Cambridge Past Paper Questions

A function f with domain x > 0 is such that f'(x) = 8(2x-3)^(-1/2) - 10x^3. It is given that the curve with equation y = f(x) passes through the po...

The variables x and y satisfy the equation a^(2y) = e^(3x+k), where a and k are constants. The graph of y against x is a straight line.

Solve the inequality |x-7|>4x+3.

The function f is defined by f(x) = tan²(x/2) for 0 < x < π.

The polynomial p(x) is defined by p(x) = ax³ – ax² – 15x + 18, where a is a constant. It is given that (x+2) is a factor of p(x).

It is given that ∫(from a to 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.

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.

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

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...

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.

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).

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))

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.

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 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 ...

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.

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.