Explanation : 

Ball Drawing from Box with black, blue, green, yellow, red, white balls.

Statement 1 – Finding the Smallest Number n

To find the smallest number n such that any n balls drawn from the box randomly must contain one full group of at least one colour, we need to consider the worst-case scenario. That is, we draw all the balls except one from each colour. This means we draw 13 black balls, 19 blue balls, 25 green balls, 27 yellow balls, 37 red balls, and 53 white balls. The total number of balls drawn in this case is 174. If we draw one more ball, we are guaranteed to have a full group of at least one colour. Therefore, the smallest number n is 175.

Statement 2 – Finding the Smallest Number m

The smallest number m such that any m balls drawn from the box randomly must contain at least one ball of each colour is 167. This is because in the worst-case scenario, we can draw 166 balls without selecting a black ball (since black balls are the least in number). Once we have drawn these 166 balls, the next ball we draw will certainly be a black ball, ensuring that we have at least one ball of each colour. Therefore, m = 166 + 1 = 167.

Explanation : 

125 identical cubes

In a cubical block, the cubes that are surrounded by other cubes from each side are the internal cubes.

The total number of cubes is 125, which means the cubical block is 5 cubes long, 5 cubes wide, and 5 cubes high (since 5*5*5 = 125).

The internal cubes are those that are not on any of the faces of the large cube. Since each face of the large cube is 5 cubes by 5 cubes, the internal cubes form a smaller cube that is 3 cubes by 3 cubes by 3 cubes (since we subtract the top and bottom layer from the height, and one layer from each side of the width and length).

So, the number of internal cubes is 3*3*3 = 27. Therefore, 27 cubes are surrounded by other cubes from each side..

Explanation : 

Calculating the 10^10 day starting from Sunday

The solution is as follows:

– We know that there are 7 days in a week, so we can divide 10^10 by 7 to get the number of complete weeks and the remainder.
– 10^10 / 7 = 1428571428 weeks and 4 days.

The remainder 4 is what we’re interested in. It tells us that on 10^10 days, it will be the 4th day of the week, counting from today.

If today is Sunday, then the 4th day from today (inclusive) is Wednesday.
(Sunday – Day 1, Monday – Day 2, Wednesday – Day 3, Thursday –  Day 4)

Therefore, the correct answer is (a) Wednesday.

Note: We have updated the answer & solution. Thursday would have been answer if question had asked after 10^10 days.

Explanation : 

The total marks for the four papers is 400 (100 marks each for P, Q, R and S).

If a student scores 99% in the examination, then the total marks scored by the student is 99% of 400 = 396 marks.

Now, we need to find out the number of ways in which 396 can be expressed as a sum of four integers, each ranging from 0 to 100.

The problem can be solved by considering all possible combinations and permutations of the scores in the four papers.

1. All four papers scored 99 marks each. This is one way.
2. Three papers scored 100 marks each and one paper scored 96 marks. This can happen in 4 ways (⁴C₃) · (¹C₁).
3. Two papers scored 100 marks each, one paper scored 98 marks and one paper scored 98 marks. This can happen in 6 ways (⁴C₂) · (²C₂).
4. Two papers scored 100 marks each, one paper scored 99 marks and one paper scored 97 marks. This can happen in 12 ways (⁴C₂ · ²C₁ · ¹C₁).
5. One paper scored 100 marks, two papers scored 99 marks each and one paper scored 98 marks. This can happen in 12 ways (⁴C₁ · ³C₂ · ¹C₁).

So, the total number of ways is 1 + 4 + 6 + 12 + 12 = 35.

Hence, the correct answer is (d) 35.

Model Answer:

Q1: Correct Answer – (a) 1 only

Explanation – 

To determine the correct answer, let’s analyze each assumption based on the information provided in the passage:

1. Global warming is causing spring to come early and for longer durations.
The passage mentions that warmer winters with less snow result in a longer lag time between spring events and a more protracted vernal window. This implies that the transition period from winter to the growing season (spring) is extended due to warmer conditions. Although the passage does not explicitly use the term “global warming,” it is reasonable to infer that warmer winters are a result of global warming. Therefore, this assumption is correct.

2. Early spring and longer period of spring is not good for bird populations.
The passage states that as the ice melts earlier, the birds don’t return, causing a delay or lengthening in springtime ecological events. This indicates that the change in timing affects the birds’ behavior, but it does not explicitly state that these changes are harmful or “not good” for bird populations. It only mentions a delay in their return, which could have various implications, not necessarily negative. Therefore, this assumption lacks sufficient support from the passage.

Based on this analysis, the correct answer is:
(a) 1 only – This choice is correct because the first assumption is supported by the passage, while the second assumption is not conclusively supported.

Explanation : 

Maximum Tiles on Rectangular Floor

First, we need to convert the measurements of the floor and the tiles into the same units. Since the tiles are measured in centimeters, we’ll convert the floor measurements to centimeters.
1 meter = 100 centimeters

So, the floor measures 400 cm in length and 220 cm in breadth.
Next, we need to find out how many tiles can fit along the length and the breadth of the floor.

For the length: 400 cm / 140 cm = 2.857 tiles. We can’t have a fraction of a tile, so we can fit 2 tiles along the length.
For the breadth: 220 cm / 60 cm = 3.666 tiles. Again, we can’t have a fraction of a tile, so we can fit 3 tiles along the breadth.

However, we can also try to fit the tiles in the other orientation.

For the length: 400 cm / 60 cm = 6.666 tiles. We can fit 6 tiles along the length.
For the breadth: 220 cm / 140 cm = 1.571 tiles. We can fit 1 tile along the breadth.

So, we can either fit 2 tiles along the length and 3 tiles along the breadth for a total of 2*3 = 6 tiles, or we can fit 6 tiles along the length and 1 tile along the breadth for a total of 6*1 = 6 tiles.

However, if we look closely, we can see that there is some space left over when we fit 2 tiles along the length and 3 tiles along the breadth. The length of the floor is 400 cm, and 2 tiles take up 2*140 cm = 280 cm. This leaves 400 cm – 280 cm = 120 cm of space, which is enough for another tile if we place it in the 60 cm orientation. Similarly, the breadth of the floor is 220 cm, and 3 tiles take up 3*60 cm = 180 cm. This leaves 220 cm – 180 cm = 40 cm of space, which is not enough for another tile.

So, by placing 2 tiles along the length and 3 tiles along the breadth, and then adding 2 more tiles in the space left over, we can fit a total of 2*3 + 2 = 8 tiles on the floor.

Therefore, the correct answer is (c) 8.

Model Answer:

Q1: Correct Answer – (d) 1, 2 and 3

Explanation –

The passage highlights several hurdles to the financialization of household savings in India, including:

Onerous bureaucracy
Skepticism of organized financial institutions
Lack of basic information about the best services and providers for each family
Difficulty in transitioning between services and providers

Now, let’s consider each of the given statements:

“A flexible environment is needed to develop solutions.”

This statement aligns with the idea that the current bureaucracy is onerous and hinders the adoption of financial savings. A more flexible environment could help address this issue.

“Households need customised solutions.”

The passage mentions that there is a lack of information about which services and providers are best for each family, implying that households have unique needs and require customized solutions.

“Innovations in financial technology are required.”

While the passage doesn’t explicitly mention financial technology, it does highlight the need for solutions to address the hurdles in transitioning between services and providers. Innovations in financial technology could potentially help in this regard.

Explanation : 

5 members A, B, C, D, E in a family

A and B are a married couple, so they are husband and wife.
D is the father of C, so C is the child of D.
E is D’s son, so E is the brother of C.
A and C are sisters, so they have the same parents.

Finally, I would use the family tree and the definitions to find the relation of E to B. For example:

E is the brother of C, who is the spouse of B, so E is the brother-in-law of B.

Therefore, the correct answer is © All four statements together are sufficient to answer the Question.

Explanation : 

D is a 3-digit number

To minimize the ratio of a number to the sum of its digits, we need to maximize the sum of the digits and minimize the number itself.

The smallest 3-digit number is 100, but this doesn’t maximize the sum of the digits.

The smallest 3-digit number with the maximum digit sum is 199 (1+9+9=19). This gives us a ratio of 10.47, which is the smallest possible for a 3-digit number.

The difference between the digit at the hundred’s place (1) and the digit at the unit’s place (9) of D is 9-1=8.

So, the correct answer is (c) 8.

Explanation :

Middle term of the sequence

The given sequence is a pattern of letters from the English alphabet starting from Z and going backwards to A. Each letter is repeated a number of times equal to its position in the reversed alphabet (Z is 1st, Y is 2nd, X is 3rd, and so on).

The total number of terms in the sequence is calculated by adding the number of repetitions for each letter, which forms an arithmetic series: 2 (for Z) + 3 (for Y) + 4 (for X) + … + 27 (for A).
The sum of this series is 377= (n(n+1)/2 -1), n=27

The middle term is the 189th term (since 377/2 = 188.5). To find out which letter this term is, we subtract the total number of terms for the letters before it from 189.

The total number of terms for the letters before I (starting from Z) is 188 (2 for Z, 3 for Y, 4 for X, …, 8 for S). So, 189 – 188 = 1, which means the 189th term is the first repetition of I. Therefore,

The middle term of the sequence is I.

Proof – A+B+C+D+E+F+G+H=27+26+25+24+23+22+21+20=188