Cambridge Past Paper Questions
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[Figure 6.1] The diagram shows a metal plate ABCD made from two parts. The part BCD is a semicircle. The part DAB is a segment of a circle with cen...
The equation of a curve is y = 2 + 3/(2x – 1).
Relative to an origin O, the position vectors of the points A, B and C are given by OA = (-2, 3, -1), OB = (-2, 6, 5) and OC = (2, -2, 6).
A function f is defined by f : x → 5 – 2 sin 2x for 0 ≤ x ≤ π.
Find the term independent of x in the expansion of (2x - 1/(4x²))⁹.
A function f is defined by f : x → 4 – 5x for x ∈ R.
The diagram shows a semicircle with centre O and radius 6 cm. The radius OC is perpendicular to the diameter AB. The point D lies on AB, and DC is ...
This question concerns properties of functions involving sine.
Points A and B lie on the curve y = x² – 4x + 7. Point A has coordinates (4, 7) and B is the stationary point of the curve. The equation of a line ...
A curve is such that dy/dx = -x² + 5x - 4.
The diagram shows a trapezium OABC in which OA is parallel to CB. The position vectors of A and B relative to the origin O are given by OA = (2, -2...
The diagram shows part of the curve y = √(5x – 1) and the normal to the curve at the point P (2, 3). This normal meets the x-axis at Q. [Figure X.X]
The polynomial p(x) is defined by p(x) = ax³ + bx² + 37x + 10 where a and b are constants. It is given that (x + 2) is a factor of p(x). It is also...
The parametric equations of a curve are x = 2e^(2t) + 4e^t, y = 5te^(2t).
The diagram shows the curve [Figure X.X] y = x² + 3x + 1 + 5 cos (x/2). The curve crosses the y-axis at the point P and the gradient of the curve a...
Use logarithms to solve the equation 5³ˣ⁻¹ = 2⁴ˣ, giving your answer correct to 3 significant figures.
It is given that x satisfies the equation |x + 1| = 4. Find the possible values of |x + 4| - |x - 4|.
The equation of a curve is y = tan ½x + 3 sin ½x. The curve has a stationary point M in the interval π < x < 2π. Find the coordinates of M, giving ...
The polynomials p(x) and q(x) are defined by p(x) = x³ + x² + ax - 15 and q(x) = 2x³ + x² + bx + 21, where a and b are constants. It is given that ...
The diagram shows the curve y = 4e⁻²ˣ and a straight line. The curve crosses the y-axis at the point P. The straight line crosses the y-axis at the...
The equation of a curve is x² + 4xy + 2y² = 7.
It is given that the variable x is such that 1.3²ˣ < 80 and |3x-1| > |3x – 10. Find the set of possible values of x, giving your answer in the form...
The parametric equations of a curve are x = 2e²ᵗ + 4eᵗ, y = 5te²ᵗ.
The diagram shows the curve [Figure 7.1] y = x² + 3x + 1 + 5 cos ½x. The curve crosses the y-axis at the point P and the gradient of the curve at P...
Find the quotient and remainder when x⁴ is divided by x² + 2x – 1.
Two variable quantities x and y are believed to satisfy an equation of the form y = C(aˣ), where C and a are constants. An experiment produced four...
The equation x³ = 3x + 7 has one real root, denoted by α.
The equation of a curve is 2x⁴ + xy³ + y⁴ = 10.
The variables x and y satisfy the differential equation dy/dx = 4 cos² y tan x, for 0 ≤ x < ½π, and x = 0 when y = ¼π. Solve this differential equa...
Let f(x) = (4x² + 9x - 8) / ((x + 2)(2x - 1))
The diagram shows the curve y = (1 + x²)e⁻ˣ/² for x ≥ 0. The shaded region R is enclosed by the curve, the x-axis and the lines x = 0 and x = 2. [F...
The equations of two lines l and m are r = 3i − j − 2k + λ(−i + j + 4k) and r = 4i + 4j – 3k + μ(2i + j − 2k) respectively.
The diagram shows a sketch of the curve y = 3 / √(9-x³) for values of x from -1.2 to 1.2. [Figure 1.1]
Showing all necessary working, solve the equation 2log₂ x = 3 + log₂ (x + 1), giving your answer correct to 3 significant figures.
By expressing the equation tan(θ + 60°) + tan(θ – 60°) = cot θ in terms of tan θ only, solve the equation for 0° < θ < 90°.
The curve with equation y = (2 – sin x) / cos x has one stationary point in the interval -π/2 < x < π/2.
The variables x and y satisfy the differential equation (x + 1) dy/dx = y(x + 2), and it is given that y = 2 when x = 1. Solve the differential equ...
The equation of a curve is x³y – 3xy³ = 2a⁴, where a is a non-zero constant.
Throughout this question the use of a calculator is not permitted. The complex number 1 – (√3)i is denoted by u.