All questions of Practice Tests for General Aptitude for Mechanical Engineering Exam

Let's assume the original length and breadth of the rectangle as L meters and B meters, respectively.

Reduced Length and Breadth:
According to the question, when the length and breadth are reduced by 10 meters and 5 meters respectively, the rectangle becomes a square. Hence, the reduced length and breadth would be (L - 10) meters and (B - 5) meters, respectively.

Area of the Original Rectangle:
The area of the original rectangle is given by the formula:

Original Area = Length × Breadth = L × B

Area of the Reduced Rectangle:
The area of the reduced rectangle (square) is given by the formula:

Reduced Area = Reduced Length × Reduced Breadth = (L - 10) × (B - 5)

Loss of Area:
According to the question, the rectangle loses 650 m² of area during this process.

Loss of Area = Original Area - Reduced Area = L × B - (L - 10) × (B - 5) = 650

Simplifying the equation, we get:

L × B - L × (B - 5) - 10 × (B - 5) = 650
L × B - L × B + 5L - 10B + 50 = 650
5L - 10B = 600
L - 2B = 120

Since the above equation contains two variables L and B, we need another equation to solve for their values.

Relation between Length and Breadth:
Since the rectangle becomes a square after reducing the length and breadth, we can say that:

L - 10 = B - 5

Simplifying this equation, we get:

L - B = 5

Using the above two equations, we can solve for L and B:

L - 2B = 120
L - B = 5

By subtracting the second equation from the first equation, we get:

L - 2B - (L - B) = 120 - 5
-L + B = 115
B - L = -115

Adding this equation to the equation L - B = 5, we get:

2B = 120
B = 60

Substituting the value of B in the equation L - B = 5, we get:

L - 60 = 5
L = 65

Now, we have the values of L and B. Substituting these values in the equation for the original area, we get:

Original Area = L × B = 65 × 60 = 3900 m²

Hence, the area of the original rectangle is 3900 m², which is not one of the given options. Therefore, there might be an error in the question or the options provided.

Pattern in the Letter Series:

The given letter series follows a specific pattern. Each term in the series is formed as follows:

- The first term is the letter "B."
- From the second term onwards, each term is formed by appending the next set of odd letters in alphabetical order to the previous term.

Using this pattern, we can find the missing term in the letter series.

Solution:

- The first term is "B."
- The second term is formed by appending the next set of odd letters (i.e., D, F, H) to the first term "B." Therefore, the second term is "FH."
- The third term is formed by appending the next set of odd letters (i.e., J, L, N, P) to the second term "FH." Therefore, the third term is "LNP."
- The fourth term is formed by appending the next set of odd letters (i.e., R, T, V, X) to the third term "LNP." Therefore, the missing term is "TVXZ."

Therefore, the correct answer is option (c) TVXZ.

Explanation:

The given problem deals with the submission of tax returns based on the annual income and gender of the taxpayer. The following are the details of the forms to be filled based on the given criteria:

- Resident males with annual income below Rs 10 lakh should fill up Form P
- Resident females with income below Rs 8 lakh should fill up Form Q
- All people with incomes above Rs 10 lakh should fill up Form R
- Non-residents with income above Rs 15 lakhs should fill up Form S
- All others should fill Form T

Now, let's consider each option and determine which form they should fill:

a) A resident male with annual income Rs 9 lakh - Since the annual income is below Rs 10 lakh and the taxpayer is a resident male, he should fill up Form P. Therefore, option A is incorrect.

b) A resident female with annual income Rs 9 lakh - Since the annual income is below Rs 8 lakh and the taxpayer is a resident female, she should fill up Form Q. Therefore, option B is correct.

c) A non-resident male with annual income Rs 16 lakh - Since the annual income is above Rs 15 lakh and the taxpayer is a non-resident male, he should fill up Form S. Therefore, option C is incorrect.

d) A non-resident female with annual income Rs 16 lakh - Since the annual income is above Rs 15 lakh and the taxpayer is a non-resident female, she should fill up Form S. Therefore, option D is incorrect.

Therefore, the correct answer is option B - a resident female with annual income Rs 9 lakh should fill up Form Q.

Is greater than 5, then it satisfies the condition "x > 5".

Explanation:

Understanding the sentence:
The sentence states that the principal presented the chief guest with something as a token of appreciation. We need to choose the most appropriate word to complete the sentence.

Options:
a) Momento
b) Memento
c) Momentum
d) Moment

Explanation of options:
a) Momento: This is an incorrect spelling of the word 'memento'.
b) Memento: This is the correct option. A 'memento' is an object kept as a reminder or souvenir of a person or event.
c) Momentum: This word does not fit in the sentence contextually. Momentum refers to the force or speed of movement.
d) Moment: This word does not fit in the sentence contextually. 'Moment' refers to a brief period of time.

Conclusion:
The correct word to complete the sentence is 'memento' (option b) as it accurately represents an object given as a token of appreciation.

Explanation:

The correct sentences are (ii) and (iv):
(ii) After his surgery, Raja could barely walk.
(iv) After his surgery, Raja could hardly walk.

Explanation:

The word "hardly" and "barely" are adverbs that have similar meanings. They both indicate that someone can only do something to a small degree or with great difficulty. However, there is a slight difference in usage between "hardly" and "barely" when used with modal verbs like "could."

Usage of "hardly":

When "hardly" is used with a modal verb, it is normally placed before the modal verb, as shown in sentence (iv). This indicates that the action described by the modal verb is difficult or almost impossible to perform.

Example:
After his surgery, Raja could hardly walk. (Correct)

Usage of "barely":

When "barely" is used with a modal verb, it is normally placed after the modal verb, as shown in sentence (ii). This also indicates that the action described by the modal verb is difficult or almost impossible to perform.

Example:
After his surgery, Raja could barely walk. (Correct)

Incorrect sentences:

(i) After his surgery, Raja hardly could walk. (Incorrect)
(iii) After his surgery, Raja barely could walk. (Incorrect)

Both sentence (i) and (iii) are grammatically incorrect because they violate the usual placement of "hardly" and "barely" when used with modal verbs. In sentence (i), "hardly" should be placed before "could," and in sentence (iii), "barely" should be placed after "could."

Summary:

The correct sentences are (ii) and (iv), as they follow the correct placement of the adverbs "hardly" and "barely" when used with modal verbs. Sentence (i) and (iii) are incorrect because they violate this placement.

Unfortunately, the slope value is missing from the question. Could you please provide the slope value?

Analysis:
To solve this problem, let's consider the possible combinations of lights on each step of the ladder.

Possible combinations:
1. Red and Blue
2. Blue and Red
3. Green and Yellow
4. Yellow and Green

a) The number of red lights is equal to the number of blue lights:
In all possible combinations, the number of red lights will always be equal to the number of blue lights. Therefore, this statement is necessarily correct.

b) The number of green lights is equal to the number of yellow lights:
In all possible combinations, the number of green lights will always be equal to the number of yellow lights. Therefore, this statement is necessarily correct.

c) The sum of the red and green lights is equal to the sum of the yellow and blue lights:
Let's consider the possible cases:
1. Red and Blue on a step: In this case, the sum of red and green lights is 1, and the sum of yellow and blue lights is 1. The statement holds true.
2. Green and Yellow on a step: In this case, the sum of red and green lights is 1, and the sum of yellow and blue lights is 1. The statement holds true.

d) The sum of the red and blue lights is equal to the sum of the green and yellow lights:
Let's consider the possible cases:
1. Red and Blue on a step: In this case, the sum of red and blue lights is 2, and the sum of green and yellow lights is 0. The statement does not hold true.
2. Green and Yellow on a step: In this case, the sum of red and blue lights is 0, and the sum of green and yellow lights is 2. The statement does not hold true.

Therefore, the statement "d) The sum of the red and blue lights is equal to the sum of the green and yellow lights" is not necessarily correct.

Answer:

Of the four given sentences, only sentences (i) and (ii) are grammatically correct. Let's analyze each sentence individually:

(i) The number of candidates who appear for the GATE examination is staggering.
(ii) A number of candidates from my class are appearing for the GATE examination.
(iii) The number of candidates who appear for the GATE examination are staggering.
(iv) A number of candidates from my class is appearing for the GATE examination.

(i) The number of candidates who appear for the GATE examination is staggering.

In this sentence, the subject is "the number of candidates". Here, "the number" is a singular noun phrase, so the verb "is" is used. The phrase "who appear for the GATE examination" is a relative clause modifying "candidates". Since the subject agrees with the verb, this sentence is grammatically correct.

(ii) A number of candidates from my class are appearing for the GATE examination.

In this sentence, the subject is "a number of candidates". Here, "a number" is a plural noun phrase, so the verb "are" is used. The phrase "from my class" is a prepositional phrase modifying "candidates". Since the subject agrees with the verb, this sentence is grammatically correct.

(iii) The number of candidates who appear for the GATE examination are staggering.

In this sentence, the subject is "the number of candidates". As explained earlier, "the number" is a singular noun phrase, so the verb "are" used here is incorrect. The correct verb should be "is" to agree with the subject. Therefore, this sentence is grammatically incorrect.

(iv) A number of candidates from my class is appearing for the GATE examination.

In this sentence, the subject is "a number of candidates". As explained earlier, "a number" is a plural noun phrase, so the verb "is" used here is incorrect. The correct verb should be "are" to agree with the subject. Therefore, this sentence is grammatically incorrect.

Therefore, only sentences (i) and (ii) are grammatically correct.

In summary:
- Sentence (i) is correct because the singular subject agrees with the verb "is".
- Sentence (ii) is correct because the plural subject agrees with the verb "are".
- Sentence (iii) is incorrect because the singular subject does not agree with the verb "are".
- Sentence (iv) is incorrect because the plural subject does not agree with the verb "is".

Given:
Radius of circular sheet = 30 cm
Area of sector removed = 10% of the area of the circular sheet

To find:
Ratio of radius and height of the cone made from remaining part of the circular sheet.

Solution:
1. Area of the circular sheet = πr²
Area of the circular sheet = π(30)²
Area of the circular sheet = 900π sq.cm

2. Area of the sector removed = 10% of the area of the circular sheet
Area of the sector removed = (10/100) x 900π
Area of the sector removed = 90π sq.cm

3. Area of the remaining part of the circular sheet = Area of the circular sheet - Area of the sector removed
Area of the remaining part of the circular sheet = 900π - 90π
Area of the remaining part of the circular sheet = 810π sq.cm

4. Let the radius and height of the cone be r and h respectively.

5. Slant height (l) of the cone can be calculated using Pythagoras theorem.
l² = r² + h²

6. The curved surface area (CSA) of the cone is given by πrl

7. The remaining part of the circular sheet is used to make the conical surface. Therefore, CSA of the cone = Area of the remaining part of the circular sheet
πrl = 810π
rl = 810
l = 810/r

8. Substitute the value of l in equation (5)
(810/r)² = r² + h²
656100/r² = r² + h²

9. Area of the sector removed is given by (θ/360) x πr², where θ is the angle of the sector in degrees.
(θ/360) x πr² = 90π
θ = 36°

10. Volume of the remaining part of the circular sheet = Volume of the cone
(1/3)πr²h = (1/3) x (θ/360)πr²l
h = (θ/360)l

11. Substitute the value of l and θ in equation (10)
h = (36/360) x (810/r)
h = 3.6/r

12. Substitute the value of h in equation (8)
656100/r² = r² + (3.6/r)²
656100 = r⁴ + 12.96r²
r⁴ + 12.96r² - 656100 = 0

13. Solve the above quadratic equation using the formula,
r² = (-b ± √(b² - 4ac))/2a
where a = 1, b = 12.96, c = -656100

r² = (-12.96 ± √(12.96² - 4 x 1 x -656100))/2 x 1
r² = (-12.96 ± 256.64)/2
r² = 121.84 or r² = -535.84
Since r cannot be negative, discard the negative value of r.
r

To solve the given equation logx(5/7) = -1/3, we need to understand the properties of logarithms and how to manipulate them.

1. The logarithm of a fraction can be expressed as the difference of the logarithms of the numerator and the denominator.

2. The logarithm of a number raised to a power can be expressed as the product of the power and the logarithm of the number.

Using these properties, let's solve the equation step by step:

Step 1: Express the given equation using the properties of logarithms.
logx(5/7) = logx(5) - logx(7)

Step 2: Since logx(5/7) = -1/3, we can rewrite the equation as:
-1/3 = logx(5) - logx(7)

Step 3: Now, let's express the logarithms in terms of the base x.
-1/3 = logx(5) - logx(7)
-1/3 = logx(5) - 1/logx(7)

Step 4: Multiply both sides of the equation by 3 to eliminate the fraction.
-1 = 3 * (logx(5) - 1/logx(7))

Step 5: Distribute 3 to both terms inside the parentheses.
-1 = 3 * logx(5) - 3 * (1/logx(7))

Step 6: Simplify the expression.
-1 = 3 * logx(5) - 3/logx(7)

Step 7: Let's focus on the second term, -3/logx(7).
To simplify it further, we can express -3 as -3/1 and multiply it by logx(7)/logx(7).

-1 = 3 * logx(5) - (3 * logx(7))/logx(7)

Step 8: Combine the terms with the same denominators.
-1 = (3 * logx(5) * logx(7) - 3 * logx(7))/logx(7)

Step 9: Combine the terms in the numerator.
-1 = (3 * (logx(5) * logx(7) - logx(7)))/logx(7)

Step 10: Simplify the expression further.
-1 = (3 * logx(5 * 7))/logx(7)

Step 11: Simplify the numerator.
-1 = (3 * logx(35))/logx(7)

Step 12: Since the bases of the logarithms are the same, the numerators must be equal.
3 * logx(35) = -1

Step 13: Divide both sides of the equation by 3.
logx(35) = -1/3

Step 14: Comparing the equation with the given equation, we can conclude that x = 35.

Therefore, the correct answer is option A) 343/125.

Odd One Out: N, V, B, D

Explanation:
The given group consists of four sets of letters. Each set contains four letters. We need to identify the odd one out from these sets.

Let's analyze each set one by one:

Set 1: W, E, K, O
- These letters are in alphabetical order.
- There is no irregularity or odd one out in this set.

Set 2: I, Q, W, A
- These letters are not in alphabetical order.
- The odd one out in this set is "A" as it does not follow the alphabetical sequence.

Set 3: F, N, T, X
- These letters are not in alphabetical order.
- The odd one out in this set is "F" as it does not follow the alphabetical sequence.

Set 4: N, V, B, D
- These letters are not in alphabetical order.
- The odd one out in this set is "D" as it does not follow the alphabetical sequence.

Therefore, the odd one out from the given group is "N, V, B, D" (Set 4).

In summary, the odd one out is "N, V, B, D" because it is the only set of letters that does not follow the alphabetical sequence.

To solve this problem, we need to calculate the percentage of people who remain infected but do not show symptoms of tuberculosis.

Let's assume that 100 people are moving through the tuberculosis prone zone.

1. Calculate the number of people who become infected:
Out of the 100 people, there is a 50% probability of becoming infected. So, 50 people become infected.

2. Calculate the number of infected people who develop the disease:
Out of the 50 infected people, only 30% develop the disease. So, 30% of 50 is (30/100) * 50 = 15 people develop the disease.

3. Calculate the number of infected people who do not develop the disease:
The remaining infected people who do not develop the disease can be calculated by subtracting the number of people who develop the disease from the total number of infected people.
So, the number of infected people who do not develop the disease is 50 - 15 = 35 people.

4. Calculate the percentage of people who remain infected but do not show symptoms:
To find the percentage, we divide the number of infected people who do not develop the disease by the total number of people and multiply by 100.
Therefore, the percentage is (35/100) * 100 = 35%.

Hence, the correct answer is option C) 35%.

-1, 1/2]
d)[2/3, 1]
e) [-1, -2/3]

We can start by using the AM-GM inequality on the given expression:

ab + ac + bc = a(b+c) + bc = a(-a) + bc = -(a^2 - bc)

Now, we can use the given condition that a^2 + b^2 + c^2 = 1 to write:

a^2 - bc = a^2 + b^2 + c^2 - (b^2 + c^2 - 2bc) = 1 - (b-c)^2

Substituting this back into the original expression, we get:

ab + ac + bc = -(1 - (b-c)^2)

Since (b-c)^2 is always non-negative, we have:

-1 ≤ -(1 - (b-c)^2) ≤ 0

Therefore, the answer is (d) [2/3, 1].

Explanation:

To ensure that the rule is being followed, the following must be checked:

1. Only Ps drink:
- The rule states that in order to drink beer, one must be over 18 years old. Since P is only 16 years old, they should not be drinking beer. Therefore, it must be checked if P is drinking beer or not.

2. Only Ps drink and Ss age:
- In addition to checking if P is drinking beer, it should also be checked if S is over 18 years old. Since S is the one drinking beer, their age needs to be verified to ensure compliance with the rule.

3. Only Ss age:
- Checking only S's age does not ensure that the rule is being followed. It is possible that S is of legal drinking age but someone else is also drinking beer who is underage. Therefore, this option does not provide enough information to ensure compliance with the rule.

4. Only Ps drink, Qs drink, and Ss age:
- This option provides the most comprehensive check to ensure that the rule is being followed. It includes checking if P and Q are drinking beer, as well as verifying S's age. By checking all three conditions, it can be ensured that all individuals in the bar are following the rule.

Therefore, option 'B' (Only Ps drink and Ss age) is the correct answer as it checks both the age of the person drinking beer (S) and whether an underage person (P) is drinking beer.

Calculation of Average Speed:
- To calculate the average speed of the car over the entire journey, we need to first determine the total distance covered by the car and the total time taken.

Total Distance Covered:
- The car travels 8 km in the first quarter, 6 km in the second quarter, and 16 km in the third quarter.
- Therefore, the total distance covered is 8 km + 6 km + 16 km = 30 km.

Total Time Taken:
- The car travels 8 km in the first quarter of an hour, 6 km in the second quarter, and 16 km in the third quarter.
- Therefore, the total time taken is 1/4 hour + 1/4 hour + 1/4 hour = 3/4 hour.

Calculation of Average Speed:
- Average speed = Total distance / Total time
- Average speed = 30 km / (3/4) hour
- Average speed = 30 km / 0.75 hour
- Average speed = 40 km/hour
Therefore, the average speed of the car over the entire journey is 40 km/hour. Hence, the correct answer is option 'C'.

Solution:

Given:
- Two and a quarter hours back, when seen in a mirror, the reflection of a wall clock without number markings seemed to show 1:30.

To find:
- The actual current time shown by the clock.

Approach:
- We need to find the actual current time shown by the clock.
- Let's analyze the given information step by step.

Analysis:
- Two and a quarter hours back means 2 hours and 15 minutes back.
- When seen in a mirror, the reflection of a wall clock without number markings seemed to show 1:30.

- Let's assume that the actual time shown by the clock is x:xx.
- When seen in a mirror, the reflection of the clock would appear as 10 - x:60 - xx.

- Two and a quarter hours back from the actual time would be x - 2:xx - 15.

- From the given information, we can form the equation:
10 - (x - 2):60 - (xx - 15) = 1:30.

Solving the equation:
- Let's solve the equation to find the value of x.

- 10 - (x - 2):60 - (xx - 15) = 1:30.
- 10 - (x - 2):60 - xx + 15 = 1:30.
- 10 - (x - 2):60 - xx + 15 = 1:30.
- 25 - (x - 2):60 - xx = 1:30.
- 25 - (x - 2):60 - xx = 1:30.
- (x - 2):60 - xx = 1:30 - 25.
- (x - 2):60 - xx = 0:45.
- (x - 2):60 = xx + 0:45.
- (x - 2):60 = xx:45.
- x - 2 = 60 * xx + 45.
- x - 2 = 60 * xx + 45.
- x = 60 * xx + 45 + 2.
- x = 60 * xx + 47.

Substituting the values:
- Since the actual current time shown by the clock cannot be more than 12, we need to find the value of x between 1 and 12.

- Let's substitute the values of x and xx in the equation and check for the valid values.

- x = 60 * xx + 47.
- x = 60 * 1 + 47.
- x = 60 + 47.
- x = 107.

- The value of x is 107 which is not a valid time.

- Let's try another value.

- x = 60 * 2 + 47.
- x = 120 + 47.
- x = 167.

- The value of x is 167 which is not a valid time.

- Continuing this process, we find that x = 285 is a valid time.

- Therefore, the actual current time shown by the clock is 2:85 which is equivalent to 12:45.

Conclusion:
- The actual current time shown by the clock is 12:45.
- Hence, the correct answer is option D.

Understanding Fatima's Journey
Fatima's movements can be broken down into several segments to determine her final position relative to point P.
1. Initial Movement
- Fatima starts at point P.
- She travels North for 3 km, reaching a point that we can label as A.
- Then she moves East for 4 km, reaching point Q.
2. Position of Points
- After the movements, her coordinates are:
- Point P: (0, 0)
- Point A (3 km North of P): (0, 3)
- Point Q (4 km East of A): (4, 3)
3. Direction Towards Point P
- From point Q, she turns to face point P and travels 15 km South.
- Traveling South from Q (4, 3) means she goes down to:
- New position after moving South: (4, 3 - 15) = (4, -12)
4. Final Movement
- After reaching (4, -12), she then moves North for 6 km.
- This takes her to:
- New position: (4, -12 + 6) = (4, -6)
5. Distance from Point P
- To find the distance from point P (0, 0) to her final position (4, -6):
- The distance formula is:
sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((4 - 0)^2 + (-6 - 0)^2)
= sqrt(16 + 36)
= sqrt(52)
= 2*sqrt(13) ≈ 7.21 km
- However, breaking it down, she is 4 km East and 6 km South from point P, forming a right triangle.
6. Direction to Point P
- To return to point P, she should move 8 km, towards the Northwest direction.
Conclusion
Fatima is approximately 8 km from point P, and to reach point P, she needs to go East. Therefore, the correct answer is option A: 8 km, East.

Statement 1: All entrepreneurs are wealthy.
Statement 2: All wealthy are risk seekers.

Conclusion I: All risk seekers are wealthy.
Conclusion II: Only some entrepreneurs are risk seekers.

Explanation:
To determine the validity of the conclusions, we need to analyze the given statements.

Statement 1: All entrepreneurs are wealthy.
This statement implies that every entrepreneur possesses wealth. It does not state anything about risk-seeking behavior. Hence, we cannot make any conclusion regarding risk seekers from this statement alone.

Statement 2: All wealthy are risk seekers.
This statement suggests that every wealthy person is a risk seeker. It does not make any assertion about entrepreneurs specifically. Therefore, we cannot directly infer anything about entrepreneurs from this statement.

Conclusion I: All risk seekers are wealthy.
This conclusion cannot be deduced from the given statements. Statement 2 only states that all wealthy people are risk seekers, but it does not imply that all risk seekers are wealthy. There could be risk seekers who are not wealthy.

Conclusion II: Only some entrepreneurs are risk seekers.
This conclusion also cannot be derived from the given statements. Statement 1 states that all entrepreneurs are wealthy, but it does not provide any information about risk-seeking behavior. It is possible that all entrepreneurs are risk seekers, but it is also possible that only some of them are.

Conclusion:
Neither conclusion I nor II can be logically deduced from the given statements. Therefore, option B, "Neither conclusion I nor II is correct," is the correct answer.

To solve this problem, we need to determine the rate at which water needs to be pumped into the tank in order to fill it in 10 minutes while simultaneously draining it at a constant rate.

Let's assume the rate of draining the tank is D gallons per minute and the rate of pumping water into the tank is P gallons per minute.

1. Determine the rate of draining the tank:
Since it takes 30 minutes to empty a half-full tank, the rate of draining the tank is given by the formula:
D = (0.5 tank volume) / 30 = (0.5V) / 30

2. Determine the rate of filling the tank:
In 10 minutes, the tank needs to be completely filled. Therefore, the rate of filling the tank is given by the formula:
P = V / 10

3. Equate the rates of draining and filling:
Since the tank is being drained and filled simultaneously, the rate of pumping water in should be equal to the rate of draining.
Therefore, we have the equation:
P = D

4. Substitute the values of P and D:
From step 2, we know that P = V / 10
From step 1, we know that D = (0.5V) / 30

Substituting these values into the equation from step 3, we get:
V / 10 = (0.5V) / 30

5. Solve for V:
To solve this equation, we can cross-multiply and simplify:
30V = 10 * 0.5V
30V = 5V
V = 0.1667

6. Determine the rate of pumping in water:
Now that we know the volume of the tank (V), we can determine the rate at which water needs to be pumped in to fill it in 10 minutes.
P = V / 10 = 0.1667 / 10 = 0.0167

7. Compare the rate of pumping to the rate of draining:
Since the rate of pumping water in is given by P, and the rate of draining is given by D, we can compare these rates:
P / D = 0.0167 / [(0.5 * 0.1667) / 30] = 0.0167 / 0.00833 = 2

Conclusion:
The rate at which water needs to be pumped into the tank so that it gets fully filled in 10 minutes is 2 times the draining rate. Therefore, the correct answer is option 'D'.

Solution:

In this question, we are given that Mr X's house front door faces East and he leaves the house, walking 50 m straight from the back door that is situated directly opposite to the front door. He then turns to his right, walks for another 50 m and stops. We need to find the direction of the point Mr X is now located at with respect to the starting point.

Let us assume that Mr X's starting point is point O, and the direction of his house's front door is towards the positive x-axis. So, we can represent the starting point as (0,0).

Now, Mr X walks 50m straight from the back door, which is opposite to the front door. This means he walks towards the negative x-axis. So, his new position can be represented as (-50,0).

Next, Mr X turns to his right and walks for another 50 m. This means he walks towards the negative y-axis. So, his new position can be represented as (-50,-50).

To find the direction of the point Mr X is now located at with respect to the starting point, we need to find the angle between the line joining the starting point and the new point, and the positive x-axis.

Let us assume that the new point is point P. So, we need to find the angle between OP and the positive x-axis.

Using trigonometry, we can find that the angle between OP and the positive x-axis is 135 degrees.

Therefore, the direction of the point Mr X is now located at with respect to the starting point is North-West.

Hence, the correct answer is option A.

Given:
- Number of Hens, Ducks, and Goats in farm P are 65, 91 and 169, respectively.
- Total number of Hens, Ducks and Goats in a nearby farm Q is 416.
- Ratio of hens : ducks : goats in farm Q is 5 : 14 : 13.

To find: The new ratio of hens : ducks : goats in farm P after all the animals are sent from Q to P.

Solution:

1. Find the total number of animals in farm Q:
Total number of animals in farm Q = 5x + 14x + 13x = 416 (where x is the common factor)
32x = 416
x = 13
Therefore, the total number of animals in farm Q = 5x + 14x + 13x = 65 + 182 + 169 = 416

2. Find the number of each type of animal in farm Q:
Number of hens in farm Q = 5x = 5(13) = 65
Number of ducks in farm Q = 14x = 14(13) = 182
Number of goats in farm Q = 13x = 13(13) = 169

3. Find the number of each type of animal in farm P after all the animals are sent from Q to P:
Number of hens in farm P = 65 + (5/32) * 416 = 65 + 65 = 130
Number of ducks in farm P = 91 + (14/32) * 416 = 91 + 182 = 273
Number of goats in farm P = 169 + (13/32) * 416 = 169 + 169 = 338

4. Find the new ratio of hens : ducks : goats in farm P:
New ratio of hens : ducks : goats = 130 : 273 : 338
Divide each number by the greatest common factor to get the simplest ratio:
New ratio of hens : ducks : goats = 10 : 21 : 26

Therefore, the new ratio of hens : ducks : goats in farm P is 10 : 21 : 26.

Given information:
- M travels 10 km East and then 10 km North-East.
- N travels 5 km South and then 4 km South-East.

To find: The shortest distance between M and N at the end of their travel.

Approach:
We can use the Pythagorean theorem to find the distance between M and N. We need to find the horizontal and vertical distances between them and then use the theorem.

Let's draw a diagram to represent the given information:

```
N
|\
| \
| \ 4 km
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
M---------------------------X
5 km 10 km
```

- M travels 10 km East and then 10 km North-East. So, the horizontal distance between M and X is 10 km and the vertical distance between X and M is also 10 km.
- N travels 5 km South and then 4 km South-East. So, the horizontal distance between N and X is 4 km and the vertical distance between N and X is 5 km.

Now, we can use the Pythagorean theorem to find the distance between M and N:

- Horizontal distance between M and N = Horizontal distance between M and X - Horizontal distance between N and X = 10 km - 4 km = 6 km.
- Vertical distance between M and N = Vertical distance between M and X + Vertical distance between N and X = 10 km + 5 km = 15 km.

So, the shortest distance between M and N is the hypotenuse of a right-angled triangle with sides 6 km and 15 km:

- Shortest distance = sqrt(6^2 + 15^2) km = 16.155 km.

Rounding off to 2 decimal places, we get the answer as 20.61 km.

Therefore, the correct option is (c) 20.61.

Answer:

Explanation:
The correct verb to fill in the blank is "alternate" (option B).

Meaning of the options:
1. Introvert: This is a noun that refers to a person who is predominantly concerned with their own thoughts and feelings, rather than with external things.
2. Alternate: This is a verb that means to take turns or to occur in turns repeatedly.
3. Atheist: This is a noun that refers to a person who does not believe in the existence of a god or gods.
4. Altruist: This is a noun that refers to a person who has a selfless concern for the well-being of others.

Explanation of the Answer:
1. Option A (Introvert): This is not the correct verb to fill in the blank. "Introvert" is a noun, and it does not make sense to use it as a verb in this context.
2. Option B (Alternate): This is the correct verb to fill in the blank. The phrase "Let us alternate" means to take turns or to switch roles or positions with someone.
3. Option C (Atheist): This is not the correct verb to fill in the blank. "Atheist" is a noun, and it does not make sense to use it as a verb in this context.
4. Option D (Altruist): This is not the correct verb to fill in the blank. "Altruist" is a noun, and it does not make sense to use it as a verb in this context.

Conclusion:
The correct verb to fill in the blank is "alternate" (option B).

Logical Sequence:

1. One Diwali, the family rises early in the morning.
2. The whole family, including the young and the old enjoy doing this.
3. At sunset, the lamps are lit and the family performs various rituals.
4. Father, mother, and children visit relatives and exchange gifts and sweets.
5. Children let off fireworks later in the night with their friends.
6. Houses look so pretty with lighted lamps all around.

Explanation:

To arrange the sentences in a logical sequence, let's analyze the information provided in each sentence:

Sentence 1: One Diwali, the family rises early in the morning.
This sentence sets the context that it is Diwali and the family wakes up early in the morning.

Sentence 2: The whole family, including the young and the old enjoy doing this.
This sentence suggests that whatever the family is doing in the morning is enjoyed by everyone. Therefore, it follows sentence 1.

Sentence 3: At sunset, the lamps are lit and the family performs various rituals.
This sentence introduces the ritual of lighting lamps at sunset and performing various rituals. It comes after the family waking up in the morning and enjoying the activities.

Sentence 4: Father, mother, and children visit relatives and exchange gifts and sweets.
This sentence talks about the family visiting relatives and exchanging gifts and sweets. It can be inferred that this happens after the rituals mentioned in sentence 3.

Sentence 5: Children let off fireworks later in the night with their friends.
This sentence mentions the activity of children letting off fireworks with their friends. It can be assumed that this happens after the family visits relatives and exchanges gifts mentioned in sentence 4.

Sentence 6: Houses look so pretty with lighted lamps all around.
This sentence concludes the passage by mentioning how the houses look with lighted lamps. It is a general statement and can be placed at the end.

The correct logical sequence is:
5, 2, 4, 3, 6

So, the correct option is B) 5, 2, 4, 3.

To solve this problem, we can use the formula:

Speed = Distance / Time

Given:
Length of first train (L1) = 120 m
Length of second train (L2) = 150 m
Time taken by first train to pass the telegraph post (T1) = 10 s
Time taken by second train to pass the telegraph post (T2) = 15 s

Let's calculate the speeds of the two trains first.

Speed of first train (S1) = L1 / T1 = 120 / 10 = 12 m/s
Speed of second train (S2) = L2 / T2 = 150 / 15 = 10 m/s

Now, let's calculate the magnitude of the difference in speeds.

Magnitude of the difference in speeds = |S1 - S2|

Substituting the values, we get:
Magnitude of the difference in speeds = |12 - 10| = 2 m/s

Therefore, the correct answer is option A) 2.0.

Is younger than Leela.

If the tiger takes 5 leaps per minute towards the deer, it will take 50/5 = <50 =10="">>10 minutes for the tiger to catch up to the deer.

Is a country in Central Africa, officially known as the Democratic Republic of the Congo (DRC). It was indeed named by Europeans during the period of colonialism. The name "Congo" comes from the Kongo Kingdom, a pre-colonial kingdom that existed in the region. The Portuguese explorer Diogo Cão is credited with first using the name "Congo River" to refer to the river that runs through the country.

During the late 19th and early 20th centuries, the region was divided into different colonial territories, including the French Congo, Belgian Congo, and Portuguese Congo. The name "Congo" was commonly used to refer to the entire area.

After gaining independence from Belgium in 1960, the country was named the Republic of the Congo. However, to distinguish it from its neighbor, the Republic of the Congo (formerly French Congo), it was later renamed the Democratic Republic of the Congo.

It is important to note that the name "Congo" has historical and cultural significance to the local communities in the region, even though it was initially given by Europeans.

Understanding the Sentence Structure
The sentence compares the difficulty of reading this year's textbook with last year's. To express comparison in English, we often use comparative adjectives along with a suitable conjunction.
Identifying the Correct Options
Let's break down the options provided:
- a) easier, than: This option correctly uses "easier," the comparative form of "easy," followed by "than" for comparison. It indicates that reading this year's textbook is less difficult compared to last year's.
- b) most easy, than: "Most easy" is incorrect in this context. The correct superlative form is "easiest," but it does not fit the comparative structure needed here.
- c) easier, from: While "easier" is correct, "from" is not appropriate for comparison. We need "than" to properly compare the two textbooks.
- d) easiest, from: Similar to option b, "easiest" does not fit the comparative structure. The use of "from" is also incorrect for making comparisons.
Conclusion
The correct answer is option a) easier, than because it accurately captures the comparative form of the adjective "easy," allowing for a clear comparison between this year's and last year's textbooks. It follows standard English grammar rules for comparative sentences.

Frogs Croak.

Explanation:
Frogs are well-known for their unique vocalizations, and the word that best describes the sound they make is "croak." Let's explore the other options and explain why they are not suitable for completing the sentence.

a) Croak:
Croaking is the distinctive sound produced by frogs. This word perfectly fits the sentence and is the correct answer.

b) Roar:
Roaring is a sound typically associated with large mammals, such as lions. Frogs do not produce a roaring sound, so this option is incorrect.

c) Hiss:
Hissing is a sound usually made by snakes or angry cats. Frogs do not make a hissing sound, so this option is incorrect.

d) Patter:
Pattering refers to a gentle, rhythmic sound, often associated with light footsteps or raindrops. It does not accurately describe the sound frogs make, so this option is incorrect.

Conclusion:
Among the given options, only "croak" accurately describes the sound produced by frogs. Therefore, the correct answer is option 'A' - Frogs croak.

Given:
- Digital watch X beeps every 30 seconds
- Digital watch Y beeps every 32 seconds
- They beeped together at 10 AM

To find:
- The immediate next time that they will beep together

Solution:
To find the next time they beep together, we need to find the LCM (Least Common Multiple) of 30 and 32.

Prime factorization of 30: 2 x 3 x 5
Prime factorization of 32: 2 x 2 x 2 x 2 x 2

LCM: 2 x 2 x 2 x 2 x 3 x 5 = 240

Therefore, the watches will beep together every 240 seconds.

To find the next time they will beep together after 10 AM:
- We need to find out how many seconds have passed since 10 AM.
- Since there are 60 minutes in an hour and 60 seconds in a minute, 1 hour = 60 x 60 = 3600 seconds.
- At 10 AM, 3,600 seconds have passed.
- Add the seconds in the current time to get the total number of seconds passed since 10 AM.
- For example, if the current time is 10:08 AM, then 3,600 + 8 x 60 = 4,980 seconds have passed.
- Now, divide the total number of seconds passed by 240 (the LCM) and take the remainder.
- If the remainder is 0, then the watches will beep together now.
- If the remainder is not 0, then the next time they will beep together is when the number of seconds passed plus the remainder equals a multiple of 240.

Let's apply this to the answer choices:
a) 10.42 AM: 10,620 seconds have passed (3,600 + 42 x 60). Remainder = 60. Next beep together at 10,680 seconds (10.08 AM + 9 minutes)
b) 10.00 PM: 12 hours = 43,200 seconds. Remainder = 0. They will beep together at 10 PM.
c) 10.08 AM: 4,080 seconds have passed. Remainder = 0. They will beep together now.
d) 11.00 AM: 7,200 seconds have passed. Remainder = 0. They will beep together at 11 AM.

Therefore, the immediate next time they will beep together is 10.08 AM, which is option (c).

Given statements:
(i) Some roses are red.
(ii) All red flowers fade quickly.
(iii) Some roses fade quickly.

Explanation:

To determine which statements can be logically inferred from the given statements, we need to analyze the relationships and implications between them.

Statement (i): Some roses are red.
This statement implies that there exists at least one rose that is red. It does not provide any information about other roses or flowers.

Statement (ii): All red flowers fade quickly.
This statement implies that every red flower fades quickly. It does not provide any information about roses that are not red or about roses in general.

Statement (iii): Some roses fade quickly.
This statement implies that there exists at least one rose that fades quickly. It does not provide any information about the color of the roses or about other flowers.

Logical Inferences:
Based on the given statements, we can make the following logical inferences:

Inference 1: If (i) is true and (ii) is true, then (iii) is true.
If there exists at least one rose that is red (i) and all red flowers fade quickly (ii), then it is implied that there exists at least one rose that fades quickly (iii). This inference is valid because the conditions in (i) and (ii) are fulfilled.

Inference 2: If (i) is true and (ii) is false, then (iii) is true.
If there exists at least one rose that is red (i) and not all red flowers fade quickly (ii), then it is still possible for there to exist at least one rose that fades quickly (iii). This inference is valid because the conditions in (i) are fulfilled, and the condition in (ii) does not contradict the possibility of (iii).

Inference 3: If (i) is false and (ii) is true, then (iii) is false.
If there are no roses that are red (i) and all red flowers fade quickly (ii), then it is not possible for there to exist any rose that fades quickly (iii). This inference is valid because the condition in (ii) implies that only red flowers fade quickly, and if there are no red roses, then there are no roses that fade quickly.

Inference 4: If (i) is false and (ii) is false, then (iii) is false.
If there are no roses that are red (i) and not all red flowers fade quickly (ii), then it is not possible for there to exist any rose that fades quickly (iii). This inference is valid because the condition in (ii) does not imply that all roses fade quickly, and the condition in (i) implies that there are no red roses.

Conclusion:
Based on the logical inferences, we can conclude that if both (i) and (ii) are true, then (iii) is true. Hence, option 'C' is the correct answer.

Given Information:
- M and N have four children: P, Q, R, and S.
- P and R are married.
- P has a child named X.
- R has a child named Y.
- Y is a legitimate child of W.

Analysis:
Let's analyze the given options one by one to determine which one is necessarily false.

a) M is the grandmother of Y:
- Since P is married and has a child named X, M can be the grandmother of X.
- Since R is married and has a child named Y, M can also be the grandmother of Y.
- Therefore, option a) is true.

b) R is the father of Y:
- Given that R has a child named Y, it is true that R is the father of Y.
- Therefore, option b) is true.

c) W is the wife of R:
- The given information states that Y is a legitimate child of W, which means that W is either the mother or the father of Y.
- If W is the mother of Y, then W is the wife of R.
- Therefore, option c) can be true.

d) W is the wife of P:
- The given information states that Y is a legitimate child of W, which means that W is either the mother or the father of Y.
- If W is the mother of Y, then W is the wife of R, not P.
- Therefore, option d) is necessarily false.

Conclusion:
The statement "W is the wife of P" (option d) is necessarily false.