Cambridge Past Paper Questions

Given that logₐ pq = 9 and logₐ p²q = 15, find the value of

The line x – 2y = 6 intersects the curve x² + xy + 10y + 4y² = 156 at the points A and B. Find the length of AB.

[Figure 6.1] Using sin15° = √2/4 (√3 - 1) and without using a calculator, find the value of sinθ in the form a + b√2, where a and b are integers.

Solutions to this question by accurate drawing will not be accepted. [Figure 7.1] The vertices of the trapezium ABCD are the points A(−5, 4), B(8, ...

[Figure 8.1] The diagram shows an isosceles triangle OBD in which OB = OD = 18 cm and angle BOD = 1.5 radians. An arc of the circle, centre O and r...

A function f is such that f(x) = 4x³ + 4x² + ax + b. It is given that 2x – 1 is a factor of both f(x) and f'(x).

Answer only one of the following two alternatives. EITHER A curve is such that y = 5x² / (1 + x²)

Solve the equation |7x + 5| = |3x – 13|.

The total surface area, A cm², of a solid cylinder with radius r cm and height 5 cm is given by A = 2πr² + 10πr. Given that r is increasing at a ra...

Solve the inequality 4x(4 – x) > 7.

Find the coefficient of x⁵ in the expansion of (2 – x)⁸.

A 4-digit number is formed by using four of the six digits 2, 3, 4, 5, 6 and 8; no digit may be used more than once in any number. How many differe...

The variables x and y are related in such a way that when lg y is plotted against lg x a straight line graph is obtained as shown in the diagram. T...

A plane, whose speed in still air is 420kmh⁻¹, travels directly from A to B, a distance of 1000km. The bearing of B from A is 230° and there is a w...

The acceleration, a ms⁻², of a particle, t s after passing through a fixed point O, is given by a = 4 – 2t, for t > 0. The particle, which moves in...

Answer only one of the following alternatives. EITHER The point A(0, 10) lies on the curve for which dy/dx = e⁻ˣ/⁴. The point B, with x-coordinate ...

Answer only one of the following alternatives. OR The diagram shows part of the curve y = 8 + e⁻ˣ/³ crossing the y-axis at P. The normal to the cur...

The diagram shows the graph of y = a sin(bx) + c for 0 ≤ x ≤ 2π, where a, b and c are positive integers.

Find the set of values of k for which the curve y = (k + 1)x² – 3x + (k + 1) lies below the x-axis.

Show that (1 + sin θ / cos θ) + (cos θ / (1 + sin θ)) = 2 sec θ.

The sets A and B are such that A = {x: cos x = 1/2, 0° ≤ x ≤ 620°}, B = {x: tan x = √3, 0° ≤ x ≤ 620°}.

The function f(x) = ax³ + 4x² + bx – 2, where a and b are constants, is such that 2x – 1 is a factor. Given that the remainder when f(x) is divided...

Solutions to this question by accurate drawing will not be accepted. The points A(-3, 2) and B(1, 4) are vertices of an isosceles triangle ABC, whe...

Find the set of values of x for which x² < 6 – 5x.

Do not use a calculator in this question. Express (4√5-2)² / (√5-1) in the form p√5 + q, where p and q are integers.

Given that logₚ X = 5 and logₚ Y = 2, find

Solve the simultaneous equations 4ˣ / 256ʸ = 1024, 3²ˣ × 9ʸ = 243.

The diagram shows a box in the shape of a cuboid with a square cross-section of side x cm. The volume of the box is 3500cm³. Four pieces of tape ar...

The table shows experimental values of two variables x and y. | x | 2 | 4 | 6 | 8 | |---|---|---|---|---| | y | 9.6 | 38.4 | 105 | 232 | It is know...

The diagram shows a river with parallel banks. The river is 40m wide and is flowing with a speed of 1.8 ms⁻¹. A canoe travels in a straight line fr...

The diagram shows a circle with centre O and a chord AB. The radius of the circle is 12 cm and angle AOB is 1.4 radians. [Figure showing a circle w...

The diagram shows part of the curve y = e^(x/3). The tangent to the curve at P (9, e³) meets the x-axis at Q. [Figure showing a curve y=e^(x/3) and...

Find the coordinates of the stationary point on the curve y = x² + 16/x.

On the axes below, sketch the curve y = 3 cos 2x-1 for 0° ≤ x ≤ 180°.

A curve is such that dy/dx = 2/√x+3 for x >-3. The curve passes through the point (6, 10).