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
51
Simplify the expression: 10^(2log(3)).
Answer:
9
Step 1: Use the power rule on the exponent: 2log(3) becomes log(3^2) = log(9). Step 2: The expression is now 10^(log(9)). Step 3: Apply the inverse property 10^(log(x)) = x. Therefore, the result is 9.
52
Solve for x: log(x+1) + log(x-1) = log(3).
Answer:
2
Step 1: Combine the left side using the product rule: log((x+1)(x-1)) = log(3). Step 2: Equate the arguments: x^2 - 1 = 3. Step 3: x^2 = 4, so x = 2 or x = -2. Since log argument must be positive, x = -2 is invalid. Thus, x = 2.
53
If log(x) = a and log(y) = b, express log(10xy) in terms of a and b.
Answer:
1 + a + b
Step 1: Expand using the product rule: log(10 * x * y) = log(10) + log(x) + log(y). Step 2: Evaluate log(10) (base 10 is implied), which is 1. Step 3: Substitute a and b for log(x) and log(y) to get 1 + a + b.
54
What is the expansion of ln(x^2 / y)?
Answer:
2*ln(x) - ln(y)
Step 1: Use the quotient rule to split the fraction: ln(x^2) - ln(y). Step 2: Use the power rule on the first term: 2*ln(x). Step 3: Combine them to get the final expanded form: 2*ln(x) - ln(y).
55
Find the value of log_sqrt(2)(8).
Answer:
6
Step 1: Let x = log_sqrt(2)(8). Convert to exponential form: (sqrt(2))^x = 8. Step 2: Express both sides with base 2: (2^(1/2))^x = 2^3, which is 2^(x/2) = 2^3. Step 3: Equate the exponents: x/2 = 3, yielding x = 6.
56
Solve for x: log4(x) = 1.5.
Answer:
8
Step 1: Convert to exponential form: 4^1.5 = x. Step 2: Write 1.5 as 3/2. So, x = 4^(3/2). Step 3: This means the square root of 4, cubed. sqrt(4) = 2, and 2^3 = 8. Thus, x = 8.
57
If ln(x) = 2.5, which represents the exact value of x?
Answer:
e^2.5
Step 1: The natural logarithm 'ln' has base 'e'. Step 2: Convert the equation log_e(x) = 2.5 into exponential form. Step 3: The base 'e' raised to the exponent 2.5 equals the argument x. Thus, x = e^2.5.
58
Simplify: log3(1/27).
Answer:
-3
Step 1: Let x = log3(1/27). In exponential form, 3^x = 1/27. Step 2: Express 1/27 as a base of 3: 1/27 = 1/(3^3) = 3^-3. Step 3: Equate the exponents in 3^x = 3^-3 to get x = -3.
59
Which of the following equals log_b(a)?
Answer:
1 / log_a(b)
Step 1: Recall the change of base formula: log_b(a) = ln(a) / ln(b). Step 2: Recall that log_a(b) = ln(b) / ln(a). Step 3: Therefore, log_b(a) is the reciprocal of log_a(b), written as 1 / log_a(b).
60
Solve for x: log5(x^2) = 2.
Answer:
5 or -5
Step 1: Convert to exponential form: 5^2 = x^2. Step 2: Evaluate the left side: 25 = x^2. Step 3: Solve for x by taking the square root. Since x is squared inside the log, x can be positive or negative. Thus, x = 5 or x = -5.