Cambridge & Pearson Edexcel Past Paper Questions

Prove that cos(θ+30°)cos(θ+60°) = (1/4)√3 - (1/2)sin 2θ.

Use logarithms to show that the equation 5⁸ʸ = 6⁷ˣ can be expressed in the form y = kx. Give the value of the constant k correct to 3 significant f...

Let f(x) = 4 sin² 3x. Find the value of f' (π/4).

A curve has equation 6e⁻ˣy² + e²ˣ – 12y + 7 = 0. Find the gradient of the curve at the point (ln 3, 2).

Sketch the graphs of y = 1+e²ˣ and y = |x-4| on the same diagram.

The polynomial p(x) is defined by p(x) = ax³ + bx² - ax + 8, where a and b are constants. It is given that (x + 2) is a factor of p(x), and that th...

The diagram shows the curves with equations y = √5x² +7 and y = 27/(2x+5) for x ≥ 0. The curves meet at the point (2, 3). Region A is bounded by th...

Express 4 sin θ sin(θ+ 60°) in the form a+Rsin(2θ-α), where a and R are positive integers and 0° < α < 90°.

The function f is defined by f(x) = tan²(½x) for 0 ≤ x < π.

It is given that ∫_a^(a³) (10 / (2x+1)) dx = 7, where a is a constant greater than 1.

A curve has parametric equations x = (e^(2t)-2) / (e^(2t)+1), y = e^(3t) + 1.

Prove that cos(θ+30°)cos(θ+60°) = ¼√3 - ½sin 2θ.

The polynomial 4x³ + ax² +5x+b, where a and b are constants, is denoted by p(x). It is given that (2x+1) is a factor of p(x). When p(x) is divided ...

Find the exact value of ∫ x² ln 3x dx. Give your answer in the form a lnb+c, where a and c are rational and b is an integer.

The equation of a curve is ln(x+y) = 3x²y. Find the gradient of the curve at the point (1,0).

Show that sec⁴θ-tan⁴θ = 1+2tan²θ.

Hence or otherwise solve the equation sec⁴2α-tan⁴ 2α = 2 tan² 2α sec² 2α for 0° < α < 180°.

By sketching a suitable pair of graphs, show that the equation 2+e-0.2x = ln(1+x) has only one root.

Show by calculation that this root lies between 7 and 9.

Use the iterative formula x_n+1 = exp(2+e-0.2x_n) - 1 to determine the root correct to 2 decimal places. Give the result of each iteration to 4 dec...

The diagram shows the curve y = sin2x(1+sin2x), for 0 ≤ x ≤ ¾π, and its minimum point M. The shaded region bounded by the curve that lies above the...

Let f(x) = (5x²+8x+5) / ((1+2x)(2+x²))

Given that z = 1+yi and that y is a real number, express 1/z in the form a+bi, where a and b are functions of y.

Show that (a-½)²+b² = ¼, where a and b are the functions of y found in part (a).

On a single Argand diagram, sketch the loci given by the equations Re(z) = 1 and |z - ½| = ½, where z is a complex number.

The complex number z is such that Re(z) = 1. Use your answer to part (b) to give a geometrical description of the locus of 1/z.

The position vector of point A relative to the origin O is OA = 8i−5j+6k. The line l passes through A and is parallel to the vector 2i+j+4k.

A large cylindrical tank is used to store water. The base of the tank is a circle of radius 4 metres. At time t minutes, the depth of the water in ...

Expand (9-3x)¹ᐟ² in ascending powers of x, up to and including the term in x², simplifying the coefficients.

By sketching a suitable pair of graphs, show that the equation cot2x = secx has exactly one root in the interval 0 < x < ½π.

The square roots of 6-8i can be expressed in the Cartesian form x+iy, where x and y are real and exact. By first forming a quartic equation in x or...

Solve the equation 5ˣ = 5ˣ⁺² – 10. Give your answer correct to 3 decimal places.

The complex number u is given by u = (cos(⅓π) + i sin(⅓π))⁴ / (cos(½π) - i sin(½π)). Find the exact value of arg u.

The variables x and y satisfy the equation ay = bˣ, where a and b are constants. The graph of lny against x is a straight line passing through the ...

Show that the equation tan³x+2 tan2x-tanx = 0 may be expressed as tan⁴x-2 tan²x-3 = 0 for tanx ≠ 0.

The parametric equations of a curve are x = tan²2t, y = cos 2t, for 0 < t < ¼π.

With respect to the origin O, the points A, B and C have position vectors given by OA = (2, 1, -3), OB = (0, 4, 1), and OC = (-3, -2, 2).

A balloon in the shape of a sphere has volume V and radius r. Air is pumped into the balloon at a constant rate of 40π starting when time t = 0 and...

Let f(x) = 2e²ˣ / (e²ˣ - 3eˣ + 2)

The complex number z satisfies |z|= 2 and 0 ≤ argz <¦π.

Let f(x) = 2x³ - 5x² +4. Show that if a sequence of values given by the iterative formula xn+1=√4/(5-2xn), converges, then it converges to a root o...

The number of bacteria in a population, P, at time t hours is modelled by the equation P = aekt, where a and k are constants. The graph of InP agai...

Find the complex number z satisfying the equation (z-3i)/(z+3i) = (2-9i)/5. Give your answer in the form x+ iy, where x and y are real.

Show that cos⁴θ - sin⁴θ - 4 sin²θcos²θ ≡ cos²2θ + cos 2θ - 1.

The lines l and m have vector equations l: r = 2i+j-3k+λ(-i+2k) and m: r = 2i+j-3k+µ(2i−j+5k). Lines l and m intersect at the point P.

The parametric equations of a curve are x = 3 sin 2t, y = tan t + cot t, for 0 < t < ¦π.

Let f(x) = 7a² / ((a-2x)(3a+x)), where a is a positive constant.

Find the quotient and remainder when x⁴ + 16 is divided by x² + 4.

A water tank is in the shape of a cuboid with base area 40000 cm². At time t minutes the depth of water in the tank is h cm. Water is pumped into t...

The diagram shows the curve y = 2sinx√2+cosx, for 0 ≤ x ≤ 2π, and its minimum point M, where x = a. [Figure 11.1]