Mathematics 2015 Jamb Examination

We can express the first equation in terms of base 3: \[ (3^3)^x \div (3^4)^{x+2y} = 3^2 \] \[ 3^{3x} \div 3^{4x+8y} = 3^2 \] Using the law of indices \( a^m \div a^n = a^{m-n} \): \[ 3^{3x - (4x + 8y)} = 3^2 \] Equating the powers: \[ 3x - 4x - 8y = 2 \] \[ -x - 8y = 2 \text{ or } x + 8y = -2 \quad \text{---(i)} \] From the second equation: \[ x + 4y = 0 \Rightarrow x = -4y \quad \text{---(ii)} \] Substitute (ii) into (i): \[ -4y + 8y = -2 \Rightarrow 4y = -2 \Rightarrow y = -1/2 \] Using \( x = -4y \): \[ x = -4(-1/2) = 2 \] Thus, \( x = 2, y = -1/2 \).

To simplify the product of surds, we multiply the numbers under the square root sign: \[ \sqrt{30} \times \sqrt{40} = \sqrt{30 \times 40} = \sqrt{1200} \] Now, we look for the largest perfect square factor of 1200, which is 400: \[ \sqrt{400 \times 3} \] \[ = \sqrt{400} \times \sqrt{3} \] \[ = 20\sqrt{3} \]

Rearrange the equation: \[ \sqrt{2x + 2} = 1 + \sqrt{x} \] Square both sides: \[ 2x + 2 = (1 + \sqrt{x})^2 \] \[ 2x + 2 = 1 + x + 2\sqrt{x} \] \[ x + 1 = 2\sqrt{x} \] Square both sides again: \[ (x + 1)^2 = (2\sqrt{x})^2 \] \[ x^2 + 2x + 1 = 4x \] \[ x^2 - 2x + 1 = 0 \] Factorize the quadratic: \[ (x - 1)^2 = 0 \Rightarrow x = 1 \text{ (twice)} \]

Let the numbers be \( x \) and \( y \). \[ x + y = 5 \Rightarrow y = 5 - x \] \[ xy = -14 \] Substitute \( y \): \[ x(5 - x) = -14 \] \[ 5x - x^2 = -14 \] \[ x^2 - 5x - 14 = 0 \] Factorize: \[ (x - 7)(x + 2) = 0 \] \[ x = 7 \text{ or } x = -2 \] When \( x = 7, y = -2 \); when \( x = -2, y = 7 \). The numbers are 7 and -2.

Using indices with base 5: \[ 5^{x-2} = (5^{-3})^{x+3} \] \[ 5^{x-2} = 5^{-3(x+3)} \] Equating the exponents: \[ x - 2 = -3x - 9 \] \[ x + 3x = -9 + 2 \] \[ 4x = -7 \Rightarrow x = -7/4 \]

The exterior angle of a regular polygon is calculated as \( 180^\circ - \text{interior angle} \): \[ \text{Exterior angle} = 180^\circ - 120^\circ = 60^\circ \] Since the sum of exterior angles is always \( 360^\circ \), the number of sides \( n \) is: \[ n = \dfrac{360^\circ}{\text{Exterior angle}} = \dfrac{360}{60} = 6 \] A regular polygon with interior angles of 120° is a hexagon with 6 sides.

Duty is calculated as a percentage of the total bill: \[ \text{Duty} = 25\% \text{ of } 80 \] \[ \text{Duty} = \dfrac{25}{100} \times 80 \] \[ \text{Duty} = 0.25 \times 80 = 20 \] The duty to be added is ₦20.

Multiply the two numbers: \[ 2.04 \times 3.7 = 7.548 \] Rounding to 2 decimal places, we look at the third decimal digit (8). Since it is 5 or greater, we round up the second decimal digit: \[ 7.548 \approx 7.55 \]

Let the Cost Price (C.P.) be 100. Markup of 30% on cost: \[ \text{Marked Price (M.P.)} = 100 + (30\% \text{ of } 100) = 130 \] A discount of 20% is placed on the M.P.: \[ \text{Discount} = 20\% \text{ of } 130 = 0.2 \times 130 = 26 \] \[ \text{Selling Price (S.P.)} = 130 - 26 = 104 \] Profit = \( S.P. - C.P. = 104 - 100 = 4 \). Profit percentage = \( \dfrac{4}{100} \times 100 = 4\% \).

To rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator \( (2 + \sqrt{3}) \): \[ \dfrac{5}{2 - \sqrt{3}} \times \dfrac{2 + \sqrt{3}}{2 + \sqrt{3}} \] \[ = \dfrac{5(2 + \sqrt{3})}{2^2 - (\sqrt{3})^2} \] \[ = \dfrac{5(2 + \sqrt{3})}{4 - 3} \] \[ = \dfrac{5(2 + \sqrt{3})}{1} = 5(2 + \sqrt{3}) \]