GATE CS 2024 Set 2

Walk, jog and sprint are the same action at rising intensity. The blank therefore needs a word for being troubled that is stronger than “bothered” but milder than “daunted” (losing one’s nerve).

• fazed means disconcerted or thrown off balance, which sits between the two.
• phased (done in stages) only sounds like “fazed”; it says nothing about feelings.
• phrased means put into words, and fused means joined together.

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

Find the sentence that needs nothing before it, then follow the words that point back.

• S opens: it introduces the event, a chunk of the Antarctic Plate colliding with the Eurasian Plate.
• P follows: “This fortuitous geological event” is that collision, which pushed the rocks up.
• R comes next: “this massive edifice” is the raised rock mass from P, now cooling.
• Q closes: “Thus” draws the conclusion that the Himalayas are an ongoing event.

A and B begin with “Thus”, which cannot open a passage, and D places R before P, before the edifice has been formed.

Answer: C

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

Three cuts are enough: cut through the centre of the cube once parallel to each pair of opposite faces. Each cut halves every piece it passes through, giving \(2 \times 2 \times 2 = 8\) identical small cubes.

Fewer cuts cannot work. A straight cut splits each existing piece into at most two, so one cut gives at most 2 pieces and two cuts at most 4. At least 3 cuts are therefore needed, and 4, 7 and 8 are all more than the minimum.

Answer: A

The last row has no crosses, so each of its cells counts the cells beside it in row 4 plus the non-cross cells above it in row 3, which reads 3, X, 6, X.

• Cell 1: 3 and X above, one cell beside, so 2.
• Cell 2: 3, X and 6 above, two cells beside, so 4.
• Cell 3: X, 6 and X above, two cells beside, so 3.
• Cell 4: 6 and X above, one cell beside, so 2.

Sum \(= 2 + 4 + 3 + 2 = 11\). The printed numbers follow the same rule; for example, the 3 in row 3 counts the 5 diagonally above it and the two cells below it.

Answer: A