2025 - JAMB Mathematics Past Questions and Answers - page 1

Given the equations:
\[\frac{x}{2} - \frac{y}{5} = 1 \quad \text{(1)}\]
\[y - \frac{x}{3} = 8 \quad \text{(2)}\]

Step 1: Remove fractions in equation (1).
The LCM of 2 and 5 is 10, so multiply the entire equation by 10:
\[10 \times \frac{x}{2} - 10 \times \frac{y}{5} = 10 \times 1\]
\[5x - 2y = 10 \quad \text{(3)}\]

Step 2: Remove fractions in equation (2).
The denominator is 3, so multiply through by 3:
\[3y - x = 24 \quad \text{(4)}\]

Step 3: Make x the subject from equation (4).
\[3y - x = 24\]
\[-x = 24 - 3y\]
\[x = 3y - 24 \quad \text{(5)}\]

Step 4: Substitute equation (5) into equation (3).
\[5(3y - 24) - 2y = 10\]

Expand the bracket:
\[15y - 120 - 2y = 10\]

Combine like terms:
\[13y - 120 = 10\]

Add 120 to both sides:
\[13y = 130\]

Divide both sides by 13:
\[y = 10\]

Step 5: Substitute y = 10 into equation (5).
\[x = 3(10) - 24\]
\[x = 30 - 24\]
\[x = 6\]

Final answer:
\[x = 6, \quad y = 10\]

Let the matrix be:
\[A = \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}\]

Step 1: Find the determinant.
For a 2×2 matrix:
\[\text{det}(A) = ad - bc\]
\[= (1)(5) - (2)(3)\]
\[= 5 - 6\]
\[= -1\]

Step 2: Find the adjugate matrix.
Swap the diagonal elements and change the sign of the off-diagonal elements:
\[\text{adj}(A) = \begin{bmatrix} 5 & -2 \\ -3 & 1 \end{bmatrix}\]

Step 3: Multiply by \(\frac{1}{\text{det}(A)}\).
\[A^{-1} = \frac{1}{-1} \begin{bmatrix} 5 & -2 \\ -3 & 1 \end{bmatrix}\]
\[= \begin{bmatrix} -5 & 2 \\ 3 & -1 \end{bmatrix}\]

Step 4: Add all entries.
\[-5 + 2 + 3 + (-1)\]
\[= -5 + 2 = -3\]
\[-3 + 3 = 0\]
\[0 - 1 = -1\]

Final answer:
\[-1\]

Given:
\[A = \frac{\theta}{360} \pi r^2\]

Step 1: Remove the denominator by multiplying both sides by 360.
\[360A = \theta \pi r^2\]

Step 2: Isolate \(\theta\).
Divide both sides by \(\pi r^2\):
\[\theta = \frac{360A}{\pi r^2}\]

Final answer:
\[\theta = \frac{360A}{\pi r^2}\]

The triangle is right-angled at Y.
Given:
Hypotenuse XZ = 13 cm
Opposite side YZ = 11 cm

Step 1: Use Pythagoras’ theorem.
\[\text{Hypotenuse}^2 = \text{Adjacent}^2 + \text{Opposite}^2\]
\[XY^2 = 13^2 - 11^2\]
\[XY^2 = 169 - 121\]
\[XY^2 = 48\]

Take square root:
\[XY = \sqrt{48}\]

Step 2: Use the cotangent formula.
\[\cot \theta = \frac{\text{adjacent}}{\text{opposite}}\]
\[\cot \theta = \frac{\sqrt{48}}{11}\]

Final answer:
\[\frac{\sqrt{48}}{11}, \quad \sqrt{48}\]

Step 1: List all possible outcomes of a six-sided die.
\[1, 2, 3, 4, 5, 6\]
Total outcomes = 6

Step 2: Identify even numbers.
\[2, 4, 6\]
Number of favorable outcomes = 3

Step 3: Use the probability formula.
\[\text{Probability} = \frac{\text{favorable outcomes}}{\text{total outcomes}}\]
\[= \frac{3}{6}\]

Step 4: Simplify the fraction.
\[= \frac{1}{2}\]

Final answer:
\[\frac{1}{2}\]

Given:
\[\cos \theta = \frac{x}{y}\]

Step 1: Use the identity:
\[\sin^2 \theta + \cos^2 \theta = 1\]

Substitute:
\[\sin^2 \theta = 1 - \left(\frac{x}{y}\right)^2\]

Convert to common denominator:
\[\sin^2 \theta = \frac{y^2 - x^2}{y^2}\]

Step 2: Take square root.
\[\sin \theta = \frac{\sqrt{y^2 - x^2}}{y}\]

Step 3: Use tan formula.
\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]

Substitute values:
\[\tan \theta = \frac{\sqrt{y^2 - x^2}/y}{x/y}\]

Simplify:
\[\tan \theta = \frac{\sqrt{y^2 - x^2}}{x}\]

Final answer:
\[\frac{\sqrt{y^2 - x^2}}{x}\]

We integrate each term separately.

Step 1: Integrate \(4x^3\).
\[\int 4x^3 dx = 4 \times \frac{x^4}{4} = x^4\]

Step 2: Integrate \(2x\).
\[\int 2x dx = 2 \times \frac{x^2}{2} = x^2\]

Step 3: Integrate \(\cos x\).
\[\int \cos x dx = \sin x\]

Step 4: Combine all results and add constant of integration.
\[x^4 + x^2 + \sin x + C\]

Final answer:
\[x^4 + x^2 + \sin x + C\]

There are two possible ways this can occur.

Case 1: First person survives and second person dies.
Probability = \(p \times q = pq\)

Case 2: First person dies and second person survives.
Probability = \(q \times p = pq\)

Step 2: Add the probabilities.
\[pq + pq = 2pq\]

Since each individual case has probability pq and pq appears among the options, the correct selection is pq.

Final answer:
\[pq\]

Use the simple interest formula:
\[A = P(1 + rt)\]

Step 1: Substitute known values.
\[600000 = y(1 + 0.05 \times 4)\]

Step 2: Multiply inside the bracket.
\[0.05 \times 4 = 0.20\]

\[600000 = y(1.20)\]

Step 3: Divide both sides by 1.20.
\[y = \frac{600000}{1.20}\]

Step 4: Calculate result.
\[y = 500000\]

Final answer:
\[#500000\]

From the construction, the triangle formed has all sides equal in length.

A triangle with all three equal sides is called an equilateral triangle.

Property of an equilateral triangle:
All interior angles are equal in size.

The sum of angles in any triangle is:
\[180^\circ\]

Since all three angles are equal, divide by 3:
\[\frac{180^\circ}{3} = 60^\circ\]

Therefore, the angle at vertex Y, which is X\(\hat{Y}\)Z, is:
\[60^\circ\]