1. Problem: How many pairs of twin primes are there between the integers 1 to 100? 1. Work: The twin prime pairs up to 100 are $ (3,5),(5,7),(11,13),(17,19),(29,31),(41,43),(59,61),(71,73) $. 2. Answer: $8$ pairs, option A. 2. Problem: What is the missing term in the series $4,12,36,\;\_,324,972$? 1. Work: The sequence multiplies by $3$ each step, so the missing term is $36\times 3=108$. 2. Answer: $108$, option C. 3. Problem: Which fraction is largest among $\tfrac{3}{13},\tfrac{2}{15},\tfrac{4}{17}$? 1. Work: Decimal approximations are $\tfrac{3}{13}\approx 0.2308$, $\tfrac{2}{15}\approx 0.1333$, $\tfrac{4}{17}\approx 0.2353$. 2. Answer: $\tfrac{4}{17}$ is largest, option C. 4. Problem: What is the HCF of $24,30$ and $42$? 1. Work: Prime factors give HCF $=2\times 3=6$. 2. Answer: $6$, option C. 5. Problem: What is the LCM of $0.6,9.6$ and $0.12$? 1. Work: Multiply by $100$ to clear decimals: LCM of $60,960,12$ is $960$, so LCM of decimals is $960/100=9.6$. 2. Answer: $9.6$, option B. 6. Problem: What is the cube root of $-5832$? 1. Work: $18^3=5832$, so cube root is $-18$. 2. Answer: $-18$, option D. 7. Problem: $272\times 425\div p^2=400$, find $p$. 1. Work: $272\times 425=115600$, so $115600/p^2=400$ gives $p^2=115600/400=289$ and $p=17$. 2. Answer: $17$, option B. 8. Problem: An amount doubles itself on simple interest in four years. What is the percent per annum rate of interest? 1. Work: Doubling means total interest for $4$ years equals principal, so $r\times 4/100=1$ gives $r=25$. 2. Answer: $25\%$, option B. 9. Problem: A train covers a distance of $200$ km with a speed of $10$ km/h. What time is taken? 1. Work: Time $=200/10=20$ hours. 2. Answer: $20$ h, option D. 10. Problem: A train covers $90$ m in passing a standing man. Find the length of the train. 1. Work: Distance covered equals train length, so length $=90$ m. 2. Answer: $90$ m, option C. 11. Problem: If speed of boat in still water is $8$ km/h and stream is $4$ km/h, find upstream speed. 1. Work: Upstream speed $=8-4=4$ km/h. 2. Answer: $4$ km/h, option A. 12. Problem: What will be the angle between the two hands of a clock at $9\!:\!50$ AM? 1. Work: Angle $=|30\times 9-5.5\times 50|=|270-275|=5$ degrees. 2. Answer: $5^\circ$, option A. 13. Problem: What will be the average of first $100$ natural numbers? 1. Work: Average $=(1+100)/2=50.5$. 2. Answer: $50.5$, option B. 14. Problem: Divide $1111$ in the ratio $8:3$. 1. Work: Sum parts $=11$, so parts are $1111\times 8/11=808$ and $1111\times 3/11=303$. 2. Answer: $808,303$, option C. 15. Problem: Present age of Karan is $5$ times Shivam; after $10$ years Karan will be $3$ times Shivam. Find present ages. 1. Work: Let Shivam $=s$, Karan $=5s$. Then $5s+10=3(s+10)$ gives $2s=20$, $s=10$, Karan $=50$. 2. Answer: Karan $50$, Shivam $10$, option B. 16. Problem: Value of $\sqrt{36.1/102.4}$? 1. Work: $36.1/102.4=361/1024=(19^2)/(32^2)$ so square root $=19/32$. 2. Answer: $19/32$, option A. 17. Problem: Article bought for $250$. Selling price for $10\%$ profit? 1. Work: SP $=250\times 1.10=275$. 2. Answer: $275$, option D. 18. Problem: Item sold for $680$ after $15\%$ discount; find marked price. 1. Work: $680=0.85\times MP$ so $MP=680/0.85=800$. 2. Answer: $800$, option D. 19. Problem: Simple interest on $8930$ at $8\%$ per annum after $5$ years? 1. Work: $SI=8930\times 8\times 5/100=8930\times 0.4=3572$. 2. Answer: $3572$, option C. 20. Problem: Varun and Syan together $3$ days, Syan and Anil $4$ days, Anil and Varun $6$ days. Find days for Anil alone. 1. Work: Rates: $V+S=1/3$, $S+A=1/4$, $A+V=1/6$. Sum gives $2(A+V+S)=1/3+1/4+1/6=3/4$ so $A+V+S=3/8$. Then $A=(3/8)-(V+S)=(3/8)-(1/3)=1/24$ so Anil alone takes $24$ days. 2. Answer: $24$, option D. 21. Problem: Convert $25$ m/s to km/h. 1. Work: Multiply by $18/5$: $25\times 18/5=90$ km/h. 2. Answer: $90$ km/h, option B. 22. Problem: Without stoppage speed $54$ km/h and with stoppage $45$ km/h; how many minutes does the train stop per hour? 1. Work: If running fraction $t$ of hour then $54t=45$ gives $t=5/6$ hour running so stoppage $=1/6$ hour $=10$ minutes. 2. Answer: $10$ minutes, option A. 23. Problem: Time for a boat to cover $64$ km along stream if boat in still water $12$ km/h and stream $4$ km/h? 1. Work: Downstream speed $=12+4=16$ km/h so time $=64/16=4$ h. 2. Answer: $4$ h, option D. 24. Problem: Angle between hands at $9\!:\!50$ (alternate listing). 1. Work: Same as Q12 gives $5^\circ$. 2. Answer: $5^\circ$, option D. 25. Problem: What day was $9$ March $2000$ if $5$ March $1999$ was Friday? 1. Work: From $5$ Mar $1999$ to $9$ Mar $2000$ count days; $1999$ not leap so days to $5$ Mar $2000$ is $365$ then plus $4$ more days gives $369$ days; $369\bmod 7=369-7\times 52=369-364=5$ shift forward $5$ days from Friday to Wednesday. 2. Answer: Wednesday, option A. 26. Problem: Last two digits of $7^{2008}$? 1. Work: $7^4\equiv 2401\equiv 1\pmod{100}$ so period $4$ for last two digits, $2008\bmod 4=0$ so last two digits equal those of $7^4$, which are $01$. 2. Answer: $01$, option C. 27. Problem: Next term in series $50,200,100,100,200,50,400,\ldots$? 1. Work: Observing factors of $25$ gives sequence $25\times 2,8,4,4,8,2,16,\ldots$ so next factor appears to be $1$ giving $25$. 2. Answer: $25$, option C. 28. Problem: Find $1.08\div 0.000108$. 1. Work: Multiply numerator and denominator by $100000$ to see $1.08/0.000108=10000$ because $0.000108\times 10000=1.08$. 2. Answer: $10000$, option C. 29. Problem: Least number which when divided by $24,32,36$ leaves remainders $19,27,31$ respectively? 1. Work: Each remainder is $5$ less than divisor, so $N+5$ divisible by $24,32,36$. LCM$=288$, so $N=288-5=283$. 2. Answer: $283$, option D. 30. Problem: How many digits are there in $\sqrt{1838736}$? 1. Work: Compute $1356^2=1838736$, so square root $=1356$ which has $4$ digits. 2. Answer: $4$, option D. 31. Problem: Find $x$ from $55\times 45+205-15\times 12=x^2$. 1. Work: $55\times 45=2475$, $15\times 12=180$, so $x^2=2475+205-180=2500$, hence $x=50$. 2. Answer: $50$, option D. 32. Problem: If average of $9$ consecutive positive integers is $55$, what is largest integer? 1. Work: Middle integer is $55$, largest is $55+4=59$. 2. Answer: $59$, option C. 33. Problem: Two numbers ratio $5:8$. If $4$ subtracted from each, ratio becomes $7:12$. Find original numbers. 1. Work: Let $5k$ and $8k$. Then $(5k-4)/(8k-4)=7/12$ gives $k=5$, so numbers $25$ and $40$. 2. Answer: $25,40$, option B. 34. Problem: Akshay is elder than Vinay and younger than Karthik, sum of Vinay and Karthik is $48$. What is Akshay's age? 1. Work: With only ordering and sum given, a reasonable assumption is Akshay is between them and take the average $48/2=24$ as a plausible value. 2. Answer: $24$, option A. 35. Problem: Express $2\tfrac{1}{4}$ in percent. 1. Work: $2\tfrac{1}{4}=2.25$ so percent $=2.25\times 100=225\%$. 2. Answer: $225$, option B. 36. Problem: Dealer sells at $20\%$ loss on cost price but uses $40\%$ less weight. What is his percentage profit or loss? 1. Work: Let cost per true kg $=100$. He sells a labeled kg at $20\%$ loss so SP $=80$ but gives only $0.6$ kg whose cost $=0.6\times 100=60$, so profit $=80-60=20$ on cost $60$ which is $20/60=1/3=33\tfrac{1}{3}\%$ profit. 2. Answer: $33\tfrac{1}{3}\%$, option D. 37. Problem: Rita bought TV with $20\%$ discount and made profit of $800$ by selling for $16800$. Find labeled price. 1. Work: Profit $800$ means cost price $=16800-800=16000$. If this equals $80\%$ of labeled price then labeled price $=16000/0.8=20000$. 2. Answer: $20000$, option D. 38. Problem: Difference of simple interest from two banks for $1000$ in two years is $20$. Find difference in rate of interest. 1. Work: Difference $=1000\times 2\times \Delta r/100=20$ gives $20\Delta r=20$ so $\Delta r=1$. 2. Answer: $1\%$, option A. 39. Problem: $8000$ becomes $12500$ in $2$ years at compound interest. What will be the sum after $3$ years? 1. Work: $(1+r)^2=12500/8000=1.5625$ so $1+r=1.25$ giving $r=25\%$. Amount after $3$ years $=8000\times 1.25^3=15625$. 2. Answer: $15625$, option B. 40. Problem: If $6$ persons working $8$ h a day earn $8400$ per week, how much will $9$ persons working $6$ h a day earn per week? 1. Work: Earnings proportional to person-hours. First group person-hours $=6\times 8=48$ gives rate $8400/48=175$ per person-hour. Second group person-hours $=9\times 6=54$ gives earnings $=54\times 175=9450$. 2. Answer: $9450$, option C. 41. Problem: A person covers $20\tfrac{1}{2}$ km in $3$ h. What distance will be covered in $5$ h? 1. Work: Speed $=20.5/3$ km/h so distance in $5$ h $=5\times 20.5/3=102.5/3\approx 34.17$ km, closest option $34$ km. 2. Answer: $34$ km, option D. 42. Problem: A $440$ m long train runs at $240$ km/h; time to pass a man running opposite at $24$ km/h? 1. Work: Relative speed $=240+24=264$ km/h $=264\times 5/18=73.333\dots$ m/s, time $=440/73.333\dots=6$ s. 2. Answer: $6$ s, option C. 43. Problem: Boatman rows $1$ km in $5$ min along stream and $6$ km in $1$ h against stream. Find speed of stream. 1. Work: Downstream speed $=1/(5/60)=12$ km/h, upstream $=6/1=6$ km/h, stream speed $=(12-6)/2=3$ km/h. 2. Answer: $3$ km/h, option A. 44. Problem: At what time between $3$ and $4$ will hands be in opposite direction? 1. Work: Solve $|30\times 3-5.5m|=180$ gives $m=49\tfrac{1}{11}$ minutes past $3$. 2. Answer: $49\tfrac{1}{11}$ min past $3$, option C. 45. Problem: What day was $5$ November $1987$ if $4$ April $1988$ was Monday? 1. Work: Days between $5$ Nov $1987$ and $4$ Apr $1988$ is $151$ days, $151\bmod 7=4$, so go back $4$ days from Monday to Thursday. 2. Answer: Thursday, option C. 46. Problem: A line of length $1.5$ m was measured as $1.55$ m by mistake. Error percent? 1. Work: Error $=0.05/1.5=1/30\approx 0.03333$ so percent $\approx 3.33\%$. 2. Answer: $3.33\%$, option C. 47. Problem: Find the wrong number in series $1,3,9,31,128,651,3913$. 1. Work: Pattern $a_{n}=a_{n-1}\times (n-1)+(n)$ yields expected $128$ should be $129$, so $128$ is wrong. 2. Answer: $128$, option A. 48. Problem: Average of first five positive even numbers divisible by $9$? 1. Work: Even numbers divisible by $9$ are multiples of $18$: $18,36,54,72,90$ whose average $=(18+36+54+72+90)/5=54$. 2. Answer: $54$, option A. 49. Problem: Least number exactly divisible by $8,9,12,15,18$ and also a perfect square? 1. Work: LCM$=2^3\times 3^2\times 5=360$. Smallest square multiple requires even exponents: $2^4\times 3^2\times 5^2=360^\times 10=3600$. 2. Answer: $3600$, option B. 50. Problem: Car covers $200$ km in $2$ h $40$ min and a jeep covers same in $2$ h. Ratio of speeds? 1. Work: Car time $=2+40/60=8/3$ h so speed car $=200/(8/3)=75$ km/h, jeep $=100$ km/h, ratio $=75:100=3:4$. 2. Answer: $3:4$, option A. 51. Problem: How to study maths easily and tips and tricks? 1. Work: Key steps are consistent practice, understanding concepts before memorizing, solving many varied problems, reviewing mistakes, and using simple heuristics for quick calculation. 2. Tips: Practice daily with a mix of easy and challenging problems; master fundamentals; use mental arithmetic tricks such as factorization and modular arithmetic; create summary sheets for formulas; time yourself on past papers; explain solutions aloud or to peers to solidify understanding. 3. Final advice: Be patient, practice regularly, and focus on understanding the why behind methods rather than rote memorization.