IB Mathematics Sl Preparation Quiz

Solving the equation 2x + 5 = 9 involves isolating x by performing algebraic operations. First, subtract 5 from both sides of the equation to get 2x = 4. Then, divide both sides by 2 to isolate x, yielding x = 2. This process demonstrates the basic principle of balancing an equation, where operations performed on one side must be done to the other side to maintain equality. The solution shows that when 2x + 5 equals 9, the value of x must be 2 to satisfy the equation.

The probability of rolling a 5 on a fair 6-sided die is calculated by recognizing that there is one favorable outcome (rolling a 5) and six possible outcomes (one for each side of the die). Probability is found by dividing the number of favorable outcomes by the total number of possible outcomes. Therefore, the probability is 1/6. This basic principle of probability is used to predict the likelihood of specific events in random trials and can be applied to a wide range of statistical problems and real-world scenarios.

The derivative of f(x) = 3x^2 is found using basic differentiation rules. The power rule for differentiation states that the derivative of x^n is n*x^(n-1). Applying this rule to 3x^2, the exponent 2 is multiplied by the coefficient 3, giving 6x. The exponent is then reduced by one, resulting in 6x^1, or simply 6x. This derivative represents the slope of the tangent line to the curve at any point, showing how the rate of change of the function varies depending on the value of x.

The maximum value of the sine function, f(x) = sin(x), is 1. This is because the sine of an angle ranges between -1 and 1 for all values of x. In trigonometry, the sine function represents the y-coordinate of a point on the unit circle, and its maximum occurs when the angle corresponds to the highest point on the circle. The periodic nature of the sine function makes it useful in modeling wave-like phenomena, such as sound waves and alternating current, where maximum and minimum values represent peaks and troughs.

In a geometric sequence, each term is found by multiplying the previous term by the common ratio. In this case, the first term is 2 and the common ratio is 3. To find the third term, multiply the first term by the common ratio twice: 2 * 3 = 6 (second term), and 6 * 3 = 18 (third term). Geometric sequences are commonly used in finance, computer science, and physics to model exponential growth or decay, and understanding how to calculate terms in these sequences is crucial for solving related problems.

To find the hypotenuse of a right-angled triangle with legs of lengths 6 and 8, use the Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse. Substituting the values, 6^2 + 8^2 = c^2 becomes 36 + 64 = c^2, or 100 = c^2. Taking the square root of both sides, c = 10. The Pythagorean theorem is a fundamental concept in geometry that applies to all right-angled triangles and is widely used in fields requiring distance or measurement calculations.

Solving the quadratic equation x^2 - 5x + 6 = 0 can be done by factoring. The factors of 6 that add up to -5 are -2 and -3. Thus, the equation can be factored as (x - 2)(x - 3) = 0. Setting each factor equal to 0, we find that x = 2 or x = 3. This method of solving quadratic equations by factoring is a common and efficient technique when the quadratic expression can be easily factored. Quadratics are fundamental in many areas of mathematics, including physics and engineering.

The inverse function of f(x) = 2x - 3 is found by swapping the variables and solving for x. Starting with y = 2x - 3, switch x and y to get x = 2y - 3. Solve for y by adding 3 to both sides and then dividing by 2, yielding y = (x + 3)/2. Therefore, the inverse function is f^(-1)(x) = (x + 3)/2. Inverse functions essentially reverse the effect of the original function, providing an important tool in solving equations and understanding relationships between variables in mathematics.

To find the value of log base 2 of 8, ask, "2 raised to what power equals 8?" Since 2^3 = 8, the answer is 3. In logarithmic terms, this is written as log2(8) = 3. Logarithms are the inverse of exponentiation, meaning they reverse the process of raising a number to a power. Understanding this relationship helps simplify complex exponential equations and is a foundational concept in algebra and higher-level mathematics, particularly in areas involving exponential growth, such as computer science and physics.

The sine rule is a trigonometric formula used to relate the sides and angles of a triangle. It states that the ratio of a side of a triangle to the sine of its opposite angle is constant for all sides of the triangle. Mathematically, a/sinA = b/sinB = c/sinC. This rule is particularly useful for solving non-right-angled triangles, where other methods, like the Pythagorean theorem, are not applicable. The sine rule allows for the calculation of unknown angles or sides when certain measurements are given, making it a versatile tool in trigonometry.