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
21
If log_3(x) + log_3(x) = 4, solve for x.
Answer:
9
Step 1: Combine the like terms on the left side: 2*log_3(x) = 4. Step 2: Divide by 2: log_3(x) = 2. Step 3: Convert to exponential form: 3^2 = x. Therefore, x = 9.
22
Find the value of log(0.0001).
Answer:
-4
Step 1: Rewrite 0.0001 as a power of 10. 0.0001 = 1/10000 = 10^-4. Step 2: The expression is log10(10^-4). Step 3: By the power rule or inverse property, the exponent is the answer: -4.
23
Simplify the expression e^(2*ln(3)).
Answer:
9
Step 1: Use the power rule for logarithms to rewrite the exponent: 2*ln(3) = ln(3^2) = ln(9). Step 2: The expression is now e^(ln(9)). Step 3: Apply the inverse property e^(ln(x)) = x. Thus, the result is 9.
24
Solve for x: log(x-2) = 1.
Answer:
12
Step 1: The logarithm has an implied base of 10. Convert to exponential form: 10^1 = x - 2. Step 2: Simplify the equation: 10 = x - 2. Step 3: Add 2 to both sides to solve for x: x = 12.
25
Evaluate log_4(64).
Answer:
3
Step 1: Let x = log_4(64). Convert to exponential form: 4^x = 64. Step 2: Identify the power of 4 that yields 64. Step 3: Since 4 * 4 * 4 = 64 (or 4^3 = 64), x = 3.
26
If log_x(16) = 2, what is the value of x?
Answer:
4
Step 1: Convert from logarithmic form to exponential form: x^2 = 16. Step 2: Take the square root of both sides to solve for x. Step 3: Since the base must be positive, x = 4.
27
Which represents the change of base formula for log_b(x)?
Answer:
log(x) / log(b)
Step 1: The change of base formula allows calculating a log with an arbitrary base using standard bases. Step 2: The formula states log_b(x) = log_c(x) / log_c(b) for any new base c. Step 3: Using base 10, this is written as log(x) / log(b).
28
Solve for x: ln(e^x) = 10.
Answer:
10
Step 1: Use the inverse property of natural logarithms and exponential functions: ln(e^x) = x. Step 2: Substitute this simplification back into the equation. Step 3: We immediately get x = 10.
29
Find the value of log2(1/32).
Answer:
-5
Step 1: Let x = log2(1/32). Convert to exponential form: 2^x = 1/32. Step 2: Express 1/32 as a base of 2. Since 32 = 2^5, 1/32 = 2^-5. Step 3: Equate exponents: 2^x = 2^-5, giving x = -5.
30
Evaluate 10^(-log(2)).
Answer:
1/2
Step 1: Use the power rule to move the negative sign: -log(2) = log(2^-1) = log(1/2). Step 2: The expression becomes 10^(log(1/2)). Step 3: Apply the inverse property 10^(log(x)) = x. The result is 1/2 or 0.5.