Gr 12 Mathematics: November Easy P(1)
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Questions and Answers

What is the common difference in an arithmetic sequence?

  • The average of the first and last term
  • The sum of the first and last term
  • A constant value added to each term to get the next term (correct)
  • The difference between the first and last term

    • What is the formula to find the nth term of an arithmetic sequence?

  • Tn = a + (n - 1)d (correct)
  • Tn = a - nd
  • Tn = a - (n - 1)d
  • Tn = a + nd

    • How do you find the arithmetic mean between two numbers?

  • By finding the difference between the two numbers
  • By finding the sum of the two numbers
  • By finding the average of the two numbers (correct)
  • By finding the product of the two numbers

    • What is the graphical representation of an arithmetic sequence?

    <p>A straight line</p> Signup and view all the answers

    How do you test if a sequence is arithmetic?

    <p>By calculating the differences between consecutive terms</p> Signup and view all the answers

    What does the gradient of the line represent in the graphical representation of an arithmetic sequence?

    <p>The common difference</p> Signup and view all the answers

    What happens to an arithmetic sequence if the common difference is positive?

    <p>The sequence increases</p> Signup and view all the answers

    What is the use of the formula Tn = a + (n - 1)d?

    <p>To find the nth term of an arithmetic sequence</p> Signup and view all the answers

    What is the general formula for the nth term of a geometric sequence?

    <p>Tn = ar^(n-1)</p> Signup and view all the answers

    What is the geometric mean between two numbers a and b?

    <p>√(ab)</p> Signup and view all the answers

    What is the characteristic of a geometric sequence when plotted on a graph?

    <p>Exponential pattern</p> Signup and view all the answers

    What is the formula to find the common ratio of a geometric sequence?

    <p>r = Tn / T(n-1)</p> Signup and view all the answers

    What is the definition of a series?

    <p>The sum of the terms of a sequence</p> Signup and view all the answers

    What is the notation used to represent the sum of terms in a sequence?

    <p>Sigma notation</p> Signup and view all the answers

    What is the formula for the sum of the first n terms of a sequence?

    <p>Sn = T1 + T2 + ... + Tn</p> Signup and view all the answers

    What is the characteristic of a geometric sequence when r > 1?

    <p>The sequence grows exponentially</p> Signup and view all the answers

    How do you test if a sequence is geometric?

    <p>By calculating the ratios between consecutive terms</p> Signup and view all the answers

    What is the formula to find the nth term of a geometric sequence?

    <p>Tn = ar^(n-1)</p> Signup and view all the answers

    What is the general formula for a finite geometric series?

    <p>$S_n = \frac{a(1 - r^n)}{1 - r}$</p> Signup and view all the answers

    What is the condition for an infinite geometric series to converge?

    <p>$-1 &lt; r &lt; 1$</p> Signup and view all the answers

    What is the formula for the sum of an infinite geometric series?

    <p>$S_\infty = \frac{a}{1 - r}$</p> Signup and view all the answers

    What is the general form of a geometric sequence?

    <p>$T_n = a \cdot r^{n-1}$</p> Signup and view all the answers

    What is the common difference in an arithmetic sequence?

    <p>the difference between two consecutive terms</p> Signup and view all the answers

    What is the method used by Karl Friedrich Gauss to find the sum of the first 100 integers?

    <p>writing the numbers in ascending and descending order and adding the corresponding pairs of terms</p> Signup and view all the answers

    What is the sum of the first n terms of an arithmetic sequence?

    <p>$S_n = \frac{n(a + l)}{2}$</p> Signup and view all the answers

    What is the formula for the nth term of an arithmetic sequence?

    <p>$T_n = a + (n - 1)d$</p> Signup and view all the answers

    What is the alternative formula for the sum of a finite geometric series when r > 1?

    <p>$S_n = \frac{a(r^n - 1)}{r - 1}$</p> Signup and view all the answers

    What is the definition of a geometric sequence?

    <p>a sequence of numbers where each term is multiplied by a constant value to obtain the next term</p> Signup and view all the answers

    What is the sum of the first 100 positive integers?

    <p>5050</p> Signup and view all the answers

    What is the formula for the sum of a finite arithmetic series, if the last term l is known?

    <p>$S_n = \frac{n}{2} (a + l)$</p> Signup and view all the answers

    What is the definition of a relation?

    <p>A rule that associates each element of one set with at least one element of another set.</p> Signup and view all the answers

    What is the definition of a function?

    <p>A rule that associates each element of one set with exactly one element of another set.</p> Signup and view all the answers

    Which of the following best describes a one-to-one function?

    <p>Each element of the domain maps to a unique element of the range.</p> Signup and view all the answers

    What is the graphical representation of a one-to-one function?

    <p>Every vertical line intersects the graph at most once.</p> Signup and view all the answers

    Which of the following best describes a many-to-one function?

    <p>Multiple elements of the domain map to the same element of the range.</p> Signup and view all the answers

    What is the graphical representation of a many-to-one function?

    <p>Every vertical line intersects the graph at most once.</p> Signup and view all the answers

    What is the definition of an inverse function?

    <p>A function that reverses the operation of a given function.</p> Signup and view all the answers

    What is the key property that a function must have for its inverse to also be a function?

    <p>It must be a one-to-one function.</p> Signup and view all the answers

    What is the inverse of a linear function?

    <p>Also a linear function</p> Signup and view all the answers

    How do you find the inverse of y = ax^2?

    <p>Interchange x and y, then solve for y</p> Signup and view all the answers

    What is the definition of a logarithm?

    <p>The exponent to which a base must be raised to yield x</p> Signup and view all the answers

    What is the shape of the graph of an exponential function?

    <p>Increasing or decreasing</p> Signup and view all the answers

    What is the intercept of the graph of a logarithmic function?

    <p>(1, 0)</p> Signup and view all the answers

    What is the product rule of logarithms?

    <p>log_a(xy) = log_a x + log_a y</p> Signup and view all the answers

    What is the domain of the inverse of y = ax^2?

    <p>x &gt;= 0</p> Signup and view all the answers

    What is the asymptote of the graph of an exponential function?

    <p>Horizontal at y = 0</p> Signup and view all the answers

    What is the range of the graph of a logarithmic function?

    <p>All real numbers</p> Signup and view all the answers

    What is log_a(a) equal to?

    <p>1</p> Signup and view all the answers

    What does the variable $P$ represent in the future value annuity formula?

    <p>Payment amount per period</p> Signup and view all the answers

    Which formula is used to calculate the present value of an annuity?

    <p>$ PV = P \frac{1 - (1 + i)^{-n}}{i} $</p> Signup and view all the answers

    What does the variable $n$ signify in the formulas?

    <p>Total number of periods</p> Signup and view all the answers

    In the context of future value annuities, what does $F$ represent?

    <p>Future value of the annuity</p> Signup and view all the answers

    What is the primary purpose of calculating the present value of an annuity?

    <p>To determine the current worth of future payments</p> Signup and view all the answers

    When calculating the future value of an annuity, what role does compound interest play?

    <p>It increases the total value by accumulating on payments</p> Signup and view all the answers

    What does the formula P = x [(1 - (1 + i)^(-n))/i] represent?

    <p>Present value of an annuity</p> Signup and view all the answers

    What is the formula to calculate the future value of an annuity?

    <p>F = x [(1 + i)^n - 1]/i</p> Signup and view all the answers

    What does the formula A = P(1 + in) represent?

    <p>Simple interest</p> Signup and view all the answers

    What is the formula to calculate the effective annual rate?

    <p>EAR = (1 + i_nominal/m)^m - 1</p> Signup and view all the answers

    What does the formula n = log(A/P)/log(1 + i) represent?

    <p>Calculating the period of an investment under compound interest</p> Signup and view all the answers

    What is the formula to calculate the remaining loan balance?

    <p>P_balance = x [(1 - (1 + i)^(-n_remaining))/i]</p> Signup and view all the answers

    What is the formula to calculate the total amount paid for an annuity?

    <p>T = n * x</p> Signup and view all the answers

    What is the formula to calculate the total interest paid for an annuity?

    <p>I = T - P</p> Signup and view all the answers

    What does the formula log_a x = log_b x / log_b a represent?

    <p>Change of base for logarithms</p> Signup and view all the answers

    What is the branch of mathematics that focuses on optimization problems and rates of change?

    <p>Differential calculus</p> Signup and view all the answers

    What is the formula to calculate the period of an investment in compound interest?

    <p>$n = rac{ ext{log}ig(rac{A}{P}ig)}{ ext{log}(1+i)}$</p> Signup and view all the answers

    Which of the following represents the domain of the logarithmic function $f^{-1}(x) = ext{log}(x)$?

    <p>$x &gt; 0$</p> Signup and view all the answers

    What type of interest is calculated on the initial principal, including interest from previous periods?

    <p>Compound Interest</p> Signup and view all the answers

    What does the intercept of the exponential function $f(x) = 10^x$ represent?

    <p>The value of the function at $x = 0$</p> Signup and view all the answers

    Which statement about an annuity is true?

    <p>An annuity is a series of equal payments made at regular intervals.</p> Signup and view all the answers

    What is the primary difference between a Future Value Annuity (FVA) and a Present Value Annuity (PVA)?

    <p>FVA accumulates money over time, while PVA pays off loans.</p> Signup and view all the answers

    Which of the following correctly describes the asymptote of the logarithmic function?

    <p>$x = 0$</p> Signup and view all the answers

    In the context of financial calculations, which log function is commonly used to determine pH levels?

    <p>$ ext{pH} = - ext{log}_{10}[ ext{H}^+]$</p> Signup and view all the answers

    What is the range of the function for the inverse exponential function?

    <p>$y ext{ is any real number}$</p> Signup and view all the answers

    What is the condition for two events to be mutually exclusive?

    <p>They cannot happen at the same time</p> Signup and view all the answers

    What is the formula for the probability of two independent events A and B?

    <p>P(A and B) = P(A) × P(B)</p> Signup and view all the answers

    What is the complementary rule in probability?

    <p>P(A) + P(not A) = 1</p> Signup and view all the answers

    What does the symbol A ∩ B represent in probability?

    <p>The intersection of sets A and B</p> Signup and view all the answers

    What is the purpose of Venn diagrams in probability?

    <p>To show the relationship between events</p> Signup and view all the answers

    What is the formula for the probability of two events A and B that are not mutually exclusive?

    <p>P(A or B) = P(A) + P(B) - P(A and B)</p> Signup and view all the answers

    What is the condition for two events to be independent?

    <p>P(A and B) = P(A) × P(B)</p> Signup and view all the answers

    What is the sample space in probability?

    <p>The set of all possible outcomes</p> Signup and view all the answers

    What is the symbol A' in probability?

    <p>The complement of event A</p> Signup and view all the answers

    What is the formula for the probability of the complement of event A?

    <p>P(A') = 1 - P(A)</p> Signup and view all the answers

    What is the limit of the function as x approaches -6?

    <p>-8</p> Signup and view all the answers

    Which operation can be performed on the function to simplify it, given that x is not equal to -6?

    <p>Cancelling the x + 6 terms</p> Signup and view all the answers

    What is the definition of the derivative as described?

    <p>The limit of the gradient of the tangent to the curve</p> Signup and view all the answers

    Which notation is NOT commonly used to express the derivative of a function?

    <p>k * f(x)</p> Signup and view all the answers

    Which rule for differentiation is used for a constant multiplied by a function?

    <p>frac{d}{dx} [k * f(x)] = k * frac{d}{dx}[f(x)]</p> Signup and view all the answers

    What happens to the limit of y as x approaches -6?

    <p>It approaches -8.</p> Signup and view all the answers

    Which rule for differentiation applies to the sum of two functions?

    <p>Sum of the individual derivatives</p> Signup and view all the answers

    Under what condition should first principles be used to find the derivative?

    <p>When explicitly requested</p> Signup and view all the answers

    What does the expression $f'(x) = rac{d}{dx}[f(x)]$ represent?

    <p>Derivative of the function</p> Signup and view all the answers

    Which of the following is true about the graphical representation of the function near x = -6?

    <p>There is a hole in the graph.</p> Signup and view all the answers

    What is the derivative of the function ( f(x) = x^3 + 2x^2 - 5x + 1 )?

    <p>( 3x^2 + 4x - 5 )</p> Signup and view all the answers

    Which of the following is NOT a valid notation for the second derivative of ( y ) with respect to ( x )?

    <p>( rac{dy}{dx} )</p> Signup and view all the answers

    What does the sign of the second derivative tell us about the original function?

    <p>The concavity of the original function.</p> Signup and view all the answers

    What is the slope of the tangent line to the function ( f(x) = x^3 - 2x^2 + 3x ) at the point ( x = 2 )?

    <p>( 5 )</p> Signup and view all the answers

    How does the coefficient ( a ) affect the shape of the cubic graph ( y = ax^3 + bx^2 + cx + d )?

    <p>It determines the shape and orientation of the graph.</p> Signup and view all the answers

    What is the y-intercept of the cubic function ( f(x) = 2x^3 - 5x^2 + 3x - 1 )?

    <p>( -1 )</p> Signup and view all the answers

    What is the gradient of the normal to the curve ( y = x^2 + 2x - 3 ) at the point ( x = 1 )?

    <p>( - rac{1}{4} )</p> Signup and view all the answers

    Which of the following statements is TRUE about the derivative of a function?

    <p>The derivative of a function represents the slope of the tangent line at a point on the function's graph.</p> Signup and view all the answers

    What is the equation of the tangent line to the function ( f(x) = x^2 - 3x + 2 ) at the point ( x = 2 )?

    <p>( y = x - 2 )</p> Signup and view all the answers

    Which of the following is a correct statement about stationary points on a graph?

    <p>A stationary point is where the graph has a maximum or minimum value.</p> Signup and view all the answers

    What does the addition rule state for mutually exclusive events?

    <p>P(A or B) = P(A) + P(B)</p> Signup and view all the answers

    How is the probability of a sequence of outcomes calculated using a tree diagram?

    <p>By multiplying the probabilities along the branches</p> Signup and view all the answers

    What are complementary events?

    <p>Events that cover all possible outcomes</p> Signup and view all the answers

    What is represented in a two-way contingency table?

    <p>Counts or percentages related to probability problems</p> Signup and view all the answers

    Which formula correctly defines the product rule for independent events?

    <p>P(A and B) = P(A) × P(B)</p> Signup and view all the answers

    What does the symbol A' represent in probability?

    <p>The complement of event A</p> Signup and view all the answers

    Which of the following describes mutually exclusive events?

    <p>They cannot happen at the same time.</p> Signup and view all the answers

    What is the complementary rule of probability?

    <p>P(not A) = 1 - P(A)</p> Signup and view all the answers

    What does the second derivative of a function tell us about the function's graph?

    <p>The concavity of the graph</p> Signup and view all the answers

    What is the purpose of using synthetic division when factoring cubic polynomials?

    <p>To find the x-intercepts</p> Signup and view all the answers

    What does the sign of the leading coefficient (a) in a cubic polynomial tell us about the graph's end behavior?

    <p>The overall shape of the graph</p> Signup and view all the answers

    How can you determine if a stationary point is a local maximum or a local minimum using the first derivative?

    <p>The first derivative is negative to the left and positive to the right of the stationary point</p> Signup and view all the answers

    What is the main purpose of applying differential calculus to optimization problems?

    <p>To find the stationary points of a function</p> Signup and view all the answers

    What is the relationship between the divisor and the root in synthetic division?

    <p>The divisor is the negative of the root</p> Signup and view all the answers

    What is the key difference between the long division method and synthetic division for factoring polynomials?

    <p>Synthetic division is a simplified version of long division, specifically designed for dividing polynomials by linear factors.</p> Signup and view all the answers

    What is the main purpose of finding the turning points of a cubic polynomial?

    <p>To identify the points where the function changes from increasing to decreasing or vice versa</p> Signup and view all the answers

    Which of the following methods can be used to solve cubic equations?

    <p>All of the above</p> Signup and view all the answers

    What is the primary application of differential calculus in understanding rates of change?

    <p>Calculating the instantaneous rate of change of a function at a specific point</p> Signup and view all the answers

    According to the Remainder Theorem, what is the remainder when the polynomial ( p(x) = x^3 - 2x^2 + 5x - 1 ) is divided by ( x - 2 )?

    <p>7</p> Signup and view all the answers

    What is the factor of the polynomial ( p(x) = x^3 + 3x^2 - 4x - 12 ) based on the Factor Theorem?

    <p>x + 2</p> Signup and view all the answers

    If (p(x) = x^3 - 5x^2 + 7x - 3) and (p(1) = 0), what is a factor of (p(x))?

    <p>x - 1</p> Signup and view all the answers

    Which of the following steps is NOT involved in solving a cubic equation?

    <p>Solving the cubic polynomial using the cubic formula.</p> Signup and view all the answers

    What is the remainder when the polynomial (p(x) = 2x^4 - 3x^3 + 5x - 1) is divided by (x + 1)?

    <p>-1</p> Signup and view all the answers

    Which of the following is NOT a factor of the polynomial (p(x) = x^3 - 6x^2 + 11x - 6)?

    <p>x - 4</p> Signup and view all the answers

    What is the remainder when the polynomial (p(x) = x^3 + 2x^2 - 5x + 1) is divided by (2x - 1)?

    <p>3</p> Signup and view all the answers

    If a polynomial (p(x)) has a root of (x = -3), which of the following is a factor of (p(x))?

    <p>x + 3</p> Signup and view all the answers

    What is the probability of getting a sum of 7 when two dice are rolled?

    <p>1/6</p> Signup and view all the answers

    What is the general formula for the (n)-th term of a geometric sequence?

    <p>T_n = ar^{n-1}</p> Signup and view all the answers

    What is the common ratio (r) of a geometric sequence if the second term is 12 and the first term is 3?

    <p>4</p> Signup and view all the answers

    Which of the following is NOT a characteristic of a geometric sequence?

    <p>The terms form a linear pattern when plotted on a graph.</p> Signup and view all the answers

    How do you test if a sequence is geometric?

    <p>Calculate the ratios between consecutive terms.</p> Signup and view all the answers

    What is the geometric mean between 4 and 9?

    <p>6</p> Signup and view all the answers

    What is the formula for the sum of the first (n) terms of a finite series?

    <p>S_n = T_1 + T_2 + T_3 + ... + T_n</p> Signup and view all the answers

    Which of the following best describes a finite series?

    <p>The sum of a specific number of terms in a sequence.</p> Signup and view all the answers

    What is the concise way to represent the sum of terms in a sequence?

    <p>Sigma notation</p> Signup and view all the answers

    What is the formula for the sum of an infinite geometric series, given that the common ratio (r) is less than 1?

    <p>S_ = a / (1 - r)</p> Signup and view all the answers

    What is the general form of the sigma notation for the sum of the first (n) terms of a sequence?

    <p>_(i=1)^n T_i</p> Signup and view all the answers

    What is the value of (S_{100}) in the arithmetic series: 1 + 2 + 3 + ... + 100?

    <p>5050</p> Signup and view all the answers

    What is the general formula for the sum of a finite arithmetic series, where (a) is the first term, (l) is the last term, and (n) is the number of terms?

    <p>(S_n = rac{n}{2}(a + l))</p> Signup and view all the answers

    Which of the following describes a relation but not a function?

    <p>A single element in set A can map to multiple elements in set B.</p> Signup and view all the answers

    Which of the following best describes a one-to-one function?

    <p>A function where every vertical line intersects the graph at most once.</p> Signup and view all the answers

    What is the key property that a function must have for its inverse to also be a function?

    <p>The function must be one-to-one.</p> Signup and view all the answers

    What is the inverse of the function (f(x) = 2x + 3)?

    <p>(f^{-1}(x) = rac{x - 3}{2})</p> Signup and view all the answers

    Which of the following is NOT a characteristic of a one-to-one function?

    <p>Every vertical line intersects the graph at most once.</p> Signup and view all the answers

    Which of the following describes a many-to-one function?

    <p>Multiple elements of the domain map to the same element of the range.</p> Signup and view all the answers

    What is the relationship between the graph of a function and the graph of its inverse?

    <p>They are reflections of each other across the line y = x.</p> Signup and view all the answers

    Given a function (f(x)), what does the notation (f^{-1}(x)) represent?

    <p>The inverse function of (f(x)).</p> Signup and view all the answers

    What is the purpose of the formula Tn = a + (n - 1)d?

    <p>To find the nth term of an arithmetic sequence.</p> Signup and view all the answers

    What is the result of plotting the terms of an arithmetic sequence against their positions?

    <p>A straight line</p> Signup and view all the answers

    What is the effect of a negative common difference on an arithmetic sequence?

    <p>The sequence decreases</p> Signup and view all the answers

    What do you need to find in order to determine if a sequence is arithmetic?

    <p>The differences between consecutive terms</p> Signup and view all the answers

    What is the arithmetic mean of two numbers?

    <p>The average of the two numbers</p> Signup and view all the answers

    What is the purpose of the steps to determine terms in an arithmetic sequence?

    <p>To find any term in the sequence</p> Signup and view all the answers

    What is the relationship between the common difference and the gradient of the line in the graphical representation of an arithmetic sequence?

    <p>The gradient is equal to the common difference</p> Signup and view all the answers

    How do you find the number of terms in an arithmetic sequence?

    <p>By setting the nth term equal to a specific value and solving for n</p> Signup and view all the answers

    What condition must the common ratio (r) satisfy for an infinite geometric series to converge?

    <p>-1 &lt; r &lt; 1</p> Signup and view all the answers

    Which formula is used to find the sum of the first (n) terms of a geometric series?

    <p>$S_n = rac{a(1 - r^n)}{1 - r}$</p> Signup and view all the answers

    What does a geometric sequence's term (T_n) equal when the common ratio (r) is 2 and the first term (a) is 3?

    <p>3 \cdot 2^{n-1}</p> Signup and view all the answers

    Which of the following describes the behavior of an infinite geometric series when (r \geq 1)?

    <p>The series diverges or sums to infinity.</p> Signup and view all the answers

    In the formula for the sum of a finite arithmetic series, what does the symbol (d) represent?

    <p>The constant difference between terms.</p> Signup and view all the answers

    How is the sum of the first 100 positive integers calculated using Karl Friedrich Gauss's method?

    <p>By pairing terms and simplifying to (2S_{100} = 101 \cdot 100)</p> Signup and view all the answers

    What is the formula to change the base of a logarithm?

    <p>log_a x = log_b x / log_b a</p> Signup and view all the answers

    What is the formula for the sum of an infinite geometric series when the common ratio (r) meets the convergence condition?

    <p>$S_\infty = rac{a}{1 - r}$</p> Signup and view all the answers

    What is the common difference (d) in the arithmetic sequence where the first term is 5 and the second term is 9?

    <p>4</p> Signup and view all the answers

    What is the graph of the inverse of an exponential function?

    <p>A curve that is a reflection about the line y = x</p> Signup and view all the answers

    What type of series sums to a finite value when the common ratio (r) is between -1 and 1?

    <p>Convergent Infinite Geometric Series</p> Signup and view all the answers

    What is the formula for calculating the accumulated amount in compound interest?

    <p>A = P(1 + i)^n</p> Signup and view all the answers

    What is the purpose of a future value annuity?

    <p>To accumulate a sum of money in the future</p> Signup and view all the answers

    What is the formula for solving for n in compound interest calculations?

    <p>n = log(A/P) / log(1 + i)</p> Signup and view all the answers

    What is the type of annuity that is used to pay off a loan over time?

    <p>Present Value Annuity (PVA)</p> Signup and view all the answers

    What is the inverse function of the linear function represented by the equation $y = ax + q$?

    <p>y = rac{1}{a}x - rac{q}{a}</p> Signup and view all the answers

    Which statement about the inverse of the function $y = ax^2$ is true if $a > 0$?

    <p>The domain of the inverse function is restricted to $x \geq 0$.</p> Signup and view all the answers

    What is the purpose of understanding the time value of money in annuities?

    <p>To understand how interest affects the accumulation or repayment of funds</p> Signup and view all the answers

    What is the formula for calculating the future value of an annuity?

    <p>FV = P * (((1 + i)^n - 1) / i) * (1 + i)</p> Signup and view all the answers

    What does the graph of the exponential function $f(x) = b^x$ look like when $b > 1$?

    <p>It increases and approaches positive infinity.</p> Signup and view all the answers

    What is the difference between the nominal and effective interest rates?

    <p>The nominal rate is the stated rate, while the effective rate is the actual rate</p> Signup and view all the answers

    What is the correct way to express the inverse of the exponential function $y = b^x$?

    <p>y = \log_b x</p> Signup and view all the answers

    What is the type of depreciation that uses a linear method?

    <p>Simple Depreciation</p> Signup and view all the answers

    Which property of inverse functions is true for both linear and quadratic functions?

    <p>The range of the original function becomes the domain of the inverse.</p> Signup and view all the answers

    What is the asymptote of the graph of the function $y = b^x$?

    <p>Horizontal asymptote at $y = 0$</p> Signup and view all the answers

    If the function $y = ax^2$ has a negative value for $a$, what is typically the restriction on the domain for the inverse function?

    <p>$x \leq 0$</p> Signup and view all the answers

    Which logarithmic property corresponds to $\log_a(xy)$?

    <p>The product rule</p> Signup and view all the answers

    Which value corresponds to $\log_a 1$ for any base $a$?

    <p>0</p> Signup and view all the answers

    What relationship exists between the domain and range of the logarithmic function $y = \log_b x$?

    <p>Domain is $x &gt; 0$ and range is $y \in \mathbb{R}$</p> Signup and view all the answers

    What is the value of the limit $\lim_{x \to -6} \frac{(x + 6)(x - 2)}{x + 6}$ ?

    <p>-8</p> Signup and view all the answers

    Which of the following is NOT a notation for the derivative of a function f(x)?

    <p>Df(x)</p> Signup and view all the answers

    What is the derivative of the function f(x) = x^3?

    <p>3x^2</p> Signup and view all the answers

    What is the derivative of the function g(x) = 5x^2 + 2x - 1?

    <p>10x + 2</p> Signup and view all the answers

    What is the derivative of the function h(x) = 4?

    <p>0</p> Signup and view all the answers

    What is the derivative of the function k(x) = 3x^4 - 2x^2 + 5x?

    <p>12x^3 - 4x + 5</p> Signup and view all the answers

    What is the derivative of the function p(x) = (x + 2)(x - 1)?

    <p>2x + 1</p> Signup and view all the answers

    What is the derivative of the function q(x) = 1/x?

    <p>-1/x^2</p> Signup and view all the answers

    What is the derivative of the function r(x) = √x?

    <p>1/(2√x)</p> Signup and view all the answers

    What is the derivative of the function s(x) = e^x?

    <p>e^x</p> Signup and view all the answers

    What does the variable ( FV ) represent in the future value annuity formula?

    <p>Future Value of the annuity</p> Signup and view all the answers

    In the present value of an annuity formula, what does the variable ( P ) indicate?

    <p>Payment amount per period</p> Signup and view all the answers

    How does the future value annuity formula account for interest?

    <p>By compounding interest on payments</p> Signup and view all the answers

    What is represented by the variable ( n ) in the formulas for present and future value of annuities?

    <p>Number of periods</p> Signup and view all the answers

    What would happen to the future value of an annuity if the interest rate increases, assuming all other factors remain constant?

    <p>It would increase</p> Signup and view all the answers

    Which of the following best describes the process of calculating the present value of an annuity?

    <p>Discounting future payments to find their current equivalent</p> Signup and view all the answers

    Which of the following statements is TRUE regarding mutually exclusive events?

    <p>The probability of the intersection of two mutually exclusive events is 0.</p> Signup and view all the answers

    What is the simplified form of the addition rule for probability when events A and B are mutually exclusive?

    <p>P(A or B) = P(A) + P(B)</p> Signup and view all the answers

    Which of the following is NOT a characteristic of complementary events?

    <p>Complementary events are always independent.</p> Signup and view all the answers

    The complementary rule in probability states that:

    <p>P(A) = 1 - P(A')</p> Signup and view all the answers

    Which of the following represents the intersection of events A and B?

    <p>A ∩ B</p> Signup and view all the answers

    What is the simplified form of the product rule for probability when events A and B are independent?

    <p>P(A and B) = P(A) * P(B)</p> Signup and view all the answers

    In the context of probability, what does the symbol 'S' represent?

    <p>Sample space</p> Signup and view all the answers

    What does the notation 'A ∪ B' represent in probability?

    <p>The union of events A and B</p> Signup and view all the answers

    Which of the following statements is TRUE regarding mutually exclusive and independent events?

    <p>Mutually exclusive and independent events are distinct concepts.</p> Signup and view all the answers

    What is the probability of event A' (the complement of event A) in terms of the probability of event A?

    <p>P(A') = 1 - P(A)</p> Signup and view all the answers

    What does the variable $n$ represent in the present value of an annuity formula?

    <p>Number of periods</p> Signup and view all the answers

    Which formula will you use to find the future value of a series of payments?

    <p>$F = x \left[\frac{(1 + i)^n - 1}{i}\right]$</p> Signup and view all the answers

    Which formula is used to calculate the effective annual rate (EAR)?

    <p>$\text{EAR} = \left(1 + \frac{i_{\text{nominal}}}{m}\right)^{m} - 1$</p> Signup and view all the answers

    What is the correct formula for calculating total interest paid on a loan?

    <p>$I = T - P$</p> Signup and view all the answers

    Which scenario best describes the use of the limits in calculus?

    <p>Solving for the rate of change of a variable</p> Signup and view all the answers

    What does the monthly payment amount $x$ represent in the present value formula?

    <p>Payment amount per period</p> Signup and view all the answers

    In the formula for calculating the number of periods $n$ using compound interest, what does $A$ stand for?

    <p>Total amount accumulated</p> Signup and view all the answers

    What is the formula to calculate a loan balance remaining?

    <p>$P_{ ext{balance}} = x \left[\frac{1 - (1 + i)^{-n_{ ext{remaining}}}}{i}\right]$</p> Signup and view all the answers

    What best describes the Achilles and the Tortoise paradox?

    <p>Achilles can never catch the tortoise in a race with a head start.</p> Signup and view all the answers

    How is simple interest calculated over time?

    <p>$A = P(1 + in)$</p> Signup and view all the answers

    What is the purpose of finding the stationary points of a cubic function?

    <p>To classify the function as a local maximum or local minimum</p> Signup and view all the answers

    What is the derivative of a function used to describe?

    <p>The rate of change of the original function</p> Signup and view all the answers

    What is the formula to find the quotient in synthetic division?

    <p>q2 = a3</p> Signup and view all the answers

    What is the notation for the second derivative of a function?

    <p>f''(x)</p> Signup and view all the answers

    What is the purpose of finding the x-intercepts of a cubic function?

    <p>To solve the polynomial equation</p> Signup and view all the answers

    What is the definition of a point of inflection?

    <p>A point where the graph changes concavity</p> Signup and view all the answers

    What is the relationship between the gradients of the tangent and the normal to a curve?

    <p>m_tangent × m_normal = -1</p> Signup and view all the answers

    What is the use of the second derivative in analyzing a function?

    <p>To identify stationary points on a graph</p> Signup and view all the answers

    What is the formula to find the remainder in synthetic division?

    <p>R = a0 + q0 * d/c</p> Signup and view all the answers

    What is the gradient of the tangent line to a curve at a point equal to?

    <p>The derivative of the function at that point</p> Signup and view all the answers

    What is the purpose of finding the y-intercept of a cubic function?

    <p>To sketch the graph of the function</p> Signup and view all the answers

    What is the effect of a negative coefficient a on the graph of a cubic function?

    <p>The graph falls to the right and rises to the left</p> Signup and view all the answers

    What is the definition of concave up?

    <p>The graph opens upwards</p> Signup and view all the answers

    How do you find the y-intercept of a cubic function?

    <p>Set x = 0 and solve for y</p> Signup and view all the answers

    What is the purpose of finding the derivative of a cubic function?

    <p>To find the stationary points of the function</p> Signup and view all the answers

    What is the definition of a local minimum?

    <p>A point where the function changes from decreasing to increasing</p> Signup and view all the answers

    What is the purpose of finding the equation of a tangent line to a curve?

    <p>To sketch the graph of the function</p> Signup and view all the answers

    What is the purpose of finding the second derivative of a cubic function?

    <p>To determine the concavity of the function</p> Signup and view all the answers

    What is the notation for the derivative of a function?

    <p>All of the above</p> Signup and view all the answers

    What is the purpose of the derivative in finding the equation of a tangent line?

    <p>To find the gradient of the line</p> Signup and view all the answers

    What mathematical principle determines the probability of independent events occurring simultaneously?

    <p>Product Rule</p> Signup and view all the answers

    What is the relationship between mutually exclusive events regarding their probabilities?

    <p>Their probabilities can be added together.</p> Signup and view all the answers

    What type of table is primarily used to explore the dependence or independence of two events?

    <p>Two-way contingency table</p> Signup and view all the answers

    In probability, what is the term for the set that includes all possible outcomes?

    <p>Sample space</p> Signup and view all the answers

    How is the probability of the complementary event of A expressed mathematically?

    <p>P(not A) = 1 - P(A)</p> Signup and view all the answers

    What does the symbol A ∩ B represent in probability?

    <p>Intersection of sets A and B</p> Signup and view all the answers

    Which of the following illustrates the Addition Rule for mutually exclusive events?

    <p>P(A or B) = P(A) + P(B)</p> Signup and view all the answers

    What is the probability calculation method used to determine outcomes along branches in a tree diagram?

    <p>Product of probabilities</p> Signup and view all the answers

    What does the Remainder Theorem state about the remainder when a polynomial is divided by a linear polynomial?

    <p>The remainder is equal to the evaluation of the polynomial at a specific point.</p> Signup and view all the answers

    Which statement correctly relates the Factor Theorem to the Remainder Theorem?

    <p>If the remainder is zero, then the corresponding factor exists.</p> Signup and view all the answers

    What is the outcome when applying the Factor Theorem to a polynomial at a root?

    <p>The polynomial will be equal to zero.</p> Signup and view all the answers

    In the context of cubic equations, what is the first step in solving them using factorization?

    <p>Identify a possible factor using the Factor Theorem.</p> Signup and view all the answers

    Which formula is used to express a polynomial after dividing it by a linear polynomial?

    <p>The polynomial can be expressed as product of its factor times quotient plus remainder.</p> Signup and view all the answers

    What does the quadratic formula allow you to find in relation to a quadratic polynomial?

    <p>The roots of the quadratic polynomial.</p> Signup and view all the answers

    What should be true for a divisor to be considered a factor of a polynomial according to the Factor Theorem?

    <p>The polynomial must evaluate to zero at the point derived from the divisor.</p> Signup and view all the answers

    Which of the following describes the relationship between the degree of the polynomial and the degree of the quotient when dividing by a linear polynomial?

    <p>The degree of the quotient is one less than that of the polynomial.</p> Signup and view all the answers

    What is the primary use of the Addition Rule in probability?

    <p>To calculate the sum of probabilities of disjoint events.</p> Signup and view all the answers

    When factoring a polynomial, what does finding a zero of the polynomial indicate?

    <p>The corresponding linear factor can divide the polynomial evenly.</p> Signup and view all the answers

    What is the formula for the sum of a finite arithmetic series?

    <p>$S_n = \frac{n}{2} (a + l)$</p> Signup and view all the answers

    What is the key property that a function must have for its inverse to also be a function?

    <p>It must be a one-to-one function</p> Signup and view all the answers

    What is the graphical representation of a one-to-one function?

    <p>Every horizontal line intersects the graph at most once</p> Signup and view all the answers

    What is the definition of a many-to-one function?

    <p>Multiple elements in set A map to the same element in set B</p> Signup and view all the answers

    What is the formula for the inverse of a linear function f(x) = ax + q?

    <p>f$^{-1}$(x) = x/a + q/a</p> Signup and view all the answers

    What is the definition of an inverse function?

    <p>A function that reverses the operation of a given function</p> Signup and view all the answers

    What is the graphical representation of the inverse of a function?

    <p>The graph of the inverse function is the reflection of the original function's graph across the line y = x</p> Signup and view all the answers

    What is the condition for a function to have an inverse that is also a function?

    <p>The function must be a one-to-one function</p> Signup and view all the answers

    What is the definition of a relation?

    <p>A rule that associates each element of one set with at least one element of another set</p> Signup and view all the answers

    What is the definition of a function?

    <p>A relation where each element in set A maps to exactly one element in set B</p> Signup and view all the answers

    What is the general formula for the nth term of a geometric sequence?

    <p>$T_n = ar^{n-1}$</p> Signup and view all the answers

    What is the common ratio (r) in the geometric sequence 2, 6, 18, 54, ...?

    <p>3</p> Signup and view all the answers

    What is the formula for the sum of the first n terms of an arithmetic sequence?

    <p>$S_n = n(a + l)/2$</p> Signup and view all the answers

    Which of the following is a characteristic of a geometric sequence when the common ratio (r) is negative?

    <p>The terms alternate in sign.</p> Signup and view all the answers

    What is the geometric mean between 4 and 9?

    <p>6</p> Signup and view all the answers

    What is the condition for an infinite geometric series to converge?

    <p>$-1 &lt; r &lt; 1$</p> Signup and view all the answers

    What is the formula for the sum of a finite geometric series?

    <p>$S_n = a(1 - r^n)/(1 - r)$</p> Signup and view all the answers

    Which of the following statements accurately describes the graphical representation of a geometric sequence?

    <p>It forms an exponential pattern.</p> Signup and view all the answers

    Which of the following is an example of a finite arithmetic series?

    <p>1 + 2 + 3 + ... + 100</p> Signup and view all the answers

    What is the formula for the sum of the first n terms of a geometric series?

    <p>$S_n = a(1 - r^n) / (1 - r)$</p> Signup and view all the answers

    What is the condition for an infinite geometric series to converge?

    <p>The common ratio (r) must be less than 1.</p> Signup and view all the answers

    What is the formula for the sum of an infinite geometric series?

    <p>$S_\infty = a/(1 - r)$</p> Signup and view all the answers

    What is the formula for the nth term of a geometric sequence?

    <p>$T_n = a \cdot r^{n-1}$</p> Signup and view all the answers

    What is the formula for the sum of an infinite geometric series that converges?

    <p>$S_\infty = a / (1 - r)$</p> Signup and view all the answers

    What is the method used by Karl Friedrich Gauss to find the sum of the first 100 integers?

    <p>Summing each pair of integers from both ends</p> Signup and view all the answers

    What is the sigma notation for the sum of the first 5 terms of the geometric sequence 3, 9, 27, 81, ...?

    <p>$\sum_{i=1}^{5} 3^{i-1}$</p> Signup and view all the answers

    Which of the following is NOT a characteristic of a series?

    <p>It is always a geometric sequence.</p> Signup and view all the answers

    What is the definition of a geometric sequence?

    <p>A sequence of numbers where each term is found by multiplying the previous term by a constant value</p> Signup and view all the answers

    What is the alternative formula for the sum of a finite geometric series when $r > 1$?

    <p>$S_n = a(r^n - 1)/(r - 1)$</p> Signup and view all the answers

    What is the sum of the first 100 positive integers?

    <p>5050</p> Signup and view all the answers

    What is the inverse of the function y = ax + q?

    <p>y = (1/a)x - (q/a)</p> Signup and view all the answers

    What is the domain of the inverse of y = ax^2?

    <p>x ≥ 0 if a &gt; 0, x ≤ 0 if a &lt; 0</p> Signup and view all the answers

    What is the definition of a logarithm?

    <p>The logarithm of a number x with a base b is the exponent y to which b must be raised to yield x.</p> Signup and view all the answers

    What is the shape of the graph of an exponential function?

    <p>Increasing or decreasing</p> Signup and view all the answers

    What is the product rule of logarithms?

    <p>log_a(xy) = log_a x + log_a y</p> Signup and view all the answers

    What is the inverse of y = b^x?

    <p>y = log_b x</p> Signup and view all the answers

    What is the asymptote of the graph of an exponential function?

    <p>Horizontal asymptote at y = 0</p> Signup and view all the answers

    What is log_a(a) equal to?

    <p>1</p> Signup and view all the answers

    What is the range of the graph of a logarithmic function?

    <p>y ∈ ℝ</p> Signup and view all the answers

    What is the formula to find the inverse of y = ax^2?

    <p>y = ±√(x/a)</p> Signup and view all the answers

    What does the variable $FV$ represent in the future value annuity formula?

    <p>Future Value of the annuity</p> Signup and view all the answers

    In the present value of an annuity formula, what does the variable $PV$ signify?

    <p>Present Value of the annuity</p> Signup and view all the answers

    Which component of the future value annuity formula primarily affects the accumulation of interest over time?

    <p>Number of periods</p> Signup and view all the answers

    How is the total value of a future value annuity calculated?

    <p>Sum of payments plus any accrued interest</p> Signup and view all the answers

    What is the role of the variable $i$ in the future and present value annuity formulas?

    <p>Interest rate per payment period</p> Signup and view all the answers

    Which formula would you use to calculate the present value required to achieve a series of future payments?

    <p>$PV = P \frac{1 - (1 + i)^{-n}}{i}$</p> Signup and view all the answers

    What is the purpose of a Future Value Annuity?

    <p>To accumulate a sum of money in the future by making regular deposits</p> Signup and view all the answers

    What is the formula to solve for n in compound interest calculations?

    <p>n = log(A / P) / log(1 + i)</p> Signup and view all the answers

    What is the type of interest calculated on the initial principal, including interest from previous periods?

    <p>Compound Interest</p> Signup and view all the answers

    What is the formula to calculate the pH level of a solution?

    <p>pH = -log[H+]</p> Signup and view all the answers

    What is the reflection of the original function about the line y = x in graphing?

    <p>The graph of the inverse</p> Signup and view all the answers

    What is the type of interest earned or paid on the principal alone over a period?

    <p>Simple Interest</p> Signup and view all the answers

    What is the formula to calculate the accumulated amount in compound interest?

    <p>A = P(1 + i)^n</p> Signup and view all the answers

    What is the purpose of an annuity?

    <p>To pay off a loan or debt over time with regular payments</p> Signup and view all the answers

    What is the type of depreciation that considers the reduced value in each period?

    <p>Compound Depreciation</p> Signup and view all the answers

    What is the formula to calculate the nominal and effective interest rates?

    <p>1 + i = (1 + i_m/m)^m</p> Signup and view all the answers

    What is the common difference of the arithmetic sequence: 2, 5, 8, 11, ...?

    <p>3</p> Signup and view all the answers

    If the first term of an arithmetic sequence is 7 and the common difference is -2, what is the 10th term?

    <p>-13</p> Signup and view all the answers

    Which of the following sequences is an arithmetic sequence?

    <p>1, 3, 5, 7, ...</p> Signup and view all the answers

    What is the arithmetic mean of 10 and 20?

    <p>15</p> Signup and view all the answers

    Given an arithmetic sequence with a first term of 3 and a common difference of 5, what is the value of the 6th term?

    <p>28</p> Signup and view all the answers

    If the common difference of an arithmetic sequence is negative, what can you say about the sequence?

    <p>The sequence is decreasing.</p> Signup and view all the answers

    The graphical representation of an arithmetic sequence is a:

    <p>Straight line</p> Signup and view all the answers

    What is the slope of the line representing an arithmetic sequence when plotted graphically?

    <p>Common difference</p> Signup and view all the answers

    What is the formula for the remainder when dividing polynomial (a(x)) by polynomial (b(x))?

    <p>a(x) - b(x) \cdot Q(x)</p> Signup and view all the answers

    What is the sign of the coefficient of the leading term of a cubic polynomial that opens downwards?

    <p>Negative</p> Signup and view all the answers

    What is the formula for (q_1) in synthetic division of (a(x) = a_3x^3 + a_2x^2 + a_1x + a_0) by (b(x) = cx - d)?

    <p>a_2 + q_2 \cdot rac{d}{c}</p> Signup and view all the answers

    How many turning points can a cubic polynomial have at most?

    <p>Two</p> Signup and view all the answers

    Which of the following correctly describes a local maximum point of a function?

    <p>A point where the function changes from increasing to decreasing.</p> Signup and view all the answers

    What is the x-intercept of the cubic polynomial (f(x) = x^3 - 2x^2 - x + 2)?

    <p>x = -1</p> Signup and view all the answers

    What is the derivative of (f(x) = ax^3 + bx^2 + cx + d)?

    <p>3ax^2 + 2bx + c</p> Signup and view all the answers

    What is the y-intercept of the cubic polynomial (f(x) = 2x^3 + 5x^2 - 3x + 1)?

    <p>(0, 1)</p> Signup and view all the answers

    Which of the following is a method for factorising cubic polynomials?

    <p>Both a and b</p> Signup and view all the answers

    What is the second derivative of the cubic polynomial (f(x) = ax^3 + bx^2 + cx + d)?

    <p>6ax + 2b</p> Signup and view all the answers

    Which of the following notations is NOT used to represent the second derivative of a function?

    <p>Df(x)</p> Signup and view all the answers

    What is the relationship between the gradients of the tangent and the normal to a curve at a given point?

    <p>The gradient of the tangent is the negative reciprocal of the gradient of the normal.</p> Signup and view all the answers

    What is the effect of the coefficient 'a' on the shape of the cubic function y = ax^3 + bx^2 + cx + d when a > 0?

    <p>The graph rises to the right and falls to the left.</p> Signup and view all the answers

    What is the y-intercept of the cubic function f(x) = ax^3 + bx^2 + cx + d?

    <p>d</p> Signup and view all the answers

    The derivative of a function describes which of the following?

    <p>The rate of change of the function</p> Signup and view all the answers

    What is the meaning of the second derivative of a function?

    <p>The rate of change of the gradient of the function</p> Signup and view all the answers

    Which of the following is a step in finding the equation of a tangent line to a function f(x) at x = a?

    <p>Calculate the derivative f'(a)</p> Signup and view all the answers

    Which of the following is the correct formula for the product rule of independent events?

    <p>P(A and B) = P(A) * P(B)</p> Signup and view all the answers

    What is the slope of the tangent line to the graph of a function at a given point?

    <p>The derivative of the function at that point</p> Signup and view all the answers

    If two events are mutually exclusive, what is the value of P(A and B)?

    <p>0</p> Signup and view all the answers

    What does the sign of the second derivative tell us about the original function?

    <p>Whether the gradient of the function is increasing, decreasing, or remaining constant</p> Signup and view all the answers

    Which of the following is the correct formula for the complementary rule?

    <p>P(not A) = 1 - P(A)</p> Signup and view all the answers

    What is the purpose of using differential operators like D or d/dx?

    <p>To indicate the operation of differentiation</p> Signup and view all the answers

    What does the symbol 'S' represent in probability?

    <p>The sample space, the set of all possible outcomes</p> Signup and view all the answers

    Which of the following is NOT a property of mutually exclusive events?

    <p>Their probabilities multiply to 0.</p> Signup and view all the answers

    What is the simplified addition rule for two mutually exclusive events?

    <p>P(A or B) = P(A) + P(B)</p> Signup and view all the answers

    What does the symbol 'A' represent in probability?

    <p>An event, a subset of the sample space</p> Signup and view all the answers

    If events A and B are independent, which of the following is true?

    <p>P(A and B) = P(A) * P(B)</p> Signup and view all the answers

    What is the definition of complementary events?

    <p>Events that are mutually exclusive and cover the entire sample space.</p> Signup and view all the answers

    What is the probability of an event not happening, represented by P(not A)?

    <p>1 - P(A)</p> Signup and view all the answers

    What is the limit of the function y = (x^2 + 4x - 12)/(x + 6) as x approaches -6?

    <p>-8</p> Signup and view all the answers

    What is the derivative of the function f(x) = x^2 using first principles?

    <p>2x</p> Signup and view all the answers

    What is the notation for the derivative of the function y = f(x)?

    <p>All of the above</p> Signup and view all the answers

    What is the general rule for differentiation?

    <p>d/dx [x^n] = nx^(n-1)</p> Signup and view all the answers

    What is the derivative of a constant?

    <p>0</p> Signup and view all the answers

    What is the derivative of a constant multiplied by a function?

    <p>k * d/dx [f(x)]</p> Signup and view all the answers

    What is the derivative of a sum of two functions?

    <p>d/dx [f(x) + g(x)] = d/dx [f(x)] + d/dx [g(x)]</p> Signup and view all the answers

    What is the formula for calculating the present value of an annuity?

    <p>$P = x \left[\frac{1 - (1 + i)^{-n}}{i}\right]$</p> Signup and view all the answers

    Which formula represents the future value of an annuity?

    <p>$F = x \left[\frac{(1 + i)^n - 1}{i}\right]$</p> Signup and view all the answers

    When should you use the rules for differentiation?

    <p>When the question does not specify how to determine the derivative</p> Signup and view all the answers

    What is the formula used to calculate the effective annual rate (EAR)?

    <p>$\text{EAR} = \left(1 + \frac{i_{nominal}}{m}\right)^m - 1$</p> Signup and view all the answers

    What is the notation for the limit of a function as x approaches a?

    <p>lim (x -&gt; a) f(x)</p> Signup and view all the answers

    What is the definition of the derivative of a function?

    <p>The derivative of a function is the gradient of the tangent to the curve at a point</p> Signup and view all the answers

    What is the formula for calculating the outstanding loan balance?

    <p>$P_{balance} = x \left[\frac{1 - (1 + i)^{-n_{remaining}}}{i}\right]$</p> Signup and view all the answers

    In the context of financial analysis, what does the variable 'x' typically represent?

    <p>The payment amount per period</p> Signup and view all the answers

    Which formula represents simple interest?

    <p>$A = P(1 + in)$</p> Signup and view all the answers

    Which formula represents compound interest?

    <p>$A = P(1 + i)^n$</p> Signup and view all the answers

    What is the formula for calculating the period of an investment using compound interest?

    <p>$n = \frac{\log(\frac{A}{P})}{\log(1 + i)}$</p> Signup and view all the answers

    What is the formula for calculating the payment amount for a future value annuity?

    <p>$x = \frac{F \cdot i}{(1 + i)^n - 1}$</p> Signup and view all the answers

    What is the formula for calculating the payment amount for a present value annuity?

    <p>$x = \frac{P \cdot i}{1 - (1 + i)^{-n}}$</p> Signup and view all the answers

    What is the formula to find the remainder when dividing a polynomial by a linear polynomial?

    <p>R = p igg( rac{d}{c} igg)</p> Signup and view all the answers

    Which statement is true regarding the Factor Theorem?

    <p>If p \left( \frac{d}{c} \right) = 0, then cx - d is a factor of p(x).</p> Signup and view all the answers

    What is the first step in solving a cubic equation using the Factor Theorem?

    <p>Identify a factor by substituting potential roots.</p> Signup and view all the answers

    Which of the following is the correct expression of a polynomial when it is divided by a linear polynomial?

    <p>p(x) = (cx - d) \cdot Q(x)</p> Signup and view all the answers

    How is the quotient polynomial related to the degree of the original polynomial in division?

    <p>It is a polynomial of one degree less than the original polynomial.</p> Signup and view all the answers

    What does the remainder represent when using the Remainder Theorem?

    <p>The value of the polynomial evaluated at a specific point.</p> Signup and view all the answers

    What is the result when p \left( \frac{d}{c} \right) is equal to zero?

    <p>The linear polynomial cx - d is a factor of p(x).</p> Signup and view all the answers

    Which method is used to solve for quadratic expressions once a cubic polynomial is factored?

    <p>The Quadratic Formula.</p> Signup and view all the answers

    What is the purpose of the Addition Rule in probability?

    <p>To calculate the probabilities of mutually exclusive events.</p> Signup and view all the answers

    In the Addition Rule, what does P(A or B) represent?

    <p>The probability that at least one of the events A or B occurs.</p> Signup and view all the answers

    What does the region inside the shape represent in probability events?

    <p>Outcomes included in the event</p> Signup and view all the answers

    How is the probability of a sequence of outcomes calculated using tree diagrams?

    <p>As the product of the probabilities along the branches of the sequence</p> Signup and view all the answers

    What type of events are characterized by not having any outcomes in common?

    <p>Mutually exclusive events</p> Signup and view all the answers

    In terms of probability, what does the complementary rule state?

    <p>P(not A) = 1 - P(A)</p> Signup and view all the answers

    Which of the following best describes a two-way contingency table?

    <p>A record keeping tool for counts or percentages in a probability problem</p> Signup and view all the answers

    What mathematical principle simplifies counting the number of outcomes in multiple sequences of events?

    <p>The Fundamental Counting Principle</p> Signup and view all the answers

    What formula represents the probability of either event A or event B occurring for mutually exclusive events?

    <p>P(A or B) = P(A) + P(B)</p> Signup and view all the answers

    In a probability context, what does the term 'independent events' refer to?

    <p>Events with no impact on the probability of each other</p> Signup and view all the answers

    What is the common difference in the arithmetic sequence 2, 5, 8, 11, ...?

    <p>3</p> Signup and view all the answers

    If the first term of an arithmetic sequence is 7 and the common difference is -3, what is the 5th term?

    <p>-5</p> Signup and view all the answers

    Which of the following sequences is an arithmetic sequence?

    <p>5, 10, 15, 20, ...</p> Signup and view all the answers

    What is the arithmetic mean between 12 and 20?

    <p>16</p> Signup and view all the answers

    If the common difference of an arithmetic sequence is negative, what happens to the sequence?

    <p>The sequence decreases</p> Signup and view all the answers

    What is the gradient of the line representing an arithmetic sequence when plotted on a graph?

    <p>The common difference</p> Signup and view all the answers

    What is the 10th term of the arithmetic sequence 3, 7, 11, 15, ...?

    <p>39</p> Signup and view all the answers

    If the first term of an arithmetic sequence is -5 and the common difference is 2, what is the formula for the nth term?

    <p>Tn = -5 + (n - 1)2</p> Signup and view all the answers

    What is the inverse of the function y = ax?

    <p>y = a/x</p> Signup and view all the answers

    What is the domain of the inverse of the function y = ax^2?

    <p>x ≥ 0 if a &gt; 0, x ≤ 0 if a &lt; 0</p> Signup and view all the answers

    What is the graph of the exponential function y = b^x when b > 1?

    <p>Rises rapidly</p> Signup and view all the answers

    What is the definition of a logarithm?

    <p>The power to which a base must be raised to yield a number</p> Signup and view all the answers

    What is the product rule of logarithms?

    <p>loga(xy) = loga(x) + loga(y)</p> Signup and view all the answers

    What is the intercept of the graph of a logarithmic function y = logb(x)?

    <p>(1, 0)</p> Signup and view all the answers

    What is the property of a function for its inverse to also be a function?

    <p>It must be one-to-one</p> Signup and view all the answers

    What is the shape of the graph of a logarithmic function?

    <p>Increasing</p> Signup and view all the answers

    What is the asymptote of the graph of an exponential function y = b^x?

    <p>Horizontal asymptote at y = 0</p> Signup and view all the answers

    What is the sum of the first 100 positive integers?

    <p>5050</p> Signup and view all the answers

    What is loga(a) equal to?

    <p>1</p> Signup and view all the answers

    Which of the following is NOT a characteristic of a one-to-one function?

    <p>Multiple elements in the domain can map to the same element in the range.</p> Signup and view all the answers

    Which of the following correctly describes the inverse of a linear function?

    <p>It is always a linear function.</p> Signup and view all the answers

    What is the relationship between the graphs of a function and its inverse?

    <p>They are reflections of each other across the line y=x.</p> Signup and view all the answers

    In the formula for the sum of a finite arithmetic series, what does the term 'l' represent?

    <p>The last term in the series.</p> Signup and view all the answers

    What is the general formula for the sum of a finite arithmetic series?

    <p>S_n = (n/2)(a + l)</p> Signup and view all the answers

    What is the condition for a function to have an inverse that is also a function?

    <p>The function must be one-to-one.</p> Signup and view all the answers

    What is the sum of the first 10 terms of the arithmetic sequence 2, 5, 8, 11, ...?

    <p>165</p> Signup and view all the answers

    In the formula S_n = (n/2)(a + l), what does 'n' represent?

    <p>The number of terms in the series.</p> Signup and view all the answers

    Which of the following is NOT a type of function?

    <p>One-to-Many</p> Signup and view all the answers

    What type of pattern do the terms of an arithmetic sequence form when plotted on a graph?

    <p>Linear pattern</p> Signup and view all the answers

    Which of the following describes how to find the common ratio in a geometric sequence?

    <p>Divide the second term by the first term</p> Signup and view all the answers

    If the common ratio of a geometric sequence is less than 1, what happens to the sequence?

    <p>It decays exponentially</p> Signup and view all the answers

    What is the value of the geometric mean between 4 and 16?

    <p>8</p> Signup and view all the answers

    What kind of graph is formed when plotting the terms of a geometric sequence against their position?

    <p>Exponential graph</p> Signup and view all the answers

    In the formula $T_n = ar^{n-1}$ for a geometric sequence, what does the variable $a$ represent?

    <p>The first term of the sequence</p> Signup and view all the answers

    What symbol is used in sigma notation to represent the summation of terms?

    <p>Σ</p> Signup and view all the answers

    What does the variable FV represent in the annuity formula?

    <p>Future Value of the annuity</p> Signup and view all the answers

    Which of the following is true regarding a finite series?

    <p>It sums a specific number of terms</p> Signup and view all the answers

    What must be true for an infinite geometric series to converge?

    <p>Its common ratio must be between -1 and 1</p> Signup and view all the answers

    What is the purpose of the present value of an annuity formula?

    <p>To determine the initial amount required to achieve a series of future payments</p> Signup and view all the answers

    What does the term $S_n$ represent in the context of series?

    <p>The sum of the first n terms</p> Signup and view all the answers

    What is the formula for the future value of an annuity?

    <p>FV = P * ((1 + i)^n - 1) / i</p> Signup and view all the answers

    What does the variable $r$ represent in a geometric sequence?

    <p>The common ratio</p> Signup and view all the answers

    What is the variable x in the future value annuity formula?

    <p>Payment amount per period</p> Signup and view all the answers

    Which condition must be met for an infinite geometric series to converge?

    <p>$-1 &lt; r &lt; 1$</p> Signup and view all the answers

    What is the main difference between a future value annuity and a present value annuity?

    <p>The direction of the payment</p> Signup and view all the answers

    In the context of a finite geometric series, what does the variable $n$ represent?

    <p>The number of terms</p> Signup and view all the answers

    What is the formula for the present value of an annuity?

    <p>PV = P * (1 - (1 + i)^(-n)) / i</p> Signup and view all the answers

    What is the formula to calculate the sum of the first $n$ terms in a finite geometric series?

    <p>$S_n = rac{a(1 - r^n)}{r - 1}$</p> Signup and view all the answers

    What is the first term of an arithmetic sequence if given the common difference $d = 3$ and the 5th term $T_5 = 15$?

    <p>9</p> Signup and view all the answers

    Which of the following sequences is an example of a divergent infinite geometric series?

    <p>A series with $r = 2$</p> Signup and view all the answers

    How do you calculate the sum $S_{100}$ of the first 100 integers using Gauss's method?

    <p>Pair the first and last term, then multiply by the number of pairs</p> Signup and view all the answers

    What is the sum of the first 10 terms of a geometric series where $a = 2$ and $r = 3$?

    <p>$2(1 - 3^{10})/(1 - 3)$</p> Signup and view all the answers

    If the common difference $d$ of an arithmetic series is negative, what can be inferred about the sequence?

    <p>The terms will decrease as $n$ increases</p> Signup and view all the answers

    What is the resulting sum if the first term of a finite arithmetic series is 5, the common difference is 2, and the last term is 20?

    <p>50</p> Signup and view all the answers

    What does the variable x represent in the present value of an annuity formula?

    <p>Payment amount per period</p> Signup and view all the answers

    What is the formula for the future value of an annuity?

    <p>F = x[(1 + i)^n - 1]/i</p> Signup and view all the answers

    What is the formula for the effective annual rate (EAR)?

    <p>EAR = (1 + i_nominal/m)^m - 1</p> Signup and view all the answers

    What is the formula for simple interest?

    <p>A = P(1 + in)</p> Signup and view all the answers

    What is the formula for compound interest?

    <p>A = P(1 + i)^n</p> Signup and view all the answers

    What is the formula for the present value of an annuity?

    <p>P = x[(1 + i)^-n - 1]/i</p> Signup and view all the answers

    What is the formula for the outstanding loan balance?

    <p>P_balance = x[(1 + i)^-n_remaining - 1]/i</p> Signup and view all the answers

    What is the formula for the total interest paid?

    <p>I = T - P</p> Signup and view all the answers

    What is the formula for the total amount paid?

    <p>T = n * x</p> Signup and view all the answers

    What is the formula for the future value of a series of payments?

    <p>F = x[(1 + i)^n - 1]/i</p> Signup and view all the answers

    What is the purpose of the concept of limits in mathematics?

    <p>To provide a foundation for calculus</p> Signup and view all the answers

    What is the graphical representation of the function y = (x + 6)(x - 2)/(x + 6)?

    <p>A straight line with a hole at x = -6</p> Signup and view all the answers

    What is the notation for the derivative of a function f(x)?

    <p>All of the above</p> Signup and view all the answers

    What is the formula for the derivative of a function f(x) using the definition of a derivative?

    <p>f'(x) = lim(h 0) [f(x + h) - f(x)]/h</p> Signup and view all the answers

    What is the general rule for differentiating a function of the form x^n?

    <p>d/dx [x^n] = nx^(n-1)</p> Signup and view all the answers

    What is the derivative of a constant?

    <p>0</p> Signup and view all the answers

    What is the rule for differentiating a function of the form k * f(x)?

    <p>d/dx [k * f(x)] = k * d/dx [f(x)]</p> Signup and view all the answers

    What is the rule for differentiating a function of the form f(x) + g(x)?

    <p>d/dx [f(x) + g(x)] = d/dx [f(x)] + d/dx [g(x)]</p> Signup and view all the answers

    When should you use the rules for differentiation?

    <p>When the question does not specify how to determine the derivative</p> Signup and view all the answers

    What is the purpose of notation in differentiation?

    <p>To indicate the operation of differentiation</p> Signup and view all the answers

    What is the derivative of (y = f(x)) with respect to (x)?

    <p>All of the above</p> Signup and view all the answers

    What is the relationship between the gradients of the tangent and the normal to a curve at a given point?

    <p>The product of the gradients is -1.</p> Signup and view all the answers

    What does the second derivative of a function indicate?

    <p>The rate of change of the gradient of the function.</p> Signup and view all the answers

    What is the correct notation for the second derivative of (y) with respect to (x)?

    <p>All of the above</p> Signup and view all the answers

    What is the effect of the coefficient (a) on the shape of the cubic function (y = ax^3 + bx^2 + cx + d)?

    <p>It determines the direction of the graph as (x) approaches infinity.</p> Signup and view all the answers

    What is the y-intercept of the cubic function (f(x) = ax^3 + bx^2 + cx + d)?

    <p>[ d ]</p> Signup and view all the answers

    What is the slope of the tangent line to the graph of (f(x)) at (x = a)?

    <p>[ f'(a) ]</p> Signup and view all the answers

    Which of the following is NOT a notation for the second derivative of (f(x))?

    <p>[ \frac{d}{dx}[f(x)] ]</p> Signup and view all the answers

    What is the gradient of the tangent line to the curve (y = x^2) at the point ((2, 4))?

    <p>[ 4 ]</p> Signup and view all the answers

    What is the equation of the tangent line to the curve (y = x^3) at the point ((1, 1))?

    <p>[ y = 3x - 1 ]</p> Signup and view all the answers

    What is a stationary point on a function?

    <p>A point where the derivative is zero.</p> Signup and view all the answers

    How is a local maximum identified in a cubic function?

    <p>When the function changes from increasing to decreasing.</p> Signup and view all the answers

    What does it mean if a function is concave down?

    <p>The graph opens downwards.</p> Signup and view all the answers

    What is a point of inflection?

    <p>A point where the function changes concavity.</p> Signup and view all the answers

    Which method can be used for dividing polynomials?

    <p>Synthetic Division.</p> Signup and view all the answers

    In the context of cubic equations, what does finding the x-intercepts require?

    <p>Solving the equation $f(x) = 0$.</p> Signup and view all the answers

    What does the term 'end behavior' refer to in cubic polynomials?

    <p>How the graph behaves as $x$ approaches positive or negative infinity.</p> Signup and view all the answers

    What is the first step to sketching a cubic graph?

    <p>Determine the y-intercept.</p> Signup and view all the answers

    How do you classify stationary points in the context of cubic functions?

    <p>Using the sign of the first derivative.</p> Signup and view all the answers

    What indicates that a cubic function is concave up?

    <p>The second derivative is positive.</p> Signup and view all the answers

    What does the Remainder Theorem allow you to determine when dividing a polynomial by a linear polynomial?

    <p>The remainder value from the function evaluated at a specific point</p> Signup and view all the answers

    Under what condition does the Factor Theorem indicate that a polynomial has a factor?

    <p>If the remainder when divided is zero</p> Signup and view all the answers

    What is the first step in solving a cubic equation using the Factor Theorem?

    <p>Identify a potential factor through substitution</p> Signup and view all the answers

    How can a polynomial be expressed after finding a factor based on the Factor Theorem?

    <p>As a product of a linear polynomial and a quadratic polynomial</p> Signup and view all the answers

    What relationship does the addition rule in probability define?

    <p>The probabilities of the events and their union</p> Signup and view all the answers

    What does the region inside the shape represent in a probability context?

    <p>Outcomes included in the event</p> Signup and view all the answers

    When dividing a polynomial by a linear polynomial, what degree is the quotient polynomial?

    <p>One degree less than the original polynomial</p> Signup and view all the answers

    What can be inferred if a polynomial evaluated at a certain point equals zero?

    <p>The specific linear divisor is a factor of the polynomial</p> Signup and view all the answers

    How is the probability of a sequence of outcomes calculated in tree diagrams?

    <p>As the product of the probabilities along the branches</p> Signup and view all the answers

    What is a characteristic of mutually exclusive events?

    <p>They cannot happen simultaneously</p> Signup and view all the answers

    What is the form of a polynomial expressed after performing polynomial long division?

    <p>The divisor multiplied by the quotient plus the remainder</p> Signup and view all the answers

    What result indicates that a divisor is not a factor of the polynomial when using polynomial division?

    <p>The remainder is non-zero</p> Signup and view all the answers

    What does the complementary rule state?

    <p>The probability of not A is $1 - P(A)$</p> Signup and view all the answers

    Which formula represents the remainder when dividing a polynomial by a linear polynomial?

    <p>R = p(d / c)</p> Signup and view all the answers

    How can two-way contingency tables be particularly useful in probability?

    <p>They assist in determining if events are dependent or independent</p> Signup and view all the answers

    What is the correct application of the addition rule for mutually exclusive events?

    <p>$P(A ext{ or } B) = P(A) + P(B)$</p> Signup and view all the answers

    According to the product rule for independent events, how is the probability of two independent events occurring together expressed?

    <p>$P(A ext{ and } B) = P(A) imes P(B)$</p> Signup and view all the answers

    What does the sample space S encompass in probability?

    <p>All possible outcomes of a probability experiment</p> Signup and view all the answers

    What is the probability of two mutually exclusive events A and B?

    <p>P(A) + P(B)</p> Signup and view all the answers

    What is the complementary rule in probability?

    <p>P(not A) = 1 - P(A)</p> Signup and view all the answers

    What is the product rule for independent events?

    <p>P(A and B) = P(A) × P(B)</p> Signup and view all the answers

    What is the condition for two events to be mutually exclusive?

    <p>P(A and B) = 0</p> Signup and view all the answers

    What is the purpose of a Venn diagram in probability?

    <p>To show the relationship between events</p> Signup and view all the answers

    What is the symbol for the sample space in probability?

    <p>S</p> Signup and view all the answers

    What is the union of two events represented by?

    <p>A ∪ B</p> Signup and view all the answers

    What is the intersection of two events represented by?

    <p>A ∩ B</p> Signup and view all the answers

    What is the probability of the complement of an event?

    <p>1 - P(A)</p> Signup and view all the answers

    What is the condition for events to be independent?

    <p>P(A and B) = P(A) × P(B)</p> Signup and view all the answers

    What is the formula for determining the period of an investment using compound interest?

    <p>$n = \frac{\log(\frac{A}{P})}{\log(1 + i)}$</p> Signup and view all the answers

    Which of the following describes the intercept of the exponential function $f(x) = 10^x$?

    <p>(0, 1)</p> Signup and view all the answers

    What does the variable $A$ represent in the simple interest formula $A = P(1 + in)$?

    <p>Future value accumulated</p> Signup and view all the answers

    In the context of logarithmic functions, what is the asymptote of the function $f^{-1}(x) = \log x$?

    <p>$x = 0$</p> Signup and view all the answers

    Which formula would you use to find the future value of an annuity?

    <p>$A = P \times \frac{(1 + i)^n - 1}{i}$</p> Signup and view all the answers

    What type of annuity is used primarily for accumulating a sum of money in the future?

    <p>Future Value Annuity</p> Signup and view all the answers

    Which statement about the range of the exponential function $f(x) = 10^x$ is correct?

    <p>$y \geq 0$</p> Signup and view all the answers

    What does the formula $\text{pH} = -\log_{10}[ ext{H}^+]$ represent?

    <p>Acidity of a solution</p> Signup and view all the answers

    Which of the following is a distinguishing feature of compound interest compared to simple interest?

    <p>Interest accrues on the principal and previously earned interest</p> Signup and view all the answers

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