Cambridge & Pearson Edexcel Past Paper Questions
Browse 35,704 questions from Cambridge O-Level & A-Level and Pearson Edexcel IGCSE & IAL exams. Click any question to practice.
Show that 2cosx cotx + 1 = cotx + 2cosx can be written in the form (a cosx - b)(cosx - sinx) = 0, where a and b are constants to be found.
Given that f(x) = 4x³ + kx + p is exactly divisible by x + 2 and f'(x) is exactly divisible by 2x-1, find the value of k and of p.
The diagram shows a circle, centre O, radius r cm. The points A and B lie on the circle such that angle AOB = 2θ radians. [Figure 11.1]
The universal set & is the set of real numbers and sets X, Y and Z are such that X= {integer multiples of 5}, Y = {integer multiples of 10}, Z = {π...
The first 3 terms in the expansion of (2 - 1/(4x))^5 are a + b/x + c/x². Find the value of each of the integers a, b and c.
Vectors a, b and c are such that a = [[2],[y]], b = [[1],[3]] and c = [[-5],[5]]
Do not use a calculator in this question. Find the positive value of x for which (4 + √5)x² + (2 - √5)x - 1 = 0, giving your answer in the form (a ...
Show that (1 - cos θ)(1 + sec θ) = sin θ tan θ.
Show that d/dx (e^(3x) √4x + 1) can be written in the form (e^(3x) (px+q)) / √4x + 1, where p and q are integers to be found.
The diagram shows part of the graph of y = 1 - 2cos 3x, which crosses the x-axis at the point A and has a maximum at the point B.
Variables x and y are such that when lgy is plotted against x², the straight line graph shown above is obtained.
A curve passes through the point (2, -4/3) and is such that dy/dx = (3x + 10)^(-1/2).
The diagram shows a badge, made of thin sheet metal, consisting of two semi-circular pieces, centres B and C, each of radius x cm. They are attache...
Find the values of x for which (x - 4)(x + 2) > 7.
[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...
Express 4x² + 8x - 5 in the form p(x+q)² + r, where p, q and r are constants to be found.
O, P, Q and R are four points such that OP = p, OQ = q and OR = 3q - 2p.
Use the Binomial Theorem to expand (a + b)⁴, giving each term in its simplest form.
Given that y = atan bx + c has period π/4 radians and passes through the points (0, -2) and (π/16, 0), find the value of each of the constants a, b...
Find how many 5-digit even numbers can be made using each of the digits 1, 2, 3, 4, 5 once only.
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.
Given that x² + 2kx + 4k − 3 = 0 has no real roots, show that k satisfies k² - 4k+3 < 0.
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.
Find ∫(x³ + x² + 1) / x² dx.
The graph of the curve y = p(4^(2x)) - q(4^x) passes through the points (0, 2) and (0.5, 14). Find the value of p and of q.
A function f is defined, for all real x, by f(x) = x - x². Find the greatest value of f(x) and the value of x for which this occurs.
Solve the equation 8 sin²A + 2 cosA = 7 for 0° ≤ A ≤ 180°.
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.
Simplify √x³y¹⁰ ÷ √x³y⁻⁶, giving your answer in the form xᵃyᵇ, where a and b are integers.
State the range of f.
(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 the axes below sketch the graphs of y = |2x – 5| and 9y = 80x – 16x².
Show that 5 + 4 tan²(x/3) = 4 sec²(x/3) + 1.
Given that y = e³ˣ / (4x² + 1), find dy/dx.
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.