GATE CS 2024 Set 1

In the first group, each word is a stronger degree of the same idea: dry, then arid (very dry), then parched (extremely dry, as if scorched).

The second group is about eating less. A diet limits food, a fast gives up food for a while, and the next, stronger step is to starve: to go without food until one suffers from it.

Reject and deny are not degrees of going without food, and feast is the opposite idea.

Answer: A

“Proportional” means \(x + y = k(x - y)\) for some fixed constant \(k\). Rearranging,

\[x + y = kx - ky \;\Rightarrow\; y(k + 1) = x(k - 1) \;\Rightarrow\; \frac{x}{y} = \frac{k + 1}{k - 1}.\]

The right-hand side involves only the constant \(k\), so \(\frac{x}{y}\) is the same for every pair that satisfies the condition: it is a constant. (Here \(k \neq \pm 1\), because \(k = 1\) would force \(y = 0\) and \(k = -1\) would force \(x = 0\).)

Answer: D

The median is the middle value once the data are sorted:

9, 10, 11, 11, 13, 14, 15, 17, 18, 69

There are ten values, an even count, so the median is the average of the 5th and 6th values: \(\frac{13 + 14}{2} = 13.5\).

The large value 69 drags the mean up to \(\frac{187}{10} = 18.7\) (option D), but the median depends only on the order of the values, so the outlier does not move it. Option C, 11, is the mode.

Answer: A

Take the coin counts as 5, 3 and 13; multiplying all three by the same factor would not change the percentage. The money held in each kind of coin is:

• ₹1 coins: 5 × ₹1 = ₹5
• ₹5 coins: 3 × ₹5 = ₹15
• ₹10 coins: 13 × ₹10 = ₹130

The total is ₹150, so the share in ₹5 coins is \(\frac{15}{150} \times 100 = 10\%\).

Option B, \(14\frac{2}{7}\%\), is the share of ₹5 coins by count (3 out of 21 coins), not by value.

Answer: C

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

Each blank completes a fixed English expression:

• (i) to keep one’s head down is to stay focused and avoid distractions, so down.
• (ii) a batter goes to the crease by heading out to bat, so out.
• (iii) to have a head for something is to have a natural talent for it, so a head for batting.
• (iv) a cool head on one’s shoulders means calm good sense, so on.

Only option C gives down, out, for, on. The other options use phrases such as “a head on batting” or “a cool head for his shoulders”, which are not idiomatic.

Answer: C

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 fold halves the sheet’s current long side, because the fold line is perpendicular to the long edge of the sheet as it is at that moment:

• Fold 1: 20 × 8 becomes 10 × 8.
• Fold 2: the long edge is now 10 cm, so 10 × 8 becomes 5 × 8.
• Fold 3: the long edge is now 8 cm, so 5 × 8 becomes 5 × 4.

The final 5 cm × 4 cm sheet has perimeter \(2(5 + 4) = 18\) cm. Halving the 20 cm side all three times would give 2.5 × 8 and a perimeter of 21 cm (option D), but by the third fold that side is no longer the long edge.

Answer: A

For AB to be a line of symmetry, every square needs a mirror image on the other side of AB, at the same distance from the line.

• Left group: the square above the line needs its image in the empty cell directly below it; the square just below the line on the left needs an image just above the line; the square two rows down needs an image two rows up. That is 3 squares.
• Right group: all three squares lie below the line, so each needs an image above it: 3 more squares.

None of these six cells is already filled, so the least number is \(3 + 3 = 6\).

Answer: A