Quantitative Aptitude

Combine the right-hand side into one logarithm: \(\log p + \log q + 2\log 3 = \log(9pq)\). Since the logarithm is one-to-one, the two arguments must be equal:

\[p^{2} + q^{2} = 9pq.\]

Now use \(p^{4} + q^{4} = (p^{2} + q^{2})^{2} - 2p^{2}q^{2}\). Squaring the relation gives \((p^{2} + q^{2})^{2} = 81p^{2}q^{2}\), so \(p^{4} + q^{4} = 79p^{2}q^{2}\) and

\[\frac{p^{4} + q^{4}}{p^{2}q^{2}} = 79.\]

Option B, 81, comes from forgetting to subtract the cross term \(2p^{2}q^{2}\).

Answer: A

Joining the two 54 cm edges makes a tube 54 cm long whose circular end has a circumference of 4 cm, so its radius is \(r = \frac{4}{2\pi} = \frac{2}{\pi}\) cm and

\[V_{\text{tube}} = \pi r^{2}h = \pi \cdot \frac{4}{\pi^{2}} \cdot 54 = \frac{216}{\pi}\ \text{cm}^{3}.\]

The sheet’s area is \(54 \times 4 = 216\ \text{cm}^{2}\). A cube with surface area \(6a^{2} = 216\) has side \(a = 6\) cm and volume \(216\ \text{cm}^{3}\).

The ratio is \(\dfrac{216/\pi}{216} = \dfrac{1}{\pi}\).

Answer: A

The three kinds of fat together supply \(5\% + 20\% + 20\% = 45\%\) of the energy in the diet:

\[0.45 \times 2000 = 900\ \text{kcal}.\]

All three fats give 9 kcal per gram, so the person eats \(\frac{900}{9} = 100\) g of fat.

Option A, 44.4 g, counts only one of the 20% fat types (\(400 \div 9\)). Carbohydrates and proteins, at 4 kcal/g, play no part in the fat total.

Answer: C

Each attempt is one order in which the four elements are mixed, with all four used every time. The number of orders of 4 different elements is

\[4! = 4 \times 3 \times 2 \times 1 = 24.\]

The wizards work independently, so each one goes through all 24 orders separately. Adding the two wizards’ attempts (48) answers a different question: the total number of attempts, not the number each wizard makes.

Answer: A

Students who like at least one of the two: \(10000 - 1500 = 8500\).

Let \(c\) students like their core branches, so \(4c\) like other branches. The 500 who like both are counted in both groups, so they are subtracted once:

\[c + 4c - 500 = 8500 \implies 5c = 9000 \implies c = 1800.\]

Check: 7,200 like other branches, and \(1800 + 7200 - 500 = 8500\). Adding the 500 instead, as if those students formed a separate group, gives \(5c = 8000\), that is 1,600 (option C).

Answer: A

The right side is a single logarithm: \(\frac{1}{2}[\ln(x) + \ln(y)] = \ln\sqrt{xy}\). Since \(\ln\) is one-to-one,

\[\frac{x + y}{2} = \sqrt{xy} \implies x - 2\sqrt{xy} + y = 0 \implies (\sqrt{x} - \sqrt{y})^{2} = 0.\]

So \(x = y\), the only case in which the arithmetic mean equals the geometric mean, and \(\frac{x}{y} + \frac{y}{x} = 1 + 1 = 2\).

Options 1 and 1/2 could never occur: for positive \(x\) and \(y\), \(\frac{x}{y} + \frac{y}{x}\) is never less than 2.

Answer: C

Look at the gaps between terms: \(9 - 6 = 3\) and \(14 - 9 = 5\). They are consecutive odd numbers, so the next gaps should be 7, 9 and 11:

\[14 + 7 = 21,\quad 21 + 9 = 30,\quad 30 + 11 = 41.\]

This reproduces the given 30 and 41, so \(x = 21\). Equivalently, the terms are \(n^{2} + 5\) for \(n = 1, 2, \ldots, 6\).

The other values break the pattern: 25, 18 and 20 leave gaps of 11 and 5, 4 and 12, or 6 and 10 around \(x\).

Answer: B

Take a convenient selling price of 99 for each item.

• 10% profit: cost price \(= \frac{99}{1.1} = 90\).
• 10% loss: cost price \(= \frac{99}{0.9} = 110\).

The total cost is \(90 + 110 = 200\) and the total selling price is 198, so the person loses \(\frac{2}{200} \times 100 = 1\%\).

The profit and the loss do not cancel because the 10% loss is taken on the larger cost price. In general, equal selling prices with \(x\%\) profit and \(x\%\) loss always give a net loss of \(\frac{x^{2}}{100}\%\).

Answer: C

Add the renewable shares (Hydro, Solar and Wind) for each year:

• 2007: \(30\% + 5\% + 5\% = 40\%\)
• 2023: \(35\% + 20\% + 10\% = 65\%\)

The total generation is the same in both years, so the shares can be compared directly. The percentage increase is

\[\frac{65 - 40}{40} \times 100 = 62.5\%.\]

Option A (25%) is the rise in percentage points, not the increase relative to the 2007 share.

Answer: D