Gr 11 Math June P1 (Hard)

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What is the value of $r$ in the equation $r^3 = 27$?

$\sqrt[3]{27}$

9/3

3 (correct)

9

Simplify the expression $\sqrt[4]{16a^4b^4}$.

2ab (correct)

2a^2b^2

4ab

4a^2b^2

What is the value of $a$ in the equation $a^{-rac{1}{2}} = rac{1}{3}$?

1/9

3

1/3

9 (correct)

Rationalize the denominator of the expression $rac{1}{\sqrt{2} + 1}$.

$rac{\sqrt{2} - 1}{1}$ Signup and view all the answers

Simplify the expression $rac{\sqrt{50}}{\sqrt{2}}$.

5$\sqrt{2}$ Signup and view all the answers

What is the value of $\sqrt[3]{216}$?

6 Signup and view all the answers

Simplify the expression $\sqrt[4]{a^4b^8}$.

ab^2 Signup and view all the answers

Rationalize the denominator of the expression $rac{1}{\sqrt[3]{2} - 1}$.

$rac{\sqrt[3]{2^2} + \sqrt[3]{2} + 1}{1}$ Signup and view all the answers

What is the value of $a$ in the equation $a^2 - 4a - 5 = 0$ using the quadratic formula?

5 Signup and view all the answers

Which of the following is a rational exponent?

$rac{1}{2}$ Signup and view all the answers

What is the final step in completing the square?

Solving for $x$ Signup and view all the answers

What is the purpose of completing the square?

To find the roots of the equation Signup and view all the answers

What is the formula to find the final amount in exponential growth or decay?

$A = Pe^{rt}$ Signup and view all the answers

What is the value of $(a^{-2})^{-3}$?

$a^{-6}$ Signup and view all the answers

Simplify the expression $\sqrt[3]{a^6}$:

$a^2$ Signup and view all the answers

What is the definition of $a^{-n}$?

$rac{1}{a^n}$ Signup and view all the answers

What is the formula to solve quadratic equations?

$x = -b \pm \sqrt{b^2 - 4ac}$ Signup and view all the answers

Solve the surd equation $\sqrt{x} = 2$:

$x = 4$ Signup and view all the answers

What is $\sqrt[n]{a}$ equal to?

$a^{1/n}$ Signup and view all the answers

What is the formula for exponential growth or decay?

$P(t) = P_0 imes (1 + r)^t$ Signup and view all the answers

What is the rule for eliminating a surd from an equation?

Raise both sides of the equation to the power corresponding to the root Signup and view all the answers

What is the purpose of converting percentages to decimals?

To use the exponential growth or decay formula Signup and view all the answers

What is the final step in solving surd equations?

Check the solution Signup and view all the answers

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

Completing the square Signup and view all the answers

What is the application of exponentials in finance?

Compound interest Signup and view all the answers

What is the result of simplifying the expression $(a^2)^{-3/2}$?

$a^{-3/2}$ Signup and view all the answers

What is the first step in solving surd equations?

Isolate the surd Signup and view all the answers

What is the formula for population growth?

$P(t) = P_0 imes (1 + r)^t$ Signup and view all the answers

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

b^2 - 4ac > 0 Signup and view all the answers

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

To identify a common factor or repeated expression Signup and view all the answers

What is the formula to find the roots of a quadratic equation?

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

What is the discriminant of a quadratic equation?

b^2 - 4ac Signup and view all the answers

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

b^2 - 4ac = 0 Signup and view all the answers

What is the purpose of determining restrictions when solving quadratic equations by substitution?

To ensure the expression is defined Signup and view all the answers

How can the quadratic equation be written given the roots?

(x - r1)(x - r2) = 0 Signup and view all the answers

What is the nature of the roots when the discriminant is less than zero?

Imaginary Signup and view all the answers

What is the purpose of verifying restrictions when solving quadratic equations by substitution?

To ensure the expression is defined Signup and view all the answers

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

Multiplying by constants 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?

Real and equal Signup and view all the answers

What is the formula to calculate the discriminant Δ of a quadratic equation?

Δ = b^2 - 4ac Signup and view all the answers

What is the first step in solving a quadratic inequality?

Write the quadratic in standard form Signup and view all the answers

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

To determine where the quadratic expression is positive or negative Signup and view all the answers

What is the graphical representation of a quadratic inequality?

A parabola Signup and view all the answers

What is the purpose of solving simultaneous equations?

To find the intersection of two lines Signup and view all the answers

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

Substitution Signup and view all the answers

What is the minimum number of independent equations required to solve for n unknown variables?

n Signup and view all the answers

What is the type of roots of a quadratic equation if the discriminant Δ is greater than 0 and is a perfect square?

Rational Signup and view all the answers

What is the method of solving quadratic inequalities that involves analyzing the sign of each factor on intervals defined by the critical values?

Sign analysis Signup and view all the answers

What is the effect of changing the value of $a$ in the function $y = \frac{a}{x + p} + q$ on the shape of the graph?

The graph is reflected about the horizontal asymptote. Signup and view all the answers

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

{$y : y > q, y \in \mathbb{R}$} Signup and view all the answers

What is the effect of changing the value of $p$ in the function $y = \frac{a}{x + p} + q$ on the graph?

The graph is shifted horizontally. Signup and view all the answers

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

$y = q$ Signup and view all the answers

What is the vertical asymptote of the function $y = \frac{a}{x + p} + q$?

$x = -p$ Signup and view all the answers

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

{$x : x \in \mathbb{R}$} Signup and view all the answers

What is the effect of changing the value of $q$ in the function $y = ab^x + q$ on the graph?

The graph is shifted vertically. Signup and view all the answers

What is the axes of symmetry of the function $y = \frac{a}{x + p} + q$?

The axes of symmetry intersect at $(-p, q)$. Signup and view all the answers

What is the effect of changing the value of $a$ in the function $y = ab^x + q$ on the shape of the graph?

The graph is reflected about the horizontal asymptote. Signup and view all the answers

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

The y-intercept is at $(0, ab^p + q)$. Signup and view all the answers

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

It determines the shape of the graph, whether it is a 'frown' or a 'smile'. Signup and view all the answers

What is the domain of a quadratic function?

{x : x ∈ ℝ} Signup and view all the answers

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

To find the solution to a system of equations Signup and view all the answers

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

The method of solving for the variables Signup and view all the answers

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

Average gradient = (y_A - y_C) / (x_A - x_C) Signup and view all the answers

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

To translate the words into algebraic expressions Signup and view all the answers

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

It determines the vertical shift of the graph. Signup and view all the answers

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

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

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

It shifts the graph vertically Signup and view all the answers

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

To understand how the natural world works Signup and view all the answers

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

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

What is the main difference between a 'smile' and a 'frown' in a parabolic function of the form $y = ax^2 + q$?

The direction of the opening Signup and view all the answers

What is the effect of a horizontal shift on a quadratic function?

It shifts the graph horizontally. Signup and view all the answers

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

It shifts the graph horizontally Signup and view all the answers

What is the formula to find the y-intercept of a quadratic function?

Set x = 0 and solve for y. Signup and view all the answers

What is the purpose of checking the solution to a system of equations?

To ensure the solution is correct Signup and view all the answers

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

{y : y ≥ q, y ∈ ℝ} Signup and view all the answers

What is the main difference between solving a system of equations using graphical methods and algebraic methods?

The method of solving for the variables Signup and view all the answers

What is the method to find the equation of a parabola from the graph?

Use the coordinates of the turning point and another point. Signup and view all the answers

What is the purpose of using problem-solving strategies in real-world applications?

To solve complex problems Signup and view all the answers

What is the value of $a$ in the equation $a^{rac{3}{2}} = 8$?

16 Signup and view all the answers

Which of the following is equivalent to $\sqrt[3]{a^2b^3}$?

$a^{rac{2}{3}}b^{rac{3}{3}}$ Signup and view all the answers

What is the value of $\sqrt[4]{625a^4}$, assuming $a > 0$?

$5a$ Signup and view all the answers

Rationalize the denominator of the expression $rac{1}{\sqrt[3]{2} + 2}$.

$rac{\sqrt[3]{2} + 2}{\sqrt[3]{2} - 2}$ Signup and view all the answers

What is the value of $(2a^2)^{rac{3}{2}}$?

$2^{rac{3}{2}}a^3$ Signup and view all the answers

Simplify the expression $\sqrt[4]{a^4b^8}$, assuming $a > 0$ and $b > 0$.

$a^2b^2$ Signup and view all the answers

Which of the following is a surd?

$\sqrt{12}$ Signup and view all the answers

What is the value of $\sqrt[3]{216a^3}$, assuming $a > 0$?

$6a$ Signup and view all the answers

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

b^2 - 4ac > 0 Signup and view all the answers

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

To ensure the values of the variables do not result in undefined expressions 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?

Exactly one real root (a repeated root) Signup and view all the answers

What is the formula to calculate the discriminant Δ of a quadratic equation?

b^2 - 4ac Signup and view all the answers

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

Expanding the brackets Signup and view all the answers

What is the method of solving quadratic equations that involves identifying a common factor or repeated expression?

Solving by substitution Signup and view all the answers

What is the type of roots of a quadratic equation if the discriminant Δ is greater than 0 and is a perfect square?

Rational roots Signup and view all the answers

What is the purpose of verifying restrictions when solving quadratic equations by substitution?

To ensure the values of the variables do not result in undefined expressions Signup and view all the answers

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

b^2 - 4ac = 0 Signup and view all the answers

What is the method of solving quadratic equations that involves making a substitution for a repeated expression?

Solving by substitution 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?

Real and equal Signup and view all the answers

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

To identify the intervals where the quadratic expression is positive or negative Signup and view all the answers

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

Substitution Signup and view all the answers

What is the minimum number of independent equations required to solve for n unknown variables?

n Signup and view all the answers

What is the purpose of rationalizing the denominator in an expression involving surds?

To simplify the expression for easier manipulation Signup and view all the answers

What is the difference between exponential growth and decay?

Exponential growth refers to an increase in quantity, while exponential decay refers to a decrease Signup and view all the answers

What is the type of roots of a quadratic equation if the discriminant Δ is greater than 0 and is a perfect square?

Rational Signup and view all the answers

What is the method of solving quadratic inequalities that involves analyzing the sign of each factor on intervals defined by the critical values?

Table of signs method Signup and view all the answers

What is the result of applying the power of a power rule to the expression $(a^2)^3$?

$a^6$ Signup and view all the answers

What is the purpose of solving surd equations?

To model real-world phenomena Signup and view all the answers

What is the purpose of solving simultaneous equations?

To find the intersection of two lines Signup and view all the answers

What is the graphical representation of a quadratic inequality?

A graph of a quadratic function Signup and view all the answers

What is the formula for the final amount in exponential growth or decay, given the initial amount, growth rate, and time period?

$P(t) = P_0 * (1 + r)^t$ Signup and view all the answers

What is the first step in solving a quadratic inequality?

Write the quadratic inequality in standard form Signup and view all the answers

What is the result of simplifying the expression $a^{rac{3}{2}}$?

$\sqrt{a^3}$ Signup and view all the answers

What is the purpose of converting percentages to decimals in exponential models?

To simplify the model for easier manipulation Signup and view all the answers

What is the formula to calculate the discriminant Δ of a quadratic equation?

Δ = b^2 - 4ac Signup and view all the answers

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

The discriminant is equal to zero Signup and view all the answers

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

Substitution method Signup and view all the answers

What is the result of simplifying the expression $(a^3)^{rac{2}{3}}$?

$a$ Signup and view all the answers

What is the final amount of $1000 invested at a 5% annual growth rate for 3 years, compounded annually?

1215.51 Signup and view all the answers

What is the solution to the equation x^2 + 5x - 6 = 0 using the quadratic formula?

x = 2 or x = -3 Signup and view all the answers

What is the value of a in the equation (a^(-3))^2 = 1/64?

2 Signup and view all the answers

What is the solution to the inequality x^2 - 4x - 5 > 0?

x < -1 or x > 5 Signup and view all the answers

What is the value of x in the equation (2x)^3 = 64?

2 Signup and view all the answers

What is the solution to the equation x^2 + 2x - 6 = 0 using completing the square?

x = 2 or x = -3 Signup and view all the answers

What is the value of a in the equation a^(-2/3) = 1/4?

16 Signup and view all the answers

What is the solution to the inequality x^2 + 2x - 3 > 0?

x < -3 or x > 1 Signup and view all the answers

What is the final amount of $500 invested at a 10% annual growth rate for 5 years, compounded semiannually?

831.19 Signup and view all the answers

What is the solution to the equation x^2 - 3x - 2 = 0 using the quadratic formula?

x = 2 or x = -1 Signup and view all the answers

When solving a system of equations using substitution, what is the purpose of making one of the variables the subject of both equations?

To reduce the number of equations and the number of variables by one Signup and view all the answers

What is the effect of $q > 0$ on the graph of $y = ax^2 + q$?

The graph is shifted vertically upwards by $q$ units Signup and view all the answers

What is the purpose of problem-solving strategies in natural sciences and engineering disciplines?

To model real-world problems and understand how the natural world works Signup and view all the answers

What is the effect of $a < 0$ on the shape of the graph of $y = ax^2 + q$?

The graph is a 'frown' and has a maximum turning point Signup and view all the answers

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

To translate the words into algebraic expressions Signup and view all the answers

What is the effect of $p > 0$ on the graph of $y = a(x + p)^2 + q$?

The graph is shifted horizontally to the left by $p$ units Signup and view all the answers

What is the purpose of solving systems of equations graphically?

To find the coordinates of the points where the two graphs intersect Signup and view all the answers

What is the purpose of checking the solution to ensure it is correct when solving a word problem?

To ensure the solution is reasonable and accurate Signup and view all the answers

What is the effect of $a > 0$ on the shape of the graph of $y = ax^2 + q$?

The graph is a 'smile' and has a minimum turning point Signup and view all the answers

What is the purpose of translating the words into algebraic expressions when solving a word problem?

To write a set of equations that describe the problem mathematically Signup and view all the answers

For the function y = ab^x + q, what is the effect of increasing the value of q on the graph?

The graph is shifted vertically upwards. Signup and view all the answers

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

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

For the function y = a/(x + p) + q, what is the effect of increasing the value of a on the shape of the graph?

The graph becomes further away from the horizontal asymptote. Signup and view all the answers

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

{x : x ∈ ℝ}. Signup and view all the answers

For the function y = ab^x + q, what is the effect of increasing the value of b on the direction of the graph?

The graph becomes an increasing function. Signup and view all the answers

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

The line y = q. Signup and view all the answers

For the function y = a/(x + p) + q, what is the effect of increasing the value of p on the graph?

The graph is shifted horizontally to the left. Signup and view all the answers

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

{y : y < q, y ∈ ℝ}. Signup and view all the answers

For the function y = ab^x + q, what is the axes of symmetry?

The lines x = -p and y = q. Signup and view all the answers

What is the effect of increasing the value of a on the y-intercept of the function y = ab^x + q?

The y-intercept increases. Signup and view all the answers

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

It changes the shape of the graph, making it a 'smile' or a 'frown'. Signup and view all the answers

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

The vertical line $x = -p$. Signup and view all the answers

What is the formula to calculate the average gradient of a curve between two points $A(x_A, y_A)$ and $B(x_B, y_B)$?

$\frac{y_A - y_B}{x_A - x_B}$ Signup and view all the answers

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

($-p, q$) Signup and view all the answers

What is the effect of changing the value of $q$ in the hyperbolic function $y = \frac{a}{x} + q$ on the graph?

It shifts the graph vertically. Signup and view all the answers

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

{x : x ∈ ℝ} Signup and view all the answers

What is the range of the hyperbolic function $y = \frac{a}{x} + q$?

{y : y ∈ ℝ} Signup and view all the answers

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

Set y = 0 and solve for x. Signup and view all the answers

What is the effect of shifting the quadratic function $f(x) = a(x + p)^2 + q$ horizontally by $m$ units?

Replace x with x - m. Signup and view all the answers

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

$\frac{f(a + h) - f(a)}{h}$ Signup and view all the answers

What is the value of $a$ in the equation $a^{rac{1}{2}} = 4^{rac{1}{3}}$?

16 Signup and view all the answers

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

7 Signup and view all the answers

If $a^2 + 4a - 5 = 0$, what is the value of $a$ using the quadratic formula?

a = 1 or a = -5 Signup and view all the answers

What is the value of $P(t)$ in the exponential growth model $P(t) = P_0 imes (1 + r)^t$ when $P_0 = 1000$, $r = 0.05$, and $t = 5$?

1276.78 Signup and view all the answers

What is the value of $(a^2)^{-3/2}$?

$a^{-3/2}$ Signup and view all the answers

Simplify the expression $a^{rac{3}{2}} imes a^{rac{5}{2}}$.

a^4 Signup and view all the answers

If $x^2 + 3x + 2 = 0$, what is the method to solve for $x$?

Factoring Signup and view all the answers

What is the value of $\sqrt[4]{a^4b^8}$?

$a^2b^4$ Signup and view all the answers

What is the growth rate in the exponential growth model $P(t) = P_0 imes (1 + r)^t$ when $P_0 = 500$, $P(t) = 540$, and $t = 2$?

0.05 Signup and view all the answers

What is the purpose of converting percentages to decimals in exponential growth or decay?

To make the calculation easier Signup and view all the answers

Solve the quadratic surd equation $x^2 + 4x + 4 = \sqrt{x + 2}$.

2 Signup and view all the answers

What is the final amount in the exponential decay model $P(t) = P_0 imes (1 - r)^t$ when $P_0 = 1000$, $r = 0.03$, and $t = 10$?

737.42 Signup and view all the answers

If $a^x = 27$, what is the value of $x$?

3 Signup and view all the answers

Simplify the expression $\sqrt[3]{a^6 imes b^3}$.

a^2b Signup and view all the answers

What is the definition of $a^{-n}$?

$1/a^n$ Signup and view all the answers

What is the formula to find the final amount in exponential growth or decay?

A = P(1 + r/n)^(nt) Signup and view all the answers

What is the value of $a$ in the equation $a^{rac{2}{3}} = 2^{rac{1}{3}}$?

16 Signup and view all the answers

What is the method to solve quadratic equations with non-unitary leading coefficients?

Dividing by the coefficient of $x^2$ Signup and view all the answers

Solve the surd equation $\sqrt[3]{x - 2} = 2$.

8 Signup and view all the answers

What is the purpose of completing the square?

To solve quadratic equations Signup and view all the answers

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

If Δ < 0 Signup and view all the answers

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

Expanding the brackets Signup and view all the answers

What is the purpose of determining restrictions when solving quadratic equations by substitution?

To ensure that the values for k do not result in undefined expressions 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?

Real and equal Signup and view all the answers

What is the method of solving quadratic equations by substitution?

Identifying a common factor and substituting it with a simpler variable Signup and view all the answers

What is the formula to calculate the discriminant Δ of a quadratic equation?

Δ = b^2 - 4ac Signup and view all the answers

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

To make the equation easier to solve Signup and view all the answers

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

If Δ > 0 Signup and view all the answers

What is the step in solving quadratic equations by substitution that involves solving the resulting simpler equation?

Solving for k Signup and view all the answers

What is the purpose of verifying restrictions when solving quadratic equations by substitution?

To ensure that the values for k do not result in undefined expressions Signup and view all the answers

What is the value of $a$ in the equation $a^{rac{2}{3}} = 4$?

64 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 and is a perfect square?

Real and equal Signup and view all the answers

Simplify the expression $\sqrt[3]{a^6b^9}$.

$a^2b^3$ Signup and view all the answers

What is the first step in solving a quadratic inequality?

Write the quadratic inequality in standard form Signup and view all the answers

Rationalize the denominator of the expression $rac{2}{\sqrt[3]{3} + 2}$.

$rac{6 + 4\sqrt[3]{3} + 2\sqrt[3]{9}}{7}$ Signup and view all the answers

What is the value of $a$ in the equation $a^{-rac{1}{2}} = 2$?

$rac{1}{4}$ Signup and view all the answers

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

To analyze the sign of each factor on intervals defined by the critical values Signup and view all the answers

Simplify the expression $\left(\sqrt[4]{a^2b^4} ight)^3$.

$a^{rac{3}{2}}b^3$ Signup and view all the answers

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

Substitution method Signup and view all the answers

What is the minimum number of independent equations required to solve for n unknown variables?

n Signup and view all the answers

What is the value of $\sqrt[4]{a^4b^4}$?

$a^2b^2$ Signup and view all the answers

What is the type of roots of a quadratic equation if the discriminant Δ is greater than 0 and is not a perfect square?

Irrational Signup and view all the answers

Rationalize the denominator of the expression $rac{3}{\sqrt[5]{2} + 1}$.

$rac{3\left(\sqrt[5]{2^4} - \sqrt[5]{2^3} + \sqrt[5]{2^2} - \sqrt[5]{2} + 1 ight)}{5}$ Signup and view all the answers

Simplify the expression $\sqrt[3]{a^3b^6}$.

$a^2b^2$ Signup and view all the answers

What is the graphical representation of a quadratic inequality?

A graph that shows the regions where the quadratic expression is positive or negative Signup and view all the answers

What is the purpose of solving simultaneous equations?

To find the values of multiple variables Signup and view all the answers

What is the method of solving quadratic inequalities that involves analyzing the sign of each factor on intervals defined by the critical values?

Sign analysis method Signup and view all the answers

What is the formula to calculate the discriminant Δ of a quadratic equation?

b^2 - 4ac Signup and view all the answers

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

To express one variable in terms of the other Signup and view all the answers

What is the effect of changing the value of $a$ in the function $y = ax^2 + q$ on the shape of the graph?

It changes the width of the graph Signup and view all the answers

What is the purpose of solving simultaneous equations?

To find the values of multiple variables Signup and view all the answers

What is the minimum number of independent equations required to solve for $n$ unknown variables?

$n$ Signup and view all the answers

What is the effect of changing the value of $p$ in the function $y = a(x+p)^2 + q$ on the graph?

It changes the horizontal shift of the graph Signup and view all the answers

What is the purpose of solving word problems?

To model real-world problems Signup and view all the answers

What is the result of solving a system of equations using the elimination method?

A system of equations with reduced variables Signup and view all the answers

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

To break down the problem into manageable steps Signup and view all the answers

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

It changes the vertical shift of the graph 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?

To understand the problem and determine the solution Signup and view all the answers

What is the effect of a change in $p$ on the graph of $y = \frac{a}{x + p} + q$?

The graph is shifted horizontally by $p$ units. Signup and view all the answers

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

${y : y < q, y \in \mathbb{R}}$ Signup and view all the answers

What is the intersection point of the axes of symmetry of the hyperbola $y = \frac{a}{x + p} + q$?

($-p, q$) Signup and view all the answers

What is the effect of $a$ on the shape of the graph of $y = \frac{a}{x + p} + q$?

The graph is reflected about the horizontal asymptote if $a < 0$. Signup and view all the answers

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

$y = q$ Signup and view all the answers

What is the vertical asymptote of the function $y = \frac{a}{x + p} + q$?

$x = -p$ Signup and view all the answers

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

${x : x \in \mathbb{R}}$ Signup and view all the answers

What is the effect of $q$ on the graph of $y = \frac{a}{x + p} + q$?

The graph is shifted vertically by $q$ units. Signup and view all the answers

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

The graph is shifted vertically upwards by q units. Signup and view all the answers

What is the characteristic of the graph of $y = \frac{a}{x + p} + q$ if $a > 0$?

The graph is increasing. Signup and view all the answers

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

The graph is reflected about the horizontal asymptote if $a < 0$. Signup and view all the answers

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

{y : y ≥ q, y ∈ ℝ} Signup and view all the answers

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

Average gradient = (y2 - y1) / (x2 - x1) Signup and view all the answers

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

x = -p Signup and view all the answers

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

The graph is a 'smile' and has a minimum turning point. Signup and view all the answers

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

Set y = 0 and solve for x. Signup and view all the answers

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

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

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

Replace x with x - m. Signup and view all the answers

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

{x : x ∈ ℝ} Signup and view all the answers

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

Replace y with y + n. Signup and view all the answers

What is the value of $a$ in the equation $a^{rac{-2}{3}} = rac{1}{8}$?

8 Signup and view all the answers

Simplify the expression $\sqrt[3]{a^6b^9}$

ab^3 Signup and view all the answers

What is the value of $\sqrt[4]{81a^4}$?

3a Signup and view all the answers

Rationalize the denominator of the expression $rac{2}{\sqrt[3]{3} + 2}$

$rac{6 + 4\sqrt[3]{3}}{7}$ Signup and view all the answers

What is the value of $a$ in the equation $a^{rac{3}{4}} = \sqrt{8}$?

16 Signup and view all the answers

Simplify the expression $\sqrt[5]{a^5b^5}$

ab Signup and view all the answers

What is the value of $\sqrt[6]{a^9b^{12}}$?

$a^{rac{3}{2}}b^2$ Signup and view all the answers

Rationalize the denominator of the expression $rac{3}{\sqrt[5]{a} + 2}$

$rac{3(\sqrt[5]{a^4} - 2\sqrt[5]{a^3} + 4\sqrt[5]{a^2} - 8\sqrt[5]{a})}{21}$ Signup and view all the answers

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

1 Signup and view all the answers

What is the value of a in the equation a^(-1/2) = 1/3?

1/9 Signup and view all the answers

What is the method used to solve the equation x^2 + 4x + 4 = 0?

Completing the square Signup and view all the answers

What is the value of x in the equation x^2 - 4x - 5 = 0 when solved using the quadratic formula?

-1 Signup and view all the answers

What is the formula to find the final amount in exponential growth or decay?

A = P(1 + r/n)^(nt) Signup and view all the answers

What is the purpose of converting percentages to decimals in exponential growth or decay?

To make calculations easier Signup and view all the answers

What is the value of √(25/4) in simplest form?

5/2 Signup and view all the answers

What is the method used to solve the equation x^2 - 2x - 3 = 0?

Quadratic formula Signup and view all the answers

What is the value of a^(-m/n) in simplest form?

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

What is the purpose of using the quadratic formula?

To solve quadratic equations Signup and view all the answers

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

b^2 - 4ac < 0 Signup and view all the answers

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

To simplify the process by identifying a common factor or repeated expression Signup and view all the answers

What is the formula to calculate the discriminant Δ of a quadratic equation?

Δ = b^2 - 4ac Signup and view all the answers

What is the type of roots of a quadratic equation if the discriminant Δ is greater than 0 and is a perfect square?

Rational roots Signup and view all the answers

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

Expanding the brackets to get the quadratic equation in the standard form Signup and view all the answers

What is the purpose of determining restrictions when solving quadratic equations by substitution?

To ensure that the values of the variable do not result in undefined expressions 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?

Real and equal roots Signup and view all the answers

What is the purpose of verifying restrictions when solving quadratic equations by substitution?

To ensure that the solutions do not result in undefined expressions Signup and view all the answers

How can the quadratic equation be written given the roots?

By writing down the equation as the product of two factors Signup and view all the answers

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

b^2 - 4ac = 0 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?

Real and equal Signup and view all the answers

What is the method of solving quadratic inequalities that involves analyzing the sign of each factor on intervals defined by the critical values?

Sign analysis Signup and view all the answers

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

To identify the regions where the quadratic expression is positive or negative Signup and view all the answers

What is the type of roots of a quadratic equation if the discriminant Δ is greater than 0 and is a perfect square?

Rational Signup and view all the answers

What is the first step in solving a quadratic inequality?

Write the quadratic inequality in standard form Signup and view all the answers

What is the graphical representation of a quadratic inequality?

A graph that lies above or below the x-axis Signup and view all the answers

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

Substitution Signup and view all the answers

What is the minimum number of independent equations required to solve for n unknown variables?

n Signup and view all the answers

What is the purpose of solving simultaneous equations?

To find the values of multiple variables Signup and view all the answers

What is the formula to calculate the discriminant Δ of a quadratic equation?

b^2 - 4ac Signup and view all the answers

What is the result of simplifying the expression $(a^2b^3)^{-rac{2}{3}}$?

$a^{-rac{4}{3}}b^{-2}$ Signup and view all the answers

Which of the following is a correct step in solving the surd equation $\sqrt{x+2} = 3$?

Square both sides of the equation to eliminate the surd. Signup and view all the answers

What is the growth rate of a population that doubles in size every 5 years?

100% per 5 years Signup and view all the answers

What is the result of simplifying the expression $a^{2/3} imes a^{1/2}$?

$a^{7/6}$ Signup and view all the answers

What is the application of exponentials in population growth models?

To model the increase in population over time Signup and view all the answers

What is the result of solving the surd equation $\sqrt[3]{x^2} = 2$?

16 Signup and view all the answers

What is the formula for the final amount in exponential growth or decay?

$P(t) = P_0 imes (1 + r)^t$ Signup and view all the answers

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

To verify that the solution is correct and not extraneous Signup and view all the answers

What is the result of simplifying the expression $\sqrt[4]{a^8b^{12}}$?

$a^2b^3$ Signup and view all the answers

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

The discriminant is greater than 0 Signup and view all the answers

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

A horizontal shift Signup and view all the answers

What is the range of the function y = a/(x+p) + q if a < 0?

All real numbers less than q Signup and view all the answers

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

A vertical shift Signup and view all the answers

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

All real numbers Signup and view all the answers

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

y = q Signup and view all the answers

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

The graph is reflected about the horizontal asymptote Signup and view all the answers

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

x = -p, y = q Signup and view all the answers

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

The graph is reflected about the horizontal asymptote Signup and view all the answers

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

x = -p Signup and view all the answers

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

A horizontal shift Signup and view all the answers

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

Add or subtract the equations to eliminate one variable. Signup and view all the answers

What is the purpose of problem-solving strategies in natural sciences and engineering disciplines?

To develop mathematical models that describe real-life contexts. Signup and view all the answers

What is the effect of a negative value of $q$ on the graph of the parabolic function $y = ax^2 + q$?

The graph is shifted vertically downwards by $q$ units. Signup and view all the answers

What is the axis of symmetry of the graph of the parabolic function $y = a(x + p)^2 + q$?

The line $x = -p$. Signup and view all the answers

What is the purpose of writing a set of equations that describe the problem mathematically when solving word problems?

To model the real-life context using algebraic expressions. Signup and view all the answers

What is the effect of a positive value of $a$ on the shape of the graph of the parabolic function $y = ax^2 + q$?

The graph becomes narrower. Signup and view all the answers

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

Use the solution to substitute back into the first equation to find the value of the other unknown variable. Signup and view all the answers

What is the purpose of translating the words into algebraic expressions when solving word problems?

To model the real-life context using algebraic expressions. Signup and view all the answers

What is the effect of a negative value of $a$ on the shape of the graph of the parabolic function $y = ax^2 + q$?

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

What is the purpose of solving systems of linear equations in real-life contexts?

To describe the relationships between different quantities in a real-life context. Signup and view all the answers

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

The graph is shifted vertically by $q$ units. Signup and view all the answers

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

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

What is the domain of a quadratic function?

{x : x ∈ ℝ} Signup and view all the answers

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

{y : y ∈ ℝ, y ≥ q} Signup and view all the answers

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

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

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

The gradient of the line passing through the two points Signup and view all the answers

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

Average gradient = (y2 - y1) / (x2 - x1) Signup and view all the answers

What is the effect of a horizontal shift on the graph of a quadratic function?

The graph is shifted horizontally by the shift value Signup and view all the answers

What is the formula for the equation of a parabola from the graph?

y = a(x - x1)(x - x2) Signup and view all the answers

What is the characteristic of the graph of a hyperbolic function?

The graph has a horizontal asymptote Signup and view all the answers

Gr 11 Math June P1 (Hard)

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Questions and Answers

What is the value of $r$ in the equation $r^3 = 27$?

Simplify the expression $\sqrt[4]{16a^4b^4}$.

What is the value of $a$ in the equation $a^{-rac{1}{2}} = rac{1}{3}$?

Rationalize the denominator of the expression $rac{1}{\sqrt{2} + 1}$.

Simplify the expression $rac{\sqrt{50}}{\sqrt{2}}$.

What is the value of $\sqrt[3]{216}$?

Simplify the expression $\sqrt[4]{a^4b^8}$.

Rationalize the denominator of the expression $rac{1}{\sqrt[3]{2} - 1}$.

What is the value of $a$ in the equation $a^2 - 4a - 5 = 0$ using the quadratic formula?

Which of the following is a rational exponent?

What is the final step in completing the square?

What is the purpose of completing the square?

What is the formula to find the final amount in exponential growth or decay?

What is the value of $(a^{-2})^{-3}$?

Simplify the expression $\sqrt[3]{a^6}$:

What is the definition of $a^{-n}$?

What is the formula to solve quadratic equations?

Solve the surd equation $\sqrt{x} = 2$:

What is $\sqrt[n]{a}$ equal to?

What is the formula for exponential growth or decay?

What is the rule for eliminating a surd from an equation?

What is the purpose of converting percentages to decimals?

What is the final step in solving surd equations?

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

What is the application of exponentials in finance?

What is the result of simplifying the expression $(a^2)^{-3/2}$?

What is the first step in solving surd equations?

What is the formula for population growth?

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

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

What is the formula to find the roots of a quadratic equation?

What is the discriminant of a quadratic equation?

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

What is the purpose of determining restrictions when solving quadratic equations by substitution?

How can the quadratic equation be written given the roots?

What is the nature of the roots when the discriminant is less than zero?

What is the purpose of verifying restrictions when solving quadratic equations by substitution?

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

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

What is the formula to calculate the discriminant Δ of a quadratic equation?

What is the first step in solving a quadratic inequality?

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

What is the graphical representation of a quadratic inequality?

What is the purpose of solving simultaneous equations?

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

What is the minimum number of independent equations required to solve for n unknown variables?

What is the type of roots of a quadratic equation if the discriminant Δ is greater than 0 and is a perfect square?

What is the method of solving quadratic inequalities that involves analyzing the sign of each factor on intervals defined by the critical values?

What is the effect of changing the value of $a$ in the function $y = \frac{a}{x + p} + q$ on the shape of the graph?

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

What is the effect of changing the value of $p$ in the function $y = \frac{a}{x + p} + q$ on the graph?

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

What is the vertical asymptote of the function $y = \frac{a}{x + p} + q$?

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

What is the effect of changing the value of $q$ in the function $y = ab^x + q$ on the graph?

What is the axes of symmetry of the function $y = \frac{a}{x + p} + q$?

What is the effect of changing the value of $a$ in the function $y = ab^x + q$ on the shape of the graph?

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

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

What is the domain of a quadratic function?

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

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

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

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

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

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

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

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

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

What is the main difference between a 'smile' and a 'frown' in a parabolic function of the form $y = ax^2 + q$?

What is the effect of a horizontal shift on a quadratic function?

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

What is the formula to find the y-intercept of a quadratic function?

What is the purpose of checking the solution to a system of equations?

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

What is the main difference between solving a system of equations using graphical methods and algebraic methods?