Mathematics/General Ability MCQs
Topic Notes: Mathematics/General Ability
MCQs and preparation resources for competitive exams, covering important concepts, past papers, and detailed explanations.
Plato
- Biography: Ancient Greek philosopher (427–347 BCE), student of Socrates and teacher of Aristotle, founder of the Academy in Athens.
- Important Ideas:
- Theory of Forms
- Philosopher-King
- Ideal State
41
Rationalize the denominator: 5 / sqrt(10).
Answer:
sqrt(10) / 2
Multiply top and bottom by sqrt(10). This gives (5 sqrt(10)) / 10. Simplify the fraction by dividing numerator and denominator by 5, resulting in sqrt(10) / 2.
42
Rationalize 2 / (sqrt(3) - 1).
Answer:
sqrt(3) + 1
Multiply by the conjugate (sqrt(3) + 1). The denominator evaluates to (sqrt(3))^2 - 1^2 = 3 - 1 = 2. The expression becomes 2(sqrt(3) + 1) / 2. Canceling the 2s leaves sqrt(3) + 1.
43
Simplify 1 / (sqrt(5) - sqrt(2)).
Answer:
(sqrt(5) + sqrt(2)) / 3
Multiply by the conjugate, (sqrt(5) + sqrt(2)). The new denominator is (sqrt(5))^2 - (sqrt(2))^2 = 5 - 2 = 3. The numerator is 1 * (sqrt(5) + sqrt(2)). The result is (sqrt(5) + sqrt(2)) / 3.
44
Rationalize the denominator of 1 / (2 + sqrt(3)).
Answer:
2 - sqrt(3)
Multiply the numerator and denominator by the conjugate of the denominator, which is (2 - sqrt(3)). The denominator becomes (2^2 - (sqrt(3))^2) = 4 - 3 = 1. The result is just the numerator: 2 - sqrt(3).
45
Rationalize the expression 10 / sqrt(5).
Answer:
2 sqrt(5)
Multiply numerator and denominator by sqrt(5). This results in (10 sqrt(5)) / 5. Divide 10 by 5 to get the final simplified form: 2 sqrt(5).
46
Simplify 3 / sqrt(3) by rationalizing the denominator.
Answer:
sqrt(3)
Multiply the top and bottom by sqrt(3). The expression becomes (3 * sqrt(3)) / (sqrt(3) * sqrt(3)) = (3 sqrt(3)) / 3. The 3s cancel out, leaving just sqrt(3).
47
Rationalize the denominator of 1 / sqrt(2).
Answer:
sqrt(2) / 2
To remove the surd from the denominator, multiply the numerator and the denominator by sqrt(2). This gives (1 * sqrt(2)) / (sqrt(2) * sqrt(2)) = sqrt(2) / 2.
48
Simplify sqrt(18) / sqrt(2).
Answer:
3
Combine under a single root: sqrt(18 / 2). This equals sqrt(9). The square root of 9 evaluates to 3.
49
Evaluate (2 sqrt(5))^2.
Answer:
20
Square both parts of the term separately. The square of 2 is 4, and the square of sqrt(5) is 5. Multiply the results: 4 * 5 = 20.
50
Calculate 10 sqrt(10) / 2 sqrt(2).
Answer:
5 sqrt(5)
Divide the outside coefficients: 10 / 2 = 5. Divide the terms inside the roots: sqrt(10) / sqrt(2) = sqrt(5). Combining them yields 5 sqrt(5).