Gr 11 Math June P1 (Mix)
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Questions and Answers

What is the definition of a rational number?

  • Any number that can be written as a decimal with an infinite number of digits.
  • Any number that can be written as a product of prime numbers.
  • Any number that can be written as a fraction with an integer in the numerator and the denominator. (correct)
  • Any number that can be written as a sum of two integers.

    • What is the value of r if r^n = a, where n ≥ 2?

  • r = a^n
  • r = a^(1/n)
  • r = √(a) (correct)
  • r = √(a^n)

    • What is the value of a^(1/n)?

  • a^(n/1)
  • √(a) (correct)
  • a^n
  • √(a^n)

    • What is the purpose of rationalising the denominator?

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

    What is the formula for the quotient of powers?

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

    What is the value of √(a^m)?

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

    What is the definition of a surd?

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

    What is the formula for the product of powers?

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

    What is the formula to eliminate a surd from an equation?

    <p>Raise both sides to the power corresponding to the root</p> Signup and view all the answers

    What is the purpose of checking the solution in surd equations?

    <p>To verify the solution is correct and not extraneous</p> Signup and view all the answers

    What is the general formula for exponential growth or decay?

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

    What does the growth rate (r) represent in the exponential growth formula?

    <p>The percentage increase or decrease per period</p> Signup and view all the answers

    What is the purpose of writing a surd in exponential notation?

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

    What is the first step in solving surd equations?

    <p>Isolate the surd</p> Signup and view all the answers

    What is the result of raising both sides of an equation to the power corresponding to the root?

    <p>The surd is eliminated</p> Signup and view all the answers

    What is the formula for compound interest?

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

    What is the purpose of factorizing in quadratic surd equations?

    <p>To solve for the variable</p> Signup and view all the answers

    What is the exponent rule used to simplify surd expressions?

    <p>Power of a power</p> Signup and view all the answers

    What is the formula for exponential growth or decay?

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

    What is the set of natural numbers?

    <p>{1, 2, 3, ...}</p> Signup and view all the answers

    What is the definition of a^0 when a is not equal to 0?

    <p>a^0 = 1</p> Signup and view all the answers

    What is the formula for completing the square?

    <p>x^2 + 2bx + b^2 = (x + b)^2</p> Signup and view all the answers

    What is the quadratic formula?

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

    What is the purpose of converting percentage rates into decimal form?

    <p>To make the calculation easier</p> Signup and view all the answers

    What is the definition of a rational number?

    <p>A fraction or a terminating or recurring decimal</p> Signup and view all the answers

    What is the formula for solving quadratic equations with non-unitary leading coefficients?

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

    What is the purpose of completing the square?

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

    What is the formula for a^m × a^n?

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

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

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

    What is the condition for real and distinct roots?

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

    What is the purpose of substitution in solving quadratic equations?

    <p>To identify a common factor or repeated expression</p> Signup and view all the answers

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

    <p>Assign a variable and write roots as two equations</p> Signup and view all the answers

    What does the discriminant (Δ) represent?

    <p>The nature of the roots</p> Signup and view all the answers

    What is the condition for imaginary roots?

    <p>Δ &lt; 0</p> Signup and view all the answers

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

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

    What is the final step in the substitution method?

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

    What is the result when the discriminant is equal to zero?

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

    What is the purpose of multiplying by constants in finding the quadratic equation?

    <p>To get other possible equations with the same roots</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>Non-real</p> Signup and view all the answers

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

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

    What is the first step in solving a quadratic inequality?

    <p>Write the quadratic in standard form</p> Signup and view all the answers

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

    <p>To determine the sign of each factor on intervals</p> Signup and view all the answers

    How many independent equations are required to solve for the values of n unknown variables?

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

    What is the method of solving simultaneous equations that involves expressing one variable in terms of the other?

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

    What is the purpose of solving simultaneous linear and non-linear equations?

    <p>To find the values of unknown variables</p> Signup and view all the answers

    What is the method of solving quadratic inequalities that involves determining where the graph of the quadratic function lies above or below the x-axis?

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

    Why is it necessary to consider restrictions when solving inequalities involving fractions?

    <p>To avoid undefined expressions</p> Signup and view all the answers

    What is the condition for a quadratic equation to have real and equal roots?

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

    What is the effect of a on the shape of the graph of a quadratic function?

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

    What is the domain of a quadratic function?

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

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

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

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

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

    How do you shift the graph of a quadratic function m units to the right?

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

    What is the formula for the average gradient of a curve between two points A and C?

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

    What is the average gradient of a curve at a point?

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

    What is the formula for the hyperbolic function y = a/x + q?

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

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

    <p>It shifts the graph vertically</p> Signup and view all the answers

    What is the horizontal asymptote of the hyperbolic function y = a/x + q?

    <p>The line y = q</p> Signup and view all the answers

    What is the first step in solving word problems?

    <p>Read the problem carefully.</p> Signup and view all the answers

    What is the purpose of solving systems of equations?

    <p>To describe real-life contexts mathematically.</p> Signup and view all the answers

    What happens to the graph of a parabolic function when q > 0?

    <p>It is shifted vertically upwards.</p> Signup and view all the answers

    What is the effect of a on the shape of a parabolic function?

    <p>It determines the shape of the graph, making it a 'smile' or a 'frown'.</p> Signup and view all the answers

    What is the purpose of substitution in solving systems of equations?

    <p>To eliminate one variable.</p> Signup and view all the answers

    What is the effect of p on the graph of a parabolic function?

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

    What is the final step in the problem-solving strategy?

    <p>Write the final answer.</p> Signup and view all the answers

    What is the purpose of elimination in solving systems of equations?

    <p>To eliminate one variable and reduce the number of equations and variables.</p> Signup and view all the answers

    What is the turning point of a parabolic function when a > 0?

    <p>A minimum turning point.</p> Signup and view all the answers

    What is the purpose of solving systems of equations graphically?

    <p>To find the coordinates of the points where the two graphs intersect.</p> Signup and view all the answers

    What is the vertical asymptote of the function y = a/x?

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

    What is the effect of a > 0 on the function y = a/x?

    <p>The function lies in the first and third quadrants.</p> Signup and view all the answers

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

    <p>It causes a horizontal shift.</p> Signup and view all the answers

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

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

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

    <p>It causes a vertical shift.</p> Signup and view all the answers

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

    <p>The graph lies above the horizontal asymptote.</p> Signup and view all the answers

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

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

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

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

    What is the effect of p on the function y = ab^{(x+p)} + q?

    <p>It causes a horizontal shift.</p> Signup and view all the answers

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

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

    What is the value of $a^{rac{m}{n}}$, where $m, n \in \mathbb{Z}$ and $n eq 0$?

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

    If $r = \sqrt[n]{a}$, what is the value of $r^n$?

    <p>$a$</p> Signup and view all the answers

    What is the value of $\sqrt[n]{ab}$, where $a, b > 0$?

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

    What is the purpose of rationalizing the denominator in a fraction?

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

    What is the value of $\left(\sqrt[n]{a} ight)^m$, where $m, n \in \mathbb{Z}$ and $n eq 0$?

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

    What is the condition for the radicand to have real roots if $n$ is even?

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

    What is the value of $rac{\sqrt[n]{a}}{\sqrt[n]{b}}$, where $a, b > 0$?

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

    What is the purpose of writing a surd in exponential notation?

    <p>To express the surd in a more compact form</p> Signup and view all the answers

    What is the purpose of writing a surd in exponential notation when solving surd equations?

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

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

    <p>The surd is eliminated, but a new equation is formed</p> Signup and view all the answers

    What is the purpose of checking the solution in surd equations?

    <p>To verify that the solution is correct</p> Signup and view all the answers

    What is the formula used to model exponential growth or decay?

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

    What is the purpose of rationalising the denominator?

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

    What is the first step in solving quadratic surd equations?

    <p>Isolate and simplify the equation</p> Signup and view all the answers

    What is the purpose of factorising in quadratic surd equations?

    <p>To solve for the variable</p> Signup and view all the answers

    What is the formula used to calculate the amount of money accumulated over time with a given interest rate?

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

    What is the condition for real and distinct roots in a quadratic equation?

    <p>Discriminant (Δ) is greater than 0</p> Signup and view all the answers

    What is the purpose of identifying the variables in exponential growth or decay problems?

    <p>To determine the initial amount, growth rate, and time period</p> Signup and view all the answers

    What is the purpose of applying the formula for exponential growth or decay?

    <p>To find the final amount</p> Signup and view all the answers

    What is the first step in completing the square?

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

    What is the formula for converting a surd into exponential notation?

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

    What is the set of whole numbers, including zero, represented by?

    <p>( \mathbb{N}_0 )</p> Signup and view all the answers

    What is the formula for solving quadratic equations with non-unitary leading coefficients?

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

    What is the result of raising both sides of an equation to the power corresponding to the root?

    <p>The surd is eliminated</p> Signup and view all the answers

    What is the purpose of factorizing in quadratic surd equations?

    <p>To eliminate the surd</p> Signup and view all the answers

    What is the formula for the product of powers?

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

    What is the condition for real and distinct roots in a quadratic equation?

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

    What is the purpose of checking the solution in surd equations?

    <p>To verify the solution</p> Signup and view all the answers

    What is the condition for a quadratic equation to have real and equal roots?

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

    What is the purpose of making a substitution in solving quadratic equations?

    <p>To simplify the equation by removing repeated expressions</p> Signup and view all the answers

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

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

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

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

    What is the step after assigning a variable and writing the roots as two equations in finding the quadratic equation given the roots?

    <p>Write down as the product of two factors</p> Signup and view all the answers

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

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

    What is the result when the discriminant is equal to zero?

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

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

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

    What is the final step in the substitution method?

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

    What is the purpose of multiplying by constants in finding the quadratic equation given the roots?

    <p>To get other possible equations with the same roots</p> Signup and view all the answers

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

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

    What is the first step in solving a quadratic inequality?

    <p>Write the quadratic inequality in standard form</p> Signup and view all the answers

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

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

    What is the method of solving simultaneous linear and non-linear equations that involves expressing one variable in terms of the other?

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

    What is the condition for a quadratic equation to have real and equal roots?

    <p>Discriminant Δ is equal to 0</p> Signup and view all the answers

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

    <p>To avoid undefined expressions</p> Signup and view all the answers

    What is the result when the discriminant is equal to zero?

    <p>The quadratic equation has real and equal roots</p> Signup and view all the answers

    How many independent equations are required to solve for the values of n unknown variables?

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

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

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

    Why is it necessary to consider restrictions when solving inequalities involving fractions?

    <p>To avoid undefined expressions</p> Signup and view all the answers

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

    <p>Solve for one variable in one of the equations.</p> Signup and view all the answers

    What is the effect of q on the graph of a quadratic function?

    <p>It shifts the graph vertically</p> Signup and view all the answers

    What is the purpose of assigning variables to unknown quantities in word problems?

    <p>To translate the words into algebraic expressions.</p> Signup and view all the answers

    What is the range of a quadratic function when a > 0?

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

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

    <p>The graph is shifted vertically upwards.</p> Signup and view all the answers

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

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

    What is the purpose of solving word problems using mathematical language?

    <p>To understand how the natural world works.</p> Signup and view all the answers

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

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

    What is the effect of a negative value of a on the shape of the graph of a quadratic function?

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

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

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

    What is the purpose of checking the solution in word problems?

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

    How do you shift the graph of a quadratic function 2 units to the right?

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

    What is the effect of a positive value of p on the graph of a quadratic function of the form y = a(x + p)^2 + q?

    <p>The graph is shifted horizontally to the right.</p> Signup and view all the answers

    What is the purpose of solving simultaneous linear equations?

    <p>To find the values of two or more variables.</p> Signup and view all the answers

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

    <p>The gradient of the line passing through A and C</p> Signup and view all the answers

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

    <p>As the value of a becomes larger, the graph becomes narrower.</p> Signup and view all the answers

    What is the formula for the average gradient of a curve?

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

    What is the purpose of using problem-solving strategies in word problems?

    <p>To understand how the natural world works.</p> Signup and view all the answers

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

    <p>It shifts the graph vertically</p> Signup and view all the answers

    What is the horizontal asymptote of the hyperbolic function y = a/x + q?

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

    What is the effect of a on the shape of the graph of y = 1/x?

    <p>The graph is stretched vertically by a factor of a.</p> Signup and view all the answers

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

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

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

    <p>The graph is shifted horizontally by p units.</p> Signup and view all the answers

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

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

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

    <p>The graph is shifted vertically by q units.</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>The graph is increasing or decreasing.</p> Signup and view all the answers

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

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

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

    <p>The graph is shifted horizontally by p units.</p> Signup and view all the answers

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

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

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

    <p>The axes of symmetry are the lines x = -p and y = q.</p> Signup and view all the answers

    What is the value of a^(m/n) when a > 0, m, n ∈ ℤ, and n ≠ 0?

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

    What is the purpose of rationalising the denominator?

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

    What is the result of raising both sides of an equation to the power corresponding to the root?

    <p>An equivalent equation with no surd</p> Signup and view all the answers

    What is the value of a^(-1/n) when a > 0, n ∈ ℤ, and n ≠ 0?

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

    What is the formula for √(a^m) when a > 0 and m, n ∈ ℤ?

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

    What is the result of multiplying both the numerator and the denominator by the conjugate of the denominator?

    <p>A rationalised denominator</p> Signup and view all the answers

    What is the formula for the nth root of a when a > 0 and n ∈ ℤ?

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

    What is the condition for real and distinct roots of a surd equation?

    <p>The discriminant Δ is greater than 0</p> Signup and view all the answers

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

    <p>To ensure that the denominator is not equal to zero</p> Signup and view all the answers

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

    <p>Δ &lt; 0</p> Signup and view all the answers

    What is the result when the discriminant Δ is equal to zero?

    <p>The quadratic equation has one real root</p> Signup and view all the answers

    What is the purpose of working backwards to find the quadratic equation given the roots?

    <p>To find the values of the coefficients a, b, and 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 = (-b ± √(b^2 - 4ac)) / 2a</p> Signup and view all the answers

    What is the purpose of expanding the brackets in finding the quadratic equation given the roots?

    <p>To write the equation in the standard form ax^2 + bx + c = 0</p> Signup and view all the answers

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

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

    What is the step that involves assigning a variable and writing the roots as two equations in finding the quadratic equation given the roots?

    <p>Assign a variable and write roots as two equations</p> Signup and view all the answers

    What is the purpose of multiplication by constants in finding the quadratic equation given the roots?

    <p>To get other possible equations with the same roots</p> Signup and view all the answers

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

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

    What is the primary goal when solving equations involving surds?

    <p>To isolate the surd on one side of the equation and then eliminate it</p> Signup and view all the answers

    What is the formula for exponential growth or decay?

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

    What is the purpose of simplifying surd expressions?

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

    When solving quadratic surd equations, what step should be taken after isolating and simplifying the equation?

    <p>Square both sides to eliminate the surd</p> Signup and view all the answers

    What is the condition to avoid when solving surd equations?

    <p>Potential extraneous solutions</p> Signup and view all the answers

    What is the purpose of writing a surd in exponential notation?

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

    What is the result of raising both sides of an equation to the power corresponding to the root?

    <p>The surd is eliminated</p> Signup and view all the answers

    What is the purpose of factorizing in quadratic surd equations?

    <p>To solve for the variable using algebraic methods</p> Signup and view all the answers

    What is the key point to remember when isolating the surd in an equation?

    <p>Ensure the surd is the only term on one side of the equation</p> Signup and view all the answers

    What is the application of exponential growth or decay in finance?

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

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

    <p>To ensure the correct application of the formula</p> Signup and view all the answers

    Which of the following is a rule of exponents?

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

    What is the method used to solve quadratic equations by rewriting them in the form of a perfect square trinomial?

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

    What is the formula for solving quadratic equations with non-unitary leading coefficients?

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

    What is the definition of a^(-n) when a is not equal to 0?

    <p>1/a^n</p> Signup and view all the answers

    What is the set of integers?

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

    What is the purpose of the quadratic formula?

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

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

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

    What is the result of raising both sides of an equation to the power corresponding to the root?

    <p>Eliminating the root</p> Signup and view all the answers

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

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

    What does the discriminant Δ represent in a quadratic equation?

    <p>The nature of the roots of the equation</p> Signup and view all the answers

    What is the first step in solving a quadratic inequality?

    <p>Write the quadratic inequality in standard form</p> Signup and view all the answers

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

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

    What is the method of solving simultaneous equations that involves expressing one variable in terms of the other?

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

    What is the condition for a quadratic equation to have real and equal roots?

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

    What is the purpose of solving quadratic inequalities?

    <p>To determine which part of the graph of a quadratic function lies above or below the x-axis</p> Signup and view all the answers

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

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

    Why is it necessary to consider restrictions when solving inequalities involving fractions?

    <p>To avoid undefined expressions</p> Signup and view all the answers

    How many independent equations are required to solve for the values of n unknown variables?

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

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

    <p>Make one of the variables the subject of both equations.</p> Signup and view all the answers

    What is the purpose of solving word problems in mathematics?

    <p>To model real-life situations and make predictions.</p> Signup and view all the answers

    What is the effect of a positive value of q on the graph of a quadratic function?

    <p>The graph is shifted vertically upwards.</p> Signup and view all the answers

    What is the final step in solving a system of linear equations using the elimination method?

    <p>Use the solution to substitute back into the original equation.</p> Signup and view all the answers

    What is the effect of a negative value of q on the graph of a quadratic function?

    <p>The graph is shifted vertically downwards.</p> Signup and view all the answers

    What is the purpose of identifying the question and determining what needs to be solved in a word problem?

    <p>To understand the context of the problem.</p> Signup and view all the answers

    What is the effect of a on the shape of the graph of a quadratic function?

    <p>The graph becomes narrower.</p> Signup and view all the answers

    What is the purpose of solving systems of linear equations?

    <p>To find the values of unknown variables.</p> Signup and view all the answers

    What is the effect of p on the graph of a quadratic function?

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

    What is the final step in solving a word problem?

    <p>Write the final answer.</p> Signup and view all the answers

    What is the effect of q on the graph of a quadratic function?

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

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

    <p>A horizontal shift of the graph by p units.</p> Signup and view all the answers

    What is the range of a quadratic function when a > 0?

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

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

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

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

    <p>The set of all real numbers greater than q.</p> Signup and view all the answers

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

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

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

    <p>The turning point is (p, q).</p> Signup and view all the answers

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

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

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

    <p>A vertical shift of the graph by q units.</p> Signup and view all the answers

    What is the formula for the average gradient of a curve?

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

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

    <p>The set of all real numbers except -p.</p> Signup and view all the answers

    What is the effect of a on the shape of the graph of a quadratic function?

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

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

    <p>The graph is reflected about the horizontal asymptote.</p> Signup and view all the answers

    What is the domain of a quadratic function?

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

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

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

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

    <p>Set x = 0 to find the y-intercept.</p> Signup and view all the answers

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

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

    What is the formula for the hyperbolic function y = a/x + q?

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

    What is the effect of the parameter b on the direction of the function y = ab^x + q?

    <p>The function is increasing if b &gt; 1 and decreasing if 0 &lt; b &lt; 1.</p> Signup and view all the answers

    What is the y-intercept of the function y = ab^x + q?

    <p>The point (0, q).</p> Signup and view all the answers

    If the exponential growth rate is 25% per annum, what is the decimal form of the growth rate?

    <p>0.05</p> Signup and view all the answers

    What is the final amount of $1000 invested at a 5% annual growth rate for 2 years, using the exponential growth formula?

    <p>$1102.50</p> Signup and view all the answers

    What is the value of x in the equation x^2 + 4x + 4 = 0, using the method of completing the square?

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

    If the equation is 2x^2 + 5x + 3 = 0, what is the value of the discriminant (Δ)?

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

    What is the value of x in the equation x^2 - 5x + 6 = 0, using the quadratic formula?

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

    If the equation is x^2 + 2x + 1 = 0, what is the value of x, using the method of completing the square?

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

    What is the final amount of $5000 invested at a 10% annual decay rate for 3 years, using the exponential decay formula?

    <p>$3825.00</p> Signup and view all the answers

    What is the value of x in the equation 3x^2 + 2x - 1 = 0, using the quadratic formula?

    <p>1/3</p> Signup and view all the answers

    If the equation is x^2 - 2x - 3 = 0, what is the value of the discriminant (Δ)?

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

    What is the value of x in the equation 2x^2 - 3x - 1 = 0, using the method of completing the square?

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

    What is the value of $a^{rac{m}{n}}$ in terms of radicals?

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

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

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

    What is the purpose of rationalizing the denominator in a fraction?

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

    What is the primary goal when solving surd equations?

    <p>To eliminate the surd term</p> Signup and view all the answers

    What is the value of $\left(a^m ight)^{rac{1}{n}}$?

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

    What is the condition for the radicand to be positive when $n$ is even?

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

    What is the purpose of converting surds into exponential notation?

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

    What is the result of raising both sides of an equation to the power corresponding to the root?

    <p>The equation becomes simpler</p> Signup and view all the answers

    What is the result of raising both sides of an equation to the power corresponding to the root?

    <p>The surd term is eliminated</p> Signup and view all the answers

    What is the formula for exponential growth or decay when the initial amount is P0, the growth rate is r, and the time period is t?

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

    What is the formula for simplifying $\sqrt[n]{a^m}$?

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

    What is the purpose of checking the solution in surd equations?

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

    What is the purpose of expressing the radicand as a product of factors in simplifying surds?

    <p>To simplify the radicand</p> Signup and view all the answers

    What is the effect of a negative growth rate in an exponential growth model?

    <p>The quantity decreases by a fixed percentage</p> Signup and view all the answers

    What is the purpose of factorizing in quadratic surd equations?

    <p>To solve for the variable</p> Signup and view all the answers

    What is the result of applying the power of a power rule to the expression (a^m)^n?

    <p>a^(mn)</p> Signup and view all the answers

    What is the condition for real and distinct roots in a quadratic equation?

    <p>The discriminant is greater than 0</p> Signup and view all the answers

    What is the purpose of writing surd equations in exponential notation?

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

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

    <p>To ensure that the denominator is not equal to zero</p> Signup and view all the answers

    What is the result when the discriminant, Δ, is equal to zero?

    <p>The quadratic equation has one real root</p> Signup and view all the answers

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

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

    What is the purpose of multiplying by constants in finding the quadratic equation given the roots?

    <p>To get other possible equations with the same roots</p> Signup and view all the answers

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

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

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

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

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

    <p>To make a substitution for a repeated expression to simplify the equation</p> Signup and view all the answers

    What is the final step in the substitution method of solving quadratic equations?

    <p>Verify the solutions against the restrictions</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</p> Signup and view all the answers

    What is the purpose of solving quadratic inequalities?

    <p>To determine which part of the graph of the quadratic function lies above or below the x-axis</p> Signup and view all the answers

    What is the method of solving simultaneous linear and non-linear equations?

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

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

    <p>To avoid undefined expressions</p> Signup and view all the answers

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

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

    What is the condition for a quadratic equation to have real and equal roots?

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

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

    <p>To determine where the quadratic function is positive or negative</p> Signup and view all the answers

    What is the method of solving quadratic inequalities that involves determining where the graph of the quadratic function lies above or below the x-axis?

    <p>Graphical method</p> Signup and view all the answers

    How many independent equations are required to solve for the values of n unknown variables?

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

    What is the method of solving simultaneous linear equations that involves expressing one variable in terms of the other?

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

    What is the general approach to solving systems of linear equations using substitution?

    <p>Make one of the variables the subject of one equation, then substitute it into the other equation.</p> Signup and view all the answers

    When solving a word problem, what is the first step according to the problem-solving strategy?

    <p>Read the problem carefully.</p> Signup and view all the answers

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

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

    What is the purpose of setting up a system of equations when solving a word problem?

    <p>To describe the problem mathematically.</p> Signup and view all the answers

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

    <p>A change in the shape of the graph, making it narrower or wider.</p> Signup and view all the answers

    What is the final step in the problem-solving strategy when solving a word problem?

    <p>Write the final answer.</p> Signup and view all the answers

    What is the effect of the parameter p on the graph of a quadratic function of the form y = a(x + p)^2 + q?

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

    What is the purpose of using a system of equations to model real-world problems?

    <p>To describe the problem mathematically.</p> Signup and view all the answers

    What is the result of solving a system of equations using substitution or elimination methods?

    <p>One equation with one unknown variable.</p> Signup and view all the answers

    What is the purpose of checking the solution when solving a word problem?

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

    What is the shape of the graph of a quadratic function when a < 0?

    <p>A frown</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</p> Signup and view all the answers

    What is the effect of q on the graph of a quadratic function?

    <p>It shifts the graph vertically</p> Signup and view all the answers

    What is the formula for 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)</p> Signup and view all the answers

    What is the range of a quadratic function when a > 0?

    <p>All real values of y greater than or equal to q</p> Signup and view all the answers

    How do you find the x-intercept(s) of a quadratic function?

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

    What is the formula for shifting a quadratic function horizontally by m units to the right?

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

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

    <p>To find the average rate of change of the curve</p> Signup and view all the answers

    What is the formula for the turning point of a quadratic function in the form f(x) = a(x + p)^2 + q?

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

    What is the impact of a on the turning point of a quadratic function?

    <p>It affects the shape of the graph but not the turning point</p> Signup and view all the answers

    What is the effect of changing the value of p in the function y = a/(x + p) + q?

    <p>A horizontal shift and a change in the vertical asymptote.</p> Signup and view all the answers

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

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

    What is the effect of changing the value of a in the function y = a/(x + p) + q?

    <p>A change in the shape of the graph and a shift in the vertical asymptote.</p> Signup and view all the answers

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

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

    What is the effect of changing the value of q in the function y = ab^(x + p) + q?

    <p>A vertical shift and a change in the horizontal asymptote.</p> Signup and view all the answers

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

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

    What is the effect of changing the value of b in the function y = ab^(x + p) + q?

    <p>A change in the direction of the graph and a shift in the vertical asymptote.</p> Signup and view all the answers

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

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

    What is the effect of changing the value of a in the function y = ab^(x + p) + q?

    <p>A change in the shape of the graph and a shift in the horizontal asymptote.</p> Signup and view all the answers

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