UPSC Prelims 2025 - CSAT (Paper II)

Explanation

We are asked to find natural numbers N < 50 that can be written as: N = p + q + r where p, q, r are distinct factors of N. This gives two main clues: Observations: If N is divisible by 3, then it might be expressible as the sum of 3 equal or related factors. Choose triplets of proper factors (excluding N itself) and check if their sum is N. Focus on numbers with many factors (highly composite numbers). These are more likely to allow such combinations. Quick Strategy: Instead of checking all numbers < 50: List highly composite numbers under 50: These include: 6, 12, 18, 24, 30, 36, 42, 48 For each such number, try adding 3 distinct proper factors (excluding N) to check if they sum to N. You’ll find that: „ 6 → 1+2+3 „ 12 → 2+4+6 „ 18 → 3+6+9 „ 24 → 4+8+12 „ 30 → 5+10+15 „ 36 → 6+12+18 „ 42 → 7+14+21 „ 48 → 8+16+24

Explanation

We are given: p, q, r are three prime numbers, all less than 20. They satisfy the condition: p – q = q – r ⇒ p + r = 2q So, q is the average of p and r, meaning the three numbers form an arithmetic progression. Now, list all primes less than 20 {2,3,5,7,11,13,17,19} Let’s find all triplets (r, q, p) in arithmetic progression with q = (p r) , + 2 and all primes < 20. Try each pair of primes (r, p) calculate q = (p r) , + 2 and check if it’s also a prime. Valid triplets (in increasing order): (3,5,7) → sum = 15 (3,7,11) → q = 7 → valid → sum = 21 (5,11,17) → q = 11 → valid → sum = 33 (3,11,19) → q = 11 → valid → sum = 33 (7,13,19) → q = 13 → valid → sum = 39 Valid distinct sums of (p + q + r) = 15, 21, 33, 39 Thus, there are 4 distinct possible values we can get for (p + q + r)

Explanation

We’re solving 1 1 1 1 p q r with p, q, r ∈ N. Since harmonic sums grow very slowly, only small values of p, q, r can satisfy this. z Assume small values: Try p = 2 and compute 1 1 1 2 q r „ You’ll get: q = 3, r = 6 q = 4, r = 4 z Try p = 3 ⇒1 1 2 3 q r Only small combo that works: q = r = 3 z List all distinct sums of p + q + r from the above: „ 2 + 3 + 6 = 11 „ 2 + 4 + 4 = 10 „ 3 + 3 + 3 = 9 There are 3 distinct possible values of p + q + r.

Explanation

We are given: Team X scored N runs in 20 overs Team Y tied the score in 10% less overs = 20 – 10% of 20 = 20 – 2 = 18 overs If Team Y's run rate was 50% higher, they would tie the score in 12 overs Let Team Y's actual run rate be r runs per over. They tied the score in 18 overs, 18r = N (Equation 1) If Team Y's run rate were 50% higher, it would be 1.5r. They would tie the score in 12 overs: 12 × 1.5r = N (Equation 2) 18r = N (same as Equation 1) Both expressions give same, consistent many solutions. So, scored by team X cannot be determined.

Explanation

If a value is increased by k %, multiply by 1 100 k If a value is decreased by k %, multiply by 1 100 k Let: x k y k 1 100 1 100 , Now apply the four successive changes to the original price p : Final Price q p x y x y p x y 2 2 Now apply the formula q p k k k k . . . 1 100 1 100 1 100 1 100 q p k k . . 1 10 1 10 2 4 2 4 q p k . 1 10 2 4 2 q p k . 10 4 2 4 2 q p k . ( ) 8 4 2 2