Gr 11 Math Nov P1 (Medium)
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Questions and Answers

What is the value of r in the equation r^n = a, where n ≥ 2?

  • √(n/a)
  • a^(1/2)
  • √[n](a) (correct)
  • a^(1/n)

What is the value of a^(-1/n)?

  • 1/(a^(1/n))
  • a^(-n)
  • 1/(√[n](a)) (correct)
  • √[n](1/a)

What is the value of a^(m/n)?

  • a^(m/n)
  • a^(m+n)
  • √[n](a^m) (correct)
  • a^m / a^n

What is the purpose of rationalising the denominator?

<p>To eliminate the surd from the denominator (D)</p> Signup and view all the answers

What is the result of multiplying √n and √n?

<p>√<a href="ab">n</a> (B)</p> Signup and view all the answers

What is the result of dividing √n by √n?

<p>√<a href="a/b">n</a> (B)</p> Signup and view all the answers

What is the result of raising √n to the power of m?

<p>√<a href="a%5Em">n</a> (C)</p> Signup and view all the answers

What is the condition for the radicand in a radical symbol when n is even?

<p>The radicand must be positive (D)</p> Signup and view all the answers

What is the rule for negative exponents?

<p>a^(-m) = 1/a^m (A)</p> Signup and view all the answers

What is the purpose of rationalizing denominators?

<p>To eliminate radicals from the denominator (A)</p> Signup and view all the answers

What is the formula for exponential growth or decay?

<p>P(t) = P_0 × (1 + r)^t (D)</p> Signup and view all the answers

What is the main goal when solving surd equations?

<p>To isolate the surd on one side of the equation (D)</p> Signup and view all the answers

What is the power of a power rule?

<p>(a^m)^n = a^(mn) (D)</p> Signup and view all the answers

What is the purpose of writing surds in exponential notation?

<p>To make manipulation easier (D)</p> Signup and view all the answers

What is the formula for fractional exponents?

<p>a^(m/n) = √(a^m) (C)</p> Signup and view all the answers

What is the final step when solving surd equations?

<p>To check the solution (C)</p> Signup and view all the answers

What is the key consideration when raising both sides of the equation to the power to eliminate the surd?

<p>To be cautious of potential extraneous solutions (D)</p> Signup and view all the answers

What is the exponential growth model used for?

<p>To model population growth and decay (B)</p> Signup and view all the answers

What is the first step in identifying the variables in an exponential growth or decay problem?

<p>Determine the initial amount, growth rate, and time period (B)</p> Signup and view all the answers

What is the formula for converting a square root to an exponent?

<p>$a^{rac{1}{2}} = \sqrt{a}$ (A)</p> Signup and view all the answers

What is the set of natural numbers represented as?

<p>$\mathbb{N}$ (A)</p> Signup and view all the answers

What is the formula for solving a quadratic equation using the completing the square method?

<p>$x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ (B)</p> Signup and view all the answers

What is the purpose of completing the square in quadratic equations?

<p>To rewrite the equation as a perfect square trinomial (D)</p> Signup and view all the answers

What is the formula for an exponential growth or decay problem?

<p>$P = P_0 imes (1 + r)^t$ (C)</p> Signup and view all the answers

What is the first step in solving a quadratic equation using the quadratic formula?

<p>Write the equation in standard form (D)</p> Signup and view all the answers

What is the definition of $a^0$?

<p>$a^0 = 1$ for $a eq 0$ (B)</p> Signup and view all the answers

What is the formula for the quadratic formula derived from?

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

What is the purpose of the quadratic formula?

<p>To solve quadratic equations (C)</p> Signup and view all the answers

What is the formula for determining the roots of a quadratic equation?

<p>x = -b ± √(b^2 - 4ac) / 2a (B)</p> Signup and view all the answers

What condition is required for a quadratic equation to have two distinct real roots?

<p>b^2 - 4ac &gt; 0 (A)</p> Signup and view all the answers

What is the purpose of making a substitution when solving a quadratic equation?

<p>To simplify the equation (D)</p> Signup and view all the answers

What is the first step in finding the quadratic equation given the roots?

<p>Write the roots as two equations (A)</p> Signup and view all the answers

What is the discriminant of a quadratic equation?

<p>b^2 - 4ac (C)</p> Signup and view all the answers

What type of roots does a quadratic equation have if the discriminant is less than zero?

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

What is the condition for a quadratic equation to have exactly one real root?

<p>b^2 - 4ac = 0 (B)</p> Signup and view all the answers

What is the purpose of identifying restrictions in the substitution method?

<p>To ensure the expression is not undefined (B)</p> Signup and view all the answers

What is the final step in solving a quadratic equation by substitution?

<p>Write the final answer (D)</p> Signup and view all the answers

What type of roots does a quadratic equation have if the discriminant is a perfect square?

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

What does a discriminant value of Δ < 0 indicate about the roots of a quadratic equation?

<p>The roots are imaginary. (C)</p> Signup and view all the answers

What is the formula for calculating the discriminant of a quadratic equation in standard form?

<p>Δ = b^2 - 4ac (C)</p> Signup and view all the answers

What is the purpose of creating a table of signs when solving quadratic inequalities?

<p>To determine the sign of each factor on intervals defined by the critical values. (C)</p> Signup and view all the answers

What is the general method for solving quadratic inequalities?

<p>Writing the quadratic inequality in standard form and then factorizing if possible. (D)</p> Signup and view all the answers

What is the purpose of determining critical values when solving quadratic inequalities?

<p>To solve the quadratic equation and find the roots. (B)</p> Signup and view all the answers

What is the result of solving a quadratic equation when the discriminant is Δ = 0?

<p>The equation has one real root. (A)</p> Signup and view all the answers

What is the graphical representation of a quadratic inequality?

<p>The graph of the quadratic function intersects the x-axis at the critical values. (B)</p> Signup and view all the answers

What is the purpose of analyzing the sign of each factor on intervals defined by the critical values?

<p>To determine the regions where the quadratic expression is positive or negative. (C)</p> Signup and view all the answers

What is the main difference between solving quadratic equations and quadratic inequalities?

<p>Quadratic equations are used to determine the roots, while quadratic inequalities are used to determine the regions where the expression is positive or negative. (B)</p> Signup and view all the answers

What is the general method for solving systems of simultaneous equations?

<p>Using the substitution method or elimination method. (D)</p> Signup and view all the answers

What is the effect of a on the shape of the graph of f(x) = a(x + p)^2 + q?

<p>It affects the shape of the graph, making it a 'smile' for a &gt; 0 and a 'frown' for a &lt; 0 (B)</p> Signup and view all the answers

What is the domain of the function f(x) = a(x + p)^2 + q?

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

What is the range of the function f(x) = a(x + p)^2 + q when a > 0?

<p>All real numbers greater than or equal to q (C)</p> Signup and view all the answers

What is the equation of the axis of symmetry for the function f(x) = a(x + p)^2 + q?

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

What is the formula to find the y-intercept of the function f(x) = a(x + p)^2 + q?

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

How do you shift the equation of a parabola 3 units to the right?

<p>Replace x with x - 3 (B)</p> Signup and view all the answers

What is the formula to find the average gradient between two points on a curve?

<p>Gradient = (y_A - y_C) / (x_A - x_C) (C)</p> Signup and view all the answers

What is the effect of p on the graph of f(x) = a(x + p)^2 + q?

<p>It causes a horizontal shift (D)</p> Signup and view all the answers

What is the effect on the graph of the function y = ab^(x+p) + q if p > 0?

<p>The graph is shifted to the left by p units. (D)</p> Signup and view all the answers

What is the turning point of the function f(x) = a(x + p)^2 + q?

<p>(-p, q) (C)</p> Signup and view all the answers

What is the formula to find the x-intercept of the function f(x) = a(x + p)^2 + q?

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

What is the range of the function y = ab^(x+p) + q if a > 0?

<p>All real numbers greater than q. (B)</p> Signup and view all the answers

What is the formula for the depreciated value using simple depreciation?

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

What is the effect of compound depreciation on the depreciation amount?

<p>The depreciation amount decreases every year. (C)</p> Signup and view all the answers

Why do we use timelines in finance?

<p>To visualize the compounding periods and changes in interest rates over time. (B)</p> Signup and view all the answers

What happens to the interest rate when amounts are compounded more than once per annum?

<p>The interest rate is divided by p. (B)</p> Signup and view all the answers

What is the effect of a negative value of b on the function f(x)?

<p>The function f(x) is not defined. (A)</p> Signup and view all the answers

What is the horizontal asymptote of the function y = ab^(x+p) + q?

<p>The line y = q. (A)</p> Signup and view all the answers

What is the average gradient of a curve between points P(a, f(a)) and Q(a + h, f(a + h))?

<p>(f(a + h) - f(a)) / h (A)</p> Signup and view all the answers

What is the main purpose of solving systems of linear equations?

<p>To model real-world problems in various fields (B)</p> Signup and view all the answers

What is the effect of q > 0 on the graph of y = a/x + q?

<p>The graph is shifted vertically upwards by q units. (C)</p> Signup and view all the answers

What is the key difference between simple and compound depreciation?

<p>The depreciation amount is calculated differently. (C)</p> Signup and view all the answers

What is the first step in solving a word problem using algebraic equations?

<p>Read the problem carefully (A)</p> Signup and view all the answers

What is the definition of a quadratic sequence?

<p>A sequence of numbers in which the second difference between any two consecutive terms is constant (D)</p> Signup and view all the answers

What is the value of the horizontal asymptote for the graph of y = a/(x + p) + q?

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

What is the formula for finding the annual depreciation rate (i) for compound decay?

<p>Rearrange A = P(1 - i)^n to solve for i. (A)</p> Signup and view all the answers

What is the effect of a > 0 on the graph of y = ab^x + q?

<p>The graph is increasing. (C)</p> Signup and view all the answers

What is the general form of the nth term of a quadratic sequence?

<p>T_n = an^2 + bn + c (A)</p> Signup and view all the answers

What is the domain of the function y = a/(x + p) + q?

<p>{x : x ∈ ℝ, x ≠ -p} (C)</p> Signup and view all the answers

What is the effect of q > 0 on the graph of a parabolic function?

<p>The graph is shifted vertically upwards by q units (B)</p> Signup and view all the answers

What is the effect of p > 0 on the graph of y = a/(x + p) + q?

<p>The graph is shifted horizontally to the right by p units. (B)</p> Signup and view all the answers

What is the main difference between the substitution and elimination methods for solving systems of linear equations?

<p>The substitution method involves solving one equation for one variable and substituting it into the other equation, while the elimination method involves adding or subtracting the equations to eliminate one variable (B)</p> Signup and view all the answers

What is the point of intersection of the axes of symmetry for the graph of y = a/(x + p) + q?

<p>(-p, q) (B)</p> Signup and view all the answers

What is the purpose of checking the solution in a problem-solving strategy?

<p>To ensure that the solution is correct and makes sense in the context of the problem (B)</p> Signup and view all the answers

What is the advantage of using algebraic equations to model real-world problems?

<p>It enables the prediction of future outcomes and trends (C)</p> Signup and view all the answers

What is the range of the function y = a/(x + p) + q?

<p>{y : y ∈ ℝ, y ≠ q} (C)</p> Signup and view all the answers

What is the equation of the vertical asymptote for the graph of y = a/(x + p) + q?

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

What is the benefit of using a problem-solving strategy?

<p>It provides a step-by-step approach to solving the problem (C)</p> Signup and view all the answers

What is the average gradient of a curve at a point P(a, f(a))?

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

What is the term for the point where the two graphs intersect in a graphical solution of a system of linear equations?

<p>Solution to the system (A)</p> Signup and view all the answers

What is the formula for the probability of the union of three events A, B, and C?

<p>P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C) (D)</p> Signup and view all the answers

What is the characteristic of mutually exclusive events?

<p>The events cannot occur simultaneously (B)</p> Signup and view all the answers

What is the purpose of tree diagrams?

<p>To organize and visualize the different possible outcomes of a sequence of events (C)</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) (B)</p> Signup and view all the answers

What is the structure of a two-way contingency table?

<p>Rows represent one event and its complement, and columns represent another event and its complement (D)</p> Signup and view all the answers

What is the formula for the probability of an event A given event B (A if B)?

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

What is the condition for two events A and B to be independent?

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

What is the purpose of using tree diagrams in probability?

<p>To visualize the different possible outcomes of a sequence of events (A)</p> Signup and view all the answers

What is the formula for calculating the probability of a specific path in a tree diagram?

<p>P(Path) = P(Event 1) × P(Event 2) × ... (A)</p> Signup and view all the answers

What is the purpose of using contingency tables in probability?

<p>To test for independence between events (C)</p> Signup and view all the answers

What is the correct approach when interest rates change or additional deposits/withdrawals are made in a compound interest problem?

<p>Break the timeline into segments and calculate each period separately (C)</p> Signup and view all the answers

What is the formula for calculating the accumulated value in a compound interest problem?

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

What is the difference between nominal and effective interest rates?

<p>Nominal interest rate is the annual interest rate, while effective interest rate is the rate that takes into account compounding (B)</p> Signup and view all the answers

What is the formula for calculating the effective interest rate?

<p>1 + i-effective = (1 + i-nominal/m)^m (D)</p> Signup and view all the answers

What is the formula for calculating the probability of two independent events?

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

What is the formula for calculating the conditional probability of event A given event B?

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

What is the concept of mutually exclusive events in probability?

<p>Two events that cannot occur together (C)</p> Signup and view all the answers

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

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

What is the concept of the sample space in probability?

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

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

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

What is the value of a^(1/n), where a > 0 and n is a positive integer?

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

What is the result of multiplying a^(m/n) by a^(p/n)?

<p>a^((m+p)/n) (C)</p> Signup and view all the answers

What is the condition for the radicand in a radical symbol when n is odd?

<p>The radicand can be positive or negative (A)</p> Signup and view all the answers

What is the purpose of simplifying surds?

<p>To express the radicand as a product of factors (C)</p> Signup and view all the answers

What is the result of dividing √n by √n?

<p>√<a href="a/b">n</a> (D)</p> Signup and view all the answers

What is the purpose of rationalizing the denominator?

<p>To eliminate the surd from the denominator (C)</p> Signup and view all the answers

What is the result of raising √n to the power of m?

<p>a^(m/n) (C)</p> Signup and view all the answers

What is the definition of a surd?

<p>A radical resulting in an irrational number (A)</p> Signup and view all the answers

What is the purpose of converting surds into exponential notation when solving surd equations?

<p>To apply the necessary exponent rules and simplify the terms. (B)</p> Signup and view all the answers

What is the result of raising both sides of the equation to the power corresponding to the root when solving surd equations?

<p>The surd is eliminated, and the equation becomes quadratic. (A)</p> Signup and view all the answers

What is the nature of the roots of a quadratic equation if the discriminant Δ is less than 0?

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

What is the key consideration when raising both sides of the equation to the power to eliminate the surd?

<p>Be cautious of potential extraneous solutions. (D)</p> Signup and view all the answers

How do you determine the critical values when solving a quadratic inequality?

<p>By setting the quadratic equation to zero (D)</p> Signup and view all the answers

What is the formula for exponential growth or decay?

<p>P(t) = P₀ × (1 + r)^t (A)</p> Signup and view all the answers

What is the purpose of creating a table of signs when solving quadratic inequalities?

<p>To analyze the sign of each factor on intervals defined by the critical values (D)</p> Signup and view all the answers

What is the purpose of rationalizing the denominator?

<p>To eliminate radicals from the denominator for easier manipulation of expressions. (A)</p> Signup and view all the answers

What is the general method for solving systems of simultaneous equations with linear and non-linear equations?

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

What is the formula for converting a surd to an exponential?

<p>a^(m/n) = √<a href="a%5Em">n</a> (B)</p> Signup and view all the answers

What is the main goal when solving surd equations?

<p>To isolate the surd term and eliminate it. (A)</p> Signup and view all the answers

What is the result of solving a quadratic equation when the discriminant Δ is equal to 0?

<p>Real and equal roots (D)</p> Signup and view all the answers

What is the purpose of determining the nature of the roots of a quadratic equation?

<p>To determine the type of roots (B)</p> Signup and view all the answers

What is the power of a power rule?

<p>(a^m)^n = a^(mn) (D)</p> Signup and view all the answers

What is the application of exponential growth models in real-world scenarios?

<p>To model population growth and decay. (D)</p> Signup and view all the answers

What is the formula for calculating the discriminant of a quadratic equation in standard form?

<p>Δ = b^2 - 4ac (B)</p> Signup and view all the answers

What is the purpose of creating a table of signs when solving quadratic inequalities?

<p>To analyze the sign of each factor on intervals defined by the critical values (D)</p> Signup and view all the answers

What is the importance of checking solutions when solving surd equations?

<p>To ensure the solution is correct and accurate. (D)</p> Signup and view all the answers

What is the graphical representation of a quadratic inequality?

<p>A region on the coordinate plane (A)</p> Signup and view all the answers

What is the condition for a quadratic equation to have exactly one real root?

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

What is the first step in solving a quadratic equation using the completing the square method?

<p>Rearrange the equation in the standard form (B)</p> Signup and view all the answers

What is the purpose of the quadratic formula?

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

What is the definition of a^0, where a is not equal to zero?

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

What is the formula for exponential growth or decay?

<p>P = P_0(1 + r)^t (D)</p> Signup and view all the answers

What is the result of multiplying √n and √n?

<p>√<a href="ab">n</a> (B)</p> Signup and view all the answers

What is the condition for a quadratic equation to have exactly one real root?

<p>The discriminant is equal to zero (B)</p> Signup and view all the answers

What is the purpose of identifying restrictions in the substitution method?

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

What is the formula for determining the roots of a quadratic equation using the quadratic formula?

<p>x = (-b ± √(b^2 - 4ac)) / 2a (D)</p> Signup and view all the answers

What is the effect of a on the shape of the graph of f(x) = a(x + p)^2 + q?

<p>It affects the width of the graph (B)</p> Signup and view all the answers

What is the purpose of creating a table of signs when solving quadratic inequalities?

<p>To analyze the sign of each factor on intervals defined by the critical values (B)</p> Signup and view all the answers

What is the formula for determining the roots of a quadratic equation in the form ax^2 + bx + c = 0?

<p>x = -b/2a ± √(b^2 - 4ac)/2a (C)</p> Signup and view all the answers

What is the condition for a quadratic equation to have exactly one real root?

<p>b^2 - 4ac = 0 (C)</p> Signup and view all the answers

What is the purpose of identifying restrictions in the substitution method?

<p>To ensure the values of the substituted variable do not result in undefined expressions (B)</p> Signup and view all the answers

What is the nature of the roots of a quadratic equation when the discriminant Δ is greater than zero?

<p>The roots are real and distinct (A)</p> Signup and view all the answers

What is the final step in solving a quadratic equation by substitution?

<p>Write the final answer (D)</p> Signup and view all the answers

What is the formula for finding the quadratic equation given the roots r_1 and r_2?

<p>x^2 - (r_1 + r_2)x + r_1r_2 = 0 (A)</p> Signup and view all the answers

What is the purpose of making a substitution when solving a quadratic equation?

<p>To make a repeated expression simpler (D)</p> Signup and view all the answers

What is the type of roots a quadratic equation has when the discriminant Δ is a perfect square?

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

What is the first step in finding the quadratic equation given the roots?

<p>Write the roots as two equations (B)</p> Signup and view all the answers

What is the condition for a quadratic equation to have two complex conjugate roots?

<p>b^2 - 4ac &lt; 0 (B)</p> Signup and view all the answers

What is the main goal of solving a system of linear equations?

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

What is the result of substituting the solution back into the original equation in the elimination method?

<p>The solution to the system of equations (C)</p> Signup and view all the answers

What is the purpose of drawing the graph of each equation in the graphical method?

<p>To find the point of intersection (A)</p> Signup and view all the answers

What is the key step in solving word problems using algebraic equations?

<p>Translating the words into algebraic expressions (D)</p> Signup and view all the answers

What is the definition of a quadratic sequence?

<p>A sequence of numbers in which the second difference is constant (D)</p> Signup and view all the answers

What is the general form of a quadratic sequence?

<p>T_n = an^2 + bn + c (B)</p> Signup and view all the answers

What is the effect of q on the graph of a quadratic function y = ax^2 + q?

<p>It shifts the graph vertically upwards (B)</p> Signup and view all the answers

What is the purpose of solving systems of equations in real-world applications?

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

What is the first step in solving a word problem using algebraic equations?

<p>Assign variables to the unknown quantities (A)</p> Signup and view all the answers

What is the purpose of checking the solution in a word problem?

<p>To ensure that the solution is reasonable (D)</p> Signup and view all the answers

What is the effect of a negative value of a on the graph of f(x) = a(x + p)^2 + q?

<p>The graph is a 'frown' and has a maximum turning point. (C)</p> Signup and view all the answers

What is the axis of symmetry for the graph of f(x) = a(x + p)^2 + q?

<p>The vertical line x = -p (A)</p> Signup and view all the answers

What is the effect of a horizontal shift of p units to the right on the graph of f(x) = a(x)^2 + q?

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

What is the formula for the average gradient between two points on a curve?

<p>(y_A - y_C) / (x_A - x_C) (D)</p> Signup and view all the answers

What is the turning point of the graph of f(x) = a(x + p)^2 + q?

<p>(-p, q) (B)</p> Signup and view all the answers

What is the range of the graph of f(x) = a(x + p)^2 + q if a > 0?

<p>{y : y ≥ q, y ∈ ℝ} (A)</p> Signup and view all the answers

What is the effect of a vertical shift of q units upwards on the graph of f(x) = a(x)^2 + q?

<p>Replace y with y + q (C)</p> Signup and view all the answers

What is the domain of the function f(x) = a(x + p)^2 + q?

<p>{x : x ∈ ℝ} (A)</p> Signup and view all the answers

What is the y-intercept of the graph of f(x) = a(x + p)^2 + q?

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

What is the formula for the gradient of the tangent to the curve at a point?

<p>(y_A - y_C) / (x_A - x_C) (C)</p> Signup and view all the answers

What is the formula for the average gradient of a curve f(x) with points P(a, f(a)) and Q(a + h, f(a + h))?

<p>Average gradient = (f(a + h) - f(a))/h (A)</p> Signup and view all the answers

What is the effect of q on the graph of y = a/x + q?

<p>A change in q causes a vertical shift. (A)</p> Signup and view all the answers

What is the domain of the function y = a/(x + p) + q?

<p>{x: x ∈ ℝ, x ≠ -p} (D)</p> Signup and view all the answers

What is the range of the function y = a/(x + p) + q?

<p>{y: y ∈ ℝ, y ≠ q} (C)</p> Signup and view all the answers

What is the vertical asymptote of the function y = a/(x + p) + q?

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

What is the effect of a on the shape of the graph of y = a/x + q?

<p>For a &gt; 0, the graph lies in the first and third quadrants. (D)</p> Signup and view all the answers

What is the horizontal asymptote of the function y = ab^x + q?

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

What is the effect of q on the graph of y = ab^x + q?

<p>A change in q causes a vertical shift. (D)</p> Signup and view all the answers

What is the axis of symmetry of the function y = a/(x + p) + q?

<p>The line x = -p (B)</p> Signup and view all the answers

What is the condition for a quadratic equation to have exactly one real root?

<p>The discriminant is Δ = 0. (D)</p> Signup and view all the answers

What is the effect of b on the direction of the function f(x) = ab^(x+p) + q, assuming a > 0?

<p>If b &gt; 1, the function is increasing, if 0 &lt; b &lt; 1, the function is decreasing, and if b ≤ 0, the function is not defined. (B)</p> Signup and view all the answers

What is the effect of a change in p on the graph of f(x) = ab^(x+p) + q?

<p>A horizontal shift occurs. (C)</p> Signup and view all the answers

What is the effect of a change in q on the graph of f(x) = ab^(x+p) + q?

<p>A vertical shift occurs. (A)</p> Signup and view all the answers

What is the domain of the function f(x) = ab^(x+p) + q?

<p>All real numbers. (A)</p> Signup and view all the answers

What is the range of the function f(x) = ab^(x+p) + q, assuming a > 0?

<p>All real numbers greater than q. (D)</p> Signup and view all the answers

What is the formula for simple depreciation?

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

What is the formula for compound depreciation?

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

What is the purpose of using timelines when working with compound interest?

<p>To visualize the compounding periods and changes in interest rates. (D)</p> Signup and view all the answers

What happens to the depreciation amount each year in simple depreciation?

<p>It remains the same. (D)</p> Signup and view all the answers

What happens to the depreciation amount each year in compound depreciation?

<p>It decreases. (B)</p> Signup and view all the answers

What is the formula for the probability of three events A, B, and C occurring?

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

What is the definition of mutually exclusive events?

<p>Events that never occur simultaneously (B)</p> Signup and view all the answers

What is the purpose of a tree diagram?

<p>To visualize the possible outcomes of a sequence of events (B)</p> Signup and view all the answers

When calculating the accumulated value, what should you do when interest rates change or additional deposits/withdrawals are made?

<p>Break the timeline into segments and calculate each period separately (B)</p> Signup and view all the answers

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

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

What is the formula for Simple Interest?

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

What is the purpose of a contingency table?

<p>To compute probabilities and test for independence between events (D)</p> Signup and view all the answers

What is the formula for Compound Interest?

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

What is the formula for the Effective Interest Rate?

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

What is the formula for the probability of an event A given event B?

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

What is the formula for the probability of independent events?

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

What is the condition for two events A and B to be independent?

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

What is the formula for the probability of dependent events?

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

What is the formula for calculating the probability of a specific path in a tree diagram?

<p>P(Event 1) × P(Event 2|Event 1) × ... (C)</p> Signup and view all the answers

What is the definition of Mutually Exclusive Events?

<p>Two events are mutually exclusive if they cannot occur at the same time. (C)</p> Signup and view all the answers

What is the purpose of the intersection rule for three events?

<p>To calculate the probability of three events occurring simultaneously (A)</p> Signup and view all the answers

What is the formula for the probability of a desired outcome in a tree diagram?

<p>P(Path 1) + P(Path 2) + ... (D)</p> Signup and view all the answers

What is the purpose of Venn Diagrams?

<p>To visualize the relationship between two events. (B)</p> Signup and view all the answers

What is the result of the intersection of two independent events?

<p>The probability of the intersection is equal to the product of the probabilities of each event. (A)</p> Signup and view all the answers

What is the purpose of calculating the accumulated value at the end of each period?

<p>To calculate the total accumulated value at the end of the investment or loan term. (C)</p> Signup and view all the answers

What is the value of a^(1/n) when n is a positive integer?

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

What is the result of simplifying (√(a) × √(b)) if a and b are positive integers?

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

What is the purpose of rationalizing the denominator in a surd expression?

<p>To eliminate the surd from the denominator (C)</p> Signup and view all the answers

What is the value of a^(-m/n) when a is a positive integer and m and n are integers?

<p>1 / a^(m/n) (A)</p> Signup and view all the answers

What is the result of raising a surd to a power?

<p>A surd of the power multiplied by the root (A)</p> Signup and view all the answers

What is the condition for a radicand to have real roots when n is even?

<p>The radicand must be positive (C)</p> Signup and view all the answers

What is the result of multiplying two surds with the same root?

<p>A surd with the same root and a product of the radicands (B)</p> Signup and view all the answers

What is the result of dividing a surd by another surd?

<p>A surd with the same root and a quotient of the radicands (B)</p> Signup and view all the answers

What is the result of raising both sides of the equation to the power corresponding to the root when solving surd equations?

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

What is the formula for determining the roots of a quadratic equation in the form ax^2 + bx + c = 0?

<p>$x = rac{-b}{2a} \pm rac{\sqrt{b^2 - 4ac}}{2a}$ (B)</p> Signup and view all the answers

What is the purpose of converting percentage rates into decimal form before using them in the exponential growth or decay formula?

<p>To ensure accuracy in the final result (A)</p> Signup and view all the answers

What type of roots does a quadratic equation have if the discriminant is greater than zero?

<p>Two distinct real roots (A)</p> Signup and view all the answers

What is the purpose of converting surds into their exponential form?

<p>To make manipulation easier (B)</p> Signup and view all the answers

What is the result of applying the formula for exponential growth or decay to find the final amount?

<p>A percentage increase or decrease (B)</p> Signup and view all the answers

What is the step in the substitution method for solving quadratic equations where you identify a common factor or repeated expression?

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

What is the formula for exponential growth or decay?

<p>P(t) = P0 × (1 + r)^t (D)</p> Signup and view all the answers

What is the main goal when solving quadratic equations using the completing the square method?

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

What is the formula for finding the quadratic equation given the roots r_1 and r_2?

<p>(x - r_1)(x - r_2) = 0 (B)</p> Signup and view all the answers

What is the key consideration when isolating the surd on one side of the equation?

<p>Ensuring the surd is the only term on one side of the equation (D)</p> Signup and view all the answers

What is the condition for a quadratic equation to have exactly one real root?

<p>b^2 - 4ac = 0 (D)</p> Signup and view all the answers

What is the purpose of the quadratic formula in solving quadratic equations?

<p>To solve quadratic equations that cannot be factored (A)</p> Signup and view all the answers

What is the purpose of checking the solution by substituting back into the original equation?

<p>To verify the solution is correct (D)</p> Signup and view all the answers

What is the result of applying the power of a power rule?

<p>(a^m)^n = a^(mn) (A)</p> Signup and view all the answers

What is the purpose of identifying restrictions in the substitution method for solving quadratic equations?

<p>To ensure that the expressions are defined (C)</p> Signup and view all the answers

What is the condition required for a quadratic equation to have two distinct real roots?

<p>The discriminant must be greater than zero (C)</p> Signup and view all the answers

What is the formula for fractional exponents?

<p>a^(m/n) = √<a href="a%5Em">n</a> (B)</p> Signup and view all the answers

What is the result of solving a quadratic equation when the discriminant is Δ = 0?

<p>One real root (C)</p> Signup and view all the answers

What type of roots does a quadratic equation have if the discriminant is a perfect square?

<p>Two distinct rational roots (B)</p> Signup and view all the answers

What is the final step in solving a quadratic equation by substitution?

<p>Write the final answer (B)</p> Signup and view all the answers

What is the condition for a quadratic equation to have two distinct real roots?

<p>The discriminant is greater than zero (D)</p> Signup and view all the answers

What is the purpose of creating a table of signs when solving quadratic inequalities?

<p>To analyze the sign of each factor on intervals defined by the critical values (A)</p> Signup and view all the answers

What is the purpose of rationalizing denominators?

<p>To eliminate radicals from the denominator (C)</p> Signup and view all the answers

What is the purpose of making a substitution when solving a quadratic equation?

<p>To make the equation easier to solve (A)</p> Signup and view all the answers

What is the general method for solving quadratic inequalities?

<p>Creating a table of signs and analyzing the sign of each factor (B)</p> Signup and view all the answers

What is the definition of a^0?

<p>a^0 = 1, for a ≠ 0 (B)</p> Signup and view all the answers

What is the main difference between solving quadratic equations and quadratic inequalities?

<p>The number of solutions (D)</p> Signup and view all the answers

What is the discriminant of a quadratic equation in the form ax^2 + bx + c = 0?

<p>b^2 - 4ac (A)</p> Signup and view all the answers

What is the nature of the roots of a quadratic equation when the discriminant Δ is greater than 0?

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

What is the effect of a on the shape of the graph of f(x) = a(x + p)^2 + q?

<p>It affects the direction of the opening of the parabola (B)</p> Signup and view all the answers

What is the purpose of determining critical values when solving quadratic inequalities?

<p>To identify the intervals where the quadratic expression is positive or negative (A)</p> Signup and view all the answers

What is the formula for calculating the discriminant of a quadratic equation in standard form?

<p>Δ = b^2 - 4ac (C)</p> Signup and view all the answers

What is the result of solving a quadratic equation when the discriminant is Δ = 0?

<p>One real root (B)</p> Signup and view all the answers

What is the purpose of creating a table of signs when solving quadratic inequalities?

<p>To identify the intervals where the quadratic expression is positive or negative (A)</p> Signup and view all the answers

What is the graphical representation of a quadratic inequality?

<p>A region on a graph above or below the x-axis (B)</p> Signup and view all the answers

What is the condition for a quadratic equation to have two distinct real roots?

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

What is the purpose of analyzing the sign of each factor on intervals defined by the critical values?

<p>To identify the intervals where the quadratic expression is positive or negative (D)</p> Signup and view all the answers

What is the general method for solving quadratic inequalities?

<p>Substitution and sign analysis (D)</p> Signup and view all the answers

What is the main difference between solving quadratic equations and quadratic inequalities?

<p>The direction of the inequality sign (A)</p> Signup and view all the answers

What is the first step in solving a system of linear equations using substitution?

<p>Solve for one of the variables in terms of the other. (C)</p> Signup and view all the answers

What is the purpose of making a substitution when solving a quadratic equation?

<p>To make the solution more manageable. (A)</p> Signup and view all the answers

What is the definition of a quadratic sequence?

<p>A sequence of numbers in which the second difference between any two consecutive terms is constant. (B)</p> Signup and view all the answers

What is the effect of q on the graph of the function y = ax^2 + q?

<p>It shifts the graph vertically upwards or downwards. (D)</p> Signup and view all the answers

What is the purpose of identifying variables in a word problem?

<p>To translate the words into mathematical equations. (B)</p> Signup and view all the answers

What is the general method for solving systems of simultaneous equations?

<p>Use either substitution or elimination. (A)</p> Signup and view all the answers

What is the purpose of solving systems of linear equations?

<p>To find the values of multiple variables. (B)</p> Signup and view all the answers

What is the result of solving a quadratic equation when the discriminant is Δ = 0?

<p>The equation has one real root. (B)</p> Signup and view all the answers

What is the purpose of solving quadratic inequalities?

<p>To find the intervals on which the inequality is true. (C)</p> Signup and view all the answers

What is the graphical representation of a quadratic inequality?

<p>A region on the coordinate plane. (B)</p> Signup and view all the answers

What is the effect of the value of a on the shape of the graph of f(x) = a(x + p)^2 + q?

<p>The graph becomes wider as a gets closer to 0. (D)</p> Signup and view all the answers

What is the axis of symmetry for the graph of f(x) = a(x + p)^2 + q?

<p>The line x = -p. (B)</p> Signup and view all the answers

What is the turning point of the graph of f(x) = a(x + p)^2 + q?

<p>(-p, q). (C)</p> Signup and view all the answers

What is the effect of the value of p on the graph of f(x) = a(x + p)^2 + q?

<p>A horizontal shift. (B)</p> Signup and view all the answers

What is the range of the function f(x) = a(x + p)^2 + q when a > 0?

<p>{y : y ≥ q, y ∈ ℝ}. (A)</p> Signup and view all the answers

What is the purpose of finding the y-intercept when sketching the graph of f(x) = a(x + p)^2 + q?

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

How do you find the x-intercept of the graph of f(x) = a(x + p)^2 + q?

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

What is the purpose of finding the domain of the function f(x) = a(x + p)^2 + q?

<p>To determine the possible values of x. (B)</p> Signup and view all the answers

What is the average gradient of the curve at point P when h = 0?

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

What is the average gradient of a curve between two points?

<p>The gradient of the line passing through the two points. (A)</p> Signup and view all the answers

What is the effect of q on the graph of y = a/x + q?

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

What is the formula for calculating the average gradient of a curve between two points A(x_A, y_A) and C(x_C, y_C)?

<p>Gradient = (y_A - y_C) / (x_A - x_C). (B)</p> Signup and view all the answers

What is the domain of the function y = a/(x + p) + q?

<p>{x : x ∈ ℝ, x ≠ -p} (D)</p> Signup and view all the answers

What is the horizontal asymptote of the function y = a/(x + p) + q?

<p>y = q (D)</p> Signup and view all the answers

What is the effect of a on the shape of the graph of y = a/x + q?

<p>It affects the direction of the graph (D)</p> Signup and view all the answers

What is the axis of symmetry of the graph of y = a/(x + p) + q?

<p>The line that intersects at (-p, q) (B)</p> Signup and view all the answers

What is the formula for the average gradient?

<p>(f(a + h) - f(a)) / h (C)</p> Signup and view all the answers

What is the vertical asymptote of the function y = a/(x + p) + q?

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

What is the effect of p on the graph of y = a/(x + p) + q?

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

What is the range of the function y = a/(x + p) + q?

<p>{y : y ∈ ℝ, y ≠ q} (B)</p> Signup and view all the answers

If b > 1, what is the nature of the function f(x)?

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

What is the effect of a change in p on the graph of y = ab^(x+p) + q?

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

What is the range of the function y = ab^(x+p) + q when a > 0?

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

What is the formula for simple depreciation?

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

What is the key characteristic of compound depreciation?

<p>Depreciation amount decreases as the asset's value decreases (A)</p> Signup and view all the answers

How do you find the annual depreciation rate (i) for simple or compound decay?

<p>Rearrange the formula to solve for i (A)</p> Signup and view all the answers

What is the purpose of timelines in compound interest?

<p>To visualize the periods and changes in compounding and interest rates (D)</p> Signup and view all the answers

What happens to the depreciation amount each year in compound depreciation?

<p>It decreases as the asset's value decreases (D)</p> Signup and view all the answers

What is the horizontal asymptote of the function y = ab^(x+p) + q?

<p>y = q (A)</p> Signup and view all the answers

What is the domain of the function y = ab^(x+p) + q?

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

What is the formula for the probability of three events A, B, and C occurring?

<p>P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C) (D)</p> Signup and view all the answers

What is the condition for two events to be independent?

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

What is the purpose of a tree diagram?

<p>To organise and visualise the different possible outcomes of a sequence of events (A)</p> Signup and view all the answers

What is the formula for the probability of a specific path in a tree diagram?

<p>P(Path) = P(Event 1) × P(Event 2) × ... (C)</p> Signup and view all the answers

What is the purpose of a contingency table?

<p>To show the counts of outcomes for two events and their complements (B)</p> Signup and view all the answers

What is the formula for the probability of an event A given event B?

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

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

<p>P(A ∩ B) = 0 (B)</p> Signup and view all the answers

What is the purpose of the independence test in a contingency table?

<p>To test for independence between events (B)</p> Signup and view all the answers

What is the formula for calculating the probability of an event A?

<p>P(A) = n(A) / n(Total) (C)</p> Signup and view all the answers

What is the purpose of drawing a tree diagram?

<p>To determine the probability of multiple events occurring (B)</p> Signup and view all the answers

What is the purpose of breaking down periods when calculating the accumulated value of an investment or loan?

<p>To adjust the principal for deposits or withdrawals and changes in interest rates (C)</p> Signup and view all the answers

What is the formula for calculating the accumulated value of an investment or loan using compound interest?

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

What is the difference between independent and dependent events?

<p>Independent events are those where the occurrence of one does not affect the probability of the other, while dependent events are those where the occurrence of one affects the probability of the other (A)</p> Signup and view all the answers

What is the formula for calculating the effective interest rate?

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

What is the purpose of the addition rule for mutually exclusive events?

<p>To calculate the probability of two mutually exclusive events (D)</p> Signup and view all the answers

What is the formula for calculating the conditional probability of event A given event B?

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

What is the purpose of creating a Venn diagram?

<p>To visualize the relationships between events and their probabilities (C)</p> Signup and view all the answers

What is the formula for calculating the probability of two events that are not mutually exclusive?

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

What is the purpose of calculating the accumulated value at the end of each period?

<p>To determine the total accumulated value at the end of the investment or loan term (D)</p> Signup and view all the answers

What is the formula for calculating the nominal interest rate?

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

What is the value of $a^{1/n}$?

<p>$\sqrt[n]{a}$ (D)</p> Signup and view all the answers

What is the result of multiplying $\sqrt[n]{a}$ and $\sqrt[n]{b}$?

<p>$\sqrt[n]{ab}$ (B)</p> Signup and view all the answers

What is the value of $a^{-1/n}$?

<p>$rac{1}{\sqrt[n]{a}}$ (D)</p> Signup and view all the answers

What is the purpose of rationalizing the denominator in a surd expression?

<p>To eliminate the surd from the denominator (C)</p> Signup and view all the answers

What is the value of $\sqrt[n]{a^m}$?

<p>$a^{m/n}$ (A)</p> Signup and view all the answers

What is the result of dividing $\sqrt[n]{a}$ by $\sqrt[n]{b}$?

<p>$\sqrt[n]{a/b}$ (C)</p> Signup and view all the answers

What is the condition for the radicand in a radical symbol when $n$ is even?

<p>The radicand must be positive (C)</p> Signup and view all the answers

What is the formula for the product of powers?

<p>$a^m imes a^n = a^{m+n}$ (A)</p> Signup and view all the answers

What is the value of $a^{mn}$ in terms of $(a^m)^n$?

<p>$a^{mn}$ (D)</p> Signup and view all the answers

Solve for $x$ in the equation $\sqrt{x} = 2$.

<p>$x = 4$ (B)</p> Signup and view all the answers

What is the value of $a^{rac{m}{n}}$ in terms of $\sqrt[n]{a^m}$?

<p>$\sqrt[n]{a^m}$ (B)</p> Signup and view all the answers

Solve for $x$ in the equation $x^2 = 16$.

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

What is the purpose of writing surds in exponential notation?

<p>To make manipulation easier (B)</p> Signup and view all the answers

Solve for $x$ in the equation $\sqrt{x+2} = 3$.

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

What is the formula for exponential growth or decay?

<p>$P(t) = P_0 imes (1 + r)^t$ (D)</p> Signup and view all the answers

What is the key consideration when raising both sides of the equation to the power to eliminate the surd?

<p>Be cautious of potential extraneous solutions (D)</p> Signup and view all the answers

What is the result of solving a quadratic equation when the discriminant is = 0?

<p>The equation has exactly one real root (A)</p> Signup and view all the answers

What is the condition for a quadratic equation to have exactly one real root?

<p>The discriminant is = 0 (B)</p> Signup and view all the answers

What is the nature of the roots of a quadratic equation if the discriminant Δ is equal to 0?

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

What is the purpose of creating a table of signs when solving quadratic inequalities?

<p>To determine where the quadratic expression is positive or negative (C)</p> Signup and view all the answers

How do you determine the critical values when solving quadratic inequalities?

<p>By setting the quadratic equation to zero (D)</p> Signup and view all the answers

What is the general method for solving quadratic inequalities?

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

What is the purpose of graphing the quadratic function when solving quadratic inequalities?

<p>To visualize the solution and identify the regions where the inequality holds true (B)</p> Signup and view all the answers

What type of roots does a quadratic equation have if the discriminant Δ is a perfect square?

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

What is the formula for calculating the discriminant of a quadratic equation in standard form?

<p>Δ = b^2 - 4ac (A)</p> Signup and view all the answers

What is the purpose of determining the sign of each factor on intervals defined by the critical values?

<p>To determine where the quadratic expression is positive or negative (A)</p> Signup and view all the answers

What is the general method for solving systems of simultaneous equations?

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

What is the main difference between solving quadratic equations and quadratic inequalities?

<p>The purpose of solving them (B)</p> Signup and view all the answers

What is the condition for a quadratic equation to have two distinct real roots?

<p>If Δ &gt; 0 (D)</p> Signup and view all the answers

What is the purpose of making a substitution when solving a quadratic equation?

<p>To simplify the equation (A)</p> Signup and view all the answers

What type of roots does a quadratic equation have if the discriminant is a perfect square?

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

What is the first step in solving a quadratic equation using the completing the square method?

<p>Write the equation in the standard form ax^2 + bx + c = 0. (B)</p> Signup and view all the answers

What is the formula for determining the roots of a quadratic equation?

<p>x = (-b ± √(b^2 - 4ac)) / 2a (B)</p> Signup and view all the answers

What is the purpose of completing the square in quadratic equations?

<p>To transform the quadratic equation into an equation that can be solved using the square root property. (A)</p> Signup and view all the answers

What is the first step in finding the quadratic equation given the roots?

<p>Write down the roots as two equations (C)</p> Signup and view all the answers

What is the purpose of identifying restrictions in the substitution method?

<p>To ensure the expression is defined (A)</p> Signup and view all the answers

What is the formula for exponential growth or decay?

<p>P = P_0(1 + r)^t (D)</p> Signup and view all the answers

What is the set of rational numbers represented as?

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

What is the formula for calculating the discriminant of a quadratic equation in standard form?

<p>Δ = b^2 - 4ac (B)</p> Signup and view all the answers

What type of roots does a quadratic equation have if the discriminant is Δ > 0?

<p>Real and distinct roots (D)</p> Signup and view all the answers

What is the definition of a^(-n)?

<p>The reciprocal of a^n (B)</p> Signup and view all the answers

What is the formula for solving a quadratic equation using the quadratic formula?

<p>x = -b ± √(b^2 - 4ac) / 2a (A)</p> Signup and view all the answers

What is the final step in solving a quadratic equation by substitution?

<p>Write the final answer (A)</p> Signup and view all the answers

What is the condition for a quadratic equation to have exactly one real root?

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

What is the purpose of the quadratic formula?

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

What is the formula for converting a square root to an exponent?

<p>√(a) = a^(1/2) (B)</p> Signup and view all the answers

What is the rule for negative exponents?

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

What is the set of real numbers represented as?

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

What is the main goal of solving word problems?

<p>To write a set of equations that describe the problem mathematically. (A)</p> Signup and view all the answers

What is the characteristic of a quadratic sequence?

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

What is the effect of q on the graph of the function y = ax^2 + q?

<p>It shifts the graph vertically by q units. (C)</p> Signup and view all the answers

What is the first step in solving a system of equations using the substitution method?

<p>Solve one equation for one variable. (B)</p> Signup and view all the answers

What is the purpose of solving systems of equations?

<p>To find the values of multiple variables. (A)</p> Signup and view all the answers

What is the graphical representation of a system of equations?

<p>Two or more lines on the coordinate plane. (B)</p> Signup and view all the answers

What is the purpose of using mathematical models in real-world applications?

<p>To model population growth. (B)</p> Signup and view all the answers

What is the general form of a quadratic sequence?

<p>Tn = an^2 + bn + c (D)</p> Signup and view all the answers

What is the purpose of solving quadratic equations?

<p>To find the roots of the equation. (A)</p> Signup and view all the answers

What is the method of solving a system of equations by making one variable the subject of both equations?

<p>Substitution method. (C)</p> Signup and view all the answers

What is the formula for the average gradient of a curve f(x) between two points P(a, f(a)) and Q(a + h, f(a + h))?

<p>(f(a + h) - f(a))/h (C)</p> Signup and view all the answers

What is the effect of q on the graph of y = a/x + q?

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

What is the domain of the function y = a/(x + p) + q?

<p>{x : x ∈ ℝ, x ≠ -p} (D)</p> Signup and view all the answers

What is the horizontal asymptote of the function y = a/(x + p) + q?

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

What is the effect of a on the shape of the graph of f(x) = a(x + p)^2 + q?

<p>It determines the shape of the graph as a 'smile' or a 'frown' (A)</p> Signup and view all the answers

What is the effect of a on the shape of the graph of y = a/(x + p) + q?

<p>Changes the quadrants in which the graph lies (B)</p> Signup and view all the answers

What is the turning point of the function f(x) = a(x + p)^2 + q?

<p>(-p, q) (D)</p> Signup and view all the answers

What is the vertical asymptote of the function y = a/(x + p) + q?

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

What is the axis of symmetry of the function f(x) = a(x + p)^2 + q?

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

What is the formula for the exponential function of the form y = ab^x + q?

<p>y = ab^x + q (B)</p> Signup and view all the answers

What is the effect of q on the graph of y = ab^x + q?

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

What is the range of the function f(x) = a(x + p)^2 + q when a > 0?

<p>{y : y ≥ q, y ∈ ℝ} (B)</p> Signup and view all the answers

How do you find the y-intercept of the function f(x) = a(x + p)^2 + q?

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

What is the axes of symmetry of the graph of y = a/(x + p) + q?

<p>Two lines that intersect at (-p, q) (A)</p> Signup and view all the answers

What is the effect of a on the shape of the graph of y = ab^x + q?

<p>Changes the direction of the graph (C)</p> Signup and view all the answers

What is the effect of p on the graph of f(x) = a(x + p)^2 + q?

<p>It shifts the graph horizontally (A)</p> Signup and view all the answers

What is the formula to find the average gradient of a curve between two points A(x_A, y_A) and C(x_C, y_C)?

<p>Gradient = (y_A - y_C) / (x_A - x_C) (D)</p> Signup and view all the answers

What is the purpose of finding the average gradient of a curve?

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

What is the domain of the function f(x) = a(x + p)^2 + q?

<p>{x : x ∈ ℝ} (B)</p> Signup and view all the answers

How do you shift the graph of f(x) = a(x + p)^2 + q by m units to the right?

<p>Replace x with x - m (B)</p> Signup and view all the answers

What is the effect of b on the direction of the function f(x) = ab^(x+p) + q, assuming a > 0?

<p>If b &gt; 1, the function is increasing; if 0 &lt; b &lt; 1, the function is decreasing. (D)</p> Signup and view all the answers

What is the effect of a change in p on the graph of the function f(x) = ab^(x+p) + q?

<p>A horizontal shift occurs. (A)</p> Signup and view all the answers

What is the range of the function f(x) = ab^(x+p) + q when a > 0?

<p>{y: y &gt; q, y ∈ ℝ} (D)</p> Signup and view all the answers

What is the formula for simple depreciation?

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

What is the characteristic of compound depreciation?

<p>The depreciation amount decreases each year. (A)</p> Signup and view all the answers

How is the annual depreciation rate (i) found for simple or compound decay?

<p>By solving for i in the formula A = P(1 - i^n) (C), By solving for i in the formula A = P(1 - in) (D)</p> Signup and view all the answers

What is the purpose of using timelines in compounding?

<p>To visualize the periods and changes in compounding and interest rates. (A)</p> Signup and view all the answers

What happens to the interest rate when compounding occurs more than once a year?

<p>It is divided by the number of compounding periods. (D)</p> Signup and view all the answers

What is the horizontal asymptote of the function f(x) = ab^(x+p) + q?

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

What is the domain of the function f(x) = ab^(x+p) + q?

<p>{x: x ∈ ℝ} (B)</p> Signup and view all the answers

What is the purpose of breaking down periods in compound interest calculations?

<p>To adjust the principal for deposits or withdrawals (B)</p> Signup and view all the answers

What is the formula for compound interest?

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

What is the definition of dependent events?

<p>Two events are dependent if the occurrence of one affects the probability of the other (A)</p> Signup and view all the answers

What is the formula for the effective interest rate?

<p>1 + i-effective = (1 + i-nominal/m)^m (B)</p> Signup and view all the answers

What is the purpose of the addition rule for mutually exclusive events?

<p>To find the probability of two events occurring together or separately (D)</p> Signup and view all the answers

What is the formula for the conditional probability of event A given event B?

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

What is the purpose of Venn diagrams in probability?

<p>To represent the intersection and union of events (C)</p> Signup and view all the answers

What is the formula for the probability of two independent events occurring?

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

What is the purpose of calculating the accumulated value at the end of each period in compound interest?

<p>To adjust the principal for the next period (D)</p> Signup and view all the answers

What is the definition of mutually exclusive events?

<p>Two events are mutually exclusive if they cannot occur at the same time (D)</p> Signup and view all the answers

What is the formula for the probability of a union of three events A, B, and C?

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

What is the probability of an event occurring given that another event has occurred, denoted by P(A|B)?

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

What is the purpose of a contingency table?

<p>To test for independence between two events (C)</p> Signup and view all the answers

What is the formula for the probability of a path in a tree diagram?

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

What is the characteristic of independent events?

<p>The occurrence of one event does not affect the probability of the other (A)</p> Signup and view all the answers

What is the formula for the probability of an event A, denoted by P(A)?

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

What is the purpose of a tree diagram?

<p>To organize and visualize the different possible outcomes of a sequence of events (B)</p> Signup and view all the answers

What is the condition for two events A and B to be independent?

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

What is the formula for calculating the probability of a desired outcome when there are multiple paths leading to it?

<p>P(Path 1) + P(Path 2) + ... (A)</p> Signup and view all the answers

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