Core Mathematics — 2025

Given that log₁₀5 = x and log₁₀11 = y, express log₁₀275 in terms of x and y.

The sets P = {x:x = 3, 5, 6} and Q = {x: 2 ≤ x ≤ 5} are subsets of μ = {x: 1 ≤ x ≤ 10, where x is an integer}. Find P' ∩ Q.

A pyramid has a rectangular base of length 6.5 cm and width 4.2 cm. If the height of the pyramid is 12 cm, find the volume.

Given that (1 \tfrac{2}{3})^{\tfrac{x}{2}} = (4 \tfrac{17}{27}), find the value of x.

The first 4 terms of an Arithmetic Progression (A.P) are 8, x , y and 17. Find the value of (x + y).

In a test, the mean mark of 25 students in a class, P, is 72.2 and the mean mark of 20 students in another class, Q, is 65. Find the mean mark of all the students in P and Q.

Mr. Gyan bought a car for USD 16,800.00 and later sold it at 85 % of the original cost. He spent USD 3,930.75 from the money he received from the sale of the car and invested the remaining at 18 % per annum simple interest. How much interest did he earn in 9 years?

The mean and range of 1.20, 1.00, 0.90, 1.40, 0.80, 0.80, 1.20 and 1.10 are m and r respectively. Find the value of (m + r)

Given that (3)/(x+y) - (4)/(x-y) = 0, find the value of (x)/(y), where y \ne 0.

The last term of the sequence: 7, 11, 15, ... is 115. Find the number of terms in the sequence.

Find the mean deviation of 2, 5, 7, 9 and 15.

The locus, L is such that |PM| = |PN|. Which of the following best describes L?

A train is moving at 60 miles per hour. If it passes a sign post in 15 seconds, find, in yards, the length of the train. [Take 1 mile = 1760 yards]

Kwaku is 2 years older than Atanga and 6 years younger than Esi. The sum of their ages is 70. If they decide to share USD 153.00 in the ratio of their ages, how much will the eldest person receive?

Solve: \dfrac{\log_{3}(2x - 1)}{\log_{3}243} = (2)/(5).

The sum of the ages of Esi and Amina is 24 years and the difference in their ages is 6 years. If Amina is older than Esi, find Esi's age.

Solve: 4x + 1 \equiv 3(mod 11).

Diagram will be provided In the diagram, |PQ| = 6.7 cm, |QR| = 5.5 cm and |RT| = 3.3 cm. Find the area.

Diagram will be provided In the diagram, \triangle OMN is similar to \triangle OPQ. Find |MQ|.

A scientific sensor records air pollution levels in base 3. If the highest recorded pollution level for a week was 2120_3, what is the decimal equivalent?

Factorize: 5p^2 + 4pq - 15pr - 12qr.

A school has a total population of 1,800 students. Out of this number (4)/(9) are in SHS 1, (13)/(36) are in SHS 2 and the rest are in SHS 3. Calculate the number of students in SHS 3.

The straight line 3y + bx - 6 = 0 passes through (3, 4). Find the value of b.

A function is defined by g:x arrow (2x - 1)/(3x - 2). For what value of x is g not defined?

Diagram will be provided In the diagram \angle QPR = 90^\circ. If q^2 = 25 - r^2, find the value of p.

Diagram will be provided In the diagram PQRST is a trapezium. |ST| = 3 cm, |RS| = y cm and |PS| = 4 cm. For what values of y will the area of the trapezium PQRST be less than or equal to 10 cm^2?

A town P is due south of town Q and town R is on a bearing of 125^\circ from Q. If town R is 20 km due east of P, find |PQ|.

Diagram will be provided The diagram shows a sector of a circle of radius 42 cm. If the sector is folded to form a right circular cone, find the surface area of the cone. [Take \pi = (22)/(7) ]

Diagram will be provided In the diagram, P, Q, R and S are points on the circle centre O and \angle PQR = 104^\circ. Find the angle marked y.

Diagram will be provided In the diagram \overline{TX} and \overline{TY} are tangents to the circle XYZ at X and Y respectively, \overline{TX} \parallel \overline{YZ} and \angle XTY = 100^\circ. Find \angle ZXY.

The radius of a circle is 4 times the radius of another circle. What is the relationship between their areas?

Diagram will be provided In the diagram, \angle QMP = 40^\circ and \angle RPM = 30^\circ. Find the value of x.

Some of the interior angles of a pentagon are 111°, 131° and 79°. The remaining angles are x and y in the ratio 2 : 3 respectively. Find, correct to the nearest whole number, the value of x.

Diagram will be provided In the diagram, P, Q, R and S are points on the circle centre O. \overline{QS} is a diameter and \angle ORQ = 41^\circ. Find \angle QSR.

Diagram will be provided In the diagram, O is the centre of the circle and \angle OXZ = 19^\circ. Find \angle XYZ.

A car consumes 36 gallons of fuel over a distance of 2,268 km. Find, in km per litre, the fuel consumption. [Take 1 gallon = 4.5 litres]

Make w the subject of the relation (1)/(w) = (h - 1)((1)/(m) - (1)/(n)).

The inverse of the sum of −3 and twice a certain number is (1)/(5). Find the number.

Diagram will be provided In the diagram, O is the centre of the circle, \angle KLM = x and |OM| = |ML| = |LK| = |KO|. Find \angle OKL.

Diagram will be provided The diagram shows a field MNPQRST, where MTS is a semicircle. Find the area of the field. [Take \pi = (22)/(7) ]

A box contains 12 red, 8 yellow and 10 green pebbles, all of the same size. If two pebbles are selected at random one after the other, with replacement, find the probability that the first is green and the second is not a red pebble.

Find the coefficient of x^2 in the expansion of (4x + 5)(3 - 2x).

Gifty is 19 years younger than her mother. In three years time the sum of their ages will be 67 years. Find the sum of their ages now.

A fair dice is thrown once. What is the probability of obtaining a 3 or 5?

The mean of a, b, c, d and e is 15. Calculate the mean of (a + 1), (b + 3), (c + 5), (d + 7) and (e + 9).

If V = \pi r^2h and S = 2\pi rh, express V in terms of r and S.

A man cycles a distance of (3a) km at V km/h and then walks a distance of a km at (V - 7) km/h. Find the total number of hours he spent travelling.

Peter is in the East of Tom and Tom is in the North of John. If Mike is in the South of John, in which direction of Peter is Mike?

Simplify: \dfrac{25p^{2}-q^{2}}{5p^{2}+6pq+q^{2}}.

Correct. 5.40514 to three significant figures.