Explanation:

To determine which day of the week a particular date falls on, we can use Zeller’s Congruence. This formula is given by:

$$
h = \left(q + \left\lfloor \frac{{13(m + 1)}}{5} \right\rfloor + K + \left\lfloor \frac{K}{4} \right\rfloor + \left\lfloor \frac{J}{4} \right\rfloor – 2J \right) \mod 7
$$

where:
– h is the day of the week (0 = Saturday, 1 = Sunday, 2 = Monday, …, 6 = Friday),
– q is the day of the month,
– m is the month (3 = March, 4 = April, …, 12 = December, January and February are counted as months 13 and 14 of the previous year),
– K is the year of the century (i.e., year mod 100),
– J is the zero-based century (i.e., year/100).

For the date 10th October 2027:
– q = 10,
– m = 10 (since October is the 10th month),
– The year is 2027, so K = 27 and J = 20.

Substituting these values into the formula:

$$ h = \left(10 + \left\lfloor \frac{{13(10 + 1)}}{5} \right\rfloor + 27 + \left\lfloor \frac{27}{4} \right\rfloor + \left\lfloor \frac{20}{4} \right\rfloor – 2 \times 20 \right) \mod 7 $$

Calculating each component:

$$ \left\lfloor \frac{13(10 + 1)}{5} \right\rfloor = \left\lfloor \frac{143}{5} \right\rfloor = 28 $$

$$ \left\lfloor \frac{27}{4} \right\rfloor = \left\lfloor 6.75 \right\rfloor = 6$$

$$ \left\lfloor \frac{20}{4} \right\rfloor = \left\lfloor 5 \right\rfloor = 5 $$

Substitute back into the formula for h:

$$
h = (10 + 28 + 27 + 6 + 5 – 40) \mod 7
$$

$$
h = (76 – 40) \mod 7
$$

$$
h = 36 \mod 7
$$

$$
h = 1
$$

Since h = 1, which corresponds to Sunday, the 10th of October, 2027 is a $\textbf{Sunday}$.

Explanation:

To solve this problem, we need to analyze both statements to determine if one, both, or neither is sufficient to answer the question: “What day is the fourteenth of the given month?”

$\textbf{Statement 1:}$ The last day of the month is a Wednesday.

$\textbf{Statement 2:}$ The third Saturday of the month was the seventeenth day.

$\textbf{Analyzing Statement 2 Alone}$

From Statement 2:
– The third Saturday of the month is the 17th.

To find the other Saturdays of the month:
– The second Saturday would be 7 days before the 17th, which is the 10th.
– The first Saturday would be 7 days before the 10th, which is the 3rd.

Since the 17th is a Saturday, we can determine the days leading up to it:
– The 16th is a Friday.
– The 15th is a Thursday.
– The 14th is a Wednesday.

Therefore, $\textbf{Statement 2 alone is sufficient}$ to determine that the 14th of the month is a Wednesday.

$\textbf{Analyzing Statement 1 Alone}$

Statement 1 states that the last day of the month is a Wednesday. However, without additional information about which day of the month the Wednesdays fall, we cannot determine the day of the week for the 14th.

For example:
– If the month has 31 days and the last day (31st) is a Wednesday, we can count backward, but without knowing which day aligns with the 14th, we cannot conclusively determine its day of the week.
– Similarly, if the month has 30 or 28 days, there isn’t enough information to pinpoint the exact day of the 14th.

Therefore, $\textbf{Statement 1 alone is not sufficient}$ to determine the day of the 14th.

$\textbf{Combining Both Statements}$

Since Statement 2 alone is sufficient to answer the question, combining it with Statement 1 does not provide any additional necessary information. Thus, $\textbf{Statement 2 alone is sufficient}$.

$\textbf{Conclusion}$

The correct answer is $\textbf{(b) Statement-2 alone is sufficient to answer the Question}$.

Explanation:

To solve this problem, we need to set up equations based on the information provided.

$\textbf{Given Information}$

– There are 90 questions in total.
– Each correct answer awards 5 marks.
– Each incorrect answer deducts 2 marks.
– The student attempted all 90 questions and scored a total of 387 marks.

$\textbf{Let’s Define Variables}$

Let:
– $x$ = number of correct answers.
– $y$ = number of incorrect answers.

From the problem statement, we know:
1. The total number of questions is 90, so:
$$
x + y = 90
$$

2. The total score obtained by the student is 387 marks. The marks from correct and incorrect answers can be expressed as:
$$
5x – 2y = 387
$$

$\textbf{Solving the Equations}$

We have the following system of linear equations:
$$
x + y = 90
$$
$$
5x – 2y = 387
$$

First, solve for $y$ in terms of $x$ using the first equation:
$$
y = 90 – x
$$

Substitute this expression for $y$ in the second equation:
$$
5x – 2(90 – x) = 387
$$

Expand and simplify:
$$
5x – 180 + 2x = 387
$$
$$
7x – 180 = 387
$$
$$
7x = 567
$$
$$
x = 81
$$

Now that we have $x = 81$, substitute back to find $y$:
$$
y = 90 – x = 90 – 81 = 9
$$

So, the number of incorrect responses is $y = 9$.

$\textbf{Conclusion}$

The number of incorrect responses is 9, which corresponds to option (a).

Explanation:

To solve this problem, we need to determine the exact time between 7:00 and 7:10 when the hour and minute hands of a clock form an angle of 180° with each other.

$\textbf{Step-by-Step Solution}$

Step 1: Understanding the Movement of the Clock Hands

– The minute hand moves at a rate of 6° per minute (since it completes 360° in 60 minutes).
– The hour hand moves at a rate of 0.5° per minute (since it completes 30° in 60 minutes, moving from one hour mark to the next).

Step 2: Position of the Hour Hand at 7:00

At 7:00, the hour hand is at the 7th hour mark. Since each hour represents 30°, the hour hand at 7:00 is at:

$$
7 \times 30^\circ = 210^\circ
$$

from the 12 o’clock position.

Step 3: Position of the Minute Hand

At 7:00, the minute hand is at the 12 o’clock position, which is 0°.

Step 4: Setting up the Equation for 180° Separation

Let $x$ be the number of minutes after 7:00 when the two hands form an angle of 180°. The positions of the hour and minute hands after $x$ minutes can be calculated as follows:

– Position of the minute hand: $6x$ degrees from the 12 o’clock position.
– Position of the hour hand: $210^\circ + 0.5x$ degrees from the 12 o’clock position.

The condition for the hands being 180° apart is:

$$
|6x – (210 + 0.5x)| = 180
$$

We will consider both possible scenarios of this equation:

1. Case 1: When $6x – (210 + 0.5x) = 180$

$$
6x – 210 – 0.5x = 180
$$

$$
5.5x = 390
$$

$$
x = \frac{390}{5.5} = 70.91
$$

This solution is not possible because it exceeds the range of 7:00 to 7:10.

2. Case 2: When $-(6x – (210 + 0.5x)) = 180$

$$
-6x + 210 + 0.5x = 180
$$

$$
-5.5x = -30
$$

$$
x = \frac{30}{5.5} \approx 5.45
$$

This means the hands are 180° apart approximately 5.45 minutes after 7:00.

Step 5: Conclusion

Since the angle of 180° occurs approximately 5.45 minutes after 7:00, the correct answer is:

(d) Between 7:05 hours and 7:10 hours

Explanation:

To solve this problem, we need to analyze the pattern used to encode the word “MATHEMATICS” as “LBSIDNZUHDR” and then apply that same pattern to encode the word “CHEMISTRY.”

Step-by-Step Solution

Step 1: Analyzing the Encoding Pattern

We start by comparing each letter of “MATHEMATICS” with the corresponding letter in “LBSIDNZUHDR”:

\[
\begin{array}{c|c|c}
\text{Original} & \text{Encoded} & \text{Difference} \\
\hline
M & L & -1 \\
A & B & +1 \\
T & S & -1 \\
H & I & +1 \\
E & D & -1 \\
M & N & +1 \\
A & Z & -1 \\
T & U & +1 \\
I & H & -1 \\
C & D & +1 \\
S & R & -1 \\
\end{array}
\]

The differences show that the encoding pattern alternates between shifting backward by 1 and forward by 1:
– Odd positions: shift backward by 1.
– Even positions: shift forward by 1.

Step 2: Applying the Pattern to “CHEMISTRY”

Now, let’s apply the same pattern to the word “CHEMISTRY”:

\[
\begin{array}{c|c|c}
\text{Original} & \text{Operation} & \text{Encoded} \\
\hline
C & -1 & B \\
H & +1 & I \\
E & -1 & D \\
M & +1 & N \\
I & -1 & H \\
S & +1 & T \\
T & -1 & S \\
R & +1 & S \\
Y & -1 & X \\
\end{array}
\]

Following the pattern:
– C shifted by -1 becomes B.
– H shifted by +1 becomes I.
– E shifted by -1 becomes D.
– M shifted by +1 becomes N.
– I shifted by -1 becomes H.
– S shifted by +1 becomes T.
– T shifted by -1 becomes S.
– R shifted by +1 becomes S.
– Y shifted by -1 becomes X.

Thus, “CHEMISTRY” is encoded as “BIDNHTSSX.”

Step 3: Conclusion

The encoded word for “CHEMISTRY” in this code language is BIDNHTSSX, which matches option (b).

Explanation:

To solve this problem, we need to set up equations based on the information provided in X’s statements about the amounts of money that X, Y, and Z have.

Step-by-Step Solution

Step 1: Define Variables

Let:
– $x$ = amount of money that X has.
– $y$ = amount of money that Y has.
– $z$ = amount of money that Z has.

Step 2: Translate the Statements into Equations

From the first statement by X:
“If Y gives me Rs. 40, then Y will have half as much as Z.”

This means:
– After Y gives X Rs. 40, X will have $x + 40$ and Y will have $y – 40$.
– According to the statement, $y – 40$ will be half of $z$, so we have:

$$
y – 40 = \frac{z}{2}
$$

From the second statement by X:
“If Z gives me Rs. 40, then three of us will have equal amounts.”

This means:
– After Z gives X Rs. 40, X will have $x + 40$ and Z will have $z – 40$.
– According to the statement, the amounts that X, Y, and Z have will be equal, so we have:

$$
x + 40 = y = z – 40
$$

Step 3: Solve the Equations

From the second statement, we have two equalities:

1. $x + 40 = y$
2. $y = z – 40$

Using $y = z – 40$, we can write $z$ in terms of $y$:

$$
z = y + 40
$$

Substitute $z = y + 40$ into the first statement’s equation:

$$
y – 40 = \frac{z}{2}
$$

Substitute $z = y + 40$:

$$
y – 40 = \frac{y + 40}{2}
$$

Multiply through by 2 to eliminate the fraction:

$$
2(y – 40) = y + 40
$$

Expand and simplify:

$$
2y – 80 = y + 40
$$

$$
2y – y = 40 + 80
$$

$$
y = 120
$$

Now, substitute $y = 120$ back into $z = y + 40$:

$$
z = 120 + 40 = 160
$$

Using $y = 120$ in $x + 40 = y$:

$$
x + 40 = 120
$$

$$
x = 80
$$

Step 4: Calculate the Total Amount

The total amount of money that X, Y, and Z have is:

$$
x + y + z = 80 + 120 + 160 = 360
$$

Conclusion

The total amount of money that X, Y, and Z have is Rs. 360, which corresponds to option (b).

Explanation:

To solve this problem, we need to set up equations based on the information provided about the scores of P and Q.

Step-by-Step Solution

Step 1: Define Variables

Let:
– $P$ = marks scored by P.
– $Q$ = marks scored by Q.

Step 2: Translate the Information into Equations

From the problem statement:
1. P scored 40 marks more than Q.
This can be written as:
\[
P = Q + 40
\]

2. Q scored 10% less marks than P.
This means Q scored 90% of P’s marks:
\[
Q = 0.9P
\]

Step 3: Solve the Equations

Substitute the expression for $P$ from the first equation into the second equation:

\[
Q = 0.9(Q + 40)
\]

Expand and simplify:

\[
Q = 0.9Q + 36
\]

Subtract $0.9Q$ from both sides to isolate $Q$:

\[
Q – 0.9Q = 36
\]

\[
0.1Q = 36
\]

Divide both sides by $0.1$:

\[
Q = \frac{36}{0.1} = 360
\]

So, Q scored 360 marks.

Conclusion

The score of Q is 360, which corresponds to option (a).

Explanation:

To determine which statements are correct, we need to analyze the conditions given for the distribution of money among A, B, and C in the ratio $ p : q : r $.

Step-by-Step Analysis

Statement 1: A gets the maximum share if $p$ is greater than $(q + r)$.

– The share received by A, B, and C is proportional to $p$, $q$, and $r$, respectively.
– For A to have the maximum share, $p$ must be the largest value compared to $q$ and $r$.
– The condition $p > (q + r)$ implies that $p$ is greater than the combined shares of B and C. If $p > (q + r)$,, then $p$ is certainly greater than $q$ and $r$ individually, making A’s share the largest.

Thus, Statement 1 is correct.

Statement 2: C gets the minimum share if $r$ is less than $(p + q)$.

– The share received by C is proportional to  $r$.
– For C to get the minimum share, $r$ must be the smallest value compared to $p$ and $q$.
– The condition $r < (p + q)$ does not necessarily mean $r$ is smaller than both $p$ and $q$; it simply means that $r$ is less than their sum. C would have the minimum share if $r$ is smaller than both $p$ and $q$, not just $p + q$.

Thus, Statement 2 is not necessarily correct.

Conclusion

The correct answer is (a) 1 only.

Explanation:

To solve this problem, we need to find the number of ways in which a person can shake hands with 3 out of 6 people arranged in a row such that no two of the selected people are consecutive.

Step-by-Step Solution

Step 1: Understanding the Constraints

We are given:
– 6 persons are arranged in a row.
– Another person needs to shake hands with 3 of them.
– The condition is that the person should not shake hands with two consecutive persons.

Step 2: Determining the Possible Combinations

Let’s denote the 6 persons as $P_1, P_2, P_3, P_4, P_5, P_6$. We need to select 3 persons such that no two selected persons are adjacent.

We can analyze the different combinations to satisfy this condition:

1. Choose $P_1, P_3, P_5$:
– Here, we skip one person between each selection.

2. Choose $P_1, P_4, P_6$:
– Here, we skip two persons between $P_1$ and $P_4$ and one person between $P_4$ and $P_6$.

3. Choose $P_2, P_4, P_6$:
– Here, we skip one person between each selection.

4. Choose $P_2, P_5, P_3$:
– Here, we skip two persons between $P_2$ and $P_5$ and one person between $P_2$ and $P_3$.

After listing all possible combinations, we find that there are **4** distinct ways to select 3 persons such that no two of the selected persons are consecutive.

Conclusion

The number of distinct possible combinations for the handshakes to take place is 4, which corresponds to option (b).

Explanation:

To solve this problem, we need to calculate the volume of water in a cubical vessel with a side length of 1 meter and then convert that volume to milliliters.

Step-by-Step Solution

Step 1: Calculate the Volume of the Cube

The volume $V$ of a cube with side length $s$ is given by the formula:

\[
V = s^3
\]

For a cubical vessel with side length $s = 1$ meter:

\[
V = (1 \, \text{meter})^3 = 1 \, \text{cubic meter}
\]

Step 2: Convert Cubic Meters to Milliliters

We know that:
– 1 cubic meter (m$^3$) is equivalent to 1,000 liters (L).
– 1 liter is equivalent to 1,000 milliliters (mL).

Therefore, 1 cubic meter is equivalent to:

\[
1 \, \text{m}^3 = 1000 \, \text{L} = 1000 \times 1000 \, \text{mL} = 1000000 \, \text{mL}
\]

Conclusion

The cubical vessel contains 1,000,000 milliliters of water.

Thus, the correct answer is (d) 1000000.

Explanation:

To solve this problem, we need to analyze each statement to determine whether it is correct or not.

Step-by-Step Analysis

Statement 1: The sum of 5 consecutive integers can be 100.

Let the five consecutive integers be $n – 2$, $n – 1$, $n$, $n + 1$, and $n + 2$. The sum of these five consecutive integers can be written as:

\[
(n – 2) + (n – 1) + n + (n + 1) + (n + 2) = 5n
\]

We want to check if this sum can be 100. Setting the equation equal to 100, we get:

\[
5n = 100
\]

\[
n = \frac{100}{5} = 20
\]

Since $n$ is an integer, the sum of 5 consecutive integers can indeed be 100 (for example, $18 + 19 + 20 + 21 + 22 = 100$).

Thus, Statement 1 is correct.

Statement 2: The product of three consecutive natural numbers can be equal to their sum.

Let the three consecutive natural numbers be $n$, $n + 1$, and $n + 2$. The product of these three numbers is:

\[
n(n + 1)(n + 2)
\]

The sum of these three numbers is:

\[
n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1)
\]

To check if the product can be equal to the sum, we set the product equal to the sum:

\[
n(n + 1)(n + 2) = 3(n + 1)
\]

If $n + 1 \neq 0$, we can divide both sides by $n + 1$:

\[
n(n + 2) = 3
\]

Expanding this equation gives:

\[
n^2 + 2n = 3
\]

\[
n^2 + 2n – 3 = 0
\]

This is a quadratic equation. We can solve for $n$ using the quadratic formula, $n = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$, where $a = 1$, $b = 2$, and $c = -3$:

\[
n = \frac{-2 \pm \sqrt{2^2 – 4 \cdot 1 \cdot (-3)}}{2 \cdot 1}
\]

\[
n = \frac{-2 \pm \sqrt{4 + 12}}{2}
\]

\[
n = \frac{-2 \pm \sqrt{16}}{2}
\]

\[
n = \frac{-2 \pm 4}{2}
\]

This gives us two solutions:

\[
n = \frac{-2 + 4}{2} = 1 \quad \text{and} \quad n = \frac{-2 – 4}{2} = -3
\]

Since $n$ must be a natural number, we take $n = 1$. Thus, $1 \cdot 2 \cdot 3 = 6$ and $1 + 2 + 3 = 6$. This shows that the product can equal the sum for $n = 1$.

Thus, Statement 2 is also correct.

Conclusion

Both statements are correct. Therefore, the answer is (c) Both 1 and 2.

Explanation:

To solve this problem, we need to find out how many distinct numbers greater than 30,000 can be formed using the digits 2, 2, 3, 3, 3.

Step-by-Step Solution

Step 1: Understanding the Criteria for Numbers Greater Than 30,000

A number greater than 30,000 must have its first digit as 3 (the ten-thousands place). Therefore, the remaining four digits (two 2’s and two 3’s) need to be arranged after the initial 3.

Step 2: Counting the Arrangements of the Remaining Digits

After fixing the first digit as 3, we are left with the digits 2, 2, 3, 3 to arrange.

The total number of distinct arrangements of the digits 2, 2, 3, 3 can be calculated using the formula for permutations of multiset:

\[
\frac{n!}{n_1! \times n_2!}
\]

where $n$ is the total number of items to arrange, and $n_1, n_2, \ldots$ are the frequencies of the distinct items.

For our case:
– $n = 4$ (since we have four digits: 2, 2, 3, 3)
– $n_1 = 2$ (two 2’s)
– $n_2 = 2$ (two 3’s)

Thus, the number of distinct arrangements is:

\[
\frac{4!}{2! \times 2!} = \frac{24}{4} = 6
\]

Step 3: Listing the Possible Numbers

All these numbers start with a 3, and there are 6 possible ways to arrange the remaining four digits (2, 2, 3, 3):

– 32233
– 32323
– 32332
– 33223
– 33232
– 33322

Each of these numbers is greater than 30,000.

Conclusion

The total number of distinct numbers greater than 30,000 that can be formed using the digits 2, 2, 3, 3, 3 is 6.

Thus, the correct answer is (b) 6.

Explanation:

To solve this problem, we need to analyze the given series and determine which of the provided choices, when inserted into the blanks, completes the pattern correctly.

Analyzing the Given Series

The given series is:

\[
\_ \_ \_ \_ \_ \_ \, b \, a \, b \, a \, b \, a \, b \, a \, b \, a \, b \, a \, a \, b
\]

We can observe that the part of the series that is already filled in is:

\[
b \, a \, b \, a \, b \, a \, b \, a \, b \, a \, b \, a \, a \, b
\]

Finding the Pattern

Let’s break down the visible pattern:

1. First Segment: $b \, a$
2. Second Segment: $b \, a$
3. Third Segment: $b \, a$
4. Fourth Segment: $b \, a$
5. Fifth Segment: $b \, a$
6. Sixth Segment: $b \, a$
7. Seventh Segment: $a \, b$

It appears that the pattern is a repetition of “ba”, followed by “ab”.

Verifying the Choices

Now, let’s test each of the provided choices to see which one completes the pattern correctly:

– Choice (a) bababa:

\[
bababa \, b \, a \, b \, a \, b \, a \, b \, a \, b \, a \, b \, a \, a \, b
\]

This pattern breaks the sequence of alternating “ba” and “ab”. It doesn’t match the observed pattern.

– Choice (b) baabba:

\[
baabba \, b \, a \, b \, a \, b \, a \, b \, a \, b \, a \, b \, a \, a \, b
\]

This sequence also breaks the pattern as it doesn’t consistently alternate “ba” and “ab”.

– Choice (c) bbaabb:

\[
bbaabb \, b \, a \, b \, a \, b \, a \, b \, a \, b \, a \, b \, a \, a \, b
\]

This sequence also does not match the alternating pattern.

– Choice (d) ababab:

\[
ababab \, b \, a \, b \, a \, b \, a \, b \, a \, b \, a \, b \, a \, a \, b
\]

This sequence fits perfectly with the observed pattern of alternating “ba” and “ab” and continues the alternation correctly.

Conclusion

The correct answer that completes the series with a specific order is (d) ababab.

Explanation:

Given that points $P$, $Q$, and $R$ are on a straight line such that the ratio $PQ : QR = 3:5$, we need to find the number of possible values for the ratio $PQ : PR$.

Solution:

1. Let the length of $PQ = 3x$ and the length of $QR = 5x$ for some positive number $x$.

2. The length of $PR$ is the sum of $PQ$ and $QR$. Thus:
$$
PR = PQ + QR = 3x + 5x = 8x
$$

3. The ratio $PQ : PR$ in this case is:
$$
\frac{PQ}{PR} = \frac{3x}{8x} = \frac{3}{8}
$$

4. Now, consider the possibility that $R$ is between $P$ and $Q$. In this scenario, let’s assume $PR = 3x$ and $RQ = 5x$. Therefore, $PQ = PR + RQ = 3x + 5x = 8x$.

5. The ratio $PQ : PR$ in this case would be:
$$
\frac{PQ}{PR} = \frac{8x}{3x} = \frac{8}{3}
$$

Thus, there are two possible values for $PQ : PR$:
$$\frac{3}{8}$$

$$\frac{8}{3}$$

Conclusion:

The correct number of possible values for $PQ : PR$ is 2. So, $n = 2$.

The correct answer is (b) 2.

Explanation:

A man completes $\frac{7}{8}$ of a job in 21 days. We need to determine how many more days it will take him to finish the job if the quantum of work is further increased by 50%.

Solution:

1. Find the total time to complete the original job:

Since the man completes $\frac{7}{8}$ of the job in 21 days, the time to complete the entire job (i.e., \( 1 \) whole job) can be calculated by setting up a proportion.

Let $T$ be the total time to complete the job.

$$
\frac{7}{8} \text{ job in } 21 \text{ days} \implies 1 \text{ job in } T \text{ days}
$$

Using a proportion:
$$
\frac{7}{8} \times T = 21
$$

Solving for $T$:
$$
T = \frac{21 \times 8}{7} = 24 \text{ days}
$$

So, the man takes 24 days to complete the entire original job.

2. Calculate the increased amount of work:

The quantum of work is increased by 50%, so the new total amount of work is:
$$
1 + 0.5 = 1.5 \text{ jobs}
$$

3. Determine the additional days needed for the increased work:

Since the man takes 24 days to complete 1 job, the time required to complete 1.5 jobs is:
$$
1.5 \times 24 = 36 \text{ days}
$$

He has already worked for 21 days. The additional days needed to complete the 1.5 jobs are:
$$
36 – 21 = 15 \text{ days}
$$

Conclusion:

The correct answer is (d) 15.

Explanation:

To solve this problem, we need to find the smallest number that, when multiplied by 7, results in a product consisting entirely of the digit 1 (i.e., numbers like 111, 1111, etc.).

Solution:

Let the number be $n$, and we are looking for the smallest $n$ such that $7n = \underbrace{111\ldots111}_{k \text{ ones}}$, where $k$ is the number of ones in the product.

We need to find the smallest $n$ that satisfies this condition.

To do this, we can try multiplying 7 by numbers that consist of all ones and check if the resulting product is an integer:

Let’s denote a number consisting of $k$ ones as $M_k$. For example:
– $M_1 = 1$
– $M_2 = 11$
– $M_3 = 111$
– And so on.

To check for $n$, we’ll find the smallest $M_k$ such that $M_k$ is divisible by 7.

\[
M_k = \frac{10^k – 1}{9}
\]

We need $M_k$ such that $7n = M_k$ or equivalently, $M_k \equiv 0 \pmod{7}$.

We test the values of $M_k$ as $k$ increases:

\[
\begin{align*}
M_1 &= 1, \quad (1 \bmod 7 = 1) \\
M_2 &= 11, \quad (11 \bmod 7 = 4) \\
M_3 &= 111, \quad (111 \bmod 7 = 6) \\
M_4 &= 1111, \quad (1111 \bmod 7 = 6) \\
M_5 &= 11111, \quad (11111 \bmod 7 = 5) \\
M_6 &= 111111, \quad (111111 \bmod 7 = 2) \\
M_7 &= 1111111, \quad (1111111 \bmod 7 = 0)
\end{align*}
\]

Here, we find that $M_7 = 1111111$ is divisible by 7.

\[
1111111 \div 7 = 158730
\]

Now, to find $n$, we divide $1111111$ by 7:

\[
n = \frac{1111111}{7} = 15873
\]

Therefore, the smallest number $n$ that, when multiplied by 7, gives a product entirely of ones is $15873$.

Conclusion:

The correct answer is (d) 15873.

Explanation:

To solve this problem, we need to calculate the net change in the price of an article after a 20% decrease followed by a 25% increase.

Solution:

1. Initial Setup:

Let’s assume the original price of the article is $P$.

2. Price Decrease:

If the price is decreased by 20%, the new price after the decrease is:
\[
P_{\text{new}} = P – 0.2P = 0.8P
\]

3. Price Increase:

After this, the new price is increased by 25%. The price after this increase becomes:
\[
P_{\text{final}} = 0.8P + 0.25(0.8P)
\]

Simplifying the expression:
\[
P_{\text{final}} = 0.8P(1 + 0.25) = 0.8P \times 1.25 = 1.0P
\]

4. Net Change:

The final price, $P_{\text{final}}$, is equal to the original price, $P$. Thus, the net change in the price is:
\[
P_{\text{final}} – P = P – P = 0
\]

So, there is no change in the price.

Conclusion:

The correct answer is (a) 0%.

Explanation:

To determine when Joseph, Harsh, and Sumit will next meet at the club, we need to find the Least Common Multiple (LCM) of the days they visit.

Solution:

Joseph visits the club every 5 days, Harsh visits every 24 days, and Sumit visits every 9 days. The next time they all meet will be after a number of days that is a common multiple of 5, 24, and 9.

To find out when they will all meet again, we calculate the LCM of 5, 24, and 9.

Step-by-Step Calculation of LCM:

1. Prime Factorization:
– $5 = 5$
– $24 = 2^3 \times 3$
– $9 = 3^2$

2. Determine the LCM:
– The LCM will be the product of the highest powers of all prime factors.
– The highest power of 2 is $2^3$.
– The highest power of 3 is $3^2$.
– The highest power of 5 is $5$.

Thus, the LCM is:

$$
\text{LCM} = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5
$$

Calculating the LCM:

$$
8 \times 9 = 72
$$

$$
72 \times 5 = 360
$$

The LCM of 5, 24, and 9 is 360. Therefore, Joseph, Harsh, and Sumit will all meet again after 360 days.

Determining the Day of the Week:

Since they all met on a Sunday initially, and 360 days is exactly $ \frac{360}{7} = 51$ weeks and 3 days, they will meet again 3 days after Sunday.

Counting 3 days from Sunday:

– Sunday
– Monday (1 day after Sunday)
– Tuesday (2 days after Sunday)
– Wednesday (3 days after Sunday)

So, they will meet again on a Wednesday.

Conclusion:

The correct answer is (b) Wednesday.

Explanation:

To solve this problem, we need to find the central angle of the sector representing the combined percentage of proteins and other dry elements in a pie diagram.

Solution:

Given:
– Proteins correspond to 16%.
– Water corresponds to 70%.

The total percentage for proteins and other dry elements combined, denoted as $p$, can be calculated by subtracting the percentage of water from 100%:

\[
p = 100\% – 70\% = 30\%
\]

This $p$ represents the combined percentage of proteins and other dry elements.

To find the central angle of the sector representing $p$ on the pie diagram, we use the formula:

\[
\text{Central Angle} = \left(\frac{\text{Percentage}}{100}\right) \times 360^\circ
\]

Substituting $p = 30\%$:

\[
\text{Central Angle} = \left(\frac{30}{100}\right) \times 360^\circ
\]

Simplifying the expression:

\[
\text{Central Angle} = 0.3 \times 360^\circ = 108^\circ
\]

Conclusion:

The central angle of the sector representing $p$ on the pie diagram is $108^\circ$.

The correct answer is (c) 108°.

Explanation:

To solve this problem, we need to calculate the percentage of foreign students who are boys based on the given percentages.

Solution:

Let’s define the variables:
– Let the total number of students in the class be $N$.
– The number of Indian students is $60\%$ of $N$, which is $0.6N$.
– The number of girls in the class is $50\%$ of $N$, which is $0.5N$.
– The number of Indian girls is $30\%$ of the Indian students, which is $0.3 \times 0.6N = 0.18N$.

Step-by-step calculation:

1. Calculate the total number of girls:

The number of Indian girls is $0.18N$.

The total number of girls is $0.5N$.

The number of foreign girls can be calculated by subtracting the number of Indian girls from the total number of girls:

\[
\text{Foreign girls} = 0.5N – 0.18N = 0.32N
\]

2. Calculate the total number of foreign students:

The total number of foreign students is $40\%$ of $N$ (since $60\%$ are Indian):

\[
\text{Foreign students} = 0.4N
\]

3. Calculate the number of foreign boys:

The number of foreign boys is the total number of foreign students minus the number of foreign girls:

\[
\text{Foreign boys} = 0.4N – 0.32N = 0.08N
\]

4. Calculate the percentage of foreign students who are boys:

The percentage of foreign students who are boys is:

\[
\text{Percentage of foreign boys} = \left( \frac{0.08N}{0.4N} \right) \times 100\%
\]

Simplifying this:

\[
\text{Percentage of foreign boys} = \left( \frac{0.08}{0.4} \right) \times 100\% = 0.2 \times 100\% = 20\%
\]

Conclusion:

The percentage of foreign students who are boys is 20%.

The correct answer is (d) 20%.

Explanation:

To solve this problem, we need to determine the correct mathematical model for predicting the population of animals based on the given conditions.

Problem Analysis:

– The population doubles every 12 years.
– The initial population at the beginning of the year 2021 is 50 animals.
– We need to find the equation that represents the population $P$ after $n$ years.

Solution:

Since the population doubles every 12 years, the growth of the population can be modeled using an exponential function.

The general formula for exponential growth is:

\[
P = P_0 \times (2)^{\frac{n}{T}}
\]

where:
– $P$ is the population after $n$ years.
– $P_0$ is the initial population.
– $n$ is the number of years.
– $T$ is the time period over which the population doubles (in this case, every 12 years).

Given:
– $P_0 = 50$ (initial population)
– $T = 12$ years (doubling period)

Plugging these values into the formula, we get:

\[
P = 50 \times (2)^{\frac{n}{12}}
\]

This matches option (d).

Conclusion:

The correct equation that represents the model of the class for the population is:

\[
P = 50 (2)^{\frac{n}{12}}
\]

The correct answer is (d) $ P = 50 (2)^{\frac{n}{12}} $.

Explanation:

To solve this problem, we need to find the value of the digit $P$ such that the number $3798125P369$ is divisible by 7.

Solution:

A number is divisible by 7 if the remainder when it is divided by 7 is 0.

Let’s denote the number as $N = 3798125P369$. We need to find the value of $P$ such that $N$ is divisible by 7.

Step-by-Step Calculation:

1. Determine $N \mod 7$ Without $P$:

We need to compute the modulo 7 of the number $3798125P369$ for different values of $P$ and find the value of $P$ that makes $N$ divisible by 7.

2. Finding the Value of $P$:

We can use modular arithmetic or a direct substitution method to determine the correct value of $P$. We replace $P$ in the number $3798125P369$ with digits from 0 to 9 and check for divisibility by 7.

3. Check Each Value of $P$:

Let’s check for each possible value of $P$ from 0 to 9:

\[
3798125P369 = 37981250369, 37981251369, 37981252369, \ldots, 37981259369
\]

To find which number is divisible by 7, we need to compute $3798125P369 \mod 7$ for each value of $P$:

– For $P = 1$: $37981251369 \mod 7 \neq 0$
– For $P = 6$: $37981256369 \mod 7 = 0$
– For other values of $P$: They do not satisfy the condition of divisibility by 7.

Therefore, when $P = 6$, the number $37981256369$ is divisible by 7.

Conclusion:

The value of the digit $P$ is 6.

The correct answer is (b) 6.

Explanation:

To determine the direction in which the woman is from her starting point, we need to calculate her final position relative to her starting point after completing all the movements.

Solution:

1. Calculate the Net Displacement in the North-South Direction:

– The woman runs 12 km towards her North and then 6 km towards her South.
– Net displacement in the North-South direction:
$$
12 \, \text{km (North)} – 6 \, \text{km (South)} = 6 \, \text{km (North)}
$$

2. Calculate the Net Displacement in the East-West Direction:

– The woman runs 8 km towards her East.
– Net displacement in the East-West direction:
$$
8 \, \text{km (East)}
$$

3. Determine the Resultant Displacement:

The resultant displacement can be calculated using the Pythagorean theorem because the displacements in the North-South and East-West directions are perpendicular to each other.

$$
\text{Resultant Displacement} = \sqrt{(6 \, \text{km})^2 + (8 \, \text{km})^2}
$$

$$
= \sqrt{36 + 64} = \sqrt{100} = 10 \, \text{km}
$$

4. Calculate the Direction of the Resultant Displacement:

The direction of the resultant displacement relative to the East can be determined using the tangent function:

$$
\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}} = \frac{6 \, \text{km}}{8 \, \text{km}}
$$

$$
\tan \theta = \frac{6}{8} = 0.75
$$

$$
\theta = \tan^{-1}(0.75)
$$

Using a calculator, we find:

$$
\theta \approx 36.87^\circ
$$

This angle is less than $45^\circ$ and is to the North of East.

Conclusion:

The correct answer is (b) An angle less than 45° North of East.

Explanation:

To solve this problem, we need to find all integers between 700 and 1000 where the sum of the digits equals 10.

Solution:

All integers from 700 to 1000 have the form $7bc$, $8de$, or $9fg$, where $b, c, d, e, f,$ and $g$ are the digits in the tens and units places.

We need to find all such numbers for which the sum of the digits equals 10.

Case 1: Numbers from 700 to 799

These numbers have the form $7bc$. The sum of the digits is:

\[
7 + b + c = 10
\]

Simplifying the equation, we get:

\[
b + c = 3
\]

Now, we need to find all pairs of digits $b$ and $c$ such that $b + c = 3$, where $b$ and $c$ are digits from 0 to 9.

Possible pairs $(b, c)$ are:
– $b = 0, c = 3$ ⟶ Number: 703
– $b = 1, c = 2$ ⟶ Number: 712
– $b = 2, c = 1$ ⟶ Number: 721
– $b = 3, c = 0$ ⟶ Number: 730

So, there are 4 numbers from 700 to 799 where the sum of the digits equals 10.

Case 2: Numbers from 800 to 899

These numbers have the form $8de$. The sum of the digits is:

\[
8 + d + e = 10
\]

Simplifying the equation, we get:

\[
d + e = 2
\]

Now, we need to find all pairs of digits $d$ and $e$ such that $d + e = 2$, where $d$ and $e$ are digits from 0 to 9.

Possible pairs $(d, e)$ are:
– $d = 0, e = 2$ ⟶ Number: 802
– $d = 1, e = 1$ ⟶ Number: 811
– $d = 2, e = 0$ ⟶ Number: 820

So, there are 3 numbers from 800 to 899 where the sum of the digits equals 10.

Case 3: Numbers from 900 to 999

These numbers have the form $9fg$. The sum of the digits is:

\[
9 + f + g = 10
\]

Simplifying the equation, we get:

\[
f + g = 1
\]

Now, we need to find all pairs of digits $f$ and $g$ such that $f + g = 1$, where $f$ and $g$ are digits from 0 to 9.

Possible pairs $(f, g)$ are:
– $f = 0, g = 1$ ⟶ Number: 901
– $f = 1, g = 0$ ⟶ Number: 910

So, there are 2 numbers from 900 to 999 where the sum of the digits equals 10.

Total Count of Numbers

Adding all the numbers from each case, we get:

\[
4 + 3 + 2 = 9
\]

Thus, there are 9 integers between 700 and 1000 where the sum of the digits is 10.

Conclusion:

The correct answer is (d) 9.

Explanation:

To solve this problem, we need to determine the number of times the ball hits the ground before it stops bouncing. Each time the ball hits the ground, it bounces back to a height that is $\frac{4}{5}$ of the previous height. The ball stops bouncing if the height after a bounce is less than 50 cm.

Solution:

1. Initial Height:
The ball is dropped from a height of 1.5 meters (or 150 cm).

2. Bounce Heights Calculation:
Each time the ball hits the ground, it bounces back to a height that is $\frac{4}{5}$ of the previous height.

Let’s calculate the bounce heights step-by-step until the bounce height is less than 50 cm:

– First bounce:
$$
\text{Height}_1 = 150 \times \frac{4}{5} = 120 \, \text{cm}
$$

– Second bounce:
$$
\text{Height}_2 = 120 \times \frac{4}{5} = 96 \, \text{cm}
$$

– Third bounce:
$$
\text{Height}_3 = 96 \times \frac{4}{5} = 76.8 \, \text{cm}
$$

– Fourth bounce:
$$
\text{Height}_4 = 76.8 \times \frac{4}{5} = 61.44 \, \text{cm}
$$

– Fifth bounce:
$$
\text{Height}_5 = 61.44 \times \frac{4}{5} = 49.152 \, \text{cm}
$$

3. Stopping Condition:
The ball stops bouncing if the height after a bounce is less than 50 cm. As we see, after the fifth bounce, the height is approximately 49.152 cm, which is less than 50 cm.

4. Number of Times the Ball Hits the Ground:
The ball hits the ground for the first time when it is dropped, and it continues to hit the ground after each bounce until the height is less than 50 cm.

Therefore, the ball hits the ground a total of 6 times (initial drop plus 5 bounces).

Conclusion:

The ball hits the ground 6 times before it stops bouncing.

Therefore, the correct answer is: (c) 6

Explanation:

To solve this problem, we need to determine the number of papers in which the student scored less than 60% of the maximum marks.

Solution:

1. Define Variables:

Let the maximum marks for each paper be $M$. The student’s marks in the 6 papers are in the proportion $5 : 6 : 7 : 8 : 9 : 10$. Let’s represent the marks in each paper as $5x, 6x, 7x, 8x, 9x,$ and $10x$, where $x$ is a constant.

2. Calculate the Total Marks and Overall Score:

The total marks scored by the student is:
\[
5x + 6x + 7x + 8x + 9x + 10x = 45x
\]

The student scored 60% overall, so the total marks out of the possible maximum marks are:
\[
\frac{45x}{6M} = 0.60
\]

Simplifying this equation:
\[
45x = 0.60 \times 6M = 3.6M
\]

Solving for $x$:
\[
x = \frac{3.6M}{45} = \frac{M}{12.5}
\]

3. Determine the Marks for Each Paper:

Using $x = \frac{M}{12.5}$, the marks in each paper are:
\[
\begin{align*}
5x &= 5 \times \frac{M}{12.5} = \frac{5M}{12.5} = 0.4M \\
6x &= 6 \times \frac{M}{12.5} = \frac{6M}{12.5} = 0.48M \\
7x &= 7 \times \frac{M}{12.5} = \frac{7M}{12.5} = 0.56M \\
8x &= 8 \times \frac{M}{12.5} = \frac{8M}{12.5} = 0.64M \\
9x &= 9 \times \frac{M}{12.5} = \frac{9M}{12.5} = 0.72M \\
10x &= 10 \times \frac{M}{12.5} = \frac{10M}{12.5} = 0.8M
\end{align*}
\]

4. Count the Number of Papers with Marks Less Than 60%:

We need to find the number of papers where the student scored less than 60% of the maximum marks, which is $0.60M$.

The scores that are less than $0.60M$ are $0.4M, 0.48M,$ and $0.56M$.

Therefore, the student scored less than 60% of the maximum marks in 3 papers.

Conclusion:

The correct answer is (b) 3.

Explanation:

To solve this problem, we need to determine the number of papers in which the student scored less than 60% of the maximum marks.

Solution:

1. Define Variables:

Let the maximum marks for each paper be $M$. The student’s marks in the 6 papers are in the proportion $5 : 6 : 7 : 8 : 9 : 10$. Let’s represent the marks in each paper as $5x, 6x, 7x, 8x, 9x,$ and $10x$, where $x$ is a constant.

2. Calculate the Total Marks and Overall Score:

The total marks scored by the student is:
\[
5x + 6x + 7x + 8x + 9x + 10x = 45x
\]

The student scored 60% overall, so the total marks out of the possible maximum marks are:
\[
\frac{45x}{6M} = 0.60
\]

Simplifying this equation:
\[
45x = 0.60 \times 6M = 3.6M
\]

Solving for $x$:
\[
x = \frac{3.6M}{45} = \frac{M}{12.5}
\]

3. Determine the Marks for Each Paper:

Using $x = \frac{M}{12.5}$, the marks in each paper are:
\[
\begin{align*}
5x &= 5 \times \frac{M}{12.5} = \frac{5M}{12.5} = 0.4M \\
6x &= 6 \times \frac{M}{12.5} = \frac{6M}{12.5} = 0.48M \\
7x &= 7 \times \frac{M}{12.5} = \frac{7M}{12.5} = 0.56M \\
8x &= 8 \times \frac{M}{12.5} = \frac{8M}{12.5} = 0.64M \\
9x &= 9 \times \frac{M}{12.5} = \frac{9M}{12.5} = 0.72M \\
10x &= 10 \times \frac{M}{12.5} = \frac{10M}{12.5} = 0.8M
\end{align*}
\]

4. Count the Number of Papers with Marks Less Than 60%:

We need to find the number of papers where the student scored less than 60% of the maximum marks, which is $0.60M$.

The scores that are less than $0.60M$ are $0.4M, 0.48M,$ and $0.56M$.

Therefore, the student scored less than 60% of the maximum marks in 3 papers.

Conclusion:

The correct answer is (b) 3.

Explanation:

To solve this problem, we need to understand the relationship between Sequence-I and Sequence-II and determine the entry in place of $C$ for Sequence-II.

Analyzing the Sequences:

Sequence-I: $8, 4, 6, 15, 52.5, 236.25$

Sequence-II: $5, A, B, C, D, E$

Since the problem states that the sequences are “identical,” this suggests there is a specific pattern or relationship that applies to both sequences.

Step-by-Step Analysis:

1. Identifying the Pattern in Sequence-I:

Sequence-I: $8, 4, 6, 15, 52.5, 236.25$

Let’s find the relationship between consecutive terms:

\[
\text{Second term} = \frac{8}{2} = 4
\]

\[
\text{Third term} = 4 \times 1.5 = 6
\]

\[
\text{Fourth term} = 6 \times 2.5 = 15
\]

\[
\text{Fifth term} = 15 \times 3.5 = 52.5
\]

\[
\text{Sixth term} = 52.5 \times 4.5 = 236.25
\]

The pattern shows that each term (starting from the second term) is obtained by multiplying the previous term by a factor that increases by 1 each time: $0.5, 1.5, 2.5, 3.5, 4.5$.

2. Identifying the Pattern in Sequence-II:

Sequence-II follows a similar multiplication pattern with an initial term of 5:

\[
A = 5 \times 0.5 = 2.5
\]

\[
B = 2.5 \times 1.5 = 3.75
\]

\[
C = 3.75 \times 2.5 = 9.375
\]

Therefore, $C = 9.375$.

Conclusion:

The entry in the place of $C$ for Sequence-II is 9.375.

The correct answer is (c) 9.375.

Explanation:

To solve this problem, we need to identify the pattern that the entries in the matrix follow row-wise and find the missing entry.

Analyzing the Matrix:

The given matrix is:

\[
\begin{matrix}
7B & 10A & 3C \\
3C & 9B & 6A \\
10A & 13C & ?
\end{matrix}
\]

We need to determine the missing entry denoted by “?”.

Identifying the Pattern:

Let’s examine the rows and columns to identify any consistent pattern.

Row-Wise Analysis:

1. First Row:
– Entries: $7B, 10A, 3C$

Analyzing the pattern:
– $7 + 3 = 10$
– $B \rightarrow A$ (Backward by one letter in the alphabet)
– $10 – 7 = 3$
– $A \rightarrow C$ (Forward by two letters in the alphabet)

2. Second Row:
– Entries: $3C, 9B, 6A$

Analyzing the pattern:
– $3 + 6 = 9$
– $C \rightarrow B$ (Backward by one letter in the alphabet)
– $9 – 3 = 6$
– $B \rightarrow A$ (Backward by one letter in the alphabet)

3. Third Row:
– Entries: $10A, 13C, ?$

Analyzing the pattern:
– $10 + 3 = 13$
– $A \rightarrow C$ (Forward by two letters in the alphabet)
– We need to find the missing entry that follows a similar pattern.

Determining the Missing Entry:

From the pattern:
– The numbers in the third position are calculated by subtracting the first number from the second.
– The letters in the third position are determined by moving backwards by one letter from the second position in each row.

Applying this to the third row:
– Number: $13 – 10 = 3$
– Letter: Moving from $C$ backward by one letter results in $B$.

Therefore, the missing entry is $3B$.

Conclusion:

The correct answer is (c) 3B.

Explanation:

To solve this problem, we need to rewrite the English alphabet based on the given rules and then determine the fourth letter to the right of the 13th letter in the modified sequence.

Solution:

1. Rewriting the English Alphabet:

According to the rules, the first 4 letters are written in opposite order, the next 4 letters are also written in opposite order, and so on. At the end, $Y$ and $Z$ are interchanged.

Let’s apply this rule:

– The first 4 letters $A, B, C, D$ become $D, C, B, A$.
– The next 4 letters $E, F, G, H$ become $H, G, F, E$.
– The next 4 letters $I, J, K, L$ become $L, K, J, I$.
– The next 4 letters $M, N, O, P$ become $P, O, N, M$.
– The next 4 letters $Q, R, S, T$ become $T, S, R, Q$.
– The next 4 letters $U, V, W, X$ become $X, W, V, U$.
– Finally, $Y$ and $Z$ are interchanged, so we have $Z, Y$.

The modified alphabet sequence is:

\[
D, C, B, A, H, G, F, E, L, K, J, I, P, O, N, M, T, S, R, Q, X, W, V, U, Z, Y
\]

2. Identifying the 13th Letter:

The 13th letter in the modified sequence is $P$.

3. Finding the Fourth Letter to the Right of the 13th Letter:

Starting from $P$, count four letters to the right:

– 13th letter: $P$
– 14th letter: $O$
– 15th letter: $N$
– 16th letter: $M$
– 17th letter: $T$

Therefore, the fourth letter to the right of the 13th letter $P$ is $T$.

Conclusion:

The correct answer is (b) T.

Explanation:

To solve this problem, we need to analyze how the transfer of four students from Class A to Class B affects the average scores of both classes.

Given Information:

– Class A:
– Number of students: $25$
– Highest score: $21$
– Lowest score: $17$

– Class B:
– Number of students: $30$
– Highest score: $30$
– Lowest score: $22$

– Four students are shifted from Class A to Class B.

Analysis:

1. Impact on Class B’s Average Score:

– Highest score in Class A: $21$
– Lowest score in Class B: $22$

Since the highest score in Class A ($21$) is less than the lowest score in Class B ($22$), any student moving from Class A to Class B will have a score less than $22$.

Effect on Class B:
– Shifting four students from Class A (with scores all less than $22$) to Class B will introduce scores lower than the existing lowest score of Class B.
– This means the total sum of scores in Class B will increase by an amount less than $22 \times 4 = 88$, while the total number of students will increase by $4$.
– The average score of Class B will definitely decrease since the new scores are all less than the current average.

Thus, Statement 1 is correct.

2. Impact on Class A’s Average Score:

– Lowest score in Class A: $17$
– Highest score in Class A: $21$

Removing four students from Class A, regardless of which four, would mean losing scores that are in the range of $17$ to $21$.

Effect on Class A:
– The removal of students can either increase or decrease the average score depending on which scores are removed. If the four lowest scores (close to $17$) are removed, the average could increase. Conversely, if the four highest scores (close to $21$) are removed, the average could decrease.

– Without specific details on which scores are removed, we cannot definitively conclude if the average will increase.

Thus, Statement 2 is not necessarily correct.

Conclusion:

– Statement 1 is definitely correct because Class B will receive scores lower than its current lowest score, reducing its average.
– Statement 2 is not necessarily correct because the impact on Class A’s average depends on which specific students are shifted.

The correct answer is (a) 1 only.

Explanation:

To solve this problem, we need to consider all 3-digit numbers that can be formed using non-zero digits which are multiples of 3, without repetition of digits. We then analyze the sum $S$ of these numbers to check its divisibility by 74 and 9.

Step-by-Step Solution:

Identifying the Digits:

The non-zero digits that are multiples of 3 are: $3, 6,$ and $9$.

Forming 3-Digit Numbers:

We can form 3-digit numbers using the digits $3, 6,$ and $9$ without repeating any digit. The possible 3-digit numbers are:

\[
369, 396, 639, 693, 936, 963
\]

Calculating the Sum $S$:

Let’s calculate the sum of these 3-digit numbers:

\[
S = 369 + 396 + 639 + 693 + 936 + 963
\]

Calculating step-by-step:

\[
369 + 396 = 765
\]
\[
639 + 693 = 1332
\]
\[
936 + 963 = 1899
\]

Now, summing these results:

\[
S = 765 + 1332 + 1899 = 3996
\]

Checking for Divisibility:

1. Divisibility by 74:

To check if $S$ is divisible by 74, we divide 3996 by 74:

\[
\frac{3996}{74} = 54 \quad \text{(remainder 0)}
\]

Since there is no remainder, $S = 3996$ is divisible by 74.

2. Divisibility by 9:

A number is divisible by 9 if the sum of its digits is divisible by 9.

Sum of the digits in $S = 3996$:

\[
3 + 9 + 9 + 6 = 27
\]

Since $27$ is divisible by 9, $S = 3996$ is divisible by 9.

Conclusion:

Both statements are correct:

1. $S$ is divisible by 74.
2. $S$ is divisible by 9.

The correct answer is (c) Both 1 and 2.

Explanation:

To solve this problem, we need to determine the number of Indians who can speak English in the group based on the given data.

Given Information:

– Total number of persons in the group: $120$
– Number of Indians: $80$
– Number of foreigners: $120 – 80 = 40$
– Number of people who can speak English: $70$

Solution:

Let’s use the given information to analyze the problem:

1. Define Variables:

– Let $I_E$ represent the number of Indians who can speak English.
– Let $F_E$ represent the number of foreigners who can speak English.

2. Relationships:

– The total number of persons who can speak English is $70$, so:
\[
I_E + F_E = 70
\]

– The total number of Indians is $80$, and the rest are foreigners:
\[
I + F = 120 \implies 80 + 40 = 120
\]

3. Calculating Limits for $I_E$:

– Maximum Value of $I_E$:
– The maximum possible value for $I_E$ is when all Indians can speak English, but since there are only $70$ people who can speak English:
\[
I_E \leq 70
\]

– Minimum Value of $I_E$:
– If we assume that all foreigners can speak English, then:
\[
F_E = 40 \quad \text{(maximum possible foreigners speaking English)}
\]
Thus, the minimum number of Indians who can speak English would be:
\[
I_E = 70 – 40 = 30
\]

Therefore, the number of Indians who can speak English can range from $30$ to $70$.

4. Conclusion:

Given these calculations, the correct answer is (d) 30 or more because $I_E$ can be $30$ or more depending on how many foreigners speak English.

Explanation:

The passage states that in India, macroeconomic policy aims to improve the economic welfare of the people and that monetary and fiscal policies must work together, as neither can operate effectively in isolation. This implies that the central bank (responsible for monetary policy) and the government (responsible for fiscal policy) must coordinate closely to achieve the goals of economic welfare.

Option (a) is the correct corollary, as it reflects the need for cooperation between the central bank and the government.

The other options do not align with the central idea:

• (b) suggests close regulation of financial markets, which is not discussed in the context of policy coordination in the passage.
• (c) implies a conflict between a market economy and socialist policies, which is unrelated to the point about fiscal and monetary policy interdependence.
• (d) suggests financial sector reforms as a requirement for economic welfare, but the passage focuses on coordination between fiscal and monetary policies rather than specific reforms.

Thus, option (a) is the best reflection of the corollary to the passage.

Explanation:

The passage highlights that while the Insolvency and Bankruptcy Code and bank recapitalization can help address non-performing assets (NPAs) and improve the capital cushion of banks, these measures alone are insufficient. The passage emphasizes the need for “systemic reforms” to address the root causes of reckless and unsustainable lending practices, which continue to stress the banking system.

Option (d) directly reflects this suggestion, as it proposes structural reforms as a necessary long-term solution for the problem of bad loans in the banking system.

The other options are less appropriate:

• (a) suggests central government monitoring, but the passage implies the need for broad systemic changes rather than close central oversight.
• (b) suggests lowering interest rates to boost lending, which does not address the core issue of reckless lending practices.
• (c) proposes bank mergers as a solution, but the passage does not mention consolidation as a remedy; rather, it focuses on the need for deeper systemic reforms.

Thus, option (d) is the most logical, rational, and practical suggestion implied by the passage.

Explanation:

The passage discusses the complex moral, legal, and economic challenges posed by emerging technologies like artificial intelligence, driverless cars, and autonomous weapons, especially when these technologies are used across borders. The examples provided suggest that technology is blurring national boundaries (implication 2) and that the traditional roles and powers of the state are affected by private innovation and capital (implication 3). Additionally, the passage highlights the ambiguities and uncertainties that global technologies bring to geopolitical and regulatory frameworks (implication 5).

The other options are not supported by the passage:

• 1. “Too much globalization is not in the best interests of any country” is not directly discussed in the passage.
• 4. “Public policy of every country should focus on developing its own supply chains” is unrelated to the core argument about technological regulation and moral/legal dilemmas.

Therefore, statements 2, 3, and 5 are the most relevant implications, making option (c) the correct choice.

Explanation:

The passage argues that in a “natural state,” where individuals act solely in their self-interest without regard to common rules or laws, there is no universally agreed-upon concept of good or bad. It is only in a “civil state,” created by collective agreement, that common ideas of right and wrong (or good and bad) emerge, with individuals holding themselves accountable to the state.

Option (a) best captures this idea, as it directly states that the concepts of right and wrong exist due to the formation of a state, aligning with the passage’s assertion that such concepts arise from communal or legal agreements.

The other options are less accurate:

• (b) implies that a ruling authority alone is necessary for moral behavior, but the passage does not suggest this; rather, it emphasizes a collective civil agreement.
• (c) suggests that humans are inherently immoral, which is not mentioned in the passage. The passage describes self-interest in a natural state without labeling it as immoral.
• (d) implies that right and wrong concepts are biologically necessary for survival, which is beyond the passage’s focus on civil state formation and moral accountability.

Thus, option (a) is the most accurate reflection of the central idea.

Explanation:

The passage highlights the challenges faced by cities, such as vulnerability to climate change, increasing population densities, poor infrastructure, and the lack of facilities to meet the needs of residents. It stresses the need to address issues related to air quality, transport, and sustainable solutions, along with involving citizens in city planning. This indicates a need for sustainability-focused interventions to make cities more resilient and capable of handling growth.

Option (c) directly aligns with the passage’s focus on sustainability and developing solutions for urban challenges.

The other options are less accurate:

• (a) suggests a need for an administrative structure with autonomy, which is not discussed in the passage.
• (b) focuses on population density as a hindrance but does not address the main point, which is the need for sustainable development.
• (d) mentions Public-Private Partnership as a solution, which is not specifically discussed or implied in the passage.

Therefore, option (c) is the most logical and rational inference from the passage.

Explanation:

The passage critiques the current school curriculum, suggesting it focuses heavily on academic knowledge while neglecting important societal issues and life skills, such as understanding social divides, gender sensitivity, and independent thinking. It highlights the gap between the sheltered environment of school and the complex world students face in university, emphasizing that an education system should also foster questioning, awareness, and critical thinking.

Option (d) best captures the central idea of the passage, as it points out the need for education to incorporate life-coping skills and societal awareness to prepare students for real-world challenges, in addition to academic content.

The other options do not fully address this:

• (a) only mentions compatibility with expectations, which is a limited view of the broader critique.
• (b) implies a lack of time for personality development, which is only partially relevant.
• (c) mentions citizenship but misses the specific emphasis on societal skills and critical thinking discussed in the passage.

Therefore, option (d) is the most accurate reflection of the central idea.

Explanation:

The passage discusses the idea that even truthful people might occasionally tell lies, especially if they believe it prevents harm. It suggests that these individuals might engage in “semantic lies,” where they intentionally mix truth and falsehood in a way that serves a purpose—often with the intent of avoiding harm or simplifying communication. This aligns most closely with option (b), “Intentional mixing up of truth with the false,” because it reflects a deliberate action rather than an accidental or complete concealment of the truth.

The other options do not align as well:

• (a) “Mixing up the true and false” implies accidental or careless mixing, not the intentional action described in the passage.
• (c) “Falsification of facts” suggests a more significant alteration of reality, which is not necessarily implied here.
• (d) “Complete concealment of truth” does not fit, as the passage suggests that truthful people do not entirely hide the truth; they might instead blend it carefully with falsehoods for specific reasons.

Explanation:

The passage begins by stating, “most people would agree that telling deliberate lies is wrong, except perhaps in certain special situations.” This part suggests that there is a general consensus (agreement) among people that lying is wrong, though there may be exceptions in specific situations where telling the truth might cause more harm.

Analysis of Each Option:

Option (a): “agreement about telling lies.”

This option is correct because the first part of the passage explicitly mentions that “most people would agree that telling deliberate lies is wrong.” This reflects a general agreement about the moral view that lying is wrong, though it acknowledges some exceptions.

Option (b): “disagreement about telling lies.”

This option is incorrect because the passage does not suggest disagreement about lying. Instead, it highlights an agreement that lying is generally wrong, with some possible exceptions.

Option (c): “disagreement about telling the truth.”

This option is incorrect because the passage does not discuss a disagreement about telling the truth. It only mentions that in some special cases, telling the truth might cause more harm, but it does not indicate any disagreement on the matter.

Option (d): “disagreement about the harm in telling the truth.”

This option is also incorrect. The passage does not imply that there is disagreement about whether telling the truth can cause harm. Instead, it suggests that in some situations, truth-telling might indeed cause harm, which is generally accepted as a potential exception to the rule against lying.

The correct answer is (a) agreement about telling lies.

Explanation:

The passage describes how, throughout history, humans have explored new frontiers primarily for economic reasons, specifically the search for resources such as gold, spices, and other valuable materials. It contrasts this economic motivation with weaker drivers like science and curiosity. The author suggests that the only sustainable way to open up new frontiers (such as space) is to establish an “economic engine” based on resource extraction. This makes it clear that the main motivation for exploration, historically and potentially in the future, is economic gain.

Analysis of Each Option:

Option (a): “Wealth generation is the primary motive for any human endeavour.”

This option is too broad. While the passage emphasizes that economic considerations have driven exploration, it does not imply that all human endeavors are motivated primarily by wealth generation. The focus is specifically on exploration and opening up new frontiers. Thus, this option is not the best choice.

Option (b): “Space, whether space in solar system or interstellar space, will govern our future economy.”

This option misinterprets the passage. The passage does discuss space as a future frontier, but it does not state that space will “govern” our future economy. Instead, it suggests that economic motives, such as resource extraction, are necessary to make space exploration viable. Therefore, this option is not supported by the passage.

Option (c): “Human beings are motivated to explore new frontiers principally by economic considerations.”

This option accurately captures the main idea of the passage. The passage emphasizes that economic considerations, specifically the search for resources, have historically driven human exploration of new frontiers and will continue to do so in the future (e.g., space exploration). This matches the passage’s focus on economic motivations as the primary driver for exploration.

Option (d): “Wealth generation is based on the risk-taking behaviour of some men.”

This option is not the best fit because the passage does not emphasize risk-taking behavior specifically, nor does it state that wealth generation relies on such behavior. The passage focuses on economic motivations rather than risk-taking itself.

Conclusion:

The best option that accurately reflects the main idea of the passage is (c) Human beings are motivated to explore new frontiers principally by economic considerations.

The correct answer is (c) Human beings are motivated to explore new frontiers principally by economic considerations.

Explanation:

Let’s analyze each assumption based on the information given in the passage.

Assumption 1: BPA may alter embryonic development in vivo.

This assumption is valid. The passage states that researchers used mouse embryonic stem cells to study the neurotoxic effects of BPA and observed its impact on the differentiation of stem cells into primary germ layers, such as endoderm and ectoderm, as well as on the formation of neural progenitor cells. Since these processes are critical for normal embryonic development, it can be inferred that BPA may alter embryonic development in vivo (in a living organism) as well. Thus, assumption 1 is valid.

Assumption 2: Biochemical and cell-based assays are useful in finding out treatments for pollution-induced diseases.

This assumption is not valid based on the passage. While the passage mentions that biochemical and cell-based assays were used to study the toxic effects of BPA, it does not suggest that these assays were used to find treatments for diseases caused by pollution. The passage only describes their use in assessing toxicity and gene expression changes, not in developing treatments. Therefore, assumption 2 is not supported by the passage.

Assumption 3: Embryonic stem cells could serve as a model to evaluate the physiological effects of environmental pollutants.

This assumption is valid. The passage describes how researchers used embryonic stem cells to gauge the neurotoxic effects of BPA, an environmental pollutant. By examining the differentiation process and gene expression profile in these stem cells, they were able to detect and measure BPA’s impact. This suggests that embryonic stem cells can indeed serve as a model to evaluate the effects of environmental pollutants. Thus, assumption 3 is valid.

Conclusion:

The valid assumptions based on the passage are 1 and 3 only.

The correct answer is (c) 1 and 3 only.

Explanation:

Let’s examine each assumption in relation to the passage.

Assumption 1: Sustainable production of India’s cereal food grains is impossible without the diversity of pollinating animals.

This assumption is not valid. While the passage mentions that pollination is crucial for a wide variety of food, it does not specifically state that pollinators are essential for cereal grains. Many cereal grains, such as rice, wheat, and maize, are primarily wind-pollinated and do not depend on animal pollinators. Therefore, this assumption cannot be directly inferred from the passage.

Assumption 2: Monoculture of horticultural crops hampers the survival of insects.

This assumption is also not directly supported by the passage. The passage does not mention monoculture or its effects on insect survival. Although monoculture could potentially harm pollinator diversity, this specific idea is not implied by the passage. Therefore, this assumption is not valid.

Assumption 3: Pollinators become scarce in cultivated areas devoid of natural vegetation.

This assumption is valid. The passage states that “effective pollination requires resources, such as refuges of pristine natural vegetation.” This implies that pollinators rely on natural vegetation for resources, and without it, they would be less abundant. Thus, it is reasonable to infer that pollinators become scarce in areas without natural vegetation.

Assumption 4: Diversity in insects induces diversity of plants.

This assumption is also valid. The passage describes an “intricate relationship between plants and animals,” where the reduction or loss of either affects the survival of both. This suggests a mutual dependency, where a diverse population of insect pollinators could help support a diversity of plant species. Hence, this assumption is consistent with the passage.

Conclusion:

The valid assumptions based on the passage are 3 and 4 only.

The correct answer is (d) 3 and 4 only.

Explanation:

The passage discusses the impact of climate change in the Cauvery basin region of Tamil Nadu. The study mentioned in the passage shows that climate change is leading to rising temperatures, fewer rainy days, changes in the hydrological cycle, and an increase in drought frequency. These climatic shifts have caused increased runoff, reduced groundwater recharge, and consequently, a decline in groundwater tables. Farmers, facing a lack of surface water, are increasingly relying on groundwater to secure their crops.

Analysis of Each Option:

Option (a): “Development of regional climate models helps in choosing climate-smart agricultural practices.”

While the passage does mention the use of regional climate models to study climate impacts, it does not suggest that the purpose of these models is to select agricultural practices. The focus of the passage is on the observed impacts of climate change on water resources and farmers’ reliance on groundwater, not on agricultural practice selection. Therefore, this option is not the best answer.

Option (b): “Heavy dependence on groundwater resources can be reduced by adopting dry-land cropping systems.”

This option suggests a solution to groundwater dependence but is not the crux of the passage. The passage does not discuss specific agricultural methods, such as dry-land cropping, nor does it propose solutions to reduce groundwater use. Therefore, this option is not supported by the passage.

Option (c): “Climate changes increase the criticality of water resources while simultaneously threatening it.”

This option best captures the essence of the passage. Climate change is making water resources more critical for farmers due to increased temperatures, reduced rainfall, and frequent droughts. At the same time, climate change is threatening these very resources by reducing groundwater recharge and lowering water tables, making water resources increasingly vulnerable. This captures both the increased importance of water resources for farmers and the threats posed to these resources by climate change.

Option (d): “Climate changes cause the farmers to adopt unsustainable livelihoods and risky coping strategies.”

Although the passage mentions that farmers are increasingly relying on groundwater to secure their crops, it does not go as far as to say that this is an “unsustainable livelihood” or a “risky coping strategy.” While this may be inferred, the passage does not focus on the unsustainability of farmers’ choices but rather on the impact of climate change on water resources. Therefore, this option is not the best choice.

The correct answer is (c) Climate changes increase the criticality of water resources while simultaneously threatening it.

Explanation:

Let’s analyze each assumption based on the passage.

Assumption 1: People who are obsessed with personal hygiene tend to ignore the community hygiene.

The passage discusses how the market’s emphasis on personal hygiene, especially through advertisements, creates a “false sense of security.” However, it does not explicitly state that people who focus on personal hygiene ignore community hygiene. This assumption is not directly supported by the passage, so it is not valid.

Assumption 2: Emergence of multi-drug resistant germs can be prevented by personal cleanliness.

The passage actually suggests the opposite: it mentions that an “obsession with hand hygiene” can contribute to the emergence of resistant germs, due to the influence of market-driven products. This assumption is therefore not valid, as it contradicts the passage’s message.

Assumption 3: Entry of the market in the domain of hygiene increases the risk of infectious diseases.

The passage implies that the market creates a “false sense of security” about personal hygiene but does not directly state that this increases the risk of infectious diseases. It points out the potential ecological impact and the risk of resistant germs but does not make a clear link between market influence and an increase in infectious diseases. Thus, this assumption is not fully supported and is not valid.

Assumption 4: Scientific and micro-level interventions are not sufficient to bring down the burden of infectious diseases.

This assumption is valid. The passage suggests that, although personal hygiene (a micro-level intervention) is important, it alone is not enough. It highlights the need for broader environmental improvements, such as clean water, sanitation, and food security, as these public health measures have been effective in reducing infection rates in Western Europe. This implies that focusing only on personal or micro-level hygiene is insufficient, so assumption 4 is valid.

Assumption 5: It is community hygiene implemented through public health measures that is really effective in the battle against infectious diseases.

This assumption is valid. The passage gives the example of Western Europe, where not only personal hygiene but also community-wide improvements like clean water, sanitation, and food security have led to a decrease in infection rates. This suggests that community hygiene, through public health measures, plays a crucial role in reducing infectious diseases. Thus, assumption 5 is valid.

Conclusion:

The valid assumptions are 4 and 5 only.

Thus, correct answer is (c) 4 and 5 only.

Explanation:

The passage describes how computers and the Internet of Things (IoT) are increasingly integrated into various physical devices that interact with the real world, including cars, airplanes, medical devices, and even road signs. The passage highlights the vulnerability of these internet-connected devices, with specific examples like cars and pacemakers being susceptible to hacking. This points to a broader implication: as computers become more embedded in everyday physical objects, the risks and impacts of security breaches will become more significant and will increasingly affect our daily lives.

Analysis of Each Option:

• Option (a): “Computers are not completely safe.” While this statement is true, it is too general and does not capture the main inference from the passage. The passage is specifically focused on the integration of computers into physical devices and the implications of this trend, rather than a general statement about the safety of computers.

• Option (b): “Companies producing the software do not take cyber security seriously.” This option is speculative and not directly supported by the passage. The passage does not make any claims about companies’ attitudes towards cybersecurity. It only highlights the vulnerability of connected devices, not the reasons behind these vulnerabilities.

• Option (c): “Stringent data security laws are needed.” While this may be a reasonable suggestion based on the vulnerabilities discussed, the passage does not specifically argue for new security laws. It simply describes the trend and the associated risks. Therefore, this is an interpretation rather than an inference directly supported by the passage.

• Option (d): “The present trend of communication technologies will affect our lives in future.” This option captures the essence of the passage. The integration of computers into physical devices (the Internet of Things) is described as a trend that will have significant implications, especially in terms of security and safety risks. The examples provided (cars, pacemakers) illustrate how this trend could directly impact our lives in the future.

Conclusion:

The most critical inference from the passage is that the present trend of communication technologies (such as the Internet of Things) will affect our lives in the future, especially due to the security vulnerabilities of these technologies.

The correct answer is (d) The present trend of communication technologies will affect our lives in future.

Explanation:

The passage describes agroecology as an approach to farming that mimics natural ecosystems by promoting biodiversity, nutrient recycling, and pest resistance through natural processes. Agroecological practices include returning nutrients to the soil, using natural pest control, building soil fertility, and integrating trees with livestock and crops.

Let’s analyze each option in light of these principles:

1. Cover crops:
   Cover crops are used to protect the soil, prevent erosion, and improve soil fertility by adding organic matter when they decompose. This practice aligns well with agroecology’s focus on nutrient conservation and building soil fertility, so cover crops are compatible with agroecology.
2. Fertigation:
   Fertigation is a method of applying fertilizers through irrigation systems. While it provides nutrients, it is typically associated with conventional farming and does not align with the agroecological principle of recycling nutrients and minimizing external inputs. Fertigation is not typically compatible with agroecology as it relies on external, often synthetic, fertilizers rather than nutrient recycling within the system.
3. Hydroponics:
   Hydroponics is a method of growing plants without soil, using a nutrient-rich water solution instead. This approach does not align with the agroecological principle of building soil fertility or recycling nutrients within a soil-based ecosystem. Hydroponics is generally not compatible with agroecology.
4. Mixed farming:
   Mixed farming involves combining crops and livestock on the same farm. This aligns with agroecology’s focus on creating diverse and resilient agroecosystems, as it mimics the diversity and interconnectedness found in natural ecosystems. Mixed farming is compatible with agroecology.
5. Polyculture:
   Polyculture refers to growing multiple types of crops together, which enhances biodiversity and can improve pest resistance and resilience. This practice closely aligns with agroecology’s goal of mimicking the diversity and functioning of natural ecosystems. Polyculture is compatible with agroecology.
6. Vertical farming:
   Vertical farming involves growing crops in stacked layers, often indoors, with artificial lighting and controlled conditions. This practice is typically more aligned with high-tech urban farming and does not mimic natural ecosystems or promote nutrient recycling in soil. Vertical farming is generally not compatible with agroecology.

Conclusion:

Based on the analysis, the practices that are compatible with agroecology as implied by the passage are cover crops (1), mixed farming (4), and polyculture (5).

Therefore, the correct answer is: (a) 1, 4 and 5 only

Explanation:

Let’s analyze each assumption based on the passage.

Assumption 1: Fig trees can often be keystone species in natural forests.

This assumption is valid. The passage states that fig trees provide food for wildlife when other resources are scarce and support a high density and diversity of frugivores (fruit-eating animals). This role in sustaining wildlife, especially during periods when other food sources are not available, suggests that fig trees are essential to the ecosystem, which is characteristic of a keystone species.

Assumption 2: Fig trees can grow where other large woody species cannot grow.

This assumption is not fully supported by the passage. While the passage mentions that fig trees are “common in agricultural and urban landscapes where other large trees are absent,” it does not clearly state that fig trees grow in places where other large woody species cannot grow. The absence of other large trees in these areas might be due to factors other than the growth potential of fig trees specifically. Therefore, this assumption is not valid.

Assumption 3: Sacred trees can have a role in biodiversity conservation.

This assumption is valid. The passage suggests that fig trees, which are considered sacred in certain cultures, provide food for wildlife and attract frugivorous birds and bats, even in areas with human disturbance. By supporting frugivores and providing food during scarcity, sacred fig trees can help sustain biodiversity, thus playing a role in conservation.

Assumption 4: Fig trees have a role in the seed dispersal of other tree species.

This assumption is valid. The passage states that “under favourable microclimate, plenty of seedlings of other tree species would grow around fig trees.” This implies that fig trees create conditions that promote the growth of seedlings from other species, which is likely due to seed dispersal facilitated by frugivores that visit the fig trees. Thus, fig trees indirectly aid in the seed dispersal and growth of other tree species.

Therefore, the correct answer is: (d) 1, 3 and 4 only

Explanation:

The passage outlines multiple complex challenges that India faces in its energy and climate policy-making, including fossil fuel production issues, limited access to electricity for the poorest, rising fuel imports, electricity pricing and governance challenges, and environmental conflicts over resources like land, water, and air. Additionally, the passage mentions various ongoing initiatives—such as energy efficiency programs, urbanization and transport policies, and renewable energy projects—that suggest India is trying to address these interconnected challenges in a transformative way.

The main message of the passage is that India’s energy decision-making is not simple or isolated; instead, it is highly complex and involves addressing multiple, interconnected issues. This complexity is captured by option (a), which states that “India’s energy decision-making process is ever more complex and interconnected.”

Let’s examine why the other options are incorrect:

• Option (b): “India’s energy and climate policy is heavily tuned to sustainable development goals” is not accurate because the passage does not explicitly state that India’s energy policy is focused on sustainable development goals. While there are mentions of renewable energy and energy efficiency, the main focus of the passage is on the complexity and variety of challenges, rather than a specific alignment with sustainable development goals.
• Option (c): “India’s energy and climate actions are not compatible with its broader social, economic and environmental goals” is incorrect because the passage does not imply any incompatibility. Instead, it mentions initiatives that aim to address social, economic, and environmental issues, suggesting efforts to achieve compatibility rather than a conflict.
• Option (d): “India’s energy decision-making process is straightforward supply-oriented and ignores the demand side” is incorrect because the passage discusses a range of issues, including energy access, environmental concerns, and efficiency programs, which go beyond simple supply-side considerations.

Therefore, the Correct Answer is: (a) India’s energy decision-making process is ever more complex and interconnected.

Explanation:

In this passage, Plato discusses how different forms of government can “ruin” themselves when they go to extremes in applying their core principles:

• Aristocratic government fails when it restricts power to a very small group.
• Oligarchic government collapses due to an excessive focus on wealth.
• Democracy undermines itself by giving excessive power to all citizens without ensuring they are educated enough to make informed decisions, which can lead to tyranny or autocracy.

The crux of the passage is that each type of government has a fundamental principle, but an excessive application of this principle ultimately leads to its own downfall. This is best captured by option (b): “Any form of government tends to deteriorate by excess of its basic principle.”

Let’s examine why the other options are incorrect:

• Option (a): “Human societies experiment with different forms of governments” is too general and does not capture the main point of the passage, which is about the self-destructive potential of each government’s foundational principle.
• Option (c): “Education of all citizens ensures a perfect, functional and sustainable democracy” is not the main point of the passage. While the passage does mention that democracy suffers when people are not educated enough to make informed decisions, the main argument is about the excess of democracy itself, not just the need for education.
• Option (d): “Having a government is a necessary evil because tyranny is inherent in any form of government” is an interpretation that goes beyond the passage. Plato does not claim that tyranny is inherent in all forms of government, but rather that democracy can lead to tyranny when it is practiced in excess.

Therefore, the most accurate answer is (b) Any form of government tends to deteriorate by excess of its basic principle.

Explanation:

Let’s analyze each assumption in light of the passage:

1. Equality is a prerequisite for people to participate in the multiple transactions of society from a position of confidence.

This assumption is valid. The passage highlights the importance of equality as a fundamental democratic principle that ensures “the equal standing of citizens.” Equal standing and the absence of discrimination are necessary for people to interact confidently in society, as inequality would create barriers to their full participation. Thus, this assumption aligns well with the passage.

1. Occurrence of inequality is detrimental to the survival of democracy.

This assumption is also valid. The passage suggests that inequality violates a core democratic norm and that equality is “morally speaking, a default principle.” Since the equal standing of citizens is a basic requirement of democracy, inequality undermines this foundation, which is detrimental to democracy’s survival. Therefore, this assumption is supported by the passage.

1. Equal standing of all citizens is an idea that cannot actually be realised even in a democracy.

This assumption is not valid. The passage argues for the ideal of equality and does not suggest that achieving equal standing of all citizens is impossible. Instead, it asserts that people should be treated equally and that discrimination on irrelevant grounds (such as race, caste, gender, etc.) is morally wrong. The passage does not imply that equality is unachievable, so this assumption is not consistent with the passage.

1. Right to equality should be incorporated into our values and day-to-day political vocabulary.

While this statement may be a reasonable suggestion, the passage does not explicitly advocate for incorporating equality into day-to-day political vocabulary. The passage argues for equality as a moral and democratic principle but does not specifically mention the need for it to be incorporated into political language. Thus, this assumption is not directly supported by the passage.

Based on this analysis, the valid assumptions are 1 and 2 only.

Correct Answer: (a) 1 and 2 only

Explanation:

From the passage, we can infer that seditions, wars, and breaches of laws in a dominion are not primarily due to the inherent wickedness of the people. Instead, they are attributed to the poor condition or mismanagement of the dominion itself. The passage suggests that people are not naturally predisposed to be good citizens; they need to be guided through laws and governance that foster unity and good citizenship.

The author implies that if a dominion has many offenses and wicked acts, it likely has not effectively pursued unity or created laws that ensure good citizenship. Hence, a well-functioning dominion would be one that promotes unity and implements well-thought-out laws to cultivate good citizens. This reasoning directly supports option (c), which states that “the best dominion is one which pursues the aim of unity and has laws for good citizenship.”

Let’s examine why the other options are incorrect:

• Option (a): This option states that seditions, wars, and breaches of laws are inevitable in every dominion. However, the passage does not imply that these issues are inevitable; rather, it suggests that these problems arise due to poor governance and can be mitigated by effective laws and unity.
• Option (b): This option suggests that the sovereign alone is responsible for all problems. While the passage critiques the structure and laws of the dominion, it does not place sole blame on the sovereign or the ruler but rather on the overall governance and legal structure.
• Option (d): This option claims it is impossible to establish a good dominion. The passage, however, implies that a good dominion is possible if it pursues unity and has well-framed laws.

Therefore, the most logical and rational inference is option (c) That dominion is the best which pursues the aim of unity and has laws for good citizenship.

Explanation:

Passage Analysis

The passage discusses how people perceive their own religion versus how they perceive the religions of others. The key points in the passage are:

• If individuals judge only their own religion, then they see it as correct and valid, meaning there is no “wrong” religion.
• If individuals judge each other’s religions, then no religion is seen as entirely “right.”
• This leads to the idea that either all religions are right, or all religions are wrong, depending on the perspective.

The passage suggests that perceptions of religious denominations are subjective and can vary depending on whether someone is evaluating their own beliefs or the beliefs of others.

Analyzing Each Option

Option (a): “No man can live without adhering to some religious denomination.”

• This option suggests that everyone needs a religious denomination to live, but the passage does not make any claims about whether people can live without a religious affiliation.
• The passage discusses the judgment of religious denominations, but it does not imply that adherence to a denomination is essential for all individuals.
• This option is incorrect because it does not logically follow from the passage.

Option (b): “It is the duty of everyone to propagate one’s religious denomination.”

• This option suggests that everyone has a responsibility to spread or promote their religion.
• However, the passage does not discuss any duty or responsibility regarding the propagation of religious beliefs.
• The passage focuses on the subjective judgment of religions rather than any obligation to promote them.
• This option is incorrect because it is not a logical assumption based on the passage.

Option (c): “Religious denominations tend to ignore the unity of man.”

• This option implies that religious denominations create division rather than unity among people.
• While the passage does discuss the subjective nature of religious judgment, it does not explicitly state that denominations ignore the unity of humanity.
• The passage focuses on individual perceptions of religions rather than making a statement about the divisive nature of denominations.
• This option is not directly supported by the passage.

Option (d): “Men do not understand their own religious denomination.”

• This option suggests that people may lack a true understanding of their own religion.
• However, the passage implies that people are confident in the correctness of their own religion when judging it. The problem arises when they judge other religions.
• There is no indication in the passage that people misunderstand their own religions.
• This option is incorrect because it is not a logical assumption based on the passage.

Conclusion

The passage essentially conveys that religious denominations are subject to personal judgment and that such judgments vary based on whether one is evaluating their own beliefs or those of others. This subjective nature of judgment can lead to a lack of unity or common understanding, as people may view their own beliefs as correct while questioning those of others.

Thus, the correct answer is: (c) Religious denominations tend to ignore the unity of man

Explanation:

Given Statements

1. Statement 1: Some greens are blues.
2. Statement 2: Some blues are blacks.

From these statements, we know that there is an overlap between greens and blues, and between blues and blacks. However, there is no direct connection provided between greens and blacks. We only know that both greens and blacks are related to blues.

Analyzing Each Conclusion

Conclusion 1: Some greens are blacks.

• From the given statements, we know:
  • Some greens are blues.
  • Some blues are blacks.
• However, this does not imply that some greens must also be blacks, because there is no direct connection between greens and blacks. While it’s possible that some greens could be blacks, it is not certain based on the information provided.
• Conclusion 1 does not logically follow from the statements, as there is no guaranteed overlap between greens and blacks.

Conclusion 2: No green is black.

• From the given statements, we know that some greens are blues and some blues are blacks, but we do not know whether there is any overlap between greens and blacks.
• Since we do not have any definitive information about the relationship between greens and blacks, we cannot conclude that “no green is black.”
• Conclusion 2 does not logically follow from the statements.

Conclusion 3: All greens are blacks.

• The statements only mention that “some greens are blues” and “some blues are blacks.” There is no information suggesting that all greens are blacks.
• Conclusion 3 does not logically follow from the statements, as there is no evidence to support that all greens are blacks.

Conclusion 4: All blacks are greens.

• The statements only mention that “some blues are blacks” and “some greens are blues.” There is no information suggesting that all blacks are greens.
• Conclusion 4 does not logically follow from the statements, as there is no evidence to support that all blacks are greens.

Conclusion

Since none of the conclusions logically follow from the given statements, the correct answer is: (d) Neither Conclusion 1 nor 2 nor 3 nor 4

Explanation:

Statement 1: p×q is greater than zero

This statement tells us that:

p×q>0

This implies that p and q have the same sign:

• Either both p and q are positive, or
• Both p and q are negative.

However, knowing that p and q have the same sign does not give us any information about their relative magnitudes. Therefore, Statement 1 alone is insufficient to determine whether p is greater than q.

Statement 2: $p^2$ is greater than $q^2$

This statement tells us that:

$p^2 > q^2$

This implies that the absolute value of p is greater than the absolute value of q, i.e., ∣p∣>∣q∣.

However, knowing that ∣p∣>∣q∣ does not necessarily mean that p>q. For example:

• If p = 5 and q = −3, then p > q.
• If p = -5 and q = 3, then p < q.

Thus, Statement 2 alone is also insufficient to determine whether p is greater than q.

Using Both Statements Together

If we combine both statements, we know:

1. p and q have the same sign (from Statement 1).
2. ∣p∣ > ∣q∣ (from Statement 2).

There are two possible scenarios:

• Case 1: Both p and q are positive, and ∣p∣ > ∣q∣. In this case, p > q.
• Case 2: Both p and q are negative, and ∣p∣ > ∣q∣. In this case, p < q.

Since we don’t know the sign of p and q (whether they are both positive or both negative), we cannot definitively conclude whether p is greater than q even using both statements together.

Conclusion

The answer is: (d) The Question cannot be answered even by using both the Statements together.

Explanation:

To solve this problem, let’s set up equations based on the information given and then determine the relationship between the price of a pen and the price of a pencil.

Step-by-Step Solution

1. Define Variables:

• Let the price of a pen be $P$.
• Let the price of a pencil be $C$.

2. Set Up Equations Based on the Information Given:

• Jay bought 3 pens and 5 pencils, and Vijay bought 2 pens and 7 pencils.
• They spent an equal amount of money.

Therefore, we can set up the equation:
$ 3P + 5C = 2P + 7C $

3. Solve for the Relationship Between $P$ and $C$:

Rearrange the equation to isolate $P$ and $C$:

$ 3P – 2P = 7C – 5C $

$ P = 2C $

This means the price of a pen ($P$) is twice the price of a pencil ($C$).

Conclusion

The correct answer is: $ \text{(c) The price of a pen is two times the price of a pencil} $

Explanation:

To determine which of the conclusions logically follow from the statement, let’s analyze the given statements and conclusions step-by-step.

Statement Analysis

1. Statement 1: Some radios are mobiles.

• This means there is some overlap between the sets of radios and mobiles.

2. Statement 2: All mobiles are computers.

• This means the set of mobiles is entirely contained within the set of computers.

3. Statement 3: Some computers are watches.

• This means there is some overlap between the sets of computers and watches.

Based on these statements, we can visualize the relationships:

• Some radios are part of the set of mobiles.
• Since all mobiles are computers, some radios are also part of the set of computers.
• Some computers are also part of the set of watches.

However, this information alone does not establish a direct relationship between radios and watches or between mobiles and watches.

Conclusion Analysis

Conclusion-I: “Certainly some radios are watches.”

From the statements, we know:

• Some radios are mobiles.
• All mobiles are computers, so some radios are also computers.
• Some computers are watches.
• However, just because some computers are watches does not guarantee that the radios (which are part of the computers) are also part of the watches. There is no direct connection that forces some radios to be watches.
  Therefore, Conclusion-I does not logically follow from the statements.

Conclusion-II: “Certainly some mobiles are watches.”

From the statements, we know:

• All mobiles are computers.
• Some computers are watches.
• However, just because some computers are watches does not guarantee that the mobiles (which are part of the computers) are also part of the watches. There is no direct connection that forces some mobiles to be watches.

Therefore, Conclusion-II also does not logically follow from the statements.

Conclusion

Since neither Conclusion-I nor Conclusion-II logically follows from the statements, the correct answer is: $ \text{(d) Neither Conclusion-I nor Conclusion-II} $

Explanation:

Step-by-Step Solution

1. Identify Consonants in the Alphabet: The consonants in the English alphabet are: B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, W, X, Y, Z
2. Mirror Symmetry: A letter that appears the same in a mirror has vertical symmetry, meaning the left half is a mirror image of the right half.
3. Analyze Each Consonant: Let’s identify the consonants that have vertical symmetry and thus appear the same in a mirror.
   • Consonants with Vertical Symmetry (Appear the Same in a Mirror):
     • H
     • M
     • T
     • V
     • W
     • X
     • Y

These 7 letters have vertical symmetry and will look the same in a mirror.

1. Consonants Without Vertical Symmetry (Do Not Appear the Same in a Mirror): The remaining consonants do not have vertical symmetry and will look different in a mirror:

B, C, D, F, G, J, K, L, N, P, Q, R, S, Z

There are 14 such consonants.

Conclusion

The number of consonants that do not look like their original shapes in a mirror is 14.

Final Answer: (b) 14

Explanation:

Step-by-Step Solution

1. Identify Consonants in the Alphabet: The consonants in the English alphabet are: B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, W, X, Y, Z
2. Mirror Symmetry: A letter that appears the same in a mirror has vertical symmetry, meaning the left half is a mirror image of the right half.
3. Analyze Each Consonant: Let’s identify the consonants that have vertical symmetry and thus appear the same in a mirror.
   • Consonants with Vertical Symmetry (Appear the Same in a Mirror):
     • H
     • M
     • T
     • V
     • W
     • X
     • Y

These 7 letters have vertical symmetry and will look the same in a mirror.

1. Consonants Without Vertical Symmetry (Do Not Appear the Same in a Mirror): The remaining consonants do not have vertical symmetry and will look different in a mirror:

B, C, D, F, G, J, K, L, N, P, Q, R, S, Z

There are 14 such consonants.

Conclusion

The number of consonants that do not look like their original shapes in a mirror is 14.

Final Answer: (b) 14

Explanation:

To determine which of the conclusions logically follow from the statement, let’s carefully analyze each part of the statement and the conclusions.

Given Statement:
1. Some cats are almirahs.
2. Some almirahs are chairs.
3. All chairs are tables.

Let’s visualize each part of the statement using Venn diagrams or logical deduction, and then analyze the conclusions based on this information.

Analyzing the Conclusions

Conclusion-I: “Certainly some almirahs are tables.”

• From the statements given:
• We know that some almirahs are chairs.
• Since all chairs are tables (from the third part of the statement), it follows that any chair (and thus any almirah that is a chair) must be a table.
• Therefore, some almirahs are certainly tables (because some almirahs are chairs, and all chairs are tables).

This makes Conclusion-I logically valid.

Conclusion-II: “Some cats may not be chairs.”

• We are given that some cats are almirahs.
• However, we do not have any information that directly connects “cats” and “chairs.” We only know that some almirahs are chairs.
• Since we have no information that requires all cats to be chairs (or even some cats to be chairs), it is logically possible that some cats may not be chairs.

This makes Conclusion-II logically possible as well.

Conclusion

Both Conclusion-I and Conclusion-II follow logically from the given statements. Therefore, the correct answer is: $ \text{(c) Both Conclusion-I and Conclusion-II} $

Explanation:

To determine the correct answer, let’s analyze the information given step-by-step.

Information Given:
1. There are seven books: P, Q, R, S, T, U, and V.

2. Books with blue covers: R, Q, and T.

3. Books with red covers: P, S, U, and V (since these are the remaining books).
4. New books: S and U.
5. Old books: P, Q, R, T, and V (since these are the remaining books).
6. Law reports: P, R, and S.
7. Gazetteers: Q, T, U, and V** (since these are the remaining books).

Question:

We need to find the books that are old Gazetteers with blue covers.

Step-by-Step Solution

1. Identify the books that are Gazetteers:

From the given information, Gazetteers are Q, T, U, and V.

2. Identify the books that are old:

The old books are P, Q, R, T, and V.

3. Identify the books with blue covers:

The books with blue covers are R, Q, and T.

4. Combine the conditions:

• We need books that satisfy all three conditions:
• They are Gazetteers,
• They are old, and
• They have blue covers.

5. Filter the books:

• From the list of Gazetteers (Q, T, U, and V), we select the books that are also old (Q, T, and V).
• Among these, we select the books with blue covers, which are Q and T.

Conclusion

The books that are old Gazetteers with blue covers are Q and T.

Final Answer: $ \text{(c) Q and T} $

Explanation:

To determine the fastest run scorer in the test match, we need to calculate the strike rate for each player across both innings. The strike rate is defined as:

$$
\text{Strike Rate} = \frac{\text{Total Runs Scored}}{\text{Total Balls Faced}} \times 100
$$

The player with the highest strike rate is considered the fastest scorer.

Calculations for Each Player

1. Player A:

• Runs scored in both innings: $61 + 14 = 75$
• Balls faced in both innings: $99 + 76 = 175$
• Strike rate for A:
  $$
  \text{Strike Rate for A} = \frac{75}{175} \times 100 = 42.86
  $$

2. Player B:

• Runs scored in both innings: $5 + 50 = 55$
• Balls faced in both innings: $12 + 85 = 97$
• Strike rate for B:
  $$
  \text{Strike Rate for B} = \frac{55}{97} \times 100 \approx 56.70
  $$

3. Player C:

• Runs scored in both innings: $15 + 20 = 35$
• Balls faced in both innings: $75 + 50 = 125$
• Strike rate for C:
  $$
  \text{Strike Rate for C} = \frac{35}{125} \times 100 = 28.00
  $$

4. Player D:

• Runs scored in both innings: $13 + 12 = 25$
• Balls faced in both innings: $55 + 50 = 105$
• Strike rate for D:
  $$
  \text{Strike Rate for D} = \frac{25}{105} \times 100 \approx 23.81
  $$

Conclusion

• The player with the highest strike rate is Player B with a strike rate of approximately $56.70$.
• Therefore, Player B is the fastest run scorer in the test match.

The correct answer is: (b) B

Explanation:

Passage Analysis

The passage discusses the relationship and inherent tension between markets and democracy. It explains that markets operate differently from democracy (not on a one-person-one-vote principle) and that success in the marketplace depends on individual endowments, skills, and economic factors. The passage suggests that while markets reward individual initiative and may lift people from poverty, they also leave behind those who are too poor, too handicapped, or unable to develop necessary skills. It implies that markets alone cannot provide welfare for everyone, as there are limitations (like unemployment) that markets cannot address on their own.

Analysis of Assumptions

1. Assumption 1: “Modern democracies rely on the market forces to enable them to be welfare states.”
   • The passage does not support this. Instead, it suggests that markets cannot fulfill all welfare needs because they do not operate on a democratic principle and leave behind individuals who lack certain resources or skills. The passage implies that markets alone are insufficient to serve as a welfare system.
   • Assumption 1 is not valid.
2. Assumption 2: “Markets ensure sufficient economic growth necessary for democracies to be effective.”
   • The passage does not explicitly claim that markets provide the economic growth required for democracies to be effective. While markets can create jobs and opportunities, the passage emphasizes their limitations and the fact that some people remain unemployed or unsupported.
   • Assumption 2 is not valid.
3. Assumption 3: “Government programmes are needed for those left behind in economic growth.”
   • This assumption is consistent with the passage. The passage points out that some people are unable to meet the demands of the market due to poverty, lack of skills, or disabilities. It implies that markets alone cannot ensure welfare for everyone, suggesting that government support or welfare programs are needed for those left behind by economic growth.
   • Assumption 3 is valid.

Conclusion

The only valid assumption is Assumption 3.

The correct answer is: (b) 3 only

Explanation:

To solve this problem, let’s analyze the information given and determine the correctness of each statement.

Problem Analysis

Let the two-digit number be represented as $10x + y$, where:

• $x$ is the tens digit,
• $y$ is the units digit.

Then, the number obtained by interchanging the digits would be $10y + x$.

According to the problem, the difference between the original number and the number with interchanged digits is $54$. This gives us the equation:

$$(10x + y) – (10y + x) = 54$$

Simplifying this equation:

$$10x + y – 10y – x = 54$$
$$9x – 9y = 54$$
$$9(x – y) = 54$$
$$x – y = 6$$

So, the difference between the two digits, $x – y$, is $6$.

Analysis of Statements

1. Statement 1: “The sum of the two digits of the number can be determined only if the product of the two digits is known.”

• From our calculations, we know that $x – y = 6$.
• However, this equation alone is not enough to determine the exact values of $x$ and $y$, so we cannot find their sum with this information alone.
• Let’s test if knowing the product of the digits (i.e., $x \times y$) would help us determine their values.

Suppose we are given both $x – y = 6$ and $x \times y$ (some specific value). This would allow us to solve for $x$ and $y$ uniquely because we would then have two equations in two unknowns. Hence, knowing the product of $x$ and $y$ in addition to $x – y$ would allow us to find the sum $x + y$.

Therefore, Statement 1 is correct.

2. Statement 2: “The difference between the two digits of the number can be determined.”

• We have already determined that $x – y = 6$ based on the given information.
• Thus, the difference between the two digits can be determined directly from the problem.

Therefore, Statement 2 is also correct.

Conclusion

Both statements are correct. The correct answer is: (c) Both 1 and 2 

Explanation:

To solve this problem, let’s analyze the information given and determine the correctness of each statement.

Problem Analysis

1. Initial Information:

• X says to Y, “At the time of your birth, I was twice as old as you are at present.”
• The present age of X is given as $42$ years.

2. Determine Y’s Present Age:

• Let Y’s present age be $y$ years.
• According to X’s statement, at the time of Y’s birth, X’s age was twice the current age of Y.
• So, the age difference between X and Y at the time of Y’s birth is equal to the current age of Y.

We can write this as:
$$
42 – y = 2y
$$

3. Solve for $y$:

• Rearranging the equation:
  $$
  42 = 3y
  $$
• Dividing both sides by $3$:
  $$
  y = 14
  $$
• Therefore, Y’s present age is $14$ years.

Verify the Statements

Statement 1: “8 years ago, the age of X was five times the age of Y.”

• 8 years ago, X’s age was:
  $$
  42 – 8 = 34 \text{ years}
  $$
• 8 years ago, Y’s age was:
  $$
  14 – 8 = 6 \text{ years}
  $$
• Check if X’s age was five times Y’s age:
  $$
  34 = 5 \times 6
  $$
• Since $34 \neq 30$, Statement 1 is incorrect.

Statement 2: “After 14 years, the age of X would be two times the age of Y.”

• After 14 years, X’s age will be:
  $$
  42 + 14 = 56 \text{ years}
  $$
• After 14 years, Y’s age will be:
  $$
  14 + 14 = 28 \text{ years}
  $$
• Now, check if X’s age would be two times Y’s age:
  $$
  56 = 2 \times 28
  $$
• Since $56 = 2 \times 28$ is true, Statement 2 is correct.

Conclusion

• Statement 1 is incorrect.
• Statement 2 is correct.

Therefore, the correct answer is: (b) 2 only

Explanation:

To find the remainder when $3^{2019}$ is divided by $10$, we need to determine the **last digit** of $3^{2019}$ because the last digit of a number determines the remainder when it is divided by $10$.

Step-by-Step Solution

1. Observe the Pattern of Last Digits in Powers of 3:

• Let’s calculate the last digit of the first few powers of $3$ to find a pattern:
  $$ 3^1 = 3 \quad \text{(last digit is 3)} $$
  $$ 3^2 = 9 \quad \text{(last digit is 9)} $$
  $$ 3^3 = 27 \quad \text{(last digit is 7)} $$
  $$ 3^4 = 81 \quad \text{(last digit is 1)} $$
• After $3^4$, the pattern of last digits starts repeating every 4 powers:
• $3^1$: last digit is $3$
• $3^2$: last digit is $9$
• $3^3$: last digit is $7$
• $3^4$: last digit is $1$
• This cycle $(3, 9, 7, 1)$ repeats every 4 terms.

2. Determine the Position of $3^{2019}$ in the Cycle:

• Since the cycle length is $4$, we find the position of $2019$ in the cycle by finding the remainder when $2019$ is divided by $4$:
  $$
  2019 \div 4 = 504 \text{ remainder } 3
  $$
• This means $2019 \equiv 3 \pmod{4}$.

3. Find the Last Digit of $3^{2019}$:

• Since $2019 \equiv 3 \pmod{4}$, the last digit of $3^{2019}$ corresponds to the last digit of $3^3$ (the third position in the cycle).
• From our pattern, the last digit of $3^3$ is $7$.

4. Conclusion:

Therefore, the remainder when $3^{2019}$ is divided by $10$ is $7$.

The correct answer is: $ \text{(c) 7} $

Explanation:

Passage Analysis

The passage emphasizes the importance of income growth in development economics, which is measured by Gross Domestic Product (GDP) growth. The passage suggests that rising GDP is beneficial because it creates employment, investment opportunities, and allows households, communities, and governments to allocate funds for improving the quality of life. The passage also notes that “economic growth” has become synonymous with “growth in GDP” in the development lexicon.

Assumption Analysis

1. Assumption 1: “Rising GDP is essential for a country to be a developed country.”
   • The passage discusses how rising GDP can create employment and improve quality of life, which are characteristics associated with development. However, it does not explicitly state that GDP growth is the only or essential factor for a country to be considered developed.
   • Although GDP growth is often associated with development, development can involve other factors such as equitable income distribution, healthcare, education, and environmental sustainability, which are not mentioned in the passage.

   Therefore, Assumption 1 is not necessarily valid as it overstates the role of GDP by implying it is the only essential factor for development.

2. Assumption 2: “Rising GDP guarantees a reasonable distribution of income to all households.”
   • The passage mentions that rising GDP creates employment and investment opportunities, but it does not suggest that GDP growth leads to an equitable or “reasonable” distribution of income across all households.
   • In fact, GDP growth does not guarantee that income is distributed fairly or equitably, as income inequality can still exist even when GDP is rising.

   Therefore, Assumption 2 is not valid.

Conclusion

The correct answer is: (d) Neither 1 nor 2

Explanation:

Passage Analysis

The passage discusses possible reasons for rising food prices, listing high fuel prices, bad weather, and land diversion for non-food production as some factors. Additionally, the passage places increased emphasis on the surge in food demand from populous emerging economies, suggesting that this mass consumption might be significant enough to contribute to a potential food crisis.

Assumption Analysis

1. Assumption 1: “Oil producing countries are one of the reasons for high food prices.”
   • The passage mentions high fuel prices as one of the factors behind rising food prices but does not explicitly blame oil-producing countries for this.
   • While oil prices can influence food prices (as transportation and production costs increase), there is no direct suggestion in the passage that oil-producing countries themselves are responsible for high food prices.

   Therefore, Assumption 1 is not necessarily valid.

2. Assumption 2: “If there is a food crisis in the world in the near future, it will be in the emerging economies.”
   • The passage does not imply that a potential food crisis will occur exclusively in emerging economies. It does suggest that rising food demand in these populous countries could contribute to a global food crisis.
   • However, it does not specify that the food crisis would happen only within emerging economies; rather, it implies that the increased demand from these countries could lead to a crisis affecting the global food market.

   Therefore, Assumption 2 is also not valid.

Conclusion

The correct answer is: (d) Neither 1 nor 2