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PSAT/NMSQT Algebra Questions
Practise algebra (mathematics) for the Preliminary SAT / National Merit Scholarship Qualifying Test: 10 multiple-choice questions, each with the correct answer and a worked explanation. Click an option to check it, or “New questions” for another set.
Algebra practice
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Q1.Solve: 4x + y = 7 and 5x + y = 6
Show answer & explanation
Answer: C — x = -1, y = 11
Elimination/substitution gives x = -1, y = 11. Check: 4(-1) + 1(11) = 7 ✓ and 5(-1) + 1(11) = 6 ✓.
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Q2.Solve: 4x + 3y = 5 and 2x + 4y = -10
Show answer & explanation
Answer: A — x = 5, y = -5
Elimination/substitution gives x = 5, y = -5. Check: 4(5) + 3(-5) = 5 ✓ and 2(5) + 4(-5) = -10 ✓.
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Q3.Find the sum of the first 21 terms of the AP 3, 12, 21, …
Show answer & explanation
Answer: B — 1953
Sₙ = n/2 × [2a + (n − 1)d] = 21/2 × [6 + 180] = 1,953.
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Q4.Solve: x² − 16x + 63 = 0
Show answer & explanation
Answer: B — 7, 9
Find two roots with sum 16 and product 63: 9 and 7. So x² − 16x + 63 = (x − 9)(x − 7) = 0.
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Q5.Find log₂ 32.
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Answer: B — 5
log_2(32) asks “2 to what power gives 32?” → 2^5 = 32, so the answer is 5.
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Q6.Find log 10,000 (base 10).
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Answer: A — 4
log_10(10000) asks “10 to what power gives 10000?” → 10^4 = 10000, so the answer is 4.
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Q7.Solve: 3x + 3y = 30 and 5x + 3y = 34
Show answer & explanation
Answer: A — x = 2, y = 8
Elimination/substitution gives x = 2, y = 8. Check: 3(2) + 3(8) = 30 ✓ and 5(2) + 3(8) = 34 ✓.
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Q8.Solve: x² − 14x + 48 = 0
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Answer: B — 6, 8
Find two roots with sum 14 and product 48: 8 and 6. So x² − 14x + 48 = (x − 8)(x − 6) = 0.
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Q9.Solve: 4x + y = 4 and 2x + 2y = -4
Show answer & explanation
Answer: B — x = 2, y = -4
Elimination/substitution gives x = 2, y = -4. Check: 4(2) + 1(-4) = 4 ✓ and 2(2) + 2(-4) = -4 ✓.
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Q10.Solve: x² − 7x + 6 = 0
Show answer & explanation
Answer: C — 1, 6
Find two roots with sum 7 and product 6: 1 and 6. So x² − 7x + 6 = (x − 1)(x − 6) = 0.
How to approach algebra questions
Factorise quadratics by finding two numbers with the right sum and product; check roots by substitution. For simultaneous equations, elimination is usually faster than substitution under time pressure.
About PSAT/NMSQT
Preliminary SAT / National Merit Scholarship Qualifying Test — conducted by College Board & National Merit Scholarship Corporation. The practice SAT that also screens for National Merit Scholarships.
Original practice questions written for this site in the style and difficulty of each exam’s syllabus. They are not past papers and the site is not affiliated with any exam body. Exam patterns, marking schemes and syllabi change — always check the latest official notification.
PSAT/NMSQT practice topics
| Topic | Section | Question source |
|---|---|---|
| Algebra | Mathematics | Calculated (new numbers every set) |
| Percentages | Quantitative aptitude | Calculated (new numbers every set) |
| English Grammar | English | Question bank |
| Vocabulary & Idioms | English | Question bank |