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
101
Write 2 * log(3) + log(4) as a single logarithm.
Answer:
log(36)
Step 1: Use the power rule to rewrite 2 * log(3) as log(3^2), which is log(9). Step 2: The expression becomes log(9) + log(4). Step 3: Use the product rule: log(A) + log(B) = log(A * B). This gives log(9 * 4) = log(36).
102
Simplify: log5(25^x).
Answer:
2x
Step 1: Use the power rule to bring the exponent to the front: x * log5(25). Step 2: Evaluate log5(25). Since 5^2 = 25, log5(25) = 2. Step 3: Multiply the result by x to get x * 2, which is 2x.
103
Evaluate log2(3) * log3(4).
Answer:
2
Step 1: Use the change of base formula: log_b(a) = ln(a)/ln(b). Step 2: Rewrite the expression: [ln(3)/ln(2)] * [ln(4)/ln(3)]. Step 3: Cancel out ln(3) to get ln(4)/ln(2), which is log2(4). Since 2^2 = 4, the answer is 2.
104
Use the change of base formula to express log2(10) in terms of common logarithms.
Answer:
log(10) / log(2)
Step 1: The change of base formula is log_b(a) = log_c(a) / log_c(b). Step 2: To change to common logarithms (base 10), we set c = 10. Step 3: Substituting a = 10 and b = 2 gives log_10(10) / log_10(2), which is written as log(10) / log(2).
105
Simplify e^(ln(7)).
Answer:
7
Step 1: Use the inverse property of exponential and logarithmic functions: a^(log_a(x)) = x. Step 2: Here, the base of the exponential is 'e', and the base of 'ln' is also 'e'. Step 3: Applying the property gives e^(ln(7)) = 7.
106
Evaluate ln(e^5).
Answer:
5
Step 1: Recall that 'ln' is logarithm base e. So the expression is log_e(e^5). Step 2: Use the power rule for logarithms: ln(e^5) = 5 * ln(e). Step 3: Since ln(e) = 1, the expression simplifies to 5 * 1 = 5.
107
What is the base of the natural logarithm (ln)?
Answer:
e
Step 1: Logarithms can have any positive base except 1. Step 2: The common logarithm 'log' defaults to base 10. Step 3: The natural logarithm 'ln' is mathematically defined as having the base 'e', Euler's number (approximately 2.718).
108
Solve for x: log_x(125) = 3.
Answer:
5
Step 1: Rewrite the equation in exponential form: x^3 = 125. Step 2: Take the cube root of both sides. Step 3: Since 5 * 5 * 5 = 125, the cube root of 125 is 5. Therefore, x = 5.
109
If log_x(81) = 4, find x.
Answer:
3
Step 1: Convert the logarithmic equation to exponential form: x^4 = 81. Step 2: Take the fourth root of both sides. Step 3: Since 3^4 = 81 (and the base of a logarithm must be positive), x = 3.
110
Find x if log5(2x + 1) = 2.
Answer:
12
Step 1: Convert to exponential form: 5^2 = 2x + 1. Step 2: Simplify the left side: 25 = 2x + 1. Step 3: Subtract 1 and divide by 2: 24 = 2x, so x = 12.