UPSC Prelims 2026 · CSAT (Paper II): How many times does 5 appear in all two-digit positive integers?
How many times does 5 appear in all two-digit positive integers?
Answer (b) & explanation
The digit 5 appears 19 times (b). Step 1: 5 in the tens place: 50, 51, …, 59 = 10 times. Step 2: 5 in the units place: 15, 25, 35, 45, 55, 65, 75, 85, 95 = 9 times (05 is not a two-digit number). Step 3: total = 10 + 9 = 19. The number 55 has two 5s, and both are counted. Option (a), 18, counts 55 once. Options (c) and (d) wrongly add 05 or 5.
Tip: count place by place. The question asks how many times the digit appears, not how many numbers contain it (that would be 18).
Answer as per the official UPSC answer key.
Same theme in UPSC Mains
Related questions from other years
Read the passage
How is deflation done? Most countries use a method called ‘double deflation’, where input and output prices are deflated separately. Consider a manufacturer importing oil for use in production. If oil prices fall, output prices do not and quantities remain the same, real value added should not change. But if the same deflator is used for inputs and outputs, as in India, it would look as if the manufacturer had become more productive.
Which of the following statements is/are correct?
- 1. Real value should not change in the instance of static output cost and unchanged quantities against falling oil prices.
- 2. Deflators are to be used separately for inputs and outputs, and this is a practice universally adopted by all economies.
Select the answer using the code given below.
Answer (d) & explanation
Neither statement is correct (d). Step 1: the passage says if oil prices fall while output prices and quantities stay the same, real value added should not change. Statement 1 changes this to "real value" and "static output cost", which distorts the idea. So 1 is not correct as worded. Step 2: statement 2 says separate deflators are used "universally" by all economies. But the passage says India uses the same deflator for both. So 2 is false. Step 3: options (a), (b) and (c) fail.
Tip: in economic passages, exact terms matter: "value added" is not "value", and "price" is not "cost". "Universally" is disproved by a single exception, here India.
Answer as per the official UPSC answer key.
Read the passage
India is starting to deploy AI for critical use cases such as weather forecasting, pest detection and control, and crop yield optimisation. However, penetration is limited to a small subset of tech-savvy farmers. In the US and in Europe, generative AI tools have started offering precision farming at scale, integrating large datasets to provide real-time agronomic insights. For at-scale integration and accessibility of AI tools in India, it would be helpful to develop Indian languages-based AI tools for smallholder farmers, partner with AgTechs to create affordable AI solutions, and disseminate AI-based advisory services through government programmes.
Which of the following statements is/are not correct?
- 1. Tech-savvy farmers will drive the AgTech companies of the future.
- 2. The development of advisory services by the government programmes for the use of AI tools in agriculture would be helpful.
- 3. In the US and Europe, AI tools have replaced traditional agricultural practices.
- 4. The integration of large datasets for use in real-time agronomic analysis is already a reality.
Select the answer using the code given below.
Answer (a) & explanation
Statements 1 and 3 are not correct (a). Step 1: statement 1 says tech-savvy farmers will drive AgTech companies. The passage only says AI use is limited to such farmers; it never says they will drive companies. Not correct. Step 2: statement 3 says AI has replaced traditional farming in the US and Europe. The passage says AI tools have "started offering" precision farming, not replaced old methods. Not correct. Step 3: statement 2 matches the suggestion of government advisory services, and statement 4 matches "integrating large datasets to provide real-time agronomic insights" already happening in the West. So both are correct, and options (b), (c) and (d) fail.
Tip: in "not correct" questions, mark each statement true or false, then pick the false ones.
Answer as per the official UPSC answer key.
Consider the following statements :
Every red is blue. Every blue is green. Every green is yellow.
Which of the following statements denoted by P, Q and R are correct?
P. Every blue is yellow.
Q. Every red is green.
R. Every red is yellow.
Select the answer using the code given below.
Answer (d) & explanation
All three, P, Q and R, are correct (d). Step 1: "Every red is blue" means red is inside blue. Likewise blue is inside green, and green is inside yellow. So red ⊂ blue ⊂ green ⊂ yellow, like circles nested one inside another. Step 2: P: blue sits inside green, which sits inside yellow, so every blue is yellow ✓. Q: red ⊂ blue ⊂ green, so every red is green ✓. Step 3: R: red sits inside all the larger circles, so every red is yellow ✓. Options (a), (b) and (c) leave one out.
Tip: "every A is B" chains pass along: if A is inside B and B inside C, then A is inside C. Draw nested circles to check.
Answer as per the official UPSC answer key.
An alloy P contains 20% copper and 80% zinc by weight. Another alloy Q contains 60% copper and 40% zinc by weight. A third alloy R is to be prepared from P and Q so that it contains equal amount of copper and zinc. In what ratio, amounts of P and Q be mixed in order to get R?
Answer (a) & explanation
P and Q must be mixed in the ratio 1 : 3 (a). Step 1: "equal copper and zinc" means R is 50% copper. Step 2: use alligation on copper: P has 20%, Q has 60%, and the mix needs 50%. P : Q = (60 − 50) : (50 − 20) = 10 : 30. Step 3: simplify to 1 : 3. Check: 1 kg P gives 0.2 kg copper; 3 kg Q gives 1.8 kg; total 2 kg copper in 4 kg = 50% ✓. Option (b) reverses the ratio. Options (c) and (d) fail.
Tip: in alligation, the amount of each item is the distance of the other item from the mean. Always check with a quick calculation.
Answer as per the official UPSC answer key.