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
111
Solve for x: log3(x - 1) = 2.
Answer:
10
Step 1: Convert the equation from logarithmic to exponential form: 3^2 = x - 1. Step 2: Simplify the exponential term: 9 = x - 1. Step 3: Add 1 to both sides to solve for x: x = 9 + 1 = 10.
112
Solve for x: log2(x) = 5.
Answer:
32
Step 1: Identify the base (2), the exponent (5), and the argument (x). Step 2: Convert the logarithmic equation to exponential form: 2^5 = x. Step 3: Calculate 2^5: 2 * 2 * 2 * 2 * 2 = 32. Thus, x = 32.
113
If log(x) = 2, what is the value of x? (Assume base 10)
Answer:
100
Step 1: A logarithm written without a base implies base 10 (common logarithm). So, log10(x) = 2. Step 2: Convert to exponential form: 10^2 = x. Step 3: Calculate 10^2, which equals 100. Therefore, x = 100.
114
Which of the following is equivalent to log(x^3)?
Answer:
3 * log(x)
Step 1: Recall the power rule for logarithms, which states that log_b(M^k) = k * log_b(M). Step 2: Apply this rule to the expression log(x^3). Step 3: The exponent 3 moves to the front, resulting in 3 * log(x).
115
Evaluate 3 * log2(2).
Answer:
3
Step 1: Recognize the value of log2(2). Since the base and argument are the same, log2(2) = 1. Step 2: Substitute this value back into the expression: 3 * 1. Step 3: The result is 3.
116
Simplify log6(72) - log6(2).
Answer:
2
Step 1: Use the quotient rule for logarithms: log_b(M) - log_b(N) = log_b(M / N). Step 2: Apply the rule: log6(72) - log6(2) = log6(72 / 2) = log6(36). Step 3: Evaluate log6(36). Since 6^2 = 36, the value is 2.
117
Simplify log10(5) + log10(2).
Answer:
1
Step 1: Use the product rule for logarithms: log_b(M) + log_b(N) = log_b(M * N). Step 2: Apply the rule: log10(5) + log10(2) = log10(5 * 2) = log10(10). Step 3: Since log_b(b) = 1, log10(10) = 1.
118
What is the value of log3(27) - log3(9)?
Answer:
1
Step 1: Evaluate each term. log3(27) = 3 (since 3^3 = 27) and log3(9) = 2 (since 3^2 = 9). Step 2: Subtract the values: 3 - 2 = 1. Step 3: Alternatively, use the quotient rule: log3(27/9) = log3(3) = 1.
119
Evaluate log2(64) + log2(4).
Answer:
8
Step 1: Evaluate each logarithm separately. log2(64) = 6 because 2^6 = 64. Step 2: log2(4) = 2 because 2^2 = 4. Step 3: Add the results: 6 + 2 = 8. Alternatively, use the product rule: log2(64 * 4) = log2(256) = 8.
120
Find the value of log(1/2)(8).
Answer:
-3
Step 1: Let x = log(1/2)(8). Exponential form is (1/2)^x = 8. Step 2: Rewrite both sides using base 2. (1/2) is 2^-1, and 8 is 2^3. Step 3: This gives (2^-1)^x = 2^3, so 2^(-x) = 2^3. Therefore, -x = 3, which means x = -3.