2024 - JAMB Mathematics Past Questions and Answers - page 7

We are given the equation: \(25^{1 - x} \times 5^{x + 2} \div (\frac{1}{125})^{x} = 625^{-1}\). Start by expressing all numbers in terms of base 5:

• \(25 = 5^2\), so \(25^{1 - x} = (5^2)^{1 - x} = 5^{2(1 - x)} = 5^{2 - 2x}\).
• \(125 = 5^3\), so \((\frac{1}{125})^x = (5^{-3})^x = 5^{-3x}\).
• \(625 = 5^4\), so \(625^{-1} = (5^4)^{-1} = 5^{-4}\).

Now, the equation becomes: \(5^{2 - 2x} \times 5^{x + 2} \div 5^{-3x} = 5^{-4}\).

Using the laws of exponents, combine the terms: \(5^{2 - 2x} \times 5^{x + 2} \div 5^{-3x} = 5^{2 - 2x + x + 2 + 3x} = 5^{4 + 2x}\).

Set the exponents equal: \(4 + 2x = -4\), so solving for x gives: \(2x = -8\), hence \(x = -4\).

The area of a circle is given by the formula: \(A = \pi r^2\), where \(r\) is the radius of the circle. Given that the radius is 4 cm and \(\pi = 3.142\), the area is:

\(A = 3.142 \times 4^2 = 3.142 \times 16 = 50.272\) cm². The closest option is 50.3 cm².

We are asked to integrate \(\frac{2x^3 + 2x}{x}\) with respect to x. First, simplify the expression: \(\frac{2x^3 + 2x}{x} = 2x^2 + 2\).

Now, integrate term by term:

• \(\int 2x^2 dx = \frac{2x^3}{3}\),
• \(\int 2 dx = 2x\).

The result of the integration is: \(\frac{2x^3}{3} + 2x + k\), where k is the constant of integration.

We are given the equation \(8x^{-2} = \frac{2}{25}\). First, rewrite the equation as \(8/x^2 = 2/25\), then cross-multiply to get:

\(8 \times 25 = 2 \times x^2\), or \(200 = 2x^2\).

Divide both sides by 2: \(100 = x^2\), and taking the square root of both sides gives \(x = 10\).