CAT Surds and Indices Bible Top 10 Must Solve Surds and Indices Questions covering 90% of Syllabus – Page 5
All 22 Surds and Indices questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 5 of 5.
For a real number \(x\), if \(\frac{1}{2}, \frac{\log _3\left(2^x-9\right)}{\log _3 4}\), and \(\frac{\log _5\left(2^x+\frac{17}{2}\right)}{\log _5 4}\) are in an arithmetic progression, then the common difference is
Answer: C. \(\log _4\left(\frac{7}{2}\right)\)
Solution — first steps
Step 1. Use change of base: the middle and third terms are log_4(2^x−9) and log_4(2^x+17/2).
Step 2. For three terms in AP, twice the middle equals first plus third.
Step 2. For three terms in AP, twice the middle equals first plus third.
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Let \(n\) be any natural number such that \(5^{n-1} \lt 3^{n+1}\). Then, the least integer value of \(m\) that satisfies \(3^{n+1} \lt 2^{n+m}\) for each such \(n\), is
Answer: 5
Solution — first steps
Step 1. Rewrite the first inequality as (5/3)^(n−1)9. Since the base exceeds 1, allowed n are 1 through 5.
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Last day solving checklist
- Name the unknown. Include its unit and state whether it is a count, amount, rate or parameter.
- Form the relationships. Translate each given condition into an equation or restriction.
- Choose the method. Use the formula, substitution, elimination, a sign chart or a bound appropriate to this concept.
- Check restrictions. Check integer counts, signs, nonzero denominators, domains and all original conditions.
- Verify the answer. Substitute into the original question; for an extreme, show a feasible example reaches it.
All 16 repeating CAT PYQs and how much each adds
| CAT PYQ | Repeating CAT PYQ | Questions | Representative | Cumulative questions | Cumulative share |
|---|---|---|---|---|---|
| 1 | Convert logarithms to powers using base rules | 7 | CAT 2021 S1 Q4 | 7 | 17.50% |
| 2 | Prime factorisation and exponent comparison | 5 | CAT 2022 S3 Q11 | 12 | 30.00% |
| 3 | Compound growth factor comparisons | 4 | CAT 2023 S1 Q16 | 16 | 40.00% |
| 4 | Exponential substitution reduces to quadratics | 4 | CAT 2023 S2 Q2 | 20 | 50.00% |
| 5 | Integer exponent bounds with common bases | 3 | CAT 2021 S2 Q15 | 23 | 57.50% |
| 6 | Modular powers and remainder cycles | 3 | CAT 2024 S1 Q10 | 26 | 65.00% |
| 7 | Products of powers use sums of exponents | 2 | CAT 2021 S3 Q20 | 28 | 70.00% |
| 8 | Symmetric reciprocal powers by recurrence identities | 2 | CAT 2023 S3 Q4 | 30 | 75.00% |
| 9 | Simplify powers into geometric series | 2 | CAT 2023 S3 Q20 | 32 | 80.00% |
| 10 | Prime powers or powers of ten and digit sums | 2 | CAT 2025 S2 Q20 | 34 | 85.00% |
| 11 | Special bases or exponent zero when a power is one | 1 | CAT 2022 S2 Q17 | 35 | 87.50% |
| 12 | Universal common divisors of power expressions | 1 | CAT 2022 S1 Q21 | 36 | 90.00% |
| 13 | Shift an affine recurrence into geometric growth | 1 | CAT 2023 S1 Q22 | 37 | 92.50% |
| 14 | Replacement iterations and a concentration threshold | 1 | CAT 2023 S2 Q14 | 38 | 95.00% |
| 15 | Prime power rule of a multiplicative function | 1 | CAT 2024 S2 Q4 | 39 | 97.50% |
| 16 | Telescoping product of logarithmic exponents | 1 | CAT 2024 S3 Q16 | 40 | 100% |
Based on CAT Actual Quant Papers.