MCQ
Algebra MCQ - Practice Questions with Answers
Solve 24 Algebra questions for RAS/RPSC preparation.
Practice questions
Q1Let x = 4 + √15. Find the value of x² + 1/x².
Rationalise 1/x = 1/(4 + √15) = 4 − √15, so x + 1/x = 8. Then x² + 1/x² = (x + 1/x)² − 2 = 64 − 2 = 62.
Q2For side lengths a = 8 and b = 3 on a rectangular frame, evaluate (a² − b²)³ / (a − b)³.
Factor a² − b² = (a − b)(a + b), so (a² − b²)³ / (a − b)³ = (a + b)³. With a + b = 11, the value is 11³ = 1331.
Q3A lens equation uses parameter y where y + 1/y = 5. Determine y⁴ + 1/y⁴.
First, y² + 1/y² = (y + 1/y)² − 2 = 25 − 2 = 23. Then y⁴ + 1/y⁴ = (y² + 1/y²)² − 2 = 23² − 2 = 529 − 2 = 527.
Q4Grain-silo depth readings x and y (metres) satisfy x² + y² = 45 and xy = 18. Taking x + y > 0, what is x³ + y³?
First (x + y)² = x² + 2xy + y² = 45 + 36 = 81, so x + y = 9. Then x³ + y³ = (x + y)((x + y)² − 3xy) = 9(81 − 54) = 9 × 27 = 243.
Q5A survey correction factor equals 6/(√7 − √3). After rationalising the denominator, which simplified form is obtained?
Multiply numerator and denominator by √7 + √3. The denominator becomes 7 − 3 = 4 and the numerator becomes 6(√7 + √3). So the expression equals 6(√7 + √3)/4 = 3(√7 + √3)/2.
You've seen 5 of 24 sample questions
Unlimited practice on Algebra comes with the RAS Test Series + Practice pack or Gate Pass.
More questions
6Rationalise and simplify the packing multiplier (√18 + √8)/(√18 − √8) for a warehouse conveyor. What is its value?
7A road-design offset equals √(12 + 2√35) metres. Writing it as √p + √q with integers p > q > 0, what is p − q?
8Two positive numbers x and y satisfy √x + √y = 5 and √x − √y = 1. What is the value of xy?
9Simplify (√5 + √3)/(√5 − √3) − (√5 − √3)/(√5 + √3).
10Three machine offsets u, v, w obey u + v + w = 0. If u = 9 and v = −4, what is u³ + v³ + w³?
11Three real numbers a, b and c satisfy a + b + c = 0 and abc = −18. What is a³ + b³ + c³?
12Rationalise the surveyor's ratio √12 / (√3 − 1) and select the fully simplified value.
13A hopper mix ratio r obeys r + 1/r = 5. Evaluate r³ + 1/r³.
14A market ledger records two unit prices p and q (in rupees) with p − q = 5 and pq = 36. What is the value of p² + q²?
15A bridge-clearance formula reduces to (√13 + √5)/(√13 − √5). After rationalising the denominator, which simplified form is obtained?