Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
It is given that x + 3 is a factor of the polynomial p(x) = 2x³ + ax² – 24x + b. The remainder when p(x) is divided by x − 2 is -15. Find the remai...
Find the coordinates of the points where the line 2y - 3x = 6 intersects the curve x²/4 + y²/9 = 5.
In the diagram, ABC is an arc of the circle centre O, radius 5 cm, and angle AOC is 1.5 radians. AD and CE are diameters of the circle and DE is a ...
Vectors i and j are vectors parallel to the x-axis and y-axis respectively. Given that a = 2i + 3j, b = i – 5j and c = 3i + 11j, find
In this question all lengths are in centimetres. The volume of a cone of height h and base radius r is given by V = ⅓πr²h. It is known that sin (π/...
The polynomial p(x) = (2x-1)(x+k)-12, where k is a constant.
A particle P is moving with a velocity of 20ms⁻¹ in the same direction as (3/4).
The diagram shows the curve y = 3x²-2x+1 and the straight line y = 2x+5 intersecting at the points P and Q. Showing all your working, find the area...
It is given that f(x) = 5e^x -1 for x ∈ R. It is given also that g(x) = x²+4 for x ∈ R.
In this question all lengths are in centimetres. A closed cylinder has base radius r, height h and volume V. It is given that the total surface are...
When lg y is plotted against x² a straight line graph is obtained which passes through the points (2, 4) and (6, 16).
It is given that y = (x²+1)(2x-3)^(1/2).
(a) On the Venn diagrams below, shade the region indicated. [Figure 1.1] (b) E = {x : 0° ≤ x ≤ 360°} P = {x: cos2x = 0.5} Q = {x: sinx = 0.5} Find...
Do not use a calculator in this question. Find the coordinates of the points of intersection of the curve y = (2x+3)²(x-1) and the line y = 3 (2x+3).
The number, B, of a certain type of bacteria at time t days can be described by B = 200e²ᵗ +800e⁻²ᵗ.
The diagram shows the right-angled triangle OAB. The point C lies on the line OB. Angle OAB = π/2 radians and angle AOB = θ radians. AC is an arc o...
A pilot wishes to fly his plane from a point A to a point B on a bearing of 055°. There is a wind blowing at 120kmh⁻¹ from the west. The plane can ...
When eʸ is plotted against 1/x, a straight line graph passing through the points (2, 20) and (4, 8) is obtained.
The diagram shows the curve y = 4+2cos 3x intersecting the line y = 5 at the points P and Q. [Figure 9.1]
The diagram shows an open container in the shape of a cuboid of width x cm, length 4x cm and height h cm. The volume of the container is 800 cm³. [...
The normal to the curve y = (x-2)(3x+1)^(1/2) at the point where x = 7/3, meets the y-axis at the point P. Find the exact coordinates of the point P.
Find the values of x for which x(6x + 7) ≥ 20.
Two variables x and y are such that y = In x / x³ for x > 0.
Without using a calculator, express (√5-3)² / (√5+1) in the form p√5+q, where p and q are integers.
The velocity-time graph represents the motion of a particle travelling in a straight line. [Figure 5.1]
The variables x, y and u are such that y = tan u and x = u³ + 1.
The diagram shows a right-angled triangle ABC with AB = 8cm and angle ABC = π/2 radians. The points D and E lie on AC and BC respectively. BAD and ...
Solutions to this question by accurate drawing will not be accepted. The points A and B have coordinates (p, 3) and (1, 4) respectively and the lin...
Given that y = sinx / lnx², find an expression for dy/dx.
Find the values of k for which the equation (k-1)x²+kx-k = 0 has real and distinct roots.
A circle has diameter x which is increasing at a constant rate of 0.01 cm s⁻¹. Find the exact rate of change of the area of the circle when x = 6 cm.