Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The velocity, v ms⁻¹, of a particle travelling in a straight line, t seconds after passing through a fixed point O, is given by v = 4/(t+1)³.
(a) The functions f and g are defined by `f(x) = 5x-2` for `x > 1`, `g(x) = 4x²-9` for `x > 0`. (b) The function h is defined by `h(x) = √(x²-1)` f...
The diagram shows the graph of a cubic curve y = f(x). [Figure X.X]
The radius, r cm, of a circle is increasing at the rate of 5 cms⁻¹. Find, in terms of π, the rate at which the area of the circle is increasing whe...
DO NOT USE A CALCULATOR IN THIS QUESTION. Find the positive solution of the equation (5+4√7)x²+(4-2√7)x-1 = 0, giving your answer in the form a+b√7...
Find the equation of the tangent to the curve y = ln(3x²-1) / (x+2) at the point where x = 1. Give your answer in the form y = mx+c, where m and c ...
The line y = 5x+6 meets the curve xy = 8 at the points A and B.
The diagram shows an isosceles triangle OAB such that OA = OB and angle AOB = θ radians. The points C and D lie on OA and OB respectively. CD is an...
A curve is such that d²y/dx² = 5 cos 2x. This curve has a gradient of 3/4 at the point (-π/12, 5π/4). Find the equation of this curve.
On the axes below, sketch the graph of y=|(x-2)(x+1)(x+2)| showing the coordinates of the points where the curve meets the axes.
The volume, V, of a sphere of radius r is given by V = 4/3 πr². The radius, r cm, of a sphere is increasing at the rate of 0.5 cms⁻¹. Find, in term...
f: x → (2x+3)² for x > 0
The diagram shows the straight line 2x+y=-5 and part of the curve xy+3 = 0. The straight line intersects the x-axis at the point A and intersects t...
The diagram shows a triangle OAB such that OA = a and OB = b. The point P lies on OA such that OP = 3/4 OA. The point Q is the mid-point of AB. The...
Variables x and y are such that, when √y is plotted against 1/x, a straight line graph passing through the points (0.5, 9) and (3, 34) is obtained....
DO NOT USE A CALCULATOR IN THIS QUESTION. p(x) = 15x³ +22x² - 15x+2
The vectors a and b are such that a = αi+j and b = 12i+βj.
Find the values of k for which the line y = kx-7 and the curve y = 3x² +8x+5 do not intersect.
A particle travels in a straight line. As it passes through a fixed point O, the particle is travelling at a velocity of 3 ms⁻¹. The particle conti...
The diagram shows part of the graphs of y = 4x^(1/2) and y = (x-3)². The graph of y = (x-3)² meets the x-axis at the point A(a, 0) and the two grap...
The function f is defined by f(x) = ln(2x+1) for x ≥ 0.
Variables x and y are such that y = sinx+e^*. Use differentiation to find the approximate change in y as x increases from π/4 to π/4+h, where h is ...
DO NOT USE A CALCULATOR IN THIS QUESTION. The point (1 - √5, p) lies on the curve y = (10+2√5) / x². Find the exact value of p, simplifying your an...
Find the values of k for which the line y=x-3 intersects the curve y = k²x² +5kx+1 at two distinct points.
The three roots of p(x) = 0, where p(x) = 2x³ + ax² + bx + c are x = 1/2, x = n and x = −n, where a, b, c and n are integers. The y-intercept of th...
Solutions to this question by accurate drawing will not be accepted. The points A and B are (4, 3) and (12, −7) respectively.
Giving your answer in its simplest form, find the exact value of
10 (a) The first 5 terms of a sequence are given below. 4 -2 1 -0.5 0.25
The circles with centres C₁ and C₂ have equal radii of length r cm. The line C₁C₂ is a radius of both circles. The two circles intersect at A and B...