Q: Direction : Study the following information carefully and answer the given question. Observe the Series I carefully to identify the logic and obtain the value of 'P'. In both of the series, the same logic is applied. Series I: 60, 68, 14, (P), -294,…

Concept & Number Series Logic: To solve this dual number series, we must first determine the mathematical pattern governing Series I. By analyzing the differences between consecutive terms, we can identify a pattern involving cubes, multipliers, and alternating signs. T_n = T_{n-1} + (-1)^n × (n-1) × n^3 (Where n represents the term's positional index in the series, starting from n=2 for the first difference.) Step-by-Step Solution: 1. Analyze Series I: • T_1 = 60 • T_2 = 68 \rightarrow 60 + 8 \rightarrow 60 + (1 × 2^3) • T_3 = 14 \rightarrow 68 - 54 \rightarrow 68 - (2 × 3^3) • T_4 (P) \rightarrow 14 + (3 × 4^3) \rightarrow 14 + (3 × 64) = 14 + 192 = 206 • Let's verify the pattern for the next term: • T_5 = 206 - (4 × 5^3) \rightarrow 206 - (4 × 125) = 206 - 500 = -294 (Matches exactly) • Hence, P = 206. 2. Evaluate Series II up to 'R': The logic remains identical. • First Term (T_1) = P - 12 = 206 - 12 = 194 • Second Term (Q) = 194 + (1 × 2^3) = 194 + 8 = 202 • Third Term (R) = 202 - (2 × 3^3) = 202 - 54 = 148 Exam Strategy & Shortcut: When you see numbers dropping and rising sharply (like 68 to 14, then up to a large positive, then down to -294), immediately suspect alternating addition/subtraction combined with squares or cubes. Finding the differences (+8, -54) quickly reveals 2^3 and a multiple of 3^3. Common Pitfall: A common mistake is failing to recognize the increasing multiplier (1, 2, 3, \dots) alongside the cubes, leading to incorrect projections for P and subsequent terms in Series II. Final Answer: Therefore, the correct answer is 148.

Q: Directions: Solve the number series to answer the following questions. $\sqrt[n]{A}$, 31, B, C, 439, B + 687, 1296 Note: (i) All the terms in the given series and 'n' are positive integer. (ii) Difference between 31 and B is X, which has 3 factors …

Concept & Number Series Pattern: Finding the unknown variables requires analyzing proper factors of a number and identifying algebraic progression patterns based on differences between terms. Step-by-Step Solution: 1. Determine X and A: X has exactly 3 proper factors (factors excluding the number itself): 13, 5, and (A - 8). Since 13 and 5 are prime, their product is 13 × 5 = 65. The proper factors of 65 are 1, 5, and 13. This gives X = 65. The third factor must be 1, so A - 8 = 1 \Rightarrow A = 9. 2. Determine B: The difference between 31 and B is X = 65. Since series terms are increasing based on the progression, B = 31 + 65 = 96. 3. Determine n: The first term is \sqrt[n]{A} = \sqrt[n]{9}. For this to be a positive integer, n = 2, making the first term 3. 4. Identify the Series Pattern: The series is 3, 31, 96, C, 439, (96 + 687 = 783), 1296. Let's calculate the differences between consecutive terms: 31 - 3 = 28; 96 - 31 = 65. 5. The differences (28, 65) follow the pattern k^3 + 1: 3^3 + 1 = 28, 4^3 + 1 = 65. 6. Determine C: The next difference should be 5^3 + 1 = 126. So, C = 96 + 126 = 222. (Verify next term: 222 + (6^3 + 1) = 222 + 217 = 439, which perfectly matches the given series). 7. Calculate Final Expression: (A + C - n) = (9 + 222 - 2) = 229. Exam Strategy & Shortcut: When told a number has exactly three proper factors and two are prime, recognize immediately that the number is the product of those distinct primes and the third factor MUST be 1. This instantly solves for A without trial and error. Common Pitfall: Assuming the 3 factors could imply a larger composite number, which leads to overcomplicating (A - 8). Remember that 1 is always a proper factor of any positive integer. Final Answer: Therefore, the correct answer is 229.

Q: Directions: Solve the number series to answer the following questions. $\sqrt[n]{A}$, 31, B, C, 439, B + 687, 1296 Note: (i) All the terms in the given series and 'n' are positive integer. (ii) Difference between 31 and B is X, which has 3 factors …

Concept & Least Common Multiple (LCM): Once the variables from the series conditions are resolved, calculating the LCM involves finding the smallest positive integer that is perfectly divisible by both numbers. Step-by-Step Solution: 1. Retrieve previously solved variables: From decoding the series conditions in the previous step, we know the difference X = 65 and the series term B = 96. 2. Substitute into the expression: We need to find the LCM of (B + 34) and X. 3. Calculate (B + 34) = 96 + 34 = 130. 4. Now, find the Least Common Multiple (LCM) of 130 and 65. 5. Since 130 is a direct multiple of 65 (65 × 2 = 130), the smallest number perfectly divisible by both is 130 itself. Exam Strategy & Shortcut: Always check if the larger number is a multiple of the smaller number before using prime factorization trees or division methods. If Y is a multiple of X, then LCM(X, Y) = Y. This skips unnecessary calculation steps. Common Pitfall: A common mistake is simply multiplying both numbers together to find the LCM, which yields 130 × 65 = 8450. This is a common multiple, but absolutely not the *least* common multiple. Final Answer: Therefore, the correct answer is 130.

Question 1

If value of $M = 30$, then for which of the following series value of $M$ not satisfied properly?  I. $18, M, 50, 97, 208, 444$ II. $-50, -49, -45, -35, -16, \frac{M}{2}$ III. $8, 5, 6.5, M, 110, 1319$

Accepted Answer

Concept & Number Series Anomalies: Test the given value in each sequence by checking successive differences (first and second order) to see if a consistent mathematical pattern breaks. Step-by-Step Solution: • Given: We need to test if M = 30 fails in any of the three series. • Testing Series I: 18, 30, 50, 97, 208, 444. • First differences: 30-18=12; 50-30=20; 97-50=47; 208-97=111; 444-208=236. • Second differences: 20-12=8; 47-20=27; 111-47=64; 236-111=125. • These are cubes: 2^3, 3^3, 4^3, 5^3. The pattern holds perfectly. • Testing Series II: -50, -49, -45, -35, -16, (M/2). • If M=30, the last term is (30/2) = 15. • First differences: 1, 4, 10, 19, 31. • Second differences: 3, 6, 9, 12. This is a perfect arithmetic progression. The pattern holds. • Testing Series III: 8, 5, 6.5, M, 110, 1319. • If M=30, let's check the transitions. • 8 × 0.5 + 1 = 5 • 5 × 1 + 1.5 = 6.5 (or 5 × 1.5 - 1 = 6.5) • There is no continuous multiplication/addition sequence using 30, 110, and 1319 that fits standard series logic (e.g., 110 × 12 - 1 = 1319 demands a much steeper curve than M=30 provides). • Since I and II perfectly accommodate M=30, III must be the one that does not. Exam Strategy & Shortcut: In "Which is NOT satisfied" questions, start with the easiest series to verify. Series II is a simple double-difference sequence. Verifying I and II quickly leaves III as the only possible answer by elimination. Common Pitfall: Wasting time trying to find the *exact* correct value for M in Series III. Once you prove I and II work perfectly, you can confidently choose "Only III". Final Answer: Therefore, the correct answer is Only III.

Question 2

In the question, three series I, II and III are given. Find the value of $x$, $y$ and $z$ to establish the correct relation among them and choose the correct option.  (i) 12, 24, 8, $x$, 6.4, 38.4 (ii) 11, 20, $y$, 50, 71, 96 (iii) 3, 4, 6, 12, $z$,…

Accepted Answer

Concept & Multi-Series Number Patterns: Solve for unknown variables by determining the unique arithmetic, geometric, or difference-based rule governing each distinct series. Step-by-Step Solution: • Series I Pattern (Alternating Multiply/Divide): 12, 24, 8, x, 6.4, 38.4. • 12 × 2 = 24 • 24 \div 3 = 8 • 8 × 4 = 32 \implies x = 32 • 32 \div 5 = 6.4 (Pattern confirmed) • Series II Pattern (Double Difference): 11, 20, y, 50, 71, 96. • First differences: 9, (y-20), (50-y), 21, 25. • Notice the last two differences are 21 and 25 (a gap of 4). • Assume an arithmetic progression for differences: 9, 13, 17, 21, 25. • 11 + 9 = 20; 20 + 13 = 33 \implies y = 33. • Check next: 33 + 17 = 50. (Pattern confirmed) • Series III Pattern (Factorial Differences): 3, 4, 6, 12, z, 156. • First differences: 1, 2, 6, (z-12), (156-z). • Notice 1=1!, 2=2!, 6=3!. The next differences must be 4! = 24 and 5! = 120. • 12 + 24 = 36 \implies z = 36. • Check next: 36 + 120 = 156. (Pattern confirmed) • Establishing Relation: • x = 32, y = 33, z = 36. • Thus, 32 < 33 < 36, which translates to x < y < z. Exam Strategy & Shortcut: Look for highly recognizable numbers like factorials (1, 2, 6, 24) or clear geometric jumps to identify the pattern type quickly without full trial and error. Common Pitfall: Misidentifying Series I as having addition/subtraction. The decimal values (6.4, 38.4) strongly hint at multiplication/division. Final Answer: Therefore, the correct answer is x < y < z.

Question 3

Direction : Study the following information carefully and answer the given question. Observe the Series I carefully to identify the logic and obtain the value of 'P'. In both of the series, the same logic is applied. Series I: 60, 68, 14, (P), -294,…

Accepted Answer

Concept & Number Series Logic: To solve this dual number series, we must first determine the mathematical pattern governing Series I. By analyzing the differences between consecutive terms, we can identify a pattern involving cubes, multipliers, and alternating signs. T_n = T_{n-1} + (-1)^n × (n-1) × n^3 (Where n represents the term's positional index in the series, starting from n=2 for the first difference.) Step-by-Step Solution: 1. Analyze Series I: • T_1 = 60 • T_2 = 68 \rightarrow 60 + 8 \rightarrow 60 + (1 × 2^3) • T_3 = 14 \rightarrow 68 - 54 \rightarrow 68 - (2 × 3^3) • T_4 (P) \rightarrow 14 + (3 × 4^3) \rightarrow 14 + (3 × 64) = 14 + 192 = 206 • Let's verify the pattern for the next term: • T_5 = 206 - (4 × 5^3) \rightarrow 206 - (4 × 125) = 206 - 500 = -294 (Matches exactly) • Hence, P = 206. 2. Evaluate Series II up to 'R': The logic remains identical. • First Term (T_1) = P - 12 = 206 - 12 = 194 • Second Term (Q) = 194 + (1 × 2^3) = 194 + 8 = 202 • Third Term (R) = 202 - (2 × 3^3) = 202 - 54 = 148 Exam Strategy & Shortcut: When you see numbers dropping and rising sharply (like 68 to 14, then up to a large positive, then down to -294), immediately suspect alternating addition/subtraction combined with squares or cubes. Finding the differences (+8, -54) quickly reveals 2^3 and a multiple of 3^3. Common Pitfall: A common mistake is failing to recognize the increasing multiplier (1, 2, 3, \dots) alongside the cubes, leading to incorrect projections for P and subsequent terms in Series II. Final Answer: Therefore, the correct answer is 148.

Question 4

Direction : Study the following information carefully and answer the given question. Observe the Series I carefully to identify the logic and obtain the value of 'P'. In both of the series, the same logic is applied. Series I: 60, 68, 14, (P), -294,…

Accepted Answer

Concept & Complex Number Series Extensions: This question requires extending the logical pattern found in Series I (alternating addition and subtraction of (n-1) × n^3) to fully flesh out Series II and then generate a completely new Series III to evaluate specific logical statements. Logic Pattern = + (1 × 2^3), - (2 × 3^3), + (3 × 4^3), - (4 × 5^3), \dots Step-by-Step Solution: 1. Complete Series II: From the previous question, we know P = 206. • T_1 = P - 12 = 194 • Q = 194 + 8 = 202 • R = 202 - 54 = 148 • S = 148 + (3 × 4^3) = 148 + 192 = 340 • T = 340 - (4 × 5^3) = 340 - 500 = -160 • U = -160 + (5 × 6^3) = -160 + 1080 = 920 • W = 920 - (6 × 7^3) = 920 - 2058 = -1138 2. Form Series III: • First Term = Average of S and U = (340 + 920/2) = (1260/2) = 630. • T_1 = 630 • T_2 = 630 + (1 × 2^3) = 630 + 8 = 638 • T_3 = 638 - (2 × 3^3) = 638 - 54 = 584 • T_4 = 584 + (3 × 4^3) = 584 + 192 = 776 3. Evaluate the Statements: • *(i) The sum of the values of 'T' and 'W' is the multiple of 9:* Sum = -160 + (-1138) = -1298. The sum of digits of 1298 is 1+2+9+8 = 20, which is not divisible by 9. (False) • *(ii) The fourth term of the third series is an odd number:* The 4th term is 776, which is an even number. (False) • *(iii) The difference between the value of 'Q' and the second term of the third series is 426:* Q = 202, 2nd term of Series III = 638. Difference = |202 - 638| = 436. The statement says 426. (False) Exam Strategy & Shortcut: For multi-layered series questions, carefully list out all calculated variables (Q, R, S, T, U, W) in a clean column. Because errors in early terms cascade and invalidate the entire question, double-check your initial arithmetic for P, S, and U before testing the multiple-choice statements. Common Pitfall: A frequent error is miscalculating the alternating signs for higher indices. For example, incorrectly adding the 6th index term instead of subtracting it will completely ruin the value of W and any statement dependent on it. Final Answer: Therefore, the correct answer is None is true.

Question 5

Directions: Solve the number series to answer the following questions.  $\sqrt[n]{A}$, 31, B, C, 439, B + 687, 1296 Note: (i) All the terms in the given series and 'n' are positive integer. (ii) Difference between 31 and B is X, which has 3 factors …

Accepted Answer

Concept & Number Series Pattern: Finding the unknown variables requires analyzing proper factors of a number and identifying algebraic progression patterns based on differences between terms. Step-by-Step Solution: 1. Determine X and A: X has exactly 3 proper factors (factors excluding the number itself): 13, 5, and (A - 8). Since 13 and 5 are prime, their product is 13 × 5 = 65. The proper factors of 65 are 1, 5, and 13. This gives X = 65. The third factor must be 1, so A - 8 = 1 \Rightarrow A = 9. 2. Determine B: The difference between 31 and B is X = 65. Since series terms are increasing based on the progression, B = 31 + 65 = 96. 3. Determine n: The first term is \sqrt[n]{A} = \sqrt[n]{9}. For this to be a positive integer, n = 2, making the first term 3. 4. Identify the Series Pattern: The series is 3, 31, 96, C, 439, (96 + 687 = 783), 1296. Let's calculate the differences between consecutive terms: 31 - 3 = 28; 96 - 31 = 65. 5. The differences (28, 65) follow the pattern k^3 + 1: 3^3 + 1 = 28, 4^3 + 1 = 65. 6. Determine C: The next difference should be 5^3 + 1 = 126. So, C = 96 + 126 = 222. (Verify next term: 222 + (6^3 + 1) = 222 + 217 = 439, which perfectly matches the given series). 7. Calculate Final Expression: (A + C - n) = (9 + 222 - 2) = 229. Exam Strategy & Shortcut: When told a number has exactly three proper factors and two are prime, recognize immediately that the number is the product of those distinct primes and the third factor MUST be 1. This instantly solves for A without trial and error. Common Pitfall: Assuming the 3 factors could imply a larger composite number, which leads to overcomplicating (A - 8). Remember that 1 is always a proper factor of any positive integer. Final Answer: Therefore, the correct answer is 229.

Question 6

Directions: Solve the number series to answer the following questions.  $\sqrt[n]{A}$, 31, B, C, 439, B + 687, 1296 Note: (i) All the terms in the given series and 'n' are positive integer. (ii) Difference between 31 and B is X, which has 3 factors …

Accepted Answer

Concept & Least Common Multiple (LCM): Once the variables from the series conditions are resolved, calculating the LCM involves finding the smallest positive integer that is perfectly divisible by both numbers. Step-by-Step Solution: 1. Retrieve previously solved variables: From decoding the series conditions in the previous step, we know the difference X = 65 and the series term B = 96. 2. Substitute into the expression: We need to find the LCM of (B + 34) and X. 3. Calculate (B + 34) = 96 + 34 = 130. 4. Now, find the Least Common Multiple (LCM) of 130 and 65. 5. Since 130 is a direct multiple of 65 (65 × 2 = 130), the smallest number perfectly divisible by both is 130 itself. Exam Strategy & Shortcut: Always check if the larger number is a multiple of the smaller number before using prime factorization trees or division methods. If Y is a multiple of X, then LCM(X, Y) = Y. This skips unnecessary calculation steps. Common Pitfall: A common mistake is simply multiplying both numbers together to find the LCM, which yields 130 × 65 = 8450. This is a common multiple, but absolutely not the *least* common multiple. Final Answer: Therefore, the correct answer is 130.

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