Integration is the reverse process of differentiation, and we use the power rule to find antiderivatives.
We need to find ∫2x dx, which means we're integrating the function 2x with respect to x.
The power rule states: ∫x^n dx = \(\frac{x^{n+1}}{n+1}\) + C
Rewrite 2x as: ∫2x^1 dx
∫2x dx = 2∫x^1 dx
\(\frac{d}{dx}\)(x² + C) = 2x ✓
Therefore, ∫2x dx = x² + C
Answer: (A) x²+C
HCF (Highest Common Factor) is the largest number that divides both given numbers without leaving a remainder.
Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors = 1, 2, 3, 4, 6, 12
The largest common factor = 12
Alternative Method - Using Prime Factorization:
24 = 2³ × 3
36 = 2² × 3²
HCF = 2² × 3 = 4 × 3 = 12
Therefore, HCF of 24 and 36 = 12
The answer is (C) 12
To convert degrees to radians, use the conversion formula that relates the two angle measurement systems.
Therefore, 45° = \(\frac{\pi}{4}\) radians.
The answer is (A) π/4
The slope of a line is the coefficient of x in the standard linear equation form y = mx + c.
The equation of a line is written as: y = mx + c, where m is the slope and c is the y-intercept.
Given equation: y = 3x + 5
Standard form: y = mx + c
Here, m = 3 (coefficient of x) and c = 5 (constant term)
Therefore, the slope is 3.
The answer is (B) 3
Factorial is the product of all positive integers from 1 up to a given number.
7! means 7 × 6 × 5 × 4 × 3 × 2 × 1
7 × 6 = 42
42 × 5 = 210
210 × 4 = 840
840 × 3 = 2520
2520 × 2 = 5040
5040 × 1 = 5040
Therefore, 7! = 5040
The answer is (B) 5040
This question asks you to find the roots of a quadratic equation using factorization.
The numbers are -2 and -3, because:
(-2) × (-3) = 6
(-2) + (-3) = -5
x - 2 = 0 → x = 2
x - 3 = 0 → x = 3
For x = 2: (2)² - 5(2) + 6 = 4 - 10 + 6 = 0 ✓
For x = 3: (3)² - 5(3) + 6 = 9 - 15 + 6 = 0 ✓
Therefore, x = 2, 3
The answer is (A).
When flipping a fair coin, the probability of any single outcome equals the number of favorable outcomes divided by total possible outcomes.
A coin has exactly 2 possible outcomes: Head or Tail.
Number of ways to get a Head = 1
Therefore, the probability of getting a head in a coin toss is \(\frac{1}{2}\) or 0.5 or 50%.
The answer is C) 1/2
Square root is the inverse operation of squaring, and asks: what number multiplied by itself equals 144?
We need to find a number that when multiplied by itself gives 144.
We test the options by squaring them:
- 11² = 11 × 11 = 121 (not 144)
- 12² = 12 × 12 = 144 (matches!)
- 13² = 13 × 13 = 169 (not 144)
- 14² = 14 × 14 = 196 (not 144)
Since 12 × 12 = 144, we have √144 = 12
Therefore, the answer is (B) 12
Differentiation is finding the rate of change of a function with respect to a variable.
We have y = 3x² and we need to use the Power Rule of differentiation.
The Power Rule states: If y = ax^n, then \(\frac{dy}{dx} = n \cdot a \cdot x^{n-1}\)
Here, a = 3 and n = 2
Therefore, the derivative of y = 3x² is 6x
Answer: A) 6x
To find f(2), substitute x = 2 into the given function and calculate the result.
Therefore, f(2) = 12
The answer is B) 12