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.
The diagram shows a square ABCD of side 10 cm. The mid-point of AD is O and BXC is an arc of a circle with centre O. [Figure 4.1]
It is given that a = sin θ – 3 cos θ and b = 3 sin θ + cos θ, where 0° ≤ θ ≤ 360°.
Relative to an origin O, the position vectors of points A and B are given by OA = i - 2j + 2k and OB = 3i + pj + qk, where p and q are constants.
The point R is the reflection of the point (−1, 3) in the line 3y + 2x = 33. Find by calculation the coordinates of R.
The volume of a solid circular cylinder of radius r cm is 250π cm³.
A function f is defined by f(x) = 5/(1-3x) for x ≥ 1.
The first and last terms of an arithmetic progression are 12 and 48 respectively. The sum of the first four terms is 57. Find the number of terms i...
The diagram shows the curve y = √(1 + 4x), which intersects the x-axis at A and the y-axis at B. The normal to the curve at B meets the x-axis at C...
Find the coordinates of the point at which the perpendicular bisector of the line joining (2, 7) to (10, 3) meets the x-axis.
Find the coefficient of x² in the expansion of (1 + x²)((x/2) - (4/x))⁶
The reflex angle θ is such that cos θ = k, where 0 < k < 1.
The diagram shows a sector of a circle with radius r cm and centre O. The chord AB divides the sector into a triangle AOB and a segment AXB. Angle ...
Prove the identity 1/cos θ - cos θ/(1 + sin θ) = tan θ.
The 1st, 2nd and 3rd terms of a geometric progression are the 1st, 9th and 21st terms respectively of an arithmetic progression. The 1st term of ea...
The diagram shows a trapezium ABCD in which BA is parallel to CD. The position vectors of A, B and C relative to an origin O are given by O A = (3/...
The equation of a curve is such that d²y/dx² = 2x – 1. Given that the curve has a minimum point at (3, –10), find the coordinates of the maximum po...
The diagram shows part of the curve y = 8 – √(4 – x) and the tangent to the curve at P (3, 7). [Figure 9.1]
Functions f and g are defined by f : x → 2x – 3, x ∈ R, g : x → x² + 4x, x ∈ R. Function h is defined by h : x → x² + 4x for x ≥ k, and it is given...
The function f is such that f'(x) = 5 – 2x² and (3, 5) is a point on the curve y = f(x). Find f(x).
In the diagram, AYB is a semicircle with AB as diameter and OAXB is a sector of a circle with centre O and radius r. Angle AOB = 2θ radians. Find a...
Find the coefficients of x² and x³ in the expansion of (2 – x)⁶.
Variables u, x and y are such that u = 2x(y – x) and x + 3y = 12. Express u in terms of x and hence find the stationary value of u.
Prove the identity (sin θ – cos θ) / (sin θ + cos θ) = (tan θ – 1) / (tan θ + 1).
A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The wheel turns in such a way that the height, h m, of ...
The point C lies on the perpendicular bisector of the line joining the points A (4, 6) and B (10, 2). C also lies on the line parallel to AB throug...
The first, second and last terms in an arithmetic progression are 56, 53 and –22 respectively. Find the sum of all the terms in the progression.
Relative to an origin O, the position vectors of points A and B are given by OA = 2i + 4j + 4k and OB = 3i + j + 4k.
The equation of a curve is y = 4 / (2x – 1).
The function f is defined by f : x → 2x² – 6x + 5 for x ∈ R.
Functions f and g are defined by f:x ↦ 10 – 3x, x ∈ R, g:x ↦ 10 / (3 – 2x), x ∈ R, x ≠ 3/2. Solve the equation ff(x) = gf(2).
A curve is such that dy/dx = 8 / (5 – 2x)². Given that the curve passes through (2, 7), find the equation of the curve.
Relative to an origin O, the position vectors of points A and B are given by OA = 2i – 5j – 2k and OB = 4i – 4j + 2k. The point C is such that AB =...
Find the term that is independent of x in the expansion of
In the diagram, triangle ABC is right-angled at C and M is the mid-point of BC. It is given that angle ABC = 1/3 π radians and angle BAM = θ radian...
The diagram shows a circle with radius r cm and centre O. The line PT is the tangent to the circle at P and angle POT = α radians. The line OT meet...
Prove the identity (1 + cos θ)/(1 – cos θ) – (1 – cos θ)/(1 + cos θ) = 4/(sin θ tan θ)
Three points have coordinates A (0, 7), B (8, 3) and C (3k, k). Find the value of the constant k for which
A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that each day a greater amount of water is lost.
The diagram shows the part of the curve y = 8/x + 2x for x > 0, and the minimum point M. [Figure 10.1]
The function f is defined by f : x → 6x − x² – 5 for x ∈ R.
Find the coefficient of x in the expansion of (2x - 1/x)^5.
The point A has coordinates (−2, 6). The equation of the perpendicular bisector of the line AB is 2y = 3x + 5.
Prove the identity (1/cos θ - tan θ)² = (1 - sin θ) / (1 + sin θ)
The diagram shows a circle with radius r cm and centre O. Points A and B lie on the circle and ABCD is a rectangle. Angle AOB = 2θ radians and AD =...
A curve has equation y = 3 + 12 / (2 - x).
The diagram shows the straight line x + y = 5 intersecting the curve y = 4/x at the points A (1, 4) and B (4, 1). Find, showing all necessary worki...
The first two terms of an arithmetic progression are 16 and 24. Find the least number of terms of the progression which must be taken for their sum...
Relative to an origin O, the position vectors of three points A, B and C are given by OA = 3i + pj – 2pk, OB = 6i + (p + 4)j + 3k and OC = (p − 1)i...
The equation of a curve is y = 8√x − 2x.
The function f is defined by f(x) = 3 tan(1/2 x) - 2, for -1/2 π ≤ x ≤ 1/2 π.