Gr 11 Math Nov P1 (Hard)

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What is the value of $r$ when $r^{3} = 64$?

4 (correct)

2

6

3

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

4a

2a

a^2

2a^2 (correct)

Rationalize the denominator of the expression $rac{2}{\sqrt{3}}$:

$rac{\sqrt{6}}{6}$

$rac{2\sqrt{3}}{3}$ (correct)

$rac{2\sqrt{6}}{6}$

$rac{\sqrt{3}}{3}$

What is the value of $a$ when $a^{3/2} = 9$?

6 Signup and view all the answers

Simplify the expression $\sqrt[3]{8a^{3}}$:

2a Signup and view all the answers

What is the value of $x$ when $x^{-2/3} = rac{1}{4}$?

8 Signup and view all the answers

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

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

What is the value of $a$ when $a^{2/3} = 4$?

32 Signup and view all the answers

What is the expression for a^(-m) in terms of positive exponents?

1/a^m Signup and view all the answers

What is the rule for raising a power to another power?

(a^m)^n = a^(mn) Signup and view all the answers

What is the purpose of rationalizing denominators?

To eliminate radicals from the denominator Signup and view all the answers

When solving surd equations, what is the first step?

Isolate the surd term Signup and view all the answers

What is the formula for exponential growth or decay?

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

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

To make manipulation easier Signup and view all the answers

When solving quadratic surd equations, what is the second step?

Square both sides of the equation Signup and view all the answers

What is the formula for a^(m/n) in terms of roots?

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

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

1 Signup and view all the answers

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

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

What is the formula for exponential growth or decay?

P = P₀e^rt Signup and view all the answers

What is the purpose of completing the square method?

To solve quadratic equations Signup and view all the answers

What is the value of a⁰, where a is not equal to 0?

1 Signup and view all the answers

What is the formula for quadratic equations?

x = (-b ± √(b² - 4ac)) / 2a Signup and view all the answers

What is the definition of irrational numbers?

Non-terminating and non-recurring decimals Signup and view all the answers

What is the formula for the nth root of a number?

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

What is the formula for rational exponents?

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

What is the purpose of converting percentages to decimal form?

To make calculations easier Signup and view all the answers

What is the definition of natural numbers?

Whole numbers, excluding 0 Signup and view all the answers

What is the formula for completing the square?

x² + bx + c = (x + b/2a)² - (b² - 4ac) / 4a² Signup and view all the answers

What is the value of x in a quadratic equation if b² - 4ac = 0?

One distinct real root Signup and view all the answers

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

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

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

x = -b ± √(b² - 4ac) / 2a Signup and view all the answers

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

b² - 4ac < 0 Signup and view all the answers

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

Identify the repeated expression Signup and view all the answers

What is the purpose of the discriminant in a quadratic equation?

To determine the nature of the roots Signup and view all the answers

What is the formula used to find the discriminant of a quadratic equation?

Δ = b² - 4ac Signup and view all the answers

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

b² - 4ac = 0 Signup and view all the answers

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

Write the final answer Signup and view all the answers

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

To ensure the values do not make the denominator equal to zero 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 for the discriminant Δ of a quadratic equation in standard form ax^2 + bx + c = 0?

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

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

To identify where the quadratic expression equals zero Signup and view all the answers

What is the final step when solving a quadratic inequality?

Summarize the solution intervals on a number line Signup and view all the answers

What is the purpose of the sign analysis in quadratic inequalities?

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

What is the condition for a system of simultaneous equations to have a unique solution?

The system has the same number of independent equations as variables Signup and view all the answers

What is the advantage of using a graphical approach when solving quadratic inequalities?

It helps to visualize the solution Signup and view all the answers

What type of quadratic inequalities require special consideration when solving?

Inequalities with fractions Signup and view all the answers

What is the first step when solving a quadratic inequality?

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

What is the purpose of solving quadratic inequalities?

To identify the regions where a quadratic expression satisfies an inequality Signup and view all the answers

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

2a Signup and view all the answers

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

A vertical shift Signup and view all the answers

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

y = q Signup and view all the answers

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

A change in the shape of the graph Signup and view all the answers

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

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

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

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

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

A horizontal shift Signup and view all the answers

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

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

What is the main advantage of using the substitution method to solve a system of linear equations?

It reduces the number of variables by one in the first step. Signup and view all the answers

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

x = -p Signup and view all the answers

What is the effect of a change in q on the exponential function y = ab^x + q?

A vertical shift 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 identify the key elements of the problem that need to be addressed. Signup and view all the answers

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

A horizontal shift Signup and view all the answers

What is the characteristic of a quadratic sequence?

The second difference is constant. Signup and view all the answers

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

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

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

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

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

x = -p Signup and view all the answers

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

Add the two equations to eliminate one of the variables. Signup and view all the answers

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

Set x = 0 Signup and view all the answers

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

To describe the problem mathematically. Signup and view all the answers

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

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

What is the general form of a quadratic sequence?

Tn = an^2 + bn + c Signup and view all the answers

What is the purpose of solving for the variables in a system of linear equations?

To find the values of the variables that satisfy both equations. Signup and view all the answers

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

$\frac{y_A - y_C}{x_A - x_C}$ 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?

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

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

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

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

To ensure that the solution is reasonable and makes sense in the context of the problem. Signup and view all the answers

How do you shift a parabola $m$ units to the right?

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

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

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

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

A vertical shift Signup and view all the answers

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

A horizontal shift to the left by $p$ units Signup and view all the answers

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

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

What is the formula for the book value or depreciated value in simple depreciation?

A = P(1 - in) Signup and view all the answers

What is the characteristic of the depreciation amount in compound depreciation?

It changes every year, decreasing as the asset's value decreases Signup and view all the answers

What is the purpose of using timelines in compound interest?

To visualize the periods and changes in compounding and interest rates 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 lies above or below the horizontal asymptote Signup and view all the answers

What is the formula for finding the annual depreciation rate ($i$) in simple decay?

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

What is the characteristic of the book value in compound depreciation?

It decreases exponentially over time Signup and view all the answers

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

y = q Signup and view all the answers

What is the purpose of modifying the interest rate and the number of compounding periods?

To adjust for compounding periods Signup and view all the answers

What is the formula for calculating the accumulated value at the end of each segment in a compound interest problem?

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

What is the difference between nominal and effective interest rates?

Nominal interest rate is the rate per year, while effective interest rate is the rate per compounding period. Signup and view all the answers

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

P(A and B) = P(A) × P(B) Signup and view all the answers

What is the definition of conditional probability?

The probability of an event occurring given that another event has occurred. Signup and view all the answers

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

To calculate the accumulated value at the end of each segment. Signup and view all the answers

What is the formula for calculating the effective interest rate?

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

What is the difference between dependent and independent events?

Dependent events are affected by the occurrence of other events, while independent events are not. Signup and view all the answers

What is the purpose of using Venn diagrams in probability theory?

To visualize and represent the relationships between events and their probabilities. Signup and view all the answers

What is the formula for calculating the probability of two mutually exclusive events?

P(A or B) = P(A) + P(B) Signup and view all the answers

What is the purpose of calculating the accumulated value at the end of each segment in a compound interest problem?

To adjust the principal for the next period and calculate the accumulated value at the end of the next segment. Signup and view all the answers

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

$P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$ Signup and view all the answers

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

$P(A) imes P(B) = P(A \cap B)$ Signup and view all the answers

What is the purpose of a tree diagram?

To organize and visualize the different possible outcomes of a sequence of events Signup and view all the answers

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

$P(A \cap B) / P(B)$ Signup and view all the answers

What is the purpose of a contingency table?

To test for independence between two events Signup and view all the answers

What is the formula for the probability of an event A in a contingency table?

$n(A) / n(Total)$ Signup and view all the answers

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

$P(A) imes P(B)$ Signup and view all the answers

What is the condition for three events A, B, and C to be mutually exclusive?

$P(A \cap B) = P(A \cap C) = P(B \cap C) = 0$ Signup and view all the answers

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

$P(Event 1) imes P(Event 2) imes \ldots$ Signup and view all the answers

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

To test whether two events are independent or dependent Signup and view all the answers

What is the value of $x$ when $x^{rac{3}{4}} = 8$?

2 Signup and view all the answers

Simplify the expression $\sqrt[3]{81a^6}$:

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

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

$rac{3\sqrt[5]{32}}{2}$ Signup and view all the answers

What is the value of $a$ when $a^{rac{2}{3}} = 9$?

27 Signup and view all the answers

Simplify the expression $\sqrt[4]{16a^8}$:

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

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

$rac{2\sqrt[3]{25}}{5}$ Signup and view all the answers

What is the value of $x$ when $x^{-rac{2}{5}} = rac{1}{9}$?

3 Signup and view all the answers

Simplify the expression $\sqrt[5]{32a^5}$:

$2a$ Signup and view all the answers

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

Δ > 0 Signup and view all the answers

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

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

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

x = -b ± √(b² - 4ac) / 2a Signup and view all the answers

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

Two complex conjugate roots Signup and view all the answers

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

To ensure that the solutions are valid Signup and view all the answers

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

$1/a^{3/2}$ Signup and view all the answers

If $P(t) = P_0 imes (1 + r)^t$, what is the value of $r$ when $P(2) = 2P_0$?

$0.5$ Signup and view all the answers

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

Δ = 0 Signup and view all the answers

What is the value of $(a^{1/2} + b^{1/2})^2$ in terms of $a$ and $b$?

$a + b + 2\sqrt{ab}$ Signup and view all the answers

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

Δ = b² - 4ac Signup and view all the answers

If $\sqrt{x} = 2$, what is the value of $x$?

$4$ Signup and view all the answers

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

Determine the restrictions for the variable x Signup and view all the answers

What is the purpose of the discriminant in a quadratic equation?

To determine the nature of the roots Signup and view all the answers

What is the value of $a^{-2/3} imes a^{1/2}$ in terms of $a$?

$a^{-1/6}$ Signup and view all the answers

If $P(t) = P_0 imes (1 + r)^t$ and $P(1) = 1.2P_0$, what is the value of $r$?

$0.2$ Signup and view all the answers

What is the purpose of converting rational exponents to radical form?

To enable the application of exponent rules Signup and view all the answers

What is the value of $(a^2 + b^2)^{-1/2}$ in terms of $a$ and $b$?

$1/\sqrt{a^2 + b^2}$ Signup and view all the answers

If $x^2 = 4$, what is the value of $x$?

$\pm 2$ Signup and view all the answers

When using the quadratic formula, what is the significance of the expression b^2 - 4ac?

It determines the number of solutions to the equation Signup and view all the answers

What is the value of $a^{3/2} imes a^{-1/2}$ in terms of $a$?

$a$ Signup and view all the answers

What is the advantage of using the completing the square method to solve quadratic equations?

It provides a more intuitive understanding of the equation's structure Signup and view all the answers

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

To ensure the expression is in its simplest form Signup and view all the answers

If $P(t) = P_0 imes (1 + r)^t$ and $P(3) = 2.5P_0$, what is the value of $r$?

$0.25$ Signup and view all the answers

When solving exponential growth or decay problems, what is the importance of identifying the initial amount, growth rate, and time period?

It is necessary for finding the final amount Signup and view all the answers

What is the definition of an irrational number?

A number that cannot be expressed as a finite decimal Signup and view all the answers

When solving quadratic equations, what is the purpose of checking solutions?

To ensure the solution satisfies the original equation Signup and view all the answers

What is the formula for finding the nth root of a number?

a^(1/n) = nth root of a Signup and view all the answers

What is the purpose of converting percentages to decimal form when solving exponential growth or decay problems?

To enable the application of the exponential formula Signup and view all the answers

When solving quadratic equations, what is the significance of the expression a^2 + b^2 = c^2?

It is a Pythagorean identity with no relevance to quadratic equations Signup and view all the answers

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

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

Which of the following is a common application of quadratic sequences in real-life scenarios?

All of the above 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?

The graph shifts vertically upwards Signup and view all the answers

Which step is essential in solving a system of equations using the substitution method?

Solving for one variable in terms of the other Signup and view all the answers

What is the purpose of solving simultaneous equations in problem-solving strategies?

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

Which of the following is a characteristic of a quadratic sequence?

The second difference between consecutive terms is constant Signup and view all the answers

What is the last step in solving a quadratic equation using the substitution method?

Using the solution to find the value of the other variable Signup and view all the answers

Which of the following is a type of problem that requires the use of mathematical language to describe real-life contexts?

Word problem Signup and view all the answers

What is the primary purpose of using a graphical approach in solving quadratic inequalities?

To visualize the solution set Signup and view all the answers

What is the purpose of solving systems of equations in problem-solving strategies?

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

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

x = -p Signup and view all the answers

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

A change in the shape of the graph in quadrants Signup and view all the answers

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

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

What is the average gradient of a curve at point P when h is not equal to 0?

2a + h Signup and view all the answers

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

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

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

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

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

x = -p Signup and view all the answers

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

A horizontal shift Signup and view all the answers

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

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

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

It changes the turning point. Signup and view all the answers

What is the formula for the x-intercept of the hyperbolic function y = a/(x + p) + q?

Set y = 0 Signup and view all the answers

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

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

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

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

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

A vertical shift Signup and view all the answers

What is the formula for the y-intercept of the hyperbolic function y = a/(x + p) + q?

Set x = 0 Signup and view all the answers

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

It changes the y-intercept. 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 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 change in p on the graph of y = a(x + p)^2 + q?

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

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

A horizontal shift to the left by p units Signup and view all the answers

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

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

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

A = P(1 - i)^n Signup and view all the answers

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

A change in the position of the horizontal asymptote Signup and view all the answers

What is the purpose of using timelines in compounding?

To visualize the compounding periods Signup and view all the answers

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

A = P(1 - in) Signup and view all the answers

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

A vertical shift Signup and view all the answers

What is the key characteristic of compound depreciation?

The depreciation amount decreases each year Signup and view all the answers

What is the purpose of finding the depreciation rate (i) in depreciation problems?

To solve for the depreciation rate in simple or compound decay Signup and view all the answers

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

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

What is the probability of the event A occurring given that event B has occurred, if P(A) = 0.3, P(B) = 0.4, and P(A and B) = 0.2?

0.67 Signup and view all the answers

When calculating the accumulated value, what is the correct formula to use for each segment?

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

In a tree diagram, what is the probability of a path that consists of two events, A and B, with probabilities 0.6 and 0.7, respectively?

0.42 Signup and view all the answers

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

P(A) + P(B) + P(C) - P(A and B) - P(A and C) - P(B and C) + P(A and B and C) Signup and view all the answers

What is the primary purpose of breaking down periods when calculating compound interest?

To adjust for changes in interest rates or deposits/withdrawals Signup and view all the answers

What is the correct formula for calculating the effective interest rate?

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

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

P(A and B) = P(A) × P(B) Signup and view all the answers

When events are dependent, what is the correct formula for calculating the probability of both events occurring?

P(A and B) = P(A) × P(B | A) Signup and view all the answers

What is the purpose of a contingency table in probability?

To test for independence between two events Signup and view all the answers

What is the formula for the probability of an event A occurring, given that event B has occurred, if the events are dependent?

P(A and B) / P(B) Signup and view all the answers

What is the correct formula for calculating the probability of the union of two mutually exclusive events?

P(A or B) = P(A) + P(B) Signup and view all the answers

What is the probability of a specific path in a tree diagram, if the probabilities of the individual events are 0.4, 0.5, and 0.6?

0.12 Signup and view all the answers

What is the purpose of using Venn diagrams in probability theory?

To visualize the relationships between events Signup and view all the answers

When calculating the accumulated value, what is the correct formula to use for the initial period?

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

What is the formula for the probability of two events, A and B, occurring simultaneously, if they are independent?

P(A) × P(B) Signup and view all the answers

What is the purpose of a tree diagram in probability?

To organize and visualize the different possible outcomes of a sequence of events Signup and view all the answers

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

To determine the total accumulated value at the end of the investment or loan term Signup and view all the answers

What is the formula for the probability of an event A occurring, given that event B has not occurred, if P(A) = 0.3, P(B) = 0.4, and P(A and B) = 0.2?

0.43 Signup and view all the answers

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

P(A and B) = P(A) × P(B) Signup and view all the answers

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

P(A | B) = P(A and B) / P(B) 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?

Non-real Signup and view all the answers

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

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

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

Δ > 0 Signup and view all the answers

What is the advantage of using a graphical approach when solving quadratic inequalities?

It helps to visualize the solution Signup and view all the answers

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

To determine where the quadratic expression equals zero Signup and view all the answers

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

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

What is the purpose of solving quadratic inequalities?

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

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

Δ = 0 Signup and view all the answers

What is the final step when solving a quadratic inequality?

To write the final solution and represent it on a number line Signup and view all the answers

What is the purpose of the sign analysis in quadratic inequalities?

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

What is the result of raising both sides of the equation √x = 2 to the power 2?

x = 4 Signup and view all the answers

What is the exponential form of the surd √(a^3)?

a^(3/2) Signup and view all the answers

In an exponential growth model, what happens to the quantity over time?

It increases by a fixed percentage each period Signup and view all the answers

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

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

What is the rule for raising a power to another power?

(a^m)^n = a^(m*n) Signup and view all the answers

What is the expression for a^(-m) in terms of positive exponents?

1 / a^m Signup and view all the answers

What is the purpose of rationalizing denominators?

To eliminate the surd from the denominator Signup and view all the answers

What is the formula for simplifying the expression a^(m/n)?

√[a^(m/n)] Signup and view all the answers

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

1 Signup and view all the answers

What is the purpose of converting percentages to decimal form?

To make calculations easier Signup and view all the answers

What is the value of x in the equation $x^{rac{2}{3}} = 25$?

5 Signup and view all the answers

Simplify the expression $\sqrt[4]{a^{12}}$:

$a^{3/4}$ Signup and view all the answers

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

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

What is the purpose of rationalizing denominators in exponential equations?

To eliminate irrational numbers 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 = P₀e^(rt) Signup and view all the answers

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

16 Signup and view all the answers

Simplify the expression $\sqrt[5]{a^3b^5}$:

$a^{1/5}b^{5/5}$ Signup and view all the answers

What is the result of completing the square for the quadratic equation ax^2 + bx + c = 0?

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

What is the value of $x$ in the equation $x^{rac{3}{2}} = 27$?

9 Signup and view all the answers

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

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

If a quadratic equation has two distinct real roots, what can be said about the discriminant?

It is greater than 0 Signup and view all the answers

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

To simplify the process significantly Signup and view all the answers

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

To make calculations easier Signup and view all the answers

Rationalize the denominator of the expression $rac{3}{\sqrt[4]{a}}$:

$rac{3\sqrt[4]{a^3}}{a}$ Signup and view all the answers

What is the definition of natural numbers?

Positive integers Signup and view all the answers

Simplify the expression $\sqrt[3]{a^2b^3}$:

$a^{1/3}b$ Signup and view all the answers

Given the roots of a quadratic equation, how can we determine the original quadratic equation?

By working backwards and using the product of two factors 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?

Complex conjugate Signup and view all the answers

What is the formula for the quadratic formula in terms of the roots?

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

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

Δ = 0 Signup and view all the answers

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

To transform the equation into a perfect square trinomial Signup and view all the answers

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

Identify restrictions Signup and view all the answers

What is the definition of irrational numbers?

Non-terminating, non-recurring decimals Signup and view all the answers

What is the formula for the nth root of a number?

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

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

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

What is the purpose of the discriminant in a quadratic equation?

To determine the nature of the roots Signup and view all the answers

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

Write the final answer Signup and view all the answers

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

Δ < 0 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?

Imaginary Signup and view all the answers

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

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

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

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

What is the last step when solving a quadratic inequality?

Write the final solution and represent it on a number line Signup and view all the answers

What is the advantage of using a graphical approach when solving quadratic inequalities?

It provides a visual representation of the solution Signup and view all the answers

What type of quadratic inequalities require special consideration when solving?

Inequalities with fractions Signup and view all the answers

What is the purpose of the sign analysis in quadratic inequalities?

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

What is the condition for a system of simultaneous equations to have a unique solution?

The system must have the same number of equations as variables Signup and view all the answers

What is the first step when solving a quadratic inequality?

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

What is the purpose of solving quadratic inequalities?

To identify the region where the quadratic function is positive or negative Signup and view all the answers

What is the average gradient of the curve y = g(x) at point P(a, g(a)) when h = 0?

2a Signup and view all the answers

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

A vertical shift of the graph Signup and view all the answers

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

The line y = q Signup and view all the answers

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

All real numbers except x = -p Signup and view all the answers

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

All real numbers except y = q Signup and view all the answers

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

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

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

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

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

A change in the shape of the graph Signup and view all the answers

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

A horizontal shift of the graph Signup and view all the answers

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

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

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

x = -p Signup and view all the answers

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

x = -p Signup and view all the answers

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

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

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

y = q Signup and view all the answers

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

(p, q) Signup and view all the answers

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

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

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

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

What is the primary purpose of solving systems of equations using substitution or elimination methods?

To find the values of the variables that satisfy both equations Signup and view all the answers

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

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

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

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

In a quadratic sequence, what is the significance of the second difference being constant?

It indicates that the sequence is quadratic Signup and view all the answers

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

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

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

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

What happens to the graph of the function y = ab^(x+p) + q when the value of p increases?

The graph shifts to the left by p units. Signup and view all the answers

When solving word problems, what is the first step in the problem-solving strategy?

Read the problem carefully Signup and view all the answers

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

The graph lies below the horizontal asymptote, y = q. Signup and view all the answers

What is the purpose of solving systems of equations graphically?

To visualize the relationship between the equations Signup and view all the answers

What is the difference between simple and compound depreciation?

Simple depreciation is calculated on the initial value, while compound depreciation is calculated on the remaining value. Signup and view all the answers

In a quadratic sequence, how can the values of a, b, and c be determined?

By solving simultaneous equations Signup and view all the answers

What is the purpose of using a timeline in determining the interest rate?

To visualize the compounding periods. Signup and view all the answers

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

To ensure a systematic approach to solving the problem Signup and view all the answers

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

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

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

The horizontal asymptote changes to y = q. Signup and view all the answers

What is the formula for compound depreciation?

A = P(1 - i)^n Signup and view all the answers

What is the purpose of solving quadratic equations?

To find the values of the variables that satisfy the equation Signup and view all the answers

What is the significance of the turning point of a parabolic function?

It is the minimum or maximum value of the function Signup and view all the answers

What is the purpose of finding the depreciation rate (i)?

To determine the depreciation rate for a given period. Signup and view all the answers

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

The graph is stretched vertically by a factor of b. Signup and view all the answers

What is the formula for simple depreciation?

A = P(1 - in) Signup and view all the answers

What is the purpose of using the formula A = P(1 - i)^n or A = P(1 - in)?

To find the book value of the asset. Signup and view all the answers

What is the probability of an event A, given the occurrence of event B, if P(A) = 0.4, P(B) = 0.3, and P(A and B) = 0.2?

0.67 Signup and view all the answers

When calculating the accumulated value, what is the purpose of breaking down the timeline into segments?

To account for changes in interest rates or additional transactions Signup and view all the answers

If three events A, B, and C are independent, what is the probability of A, B, and C occurring simultaneously, given P(A) = 0.2, P(B) = 0.3, and P(C) = 0.4?

0.024 Signup and view all the answers

What is the formula for calculating the effective interest rate?

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

In a tree diagram, what does the probability of a path represent?

The probability of a sequence of events Signup and view all the answers

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

P(A) × P(B) Signup and view all the answers

What is the purpose of a contingency table?

To test for independence between events Signup and view all the answers

What is the formula for the probability of three events A, B, and C occurring simultaneously, if they are not independent?

P(A) × P(B|A) × P(C|A and B) Signup and view all the answers

What is the formula for calculating the accumulated value using the compound interest formula?

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

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

P(A and B) = P(A) × P(B) Signup and view all the answers

What is the definition of mutually exclusive events?

Events that cannot occur at the same time Signup and view all the answers

What is the formula for the probability of an event A, given the occurrence of event B, in terms of counts?

n(A and B) / n(B) Signup and view all the answers

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

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

What is the purpose of the complement rule in probability?

To find the probability of an event not occurring Signup and view all the answers

What is the purpose of the general addition rule?

To calculate the probability of the union of three events Signup and view all the answers

What is the formula for calculating the accumulated value at the end of each segment in a break-even period?

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

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

P(A) + P(B) + P(C) - P(A and B) - P(A and C) - P(B and C) + P(A and B and C) Signup and view all the answers

What is the purpose of a tree diagram in calculating probabilities?

To calculate the probability of a sequence of events Signup and view all the answers

What is the definition of dependent events?

Events that affect the probability of each other Signup and view all the answers

What is the formula for calculating the probability of two events A and B occurring, when they are not mutually exclusive?

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

What is the value of $x$ when $x^{rac{3}{2}} = 27$?

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

Simplify the expression $\sqrt[3]{216a^3}$

$6a Signup and view all the answers

Rationalize the denominator of the expression $rac{2}{\sqrt{5} + 1}$

$rac{2(\sqrt{5} - 1)}{4}$ Signup and view all the answers

What is the value of $a$ when $a^{rac{2}{3}} = 16$?

32 Signup and view all the answers

Simplify the expression $\sqrt[4]{81a^4}$

$3a$ Signup and view all the answers

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

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

What is the value of $x$ when $x^{rac{4}{5}} = 32$?

64 Signup and view all the answers

Simplify the expression $\sqrt[5]{1024a^5}$

$4a$ Signup and view all the answers

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

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

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

To identify the repeated expression Signup and view all the answers

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

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

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

Take the square root of both sides Signup and view all the answers

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

To make the growth rate more intuitive Signup and view all the answers

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

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

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

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

What is the definition of irrational numbers?

Non-terminating, non-recurring decimals Signup and view all the answers

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

To check the solutions against any restrictions Signup and view all the answers

What is the formula for exponential growth or decay?

P = P_0(1 + r)^t 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?

The roots are complex conjugates Signup and view all the answers

What is the purpose of the discriminant in a quadratic equation?

To determine the number of solutions Signup and view all the answers

What is the formula for the nth root of a number?

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

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

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

What is the definition of natural numbers?

Positive integers, excluding 0 Signup and view all the answers

What is the purpose of the discriminant in a quadratic equation?

To determine the nature of the roots Signup and view all the answers

What is the purpose of rationalizing denominators?

To eliminate surds Signup and view all the answers

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

Identify the repeated expression 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 the quadratic formula?

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

What is the nature of the roots of a quadratic equation if the discriminant Δ is equal to 1?

Real and unequal Signup and view all the answers

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

To determine the intervals where the function is positive or negative Signup and view all the answers

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

Δ = 0 Signup and view all the answers

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

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

What is the final step when solving a quadratic inequality?

Write the final solution and represent it on a number line Signup and view all the answers

What is the purpose of the sign analysis in quadratic inequalities?

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

What type of quadratic inequalities require special consideration when solving?

Inequalities involving fractions Signup and view all the answers

What is the advantage of using a graphical approach when solving quadratic inequalities?

It provides a visual representation of the solution Signup and view all the answers

What is the condition for a system of simultaneous equations to have a unique solution?

The number of equations is equal to the number of variables Signup and view all the answers

What is the purpose of solving quadratic inequalities?

To determine the intervals where a quadratic function is positive or negative Signup and view all the answers

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

Solve for one of the variables in the first equation and substitute it into the second equation. Signup and view all the answers

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

To translate the given information into algebraic expressions. Signup and view all the answers

What is the characteristic of a quadratic sequence?

The second difference between consecutive terms is constant. Signup and view all the answers

What is the effect of a positive value of q 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 solving a system of linear equations?

To determine the values of the variables that satisfy both equations. Signup and view all the answers

What is the final step in solving a word problem?

Check the solution to ensure it is correct. Signup and view all the answers

What is the advantage of using the elimination method to solve a system of linear equations?

It eliminates the need for substitution. Signup and view all the answers

What is the purpose of finding the general term of a quadratic sequence?

To determine the values of a, b, and c. Signup and view all the answers

What is the characteristic of a quadratic function?

The graph is a parabola. Signup and view all the answers

What is the purpose of solving quadratic equations?

To model real-world problems that involve quadratic relationships. Signup and view all the answers

What is the correct form of the exponential expression $a^{rac{3}{2}}$?

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

When solving surd equations, what step should be taken after eliminating the surd?

Solve the resulting equation. Signup and view all the answers

What is the purpose of using exponential laws when solving surd equations?

To apply the necessary exponent rules to simplify the terms. Signup and view all the answers

What is the formula for exponential growth or decay when the initial amount is $P_0$ and the growth rate is $r$?

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

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

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

What is the correct form of the expression $a^{-rac{2}{3}}$?

$rac{1}{a^{rac{2}{3}}}$ Signup and view all the answers

What is the purpose of using fractional exponents when solving surd equations?

To represent roots and powers using fractional exponents. Signup and view all the answers

What is the correct form of the expression $a^{rac{3}{4}}$?

$\sqrt[4]{a^3}$ Signup and view all the answers

What is the purpose of using exponent laws when solving exponential problems?

To apply the necessary exponent rules to simplify the terms. Signup and view all the answers

What is the correct form of the expression $(a^2)^{rac{1}{3}}$?

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

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

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

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

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

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

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

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

A horizontal shift Signup and view all the answers

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

(p, q) Signup and view all the answers

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

x = -p Signup and view all the answers

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

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

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

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

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

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

What is the equation of the horizontal asymptote for the graph of y = ab^x + q?

y = q Signup and view all the answers

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

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

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

A vertical shift Signup and view all the answers

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

The graph is shifted up by q units Signup and view all the answers

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

y = x + c, y = -x + c Signup and view all the answers

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

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

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

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

What is the turning point of the graph of y = a(x + p)^2 + q when a < 0?

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

What is the purpose of sketching the graph of y = a(x + p)^2 + q?

To visualize the characteristics of the graph Signup and view all the answers

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

A change in the shape of the graph Signup and view all the answers

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

2a Signup and view all the answers

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

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

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

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

What is the formula for the depreciated value of an asset using compound depreciation?

A = P(1 - i)^n Signup and view all the answers

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

The graph is shifted vertically by q units. 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 purpose of using a timeline in compound interest problems?

To visualize the periods and changes in compounding and interest rates Signup and view all the answers

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

i = (P - A) / (Pn) Signup and view all the answers

What is the characteristic of the graph of y = ab^(x+p) + q when a < 0?

The graph lies below the horizontal asymptote 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 scaled vertically by a factor of a. Signup and view all the answers

What is the purpose of finding the depreciation rate (i) in compound depreciation?

To find the book value of the asset Signup and view all the answers

If the interest rate changes during a period, what should be done to calculate the accumulated value?

Break the period into segments and calculate the accumulated value for each segment separately. Signup and view all the answers

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

$P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)$ Signup and view all the answers

What is the formula for the effective interest rate?

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

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

$P(A ∩ B) = P(A) × P(B)$ Signup and view all the answers

What is the formula for the total accumulated value?

The sum of the values from each period, taking into account changes in interest rates or additional transactions. Signup and view all the answers

What is the definition of dependent events?

Two events are dependent if the occurrence of one affects the probability of the other. Signup and view all the answers

What is the purpose of drawing a tree diagram?

To organize and visualize the possible outcomes of a sequence of events Signup and view all the answers

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

$P(A ∩ B) / P(B)$ Signup and view all the answers

What is the formula for simple interest?

A = P(1 + in) Signup and view all the answers

What is the formula for conditional probability?

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

What is the purpose of a contingency table?

To test for independence between events Signup and view all the answers

What is the formula for calculating the probability of an event A using a contingency table?

$n(A) / n(Total)$ Signup and view all the answers

What is the purpose of Venn diagrams?

To visualize the relationships between events and their probabilities. Signup and view all the answers

What is the formula for the compound interest?

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

What is the condition for two events, A and B, to be mutually exclusive?

$P(A ∩ B) = 0$ Signup and view all the answers

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

To adjust the principal amount for the next period. Signup and view all the answers

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

To calculate the probability of a specific sequence of events Signup and view all the answers

What is the formula for the addition rule for mutually exclusive events?

P(A ∪ B) = P(A) + P(B) Signup and view all the answers

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

$P(Desired Outcome) = P(Path 1) + P(Path 2) + …$ Signup and view all the answers

What is the advantage of using a contingency table over a tree diagram?

It is more useful for testing independence between events Signup and view all the answers

Gr 11 Math Nov P1 (Hard)

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Questions and Answers

What is the value of $r$ when $r^{3} = 64$?

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

Rationalize the denominator of the expression $rac{2}{\sqrt{3}}$:

What is the value of $a$ when $a^{3/2} = 9$?

Simplify the expression $\sqrt[3]{8a^{3}}$:

What is the value of $x$ when $x^{-2/3} = rac{1}{4}$?

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

What is the value of $a$ when $a^{2/3} = 4$?

What is the expression for a^(-m) in terms of positive exponents?

What is the rule for raising a power to another power?

What is the purpose of rationalizing denominators?

When solving surd equations, what is the first step?

What is the formula for exponential growth or decay?

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

When solving quadratic surd equations, what is the second step?

What is the formula for a^(m/n) in terms of roots?

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

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

What is the formula for exponential growth or decay?

What is the purpose of completing the square method?

What is the value of a⁰, where a is not equal to 0?

What is the formula for quadratic equations?

What is the definition of irrational numbers?

What is the formula for the nth root of a number?

What is the formula for rational exponents?

What is the purpose of converting percentages to decimal form?

What is the definition of natural numbers?

What is the formula for completing the square?

What is the value of x in a quadratic equation if b² - 4ac = 0?

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

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

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

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

What is the purpose of the discriminant in a quadratic equation?

What is the formula used to find the discriminant of a quadratic equation?

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

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

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

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

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

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

What is the final step when solving a quadratic inequality?

What is the purpose of the sign analysis in quadratic inequalities?

What is the condition for a system of simultaneous equations to have a unique solution?

What is the advantage of using a graphical approach when solving quadratic inequalities?

What type of quadratic inequalities require special consideration when solving?

What is the first step when solving a quadratic inequality?

What is the purpose of solving quadratic inequalities?

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

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

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

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

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

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

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

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

What is the main advantage of using the substitution method to solve a system of linear equations?

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

What is the effect of a change in q on the exponential function y = ab^x + q?

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

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

What is the characteristic of a quadratic sequence?

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

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

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

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

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

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

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

What is the general form of a quadratic sequence?

What is the purpose of solving for the variables in a system of linear equations?

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

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

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

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

How do you shift a parabola $m$ units to the right?