Question 1

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

Accepted 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.

Question 2

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?

Accepted 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.

Question 3

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

Accepted 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.

Question 4

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

Accepted 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.

Question 5

Rohit goes to a restaurant for lunch at about 1 PM. When he enters the restaurant, he notices that the hour and minute hands on the wall clock are exactly coinciding. After about an hour, when he leaves the restaurant, he notices that the clock hands are again exactly coinciding. How much

Accepted Answer

65 5/11. The hour and minute hands coincide once every 12/11 hours, i.e. 720/11 = 65 5/11 minutes. So the time Rohit spent is 65 5/11 minutes, not 66 5/13.

Question 6

a + a + a +... + a (n times) = a² b and b + b + b +... + b (m times) = ab², where a, b, n and m are natural numbers. What is the value of (m + m + m +... + m (n times))(n + n + n +... + n (m times))?

Accepted Answer

a⁴b⁴. The expressions given simplify to a²b and ab², respectively. When calculating (m times b) and (n times a), we find that multiplying these results leads to a⁴b⁴, as each term contributes to the overall exponentiation of a and b.

Question 7

A three-member committee has to be formed from a group of 9 people. How many such distinct committees can be formed?

Accepted Answer

84. The number of ways to form a committee of 3 members from 9 people is calculated using the combination formula, which is C(n, r) = (n!)/(r!(n-r)!). Here, n = 9 and r = 3, leading to C(9, 3) = (9!)/(3!(9-3)!) = 84.

Question 8

For non-negative integers, a, b, c, what would be the value of a + b + c if log a + log b + log c = 0?

Accepted Answer

3. The equation log a + log b + log c = 0 implies that a * b * c = 1, since the logarithm of a product is the sum of the logarithms. For non-negative integers, the only values that satisfy this condition are a = 1, b = 1, and c = 1, leading to a + b + c = 3.

Question 9

In manufacturing industries, loss is usually taken to be proportional to the square of the deviation from a target. If the loss is Rs. 4900 for a deviation of 7 units, what would be the loss in Rupees for a deviation of 4 units from the target?

Accepted Answer

1600. Loss is proportional to deviation squared: 4900 = c*7^2 gives c = 100. For a deviation of 4, loss = 100*4^2 = 1600 rupees. The stored answer 400 is wrong.

Question 10

A faulty wall clock is known to gain 15 minutes every 24 hours. It is synchronized to the correct time at 9 AM on 11th July. What will be the correct time to the nearest minute when the clock shows 2 PM on 15th July of the same year?

Accepted Answer

12:58 PM. The faulty clock gains 15 minutes every 24 hours, which means in the 4 days from 9 AM on July 11 to 2 PM on July 15, it gains a total of 1 hour (15 minutes x 4). Therefore, when the faulty clock shows 2 PM, the actual time is 1 hour earlier, resulting in 12:58 PM.

Question 11

Given that (log P)/(y-z)=(log Q)/(z-x)=(log R)/(x-y)=10 for x ≠ y ≠ z, what is the value of the product PQR?

Accepted Answer

1. The equations imply that log P = 10(y-z), log Q = 10(z-x), and log R = 10(x-y). Adding these gives log P + log Q + log R = 10[(y-z) + (z-x) + (x-y)] = 10(0) = 0, which means log(PQR) = 0, leading to PQR = 10⁰ = 1.

Question 12

Suresh wanted to lay a new carpet in his new mansion with an area of 70 × 55 sq. mts. However an area of 550 sq. mts. had to be left out for flower pots. If the cost of carpet is Rs. 50 per sq. mts., how much money (in Rs.) will be spent by Suresh for the carpet now?

Accepted Answer

Rs. 1,65,000. Carpet area = 70x55 = 3850 sq m, minus 550 for flower pots = 3300 sq m. At Rs 50/sq m the cost = 3300 x 50 = Rs 1,65,000.

Question 13

A retaining wall with measurements 30m × 12m × 6m was constructed with bricks of dimensions 8cm × 6cm × 6cm. If 60% of the wall consists of bricks, the number of bricks used for the construction is ______ lakhs.

Accepted Answer

45. Wall volume = 30*12*6 = 2160 m^3; bricks occupy 60% = 1296 m^3. One brick = 0.08*0.06*0.06 = 0.000288 m^3. Number of bricks = 1296/0.000288 = 4,500,000 = 45 lakhs (option 2), not 75.

Question 14

Population of state X increased by x% and the population of state Y increased by y% from 2001 to 2011. Assume that x is greater than y. Let P be the ratio of population of state X to state Y in a given year. The percentage increase in P from 2001 to 2011 is ________.

Accepted Answer

100(x−y) / (100+y). The correct option is derived from the formula for percentage change in ratios, which accounts for the different growth rates of the two populations. Since state X's population grows faster than state Y's, the change in their ratio reflects the difference in their percentage increases, adjusted for the initial size of state Y's population.

Question 15

An oil tank can be filled by pipe X in 5 hours and pipe Y in 4 hours, each pump working on its own. When the tank is full and the drainage hole is open, the oil is drained in 20 hours. If initially the tank was empty and someone started the two pumps together but left the drainage hole ope

Accepted Answer

2.50. Fill rates: X = 1/5, Y = 1/4 per hour; drain = 1/20 per hour. Net rate = 1/5 + 1/4 - 1/20 = (4+5-1)/20 = 8/20 = 2/5 tank per hour. Time = 1/(2/5) = 2.5 hours (index 2).

Question 16

Two identical cube shaped dice each with faces numbered 1 to 6 are rolled simultaneously. The probability that an even number is rolled out on each dice is:

Accepted Answer

1/4. Each die shows an even number (2,4,6) with probability 3/6 = 1/2. For both dice independently, the probability is 1/2 * 1/2 = 1/4.

Question 17

Both the numerator and the denominator of 3/4 are increased by a positive integer, x, and those of 15/17 are decreased by the same integer. This operation results in the same value for both the fractions. What is the value of x?

Accepted Answer

3. (3+x)(17-x) = (15-x)(4+x) gives 51+14x-x^2 = 60+11x-x^2, so 3x = 9 and x = 3. The stored answer 2 is incorrect.

Question 18

A survey of 450 students about their subjects of interest resulted in the following outcome.

• 150 students are interested in Mathematics.
• 200 students are interested in Physics.
• 175 students are interested in Chemistry.
• 50 students are interested in Mathematics and Physics.
• 60 st

Accepted Answer

45. To find the number of students interested in Humanities, we first calculate the total number of students interested in at least one of the subjects using the principle of inclusion-exclusion. After accounting for overlaps among the subjects, we find that 405 students are interested in Mathematics, Physics, or Chemistry, leaving 45 students who are interested in Humanities.

Question 19

In a partnership business the monthly investment by three friends for the first six months is in the ratio 3: 4: 5. After six months, they had to increase their monthly investments by 10%, 15% and 20%, respectively, of their initial monthly investment. The new investment ratio was kept con

Accepted Answer

63: 86: 110. The correct option reflects the total investments made by each partner over the year, accounting for both their initial investments and the increases after six months. By calculating the total contributions based on the given ratios and the percentage increases, we find that the final ratio of their shares in the profit aligns with 63: 86: 110.

Question 20

In how many ways can cells in a 3 × 3 grid be shaded, such that each row and each column have exactly one shaded cell? An example of one valid shading is shown.

Accepted Answer

6. The problem is equivalent to finding the number of permutations of 3 distinct objects, as each shaded cell must occupy a unique position in each row and column. This can be calculated as 3! (3 factorial), which equals 6, representing all possible arrangements of shaded cells.

Question 21

A remote village has exactly 1000 vehicles with sequential registration numbers starting from 1000. Out of the total vehicles, 30% are without pollution clearance certificate. Further, even- and odd-numbered vehicles are operated on even- and odd-numbered dates, respectively.

If 100 vehic

Accepted Answer

30. The pollution-certificate rate is 30% across all vehicles, and the even-date operating rule does not change that proportion within the even-numbered subset. For 100 randomly chosen vehicles, expected number without certificate = 0.30*100 = 30.

Question 22

If y = 5x² + 3, then the tangent at x = 0, y = 3

Accepted Answer

is parallel to the x-axis. The tangent line at a point on a curve is parallel to the x-axis when the slope of the tangent is zero. Since the derivative of the function at x = 0 is zero, the tangent line at that point is indeed horizontal, confirming it is parallel to the x-axis.

Question 23

A foundry has a fixed daily cost of Rs 50,000 whenever it operates and a variable cost of Rs 8000, where Q is the daily production in tonnes. What is the cost of production in Rs per tonne for a daily production of 100 tonnes?

Accepted Answer

580. To find the cost of production per tonne, first calculate the total cost by adding the fixed cost (Rs 50,000) to the variable cost (Rs 8,000 multiplied by 100 tonnes), which totals Rs 58,000. Dividing this total cost by the daily production of 100 tonnes gives a cost of Rs 580 per tonne.

Question 24

One percent of the people of country X are taller than 6 ft. Two percent of the people of country Y are taller than 6 ft. There are three as many people in country X as in country Y. Taking both countries together, what is the percentage of people taller than 6 ft?

Accepted Answer

1.25. Let Y have N people, X have 3N. Tall = 0.01(3N) + 0.02(N) = 0.05N out of 4N total. Percentage = 0.05N/4N x 100 = 1.25%.

Question 25

The monthly rainfall chart based on 50 years of rainfall in Agra is shown in the following figure. Which of the following are true? (k percentile is the value such that k percent of the data fall below that value)
(i) On average, it rains more in July than in December
(ii) Every year, the

Accepted Answer

(i) and (iii). Option (i) is correct because historical data typically shows that July experiences higher rainfall due to the monsoon season in Agra, while December usually has much lower precipitation. Option (iii) is also correct as July's rainfall is more consistent and predictable due to seasonal patterns, whereas February can have more variability in rainfall.

Question 26

In any given year, the probability of an earthquake greater than M magnitude 6 occurring in the Garhwal Himalayas is 0.04. The average time between successive occurrences of such earthquakes is ______ years.

Accepted Answer

25. The average time between successive occurrences of an event can be calculated using the formula 1 divided by the probability of the event. In this case, with a probability of 0.04, the average time is 1/0.04, which equals 25 years.

Question 27

A cube is built using 64 cubic blocks of side one unit. After it is built, one cubic block is removed from every corner of the cube. The resulting surface area of the body (in square units) after the removal is ___________.

Accepted Answer

96. Removing a block from each of the 8 corners of the cube increases the surface area because each corner removal exposes 3 new faces of the cube. The original surface area of the cube is 6 times the area of one face (6 units), and after removing the corners, the total surface area increases to 96 square units.

Question 28

A shaving set company sells 4 different types of razors, Elegance, Smooth, Soft and Executive. Elegance sells at Rs. 48, Smooth at Rs. 63, Soft at Rs. 78 and Executive at Rs. 173 per piece. The table below shows the numbers of each razor sold in each quarter of a year.

Quarter | Product |

Accepted Answer

Executive. The Executive razor, despite its higher price, sold a significant number of units, leading to the highest total revenue contribution compared to the other products. Its price point of Rs. 173 means that even with fewer sales, it can generate more revenue than the lower-priced razors.

Question 29

If f(x) = 2x⁷ + 3x - 5, which of the following is a factor of f(x)?

Accepted Answer

(x-1). The factor (x-1) is correct because substituting x=1 into the function f(x) yields zero, indicating that (x-1) is a root of the polynomial and thus a factor.

Question 30

In a process, the number of cycles to failure decreases exponentially with an increase in load. At a load of 80 units, it takes 100 cycles for failure. When the load is halved, it takes 10000 cycles for failure. The load for which the failure will happen in 5000 cycles is _________.

Accepted Answer

46.02. The relationship between load and cycles to failure is exponential, meaning that as the load decreases, the number of cycles until failure increases significantly. Given the data points, we can derive the load corresponding to 5000 cycles by interpolating between the known values of 100 cycles at 80 units and 10000 cycles at 40 units, leading to the conclusion that the load for 5000 cycles is approximately 46.02 units.

Question 31

Type I error in hypothesis testing is

Accepted Answer

rejection of the null hypothesis when it is true and should be accepted. A Type I error occurs when a true null hypothesis is incorrectly rejected, leading to a false conclusion that there is an effect or difference when none exists.

Question 32

The sum of the digits of a two digit number is 12. If the new number formed by reversing the digits is greater than the original number by 54, find the original number.

Accepted Answer

39. With tens t and units u, t+u=12 and 9(u-t)=54 gives u-t=6, so u=9, t=3 and the original number is 39 (its reverse 93 is the larger one). The stored 93 is the reversed number, not the original.

Question 33

A square pyramid has a base perimeter x, and the slant height is half of the perimeter. What is the lateral surface area of the pyramid?

Accepted Answer

0.25 x². Base side = x/4 and slant height l = x/2. Lateral surface area = (1/2)*perimeter*slant = (1/2)*x*(x/2) = x^2/4 = 0.25 x^2. The stored 0.75 x^2 is wrong; correct is 0.25 x^2.

Question 34

A nanth takes 6 hours and Bharath takes 4 hours to read a book. Both started reading copies of the book at the same time. After how many hours is the number of pages to be read by A nanth, twice that to be read by Bharath? Assume A nanth and Bharath read all the pages with constant pace.

Accepted Answer

3. Pages left for Ananth = 1 - t/6, for Bharath = 1 - t/4. Setting 1 - t/6 = 2(1 - t/4) gives t/3 = 1, so t = 3 hours. Stored answer 2 is wrong.

Question 35

A student was supposed to multiply a positive real number p with another positive real number q. Instead, the student divided p by q. If the percentage error in the student’s answer is 80%, the value of q is

Accepted Answer

√5. The percentage error is calculated based on the difference between the correct answer (p*q) and the incorrect answer (p/q). Given that the percentage error is 80%, we can set up the equation to find q, leading to the conclusion that q must equal √5 to satisfy the condition.

Question 36

If the sum of the first 20 consecutive positive odd numbers is divided by 20², the result is

Accepted Answer

1. The sum of the first 20 consecutive positive odd numbers equals 20^2 = 400. Dividing by 20^2 = 400 gives 1. The stored answer 1/2 is wrong; the result is 1.

Question 37

In a sample of 100 heart patients, each patient has 80% chance of having a heart attack without medicine X. It is clinically known that medicine X reduces the probability of having a heart attack by 50%. Medicine X is taken by 50 of these 100 patients. The probability that a randomly selec

Accepted Answer

20%. The probability of a patient having a heart attack without medicine X is 80%. With medicine X, this risk is reduced by 50%, resulting in a 40% chance of having a heart attack for those taking the medicine. Since 50 out of 100 patients take the medicine, the overall probability of a randomly selected patient taking medicine X and having a heart attack is 50% of 40%, which equals 20%.

Question 38

A bag contains Violet (V), Yellow (Y), Red (R), and Green (G) balls. On counting them, the following results are obtained:
(i) The sum of Yellow balls and twice the number of Violet balls is 50.
(ii) The sum of Violet and Green balls is 50.
(iii) The sum of Yellow and Red balls is 50.
(iv)

Accepted Answer

(A) Pie chart with V 10%, R 20%, Y 30%, G 40%. Option A is correct because it accurately reflects the ratios derived from the given conditions about the number of balls, satisfying all equations related to the counts of Violet, Yellow, Red, and Green balls.

Question 39

Three villages P, Q, and R are located in such a way that the distance PQ = 13 km, QR = 14 km, and RP = 15 km, as shown in the figure. A straight road joins Q and R. It is proposed to connect P to this road QR by constructing another road. What is the minimum possible length (in km) of thi

Accepted Answer

12.0. Triangle sides 13,14,15: Heron area = sqrt(21*8*7*6)=84. Altitude to QR (=14) is 2*84/14=12 km. The minimum connecting road is 12.0 km, not 12.5.

Question 40

If two distinct non-zero real variables x and y are such that (x + y) is proportional to (x − y) then the value of x/y

Accepted Answer

is a constant. The relationship given indicates that the ratio of x to y remains unchanged regardless of the specific values of x and y, meaning that x/y is a constant value for any pair of distinct non-zero real numbers satisfying the proportionality condition.

Question 41

Consider the following sample of numbers: 9, 18, 11, 14, 15, 17, 10, 69, 11, 13. The median of the sample is

Accepted Answer

13.5. Sorted: 9,10,11,11,13,14,15,17,18,69. With 10 values the median is the mean of the 5th and 6th: (13+14)/2 = 13.5.

Question 42

An unbalanced dice (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is 90% the probability that the face value is even. The probability of getting any even numbered face is the same.

If the probability that the face is even given that it is greate

Accepted Answer

0.468. The correct option is derived from the relationship between the probabilities of odd and even outcomes, along with the conditional probability given that the outcome is greater than 3. By using the provided probabilities and the total probability theorem, we can calculate that the probability of rolling a number greater than 3 aligns closely with 0.468.

Question 43

If 137 + 276 = 435 how much is 731 + 672?

Accepted Answer

1623. Since 137+276=435 holds in octal, the system is base 8. Computing 731+672 in octal: 1+2=3, 3+7=12 (write 2 carry 1), 7+6+1=16 (write 6 carry 1), giving 1623 (octal). That is option index 2, not the stored '1513'.

Question 44

5 skilled workers can build a wall in 20 days; 8 semi-skilled workers can build a wall in 25 days; 10 unskilled workers can build a wall in 30 days. If a team has 2 skilled, 6 semi-skilled and 5 unskilled workers, how long will it take to build the wall?

Accepted Answer

15 days. Per-worker rates: skilled 1/(5*20)=1/100, semi 1/(8*25)=1/200, unskilled 1/(10*30)=1/300 walls/day. Team rate = 2/100 + 6/200 + 5/300 = 6/300+9/300+5/300 = 20/300 = 1/15. So the wall takes 15 days, not 18. Correct option: '15 days'.

Question 45

Given digits 2, 2, 3, 3, 3, 4, 4, 4, 4 how many distinct 4 digit numbers greater than 3000 can be formed?

Accepted Answer

51. Using digits with at most two 2s, three 3s, four 4s and requiring the number to exceed 3000, exhaustive enumeration gives 51 distinct four-digit numbers. The stored 54 is wrong; the correct option is 1 (51).

Question 46

A tourist covers half of his journey by train at 60 km/h, half of the remainder by bus at 30 km/h and the rest by cycle at 10 km/h. The average speed of the tourist in km/h during his entire journey is

Accepted Answer

24. For distance D: D/2 at 60, D/4 at 30, D/4 at 10. Time = D/120 + D/120 + 3D/120 = 5D/120 = D/24, so average speed = D/(D/24) = 24 km/h. The answer is 24 (option C), not the stored 18.

Question 47

The current erection cost of a structure is Rs. 13,200. If the labour wages per day increase by 1/5 of the current wages and the working hours decrease by 1/24 of the current period, then the new cost of erection in Rs. is

Accepted Answer

15,180. The new cost of erection is calculated by adjusting the current cost based on the increased labor wages and decreased working hours. The increase in wages raises the overall cost, while the reduction in hours slightly offsets this increase, resulting in a final cost of Rs. 15,180.

Question 48

What will be the maximum sum of 44, 42, 40,..... ?

Accepted Answer

506. The sequence 44,42,40,... is an AP with d=-2; the sum is maximized by including all positive terms down to 2. Sum = 2+4+...+44 = 2(1+2+...+22) = 2(253) = 506, so the answer is 506, not 500.

Question 49

What is the average of all multiples of 10 from 2 to 198?

Accepted Answer

100. The average of the multiples of 10 between 2 and 198 can be calculated by identifying the first multiple (10) and the last multiple (190), then finding the average of these two values, which is 100. This is because the multiples of 10 form an arithmetic sequence, and the average of the first and last terms gives the overall average.

Question 50

The value of √(12 + √(12 + √(12 +...))) is

Accepted Answer

4.000. The expression can be set equal to x, leading to the equation x = √(12 + x). Squaring both sides gives x² = 12 + x, which simplifies to x² - x - 12 = 0. Solving this quadratic equation yields x = 4 as the positive solution, confirming that the value is 4.000.

GATE General Aptitude: Quantitative Aptitude questions with solutions

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