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
61
Simplify: log(8) + log(125).
Answer:
3
Step 1: Use the product rule: log(8) + log(125) = log(8 * 125). Step 2: Multiply the arguments: 8 * 125 = 1000. Step 3: Evaluate log(1000). Since 10^3 = 1000, log(1000) = 3.
62
If log2(x) = 4, then x^2 equals:
Answer:
256
Step 1: Solve for x by converting to exponential form: x = 2^4. Step 2: Calculate x: x = 16. Step 3: Find x^2: 16^2 = 256.
63
Evaluate: log2(32) / log2(8).
Answer:
5/3
Step 1: Evaluate the numerator: log2(32) = 5 (since 2^5 = 32). Step 2: Evaluate the denominator: log2(8) = 3 (since 2^3 = 8). Step 3: Divide the results: 5 / 3.
64
If log(x) = 1.5, what is log(x^2)?
Answer:
3
Step 1: Use the power rule for logarithms: log(x^2) = 2 * log(x). Step 2: Substitute the known value of log(x) into the expression: 2 * 1.5. Step 3: Calculate the product to get 3.
65
Solve for x: 2^x = 3^(x-1).
Answer:
ln(3) / ln(1.5)
Step 1: Take ln of both sides: x*ln(2) = (x-1)*ln(3). Step 2: Distribute: x*ln(2) = x*ln(3) - ln(3). Step 3: Group x terms: x*ln(3) - x*ln(2) = ln(3), which is x*ln(3/2) = ln(3). Therefore, x = ln(3) / ln(1.5).
66
Simplify log_a(b) * log_b(a).
Answer:
1
Step 1: Apply the change of base formula to both terms: [ln(b) / ln(a)] * [ln(a) / ln(b)]. Step 2: Notice that the numerators and denominators cancel each other out completely. Step 3: This leaves 1 * 1 = 1. This is a standard identity.
67
Evaluate 10^(log(5)).
Answer:
5
Step 1: Recognize the base of 'log' is 10. The expression is 10^(log10(5)). Step 2: Use the inverse property a^(log_a(x)) = x. Step 3: Since the bases match (10), the expression evaluates directly to the argument, 5.
68
What is the base of the logarithm in the expression log(x) if no base is written?
Answer:
10
Step 1: In mathematics, particularly in standard algebra, a logarithm written without a specified base implies the common logarithm. Step 2: The common logarithm uses base 10. Step 3: Therefore, log(x) defaults to log10(x).
69
Find x if log2(log3(x)) = 1.
Answer:
9
Step 1: Start with the outer logarithm. Convert to exponential form: 2^1 = log3(x), so log3(x) = 2. Step 2: Now solve the inner logarithm by converting to exponential form again: 3^2 = x. Step 3: Calculate 3^2, yielding x = 9.
70
Express 3ln(2) - ln(4) as a single natural logarithm.
Answer:
ln(2)
Step 1: Use the power rule to rewrite 3ln(2) as ln(2^3), which is ln(8). Step 2: The expression is now ln(8) - ln(4). Step 3: Use the quotient rule: ln(8) - ln(4) = ln(8/4) = ln(2).