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
31
If log(5) ≈ 0.699, what is the approximate value of log(50)?
Answer:
1.699
Step 1: Express 50 as a product of 5 and 10: log(50) = log(5 * 10). Step 2: Use the product rule: log(5) + log(10). Step 3: Substitute known values: 0.699 + 1 = 1.699.
32
Simplify log_a(a^x).
Answer:
x
Step 1: This expression asks 'to what power must base a be raised to equal a^x?'. Step 2: The power is clearly x. Step 3: Alternatively, use the power rule: x * log_a(a) = x * 1 = x.
33
Solve for x: log2(2x) = 4.
Answer:
8
Step 1: Convert to exponential form: 2^4 = 2x. Step 2: Evaluate 2^4 to get 16. The equation is 16 = 2x. Step 3: Divide both sides by 2 to solve for x, giving x = 8.
34
Express log(x^3 * y^2) in terms of log(x) and log(y).
Answer:
3 * log(x) + 2 * log(y)
Step 1: Apply the product rule: log(A*B) = log(A) + log(B), which gives log(x^3) + log(y^2). Step 2: Apply the power rule to each term: log(x^k) = k*log(x). Step 3: This transforms log(x^3) to 3*log(x) and log(y^2) to 2*log(y). Total is 3*log(x) + 2*log(y).
35
Find the value of log(200) - log(2).
Answer:
2
Step 1: Use the quotient property of logarithms: log(A) - log(B) = log(A / B). Step 2: Apply the property: log(200 / 2) = log(100). Step 3: Since 10^2 = 100, log(100) = 2.
36
Solve for x: log3(x) = -1.
Answer:
1/3
Step 1: Convert from logarithmic to exponential form: 3^-1 = x. Step 2: A negative exponent signifies the reciprocal of the base. Step 3: Therefore, x = 1 / 3^1 = 1/3.
37
Evaluate log5(5^7).
Answer:
7
Step 1: Use the power rule to bring the exponent to the front: 7 * log5(5). Step 2: Use the identity log_a(a) = 1. So, log5(5) = 1. Step 3: Multiply the values: 7 * 1 = 7. Alternatively, use the inverse property log_a(a^x) = x.
38
Which of the following is equivalent to log(A / B)?
Answer:
log(A) - log(B)
Step 1: Review the quotient rule for logarithms. Step 2: The quotient rule states that the logarithm of a division is the difference of the logarithms. Step 3: Therefore, log(A / B) translates exactly to log(A) - log(B).
39
If log_x(64) = 2, find x.
Answer:
8
Step 1: Convert the logarithmic expression into exponential form: x^2 = 64. Step 2: Take the square root of both sides to solve for x. Step 3: Since x must be a positive base, x = 8.
40
Calculate log_10(0.001).
Answer:
-3
Step 1: Let x = log10(0.001). Convert to exponential form: 10^x = 0.001. Step 2: Express 0.001 as a fraction: 1/1000, which is 10^-3. Step 3: Equating exponents in 10^x = 10^-3 gives x = -3.