Gr12 Mathematics: June Exam Mix P(1)
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Questions and Answers

What is the process of determining the derivative of a given function called?

  • Differentiation from first principles (correct)
  • First principle of differentiation
  • Derivative definition
  • Gradient calculation

What is the definition of the derivative of a function f(x)?

  • lim(h → 0) [f(x) + f(x + h)]/h
  • lim(h → 0) [f(x + h) - f(x)]/h (correct)
  • lim(h → 0) [f(x) - f(x + h)]/h
  • lim(h → 0) [f(x + h) + f(x)]/h

What is the general rule for differentiation?

  • d/dx [x^n] = nx^(n+1), where n ∈ ℝ and n ≠ 0
  • d/dx [x^n] = nx^(n-1), where n ∈ ℝ and n ≠ 0 (correct)
  • d/dx [x^n] = nx^(2n-1), where n ∈ ℝ and n ≠ 0
  • d/dx [x^n] = nx^n, where n ∈ ℝ and n ≠ 0

What is the derivative of a constant?

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

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

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

What is the derivative of a sum?

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

What is the derivative of a difference?

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

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

<p>f'(x) = y' = dy/dx = df/dx = d/dx[f(x)] = Df(x) = D_xy (A)</p> Signup and view all the answers

What does the symbol d/dx mean?

<p>y differentiated with respect to x (A)</p> Signup and view all the answers

Why is the equation of a tangent to a curve important?

<p>It helps to find the gradient of a tangent to a curve at a point (C)</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 (A)</p> Signup and view all the answers

What is the simplified form of the function y = (x^2 + 4x - 12)/(x + 6) for x ≠ -6?

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

What is the concept illustrated by Achilles and the Tortoise paradox?

<p>The concept of limits (D)</p> Signup and view all the answers

What is the graphical representation of the function y = (x^2 + 4x - 12)/(x + 6)?

<p>A straight line with a hole at x = -6 (D)</p> Signup and view all the answers

What is the value of y = (x^2 + 4x - 12)/(x + 6) when x = -6?

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

What is the main focus of differential calculus?

<p>Solving optimization problems (D)</p> Signup and view all the answers

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

<p>To determine the concavity of a function (C)</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>m_tangent × m_normal = -1 (C)</p> Signup and view all the answers

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

<p>To identify the local maximum and local minimum of a function (D)</p> Signup and view all the answers

How do you find the y-intercept of a cubic function f(x) = ax^3 + bx^2 + cx + d?

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

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

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

What is the derivative of a function used for?

<p>To find the gradient of the equation of a tangent line (B)</p> Signup and view all the answers

What is concavity, in the context of graphs?

<p>The rate of change of the gradient of a curve (A)</p> Signup and view all the answers

What is the purpose of finding the normal to a curve?

<p>To find the line perpendicular to the tangent line (C)</p> Signup and view all the answers

What is the notation for the second derivative of a function f(x)?

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

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

<p>To find the gradient of a curve at a given point (D)</p> Signup and view all the answers

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

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

What is the common characteristic of an arithmetic sequence?

<p>The sequence forms a linear pattern when plotted on a graph (B)</p> Signup and view all the answers

How do you find the geometric mean between two numbers a and b?

<p>Geometric Mean = √(ab) (B)</p> Signup and view all the answers

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

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

What is a series in mathematics?

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

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

<p>S(n) = T1 + T2 + ... + Tn (C)</p> Signup and view all the answers

What is an infinite series?

<p>The sum of infinitely many terms of a sequence (A)</p> Signup and view all the answers

How do you test if a sequence is geometric?

<p>Verify if the ratio between consecutive terms is constant (B)</p> Signup and view all the answers

What is the general form of sigma notation?

<p>∑(i=m to n) Ti = Tm + Tm+1 + ... + Tn-1 + Tn (B)</p> Signup and view all the answers

What happens when you plot the terms of a geometric sequence?

<p>The points form an exponential curve (A)</p> Signup and view all the answers

What is the condition for cx - d to be a factor of p(x) according to the Factor Theorem?

<p>p(d/c) = 0 (B)</p> Signup and view all the answers

What is the formula for the n-th term of an arithmetic sequence?

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

What is the purpose of the Quadratic Formula in solving cubic equations?

<p>To find the quadratic polynomial (A)</p> Signup and view all the answers

What is the common difference in an arithmetic sequence?

<p>The difference between consecutive terms (C)</p> Signup and view all the answers

What is the graphical representation of an arithmetic sequence?

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

How do you test if a sequence is arithmetic?

<p>By verifying if the difference between consecutive terms is constant (A)</p> Signup and view all the answers

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

<p>Find a factor by trial and error (B)</p> Signup and view all the answers

What is the relationship between the roots of a polynomial and its factors?

<p>A factor is a root of the polynomial (A)</p> Signup and view all the answers

What is the arithmetic mean of two numbers?

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

What is the purpose of the Factor Theorem in solving cubic equations?

<p>To factorize the polynomial (A)</p> Signup and view all the answers

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

<p>$\sum$ (sigma) (D)</p> Signup and view all the answers

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

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

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

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

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

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

What is the definition of an arithmetic sequence?

<p>A sequence where each term is added by a constant value (B)</p> Signup and view all the answers

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

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

Who is credited with a method for finding the sum of an arithmetic series?

<p>Karl Friedrich Gauss (A)</p> Signup and view all the answers

What is the sum of the first n terms of a sequence denoted as?

<p>$S_n$ (A)</p> Signup and view all the answers

What is the term used to describe the constant value in an arithmetic sequence?

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

What is the term used to describe the constant value in a geometric sequence?

<p>Common ratio (D)</p> Signup and view all the answers

What is the general formula for a finite arithmetic series?

<p>$S_n = \frac{n}{2} (2a + (n - 1) d)$ (D)</p> Signup and view all the answers

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

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

What is the general form of the inverse of y = ax^2?

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

What is the definition of an exponent?

<p>The number of times a base number is multiplied by itself (C)</p> Signup and view all the answers

What is the inverse of the function y = b^x?

<p>y = log_b(x) (B)</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 a given value (D)</p> Signup and view all the answers

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

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

What is the domain of the logarithmic function y = log_b(x)?

<p>All positive real numbers (B)</p> Signup and view all the answers

What is the result of applying the product rule of logarithms to log_a(xy)?

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

What is the result of applying the power rule of logarithms to log_a(x^b)?

<p>b × log_a(x) (D)</p> Signup and view all the answers

What is the change of base formula for logarithms?

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

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

<p>A reflection of the original function about the line y = x (C)</p> Signup and view all the answers

What is the domain of the logarithmic function f(x) = log x?

<p>x &gt; 0 (B)</p> Signup and view all the answers

What is an application of logarithms in real-life scenarios?

<p>Calculating the pH levels of a solution (C)</p> Signup and view all the answers

What is the asymptote of the exponential function f(x) = 10^x?

<p>y = 0 (B)</p> Signup and view all the answers

What is the formula to calculate the population growth, given a constant rate?

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

What is a point of inflection?

<p>The point where the graph changes concavity. (D)</p> Signup and view all the answers

Which method is NOT typically used for dividing polynomials?

<p>Polynomial Interpolation (D)</p> Signup and view all the answers

Which equation represents the division of a polynomial by another polynomial?

<p>a(x) = b(x) * Q(x) + R(x) (C)</p> Signup and view all the answers

To find the x-intercepts of a cubic polynomial, which equation should be solved?

<p>f(x) = 0 (B)</p> Signup and view all the answers

What does the Remainder Theorem state regarding a polynomial and a linear divisor?

<p>The remainder is found by evaluating the polynomial at the root of the divisor. (B)</p> Signup and view all the answers

Which of the following measures the instantaneous rate of change of a function?

<p>The derivative (B)</p> Signup and view all the answers

In cubic polynomial functions, the nature of the leading coefficient 'a' determines which characteristic?

<p>The end behavior of the graph (D)</p> Signup and view all the answers

What is the role of stationary points in the context of graph sketching?

<p>They help determine local maxima and minima. (A)</p> Signup and view all the answers

Which formula can be used for finding the coefficient of a cubic polynomial through synthetic division?

<p>R = a_0 + q_0 imes d/c (B)</p> Signup and view all the answers

Why is the concept of limits essential in calculus?

<p>To understand the behavior of functions as the input values approach a specific point (A)</p> Signup and view all the answers

What does the graph of the function y = (x^2 + 4x - 12)/(x + 6) represent?

<p>A straight line with a hole at x = -6 (A)</p> Signup and view all the answers

What is the significance of Zeno's paradox involving Achilles and the tortoise?

<p>It highlights the concept of limits in calculus (A)</p> Signup and view all the answers

Why is the function y = (x^2 + 4x - 12)/(x + 6) not defined at x = -6?

<p>Because the denominator is zero at x = -6 (A)</p> Signup and view all the answers

What is the main focus of differential calculus?

<p>To optimize functions and find rates of change (D)</p> Signup and view all the answers

What do we examine when finding the limit of a function?

<p>The behavior of the function as the input values approach a specific point (C)</p> Signup and view all the answers

What is the condition for cx - d to be a factor of p(x) according to the Factor Theorem?

<p>p(d/c) = 0 (A)</p> Signup and view all the answers

What is the purpose of the Quadratic Formula in solving cubic equations?

<p>To find the roots of a quadratic equation (C)</p> Signup and view all the answers

What is the general form of an arithmetic sequence?

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

What is the relationship between the roots of a polynomial and its factors?

<p>If a polynomial has a root, then it must have a corresponding factor (D)</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 trial and error (A)</p> Signup and view all the answers

What is the purpose of the Factor Theorem in solving cubic equations?

<p>To find one factor of a cubic polynomial (B)</p> Signup and view all the answers

What is the graphical representation of an arithmetic sequence?

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

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

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

How do you test if a sequence is arithmetic?

<p>By finding the common difference (A)</p> Signup and view all the answers

What is the common characteristic of a geometric sequence?

<p>Each term is multiplied by a constant value (D)</p> Signup and view all the answers

What is the common characteristic of an arithmetic sequence?

<p>Each term is the sum of the previous term and a constant (D)</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) (B)</p> Signup and view all the answers

What is the purpose of sigma notation?

<p>To represent the sum of terms in a sequence (B)</p> Signup and view all the answers

What is the arithmetic mean of two numbers?

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

What is the general form of sigma notation?

<p>∑(i=m to n) Ti = Tm + T(m+1) + ... + T(n-1) + Tn (A)</p> Signup and view all the answers

What is the difference between a finite series and an infinite series?

<p>A finite series has a fixed number of terms, while an infinite series has an infinite number of terms (B)</p> Signup and view all the answers

What is the graphical representation of a geometric sequence?

<p>An exponential graph (A)</p> Signup and view all the answers

How do you test if a sequence is geometric?

<p>By checking if the ratio between consecutive terms is constant (C)</p> Signup and view all the answers

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

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

What is the importance of finding the common ratio in a geometric sequence?

<p>To find the nth term of the sequence (C)</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 of the tangent and the normal is -1. (A)</p> Signup and view all the answers

What is the formula for finding the remainder (R) when a polynomial (p(x)) is divided by (cx - d)?

<p>(R = p(d/c)) (D)</p> Signup and view all the answers

Which of the following steps is NOT involved in drawing the graph of a cubic polynomial?

<p>Find the points of inflection by solving (f'(x) = 0) (B)</p> Signup and view all the answers

What is the purpose of using synthetic division in factorising cubic polynomials?

<p>To find the x-intercepts of the polynomial. (B)</p> Signup and view all the answers

What does the term "point of inflection" represent in a cubic graph?

<p>The point where the graph changes concavity. (C)</p> Signup and view all the answers

Which of the following methods is NOT used for factorising cubic polynomials?

<p>Quadratic formula (D)</p> Signup and view all the answers

What is the condition for (cx - d) to be a factor of (p(x)) according to the Factor Theorem?

<p>(p(d/c) = 0) (C)</p> Signup and view all the answers

What does the sign of (a) in the cubic polynomial (f(x) = ax^3 + bx^2 + cx + d) tell us about the graph?

<p>The end behavior of the graph. (B)</p> Signup and view all the answers

If the first derivative of a cubic function is zero at (x = 2), what can you conclude?

<p>The function has a turning point at (x = 2). (C)</p> Signup and view all the answers

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

<p>(q_2 = a_3, q_1 = a_2 + q_2 \cdot (d/c), q_0 = a_1 + q_1 \cdot (d/c), R = a_0 + q_0 \cdot (d/c)) (C)</p> Signup and view all the answers

Which of the following is NOT a method for solving cubic equations?

<p>Quadratic Formula (C)</p> Signup and view all the answers

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

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

What is the definition of the logarithm of a number x with base b?

<p>The exponent to which b must be raised to yield x (A)</p> Signup and view all the answers

What is the shape of the graph of the logarithmic function y = log_b x?

<p>Increasing and concave down (C)</p> Signup and view all the answers

What is the product rule of logarithms?

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

What is the inverse of the function y = b^x?

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

What is the range of the exponential function y = b^x?

<p>y &gt; 0 (A)</p> Signup and view all the answers

What is the change of base formula for logarithms?

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

What is the value of log_a 1?

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

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

<p>A curve that increases rapidly (B)</p> Signup and view all the answers

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

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

Which of the following is NOT a requirement for the sum of an infinite geometric series to exist (i.e., to converge)?

<p>The common ratio (r) must be greater than 0. (A), The first term (a) must be zero. (E)</p> Signup and view all the answers

Which of the following sequences is NOT an arithmetic sequence?

<p>4, 8, 12, 16, 20, ... (B)</p> Signup and view all the answers

What is the sum of the first 5 terms of the geometric series (2 + 6 + 18 + 54 + \cdots)?

<p>242 (D)</p> Signup and view all the answers

In the general form of sigma notation, what does the index of summation represent?

<p>The position of each term in the sequence. (A)</p> Signup and view all the answers

What is the formula for the nth term of the arithmetic sequence (1, 4, 7, 10, 13, ...)?

<p>(T_n = 1 + 3(n - 1)) (A)</p> Signup and view all the answers

If an infinite geometric series converges, what can we say about its common ratio (r)?

<p>(|r| &lt; 1) (A)</p> Signup and view all the answers

Which of the following is the correct formula for the sum of the first (n) terms of an arithmetic series?

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

Which of these options best describes the use of Karl Friedrich Gauss's method for calculating the sum of the first 100 positive integers?

<p>It involves pairing the integers and finding their average. (D)</p> Signup and view all the answers

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

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

What does the derivative of a function indicate about the function at a specific point?

<p>The rate of change of the function's values. (C)</p> Signup and view all the answers

Which of the following notations represents the derivative of a function f(x)?

<p>D_xf (C), D_xy (D)</p> Signup and view all the answers

Which statement is true regarding the differentiation rules?

<p>The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. (A)</p> Signup and view all the answers

What is the limit expression that defines the derivative of a function?

<p>lim (f(a + h) - f(a)) / h as h approaches 0 (D)</p> Signup and view all the answers

When should one use differentiation from first principles?

<p>When derivatives are explicitly requested. (A)</p> Signup and view all the answers

Which of the following correctly applies the derivative of a constant multiplied by a function?

<p>If k is a constant and f(x) is a function, then d(kf(x))/dx = kf'(x). (B)</p> Signup and view all the answers

In the context of differentiation, what does the term 'gradient function' refer to?

<p>The slope of a curve at a specific point. (C)</p> Signup and view all the answers

What is represented by the notation $rac{dy}{dx}$?

<p>The derivative of y with respect to x. (B)</p> Signup and view all the answers

Which of the following is NOT a concept related to differentiation?

<p>Calculating the area under a curve. (A)</p> Signup and view all the answers

What establishes the relationship between the gradient of a curve and the gradient of its tangent?

<p>The gradient of the tangent equals the derivative of the curve at that point. (B)</p> Signup and view all the answers

If a population doubles in size after 5 years, what is the annual growth rate (rounded to the nearest percent)?

<p>14% (B)</p> Signup and view all the answers

What is the pH level of a solution with a hydrogen ion concentration of ( 10^{-7} ) moles per liter?

<p>7 (C)</p> Signup and view all the answers

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

<p>( f^{-1}(x) = \log_2 x ) (D)</p> Signup and view all the answers

Which of the following statements is NOT true about the graph of ( f(x) = \log x )?

<p>The graph is symmetric about the y-axis. (B)</p> Signup and view all the answers

A radioactive substance decays to half its original amount in 10 years. What is the decay rate (rounded to the nearest percent)?

<p>6.9% (A)</p> Signup and view all the answers

What is the formula for the sum of the first n terms of an arithmetic series when the last term is known?

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

Which statement is true about the properties of functions and relations?

<p>Functions strictly require one output for each input. (B)</p> Signup and view all the answers

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

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

To find the inverse of a linear function defined by $y = ax + q$, which of the following steps is first?

<p>Interchange x and y. (A)</p> Signup and view all the answers

In the context of inverse functions, what does the notation $f^{-1}(x)$ represent?

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

If a function fails the horizontal line test, what can be concluded about its inverse?

<p>The inverse cannot be uniquely defined. (D)</p> Signup and view all the answers

What does the derived formula for the sum of the first n terms of an arithmetic series, $S_n = \frac{n}{2}(2a + (n - 1)d)$, represent?

<p>The sum of terms in an arithmetic sequence. (D)</p> Signup and view all the answers

What is the significance of interchanging x and y when finding the inverse of a function?

<p>It allows the output to become the new input. (D)</p> Signup and view all the answers

How is a one-to-one function graphically represented?

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

Which of the following statements is true concerning many-to-one functions?

<p>Every element in the range is connected to at least one element of the domain. (C)</p> Signup and view all the answers

What happens to the function $y = \frac{x^2 + 4x - 12}{x + 6}$ as $x$ approaches -6?

<p>The function approaches -8. (B)</p> Signup and view all the answers

What concept does the Achilles and the Tortoise paradox illustrate?

<p>The concept of limits in calculus. (B)</p> Signup and view all the answers

Which statement is true regarding the function $y = \frac{x^2 + 4x - 12}{x + 6}$?

<p>It has a hole at $x = -6$. (C)</p> Signup and view all the answers

What is the simplified form of the function when $x \neq -6$?

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

What is the graphical representation of $y = \frac{x^2 + 4x - 12}{x + 6}$?

<p>A straight line with a hole at $x = -6$. (D)</p> Signup and view all the answers

What is the key focus of differential calculus as introduced?

<p>The optimization of functions and rates of change. (B)</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>They are multiplicative inverses. (B)</p> Signup and view all the answers

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

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

If a cubic function has a positive leading coefficient (a > 0), what is the general shape of its graph?

<p>Rises to the right and falls to the left. (D)</p> Signup and view all the answers

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

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

How do you find the y-intercept of a cubic function f(x) = ax^3 + bx^2 + cx + d?

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

What is the relationship between the derivative of a function and the tangent line to its graph at a given point?

<p>The derivative is the slope of the tangent line. (C)</p> Signup and view all the answers

If the second derivative of a function is positive, what does this indicate about the graph of the original function?

<p>The graph is concave up. (A)</p> Signup and view all the answers

Which of the following is a step involved in finding the equation of a tangent line to a curve at a given point?

<p>Calculate the y-value at the given point. (A)</p> Signup and view all the answers

What is the purpose of finding the normal to a curve?

<p>To find a line perpendicular to the tangent at a given point. (B)</p> Signup and view all the answers

Which of the following is NOT a use of the derivative?

<p>Determining the area under a curve. (A)</p> Signup and view all the answers

What indicates a point of inflection in a function's graph?

<p>Where the second derivative changes from positive to negative or vice versa (C)</p> Signup and view all the answers

Which method is NOT typically used for factorizing cubic polynomials?

<p>Completing the square (D)</p> Signup and view all the answers

How can the y-intercept of a cubic function be determined?

<p>By solving the equation for x = 0 (B)</p> Signup and view all the answers

What is the remainder when dividing a polynomial p(x) by a linear polynomial cx - d using the Remainder Theorem?

<p>The value of p at $x = rac{d}{c}$ (D)</p> Signup and view all the answers

Which aspect is NOT determined when sketching a cubic graph?

<p>The absolute maximum value (D)</p> Signup and view all the answers

What do the stationary points of a function indicate?

<p>Points where the derivative is zero (A)</p> Signup and view all the answers

Which property is essential for a polynomial's quotient when dividing by a linear polynomial?

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

Which of the following methods can be used to determine points of inflection?

<p>Setting the second derivative to zero and checking for sign change (A)</p> Signup and view all the answers

What is indicated by the shape of the cubic graph when the coefficient a is negative?

<p>The graph opens downwards (C)</p> Signup and view all the answers

Which statement about the average and instantaneous rates of change is true?

<p>Instantaneous rate of change is the derivative at a point (C)</p> Signup and view all the answers

What confirms that a polynomial ( p(x) ) has a factor ( cx - d )?

<p>( p\left( \frac{d}{c} \right) = 0 ) (C)</p> Signup and view all the answers

Which of the following describes the first step to solve cubic equations using the Factor Theorem?

<p>Substitute potential roots into the polynomial. (A)</p> Signup and view all the answers

How can a polynomial ( p(x) ) be expressed once a factor ( cx - d ) is found?

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

What is the general formula for finding the n-th term ( T_n ) of an arithmetic sequence?

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

What indicates that a sequence is arithmetic?

<p>The difference between consecutive terms is constant. (D)</p> Signup and view all the answers

What is necessary to use the Quadratic Formula on a cubic polynomial after the factor is found?

<p>Obtain a quadratic polynomial by dividing the cubic by the factor. (D)</p> Signup and view all the answers

Which formula is used to calculate the arithmetic mean of two numbers?

<p>( ext{Arithmetic Mean} = \frac{ ext{First Term} + ext{Second Term}}{2} ) (C)</p> Signup and view all the answers

What is the outcome when you substitute ( x = \frac{d}{c} ) into a polynomial ( p(x) ) where ( cx - d ) is a factor?

<p>The result is zero. (D)</p> Signup and view all the answers

What does the gradient of the line representing an arithmetic sequence indicate?

<p>It represents the common difference of the sequence. (A)</p> Signup and view all the answers

What is the range of the exponential function defined by $f(x) = 10^x$?

<p>y &gt; 0 (B)</p> Signup and view all the answers

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

<p>x &gt; 0 (A)</p> Signup and view all the answers

In the formula for population growth, what does the variable 'n' represent?

<p>Number of growth periods (B)</p> Signup and view all the answers

What is the y-intercept of the exponential function $f(x) = 10^x$?

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

For a logarithmic function, what is the asymptote?

<p>x = 0 (D)</p> Signup and view all the answers

How is the common ratio in a geometric sequence determined?

<p>By calculating the ratio between any two consecutive terms. (C)</p> Signup and view all the answers

What characteristic does a geometric sequence exhibit when the common ratio is greater than 1?

<p>The sequence grows exponentially. (C)</p> Signup and view all the answers

What is the formula for finding the $n$-th term of an arithmetic sequence?

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

Which of the following sums describes a finite series?

<p>$S_n = T_1 + T_2 + ext{up to } T_n$ (C)</p> Signup and view all the answers

When plotted on a graph, what does a geometric sequence represent?

<p>An exponential curve. (C)</p> Signup and view all the answers

What happens to the terms of a geometric sequence when the common ratio is negative?

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

Which expression represents the geometric mean of two numbers $a$ and $b$?

<p>$ ext{Geometric Mean} = ext{sqrt}(ab)$ (A)</p> Signup and view all the answers

In sigma notation, what does the symbol $ ext{Σ}$ represent?

<p>The summation of a sequence of terms. (A)</p> Signup and view all the answers

To verify if a sequence is arithmetic, what should be checked?

<p>If the difference between all consecutive terms is constant. (A)</p> Signup and view all the answers

What is depicted when plotting the terms of an arithmetic sequence?

<p>A straight line with a slope. (C)</p> Signup and view all the answers

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

<p>6x + 2 (B)</p> Signup and view all the answers

What is the derivative of the function f(x) = 5?

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

What is the derivative of the function f(x) = 4x^3 - 2x^2 + 7x?

<p>12x^2 - 4x + 7 (B)</p> Signup and view all the answers

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

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

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

<p>3x^2 - 4x + 1 (A)</p> Signup and view all the answers

What is the derivative of the function f(x) = \frac{1}{x^2}?

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

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

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

What is the equation of the tangent to the curve y = x^3 - 2x + 1 at the point (1, 0)?

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

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

<p>The derivative represents the instantaneous rate of change of the function at that point. (A)</p> Signup and view all the answers

What is the derivative of the function f(x) = \sqrt{x}?

<p>1/(2\sqrt{x}) (B)</p> Signup and view all the answers

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

<p>$S_n = rac{n}{2}(2a + (n - 1)d)$ (B), $S_n = rac{n}{2}(a + l)$ (D)</p> Signup and view all the answers

In a one-to-one function, which of the following statements is true?

<p>Each element in the domain maps to a unique element in the range. (C)</p> Signup and view all the answers

Which of the following is necessary for a function to have an inverse that is also a function?

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

How do you find the inverse of a linear function given as $y = ax + q$?

<p>Interchange x and y, then solve for y: $x = ay + q$. (A)</p> Signup and view all the answers

What is true about the graph of an inverse function?

<p>It is the reflection of the original function's graph across the line $y = x$. (B)</p> Signup and view all the answers

For the equation $f(x) = 2x + 3$, what is its inverse function?

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

Which of the following statements about arithmetic series is true?

<p>The sum can be found using $S_n = rac{n}{2}(a + l)$. (B)</p> Signup and view all the answers

What is represented by the notation $f^{-1}(x)$?

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

In which scenario does a horizontal line test indicate that a function is one-to-one?

<p>If the horizontal line intersects the graph at most once. (B)</p> Signup and view all the answers

What is the correct expression for the inverse of the linear function given by the equation $y = ax + q$?

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

What is the restriction on the domain when finding the inverse of the quadratic function $y = ax^2$?

<p>$x &lt; 0$ if $a &gt; 0$ (D)</p> Signup and view all the answers

Which statement is true regarding the graph of the exponential function $f(x) = b^x$ when $b > 1$?

<p>The graph is increasing and always remains above the x-axis. (B)</p> Signup and view all the answers

If $y = b^x$, what is the correct form for expressing its inverse?

<p>$y = ext{log}_b x$ (A)</p> Signup and view all the answers

For the logarithmic function $y = ext{log}_b x$, which of the following is true regarding its domain?

<p>$x ext{ must be greater than zero}$ (C)</p> Signup and view all the answers

What does the product rule of logarithms state?

<p>$ ext{log}_a(xy) = ext{log}_a x + ext{log}_a y$ (A)</p> Signup and view all the answers

What is the shape of the graph of a logarithmic function $y = ext{log}_b x$?

<p>It is increasing and approaches negative infinity as x approaches zero. (A)</p> Signup and view all the answers

If $x = b^y$, which of the following describes the relationship of $b$, $x$, and $y$?

<p>$x$ is the output of raising $b$ to the power of $y$. (C)</p> Signup and view all the answers

What is the range of the exponential function $f(x) = b^x$ when $b > 0$?

<p>$(0, ext{inf})$ (D)</p> Signup and view all the answers

What is the formula for the sum of the first n terms of a finite geometric series where the common ratio (r) is greater than 1?

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

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

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

What is the general formula for a finite arithmetic series, where a is the first term, d is the common difference, and n is the number of terms?

<p>$S_n = \frac{n}{2} [a + (n - 1)d]$ (A)</p> Signup and view all the answers

What is the formula for the nth term of a geometric sequence with the first term 'a' and the common ratio 'r'?

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

What is the sum of the first 100 natural numbers, using the method of Gauss?

<p>5050 (A)</p> Signup and view all the answers

What does the symbol 'Σ' represent in sigma notation?

<p>The sum of terms in a sequence (A)</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>$\sum_{i=1}^{n} T_i$ (C)</p> Signup and view all the answers

What is the common ratio (r) in a geometric sequence where the second term is 6 and the fifth term is 48?

<p>2 (A)</p> Signup and view all the answers

Which of the following statements correctly describes the convergence of an infinite geometric series?

<p>An infinite geometric series converges if the absolute value of the common ratio (r) is less than 1. (D)</p> Signup and view all the answers

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

<p>$2x$ (C)</p> Signup and view all the answers

If y = f(x) and f'(x) = nx^(n-1), what is the value of n?

<p>any real number (C)</p> Signup and view all the answers

What is the derivative of the function f(x) = k, where k is a constant?

<p>$0$ (D)</p> Signup and view all the answers

What is the equation of the tangent to the curve y = x^2 at the point (1, 1)?

<p>y = 2x - 1 (A)</p> Signup and view all the answers

If f'(x) = 2x, what is the function f(x)?

<p>$x^2 + C$ (C)</p> Signup and view all the answers

What is the derivative of the function f(x) = kf(x), where k is a constant?

<p>$kf'(x)$ (B)</p> Signup and view all the answers

What is the derivative of the function f(x) = f(x) + g(x)?

<p>$f'(x) + g'(x)$ (C)</p> Signup and view all the answers

What is the derivative of the function f(x) = f(x) - g(x)?

<p>$f'(x) - g'(x)$ (B)</p> Signup and view all the answers

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

<p>all of the above (D)</p> Signup and view all the answers

What is the process of finding the derivative of a function from first principles?

<p>differentiation from first principles (D)</p> Signup and view all the answers

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

<p>To determine the concavity of the function (C)</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>Their product is equal to -1 (B)</p> Signup and view all the answers

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

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

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

<p>To identify local maxima and minima (B)</p> Signup and view all the answers

What is the notation for the second derivative of a function f(x)?

<p>d^2y/dx^2 (A), f''(x) (D)</p> Signup and view all the answers

What is concavity, in the context of graphs?

<p>The rate of change of the gradient (A)</p> Signup and view all the answers

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

<p>To describe the rate of change at a specific point (A)</p> Signup and view all the answers

What is the process of finding the equation of a tangent line to a curve at a point?

<p>Differentiate the function, substitute the x-coordinate, and use the point-slope form (A)</p> Signup and view all the answers

What is the purpose of finding the second derivative of a function in the context of sketching cubic graphs?

<p>To determine the concavity of the graph (A)</p> Signup and view all the answers

What is the formula for the remainder when dividing a polynomial p(x) by cx - d?

<p>R = p(d/c) (C)</p> Signup and view all the answers

What is the relationship between the second derivative and the concavity of a function?

<p>If f''(x) &gt; 0, the graph is concave up (B), If f''(x) &lt; 0, the graph is concave down (D)</p> Signup and view all the answers

What is the purpose of finding the normal to a curve?

<p>To find the perpendicular line to the curve at a specific point (C)</p> Signup and view all the answers

What is the condition for cx - d to be a factor of p(x) according to the Factor Theorem?

<p>p(d/c) = 0 (D)</p> Signup and view all the answers

What is the purpose of synthetic division in factorising cubic polynomials?

<p>To find the remainder of the division (B)</p> Signup and view all the answers

What is the general form of a cubic polynomial?

<p>ax^3 + bx^2 + cx + d (C)</p> Signup and view all the answers

What is the concept of optimisation problems in the context of differential calculus?

<p>Finding the maximum or minimum of a function (C)</p> Signup and view all the answers

What is the purpose of finding the turning points of a cubic graph?

<p>To determine the shape of the graph (D)</p> Signup and view all the answers

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

<p>They are perpendicular (A)</p> Signup and view all the answers

What is the purpose of finding the point of inflection of a cubic graph?

<p>To determine the concavity of the graph (D)</p> Signup and view all the answers

What is the application of differential calculus in solving optimisation problems?

<p>Finding the maximum or minimum of a function (A)</p> Signup and view all the answers

If (p(x) = x^3 - 6x^2 + 11x - 6) and (cx - d = x - 1), what can be concluded about (p(x))?

<p>(x - 1) is a factor of (p(x)) (C)</p> Signup and view all the answers

What is the relationship between the factor (cx - d) and the root of the polynomial (p(x))?

<p>(cx - d) is a factor of (p(x)) if and only if (p(d/c) = 0) (C)</p> Signup and view all the answers

If a polynomial (p(x)) has a root at (x = rac{d}{c}), what can be concluded about the factorization of (p(x))?

<p>(p(x) = (cx - d) \cdot Q(x)) where (Q(x)) is a quadratic polynomial (D)</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 of the polynomial by trial and error (D)</p> Signup and view all the answers

What is the graphical representation of an arithmetic sequence?

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

What is the common characteristic of an arithmetic sequence?

<p>The difference between consecutive terms is constant (D)</p> Signup and view all the answers

What is the purpose of the Factor Theorem in solving cubic equations?

<p>To find the factors of the polynomial (D)</p> Signup and view all the answers

If a polynomial (p(x)) has a factor (cx - d), what can be concluded about the root of the polynomial?

<p>The root is (x = d/c) (D)</p> Signup and view all the answers

What is the relationship between the roots of a polynomial and its factors?

<p>A polynomial has a root at (x = a) if and only if (x - a) is a factor (D)</p> Signup and view all the answers

What is the common difference in an arithmetic sequence?

<p>The difference between consecutive terms (A)</p> Signup and view all the answers

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

<p>$S_n = rac{n}{2}(a + l)$ (B), $S_n = rac{n}{2}(2a + (n - 1)d)$ (D)</p> Signup and view all the answers

What is a necessary condition for a function to have an inverse function?

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

What is the graph of the inverse function like, compared to the graph of the original function?

<p>It is a reflection across the line y = x. (A)</p> Signup and view all the answers

What is the process of finding the inverse of a linear function?

<p>Interchange x and y, solve for x, and then substitute back into the original function. (C)</p> Signup and view all the answers

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

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

What is the purpose of the horizontal line test in determining if a function has an inverse?

<p>To determine if the function is one-to-one. (D)</p> Signup and view all the answers

What is the key property of an inverse function?

<p>It undoes the operation of the original function. (D)</p> Signup and view all the answers

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

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

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

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

What is the notation for the inverse of a function f(x)?

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

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

<p>They are symmetrical about the line y = x. (C)</p> Signup and view all the answers

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

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

What is the definition of an infinite geometric series?

<p>A series with an infinite number of terms (A)</p> Signup and view all the answers

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

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

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

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

What is the definition of a finite geometric series?

<p>A series with a finite number of terms (D)</p> Signup and view all the answers

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

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

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

<p>$\sum_{i=m}^{n} T_i$ (C)</p> Signup and view all the answers

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

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

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

<p>Domain: ℝ, Range: ℝ+ (B)</p> Signup and view all the answers

What is the purpose of Karl Friedrich Gauss's method for finding the sum of an arithmetic series?

<p>To find the sum of an arithmetic series (A)</p> Signup and view all the answers

What is the logarithmic function equivalent to the exponential function y = 2^x?

<p>y = log₂x (C)</p> Signup and view all the answers

What is the property of logarithms that states loga(xy) = loga(x) + loga(y)?

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

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

<p>It rises rapidly if b &gt; 1 and falls rapidly if 0 &lt; b &lt; 1 (D)</p> Signup and view all the answers

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

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

What is the logarithmic value of loga(a) equal to?

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

What is the purpose of finding the inverse of a function?

<p>To solve equations involving the original function (A)</p> Signup and view all the answers

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

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

What is the special logarithmic value of loga(1) equal to?

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

A population of bacteria doubles every 3 hours. Assuming exponential growth, how long will it take for the population to increase 16 times its initial size?

<p>48 hours (D)</p> Signup and view all the answers

A radioactive substance has a half-life of 10 days. After 30 days, what fraction of the original substance remains?

<p>1/8 (D)</p> Signup and view all the answers

The pH of a solution is 4. What is the concentration of hydrogen ions, [H+], in the solution?

<p>10^-4 M (A)</p> Signup and view all the answers

A loan of $10,000 is taken out at an annual interest rate of 5%, compounded monthly. If the loan is to be repaid in 5 years, what is the monthly payment amount (rounded to the nearest dollar)?

<p>$189 (C)</p> Signup and view all the answers

If the graph of an exponential function (f(x) = a^x) passes through the point ((2, 9)), what is the value of (a)?

<p>3 (D)</p> Signup and view all the answers

What happens to the sequence when the common ratio, r, in a geometric sequence is less than 1 but greater than 0?

<p>The sequence decays exponentially. (A)</p> Signup and view all the answers

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

<p>An exponential curve. (D)</p> Signup and view all the answers

How can you verify that a sequence is arithmetic?

<p>The differences between consecutive terms are constant. (B)</p> Signup and view all the answers

What is the purpose of sigma notation in mathematics?

<p>To denote the sum of a series of terms in a compact form. (C)</p> Signup and view all the answers

What characterizes the growth behavior of a geometric sequence when the common ratio is negative?

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

In the context of finite series, what does the notation S_n specifically represent?

<p>The sum of the first n terms of a sequence. (C)</p> Signup and view all the answers

Given the formula for the nth term of a geometric sequence, T_n = ar^{n-1}, how would you identify the first term?

<p>It is represented by the symbol a. (A)</p> Signup and view all the answers

What is the geometric mean of two numbers a and b, as defined in mathematics?

<p>The value that forms a geometric sequence between a and b. (B)</p> Signup and view all the answers

How would you define an infinite series in mathematical terms?

<p>A series that sums all terms indefinitely. (C)</p> Signup and view all the answers

What condition must the common ratio r satisfy for a geometric series to converge?

<p>r must be less than 1 in absolute value. (A)</p> Signup and view all the answers

The function y = (x^2 + 4x - 12)/(x + 6) has a discontinuity at x = -6. What is the nature of this discontinuity?

<p>Removable discontinuity, because the function can be simplified to a continuous function for x ≠ -6 (D)</p> Signup and view all the answers

Which of the following statements accurately describes the concept of limits in the context of the Achilles and the Tortoise paradox?

<p>The paradox shows that even in situations where one object is constantly gaining on another, the limit of their distance can still be zero. (C)</p> Signup and view all the answers

The graph of the function y = (x^2 + 4x - 12)/(x + 6) is a straight line with a hole at x = -6. What does the existence of this hole indicate about the limit of the function as x approaches -6?

<p>The limit exists and is equal to -8, even though the function is not defined at x = -6. (C)</p> Signup and view all the answers

Which of the following statements about the limit of the function y = (x^2 + 4x - 12)/(x + 6) as x approaches -6 is TRUE?

<p>The limit exists and is equal to -8, because as x approaches -6, the function approaches -8. (B)</p> Signup and view all the answers

If the tortoise gets a head start in the race, Achilles appears to never overtake it. This is because:

<p>The paradox highlights the concept of infinite divisibility and the fact that Achilles must repeatedly cover smaller and smaller distances to catch up. (C)</p> Signup and view all the answers

Why is the concept of limits important in calculus?

<p>Limits are used to find the derivative of a function. (B)</p> Signup and view all the answers

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