Unit3 - Subjective Questions

1. Arithmetic Series (A.P.):
An Arithmetic Series is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant is called the common difference ().

• Formula: The term is given by .
• Example:
  • Here, the difference between consecutive terms is $5$ ().

2. Geometric Series (G.P.):
A Geometric Series is a sequence where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio ().

• Formula: The term is given by .
• Example:
  • Here, each term is multiplied by $3$ to get the next term ().

Answer: The missing term is 336.

Detailed Logic:
This series is based on the pattern of cubes of numbers minus the number itself ().

1. 1st term:
2. 2nd term:
3. 3rd term:
4. 4th term:
5. 5th term:
6. 6th term:

Finding the 7th term:
Following the pattern, for :

Thus, the next number in the series is $336$.

EJOTY Concept:
To solve alphabet-based problems quickly, it is essential to remember the positional values of letters (A=1, B=2... Z=26). The EJOTY rule is a mnemonic technique to remember the positions of letters that are multiples of 5.

Positions:

• E = 5
• J = 10
• O = 15
• T = 20
• Y = 25

Application & Time Management:
Instead of counting from A every time, a student can use EJOTY as reference points.

• Example: To find the position of 'S', one knows 'T' is 20, so 'S' (which comes before T) must be .
• Example: To find 'Q', one knows 'O' is 15, so 'P' is 16 and 'Q' is 17.

This technique significantly reduces calculation time in exams involving forward and backward letter counting.

Answer: 'DIAGRAM' will be written as 'BGYEPYK'.

Logic Explanation:
The coding follows a backward shift pattern where each letter is shifted back by 2 positions ().

Check 'MIND':

• M (13) = K (11)
• I (9) = G (7)
• N (14) = L (12)
• D (4) = B (2)

Check 'ARGUE':

• A (1) = Y (25) [Cyclic property]
• R (18) = P (16)
• G (7) = E (5)
• U (21) = S (19)
• E (5) = C (3)

Applying to 'DIAGRAM':

1. D = B
2. I = G
3. A = Y
4. G = E
5. R = P
6. A = Y
7. M = K

Result: BGYEPYK

Definition:
A Continuous Pattern Series consists of a series of small letters that follow a specific repetitive pattern. Some letters are missing, and the candidate must identify the repeated block to fill the blanks.

Solving the Example:
Series:

1. Observation: Looking at the series, we see terms involving 'a', 'b', 'd', 'm'.
2. Pattern Identification: Let's analyze the sequence near the end: dm appears. Let's try to group them. It looks like the pattern might be abdm.
3. Testing the Grouping:
   • Group 1: insert 'm' abmd (Wait, pattern might be abmd? Let's check).
   • Actually, let's look at the available letters: a, b, d, m. The sequence seems to be repeating abdm.
   • Let's try breaking it into groups of 4: / / /
   • Wait, let's look closer at the letters provided: , then we see .
   • Let's try the pattern abdm.
     • 1st block: implies 'm' is missing? No, usually typical patterns are things like , , etc.

Correction/Refined Logic: Let us look at the structure: .
Let's assume the pattern is abmd.

• / / (This doesn't fit).

Let's look for a pattern like mbad or abdm.
If we look at the segment dm, it usually follows b. If we look at b, it usually follows a.
Pattern: abmd
Sequence: / / ... (Wait, the question text has dm).

Let's assume the pattern is abmd repeated.
String:
Total letters = 12-14? Let's count indices.
Let's try logic: adbm ? No.

Standard Solution Strategy:

1. Count total positions (including blanks). Say there are 16 positions.
2. Divide into groups of 4.

Correct Logic for this specific string:
Let the pattern be mbad repeated? No.
Let the pattern be abdm repeated cyclically?
Let's try: substituting options. (Since this is subjective, we derive the pattern).

The pattern is likely abmd.
Wait, there is a dm in the string (). This suggests comes before .
So pattern could be abdm.
Let's fill: / / ... wait the string has dm together.
So is followed by . Pattern: abdm is wrong. Pattern is abdm? No, is before .

Correct Pattern: abmd
Block 1:
Block 2: (This rearranges letters? No, standard series repeat).

Let's assume the question meant to exemplify Repetitive Series.
Let pattern = abmd.
Sequence: / (Wait, there is a gap between b and d in second group).

Let's propose a valid pattern based on typical exam questions:
Pattern: mbad
Series:

(Self-Correction for the final answer): Since the specific string in the prompt is ambiguous without multiple choice options to force a specific complex pattern, the answer should focus on the method of solving.

Detailed Answer:

1. Count the total number of elements (letters + blanks). E.g., if 12, groups are or . If 16, groups are .
2. Compare the groups. Look for known letters in the same positions in different groups.
3. Fill the blanks to make the groups identical or cyclic.

Example Solution assuming pattern 'abmd':
If the series is broken into groups of 4: abmd
| |
(This fits if the input string had gaps perfectly aligned).

1. Letter Coding:
In this method, letters of a word are replaced by other letters according to a specific logical rule or pattern (e.g., shifting positions forward or backward).

• Example: If MYSTERY is coded as NZTUFSZ.
  • Logic: Each letter is shifted (MN, YZ).
  • You must decipher the logic to code a new word.

2. Direct Letter Coding:
In this method, the code letters are assigned directly without a specific mathematical or positional logic. The code for a new word is found simply by substituting the corresponding letters from the example words provided.

• Example: If SHE is coded as 532 and HEAL is coded as 3247.
  • Question: Code SEAL.
  • Logic: Look up the values. S=5, E=2, A=4, L=7.
  • Answer: 5247.
  • Here, 'E' is 2 in both words, confirming it's direct substitution, not positional shifting.

Answer: The wrong term is 34.

Detailed Explanation:
The series follows the pattern of squares of natural numbers minus 1 (), starting from .

5. (The series has 34 here)

Alternatively, this can be viewed as adding odd numbers:

In both logical approaches, the term should be 35. Therefore, 34 is the wrong term.

Concept:
In "Pairs of Letters" questions, you are asked to find the number of pairs of letters in a word that have as many letters between them in the word as they have between them in the standard English alphabet (both forward and backward).

Solving for "EDUCATION":
We compare every letter with every subsequent letter, checking the count.

Forward Direction:

1. D E: In the word, they are consecutive (D, E). In the alphabet, they are consecutive (D, E). Gap = 0. (Pair 1)
2. D I: D(4)... I(9). Gap should be 4 letters. In word: U-C-A-T (4 letters). Match! (Pair 2)
3. A D: No (backward logic check later).
4. A E: No.

Let's list them systematically:

• D to E (Forward): Consecutive.
• D (4) ... I (9): D, e, f, g, h, I. (In word: D, u, c, a, t, I). Count is 4. Match.
• N (14) to O (15) (Backward): Consecutive. (Pair 3)
• A (1) ... D (4) (Backward): A, c, u, D (Wait: A, T, C, U, D). Letters between A and D in alphabet are B, C (2 letters). In word: T, C, U (3 letters). No Match.
• A (1) ... E (5) (Backward): A, T, C, U, D, E. Between A and E in alphabet: B, C, D (3 letters). In word: T, C, U, D (4 letters). No match.

Let's re-verify strictly:
Word: E D U C A T I O N
Values: 5 4 21 3 1 20 9 15 14

Pairs:

1. D (4) - E (5): (Forward - No gap). Match.
2. N (14) - O (15): (Backward - No gap). Match.
3. D (4) - I (9): Gap of 4 (5,6,7,8). In word: U,C,A,T (4 letters). Match.
4. A (1) - E (5): Gap of 3 (2,3,4). In word: C,U,D (3 letters). Wait! Looking backwards from A: A(1), C(3), U(21), D(4), E(5). Between A and E are C,U,D. That is 3 letters. Alphabet has B,C,D (3 letters). Match.
5. A (1) - D (4): Gap of 2 (2,3). In word: C,U (2 letters). Looking backwards from A: A, C, U, D. Match.

Total Pairs: 5 pairs (D-E, N-O, D-I, A-E, A-D).
(Note: Exact count depends on strict verification, usually 4-5 pairs in this word).

Definition:
A Mixed Alphanumeric Series is a sequence composed of both letters and numbers arranged in a specific pattern. The logic for the letters and the numbers may be independent or related.

Solution Logic:

1. First Letter Pattern:

• Logic: Backward shift by 2.

2. Third Letter Pattern:

• Logic: Forward shift by 3.

3. Number (Middle Term) Pattern:
Series: $1, 2, 6, 21, 88, 445$

• Logic: where increases by 1 each step.

This demonstrates how three simultaneous patterns form the single alphanumeric series.

Concept:
Language Coding involves sentences in an artificial language translated into English (or another language). The code for a specific word is not necessarily in the same position as the word in the sentence. The codes are deciphered by finding common words between two sentences and eliminating the common codes.

Solution:

1. Compare Sentence 1 & 2:

   • 'pit dar na' = 'you are good'
   • 'dar tok pa' = 'good and bad'
   • Common Word: 'good'
   • Common Code: 'dar'
   • Therefore, 'good' = 'dar'.

2. Compare Sentence 1 & 3:

   • 'pit dar na' = 'you are good'
   • 'tim na tok' = 'they are bad'
   • Common Word: 'are'
   • Common Code: 'na'
   • Therefore, 'are' = 'na'.

3. Compare Sentence 2 & 3:

   • 'dar tok pa' = 'good and bad'
   • 'tim na tok' = 'they are bad'
   • Common Word: 'bad'
   • Common Code: 'tok'
   • Therefore, 'bad' = 'tok'.

4. Find 'they':

   • In Sentence 3 ('tim na tok' = 'they are bad'), we know 'na' = 'are' and 'tok' = 'bad'.
   • The remaining word is 'they' and the remaining code is 'tim'.

Answer: The code for 'they' is 'tim'.

Rules for Dictionary Order:

1. Compare the first letters of the words. If they are different, arrange them alphabetically (A to Z).
2. If the first letters are the same, move to the second letter, and so on.
3. If one word is a prefix of another (e.g., 'Bat' and 'Batch'), the shorter word comes first.

Arrangement:

• All words start with Se.
• Look at the 3rd letter: l (Select), l (Seldom), n (Send), l (Selfish), l (Seller).
• 'n' comes after 'l', so 'Send' (3) is last.
• Now compare the 'l' words using the 4th letter:
  • Select: e
  • Seldom: d
  • Selfish: f
  • Seller: l
• Alphabetical order of 4th letters: d, e, f, l.
  1. Seldom (2)
  2. Select (1)
  3. Selfish (4)
  4. Seller (5)

Final Order: 2, 1, 4, 5, 3.
(Seldom, Select, Selfish, Seller, Send).

Answer: The color of human blood is Yellow.

Logic and Explanation:

1. Identify the Real Fact: The actual color of human blood is Red.
2. Apply Substitution: The question states "Red is called Yellow".
3. Stop There: In substitution coding (A is called B), the answer is the immediate substitute code (B).

Common Error to Avoid:
Students often chain the logic (e.g., Blood is Red Red is Yellow Yellow is Green...). This is incorrect. You must only perform one step of substitution based on the real-world fact. Since Red is renamed to Yellow, the answer is Yellow.

Concept:
Two letters are considered opposite or complementary if the sum of their positional values is 27. For example, A (1) and Z (26) .

Common Pairs:

• A Z
• B Y
• C X
• D W
• M N

Memorization Mnemonics:
Mnemonics are used to recall these quickly:

• AZ: Azad (or A to Z)
• BY: Boy
• CX: Crux (or Xerox)
• DW: Dew (or Down Wards)
• EV: Evening (or Love - V and E)
• FU: Full (or U and F)
• GT: GT Road
• HS: High School
• IR: Indian Railways
• JQ: Jungle Queen
• KP: Kanpur (or PK movie)
• LO: Love (L and O)
• MN: Man

Answer: The next term is 252.

Solution (Method of Differences):
If the difference between terms is not constant, we check the difference of the differences.

Stage 1 (Differences):

Stage 2 (Difference of Differences):

Analysis of Stage 2:
The second differences are $14, 20, 26$. These form an Arithmetic Progression with a common difference of 6 ().

Calculating Next Terms:

1. Next term in Stage 2: .
2. Next term in Stage 1: .
3. Next term in Series: .

Alternative Logic:
Pattern:

Both methods yield 252.

Method:
To find the letter exactly midway between two given letters (say, and ), calculate the average of their positional values.

Formula:

Note: This works only if the sum is even. If the sum is odd, there is no single middle letter (there are two).

Example:
Find the letter midway between the 4th letter from the left (D) and the 18th letter from the left (R).

• Positional Value of D = 4
• Positional Value of R = 18
• Average =
• The 11th letter is K. So, K is the middle letter.

Answer: BAT = 46.

Logic:

1. Analyze Z = 52:

   • The standard position of Z is 26.
   • .
   • Hypothesis: Code = Sum of positional values 2.

2. Verify with ACT = 48:

   • A = 1
   • C = 3
   • T = 20
   • Sum = .
   • Apply logic: . (Verified).

3. Calculate for BAT:

   • B = 2
   • A = 1
   • T = 20
   • Sum = .
   • Code = .

Conclusion: The coding logic is to sum the positional values of the letters and multiply the result by 2.

Definition:
Word Formation questions test the candidate's ability to identify whether a specific word can be constructed using only the letters available in a given "source" or "parent" word. The rule is usually that you cannot use a letter more times than it appears in the source word (unless specified otherwise).

Example:
Source Word: ADMINISTRATION

Question: Which of the following words cannot be formed?

1. MIND
2. RATION
3. MINISTER

Analysis:

1. MIND: M, I, N, D are all present. (Possible)
2. RATION: R, A, T, I, O, N are all present. (Possible)
3. MINISTER:
   • M - Yes
   • I - Yes
   • N - Yes
   • I - Yes
   • S - Yes
   • T - Yes
   • E - No. The letter 'E' is not present in "ADMINISTRATION".

Conclusion: Therefore, MINISTER cannot be formed.

Answer: ACID is written as 1D3E.

Logic:
The word is coded by treating vowels and consonants differently.

1. Consonants: Shifted forward by .

   • R S
   • Q R
   • S T
   • T U

2. Vowels: Replaced by their sequential number (A=1, E=2, I=3, O=4, U=5).

   • E 2
   • U 5

Checking REQUEST:

• R (Consonant +1) S
• E (Vowel #2) 2
• Q (Consonant +1) R
• U (Vowel #5) 5
• E (Vowel #2) 2
• S (Consonant +1) T
• T (Consonant +1) U
• Result: S2R52TU (Matches).

Applying to ACID:

• A (Vowel #1) 1
• C (Consonant +1) D
• I (Vowel #3) 3
• D (Consonant +1) E

Final Code: 1D3E.

Strategy:
To solve Alphabet Classification (Odd One Out) problems, check for relationships between the letters in the pairs. Common relationships include:

1. Positional difference: Is the gap between letters constant?
2. Opposite pairs: Do the letters sum to 27?
3. Vowel/Consonant composition.

Problem Analysis:
Check if they are Opposite Pairs (Sum of positions = 27):

• DW: D(4) + W(23) = 27 (Yes, Dew)
• HS: H(8) + S(19) = 27 (Yes, High School)
• MN: M(13) + N(14) = 27 (Yes, Man)
• GT: G(7) + T(20) = 27 (Yes, GT Road)
• KP: K(11) + P(16) = 27 (Yes, Kanpur)
• JQ: J(10) + Q(17) = 27 (Yes, Jungle Queen)
• HR: H(8) + R(18) = 26 (No)

Conclusion: HR is the odd one out because it is not a complementary pair (Opposite of H is S, opposite of R is I).

Answer: The next term is 47U15.

Derivation of Logic:

1. First Element (Numbers): $2, 7, 14, 23, 34$

   • Difference pattern: .
   • Next difference must be .
   • .

2. Middle Element (Letters):

   • Pattern: Backward alphabetic order ().
   • Previous term: V. Next term: .

3. Last Element (Numbers): $5, 7, 9, 11, 13$

   • Pattern: Consecutive odd numbers ().
   • Next term: .

Combining Elements:
The next term is 47U15.