain chemical reaction the amount, x grams, of a substance is decreasing. The differential equation relating x and t, the time in seconds s...

The positive constant a is such that ∫(from 0 to a) x e^(-x/2) dx = 2.

Let f(x) = (12x² + 4x - 1) / ((x−1)(3x + 2)).

The point P has position vector 3i – 2j + k. The line l has equation r = 4i + 2j + 5k + μ(i + 2j + 3k).

Showing all necessary working, solve the equation 3|2x – 1| = 2x, giving your answers correct to 3 significant figures.

Showing all necessary working, solve the equation cot θ + cot(θ + 45°) = 2, for 0° < θ < 180°.

In the diagram, the tangent to a curve at the point P with coordinates (x, y) meets the x-axis at T. The point N is the foot of the perpendicular f...

The equation of a curve is x²(x + 3y) – y³ = 3.

The diagram shows a triangle ABC in which AB = AC = a and angle BAC = θ radians. Semicircles are drawn outside the triangle with AB and AC as diame...

Throughout this question the use of a calculator is not permitted. The complex numbers −3√3 + i and √3 + 2i are denoted by u and v respectively.

The diagram shows the curve y = (x + 1)e^(-x/3) and its maximum point M. [Figure X.X]

Let f(x) = (x - 4x²) / ((3-x)(2 + x²))

Two lines l and m have equations r = 2i – j + k + s(2i + 3j – k) and r = i + 3j + 4k + t(i + 2j + k) respectively.

Expand 4 / √(4-3x) in ascending powers of x, up to and including the term in x², simplifying the coefficients.

Showing all necessary working, solve the equation 5^(2x) = 5^x + 5. Give your answer correct to 3 decimal places.

Showing all necessary working, find the value of ∫(from 0 to π/3) x cos 3x dx, giving your answer in terms of π.

The curve with equation y = (ln x)/(3+x) has a stationary point at x = p.

5

6

The equation of a curve is 2x³ – y³ – 3xy² = 2a³, where a is a non-zero constant.

9

The points A and B have position vectors 2i + j + 3k and 4i + j + k respectively. The line l has equation r = 4i + 6j + μ(i + 2j – 2k).

A particle P is projected vertically upwards with speed 24 m s¯¹ from a point 5 m above ground level. Find the time from projection until P reaches...

The diagram shows three coplanar forces acting at the point O. The magnitudes of the forces are 6N, 8 N and 10 N. The angle between the 6 N force a...

A particle P of mass 8 kg is on a smooth plane inclined at an angle of 30° to the horizontal. A force of magnitude 100 N, making an angle of θ° wit...

A particle P moves in a straight line starting from a point O. At time t s after leaving O, the displacement s m from O is given by s = t³ – 4t² + ...

A sprinter runs a race of 200 m. His total time for running the race is 20 s. He starts from rest and accelerates uniformly for 6 s, reaching a spe...

A car has mass 1250 kg.