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Question language / प्रश्न भाषा

If 17²⁰⁰ is divided by 18, the remainder is: Explanation Series

यदि 17²⁰⁰ को 18 से विभाजित किया जाए, तो शेषफल क्या होगा? व्याख्या: 17 = 18 − 1 Therefore, 17 ≡ −1 (mod 18) So, 17²⁰⁰ ≡ (−1)²⁰⁰ Since 200 is even, (−1)²⁰⁰ = 1 Hence remainder = 1.

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A

17

B

16

C

1

✓

D

2

Correct Answerसही उत्तर

C. 1

Explanationव्याख्या

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If the unit digit in (549 × 56 × 28* × 684) is 8, then what digit will be there in place of *? We only need unit digits: 549 → 9 56 → 6 28* → * 684 → 4 Therefore, 9 × 6 × * × 4 First, 9 × 6 × 4 = 216 So we need the unit digit of: 216 × * Since 216 ends in 6, we need: 6 × * to end in 8. Check options: 6 × 4 = 24 → unit digit 4 6 × 3 = 18 → unit digit 8 6 × 2 = 12 → unit digit 2 6 × 1 = 6 Therefore, = 3 Divisibility and Remainderयदि (549 × 56 × 28 × 684) का इकाई अंक 8 है, तो * के स्थान पर कौन-सा अंक होगा?* Explanation व्याख्या: Here 28* represents a three-digit number whose unit digit is *.Same topic · Same subject · Same exam

[(946 + 157)² + (946 − 157)²] [(946 × 946) + (157 × 157)] = ? [(946 + 157)² + (946 − 157)²] [(946 × 946) + (157 × 157)] = ? Explanation Therefore, = 2a² + 2b² = 2(a²+b²) The denominator is: a²+b² Therefore, the complete fraction becomes: 2(a²+b²) (a²+b²) Cancel (a²+b²): = 2 Hence, answer = B. 2व्याख्या: Let: a = 946 b = 157 The numerator is: (a+b)² + (a−b)² Expand: (a+b)² = a² + 2ab + b² (a−b)² = a² − 2ab + b² Adding: (a+b)² + (a−b)² = a² + 2ab + b² + a² − 2ab + b² The +2ab and −2ab cancel.Same topic · Same subject · Same exam

217 × 217 + 183 × 183 = ? 217 × 217 + 183 × 183 = ? Explanationव्याख्या: 217² = 47089 183² = 33489 Therefore, 47089 + 33489 = 80578 A quicker method: 217 = 200+17 183 = 200−17 Therefore, (200+17)² + (200−17)² Using: (a+b)²+(a−b)² = 2(a²+b²) = 2(200² + 17²) = 2(40000+289) = 2×40289 = 80578Same topic · Same subject · Same exam

Which of the following fractions is less than 7 8 and greater than 1 3? Explanation 3 < fraction < 7 8 Check A: 1 4 < 1 3, so incorrect. Check B: 23 24 > 7 8, so incorrect. Check C: 11 12 > 7 8, so incorrect. Check D: Compare 17 24 with 1 3: 17 × 3 = 51 24 × 1 = 24 51 > 24 Therefore, 17 24 > 1 3 Now compare 17 24 with 7 8: 17 × 8 = 136 24 × 7 = 168 136 < 168 Therefore, 17 24 < 7 8 Hence, 1 3 < 17 24 < 7 8 Answer = D Basic Arithmeticनिम्नलिखित में से कौन-सी भिन्न 7 8 से कम और 1 3 से अधिक है? व्याख्या: We need: 1Same topic · Same subject · Same exam

1 2 + 1 6 + 1 12 + 1 20 + 1 30 + …… + 1 [n(n+1)] = ? 1 2 + 1 6 + 1 12 + 1 20 + 1 30 + …… + 1 [n(n+1)] = ? Explanation 2 = 1 (1×2) 1 6 = 1 (2×3) 1 12 = 1 (3×4) 1 20 = 1 (4×5) Therefore, the general term is: 1 [r(r+1)] Now, 1 [r(r+1)] = 1 r − 1 (r+1) Therefore, 1 (1×2) = 1 − 1 2 1 (2×3) = 1 2 − 1 3 1 (3×4) = 1 3 − 1 4 1 (4×5) = 1 4 − 1 5 ... 1 [n(n+1)] = 1 n − 1 (n+1) Hence, S = (1 − 1 2) (1 2 − 1 3) (1 3 − 1 4) ... (1 n − 1 (n+1)) All the middle terms cancel: −1 2 + 1 2 = 0 −1 3 + 1 3 = 0 −1 4 + 1 4 = 0 Therefore, S = 1 − 1 (n+1) Taking LCM: S = [(n+1) − 1] (n+1) = n (n+1) Hindi: [r(r+1)] = 1 r − 1 (n+1) = n (n+1)व्याख्या: Observe: 1 प्रत्येक पद को इस प्रकार लिख सकते हैं: 1 (r+1) अतः सभी बीच के पद कट जाते हैं और बचता है: 1 − 1Same topic · Same subject · Same exam

Which one of the following will divide (3²⁵ + 3²⁶ + 3²⁷ + 3²⁸) completely? Explanation Therefore, the expression contains: 2 × 3 × 5 = 30 Hence it is divisible by 30.निम्नलिखित में से कौन (3²⁵ + 3²⁶ + 3²⁷ + 3²⁸) को पूर्णतः विभाजित करेगा? व्याख्या: Take 3²⁵ common: 3²⁵ + 3²⁶ + 3²⁷ + 3²⁸ = 3²⁵(1 + 3 + 9 + 27) = 3²⁵ × 40 Now, 40 = 2³ × 5 And 3²⁵ contains factor 3.Same topic · Same subject · Same exam

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