CAT Surds and Indices Bible Top 10 Must Solve Surds and Indices Questions covering 90% of Syllabus – Page 4
All 22 Surds and Indices questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 4 of 5.
If \(x\) and \(y\) are positive real numbers such that \(\log _x\left(x^2+12\right)=4\) and \(3 \log _y x=1\), then \(x+y\) equals
Answer: D. 10
Solution — first steps
Step 1. log_x(x²+12)=4 means x^4=x²+12.
Step 2. Let u=x²>0. Then u²−u−12=0=(u−4)(u+3), so u=4 and x=2.
Step 2. Let u=x²>0. Then u²−u−12=0=(u−4)(u+3), so u=4 and x=2.
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Let n be the least positive integer such that 168 is a factor of 1134n. If m is the least positive integer such that 1134n is a factor of 168m, then m + n equals:
Answer: C. 15
Solution — first steps
Given 1134n and 168m conditions, solving for n and m, we get m+n = 15.
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Let a, b, m and n be natural numbers such that a > 1 and b > 1. If a
m
b
n
144
145
, then the largest possible value of n-m is
m
b
n
144
145
, then the largest possible value of n-m is
Answer: D. 579
Solution — first steps
Ans 4
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For some positive real number \(x\), if \(\log _{\sqrt{3}}(x)+\frac{\log _x(25)}{\log _x(0.008)}=\frac{16}{3}\), then the value of \(\log _3\left(3 x^2\right)\) is
Answer: 7
Solution — first steps
Step 1. All logarithms are defined only for x>0,x≠1.
Step 2. Change base:log_(√3)x=2log_3x, and log_x25/log_x0.008=log_5(25)/log_5(0.008)=2/(−3).
Step 2. Change base:log_(√3)x=2log_3x, and log_x25/log_x0.008=log_5(25)/log_5(0.008)=2/(−3).
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Let \( n \) and \( m \) be two positive integers such that there are exactly 41 integers greater than \( 8^m \) and less than \( 8^n \), which can be expressed as powers of 2. Then, the smallest possible value of \( n + m \) is:
Answer: C. 16
Solution — first steps
Answer: C. 16 Detailed Solution: Given that there are 41 integers between \( 8^m \) and \( 8^n \), which are powers of 2.
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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.