Cambridge Past Paper Questions

The polynomial p(x) is defined by p(x) = 2x³ + 5x² + ax + 2a, where a is an integer.

Given that y = ln x / x², find the exact value of dy/dx when x = e.

A curve has equation e^(2x) cos 2y + sin y = 1. Find the exact gradient of the curve at the point (0, 1/6 π).

The polynomial p(x) is defined by p(x) = 2x³ + ax² – 3x – 4, where a is a constant. It is given that (x - 4) is a factor of p(x).

The diagram shows the curve y = 3e^(2x−1). The shaded region is bounded by the curve and the lines x = a, x = a + 1 and y = 0, where a is a constan...

The diagram shows the curves y = √2π – 2x and y = sin²x for 0 ≤ x ≤ π. The shaded region is bounded by the two curves and the line x = 0. Find the ...

Given that y = In x x2', find the exact value of dy dx when x = e.

A curve has equation e2x cos 2y + sin y = 1. Find the exact gradient of the curve at the point (0, 1/6π).

The diagram shows the curve y = 3e2x−1. The shaded region is bounded by the curve and the lines x = a, x = a + 1 and y = 0, where a is a constant. ...

Solve the equation 2(3^(2x−1)) = 4^(x+1), giving your answer correct to 2 decimal places.

Solve the equation 2 cot 2x + 3 cotx = 5, for 0° < x < 180°.

The variables x and y satisfy the differential equation `dy/dx = xy / (1 + x²)` and y = 2 when x = 0. Solve the differential equation, obtaining a ...

The polynomial ax³ – 10x² + bx + 8, where a and b are constants, is denoted by p(x). It is given that (x – 2) is a factor of both p(x) and p'(x).

Let `I = ∫(from 0 to 3) (27 / (9 + x²)²) dx`.

The complex number u is defined by u = `(√2 – a√2i) / (1 + 2i)`, where a is a positive integer.

The equation of a curve is x³ + y³ + 2xy + 8 = 0.

In the diagram, OABCDEFG is a cuboid in which OA = 2 units, OC = 4 units and OG = 2 units. Unit vectors i, j and k are parallel to OA, OC and OG re...

The curve y = x√sinx has one stationary point in the interval 0 < x < π, where x = a (see diagram). [Figure 10.1]

Solve the equation ln(e2x + 3) = 2x + ln 3, giving your answer correct to 3 decimal places.

Solve the equation 3 cos 2θ = 3 cos θ + 2, for 0° ≤ θ ≤ 360°.

The polynomial ax³ + x² + bx + 3 is denoted by p(x). It is given that p(x) is divisible by (2x – 1) and that when p(x) is divided by (x + 2) the re...

The equation of a curve is y = cos³x√sinx. It is given that the curve has one stationary point in the interval 0 < x < ½π. Find the x-coordinate of...

The variables x and y satisfy the differential equation dy/dx = xey-x, and y = 0 when x = 0.

The equation of a curve is x³ + 3x²y – y³ = 3.

Let f(x) = x² + 9x / (3x−1)(x² + 3)

The lines l and m have vector equations r = −i + 3j + 4k + λ(2i – j – k) and r = 5i + 4j + 3k + µ(ai + bj + k) respectively, where a and b are cons...

The complex number −1 + √7i is denoted by u. It is given that u is a root of the equation 2x³ + 3x² + 14x + k = 0, where k is a real constant.

Find, in terms of a, the set of values of x satisfying the inequality 2|3x + a| < |2x + 3a|, where a is a positive constant.

Solve the equation cos(θ – 60°) = 3 sin θ, for 0° ≤ θ ≤ 360°.

The curve y = e⁻⁴ˣ tan x has two stationary points in the interval 0 < x < ½π.

The complex number 3 – i is denoted by u.

The parametric equations of a curve are x = 1/cos t, y = ln tan t, where 0 < t < ½π.

Let f(x) = (5x² + 8x - 3) / ((x-2)(2x² + 3)).

At time t days after the start of observations, the number of insects in a population is N. The variation in the number of insects is modelled by a...

With respect to the origin O, the point A has position vector given by OA = i + 5j + 6k. The line l has vector equation r = 4i + k + λ(−i + 2j + 3k).

The constant a is such that ∫(from 1 to a) x² ln x dx = 4.

A car starts from rest and moves in a straight line with constant acceleration for a distance of 200m, reaching a speed of 25 ms¯¹. The car then tr...

Two particles P and Q, of masses 0.5 kg and 0.3 kg respectively, are connected by a light inextensible string. The string is taut and P is vertical...

A crate of mass 300kg is at rest on rough horizontal ground. The coefficient of friction between the crate and the ground is 0.5. A force of magnit...

Three coplanar forces of magnitudes 20N, 100N and F N act at a point. The directions of these forces are shown in the diagram. Given that the three...

Two racing cars A and B are at rest alongside each other at a point O on a straight horizontal test track. The mass of A is 1200kg. The engine of A...

A particle starts from a point O and moves in a straight line. The velocity vms¯¹ of the particle at time ts after leaving O is given by v = k(3t² ...