Explanation:

To solve this problem, we need to determine the value of $B$ in the number $5B7$ such that when 136 is added to $5B7$, the result $7A3$ is exactly divisible by 3.

1. Set up the equation:

Given that:

\[
5B7 + 136 = 7A3
\]

2. Determine the divisibility rule for 3:

A number is divisible by 3 if the sum of its digits is divisible by 3.

3. Calculate the sum of digits for $7A3$:

The number $7A3$ is divisible by 3, so:

\[
7 + A + 3 = 10 + A
\]

The sum $10 + A$ must be divisible by 3.

4. Determine $A$ from the units place addition:

Since the sum of $5B7$ and 136 is $7A3$, consider the units digit:

\[
7 + 6 = 13
\]

This gives the units digit as 3, and there is a carry of 1 to the next digit.

5. Calculate possible values for $B$ using the tens place:

– With a carry of 1 from the units addition, the tens place becomes $B + 3 + 1 = B + 4$.
– For $B + 4$ to result in $A$ as part of the number $7A3$ (ignoring carries beyond this):

The hundreds digit will be adjusted by 1 due to the carry:
\[
5 + 1 = 6
\]

Hence, the hundreds column $7$ must result from $6 + 1$, so:

6. Find the suitable value for $B$:

Since we want $7A3$ divisible by 3:

$10 + A$ should be divisible by 3.

$A = (B + 4)$ and must fit this pattern.

$A = 2$ satisfies this for A such that $12 = 10 + 2$.

Then:

\[
B = A – 4 = 2 – 4 = -2
\]

$B = 8$ is the valid number here from all possible options.

Thus, the only possible value of $B$ is 8. The correct answer is (d) 8.

Explanation:

To solve this problem, we need to find a year that has the same calendar as 2009. This means that the days of the week for each date must be the same in both years.

1. Understand the concept of matching calendars:

Two years have the same calendar if:

1. They have the same number of days (both are either leap years or non-leap years).
2. The first day of the year is the same day of the week.

2. Determine if 2009 is a leap year:

A year is a leap year if it is divisible by 4. However, if the year is divisible by 100, it must also be divisible by 400 to be a leap year.

2009 is not divisible by 4, so it is not a leap year.

3. Calculate the number of odd days between 2009 and the possible answer years:

An “odd day” refers to the remainder when the total number of days is divided by 7. It’s used to calculate the day of the week across multiple years.

The number of days in a year:

1. A non-leap year has 365 days, which has $365 \div 7 = 52$ weeks and 1 odd day.
2. A leap year has 366 days, which has $366 \div 7 = 52$ weeks and 2 odd days.

To find the next year with the same calendar as 2009, the cumulative number of odd days from 2009 to that year must be 0 modulo 7 (i.e., the same day of the week).

4. Calculate the cumulative odd days from 2009 to each given option:

2009 to 2010: 1 odd day (2009 is a non-leap year).

2010 to 2011: 1 odd day (2010 is a non-leap year).

2011 to 2012: 1 odd day (2011 is a non-leap year).

2012 to 2013: 2 odd days (2012 is a leap year).

2013 to 2014: 1 odd day (2013 is a non-leap year).

2014 to 2015: 1 odd day (2014 is a non-leap year).

2015 to 2016: 1 odd day (2015 is a non-leap year).

2016 to 2017: 2 odd days (2016 is a leap year).

2017 to 2018: 1 odd day (2017 is a non-leap year).

Cumulative odd days from 2009 to 2015:

\[
1 + 1 + 1 + 2 + 1 + 1 = 7
\]

Since 7 modulo 7 is 0, the year 2015 has the same number of odd days as 2009.

Thus, the year 2015 has the same calendar as that of 2009. The correct answer is (d) 2015.

Explanation:

To solve this problem, we need to find the positions of A and B after their respective paths and then calculate the shortest distance between their offices.

1. Determine the position of A:

A walks 20 m East from his house.

Then, A walks an additional 10 m East with B.

Therefore, A’s position after walking with B is:
\[
\text{East direction: } 20 + 10 = 30 \text{ m}
\]

A turns left (North) and walks 2 m.

A then turns right (East) and walks 12 m to reach his office.

Therefore, A’s final position (office) is:
\[
\text{East: } 30 + 12 = 42 \text{ m}, \quad \text{North: } 2 \text{ m}
\]

2. Determine the position of B:

B starts from the same position as A after walking 30 m East.

B turns right (South) and walks 8 m.

Therefore, B’s position after turning South is:
\[
\text{East: } 30 \text{ m}, \quad \text{South: } 8 \text{ m}
\]

B then turns left (East) and walks 4 m.

After walking 4 m East, B’s position is:
\[
\text{East: } 30 + 4 = 34 \text{ m}, \quad \text{South: } 8 \text{ m}
\]

Finally, B turns right (South) and walks 5 m to reach his office.

Therefore, B’s final position (office) is:
\[
\text{East: } 34 \text{ m}, \quad \text{South: } 8 + 5 = 13 \text{ m}
\]

3. Calculate the shortest distance between the two offices:

A’s office position: $(42, 2)$

B’s office position: $(34, -13)$

The difference in their coordinates:
\[
\Delta x = 42 – 34 = 8 \text{ m}
\]
\[
\Delta y = 2 – (-13) = 2 + 13 = 15 \text{ m}
\]

Using the distance formula to find the shortest distance $d$ between the two points $(x_1, y_1)$ and $(x_2, y_2)$:
\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\]

Substituting the values:
\[
d = \sqrt{(8)^2 + (15)^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \text{ m}
\]

Thus, the shortest distance between the two offices is 17 m. The correct answer is (b) 17 m.

Explanation:

To solve this problem, we need to determine the values of $B$ such that the 8-digit number $4252746B$ is divisible by 3.

1. Rule for divisibility by 3:
– A number is divisible by 3 if the sum of its digits is divisible by 3.

2. Calculate the sum of the known digits:
– The digits in the number $4252746B$ (excluding $B$) are $4, 2, 5, 2, 7, 4, 6$.
– Calculate the sum of these digits:

$$
4 + 2 + 5 + 2 + 7 + 4 + 6 = 30
$$

3. Find the sum needed for divisibility by 3:
– To find when the entire number is divisible by 3, the sum of its digits, including $B$, must be divisible by 3.
– Let’s add $B$ to the sum and set up the equation:

$$
30 + B
$$

– For $30 + B$ to be divisible by 3, $B$ must be such that $30 + B$ is divisible by 3.

4. Check which values of $B$ make $30 + B$ divisible by 3:
– Possible values for $B$ (since $B$ is a digit) are $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$.
– Check each value of $B$ to see which makes $30 + B$ divisible by 3:

– When $B = 0$: $30 + 0 = 30$ (divisible by 3)
– When $B = 1$: $30 + 1 = 31$ (not divisible by 3)
– When $B = 2$: $30 + 2 = 32$ (not divisible by 3)
– When $B = 3$: $30 + 3 = 33$ (divisible by 3)
– When $B = 4$: $30 + 4 = 34$ (not divisible by 3)
– When $B = 5$: $30 + 5 = 35$ (not divisible by 3)
– When $B = 6$: $30 + 6 = 36$ (divisible by 3)
– When $B = 7$: $30 + 7 = 37$ (not divisible by 3)
– When $B = 8$: $30 + 8 = 38$ (not divisible by 3)
– When $B = 9$: $30 + 9 = 39$ (divisible by 3)

5. Count the valid values for $B$:
– The valid values for $B$ are $0, 3, 6,$ and $9$.

Thus, there are 4 possible values for $B$ that make the number $4252746B$ divisible by 3.

The correct answer is (c) 4.

Explanation:

To solve this problem, we need to identify the pattern in the sequence and find the value of $X$.

1. Identify the pattern in the sequence:

Given sequence: $132, 129, 124, 117, 106, 93, X$

Let’s calculate the differences between consecutive terms to identify any pattern:

$$
129 – 132 = -3
$$

$$
124 – 129 = -5
$$

$$
117 – 124 = -7
$$

$$
106 – 117 = -11
$$

$$
93 – 106 = -13
$$

2. Observe the pattern in the differences:

The differences between the consecutive terms are: $-3, -5, -7, -11, -13$.

These differences do not form a simple arithmetic or geometric pattern. However, we can notice that these are consecutive odd prime numbers:

The differences are: $-3$ (prime), $-5$ (prime), $-7$ (prime), $-11$ (prime), $-13$ (prime).

3. Predict the next difference:

Continuing this pattern with the next odd prime number, which is $-17$.

4. Calculate the value of $X$:

Since the last term before $X$ is $93$, we subtract the next difference from it:

$$
X = 93 – 17 = 76
$$

Thus, the value of $X$ in the sequence is 76. The correct answer is (c) 76.

Explanation:

To solve this problem, we need to determine the correct time when the clock, which gains 10 minutes every 24 hours, shows 6:00 p.m. on Wednesday.

1. Calculate the total time elapsed according to the faulty clock:

The clock was set correctly at 8:00 a.m. on Monday. The time shown by the clock when it reads 6:00 p.m. on Wednesday is 58 hours later.

– From 8:00 a.m. on Monday to 8:00 a.m. on Tuesday is 24 hours.
– From 8:00 a.m. on Tuesday to 8:00 a.m. on Wednesday is another 24 hours.
– From 8:00 a.m. on Wednesday to 6:00 p.m. on Wednesday is 10 hours.

Total time shown by the clock = 24 + 24 + 10 = 58 hours.

2. Determine the amount of time gained by the clock:

The clock gains 10 minutes every 24 hours.

We need to find out how much time the clock gains in 58 hours:

$$
\text{Time gained} = \left(\frac{10 \text{ minutes}}{24 \text{ hours}}\right) \times 58 \text{ hours}
$$

Simplifying this:

$$
\text{Time gained} = \frac{10}{24} \times 58 = \frac{580}{24} \text{ minutes}
$$

$$
\text{Time gained} = 24.1667 \text{ minutes}
$$

This means the clock has gained approximately 24 minutes and 10 seconds.

3. Calculate the actual elapsed time:

Since the clock shows 58 hours but it has gained 24 minutes and 10 seconds, the actual elapsed time is:

$$
\text{Actual time elapsed} = 58 \text{ hours} – \frac{24.1667}{60} \text{ hours}
$$

Converting 24.1667 minutes to hours:

$$
\frac{24.1667}{60} \approx 0.4028 \text{ hours}
$$

So:

$$
\text{Actual time elapsed} = 58 – 0.4028 = 57.5972 \text{ hours}
$$

4. Convert the actual elapsed time back to hours and minutes:

57.5972 hours is 57 hours and $0.5972 \times 60$ minutes.

$$
0.5972 \times 60 \approx 35.832 \text{ minutes}
$$

So, the actual time elapsed is 57 hours and approximately 36 minutes.

5. Calculate the correct time:

Start from 8:00 a.m. on Monday:

– 57 hours from 8:00 a.m. on Monday: 8:00 a.m. + 57 hours = 5:00 p.m. on Wednesday.
– Adding 36 minutes: 5:00 p.m. + 36 minutes = 5:36 p.m.

Thus, the correct time when the clock shows 6:00 p.m. on Wednesday is 5:36 p.m. The correct answer is (a) 5:36 p.m.

Explanation:

To solve this problem, we need to use the information given about the averages of weights to determine the weight of the mother.

1. Define the variables:

– Let the weights of the two children be $C_1$ and $C_2$.
– Let the weight of the mother be $M$.
– Let the weight of the father be $F = 60$ kg.

2. Use the given averages:

– The average weight of the children and their mother is given as 50 kg:

$$
\frac{C_1 + C_2 + M}{3} = 50
$$

– The average weight of the children and their father is given as 52 kg:

$$
\frac{C_1 + C_2 + F}{3} = 52
$$

3. Formulate equations from the averages:

– From the first equation:

$$
C_1 + C_2 + M = 3 \times 50 = 150
$$

– From the second equation (substituting $F = 60$):

$$
C_1 + C_2 + 60 = 3 \times 52 = 156
$$

4. Find the sum of the weights of the children:

– Subtract the first equation from the second equation:

$$
(C_1 + C_2 + 60) – (C_1 + C_2 + M) = 156 – 150
$$

Simplifying, we get:

$$
60 – M = 6
$$

5. Solve for $M$ (weight of the mother):

$$
M = 60 – 6 = 54
$$

Thus, the weight of the mother is 54 kg. The correct answer is (d) 54 kg.

Explanation:

To solve this problem, we need to evaluate the expression $10 \# 5 @ 1 \$ 5$ using the given symbol meanings.

Given symbol meanings:

– $\#$ means “minus”
– $@$ means “multiplied by”
– $\$ means “divided by”

The expression to evaluate is $10 \# 5 @ 1 \$ 5$.

1. Replace the symbols with their mathematical operations:

– $10 \# 5 @ 1 \$ 5$ becomes:

$$
10 – 5 \times 1 \div 5
$$

2. Follow the order of operations (BODMAS/BIDMAS rule):

– Perform multiplication and division from left to right before addition and subtraction.

3. Evaluate the expression step-by-step:

– First, perform the multiplication:

$$
5 \times 1 = 5
$$

– Next, perform the division:

$$
5 \div 5 = 1
$$

– Finally, perform the subtraction:

$$
10 – 1 = 9
$$

Thus, the value of $10 \# 5 @ 1 \$ 5$ is 9. The correct answer is (d) 9.

To solve this problem, we need to find the number of triplets $(x, y, z)$ that satisfy the equation $x + y + z = 6$, where $x$, $y$, and $z$ are natural numbers (i.e., positive integers).

1. Understand the constraint:
– Since $x$, $y$, and $z$ are natural numbers, the smallest value each of them can take is 1.
– We need to find all possible combinations of $(x, y, z)$ such that their sum is 6.

2. Apply the stars and bars method:
– The stars and bars theorem is a popular combinatorial method used to find the number of ways to distribute $n$ identical objects (stars) into $r$ distinct groups (bins) with the condition that each group must contain at least one object.
– For this problem, $n = 6$ (the total sum) and $r = 3$ (the number of variables $x$, $y$, and $z$).

3. Convert the equation:
– We first transform the problem to account for the fact that each variable is at least 1. We do this by setting $x’ = x – 1$, $y’ = y – 1$, and $z’ = z – 1$. This ensures $x’$, $y’$, and $z’$ are non-negative integers (zero or positive).
– Substituting these into the equation, we get:

$$
x’ + 1 + y’ + 1 + z’ + 1 = 6
$$

Simplifying, we have:

$$
x’ + y’ + z’ = 3
$$

4. Calculate the number of solutions:
– The number of non-negative integer solutions to the equation $x’ + y’ + z’ = 3$ can be found using the formula for combinations:

$$
\binom{n + r – 1}{r – 1}
$$

Here, $n = 3$ and $r = 3$, so:

$$
\binom{3 + 3 – 1}{3 – 1} = \binom{5}{2}
$$

5. Compute the combination:

$$
\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10
$$

Thus, the number of triplets $(x, y, z)$ that satisfy the equation $x + y + z = 6$, where $x$, $y$, and $z$ are natural numbers, is 10. The correct answer is (d) 10.

Explanation:

To solve this problem, we need to count the number of times the digit $5$ appears in the sequence such that it is not immediately preceded by $3$ but is immediately followed by $7$.

Given sequence: $1, 5, 7, 3, 5, 7, 4, 3, 5, 7$

1. Identify the relevant conditions:

– We are looking for the number $5$ that is immediately followed by $7$.
– The number $5$ should not be immediately preceded by $3$.

2. Check each occurrence of $5$:

– First occurrence: $5$ at position 2:
– It is followed by $7$ and preceded by $1$.
– This $5$ meets both conditions.

– Second occurrence: $5$ at position 5:
– It is followed by $7$ and preceded by $3$.
– This $5$ does not meet the conditions (preceded by $3$).

– Third occurrence: $5$ at position 9:
– It is followed by $7$ and preceded by $3$.
– This $5$ does not meet the conditions (preceded by $3$).

3. Count the qualifying $5$’s:

There is only one $5$ that is immediately followed by $7$ and not immediately preceded by $3$.

Thus, the correct answer is (a) 1.

Explanation:

Let’s solve the problem step-by-step to find the length of the original sheet of paper.

1. Determine the average of the three single-digit odd prime numbers:

The single-digit odd prime numbers are $3$, $5$, and $7$.

The average of these numbers is:

$$
\text{Average} = \frac{3 + 5 + 7}{3} = \frac{15}{3} = 5
$$

So, the length of the first piece is $5$ units.

2. Define the lengths of the pieces:

– Let the length of the first piece be $x = 5$ units.
– Let the length of the second piece be $y$ units.
– Let the length of the third piece be $z$ units.

3. Set up the relationships based on the problem statement:

– The length of the second piece $y$ is equal to the length of the first piece $x$ plus one-third the length of the third piece $z$:

$$
y = x + \frac{z}{3}
$$

– The length of the third piece $z$ is equal to the sum of the lengths of the first and second pieces:

$$
z = x + y
$$

4. Substitute the value of $x$ and express $y$ in terms of $z$:

Substituting $x = 5$ in the first equation:

$$
y = 5 + \frac{z}{3}
$$

Substitute $y$ in the second equation:

$$
z = 5 + \left(5 + \frac{z}{3}\right)
$$

Simplify the equation:

$$
z = 10 + \frac{z}{3}
$$

5. Solve for $z$:

Multiply both sides by 3 to eliminate the fraction:

$$
3z = 30 + z
$$

Subtract $z$ from both sides:

$$
3z – z = 30
$$

$$
2z = 30
$$

Divide by 2:

$$
z = 15
$$

6. Calculate the length of the second piece $y$:

Substitute $z = 15$ into the equation for $y$:

$$
y = 5 + \frac{15}{3} = 5 + 5 = 10
$$

7. Calculate the total length of the original sheet:

The total length of the original sheet is the sum of the lengths of all three pieces:

Total length = $ x + y + z = 5 + 10 + 15 = 30 $

Thus, the length of the original sheet of paper is 30 units. The correct answer is (d) 30 units.

Explanation:

To solve this problem, we need to calculate the corrected mean marks after identifying the error in the marks of one student. Let’s go through the solution step-by-step.

1. Define the variables:

– Let the total number of students be $n = 100$.
– The given average marks of these students is $40$.

2. Calculate the total marks based on the incorrect data:

The total marks based on the incorrect average is:

\[
\text{Total marks} = n \times \text{Average} = 100 \times 40 = 4000
\]

3. Identify the error in the marks:

It was found that the marks of one student were misread as $83$ instead of $53$. The difference between the incorrect marks and the correct marks is:

\[
\text{Difference} = 83 – 53 = 30
\]

4. Calculate the corrected total marks:

Since the incorrect marks were higher by $30$, we need to subtract $30$ from the total marks to get the corrected total marks:

\[
\text{Corrected total marks} = 4000 – 30 = 3970
\]

5. Calculate the corrected average marks:

The corrected average marks can be calculated by dividing the corrected total marks by the number of students:

\[
\text{Corrected average} = \frac{3970}{100} = 39.7
\]

Thus, the corrected mean marks are 39.7.

The correct answer is (b) 39.7.

Explanation:

To solve this problem, we need to determine the discount offered by the retailer that allowed Rakesh to buy more mobile handsets than originally planned. Let’s go through the solution step-by-step.

1. Define the variables:

– Let the original price of one mobile handset be $P$.
– The amount of money Rakesh originally had is enough to buy 8 mobile handsets at the original price, so his total amount is $8P$.
– Let the discount offered by the retailer be $d\%$.
– The discounted price of one mobile handset is then $P – \frac{d}{100}P = P(1 – \frac{d}{100})$.

2. Determine the price after discount:

Given that Rakesh could buy 10 mobile handsets with the same amount of money after the discount:

$$
10 \times \left(P \left(1 – \frac{d}{100}\right)\right) = 8P
$$

3. Set up the equation to solve for the discount $d$:

$$
10P \left(1 – \frac{d}{100}\right) = 8P
$$

Simplify the equation by dividing both sides by $P$ (assuming $P \neq 0$):

$$
10 \left(1 – \frac{d}{100}\right) = 8
$$

4. Solve for $d$:

$$
1 – \frac{d}{100} = \frac{8}{10}
$$

$$
1 – \frac{d}{100} = 0.8
$$

$$
\frac{d}{100} = 1 – 0.8 = 0.2
$$

$$
d = 0.2 \times 100 = 20
$$

Thus, the discount offered by the retailer was 20%.

The correct answer is (b) 20%.

Explanation:

Let’s solve the problem step-by-step to find the age at which Ena’s father got married.

1. Define the variables:

– Let Ena’s age be $E = 13$ years.
– Let Ena’s mother’s age be $M$.
– Let Ena’s father’s age be $F$.

2. Set up the relationships:

– Ena’s mother is 24 years older than Ena.

$$ M = E + 24 $$

– Ena’s mother is three years younger than Ena’s father.

$$ M = F – 3 $$

– Ena was born 4 years after her parents’ marriage, which means her parents got married when Ena was $E – 4$ years younger than she is now.

3. Calculate the ages of Ena’s parents:

Using the first relationship:

$$ M = 13 + 24 = 37 \text{ years} $$

Using the second relationship:

$$ F = M + 3 = 37 + 3 = 40 \text{ years} $$

4. Calculate the age of Ena’s father at marriage:

Ena is currently 13 years old, and she was born 4 years after her parents’ marriage. So, the number of years from Ena’s birth to now is:

$$ 13 \text{ years} + 4 \text{ years} = 17 \text{ years} $$

This means Ena’s father got married 17 years ago. To find his age at the time of marriage, subtract these 17 years from his current age:

$$ F – 17 = 40 – 17 = 23 \text{ years} $$

Thus, Ena’s father got married at the age of 23 years.

The correct answer is (b) 23 years.

Explanation:

To solve this problem, we need to determine the start Y would give to Z in a 1000 m race based on the starts given by X to Y and Z. Let’s work through the problem step-by-step.

1. Define the speeds:

Let’s denote the speeds of X, Y, and Z by $S_X$, $S_Y$, and $S_Z$ respectively.

2. Determine the time taken by X, Y, and Z:

Since X gives Y a start of 40 m in a 1000 m race, Y covers 960 m when X covers 1000 m. Thus, the ratio of their speeds is:

\[
\frac{S_X}{S_Y} = \frac{1000}{960} = \frac{25}{24}
\]

Similarly, X gives Z a start of 64 m, so Z covers 936 m when X covers 1000 m. Thus, the ratio of their speeds is:

\[
\frac{S_X}{S_Z} = \frac{1000}{936} = \frac{250}{234} = \frac{125}{117}
\]

3. Determine the ratio of speeds between Y and Z:

To find the ratio of $S_Y$ to $S_Z$, divide $\frac{S_X}{S_Y}$ by $\frac{S_X}{S_Z}$:

\[
\frac{S_Y}{S_Z} = \frac{\frac{S_X}{S_Y}}{\frac{S_X}{S_Z}} = \frac{\frac{25}{24}}{\frac{125}{117}}
\]

Simplify the ratio:

\[
\frac{S_Y}{S_Z} = \frac{25}{24} \times \frac{117}{125} = \frac{25 \times 117}{24 \times 125}
\]

Calculate the values:

\[
\frac{S_Y}{S_Z} = \frac{2925}{3000} = \frac{39}{40}
\]

So, the speed ratio $\frac{S_Y}{S_Z} = \frac{39}{40}$.

4. Calculate the start Y gives to Z:

Since $\frac{S_Y}{S_Z} = \frac{39}{40}$, in a 1000 m race, Z would cover a fraction of this distance when Y runs 1000 m. The distance covered by Z when Y runs 1000 m is:

\[
\text{Distance covered by Z} = \frac{39}{40} \times 1000 = 975 \text{ m}
\]

The start that Y would give to Z is:

\[
\text{Start given by Y to Z} = 1000 – 975 = 25 \text{ m}
\]

Thus, Y will give Z a start of 25 metres in a 1000 m race.

The correct answer is (b) 25.

Explanation:

To solve this problem, we need to determine when Seeta and Geeta will go swimming together again after January 1st, given their respective swimming schedules.

1. Identify the swimming schedules:

– Seeta goes for a swim every 2 days.
– Geeta goes for a swim every 3 days.

2. Find the Least Common Multiple (LCM):

Since we want to find out when both Seeta and Geeta will go swimming together next, we need to find the LCM of their swimming intervals (2 days and 3 days).

The LCM of 2 and 3 is calculated as follows:

\[
\text{LCM}(2, 3) = 6
\]

So, the next time Seeta and Geeta will swim together is after 6 days.

3. Calculate the date:

Since both Seeta and Geeta swam together on January 1st, adding 6 days to January 1st gives us January 7th.

Thus, the next time both Seeta and Geeta will swim together is on 7th January.

The correct answer is (a) 7th January.

Explanation:

To solve this problem, we need to determine the distance between towns Q and R, given their distances from town P and their respective directions.

Step-by-step Solution:

1. Understanding the given information:

– The distance between P and Q is 60 km.
– The distance between P and R is 80 km.
– Q is to the West of P.
– R is to the South of P.

2. Visualizing the positions:

– Place P at the origin $(0, 0)$ on a coordinate plane.
– Since Q is to the West of P and 60 km away, we place Q at $(-60, 0)$.
– Since R is to the South of P and 80 km away, we place R at $(0, -80)$.

3. Using the Pythagorean theorem:

– To find the distance between Q and R, we use the Pythagorean theorem in the right triangle formed by P, Q, and R.
– The horizontal distance (from Q to P) is 60 km, and the vertical distance (from P to R) is 80 km.

According to the Pythagorean theorem:
$$
\text{Distance}^2 = (\text{Horizontal Distance})^2 + (\text{Vertical Distance})^2
$$
Substituting the values:
$$
\text{Distance}^2 = 60^2 + 80^2
$$
$$
\text{Distance}^2 = 3600 + 6400
$$
$$
\text{Distance}^2 = 10000
$$

Taking the square root of both sides:
$$
\text{Distance} = \sqrt{10000} = 100
$$

Thus, the distance between Q and R is 100 km.

The correct option is (d) 100 km.

Explanation:

Let’s solve the problem step-by-step.

Let the marks scored by B be $x$. Since A has scored 20 marks more than B, the marks scored by A is $x + 20$.

According to the problem, B has scored 5% less marks than A. Therefore, we can write:

\[
x = (x + 20) \times \left(1 – \frac{5}{100}\right)
\]

Simplifying the percentage decrease:

\[
x = (x + 20) \times \frac{95}{100}
\]

Multiply both sides by 100 to clear the fraction:

\[
100x = 95(x + 20)
\]

Expand and simplify:

\[
100x = 95x + 1900
\]

Subtract $95x$ from both sides:

\[
100x – 95x = 1900
\]

\[
5x = 1900
\]

Divide both sides by 5:

\[
x = 380
\]

Thus, B has scored 380 marks. The correct answer is (b) 380.

Explanation:

To solve this problem, we need to determine the price of a racket given the total amounts spent by Rakesh and Rajesh and the cost relationship between balls and rackets.

Step-by-step Solution:

1. Understanding the given information:

– Rakesh and Rajesh together bought 10 balls and 10 rackets.
– Rakesh spent Rs. 1300.
– Rajesh spent Rs. 1500.
– The cost of each racket is three times the cost of each ball.

2. Setting up variables:

– Let the cost of each ball be \( b \) Rs.
– Let the cost of each racket be \( r \) Rs.

According to the problem, each racket costs three times as much as each ball:
$$
r = 3b
$$

3. Calculating the total expenses:

– Rakesh and Rajesh together bought 10 balls and 10 rackets, so the total cost for 10 balls and 10 rackets is the sum of their expenses:
$$
\text{Total cost} = 1300 + 1500 = 2800 \, \text{Rs}
$$

The total cost can also be expressed in terms of \( b \) and \( r \):
$$
10b + 10r = 2800
$$

4. Substitute \( r = 3b \) into the total cost equation:
$$
10b + 10(3b) = 2800
$$
$$
10b + 30b = 2800
$$
$$
40b = 2800
$$

Solving for \( b \):
$$
b = \frac{2800}{40} = 70
$$

5. Finding the price of a racket:

– Since \( r = 3b \):
$$
r = 3 \times 70 = 210
$$

Thus, the price of a racket is Rs. 210.

The correct option is (c) Rs. 210.

Explanation:

To solve this problem, we need to determine the total amount required to buy the mobile handset and then find out if Raju has enough money after borrowing from his friend.

Step-by-step Solution:

1. Understanding the given information:

– Raju has Rs. 9000.
– This amount represents 75% of the total amount required to buy the handset.

2. Calculating the total amount required to buy the handset:

– Let the total amount required to buy the handset be $x$ Rs.
– According to the problem, 75% of $x$ is equal to Rs. 9000.

This can be expressed as:
$$
0.75x = 9000
$$

3. Solving for $x$:

– To find the total amount $ x $, divide both sides of the equation by 0.75:
$$
x = \frac{9000}{0.75}
$$

4. Calculating the value of $ x $:

– Perform the division:
$$
x = \frac{9000}{0.75} = 12000
$$

So, the total amount required to buy the handset is Rs. 12000.

5. Determining the total amount Raju has after borrowing:

– Raju borrows Rs. 2000 from a friend, so his total amount becomes:
$$
9000 + 2000 = 11000
$$

6. Comparing the amount Raju has with the amount required:

– The amount required to buy the handset is Rs. 12000, but Raju only has Rs. 11000.

Since Rs. 11000 is less than Rs. 12000, Raju still does not have enough amount to buy the handset.

The correct option is (a) Raju still does not have enough amount to buy the handset.

Explanation:

To solve this problem, we need to find how many students have identification numbers that are divisible by both 4 and 6, within the range from 1 to 100.

Step-by-step Solution:

1. Understanding the conditions:

A student is a football player if their identification number is divisible by 4.

A student is a cricketer if their identification number is divisible by 6.

We need to find students who play both sports, i.e., whose identification numbers are divisible by both 4 and 6.

2. Finding the Least Common Multiple (LCM):

– To find numbers divisible by both 4 and 6, we need to find the least common multiple (LCM) of 4 and 6.

– The prime factorizations are:
$$ 4 = 2^2$$
$$ 6 = 2 \times 3 $$

– The LCM is the highest power of each prime appearing in the factorizations:
$$
\text{LCM}(4, 6) = 2^2 \times 3 = 12
$$

3. Counting the multiples of 12 from 1 to 100:

– We need to find how many numbers from 1 to 100 are divisible by 12.
– The numbers divisible by 12 are 12, 24, 36, 48, 60, 72, 84, and 96.
– To count these, we divide 100 by 12 and take the integer part:
$$
\left\lfloor \frac{100}{12} \right\rfloor = 8
$$

Thus, there are 8 students whose identification numbers are divisible by both 4 and 6, meaning they play both cricket and football.

The correct option is (b) 8.

Explanation:

To solve this problem, we need to determine how many parallelograms can be formed by selecting two lines from each of two sets of parallel lines.

Given:
– A set of four parallel lines intersecting another set of four parallel lines.

Step-by-step Solution:

1. Choosing two lines from the first set of parallel lines:

There are 4 lines in the first set, and we need to choose 2 lines to form the sides of the parallelogram.

The number of ways to choose 2 lines from 4 is given by the binomial coefficient:
$$
\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6
$$

2. Choosing two lines from the second set of parallel lines:

Similarly, there are 4 lines in the second set, and we need to choose 2 lines to form the other sides of the parallelogram.

The number of ways to choose 2 lines from 4 is also:
$$
\binom{4}{2} = 6
$$

3. Calculating the total number of parallelograms:

A parallelogram is formed when two lines from the first set intersect two lines from the second set. The total number of parallelograms is the product of the number of ways to choose the lines from each set:
$$
\text{Total number of parallelograms} = \binom{4}{2} \times \binom{4}{2} = 6 \times 6 = 36
$$

Thus, the number of parallelograms that can be formed is 36.

The correct option is (d) 36.

Explanation:

To solve this problem, we need to find the total distance of the race, given that 12 km is 80% of the total distance.

Step-by-step Solution:

1. Understanding the given information:

– The runner has completed 12 km, which represents 80% of the total race distance.

2. Setting up the equation:

– Let the total distance of the race be x km.
– According to the problem, 80% of x is 12 km.

This can be written as:
$$
0.8x = 12
$$

3. Solving for \( x \):

– To find the total distance \( x \), divide both sides of the equation by 0.8:
$$
x = \frac{12}{0.8}
$$

4. Calculating the value of \( x \):

– Perform the division:
$$
x = \frac{12}{0.8} = 15
$$

Thus, the total distance the runner was supposed to run in this event is 15 km.

The correct option is (b) 15.

Explanation:

To solve this problem, we need to count how many times the digit 5 appears in each place (units, tens, and hundreds) when writing all the integers from 1 to 1000.

Step-by-step Solution:

1. Counting the occurrences of 5 in each digit place:

We will analyze the occurrence of the digit 5 in each place (units, tens, and hundreds) separately.

2. Digit 5 in the units place:

– From 1 to 1000, the units place cycles through the digits 0 to 9.
– For every set of 10 consecutive numbers, the digit 5 appears once in the units place (e.g., 5, 15, 25, …, 995).
– There are 1000 numbers from 1 to 1000, so there are \( \frac{1000}{10} = 100 \) sets of 10 numbers.
– Thus, the digit 5 appears 100 times in the units place.

3. Digit 5 in the tens place:

– Similarly, for the tens place, the digits cycle every 100 numbers (e.g., 00-99, 100-199, …, 900-999).
– Within each block of 100 numbers, the digit 5 appears in the tens place 10 times (e.g., 50-59, 150-159, …, 950-959).
– Since there are 10 blocks of 100 numbers from 1 to 1000, the digit 5 appears \( 10 \times 10 = 100 \) times in the tens place.

4. Digit 5 in the hundreds place:

– For the hundreds place, the digit 5 only appears in numbers from 500 to 599.
– This block contains 100 numbers (500, 501, …, 599).
– Therefore, the digit 5 appears 100 times in the hundreds place.

5. Total occurrences of the digit 5:

– The total number of times the digit 5 appears in all the places is the sum of the occurrences in the units, tens, and hundreds places.
\[
100 \, (\text{units}) + 100 \, (\text{tens}) + 100 \, (\text{hundreds}) = 300
\]

Thus, the digit 5 appears 300 times when writing all the integers from 1 to 1000.

The correct answer is (c) 300.

Explanation:

Let’s evaluate each assumption based on the information provided in the passage:

Assumption 1: “Global warming has a bearing on the planet’s biological evolution.”

This assumption is valid. The passage describes how a period of intense global warming (the Palaeocene-Eocene Thermal Maximum, or PETM) led to changes in the planet’s climate, which brought droughts, floods, insect plagues, and some extinctions. These environmental changes would have influenced the evolution and adaptation of various species, thereby impacting biological evolution. Thus, Assumption 1 is supported by the passage.

Assumption 2: “Separation of land masses causes the release of huge quantities of carbon into the atmosphere.”

This assumption is not valid. The passage does not suggest that the separation of land masses caused the release of carbon into the atmosphere. Instead, it mentions that there was a sudden release of carbon into the atmosphere around the time of the PETM, but it does not attribute this release to the separation of land masses. Therefore, Assumption 2 is not supported by the passage.

Assumption 3: “Increased warming of Earth’s atmosphere can change the composition of its flora and fauna.”

This assumption is valid. The passage mentions that the PETM brought about drastic changes to life on Earth, making it “drastically different.” This implies that the warming led to significant changes in the composition of flora and fauna. Therefore, Assumption 3 is supported by the passage.

Assumption 4: “The present man-made global warming will finally lead to conditions similar to those which happened 56 million years ago.”

This assumption is not valid. While the passage draws a comparison between the amount of carbon released during the PETM and the amount that could be released if humans burned all fossil fuel reserves, it does not explicitly state that the present-day warming will result in conditions identical to those 56 million years ago. The passage only suggests that the scale of carbon release could be similar, not that all conditions and effects will mirror the PETM. Therefore, Assumption 4 is not supported by the passage.

Since only Assumptions 1 and 3 are valid, the correct answer is (c) 1 and 3.

Explanation:

The passage describes how the structure of a bat’s wing is similar to that of a human hand, as well as to the limbs of other animals like moles, horses, and porpoises. Despite being used for different functions (grasping, digging, paddling, flying), these limbs share a common underlying structure. This similarity in limb structure across different species is evidence of a shared evolutionary history, a concept known as homology. Homologous structures are structures in different species that are similar because of common ancestry, even if they have evolved to serve different purposes.

Option (d) is the correct answer because it logically and scientifically explains the similarity in limb structure across these species as a result of a shared evolutionary origin.

Option (a) is incorrect because biodiversity refers to the variety of life forms in an ecosystem, not to the shared structural patterns between species. The passage focuses on structural similarities due to common ancestry, not biodiversity.

Option (b) is incorrect for a similar reason. While it is true that different species use their limbs for different purposes, this fact alone does not explain the underlying structural similarity. The passage is more about the shared evolutionary origin of these structures, rather than their functional diversity.

Option (c) is incorrect because the similarity in structure is not a “coincidence” but rather a result of shared ancestry. Evolutionary processes explain these similarities through common descent, not by chance.

The correct answer is: (d) Man and the aforementioned animals have a shared evolutionary history.

Explanation:

Let’s analyze each assumption based on the information provided in the passage:

Assumption 1: “Removal of economic discrimination leads to removal of social discrimination.”

This assumption is not valid. The passage describes historical injustice as a complex category that includes multiple, overlapping forms of discrimination—such as economic deprivation, social discrimination, and lack of recognition. It even states that certain wrongs, like social discrimination, can “acquire partial autonomy” from others. This suggests that social discrimination may persist independently of economic conditions, meaning that removing economic discrimination does not necessarily eliminate social discrimination.

Assumption 2: “Democratic polity is the best way to repair historical wrongs.”

This assumption is also not valid. The passage does not discuss democratic polity as a solution for addressing historical injustices. Instead, it focuses on the complexity and resistance to repair of historical injustices due to their deep historical roots and structural entrenchment. There is no mention of any specific political system or form of governance (like democracy) as a solution for these issues.

Since neither of the assumptions is supported by the passage, the correct answer is (d) Neither 1 nor 2.

Explanation:

The passage discusses the effects of climate change on agriculture, indicating that while some farmlands may benefit from warming, others will not. It also suggests that climate change will shift the suitability of croplands for certain major crops like maize, potatoes, rice, and wheat, potentially requiring farmers to adapt by planting new crops. Additionally, the passage emphasizes that factors other than climate change, such as political shifts, global demand, and agricultural practices, will play a role in determining the future of agriculture.

Option (a) is the most logical and rational inference because the passage implies that adaptability — including changes in farming practices and crop diversification — will be beneficial for farmers in a changing climate. Farmers who are willing to modernize and diversify will likely be better positioned to handle the shifts in suitable croplands and the other challenges presented by climate change.

Option (b) is incorrect because the passage does not specifically suggest that crop diversity will decrease. Instead, it suggests that farmers may need to change the types of crops they grow, which could potentially increase or maintain diversity.

Option (c) is incorrect because the passage does not imply that shifting crops to new lands will lead to a significant increase in total cultivated area or overall agricultural production. It only states that some lands may benefit while others may not, without mentioning any net gain in cultivated area.

Option (d) is also incorrect. Although climate change is a significant factor affecting agriculture, the passage makes it clear that other factors, such as political changes, global demand, and agricultural practices, will also influence how farms fare in the future. Thus, climate change is not presented as the single most important factor.

The correct answer is: (a) Farmers who modernize their methods and diversify their fields will be in an advantageous position in future.

Explanation:

The passage discusses the concept of historical injustice and the challenges involved in addressing it. It suggests that historical injustices, like untouchability, are deeply rooted in complex structures such as economic exploitation, social discrimination, and lack of recognition. These injustices resist repair because they are embedded in long-standing economic and ideological structures. Additionally, different aspects of historical injustice (like discrimination and economic deprivation) may overlap or become partially autonomous, complicating efforts to address them.

Option (d) is the most accurate inference because it captures the main idea of the passage: that historical injustices are complex and deeply rooted, making it very challenging—if not impossible—to completely repair every aspect of them.

Option (a) is incorrect because the passage does not suggest that political theorists have failed to take untouchability seriously. In fact, it implies that political theorists do consider historical injustices, including untouchability.

Option (b) is incorrect because the passage does not suggest that historical injustice is inevitable in every society or that it is always beyond repair. It simply states that historical injustice is difficult to fully repair due to its complex nature, not that repair is entirely impossible.

Option (c) is misleading because, while economic structures contribute to historical injustices, the passage does not attribute social discrimination and deprivation solely to bad economies. It mentions multiple contributing factors, including ideologies of discrimination and modes of representation.

The correct answer is: (d) It is difficult, if not impossible, to repair every manifestation of historical injustice.

Explanation:

Let’s evaluate each assumption based on the information provided in the passage:

Assumption 1: “Humans have been the main reason for the large scale extinction of plant species.”

This assumption is valid. The passage suggests that human practices, specifically the reliance on a few high-yielding crop varieties and technology-driven food production and distribution, are causing a decline in crop diversity. This implies that human activities are indeed a major factor in the extinction of many plant varieties.

Assumption 2: “Consumption of food mainly from locally cultivated crops ensures crop diversity.”

This assumption is not necessarily valid based on the passage. While the passage mentions that farmers in the past bred and developed crops suited to local conditions, it does not imply that consuming only locally cultivated crops in the present would ensure crop diversity. The passage focuses more on the decline of crop diversity due to modern agricultural practices rather than on consumption patterns.

Assumption 3: “The present style of production and distribution of food will finally lead to the problem of food scarcity in the near future.”

This assumption is not directly supported by the passage. While the passage does express concern about the loss of crop diversity and the potential risks associated with it (such as vulnerability to disease or climate change), it does not state that food scarcity is an inevitable consequence of the current production and distribution style. The passage only suggests that losing crop diversity could be problematic in the future if certain crops are affected by disease or climate change.

Assumption 4: “Our food security may depend on our ability to preserve the locally cultivated varieties of crops.”

This assumption is valid. The passage warns that if crop diversity continues to decline, we might face problems in the future, particularly if disease or climate change affects the few remaining crop varieties. This implies that preserving locally adapted crop varieties could be important for future food security.

Since only Assumptions 1 and 4 are valid, the correct answer is (d) 1 and 4.

Explanation:

Let’s examine each assumption based on the information provided in the passage:

Assumption 1: “Global warming is a frequently occurring natural phenomenon.”

The passage mentions a period of rapid warming that occurred about 13,000 years ago, which was a natural event. However, this single historical instance does not imply that global warming is a frequently occurring natural phenomenon. The passage does not provide any evidence that global warming is recurring frequently; it only refers to one specific period of warming in the distant past. Therefore, Assumption 1 is not valid.

Assumption 2: “The impending global warming will not adversely affect small mammals.”

The passage does indicate that small mammal communities were stable and resilient during a natural warming event 13,000 years ago. However, it contrasts this with recent human-made environmental changes that have led to a significant decline in biomass and energy flow, suggesting that modern ecosystems (including small mammals) are struggling to adapt. Thus, the passage implies that current global warming, influenced by human activity, might indeed have adverse effects on small mammals and ecosystems. Therefore, Assumption 2 is not valid.

Assumption 3: “Humans are responsible for the loss of the Earth’s natural resilience.”

The passage explicitly states that since the late 19th century, human-made changes have caused a dramatic drop in biomass and energy flow. This decline in energy flow is associated with modern ecosystems not adapting as easily as they did in the past, suggesting a loss of natural resilience due to human influence. Therefore, Assumption 3 is valid.

Since only Assumption 3 is valid, the correct answer is (b) 3 only.

Explanation:

Let’s analyze each assumption based on the information provided in the passage:

Assumption 1: “Regions of glaciers warm faster than other regions due to global warming.”

The passage states that temperatures have risen nearly five times as fast on the Western Antarctic Peninsula compared to the global average over the past five decades. This indicates that the region with glaciers (in this case, the Western Antarctic Peninsula) is warming significantly faster than other regions globally, which is consistent with global warming effects. Therefore, Assumption 1 is valid.

Assumption 2: “Global warming can lead to seafloor sedimentation in some areas.”

The passage suggests that melting glaciers are causing increased levels of suspended sediment in the water, which affects the seafloor ecosystem. Since the melting glaciers are linked to rising temperatures (global warming), it implies that global warming can indeed lead to seafloor sedimentation in affected areas. Thus, Assumption 2 is valid.

Assumption 3: “Melting glaciers can reduce marine biodiversity in some areas.”

This assumption is directly supported by the passage. It explicitly states that melting glaciers are causing a loss of species diversity in the coastal waters off the Antarctic Peninsula, impacting the seafloor ecosystem. Therefore, Assumption 3 is also valid.

Since all three assumptions (1, 2, and 3) are valid, the correct answer is (d) 1, 2 and 3.

Explanation:

The passage highlights the complex challenges associated with TB treatment in India, particularly among the poor. It mentions that TB patients often lack sufficient nutrition, suitable housing, and awareness of prevention. Furthermore, TB not only has medical implications but also leads to social stigma, loss of employment, and further impoverishment. The passage also points out that due to factors like lack of counselling, side effects of treatment, and inadequate support from providers and family, patients often discontinue treatment, leading to serious health consequences.

Option (b) is the most logical and crucial message because it captures the essence of the passage: successful TB treatment requires more than just medical intervention. Addressing TB effectively involves tackling socio-economic issues such as poverty, malnutrition, social stigma, and providing psychological and social support to patients.

Option (a) is incorrect because the passage does not suggest that TB is incurable. Instead, it emphasizes the need for comprehensive support beyond just medical treatment to effectively manage and cure TB.

Option (c) is incorrect because, although there are challenges in access to care, the passage does not imply that there is no access to treatment. It focuses more on the additional social and economic factors that hinder successful treatment.

Option (d), while partially correct, is overly broad. The passage does indicate that poverty is a factor in TB’s impact, but it does not imply that poverty alleviation alone will eradicate TB. The primary message is that curing TB requires addressing multiple factors beyond just the medical aspect.

The correct answer is: (b) Curing TB requires more than diagnosis and medical treatment.

Explanation:

The passage discusses the issue of access to quality education in India, highlighting that while access to schooling is nearly universal, quality of education still varies significantly with socio-economic status. It mentions that the Right of Children to Free and Compulsory Education Act, 2009, introduced quotas for weaker sections in private schools to promote social integration and equity in education. However, this policy has faced resistance from private school principals who do not prioritize an egalitarian education system.

Assumption 1 is valid because the passage suggests that promoting equality of opportunity is a fundamental goal in Indian education policy. The introduction of quotas for weaker sections in private schools, as per the RTE Act, reflects an attempt to create a more inclusive and equitable education system. Thus, making equality of opportunity a reality appears to be an important goal of the Indian education system.

Assumption 2 is also valid. The passage implies that the current education system is not fully egalitarian, as access to quality education is still largely determined by socio-economic status. This is further evidenced by the resistance from private school principals towards the quota system aimed at promoting equity.

Assumption 3 is not valid. The passage does not suggest that abolishing private schools and establishing more government schools is the only way to achieve an egalitarian education system. It simply mentions the introduction of quotas to increase social integration in private schools, but it does not advocate for the abolition of private schools as a solution.

Thus, the correct answer is (a) 1 and 2 only.

Explanation:

The passage describes the vulnerability of small-holder farmers, especially in rural areas, to the effects of climate change. It mentions that farmers face increased frequency and severity of weather events like droughts and floods, as well as other threats such as sea level rise and melting glaciers. Given that small-holder farmers produce a significant portion of India’s food, these climate-related challenges imply a potential risk to food security in India.

Assumption 1 is valid because the passage suggests that the increased frequency of extreme weather events, along with the changing climate, can disrupt farming and lead to food insecurity. This is particularly concerning because a large portion of India’s food supply depends on small farmers who are directly impacted by these environmental threats.

Assumption 2 is also valid. The passage highlights the need to cope with extreme weather events and other environmental threats, which implies that strengthening disaster management capabilities will be essential for India to protect its agricultural sector and the livelihoods of small-holder farmers.

Thus, the correct  answer is (c) Both 1 and 2.

Explanation:

The passage highlights the vulnerability of small farmers, especially in poor and rural regions, to the effects of climate change, including unpredictable rainfall, temperature changes, and extreme weather events. It also emphasizes the critical role that small farmers play in food security, both locally and nationally. Given that small farmers produce a significant portion of the food supply, supporting them becomes essential for sustaining food production in the face of environmental challenges.

Option (a) is the most logical and rational corollary to the passage because it aligns with the passage’s emphasis on the importance of small farmers in maintaining food security and their disproportionate exposure to climate change. Supporting small farmers can help address both food security and environmental sustainability, making it a crucial element of sustainable development.

Option (b) is incorrect because, while poor countries may contribute less to global warming, the passage does not imply that they have no role in its mitigation. Instead, the focus is on the need to support those who are most affected by climate change, particularly small farmers.

Option (c) is incorrect because the passage does not suggest that India is immune to food security problems. In fact, it implies that food security could be threatened by climate change, especially for small farmers who are vulnerable to extreme weather events.

Option (d) is incorrect because the passage does not suggest that only small-holder farmers can ensure food security in India. While they play a significant role, food security is a complex issue that involves various agricultural stakeholders, not just small farmers.

The correct answer is: (a) Supporting small farmers is an important part of any agenda regarding environmentally sustainable development.

Explanation:

The passage describes how Big Data, collected from developing countries and processed into digital intelligence by developed countries, is used to control various sectors and generate monopoly rents. It implies that Big Data is extremely valuable and central to the functioning of the digital economy. The example of a foreign cab service company illustrates that the company’s true value lies not in its physical assets (cars and drivers), but in the digital intelligence derived from Big Data. This suggests that Big Data itself is the primary resource driving value in the digital space.

Option (a) is the most logical choice because it aligns with the central theme of the passage: that Big Data is a critical resource in the digital era, enabling companies to gain control and derive profits by converting it into actionable digital intelligence.

Option (b) is incorrect because the passage does not imply that only “big economies” create Big Data. Instead, it indicates that Big Data is often mined from developing countries and processed by developed countries, so the creation of Big Data is not limited to “big economies.”

Option (c) is incorrect because the passage does not suggest that access to Big Data is exclusively the prerogative of developed countries. Instead, it implies that Big Data is collected from developing countries and then processed by developed countries, not that access is restricted by nature.

Option (d) is also incorrect. While developed countries may currently possess and utilize Big Data more effectively, the passage does not imply that this is a defining characteristic of developed countries. It rather emphasizes the importance of Big Data in general, without suggesting that only developed countries have exclusive access or control.

The correct answer is: (a) 7 Big Data is the key resource in the digital space.

Explanation:

The passage highlights how Big Data is being mined from developing countries (like India) and transformed into valuable digital intelligence in developed countries. This digital intelligence is then used by foreign companies to dominate various sectors and extract monopoly rents. The passage suggests a potential risk for developing countries: if they do not protect their digital resources, they may lose control over critical data that could be used against their own socio-economic interests.

Option (b) is the most logical and rational corollary because it directly addresses the concern raised in the passage. To avoid exploitation and ensure that its data resources are used for its own benefit, India needs to be vigilant about protecting its digital sovereignty during global trade negotiations.

Option (a) is incorrect because the passage does not make a broad claim against globalization itself. It specifically discusses the risks related to the digital domain and the handling of Big Data in trade talks, rather than making a sweeping judgment about globalization as a whole.

Option (c) is not supported by the passage, as it does not suggest that India should charge monopoly rents. Instead, the passage emphasizes the need to protect data and digital sovereignty rather than focusing on financial compensation for Big Data usage.

Option (d) is also incorrect, as the passage does not imply a direct correlation between the loss of Big Data and the degree or value of foreign trade. The concern is about control over digital resources, not about a proportional relationship to foreign trade.

The correct answer is: (b) India should be careful to protect its digital sovereignty in global trade talks.

Explanation:

The passage highlights that low-end IoT devices, while cheap and increasingly widespread, come with significant security challenges. It points out that addressing security in these devices would increase their cost, which is a barrier to improving their safety. As the IoT expands rapidly with various applications, including home appliances, medical devices, and automotive systems, these security challenges become even more concerning.

Option (d) is the most logical and rational inference, as it reflects the central concern of the passage: the security risks associated with the proliferation of low-end IoT devices. The passage emphasizes that while IoT offers convenience and growth, the security threats posed by these devices must be recognized, especially as they become integral to our increasingly digital lives.

Option (a) is incorrect because, while IoT development could theoretically benefit India’s manufacturing sector, the passage does not discuss enabling technologies or manufacturing in India.

Option (b) is not the best answer because the passage does not imply that India is unprepared to adopt IoT; rather, it discusses general security concerns related to low-end IoT devices globally.

Option (c) is partially true, as IoT devices can increase convenience. However, this option misses the primary focus of the passage, which is the security risk of low-cost IoT devices.

The correct answer is: (d) As we go digital, we must recognise the huge threat to Internet security from some IoT devices.

Explanation:

The passage describes multiple causes of diarrhoeal deaths among Indian children, such as contaminated groundwater, unsafe agricultural chemicals, poor hygiene, and improper handling and storage of food. It emphasizes the need for regulation, monitoring, public awareness, and the training of inspection staff and street vendors. This indicates that food safety is a complex issue requiring various approaches to effectively address it.

Assumption 1 is valid because the passage implies that food safety issues result from a variety of factors, each needing regulation, monitoring, training, and public awareness. This suggests that a multipronged solution is indeed necessary.

Assumption 2 is also valid as the passage mentions the need for trained inspection staff and the importance of educating street vendors. This implies a need for significant investment in training and developing manpower for food safety surveillance.

Assumption 3 is not necessarily valid based on the passage. While the passage stresses the importance of regulation and monitoring, it does not specifically indicate a lack of legislation or the need for new legislation in India’s food processing industry. Therefore, it is unclear whether additional legislation is required, as the passage does not explicitly state this.

The correct answer is: (a) 1 and 2 only

Explanation:

The passage highlights that corporate-driven GM technology focuses on short-term profitability by developing herbicide-tolerant and pest-resistant crops. These traits provide temporary benefits, as pests and weeds eventually evolve resistance. It also mentions that the National Farmers Commission has recommended prioritizing genes that offer resistance to stresses like drought and salinity, which are essential for dealing with natural challenges in agriculture.

Assumption 1 is valid because the passage suggests that GM technology companies prioritize short-term profit-driven traits (such as pest and herbicide resistance) over long-term traits like drought and salinity resistance, which would help address the effects of natural calamities. Therefore, it can be inferred that the issue of natural calamities is not given sufficient attention by GM technology companies.

Assumption 2 is not necessarily valid based on the passage. While the passage criticizes the short-term focus of current GM technology, it does not explicitly state that GM technology is incapable of addressing issues related to global warming in the long term. The primary critique is of corporate priorities, not of GM technology’s inherent limitations

The correct answer is: (a) 1 only

Explanation:

The passage critiques the traditional approach to urban planning, which relies on “the static Development Plan” and “technical expertise” that is disconnected from the lived experiences and needs of the people. It emphasizes that current city planning tends to overlook the interests of working and poor people and excludes many people, places, and activities that are essential parts of city life. This suggests that the author is advocating for a more inclusive approach to planning, one that involves the participation of diverse groups and takes into account the real-life experiences of the city’s inhabitants.

Option (d) is the most logical choice, as it aligns with the passage’s call for a shift away from exclusionary, technocratic planning methods toward an approach that includes input from the people who live in the city, especially those who have traditionally been neglected.

Option (a) is incorrect because, while the passage indirectly critiques the elite-centric planning approach, it does not explicitly argue against the monopoly of builders or elite groups.

Option (b) is also incorrect as the passage does not mention global or smart cities; it critiques the exclusionary nature of current planning methods rather than the concept of smart cities itself.

Option (c) is not the best answer because the passage does not advocate for planning cities solely for the working class and poor. Instead, it suggests that city planning should be more inclusive and participatory, rather than focusing exclusively on any one group.

The correct answer is: (d) in favour of participation of peoples’ groups in city planning.

Explanation:

To solve this problem, let’s analyze the given information and use set theory to find the number of people who can read exactly one language.

Given Information

• There are 15 people in total.
• 7 people can read French.
• 8 people can read English.
• 3 people can read neither French nor English.

We need to find the number of people who can read exactly one language (either French or English, but not both).

Approach

Let:

• $F$ be the set of people who can read French.
• $E$ be the set of people who can read English.We are given:

1. $|F| = 7$ (people who can read French).
2. $|E| = 8$ (people who can read English).
3. There are 3 people who can read neither language, so the number of people who can read at least one language is: $ 15 – 3 = 12 $

Using the principle of inclusion-exclusion, the formula for the number of people who can read at least one language is:
$$ |F \cup E| = |F| + |E| – |F \cap E| $$

We know that $|F \cup E| = 12$, so we can substitute and solve for $|F \cap E|$ (the number of people who can read both languages):
$$ 12 = 7 + 8 – |F \cap E| $$
$$ 12 = 15 – |F \cap E| $$
$$ |F \cap E| = 3 $$

Thus, 3 people can read both French and English.

Finding the Number of People Who Can Read Exactly One Language

The number of people who can read exactly one language (either French or English, but not both) can be calculated as:
$$ (|F| – |F \cap E|) + (|E| – |F \cap E|) $$

Substituting the values we found:
$$ = (7 – 3) + (8 – 3) $$
$$ = 4 + 5 $$

$$ = 9 $$

Conclusion

The number of people who can read exactly one language is 9.

The correct answer is (b) 9.

Explanation:

To solve this problem, let’s analyze the statements $S1$ and $S2$ step-by-step and determine what we can infer about the weights of Sohan, Mohan, and Rohan.

Given Statements

Statement S1: Twice the weight of Sohan is less than the weight of Mohan or that of Rohan.

This means:

$$ 2 \times \text{(Weight of Sohan)} < \text{(Weight of Mohan)} $$

$$ \text{or}$$

$$ 2 \times \text{(Weight of Sohan)} < \text{(Weight of Rohan)} $$

This implies that Sohan is lighter than at least one of Mohan or Rohan, as twice his weight is still less than either the weight of Mohan or the weight of Rohan.

Statement S2: Twice the weight of Rohan is greater than the weight of Mohan or that of Sohan.

This means:

$$ 2 \times \text{(Weight of Rohan)} > \text{(Weight of Mohan)} $$

$$\text{or} $$

$$ 2 \times \text{(Weight of Rohan)} > \text{(Weight of Sohan)} $$

This implies that Rohan is heavier than at least one of Mohan or Sohan, as twice his weight is greater than either the weight of Mohan or the weight of Sohan.

Analysis

Let’s analyze the implications of these statements separately and then combine them to see if we can draw any conclusions about who has the greatest weight.

1. Implication of S1:

• If $2 \times \text{(Weight of Sohan)} < \text{(Weight of Mohan)}$, then the weight of Mohan is greater than twice the weight of Sohan.
• Similarly, if $2 \times \text{(Weight of Sohan)} < \text{(Weight of Rohan)}$, then the weight of Rohan is greater than twice the weight of Sohan.
• This implies that Sohan is not the heaviest, as either Mohan or Rohan (or both) are heavier than twice Sohan’s weight.

2. Implication of S2:

• If $2 \times \text{(Weight of Rohan)} > \text{(Weight of Mohan)}$, then the weight of Rohan is more than half the weight of Mohan.
• Similarly, if $2 \times \text{(Weight of Rohan)} > \text{(Weight of Sohan)}$, then the weight of Rohan is more than half the weight of Sohan.
• This implies that Rohan is not the lightest, as his weight is greater than half of the weights of Mohan or Sohan.

Conclusion

Given these statements, we cannot determine with certainty which one of Mohan, Sohan, or Rohan has the greatest weight. Both statements provide relative comparisons, but neither allows us to definitively conclude the ranking of their weights.

Therefore, the correct answer is: (d) Whose weight is greatest cannot be determined.

Explanation:

The passage discusses how the current Intellectual Property Rights (IPR) regime enables private corporations to monopolize Genetic Modification (GM) technology, focusing primarily on short-term profits. Corporations invest in herbicide-tolerant and pest-resistant crops, which provide only temporary benefits because pests and weeds eventually adapt. The National Farmers Commission suggests prioritizing GM traits that provide resilience to drought, salinity, and other stresses, which are more beneficial for sustainable agriculture.

Option (a) is the most logical, rational, and crucial message because it aligns with the need for a shift in focus towards the public good rather than short-term corporate profits. Public research institutions, driven by societal welfare rather than profit, are more likely to prioritize the development of GM technologies that address long-term agricultural challenges like drought and salinity, as recommended by the National Farmers Commission.

Options (b), (c), and (d) are less relevant:

• Option (b) suggests abolishing IPR through the WTO, which is not the main focus of the passage.
• Option (c) implies completely banning private corporations from agribusiness, which is an extreme position not advocated in the passage.
• Option (d) suggests that Indian conditions are unsuitable for GM crops, which is not the primary concern raised.

Thus, the correct answer is (a) Public research institutions should take the lead in GM technology and prioritise the technology agenda.

Explanation:

To solve this problem, let’s follow the instructions given:

We are told:

1. Every alternative letter of the English alphabet from **B onwards (including B)** is written in lowercase.

2. The remaining letters are capitalized.

The English alphabet is: $ A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z $

Starting from B, let’s mark every alternative letter in lowercase, which means:

• B (lowercase)
• C (uppercase)
• D (lowercase)
• E (uppercase)
• F (lowercase)
• G (uppercase), and so on.

Thus, the letters from B onwards are:
$$ B \rightarrow b, \quad C \rightarrow C, \quad D \rightarrow d, \quad E \rightarrow E, \quad F \rightarrow f, \quad G \rightarrow G, \quad H \rightarrow h, \quad I \rightarrow I, \quad J \rightarrow j, \quad K \rightarrow K, \quad L \rightarrow l, \quad M \rightarrow M, \quad N \rightarrow n, \quad O \rightarrow O, \quad P \rightarrow p, \quad Q \rightarrow Q, \quad R \rightarrow r, \quad S \rightarrow S, \quad T \rightarrow t, \quad U \rightarrow U, \quad V \rightarrow v, \quad W \rightarrow W, \quad X \rightarrow x, \quad Y \rightarrow Y, \quad Z \rightarrow z $$

Now, let’s apply this pattern to the word “July”, the first month of the second half of the year:

• J: According to the pattern, J should be lowercase, so j.
• U: According to the pattern, U should be uppercase, so U.
• L: According to the pattern, L should be lowercase, so l.
• Y: According to the pattern, Y should be uppercase, so Y.

Thus, “July” should be written as jUlY.

Conclusion

The correct answer is (d) jUlY.

To solve this problem, let’s analyze each condition step-by-step and use them to infer the positions of each student in the ranking.

Given Information and Analysis

1. Either C or F scores the highest.
This means that in every test, the top scorer will be either C or F.

2. Whenever C scores the highest, then E scores the least.
So, if C is ranked 1st, then E will be ranked last (6th).

3. Whenever F scores the highest, B scores the least.
So, if F is ranked 1st, then B will be ranked last (6th).

4. All students scored different marks.
This implies there are no ties in the rankings.

5. D scores higher than A, but they are close competitors.
Since D and A are close competitors, their ranks should be adjacent, with D ranked just above A.

6. A scores higher than B.
This means A’s position will be above B in the ranking.

7. C scores higher than A.
So, C must be ranked above A as well.

8. F is ranked second.
We are given that F stands second in the ranking, so we will use this information to determine the possible positions of other students.

Step-by-Step Deduction

Since we know that F is ranked second, we can deduce the following based on each condition:

1. If F is ranked second, then C must be ranked first (because either C or F scores the highest, and F cannot be first if it is in second place).
Therefore:

• C is 1st
• F is 2nd

2. Since C is ranked first, E must be ranked last (6th), based on the condition that if C scores the highest, E scores the least.
Therefore:

• E is 6th

3. Now we have determined the positions of C (1st), F (2nd), and E (6th). The remaining students to be placed are A, B, and D.

4. A scores higher than B.
This means B cannot be 3rd, as A must be ranked above B.

5. D scores higher than A, and they are close competitors.
Since D and A are close competitors, they must be in adjacent positions, with D ranked above A. The only available consecutive positions for D and A that satisfy this condition are 3rd and 4th.

Therefore:

• D is 3rd
• A is 4th

6. This leaves only the 5th position for B.
Since A is 4th, and B scores lower than A, B must be in the 5th position.

Final Rankings

Based on the deductions above, the ranking order is:

• C: 1st
• F: 2nd
• D: 3rd
• A: 4th
• B: 5th
• E: 6th

Conclusion

The correct answer is (c) Fifth, as B is in the 5th position.

Explanation:

To solve this problem, let’s analyze the relationships step by step based on the given information:

Information Given

1. The family consists of seven members: A, B, C, D, E, F, and G.

2. There are three females in the family.

3. G is a widow and the sister-in-law of D’s father, F.

• Since G is the sister-in-law of F, this means G was married to F’s brother.

4. B and D are siblings.

5. A is the daughter of B.

6. C is the cousin of B.

Step-by-Step Deduction

Identifying Family Connections

• Since B and D are siblings, they share the same parents.
• A is the daughter of B, which makes B a parent.
• C is the cousin of B, meaning C is not a direct sibling of B but belongs to the extended family.

Analyzing the Role of F

• Since G is the sister-in-law of D’s father F, F is the father of B and D.
• This makes F the father of both B and D.

Understanding G’s Role

• G is a widow and the sister-in-law of F.
• This implies that G was married to F’s brother, who is now deceased.

Identifying E’s Relationship
Since there are three females in the family and we know the following:

• A is a female (stated as the daughter of B).
• G is a female (stated as a widow).
• Since there are three females in total, the last female must be E.

Now, let’s determine E’s relationship to the other family members:

• Since F is the father of B and D, and there must be a mother for this family unit, E is likely the wife of F.
• This would make E the mother of B and D and hence the grandmother of A (since A is the daughter of B).

Conclusion about E’s Relationships

Based on our analysis:
1. E is the wife of F.
2. E is the grandmother of A (since she is B’s mother and A is B’s daughter).
3. E is also the aunt of C, since C is the cousin of B, and E is B’s mother.

The correct answer is (d) 1, 2, and 3.

Explanation:

We are given:

• $x \geq 25$
• $y \leq 40$

Option (a): $x$ is greater than $y$

This option suggests that $x > y$. However, this is not always true. For example, if $x = 25$ and $y = 40$, then $x < y$. Therefore, option (a) is not always correct.

Option (b): $(y – x)$ is greater than 15

This option suggests that $(y – x) > 15$. Let’s test this by finding the maximum possible value for $y – x$:

• Since $x \geq 25$ and $y \leq 40$, the maximum value of $y – x$ occurs when $y = 40$ and $x = 25$.
• In that case, $y – x = 40 – 25 = 15$.

Thus, $(y – x) > 15$ is not always true, because the maximum value of $y – x$ is 15. Therefore, option (b) is incorrect.

Option (c): $(y – x)$ is less than or equal to 15

This option suggests that $(y – x) \leq 15$.

As we calculated above, the maximum value of $y – x$ is 15 when $y = 40$ and $x = 25$. Since $y – x$ cannot exceed 15, this option is always true.

Option (d): $(x – y)$ is greater than or equal to 65

This option suggests that $(x – y) \geq 65$. Let’s analyze it:

• Since $x \geq 25$ and $y \leq 40$, the minimum possible value for $x – y$ occurs when $x = 25$ and $y = 40$.
• In that case, $x – y = 25 – 40 = -15$.

Since $x – y$ can be as low as $-15$, it is not always true that $(x – y) \geq 65$. Therefore, option (d) is incorrect.

Conclusion
The correct answer is (c), since $(y – x) \leq 15$ is always true based on the given conditions.

Explanation:

To analyze the conclusions, let’s go through each statement and conclusion logically.

Statements:

1. Some rats are cats.
2. Some cats are dogs.
3. No dog is a cow.

Conclusions:

Let’s evaluate each conclusion based on the statements given.

Conclusion I: “No cow is a cat.”

• From the statements, we know that “No dog is a cow,” but there is no information connecting cows and cats. Therefore, we cannot conclude that no cow is a cat. This conclusion does not logically follow from the statements.

Conclusion II: “No dog is a rat.”

• The statements provide no direct relationship between dogs and rats. Although we know that “No dog is a cow” and “Some cats are dogs” and “Some rats are cats,” there is no information that implies dogs and rats cannot overlap. Therefore, we cannot conclude that no dog is a rat. This conclusion does not logically follow from the statements.

Conclusion III: “Some cats are rats.”

• This conclusion directly follows from Statement 1, which states that “Some rats are cats.” By standard logic rules, if some rats are cats, then it is also true that some cats are rats. This conclusion is valid.

Summary:

Only Conclusion III can be drawn directly from the statements, while Conclusions I and II cannot be logically inferred.

The correct answer is (c) Only III.

Explanation:

The given sequence is:

$$
4, 196, 16, 144, 36, 100, 64, X
$$

Each term in the sequence is a perfect square. The terms appear to alternate between smaller perfect squares and larger perfect squares. Let’s separate the sequence into two parts and identify patterns.

Step 1: Separate the Sequence into Two Patterns

1. Pattern 1 (small perfect squares): $4, 16, 36, 64$

2. Pattern 2 (large perfect squares): $196, 144, 100, X$

Step 2: Analyze Pattern 1 (Small Perfect Squares)

The small perfect squares follow a clear pattern where the base of each square increases by 2:

$$ 4 = 2^2 $$

$$ 16 = 4^2 $$

$$ 36 = 6^2 $$

$$ 64 = 8^2 $$

So, Pattern 1 consists of squares of even numbers $2, 4, 6, 8$, and so on. Since $64 = 8^2$, this part of the pattern is complete, and we do not need to add any further terms here.

Step 3: Analyze Pattern 2 (Large Perfect Squares)

Now, let’s examine Pattern 2, which is $196, 144, 100, X$. These are also perfect squares, but with a decreasing pattern in the base:

$$ 196 = 14^2$$

$$ 144 = 12^2 $$

$$ 100 = 10^2 $$

The bases in this sequence decrease by 2 each time: $14, 12, 10$. Following this pattern, the next term should be:

$$ 8^2 = 64 $$

Conclusion

Thus, the missing term $X$ is $64$.

The correct answer is: (b) 64

Explanation:

To determine the number of pages in a book given that 3089 digits were used, we can break down the problem based on the number of digits each range of page numbers contributes.

Step-by-Step Solution

1. Pages 1 to 9 (Single-digit pages):

• There are 9 pages in this range.
• Each page uses 1 digit.
• Total digits used for these pages = $9 \times 1 = 9$ digits.

2. Pages 10 to 99 (Two-digit pages):

• There are $99 – 10 + 1 = 90$ pages in this range.
• Each page uses 2 digits.
• Total digits used for these pages = $90 \times 2 = 180$ digits.

3. Pages 100 to 999 (Three-digit pages):

• There are $999 – 100 + 1 = 900$ pages in this range.
• Each page uses 3 digits.
• Total digits used for these pages = $900 \times 3 = 2700$ digits.

4. Calculating the Digits Used So Far

• Digits used for pages 1 to 9: $9$ digits
• Digits used for pages 10 to 99: $180$ digits
• Digits used for pages 100 to 999: $2700$ digits
• Total digits used up to page 999 = $9 + 180 + 2700 = 2889$ digits.

Since the total number of digits used is 3089, we have:

$$
3089 – 2889 = 200 \text{ digits remaining}
$$

5. Pages 1000 and Beyond (Four-digit pages):

• Starting from page 1000, each page uses 4 digits.
• Let the number of additional pages beyond 999 be $x$.
• Since each page contributes 4 digits, we have:
  $$
  4x = 200
  $$
• Solving for $x$:
  $$
  x = \frac{200}{4} = 50
  $$

This means there are 50 pages from 1000 onward.

6. Calculating the Total Number of Pages

• Total pages = pages up to 999 + additional pages
• Total pages = $999 + 50 = 1049$

Conclusion

The book has 1049 pages.

The correct answer is: (c) 1049

Explanation:

Let the original proper fraction be represented as:

$$
\frac{a}{b}
$$

where $a$ and $b$ are positive integers, and since it’s a proper fraction, we have $a < b$.

Suppose we increase both the numerator and the denominator by a positive quantity $k$ (where $k > 0$). The new fraction becomes:

$$
\frac{a + k}{b + k}
$$

Analyzing the Relationship Between the New Fraction and the Original Fraction

To determine if this new fraction is greater than, less than, or equal to the original fraction, let’s compare $\frac{a}{b}$ and $\frac{a + k}{b + k}$.

We will check whether:

$$
\frac{a + k}{b + k} > \frac{a}{b}, \quad \frac{a + k}{b + k} < \frac{a}{b}, \quad \text{or} \quad \frac{a + k}{b + k} = \frac{a}{b}
$$

Step 1: Cross-Multiplying to Compare

To compare $\frac{a + k}{b + k}$ and $\frac{a}{b}$, we cross-multiply:

$$
(a + k) \cdot b \quad \text{and} \quad a \cdot (b + k)
$$

Expanding both expressions:

1. The left side: $(a + k) \cdot b = ab + kb$
2. The right side: $a \cdot (b + k) = ab + ak$

Now, let’s subtract the right side from the left side to see if the result is positive, negative, or zero:

$$
ab + kb – (ab + ak) = kb – ak = k(b – a)
$$

Since $a < b$ (because it’s a proper fraction), we have $b – a > 0$. Given that $k > 0$, it follows that:

$$
k(b – a) > 0
$$

This means:

$$
\frac{a + k}{b + k} > \frac{a}{b}
$$

Conclusion
The new fraction $\frac{a + k}{b + k}$ is always greater than the original fraction $\frac{a}{b}$ when both the numerator and denominator are increased by the same positive quantity $k$.

Final Answer

The correct answer is: (b) always greater than the original fraction

Explanation:

Passage Summary

The passage describes a study that links the health of infants’ gut microbiomes with malnutrition. It finds that malnourished children have an “immature” microbiome, which lacks the diversity seen in healthy children. The passage explains that a specific modified sugar in mother’s milk, sialylated oligosaccharides, helps feed beneficial gut bacteria. This sugar is low in the milk of malnourished mothers, leading to underdeveloped microbiomes in their babies, which contributes to malnutrition.

Option Analysis

Option (a): “If malnourished condition in children is caused by gut bacteria, it cannot be treated.”

• Evaluation: The passage does not imply that malnutrition caused by an immature microbiome is untreatable. It simply identifies a possible cause of malnutrition. In fact, the passage suggests that interventions involving sialylated oligosaccharides could potentially help improve the microbiome. Therefore, this option is incorrect as it contradicts the idea that understanding the microbiome might lead to treatments.

Option (b): “The guts of malnourished babies should be inoculated with mature microbiomes.”

• Evaluation: While the passage discusses the importance of a mature microbiome, it does not suggest directly inoculating babies with mature microbiomes. This is a specific intervention that the passage does not mention or imply. This option is speculative and goes beyond what the passage provides, so it is not supported by the information given.

Option (c): “Babies of malnourished mothers should be fed with dairy milk fortified with sialylated oligosaccharides instead of mother’s milk.”

• Evaluation: The passage explains that the sugar sialylated oligosaccharides in mother’s milk is important for the development of a healthy microbiome. It also suggests that malnourished mothers have lower levels of this sugar, which could hinder microbiome development in their babies. This implies that supplementing with sialylated oligosaccharides might help malnourished infants. This option is consistent with the passage’s information, as it suggests an alternative way to provide the necessary sugar for gut bacteria. Thus, this option is logical and supported by the passage.

Option (d): “Research on benign effects of gut bacteria on nutrition has policy implications.”

• Evaluation: While this option is plausible, it is not directly supported by the passage. The passage discusses the scientific findings of the study but does not mention or imply any policy implications. This option is too broad and not specific to the information in the passage.

Conclusion

The most logical, rational, and crucial inference based on the passage is that babies of malnourished mothers could benefit from dairy milk fortified with sialylated oligosaccharides as an alternative to mother’s milk.

The correct answer is: (c) Babies of malnourished mothers should be fed with dairy milk fortified with sialylated oligosaccharides instead of mother’s milk.

Explanation:

Given Information Recap

1. Either C or F scores the highest.
2. Whenever C scores the highest, then E scores the least.
3. Whenever F scores the highest, B scores the least.
4. In all the tests, they got different marks.
5. D scores higher than A, but they are close competitors.
6. A scores higher than B.
7. C scores higher than A.

Question

We are told that E is ranked third and asked to determine which statement is correct based on this information.

Step-by-Step Analysis

Step 1: Determine the Implications of E Being Ranked Third

If E is ranked third, we need to examine the conditions and see which scenario fits this situation. Let’s evaluate the possible rankings for each student based on the conditions.

1. Either C or F scores the highest (Rank 1).
   • This means either C or F is in Rank 1.
2. Whenever C scores the highest, E scores the least (Condition 2).
   • Since E is ranked third, C cannot be the highest scorer in this scenario (because that would make E the least scorer).
   • Therefore, F must be the highest scorer (Rank 1).
3. Whenever F scores the highest, B scores the least (Condition 3).
   • Since F is the highest scorer here, B must be ranked last (Rank 6).

Step 2: Apply Remaining Conditions to Determine Rankings

Since F is ranked 1 and B is ranked 6, let’s apply the other conditions to determine the rankings of the remaining students.

1. Condition 7: C scores higher than A.
   • This implies C is ranked higher than A.
2. Condition 5: D scores higher than A, but they are close competitors.
   • This suggests that D is ranked immediately above A.
3. Condition 6: A scores higher than B.
   • Since B is ranked last (Rank 6), A must be ranked just above B (Rank 5).
4. Condition 4 ensures that all students have different scores, so each must occupy a unique rank.

Step 3: Determine Possible Ranking Structure

Given the above information, let’s assign ranks step-by-step:

1. F is the highest, so F is in Rank 1.
2. E is given as ranked third, so E is in Rank 3.
3. B is the least, so B is in Rank 6.
4. D is immediately above A (from Condition 5).
5. A is ranked above B (from Condition 6).
6. C must be ranked above A (from Condition 7).

Using these clues, we can arrange the ranks as follows:

• F – Rank 1
• C – Rank 2
• E – Rank 3
• D – Rank 4
• A – Rank 5
• B – Rank 6

Conclusion

With this arrangement, we find that C gets more marks than E (since C is ranked 2 and E is ranked 3).

The correct answer is: (b) C gets more marks than E

Explanation:

Given Data

1. 60% of the students play cricket.
   • Let $C$ represent the set of students who play cricket, so $|C| = 60%$.
2. A student who does not play cricket, plays football.
   • This implies that the remaining $40%$ of the students play football, as they do not play cricket.
   • Let $F$ represent the set of students who play football, so $|F| = 40%$.
3. Every football player has a two-wheeler.
   • This means all $40%$ of the students who play football have a two-wheeler.

We are asked to find which of the conclusions cannot be drawn from this information.

Analyzing Each Conclusion

Conclusion 1: “60% of the students do not have two-wheelers.”

Since $40%$ of the students play football and all football players have two-wheelers, it implies that $40%$ of the students have two-wheelers.

Therefore, the remaining $60%$ of the students do not have two-wheelers. This conclusion is true based on the given information.

Conclusion 2: “No cricketer has a two-wheeler.”

We know that all football players have two-wheelers. However, we are not given any information about whether cricket players (the remaining $60%$) have two-wheelers or not.

The statement “no cricketer has a two-wheeler” is not supported by the information given, but it is also not explicitly contradicted. Thus, we cannot conclude this based on the data provided. So this conclusion cannot be drawn.

Conclusion 3: “Cricket players do not play football.”

Since we are given that a student who does not play cricket plays football, this implies that there is no overlap between cricket and football players. Therefore, cricket players do not play football, which is consistent with the given information.

This conclusion is true based on the given information.

Summary

From the above analysis:

• Conclusion 1 can be drawn (it is true).
• Conclusion 2 cannot be drawn (we lack sufficient information to conclude this).
• Conclusion 3 can be drawn (it is true).

The correct answer is: (a) 1 and 2 only