Mathematics 2020 Jamb Examination

To evaluate \(\log_{10}4.5\), we first express 4.5 as a fraction:\[4.5 = \frac{9}{2} = \frac{3^2}{2}\]Using the laws of logarithms:\[\log_{10}\left(\frac{3^2}{2}\right) = \log_{10}(3^2) - \log_{10}2\]\[= 2\log_{10}3 - \log_{10}2\]Substitute the given values:\[= 2(0.4771) - 0.3010\]\[= 0.9542 - 0.3010 = 0.6532\]

We begin by factorising the numerator and the denominator. The numerator \(324 - 4x^2\) can have 4 factored out:\[4(81 - x^2)\]This is a difference of two squares:\[4(9 - x)(9 + x)\]The denominator \(2x + 18\) can have 2 factored out:\[2(x + 9)\]Now, substitute these back into the fraction:\[\frac{4(9 - x)(9 + x)}{2(x + 9)}\]The term \((9 + x)\) is the same as \((x + 9)\), so they cancel out:\[\frac{4(9 - x)}{2} = 2(9 - x)\]To match the options, we can factor out a negative sign from \((9 - x)\):\[2[-(x - 9)] = -2(x - 9)\]

To find the angle of the sector for meat, we first calculate the total mass of all ingredients:\[\text{Total mass} = 75 + 40 + 105 + 20 = 240\text{g}\]The angle of a sector in a pie chart is proportional to the total angle of a circle, which is 360^o:\[\text{Angle} = \frac{\text{Mass of meat}}{\text{Total mass}} \times 360^o\]\[\text{Angle} = \frac{105}{240} \times 360^o\]Dividing 360 and 240 by 120:\[\text{Angle} = \frac{105}{2} \times 3^o = \frac{315}{2}^o = 157.5^o\]

The perimeter of a sector consists of the two radii and the arc length \(L\):\[\text{Perimeter} = 2r + L\]First, calculate the arc length using \(L = \frac{\theta}{360} \times 2\pi r\):\[L = \frac{48}{360} \times 2 \times \frac{22}{7} \times 10.5\]\[L = \frac{2}{15} \times 2 \times 22 \times 1.5\]\[L = \frac{2}{15} \times 66 = 8.8\text{cm}\]Now add the two radii:\[\text{Perimeter} = 2(10.5) + 8.8 = 21 + 8.8 = 29.8\text{cm}\]

We evaluate each term separately using indices laws:
\(\frac{27}{5}\) remains as it is.
\(3^{-3} = \frac{1}{3^3} = \frac{1}{27}\)
\((\frac{1}{5})^{-1} = 5^1 = 5\)
Now multiply them together:\[\frac{27}{5} \times \frac{1}{27} \times 5\]The 27s cancel each other out, and the 5s cancel each other out:\[\frac{27}{27} \times \frac{5}{5} = 1 \times 1 = 1\]

First, find the total number of bottles in the crate:\[\text{Total} = 10 + 8 + 6 = 24\]The bottles that are "Not Coca-Cola" are Fanta and Sprite:\[\text{Not Coca-Cola} = 8 + 6 = 14\]The probability is the number of successful outcomes divided by the total number of outcomes:\[P(\text{Not Coca-Cola}) = \frac{14}{24}\]Simplifying the fraction by dividing both numerator and denominator by 2:\[\frac{14}{24} = \frac{7}{12}\]

Since PQRS is quadrilateral

2y + 2x + QPS = 360\(^o\)

i.e. (y + x) + QPS = 360\(^o\) 

QPS = 360\(^o\) - 2 (y + x)

But x + y + 68\(^o\) = 180\(^o\)

There; x + y = 180\(^o\) - 68\(^o\) = 112\(^o\)

QPS = 360 - 2(112\(^o\))

= 360\(^o\) - 224 = 136\(^o\)

The gradient \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Using the points (-2, 0) and (0, -4):\[m = \frac{-4 - 0}{0 - (-2)}\]\[m = \frac{-4}{2} = -2\]

Parallel lines have the same gradient. First, find the gradient of the line \(7x + 5y = 12\) by putting it in the form \(y = mx + c\):\[5y = -7x + 12 \implies y = -\frac{7}{5}x + \frac{12}{5}\]The gradient \(m = -\frac{7}{5}\). Now use the point-slope form with point (5, 7):\[y - y_1 = m(x - x_1)\]\[y - 7 = -\frac{7}{5}(x - 5)\]Multiply through by 5:\[5(y - 7) = -7(x - 5)\]\[5y - 35 = -7x + 35\]Rearrange to the standard form:\[7x + 5y = 35 + 35\]\[7x + 5y = 70\]

The formula for Simple Interest (I) is:\[I = \frac{P \times R \times T}{100}\]Where \(P = 225\), \(R = 4\), \(I = 27\), and \(T = x\). Substituting the values into the formula:\[27 = \frac{225 \times 4 \times x}{100}\]\[27 = \frac{900 \times x}{100}\]\[27 = 9x\]\[x = \frac{27}{9} = 3\]Therefore, the time is 3 years.