Cambridge Past Paper Questions

A circle, centre O and radius r cm, has a sector OAB of fixed area 10 cm². Angle AOB is θ radians and the perimeter of the sector is P cm.

The normal to the curve `y = tan(3x + π/2)` at the point P with coordinates `(p, -1)`, where `0 < p < π/6`, meets the x-axis at the point A and the...

The diagram shows the graph of y = a sinbx+c, where a, b and c are integers, for -180° ≤ x ≤ 180°. Find the values of a, b and c.

Given that x = sec²θ and y+2 = cot²θ, find y in terms of x.

Variables x and y are such that, when lg(2y+1) is plotted against x², a straight line graph passing through the points (1, 1) and (2, 5) is obtained.

The first three terms, in ascending powers of x, in the expansion of (1 + x/6)^12 (2−3x)^3 can be written in the form 8+px+qx², where p and q are c...

The polynomial p(x) = 6x³ + ax² + 6x + b, where a and b are integers, is divisible by 2x−1. When p(x) is divided by x−2, the remainder is 120.

The normal to the curve y = ln(3x² + 2)/(x + 1), at the point A on the curve where x = 0, meets the x-axis at point B. Point C has coordinates (0, ...

In this question all lengths are in kilometres and time is in hours. A particle P moves in a straight line such that its displacement, s, from a fi...

Variables x and y are such that when √y is plotted against x², a straight line passing through the points (9, 8) and (16, 1) is obtained. Find y as...

The polynomial p(x) = mx³ - 17x²+nx+6 has a factor x-3. It has a remainder of -12 when divided by x + 1. Find the remainder when p(x) is divided by...

Variables x and y are such that y = (1 + sin 3x)⁴ / √x. Use differentiation to find the approximate change in y when x increases from 1.9 to 1.9+h,...

In this question, i is a unit vector due east and j is a unit vector due north. Distances are measured in kilometres and time is measured in hours....

In this question all lengths are in metres. [Figure 9.1] The diagram shows a water container in the shape of a triangular prism. The depth of water...

The diagram shows part of the curves y = eˣᐟ² and y = cos 5x and part of the line x = π/4. The curves intersect at A. The curve y = cos 5x cuts the...

DO NOT USE A CALCULATOR IN THIS QUESTION. A curve has equation y = 6+ √x / 3+√x where x ≥ 0. Find the exact value of y when x = 6. Give your answer...

The diagram shows the graphs of y = |f(x)| and y = g(x), where y = f(x) and y = g(x) are straight lines. Solve the inequality |f(x)|≤g(x). [Figure 2]

Find the possible values of k for which the equation kx² + (k+5)x-4 = 0 has real roots.

Variables x and y are related by the equation y = 1+2/x+1/x² where x > 0. Use differentiation to find the approximate change in x when y increases ...

Differentiate y = (e^(4x) tanx) / lnx with respect to x.

The function f is defined by f(x) = 3 sin²x-2cosx for 2≤x≤4, where x is in radians.

In this question all lengths are in centimetres. [Figure 9]

The diagram shows part of the line y = 1 and one complete period of the curve y = 1+cosx, where x is in radians. The line PQ is a tangent to the cu...

A curve is such that d²y/dx² = ((√x+1)/√x)². Given that the gradient of the curve is 4/3 at the point (1, −1), find the equation of the curve.

1 (a) Write 5x²-14x+8 in the form a(x+b)² +c, where a, b and c are constants to be found.

2 The polynomial p is such that p(x) = ax³ +7x²+bx+c, where a, b and c are integers.

3 The points A and B have coordinates (2,5) and (10, -15) respectively. The point P lies on the perpendicular bisector of the line AB. The y-coordi...

4 The diagram shows the velocity-time graph for a particle travelling in a straight line with velocity, vms⁻¹, at time t seconds. When t = 30 the v...

5 DO NOT USE A CALCULATOR IN THIS QUESTION. In this question, all lengths are in centimetres.

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7 (a) A team of 8 people is to be chosen from a group of 15 people.

8 A curve has the equation y = (3x-4)¹ᐟ³ / (2x+1).

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10 It is given that y = (3x+1)²ln(3x+1).

The diagram shows the graph of y = a cos bx+c. Find the values of the constants a, b and c.