Explanation : 

There are 5 tasks and 5 persons p q r s t

There are 5 tasks to be assigned to 5 people.

Task-1 cannot be assigned to P or Q. So, it can be assigned to R, S or T in 3 ways.

Task-2 must be assigned to either R or S. If Task-1 was assigned to R or S, then Task-2 can only be assigned to the remaining person. If Task-1 was assigned to T, then Task-2 can be assigned to either R or S. So, Task-2 can be assigned in 2 ways.

The remaining 3 tasks can be assigned to the remaining 3 people in 3! = 6 ways.

So, the total number of ways to assign the tasks is 3 * 2 * 6 = 24.

Explanation : 

8-Digit Number Arrangement – 11223344

The number 11223344 has 4 odd digits (1,1,3,3) and 4 even digits (2,2,4,4).

In an 8-digit number, there are 4 odd positions (1st, 3rd, 5th, 7th) and 4 even positions (2nd, 4th, 6th, 8th).

The 4 odd digits can be arranged in the 4 odd positions in 4!/2!2! = 6 ways (4 factorial divided by 2 factorial times 2 factorial, because there are 2 pairs of identical digits).

Similarly, the 4 even digits can be arranged in the 4 even positions in 4!/2!2! = 6 ways.

Therefore, the total number of distinct 8-digit numbers is 6 * 6 = 36.

So, the correct answer is (c) 36.

Explanation : 

Raj has 10 pairs of red, 9 pairs of white, 8 pairs of black shoes.

In the worst-case scenario, Raj would pick one shoe from each pair without getting a matching pair.

There are ten pairs of red shoes, nine pairs of white shoes, and eight pairs of black shoes, which means there are a total of 27 pairs of shoes.

If he picks one shoe from each pair, that’s 27 attempts.

Then, he would have to start picking a second shoe from each pair. In the worst-case scenario, he would pick all the second shoes from the white and black pairs before picking a second shoe from a red pair.

That’s an additional 17 attempts (9 white pairs + 8 black pairs).

So, in total, the maximum number of attempts he would have to make is 27 (first shoes) + 17 (second shoes) = 44 attempts.

But, he still hasn’t picked a matching pair of red shoes. He would have to pick one more shoe (the matching red one) to get a pair.

So, the total maximum number of attempts is 44 + 1 = 45 attempts.

Hence, the correct answer is (d) 45.

Explanation : 

Unit Digit of a Power 57242⁹⁴⁵

To find the unit digit of 57242⁹⁴⁵, we can focus solely on the unit digit of the base number, which is 2, since the unit digit of a power depends only on the unit digit of the base.

Calculating Power

First, calculate the product 9 x 7 x 5 x 3 x 1 = 945.

Pattern in Powers of 2

Next, observe the pattern in the powers of 2:

• 2¹ = 2 (unit digit is 2)
• 2² = 4 (unit digit is 4)
• 2³ = 8 (unit digit is 8)
• 2⁴ = 16 (unit digit is 6)
• 2⁵ = 32 (unit digit is 2)
• 2⁶ = 64 (unit digit is 4)
• 2⁷ = 128 (unit digit is 8)
• 2⁸ = 256 (unit digit is 6)
• 2⁹ = 512 (unit digit is 2)

The unit digits repeat every four powers (2, 4, 8, 6). This is known as the cyclicity of 2, which is 4.

Finding Relevant Power

To find the relevant power in the cycle for 2⁹⁴⁵, calculate the remainder of 945 divided by 4:

945 ÷ 4 = 236 remainder 1

This remainder tells us that 2⁹⁴⁵ corresponds to the first number in the cycle of unit digits of 2, which is 2.

Thus, the unit digit of 57242⁹⁴⁵ is 2.

Therefore, the correct answer is 2.

Explanation :

Five candidates P, Q, R, S and T – Job Selection

The statements provide the following information:

– One of P and Q was selected for the job. This statement is true, which means either P or Q was selected.

– At least one of R and S was selected for the job. This statement is false, which implies neither R nor S was selected.

– At most two of R, S and T were selected for the job. This statement is true. But we already know from the second statement that neither R nor S was selected, so at most two people could be selected from R, S, and T. Therefore, at most, T was selected for the job.

So, the possible selections for the job could be P and T, or Q and T but no more than 2 individuals were selected. Therefore, the statement “At least four were selected for the job” is incorrect.

We also know that S was not selected for the job based on the second statement. Therefore, the statement “S was selected for the job” is also false.

Hence, the correct answer is (d) Neither 1 nor 2.

Explanation : 

Number n is formed by writing 9 for 99 times

Group the given number into sets of 3 starting from the right, or the units place. From the rightmost group of 3 digits apply the subtraction and addition operations alternatively and find the result. If the result is either a 0 or it can be divided by 13 completely without leaving a remainder, then the number is divisible by 13.

For example, in the number 1,139,502 applying the subtraction and addition operations alternatively from the rightmost group of 3 digits, we get 502 – 139 + 1 = 364. 364/13 gives 28 as quotient and 0 as remainder. Therefore, 1139502 is divisible by 13.

Similarly,

To determine if the number formed by writing the digit 9 ninety-nine times is divisible by 13

We can use a method that involves grouping the digits and applying alternating addition and subtraction operations. Here’s a step-by-step explanation:

1. Form the Number

The number in question is composed of the digit 9 repeated 99 times.

2. Group the Digits

We group the digits into sets of three, starting from the rightmost digit. This results in 33 groups of ‘999’.

3. Apply Operations

Starting from the rightmost group, we apply alternating subtraction and addition operations to these groups. This method is based on a divisibility rule for 13, which states that if you take the last three digits of a number, subtract the next three digits, add the next three, and so on, and if the resulting number is divisible by 13, then the original number is also divisible by 13.

4. Calculate

For our number, the calculation would look like this:

999 – 999 + 999 – 999 + … + 999 (33 times in total).

5. Simplify the Calculation

Since each ‘999’ is either added or subtracted, and there are an odd number of these groups (33), the calculation simplifies to just one ‘999’ (because 32 of them will cancel each other out).

6. Check Divisibility

Now, we check if 999 is divisible by 13:

999 ÷ 13 = 76 remainder 11.

Since there is a remainder when 999 is divided by 13, the original number (composed of 99 nines) is not divisible by 13.

Model Answer:

Q1: Correct Answer – (d) Liberty

Explanation – 
The passage emphasizes that the ultimate goal of the state is to free each man from fear so that he may live and act with full security and without injury to himself or his neighbour. It also mentions that the state aims to lead men to live by and exercise a free reason. Both these points suggest that the state’s ultimate goal is to provide liberty to its citizens. While personal safety, health of body and mind, and communal harmony are part of this goal, they are not the ultimate goal as per the passage.

Model Answer:

Science by itself is not enough

Q1: Correct Answer – (b) It is not desirable to have Governments managed by empirical statesmen unless well mixed with others who are grounded in learning and reflect wisdom.

Explanation –
The passage emphasizes the importance of philosophy in guiding and coordinating scientific pursuits, and draws a parallel to statesmanship in politics. It suggests that just as science needs philosophy to avoid superficiality, government suffers without philosophy, becoming a destructive bedlam when divorced from science and philosophy. Therefore, the statement that best reflects the message of the passage is that it is not desirable to have governments managed by empirical statesmen unless well mixed with others who are grounded in learning and reflect wisdom. This option encapsulates the idea that both science and government need philosophy (learning and wisdom) to guide them and make them purposeful and useful.

Model Answer:

Q1: Correct Answer – (b) Unemployment must be seen as a function of rising education and aspirations of young Indians.

Explanation – 
The passage discusses how the unemployment rate in India is influenced by the rising levels of education and the challenges faced by younger Indians in finding suitable employment. It doesn’t talk about the quality of education, lack of labour-intensive industries, or the need for self-employment. Therefore, option (b) is the most accurate reflection of the author’s intent.

Model Answer:

Q1: Correct Answer – (c) Both 1 and 2

Explanation – 
The passage clearly states that the growth of the digital economy should not compromise the personal data of citizens, implying that privacy has economic value. It also mentions that innovation will not occur in an environment where personal information is compromised, suggesting a link between privacy and innovation. Therefore, both assumptions are valid.