Cambridge Past Paper Questions

[Figure 2(a) showing an empty rectangle labelled E for a Venn diagram.] It is given that the elements of set E are the letters of the alphabet, the...

The diagram shows 3 circles with centres A, B and C, each of radius 5 cm. Each circle touches the other two circles. Angle BAC is θ radians. [Figur...

O, P, Q and R are four points such that OP = p, OQ = q and OR = 3q - 2p.

The diagram shows part of the curve y = 3x - x³ and the lines y = 3x and 2y = 27 - 3x. The curve and the line y = 3x meet the x-axis at O and the c...

A curve has equation y = (2x-5)/(x-1) - 12x.

Variables x and y are related by the equation y = (5x-1) / (3-x).

A team of 3 people is to be selected from 7 women and 6 men. Find the number of different teams that could be selected if there must be more women ...

Do not use a calculator in this question. The polynomial p(x) = 2x³ – 3x² + qx + 56 has a factor x – 2.

The coordinates of three points are A(−2, 6), B(6, 10) and C(p, 0).

The diagram shows a sector of a circle with centre O and radius 5 cm. The length of the arc AB is 7 cm. Angle AOB is θ radians. [Figure 6.1]

The matrix A is (4 5 / 3 2) and the matrix B is (4 2 / 1 3).

Find the coordinates of the points of intersection of the curve 4 + 5/y + 3/x = 0 and the line y = 15x + 10.

Solve the equation

The line y = kx – 5, where k is a positive constant, is a tangent to the curve y = x² + 4x at the point A.

It is given that p(x) = x³ + ax² + bx – 48. When p(x) is divided by x – 3 the remainder is 6. Given that p'(1) = 0, find the value of a and of b.

(a) [Figure OABC diagram] The diagram shows a figure OABC, where OA = a, OB = b and OC = c. The lines AC and OB intersect at the point M where M is...

[Figure Circle with rectangle ABCD] The diagram shows a circle, centre O, radius 12 cm. The points A and B lie on the circumference of the circle a...

Show that the curve y = (3x² + 8)² has only one stationary point. Find the coordinates of this stationary point and determine its nature.

On each of the Venn diagrams below, shade the region which represents the given set. [Figure 1.1 - Three Venn diagrams are shown, with overlapping ...

It is given that y = (5x²+4)^(1/2) / (x+1). Showing all your working, find the exact value of dy/dx when x = 3.

Vectors i and j are unit vectors parallel to the x-axis and y-axis respectively.

The first 3 terms in the expansion of (3 - x/6)^n are 81 + ax + bx². Find the value of each of the constants n, a and b.

A particle P moves in a straight line, such that its displacement, x m, from a fixed point O, t s after passing O, is given by x = 4 cos (3t) – 4.

It is given that y = A(10^bx), where A and b are constants. The straight line graph obtained when lgy is plotted against x passes through the point...

(a) A 5-digit number is to be formed from the seven digits 1, 2, 3, 5, 6, 8 and 9. Each digit can only be used once in any 5-digit number. Find the...

(a) Given that A = [-3 1; -1 2; 4 5], B = [-2 0; 1 3] and C = AB, (b) The matrix X = [5 -12; 4 -7]

The diagram shows a circle, centre O, radius 8 cm. The points A, B, C and D lie on the circumference of the circle such that AB is parallel to DC. ...

The curve y = f(x) passes through the point (1/2, 7/2) and is such that f'(x) = e^(2x-1).

Find the equation of the curve which passes through the point (2, 17) and for which dy/dx = 4x³ + 1.

The variables x and y are such that y = ln(x² + 1).

Four cinemas, P, Q, R and S each sell adult, student and child tickets. The number of tickets sold by each cinema on one weekday were P: 90 adult, ...

The curve 3x² + xy - y² + 4y – 3 = 0 and the line y = 2(1 – x) intersect at the points A and B.