GATE questions with solutions

Q1. If ‘→’ denotes increasing order of intensity, then the meaning of the words [simmer → seethe → smolder] is analogous to [break → raze → ________]. Which one of the given options is appropriate to fill the blank?

1. obfuscate
2. obliterate
3. fracture
4. fissure

Answer: obliterate

The correct option 'obliterate' fits the analogy because it represents a more intense or complete form of destruction, similar to how 'seethe' is a more intense state than 'simmer' and 'smolder' represents a lingering state. Each pair reflects a progression in intensity.

Q2. Which one of the given options is a possible value of x in the following sequence? 3, 7, 15, x, 63, 127, 255

1. 35
2. 40
3. 45
4. 31

Answer: 31

The sequence follows a pattern where each term is one less than a power of 2: 3 (2² - 1), 7 (2³ - 1), 15 (2⁴ - 1), 31 (2⁵ - 1), 63 (2⁶ - 1), 127 (2⁷ - 1), and 255 (2⁸ - 1). Thus, 31 is the only option that fits this pattern.

Q3. On a given day, how many times will the second-hand and the minute-hand of a clock cross each other during the clock time 12:05:00 hours to 12:55:00 hours?

1. 51
2. 49
3. 50
4. 55

Answer: 49

The second-hand and minute-hand of a clock cross each other 49 times in an hour because they align approximately every 1 minute and 5.45 seconds, resulting in 49 crossings within the 50-minute span from 12:05 to 12:55.

Q4. In the given text, the blanks are numbered (i)–(iv). Select the best match for all the blanks. From the ancient Athenian arena to the modern Olympic stadiums, athletics (i) the potential for a spectacle. The crowd (ii) with bated breath as the Olympian artist twists his body, stretching the javelin behind him. Twelve strides in, he begins to cross-step. Six cross-steps (iii) in an abrupt stop on his left foot. As his body (iv) like a door turning on a hinge, the javelin is launched skyward at a precise angle.

1. (i) hold (ii) waits (iii) culminates (iv) pivot
2. (i) holds (ii) wait (iii) culminates (iv) pivot
3. (i) hold (ii) wait (iii) culminate (iv) pivots
4. (i) holds (ii) waits (iii) culminate (iv) pivots

Answer: (i) holds (ii) waits (iii) culminate (iv) pivots

The correct option uses the singular verb "holds" to agree with the singular subject "athletics," while "waits" matches the singular subject "the crowd." The plural form "culminate" fits with the plural subject "six cross-steps," and "pivots" correctly aligns with the singular subject "body."

Q5. Three distinct sets of indistinguishable twins are to be seated at a circular table that has 8 identical chairs. Unique seating arrangements are defined by the relative positions of the people. How many unique seating arrangements are possible such that each person is sitting next to their twin?

1. 12
2. 14
3. 10
4. 28

Answer: 12

To find the unique seating arrangements, we can treat each pair of twins as a single unit or block. With three pairs, we have three blocks to arrange around the circular table, which can be done in (3-1)! = 2! = 2 ways. Each pair can be arranged internally in 2 ways, leading to a total of 2 * 2 * 2 = 8 arrangements. Thus, the total unique arrangements are 2 * 8 = 16, but since we need to account for indistinguishable twins, we divide by 2 for the pairs, resulting in 12 unique arrangements.

Q6. During a half-moon phase, the Earth-Moon-Sun form a right triangle. If the Moon-Earth-Sun angle at this half-moon phase is measured to be 89.85°, the ratio of the Earth-Sun and Earth-Moon distances is closest to

1. 328
2. 382
3. 238
4. 283

Answer: 382

In a right triangle formed by the Earth, Moon, and Sun during the half-moon phase, the angle of 89.85° indicates that the Earth-Sun distance is significantly larger than the Earth-Moon distance. Using the sine function, the ratio of the distances can be calculated, leading to the conclusion that the closest ratio is approximately 382.

Q7. The smallest positive root of the equation x⁵ - 5x⁴ - 10x³ + 50x² + 9x - 45 = 0 lies in the range

1. 0 < x ≤ 2
2. 2 < x ≤ 4
3. 6 ≤ x ≤ 8
4. 10 ≤ x ≤ 100

Answer: 0 < x ≤ 2

The correct option is right because evaluating the polynomial at various points shows that it changes sign between 0 and 2, indicating a root exists in that interval. Specifically, the function is negative at 0 and positive at 2, confirming the presence of a root in the specified range.

Q8. The second-order differential equation in an unknown function u: u(x,y) is defined as ∂²u/∂x² = 2. Assuming g: g(x), f: f(y), and h: h(y), the general solution of the above differential equation is

1. u = x² + f(y) + g(x)
2. u = x² + x f(y) + h(y)
3. u = x² + x f(y) + g(x)
4. u = x² + f(y) + y g(x)

Answer: u = x² + x f(y) + h(y)

Integrating d2u/dx2 = 2 twice with respect to x gives u = x^2 + x*f(y) + h(y), where the two integration 'constants' are arbitrary functions of y. This is option idx 1; the stored idx 3 with y*g(x) is incorrect.

Q9. The probability that a student passes only in Mathematics is 1/3. The probability that the student passes only in English is 4/9. The probability that the student passes in both of these subjects is 1/6. The probability that the student will pass in at least one of these two subjects is

1. 17/18
2. 11/18
3. 14/18
4. 1/18

Answer: 17/18

P(at least one) = P(only Math) + P(only English) + P(both) = 1/3 + 4/9 + 1/6 = 6/18 + 8/18 + 3/18 = 17/18 (option 0), not the stored 11/18.

Q10. The three-dimensional state of stress at a point is given by σ = [[10, 0, 0],[0, 40, 0],[0, 0, 0]] MPa. The maximum shear stress at the point is

1. 20 MPa
2. 15 MPa
3. 5 MPa
4. 25 MPa

Answer: 20 MPa

The matrix is diagonal, so principal stresses are 40, 10 and 0 MPa. Maximum shear stress = (sigma_max - sigma_min)/2 = (40 - 0)/2 = 20 MPa, not 25 MPa.

Q11. A 2 m wide strip footing is founded at a depth of 1.5 m below the ground level in a homogeneous pure clay bed. The clay bed has unit cohesion of 40 kPa. Due to seasonal fluctuations of water table from peak summer to peak monsoon period, the net ultimate bearing capacity of the footing, as per Terzaghi’s theory, will

1. remain the same
2. decrease
3. increase
4. become zero

Answer: remain the same

The net ultimate bearing capacity of the footing remains the same because it is determined by the soil's cohesion and the depth of the footing, which are not affected by seasonal fluctuations in the water table in a homogeneous clay bed.

Q12. Consider the statements P and Q. P: Soil particles formed by mechanical weathering, and close to their origin are generally subrounded. Q: A activity of the clay physically signifies its swell potential. Which one of the following options is CORRECT?

1. Both P and Q are TRUE
2. P is TRUE and Q is FALSE
3. Both P and Q are FALSE
4. P is FALSE and Q is TRUE

Answer: P is FALSE and Q is TRUE

Statement P is incorrect because soil particles formed by mechanical weathering are typically angular rather than subrounded, as they have not undergone significant transport. Statement Q is correct as the physical activity of clay, such as its ability to swell, is a key indicator of its swell potential.