CAT Functions and Domains Bible Top 10 Must Solve Functions and Domains Questions covering 90% of Syllabus – Page 4
All 39 Functions and Domains questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 4 of 11.
The number of distinct real values of \(x\), satisfying the equation \(\max \{x, 2\}-\min \{x, 2\}=|x+2|-|x-2|\), is
Answer: 2
Solution — first steps
Step 1. The left side max(x,2)−min(x,2) is
Step 2. x−2
Step 2. x−2
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For any non-zero real number \(x\), let \(f(x)+2 f\left(\frac{1}{x}\right)=3 x\). Then, the sum of all possible values of \(x\) for which \(f(x)=3\), is
Answer: B. -3
Solution — first steps
Step 1. Write A=f(x), B=f(1/x). The original equation is A+2B=3x.
Step 2. Replacing x by 1/x gives 2A+B=3/x.
Step 2. Replacing x by 1/x gives 2A+B=3/x.
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The sum of all distinct real values of \(x\) that satisfy the equation \(10^x+\frac{4}{10^x}=\frac{17}{2}\), is
Answer: A. \(2 \log _{10} 2\)
Solution — first steps
Step 1. Let t=10^x>0. The equation becomes t+4/t=17/2.
Step 2. Multiply by 2t: 2t²−17t+8=0.
Step 2. Multiply by 2t: 2t²−17t+8=0.
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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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The number of integer solutions of equation \( 2|x|\left(x^2+1\right)=5 x^2 \) is
Answer: 3
Solution — first steps
Step 1. Because the left side contains |x|, set t=|x|≥0 and note x²=t².
Step 2. The equation becomes 2t(t²+1)=5t².
Step 2. The equation becomes 2t(t²+1)=5t².
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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 14 repeating CAT PYQs and how much each adds
| CAT PYQ | Repeating CAT PYQ | Questions | Representative | Cumulative questions | Cumulative share |
|---|---|---|---|---|---|
| 1 | Reject nonphysical roots in quantity and time problems | 11 | CAT 2021 S1 Q16 | 11 | 21.15% |
| 2 | Logarithm base and argument restrictions | 10 | CAT 2021 S1 Q4 | 21 | 40.38% |
| 3 | Modulus branches and sign consistency | 6 | CAT 2021 S2 Q16 | 27 | 51.92% |
| 4 | Denominator zeros and composite or reciprocal inputs | 5 | CAT 2021 S1 Q1 | 32 | 61.54% |
| 5 | Radical arguments and square root signs | 4 | CAT 2023 S1 Q3 | 36 | 69.23% |
| 6 | Exponential substitutions zero bases and exponent conditions | 4 | CAT 2021 S2 Q15 | 40 | 76.92% |
| 7 | Feasible output range of rational expressions | 3 | CAT 2021 S2 Q13 | 43 | 82.69% |
| 8 | Piecewise function or region interval membership | 2 | CAT 2022 S3 Q6 | 45 | 86.54% |
| 9 | Positivity and nonzero roots in reciprocal identities | 2 | CAT 2023 S3 Q4 | 47 | 90.38% |
| 10 | Select a difference using its given sign | 1 | CAT 2021 S2 Q3 | 48 | 92.31% |
| 11 | Positive substitution needed for AM GM | 1 | CAT 2022 S3 Q2 | 49 | 94.23% |
| 12 | Given work time interval | 1 | CAT 2023 S3 Q8 | 50 | 96.15% |
| 13 | Select the negative shared root | 1 | CAT 2024 S1 Q20 | 51 | 98.08% |
| 14 | Open and closed floor band endpoints | 1 | CAT 2025 S1 Q22 | 52 | 100% |
Based on CAT Actual Quant Papers.