Gr 10 Math Nov P1 Mix
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Questions and Answers

Which of the following sets includes only non-negative numbers?

  • Natural Numbers (correct)
  • Rational Numbers
  • Integers
  • Imaginary Numbers

What is the correct symbol for the set of whole numbers?

  • N0 (correct)
  • N
  • Q
  • Z

Which of the following numbers is considered irrational?

  • sqrt{2} (correct)
  • 0.75
  • 1/3
  • 3

Which statement about integers is true?

<p>They include both positive and negative numbers. (B)</p> Signup and view all the answers

Which subset of real numbers can be represented as a fraction?

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

What characterizes irrational numbers?

<p>Their decimal expansions are non-repeating and non-terminating. (B)</p> Signup and view all the answers

What is the symbol for rational numbers?

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

Which of the following numbers is classified as imaginary?

<p>sqrt{-1} (A)</p> Signup and view all the answers

What is the product of the binomial extit{(A + B)} and the trinomial extit{(C + D + E)}?

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

Which of the following describes a constant in algebra?

<p>A term without a variable (B)</p> Signup and view all the answers

What is the formula for multiplying two binomials extit{(ax + b)(cx + d)}?

<p>acx^2 + adx + bcx + bd (C)</p> Signup and view all the answers

What is the identity used for factorising the difference of two squares?

<p>a^2 - b^2 = (a + b)(a - b) (B)</p> Signup and view all the answers

Which of the following correctly simplifies the expression extit{a/b × c/d}?

<pre><code>extit{ac/bd} (B) </code></pre> Signup and view all the answers

What does the term 'exponent' refer to?

<p>A number raised to a power (D)</p> Signup and view all the answers

Which process involves grouping terms with common factors?

<p>Factorising by grouping (C)</p> Signup and view all the answers

In the context of equations, what does it mean for two expressions to be equal?

<p>Both expressions represent the same quantity (C)</p> Signup and view all the answers

What is the first step in simplifying an algebraic fraction?

<p>Identify and factor out common factors (A)</p> Signup and view all the answers

What is the resulting form of the difference of two cubes, represented by extit{x^3 - y^3}?

<p>x^3 - y^3 = (x - y)(x^2 - xy + y^2) (A)</p> Signup and view all the answers

Which statement best defines an irrational number?

<p>A number with a non-repeating, non-terminating decimal expansion. (C)</p> Signup and view all the answers

What is the first step in converting a terminating decimal into a rational number?

<p>Identify the integer part. (B)</p> Signup and view all the answers

When rounding a number, if the digit after the required decimal place is less than 5, what should be done?

<p>Leave the last digit unchanged. (B)</p> Signup and view all the answers

Which expression represents the process of multiplying two linear binomials?

<p>acx^2 + adx + bcx + bd (C)</p> Signup and view all the answers

What type of number is \sqrt{4}?

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

How can an irrational number be represented more simply?

<p>By rounding it off. (A)</p> Signup and view all the answers

What is the correct definition of a binomial?

<p>An expression with two terms. (D)</p> Signup and view all the answers

What should you identify first when estimating surds?

<p>Perfect squares or cubes (C)</p> Signup and view all the answers

Which of the following is an example of a terminating decimal?

<p>2.75 (D)</p> Signup and view all the answers

What do you call the numerical factor in a term?

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

What is one method used to solve linear equations?

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

How many solutions can a linear equation have?

<p>At most one solution (D)</p> Signup and view all the answers

Which is true about quadratic equations?

<p>They can have at most one or two solutions (C)</p> Signup and view all the answers

What is the first step in solving simultaneous equations using substitution?

<p>Express one variable in terms of the other (B)</p> Signup and view all the answers

What is a necessary condition for solving quadratic equations effectively?

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

Which of the following is not a step in the method for solving quadratic equations?

<p>Combine like terms (D)</p> Signup and view all the answers

When solving by elimination, what is the first step?

<p>Make the coefficients of a variable the same (A)</p> Signup and view all the answers

What does it mean if a quadratic equation has no solutions?

<p>The quadratic does not intersect the x-axis (B)</p> Signup and view all the answers

Which step involves ensuring the quadratic is in the correct form?

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

Which of the following is true regarding solving equations?

<p>Operations must be performed on both sides to maintain balance (D)</p> Signup and view all the answers

What does the solution of a system of simultaneous linear equations represent?

<p>The intersection point of the graphs (C)</p> Signup and view all the answers

Which step is NOT part of the problem solving strategy for word problems?

<p>Translate the words into a graphical representation (C)</p> Signup and view all the answers

What is a literal equation?

<p>An equation with several letters or variables (B)</p> Signup and view all the answers

When solving the inequality $2x + 2 e 1$, what happens if both sides are multiplied by -1?

<p>The inequality is reversed (D)</p> Signup and view all the answers

In the formula for a linear sequence $T_n = dn + c$, what does the variable $d$ represent?

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

To find the common difference $d$ in a sequence, you calculate $T_n - T_{n-1}$. What does $T_n$ denote?

<p>The $n$th term in the sequence (C)</p> Signup and view all the answers

When rearranging a literal equation to isolate an unknown variable, which principle is applied?

<p>Perform the opposite operation on both sides (A)</p> Signup and view all the answers

What is the primary method used to solve linear inequalities?

<p>Using similar operations as linear equations (B)</p> Signup and view all the answers

What term describes an ordered list of items, usually numbers?

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

What is the result of $(a^m)^n$ according to the power of a power rule?

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

In the equation $v = \frac{D}{t}$, what does $v$ represent?

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

Which expression is equivalent to $a^{m/n} \times a^{p/q}$ using the multiplication of exponents rule?

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

If $a^{m/n} = a^{p/q}$, what does it imply if the bases are the same?

<p>$m/n = p/q$ (D)</p> Signup and view all the answers

What is a linear sequence defined by?

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

Which of the following represents the zero exponent law?

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

What is the simplified form of $ rac{a^m}{a^n}$ using the division of exponents rule?

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

What does the constant 'm' represent in the equation of a straight line?

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

Which method is typically used to solve exponential equations when bases cannot easily be made the same?

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

How does the value of 'c' affect a straight line graph?

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

What is the domain of a linear function of the form y = mx + c?

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

When simplifying $\left(\frac{a}{b}\right)^n$, which exponent law is applied?

<p>Raising a quotient to a power (B)</p> Signup and view all the answers

How do you calculate the y-intercept of a straight-line graph?

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

Which expression represents the prime factorization of an exponential expression?

<p>Express each base as a prime number (C)</p> Signup and view all the answers

What effect does a positive value of 'a' have on a parabola?

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

What does a negative exponent indicate for a number $a$?

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

When rewriting an expression with fractional exponents like $\sqrt[n]{a}$, what is the correct conversion?

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

What happens to the graph of the function y = ax^2 + q when q < 0?

<p>The graph shifts downwards (B)</p> Signup and view all the answers

What kind of points are needed to sketch a straight-line graph effectively?

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

What is the range of a linear function of the form f(x) = mx + c?

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

What occurs when 'm' is set to 0 in the equation of a straight line?

<p>The line becomes horizontal (B)</p> Signup and view all the answers

What happens to the graph of a parabola as the value of $a$ increases beyond 1?

<p>The graph gets narrower. (D)</p> Signup and view all the answers

For which value of $a$ will the graph of the parabola open downwards?

<p>Any negative value of $a$ (C)</p> Signup and view all the answers

Where is the y-intercept of the function $y = ax^2 + q$ calculated?

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

If $q < 0$, what can be concluded about the range of the parabola when $a > 0$?

<p>The range is $[q; ext{infinity})$. (A)</p> Signup and view all the answers

In the function $y = rac{a}{x} + q$, what is the domain of the function?

<p>$ ext{All real numbers except } x=0$ (B)</p> Signup and view all the answers

Which statement accurately describes the turning point of the quadratic function $f(x) = ax^2 + q$ when $a < 0$?

<p>The turning point is a maximum. (B)</p> Signup and view all the answers

How is the horizontal asymptote defined in the hyperbolic function $y = rac{a}{x} + q$?

<p>At $y = q$ (B)</p> Signup and view all the answers

What effect does a negative value of $a$ have on the hyperbola $y = rac{a}{x} + q$?

<p>The graph lies in the second and fourth quadrants. (D)</p> Signup and view all the answers

Which of the following best describes the axis of symmetry for the function $f(x) = ax^2 + q$?

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

What does the parameter $q$ determine in trigonometric functions?

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

Which formula represents accumulated amount with simple interest?

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

In which scenario is compound interest MORE beneficial than simple interest?

<p>Investing money over time (C)</p> Signup and view all the answers

What key variable is needed to calculate simple interest?

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

How does inflation affect prices over time?

<p>Inflation increases the price of goods and services (C)</p> Signup and view all the answers

What is the primary effect of compound interest on an investment?

<p>Causes exponential growth (B)</p> Signup and view all the answers

What is required to identify asymptotes in functions like tangents?

<p>Setting the denominator to zero (B)</p> Signup and view all the answers

What does the principal amount represent in the context of loans and investments?

<p>The original amount invested or borrowed (C)</p> Signup and view all the answers

Which of these scenarios would utilize the compound interest formula?

<p>Calculating the future value of a savings account (B)</p> Signup and view all the answers

What is meant by the term 'currency strength'?

<p>The value of a currency in relation to another (A)</p> Signup and view all the answers

What is the general form of a function representing exponential growth?

<p>y = ab^x + q, where b &gt; 1 (C)</p> Signup and view all the answers

Which characteristic is true for the sine function?

<p>The x-intercepts are located at (180°, 0) (B)</p> Signup and view all the answers

What effect does a positive value of q have on the graph of an exponential function?

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

In the function y = a sin θ + q, what does the parameter a determine?

<p>The amplitude of the wave (B)</p> Signup and view all the answers

For the function y = ab^x + q, what does the value of b represent?

<p>The growth or decay factor (D)</p> Signup and view all the answers

What is the range of the sine function?

<p>[-1, 1] (D)</p> Signup and view all the answers

Which type of asymptote do exponential functions have?

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

What happens to the graph of y = ab^x + q when a is negative?

<p>The graph curves downwards. (D)</p> Signup and view all the answers

In functions of the form y = cos θ, where are the maximum turning points located?

<p>(0°, 1) and (360°, 1) (B)</p> Signup and view all the answers

What are the x-intercepts of the cosine function?

<p>(90°, 0) and (270°, 0) (C)</p> Signup and view all the answers

What does the probability of the union of two events, A and B, account for?

<p>It sums the probabilities of both events plus their intersection. (B)</p> Signup and view all the answers

What is the probability of two mutually exclusive events occurring together?

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

Which statement is true about complementary events?

<p>Their probabilities sum to 1. (B)</p> Signup and view all the answers

What does the identity P(A ∪ B) = P(A) + P(B) - P(A ∩ B) correct?

<p>It subtracts the overlap of two events. (D)</p> Signup and view all the answers

What is the visual representation for mutually exclusive events?

<p>Two separate circles. (A)</p> Signup and view all the answers

Which statement is a consequence of complementary events?

<p>They cannot co-occur. (D)</p> Signup and view all the answers

What is represented by the area outside event A in a Venn diagram of sample space S?

<p>The complement of event A, denoted as A'. (B)</p> Signup and view all the answers

What is the probability relationship for two mutually exclusive events A and B?

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

Which identity demonstrates that the union of an event and its complement covers the sample space?

<p>A ∪ A' = S (D)</p> Signup and view all the answers

What can be inferred if two events A and B have a probability of intersection equal to zero?

<p>They are mutually exclusive events. (C)</p> Signup and view all the answers

What is the period of the cosine function?

<p>360° (A)</p> Signup and view all the answers

What effect does a negative value of the coefficient 'a' have on the parabola's graph?

<p>It changes the graph from a smile to a frown. (D)</p> Signup and view all the answers

What is the range of the tangent function, y = tan θ?

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

Which statement about the y-intercept of the tangent function is correct?

<p>It can be found by solving y = a tan(0°) + q. (D)</p> Signup and view all the answers

How can the cosine graph be related to the sine graph?

<p>The cosine graph is a phase shift of the sine graph by 90°. (C)</p> Signup and view all the answers

What effect does the value of 'q' have on the tangent function graph?

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

What is the correct expression for the domain of the tangent function, y = a tan θ + q?

<p>{θ : 0° ≤ θ ≤ 360°, θ ≠ 90°, 270°} (D)</p> Signup and view all the answers

How can the value of 'a' in the parabola equation y = ax² + q affect its shape?

<p>It dictates the direction and width of the parabola. (B)</p> Signup and view all the answers

In a hyperbola, which statement is true regarding the variable 'a'?

<p>It influences the shape and direction of the hyperbola. (C)</p> Signup and view all the answers

How do you determine the value of 'q' when finding the equation of a parabola?

<p>By identifying the y-intercept point. (B)</p> Signup and view all the answers

What is the relationship between local purchasing and the economy?

<p>It strengthens the local currency. (B)</p> Signup and view all the answers

What does a probability of 1 represent?

<p>An event will occur every time. (B)</p> Signup and view all the answers

How can probabilities be represented?

<p>As a real number, percentage, or fraction. (C)</p> Signup and view all the answers

What does relative frequency measure?

<p>How often an event occurs in experimental trials. (A)</p> Signup and view all the answers

Which statement is true about the union of two sets?

<p>It contains elements that are in at least one of the two sets. (A)</p> Signup and view all the answers

What is the formula for calculating theoretical probability?

<p>$ P(E) = \frac{n(E)}{n(S)} $ (A)</p> Signup and view all the answers

What does a probability of 0 indicate?

<p>An event will never occur. (C)</p> Signup and view all the answers

Which definition applies to the intersection of two sets?

<p>All elements that are in both sets. (C)</p> Signup and view all the answers

What key concept does a Venn diagram illustrate?

<p>Relationships between sets. (B)</p> Signup and view all the answers

What does the formula ( f = \frac{p}{t} ) signify?

<p>The probability of an event occurring based on trials. (B)</p> Signup and view all the answers

What defines a rational number?

<p>A number that can be expressed as a fraction of two integers (C)</p> Signup and view all the answers

Which of the following sets includes all natural numbers and zero?

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

Which number set does not include any negative numbers?

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

Which of the following statements about irrational numbers is true?

<p>Irrational numbers have non-repeating, non-terminating decimals. (B)</p> Signup and view all the answers

What is the relationship between rational and irrational numbers in the real number system?

<p>Real numbers include both rational and irrational numbers. (D)</p> Signup and view all the answers

In the real number hierarchy, which set is immediately below the set of integers?

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

Which of the following would be classified as an irrational number?

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

What type of number is represented by the symbol $ rac{-5}{2}$?

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

What characterizes a rational number in decimal form?

<p>It can be expressed as a fraction with integer numerator and denominator. (B)</p> Signup and view all the answers

Which step is essential when converting a recurring decimal into a rational number?

<p>Multiply by a power of 10 that matches the length of the repeating cycle. (D)</p> Signup and view all the answers

What happens to an irrational number when it is rounded off?

<p>It becomes a rational number with a decimal expansion. (A)</p> Signup and view all the answers

When estimating surds, what is the first step?

<p>Identify the perfect powers surrounding the surd. (A)</p> Signup and view all the answers

What is crucial for identifying when to round a number up?

<p>If the next digit is 5 or greater. (B)</p> Signup and view all the answers

Which expression represents the general formula for multiplying two linear binomials?

<p>(ax + b)(cx + d) = acx^2 + adx + bcx + bd (B)</p> Signup and view all the answers

What defines a surd?

<p>The n-th root of a number that cannot be simplified to a rational number. (A)</p> Signup and view all the answers

In the rounding process, what adjustment is necessary if the digit being rounded is 9?

<p>Change it to 0 and increase the preceding digit by 1. (B)</p> Signup and view all the answers

Which of the following is true about irrational numbers compared to rational numbers?

<p>Irrational numbers cannot be expressed as fractions. (A)</p> Signup and view all the answers

What is the maximum number of solutions for a linear equation?

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

Which step is NOT part of solving a linear equation?

<p>Divide by variable terms (A)</p> Signup and view all the answers

When solving a quadratic equation, what form should it be rewritten in first?

<p>Form with zero on one side (B)</p> Signup and view all the answers

What characterizes a quadratic equation compared to a linear equation?

<p>Can have two solutions (B)</p> Signup and view all the answers

In the method of solving simultaneous equations by substitution, what is the first action taken?

<p>Express one variable in terms of another (B)</p> Signup and view all the answers

Which of the following methods is used to simplify two equations to find values for two variables?

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

What is the purpose of checking the answer after solving an equation?

<p>To verify the solution satisfies the original equation (A)</p> Signup and view all the answers

What happens when operations are performed on one side of an equation?

<p>It must be done on the other side too (C)</p> Signup and view all the answers

What is a common feature of quadratic equations with no solutions?

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

When solving by elimination, what is the goal regarding the coefficients?

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

What is the first term in the expansion of the binomial extit{(A + B)(C + D + E)}?

<p>A(C + D + E) (C)</p> Signup and view all the answers

What does the identity для factorising the difference of two squares state?

<p>$a^2 - b^2 = (a + b)(a - b)$ (C)</p> Signup and view all the answers

Which of the following steps is NOT part of simplifying an algebraic fraction?

<p>Identify the greatest common divisor (C)</p> Signup and view all the answers

When multiplying two binomials extit{(ax + b)(cx + d)}, what term represents the middle term of the expansion?

<p>$adx + bcx$ (C)</p> Signup and view all the answers

Which expression correctly represents the sum of two cubes?

<p>$x^3 + y^3 = (x + y)(x^2 - xy + y^2)$ (A)</p> Signup and view all the answers

What is a key step in factorising a quadratic trinomial of the form $ax^2 + bx + c$?

<p>Generate all possible pairs of factors for $a$ and $c$ (A)</p> Signup and view all the answers

What is the result of multiplying the fractions $ rac{a}{b} imes rac{c}{d}$?

<p>$ rac{ac}{bd}$ (B)</p> Signup and view all the answers

To simplify the expression $ rac{x^2 + 2x}{x}$, which of the following is the first step?

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

What is the equivalent expression for extit{(A + B)(C + D + E)}?

<p>$AC + AD + AE + BC + BD + BE$ (C)</p> Signup and view all the answers

What does the solution of a system of simultaneous equations represent in graphical terms?

<p>The coordinates of the intersecting point (C)</p> Signup and view all the answers

In solving linear inequalities, what happens to the inequality sign when both sides are divided by a negative number?

<p>The sign is switched (D)</p> Signup and view all the answers

Which of the following steps is NOT part of the problem-solving strategy for word problems?

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

What does the variable 'd' represent in the general formula for a linear sequence $T_n = dn + c$?

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

To change the subject of the formula in a literal equation effectively, what is the initial action you perform?

<p>Add or subtract terms to isolate the variable (A)</p> Signup and view all the answers

What is the outcome of taking the square root of both sides of an equation involving a variable?

<p>There will be both positive and negative answers (A)</p> Signup and view all the answers

In a linear sequence, if the common difference is 5, what will be the value of the third term $T_3$ if the first term $T_1$ is 10?

<p>20 (D)</p> Signup and view all the answers

When isolating an unknown variable that appears in two or more terms of a literal equation, what should be performed?

<p>Factor out the variable (D)</p> Signup and view all the answers

How do you determine the common difference in a linear sequence?

<p>By calculating $T_n$ minus $T_{n-1}$ (D)</p> Signup and view all the answers

In the context of solving linear equations, what does it mean when two expressions are said to be equal?

<p>They give the same solution (C)</p> Signup and view all the answers

What is the result of simplifying the expression $(ab)^{m/n}$?

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

Which exponent law applies when dividing two expressions with the same base?

<p>Division of Exponents (C)</p> Signup and view all the answers

When simplifying the expression $ rac{a^3}{a^5}$, what is the result?

<p>$a^{-2}$ (C)</p> Signup and view all the answers

Which of the following represents the rule for simplifying an expression with a zero exponent?

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

What is the first step in solving the exponential equation $5^x = 5^3$?

<p>Set the exponents equal to each other (C)</p> Signup and view all the answers

To simplify the expression $ rac{(a^2b^3)^4}{(ab)^5}$, what is the first action you should take?

<p>Use the power of a product rule (D)</p> Signup and view all the answers

Which method can be used to solve the exponential equation $2^x = 8$?

<p>Equating the exponents by rewriting 8 as a power of 2 (B)</p> Signup and view all the answers

What transformation is needed for the expression $ rac{1}{a^{-3}}$ to express it with a positive exponent?

<p>Convert negative exponent to positive (C)</p> Signup and view all the answers

In rational exponents, what is the value of $a^{1/2}$?

<p>$ ext{sqrt}(a)$ (C)</p> Signup and view all the answers

What does the coefficient 'm' represent in a linear function of the form $y = mx + c$?

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

How does the value of 'c' affect the graph of a linear function?

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

What characteristic is used to find the x-intercept of a linear function?

<p>Set $y$ to 0 (B)</p> Signup and view all the answers

In a linear sequence, what does the term 'common difference' refer to?

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

When $a < 0$ in a quadratic function $y = ax^2 + q$, what shape does the graph take?

<p>It opens downwards like an upside-down U (A)</p> Signup and view all the answers

What does the term $T_n$ represent in the general formula for a linear sequence $T_n = dn + c$?

<p>The nth term in the sequence (D)</p> Signup and view all the answers

If the value of $m$ in the function $y = mx + c$ equals 0, what is the nature of the graph?

<p>A line that is horizontal (B)</p> Signup and view all the answers

Which of the following describes the effect of $q$ in the quadratic function $y = ax^2 + q$?

<p>It acts as a vertical shift (A)</p> Signup and view all the answers

What are the two points needed to plot a straight-line graph using the dual intercept method?

<p>X-intercept and y-intercept (A)</p> Signup and view all the answers

What does a positive value of 'm' indicate about the linear graph $y = mx + c$?

<p>The graph slopes upwards to the right (D)</p> Signup and view all the answers

What happens to the graph of a parabola when the value of ( a ) is negative?

<p>The graph opens downwards with a maximum turning point. (B)</p> Signup and view all the answers

How does the value of ( q ) affect the graph of the parabola ( y = ax^2 + q )?

<p>It determines the vertical shift of the graph. (C)</p> Signup and view all the answers

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

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

Which of the following describes the y-intercept of the function ( y = ax^2 + q )?

<p>It is located at ( (0, q) ). (C)</p> Signup and view all the answers

What are the asymptotes of the hyperbolic function ( y = \frac{a}{x} + q )?

<p>Horizontal asymptote at ( y = q ) and vertical asymptote at ( x = 0 ). (D)</p> Signup and view all the answers

For the hyperbolic function ( y = \frac{a}{x} + q ), which of the following statements is true when ( a < 0 )?

<p>The graph lies in the second and fourth quadrants. (D)</p> Signup and view all the answers

What is the domain of the hyperbolic function ( y = \frac{a}{x} + q )?

<p>{x: x \in \mathbb{R}, x \neq 0} (A)</p> Signup and view all the answers

In the context of parabolic graphs, what is indicated by the sign of ( a )?

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

Which characteristic is true for all parabolic functions in the form ( y = ax^2 + q )?

<p>They all possess an axis of symmetry. (A)</p> Signup and view all the answers

What characterizes the range of exponential functions when the constant $a$ is greater than zero?

<p>The range is greater than $q$. (B)</p> Signup and view all the answers

Which statement about the y-intercept of the function $y = ab^x + q$ is true?

<p>The y-intercept is found by letting $x = 0$. (A)</p> Signup and view all the answers

What is the effect of a negative value of $a$ on the graph of the function $y = a imes ext{sin}( heta) + q$?

<p>The graph reflects about the x-axis. (A)</p> Signup and view all the answers

For the function represented as $y = rac{a}{x} + q$, how is the vertical asymptote defined?

<p>It is the line $x = 0$. (A)</p> Signup and view all the answers

How does the value of $b$ affect the function $y = ab^x + q$?

<p>It affects the rate of growth or decay. (C)</p> Signup and view all the answers

What is the domain of the sine function defined as $y = ext{sin}( heta)$?

<p>The interval $[0°, 360°]$. (D)</p> Signup and view all the answers

When the function is represented as $y = a imes ext{cos}( heta) + q$, what is true about the amplitude when $|a| > 1$?

<p>The amplitude experiences vertical stretch. (D)</p> Signup and view all the answers

What will the vertical asymptote be in the function $y = rac{a}{x} + q$?

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

For an exponential function $y = ab^x + q$, if $b$ is between 0 and 1, what type of behavior does the function exhibit?

<p>Exponential decay. (D)</p> Signup and view all the answers

What is the proper definition of theoretical probability?

<p>A real number between 0 and 1 representing event likelihood (B)</p> Signup and view all the answers

Which of the following statements about relative frequency is true?

<p>It is calculated by the formula \ f = \frac{p}{t} (B)</p> Signup and view all the answers

How is the union of two sets represented?

<p>A ( A \cup B ) (D)</p> Signup and view all the answers

What characterizes the intersection of two events in probability?

<p>Outcomes that are in both sets (B)</p> Signup and view all the answers

In the context of probability, what does a probability of 0 indicate?

<p>The event will never occur (B)</p> Signup and view all the answers

What do Venn diagrams help to represent in probability?

<p>The relationships between different sets (D)</p> Signup and view all the answers

What aspect of a trigonometric function does the variable $q$ influence?

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

Which formula correctly represents the calculation of accumulated amount using simple interest?

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

If an event has a theoretical probability of 0.5, what can be inferred about its occurrence?

<p>It will occur half the time (C)</p> Signup and view all the answers

What is the primary difference between simple interest and compound interest?

<p>Simple interest earns interest only on the principal, while compound interest earns interest on both the principal and the accumulated interest. (C)</p> Signup and view all the answers

What does the sample space consist of in probability theory?

<p>The list of all outcomes that can occur in a random experiment (D)</p> Signup and view all the answers

Considering the conversion formula, what does 'Amount in new currency' represent?

<p>Original currency divided by exchange rate (C)</p> Signup and view all the answers

When calculating the y-intercept of a function, what must you set $x$ equal to?

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

What is the outcome of an experiment with a probability of 1?

<p>The event is certain to occur (B)</p> Signup and view all the answers

Which of the following statements about inflation is true?

<p>Inflation is the average annual increase in the price of goods and services. (C)</p> Signup and view all the answers

In a hire purchase agreement, how is interest typically calculated?

<p>At a simple interest rate on the remaining amount after the initial deposit. (C)</p> Signup and view all the answers

Which of the following explains how population growth is calculated?

<p>By using the compound interest formula, similar to investments. (A)</p> Signup and view all the answers

What do asymptotes indicate in functions like hyperbolas and tangents?

<p>They are lines that the curve approaches but never touches. (C)</p> Signup and view all the answers

Which variable influences the amplitude and reflection in trigonometric functions?

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

What does the identity $P(A \u222A B) = P(A) + P(B) - P(A \cap B)$ help to correct?

<p>It removes double-counting in probabilities. (A)</p> Signup and view all the answers

Which statement is true about mutually exclusive events?

<p>Their union is the sum of individual probabilities. (A)</p> Signup and view all the answers

What is the complement of an event A, represented as A'?

<p>The set containing all outcomes not in A. (A)</p> Signup and view all the answers

What happens to the probabilities of complementary events?

<p>They always sum to 1. (C)</p> Signup and view all the answers

If events A and B are mutually exclusive, what is true about their intersection?

<p>It is empty. (C)</p> Signup and view all the answers

In the context of Venn diagrams, what does the area of overlap represent?

<p>The joint probability of A and B. (A)</p> Signup and view all the answers

What is the graphical representation of the union of two events A and B?

<p>The combined area of both circles A and B, including the overlap. (A)</p> Signup and view all the answers

Which identity correctly describes the relationship between complementary events and the sample space?

<p>A / A' = S. (D)</p> Signup and view all the answers

What does the notation P(S) = 1 signify in probability?

<p>It indicates the total probability of all events occurring. (B)</p> Signup and view all the answers

Why is it necessary to subtract P(A \cap B) from the equation for the union of two events?

<p>To correct for double counting of shared outcomes. (A)</p> Signup and view all the answers

What is the period of the cosine and sine functions?

<p>360° (A)</p> Signup and view all the answers

What effect does a positive value of $q$ have in the function $y = a an heta + q$?

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

Which of the following describes the domain of the tangent function $y = an heta$?

<p>{ \theta : 0° \leq \theta \leq 360°, \theta \neq 90° } (B)</p> Signup and view all the answers

How can the equation of a parabola in the form $y = ax^2 + q$ be determined?

<p>By obtaining both $a$ and $q$ from the y-intercept. (A)</p> Signup and view all the answers

What is the range of the tangent function $y = an \theta$?

<p>{ f(\theta) : f(\theta) \in \mathbb{R} } (A)</p> Signup and view all the answers

What occurs when the graph of the cosine function is shifted to the right by 90°?

<p>It becomes a sine function. (A)</p> Signup and view all the answers

When determining the equation of a hyperbola, what is the first step?

<p>Identify the direction of the hyperbola. (D)</p> Signup and view all the answers

Which characteristic is true for the y-intercept of the function $y = a an \theta + q$?

<p>It is equal to $q$. (A)</p> Signup and view all the answers

What defines the range of a parabola represented by $y = ax^2 + q$?

<p>The value of $a$ determines the minimum or maximum y-value. (A)</p> Signup and view all the answers

What is the significance of asymptotes in the function $y = a \tan \theta + q$?

<p>They signify where the function will never touch or cross. (B)</p> Signup and view all the answers

Which of the following statements about natural numbers is false?

<p>Natural numbers include zero. (B)</p> Signup and view all the answers

What is a defining characteristic of irrational numbers?

<p>They cannot be expressed as a simple fraction. (C)</p> Signup and view all the answers

Which of the following sets does not belong to the real number system?

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

Which of the following is an example of a rational number?

<p>$ rac{2}{5}$ (C), $ rac{1}{ rac{1}{2}}$ (D)</p> Signup and view all the answers

Which statement best describes the relationship between whole numbers and integers?

<p>Whole numbers include all integers except negative ones. (B)</p> Signup and view all the answers

Which of the following is true about the set of rational numbers?

<p>Rational numbers can be expressed as the quotient of two integers. (C)</p> Signup and view all the answers

Which of the following examples is not a real number?

<p>$ rac{2}{0}$ (D)</p> Signup and view all the answers

How are natural numbers classified within the real number system?

<p>They are a subset of rational numbers. (C)</p> Signup and view all the answers

What characterizes rational numbers when expressed as decimals?

<p>They include both terminating and repeating decimals. (A)</p> Signup and view all the answers

Which of the following best describes a surd?

<p>The n-th root of a number that cannot be simplified to a rational number. (A)</p> Signup and view all the answers

Which method accurately describes converting a recurring decimal into a rational number?

<p>Multiply the decimal by a power of ten to align the repeating part. (B)</p> Signup and view all the answers

What is the first step to round off a decimal number to a specific decimal place?

<p>Identify the required decimal place and the next digit. (A)</p> Signup and view all the answers

In multiplying two binomials, what does the term 'ad' represent?

<p>One of the cross-products in the expansion. (A)</p> Signup and view all the answers

What is the outcome when rounding off a digit that is 9?

<p>The preceding digit increases by 1, and the 9 becomes 0. (C)</p> Signup and view all the answers

Which of the following is true about irrational numbers?

<p>They have non-repeating, non-terminating decimal expansions. (D)</p> Signup and view all the answers

When estimating the value of a surd, what should be identified first?

<p>The nearest perfect squares or cubes surrounding the surd. (C)</p> Signup and view all the answers

What distinguishes a decimal from an irrational number?

<p>Rational numbers are represented as fractions. (A)</p> Signup and view all the answers

In which situations will rounding off a decimal results in a loss of precision?

<p>When rounding irrational numbers. (B)</p> Signup and view all the answers

What is the result of applying the power of a power rule to $(3^2)^4$?

<p>3^{12} (B)</p> Signup and view all the answers

What expression simplifies to $a^m imes a^n$ according to the multiplication of exponents rule?

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

Which of the following represents the correct simplification of $ rac{a^3}{a^5}$ using the division of exponents rule?

<p>a^{-2} (C)</p> Signup and view all the answers

How can the expression $(xy)^3$ be expressed using the raising a product to a power rule?

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

In which scenario can logarithms be used to solve an exponential equation?

<p>When the bases are different (D)</p> Signup and view all the answers

What is the correct expression for the zero exponent rule?

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

What must be the same for the exponents to be equated in an exponential equation?

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

Which of the following is a step in simplifying an expression with rational exponents?

<p>Use prime factorization where necessary (C)</p> Signup and view all the answers

Which of the following statements about negative exponents is true?

<p>They indicate fractions (A)</p> Signup and view all the answers

Which expression corresponds to the law of rational exponents for powering a power?

<p>a^{(m/n) imes (p/q)} (D)</p> Signup and view all the answers

What does the solution to a system of simultaneous linear equations represent?

<p>The coordinates of the intersection point of the two graphs (D)</p> Signup and view all the answers

What is the first step when using a problem-solving strategy for word problems?

<p>Read the whole question (B)</p> Signup and view all the answers

In the context of literal equations, what is meant by changing the subject of the formula?

<p>Rearranging an equation to isolate a specific variable (D)</p> Signup and view all the answers

When solving linear inequalities, what must be done if both sides are divided by a negative number?

<p>The inequality sign is reversed (C)</p> Signup and view all the answers

What does the variable 'd' represent in the formula for a linear sequence $T_n = dn + c$?

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

Which of the following correctly defines a linear inequality?

<p>An inequality with variables where the highest exponent is 1 (A)</p> Signup and view all the answers

What is the formula for the common difference 'd' in a linear sequence?

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

When rearranging a literal equation, which technique helps isolate the unknown variable?

<p>Performing opposite operations on both sides (C)</p> Signup and view all the answers

What does the term 'term' refer to in the context of sequences?

<p>An individual item in the sequence (C)</p> Signup and view all the answers

In a linear equation, what does the solution represent geometrically?

<p>The coordinates of a point (A)</p> Signup and view all the answers

What will be the result of multiplying a monomial by a binomial represented by the equation $a(x + y)$?

<p>ax + ay (C)</p> Signup and view all the answers

What is the correct expression to factorize a trinomial of the form $ax^2 + bx + c$?

<p>$(x + m)(x + n)$ where $mn = c$ and $m + n = b$. (A)</p> Signup and view all the answers

What identity is used to factorize the difference of two cubes $x^3 - y^3$?

<p>$x^3 - y^3 = (x - y)(x^2 + xy + y^2)$ (A)</p> Signup and view all the answers

Which method involves grouping terms with common factors to simplify an expression?

<p>Factorising by pairs (A)</p> Signup and view all the answers

When multiplying two binomials $(ax + b)(cx + d)$, which components do you derive from $ad$ in the expansion?

<p>The constant term of the resulting polynomial (C)</p> Signup and view all the answers

Which of the following describes the correct procedure for simplifying the algebraic fraction $ rac{a}{b} imes rac{c}{d}$?

<p>Multiply the numerators and denominators independently (D)</p> Signup and view all the answers

For the expression $(A + B)(C + D + E)$, what does the result look like using the distributive property?

<p>$AC + AD + AE + BC + BD + BE$ (A)</p> Signup and view all the answers

To determine the common factors in the expression $3x^2 + 6x$, which of these is primarily extracted?

<p>$3x$ (C)</p> Signup and view all the answers

When applying the laws of exponents, what is the result of $a^m imes a^n$?

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

In factorisation, what does the term 'common factor' refer to?

<p>A factor shared by all terms in an expression (B)</p> Signup and view all the answers

What is the effect of the parameter $q$ in the function $y = ab^x + q$?

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

Which of the following describes the x-intercepts of the function $y = a imes ext{sin}( heta) + q$?

<p>They are found by setting $y = a imes ext{sin}( heta) + q$ to zero. (B)</p> Signup and view all the answers

How is the vertical shift of the sine function determined when the function is represented as $y = a imes ext{sin}( heta) + q$?

<p>By the value of $q$ alone. (D)</p> Signup and view all the answers

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

<p>It is given by the line $y = q$. (C)</p> Signup and view all the answers

For the function $y = a imes ext{cos}( heta) + q$, what happens when $a < 0$?

<p>The graph is reflected about the x-axis. (D)</p> Signup and view all the answers

What describes the rate of growth or decay in the function $y = ab^x$ when $b > 1$?

<p>It grows exponentially. (A)</p> Signup and view all the answers

In the sine function $y = a imes ext{sin}( heta) + q$, what effect does $|a| > 1$ have?

<p>It stretches the graph vertically. (D)</p> Signup and view all the answers

Which option best describes the domain of the exponential function $y = ab^x + q$?

<p>It encompasses all real values of $x$. (D)</p> Signup and view all the answers

For the sine function, what are the specific x-intercepts within the interval $[0°, 360°]$?

<p>At $0°, 180°, and 360°$. (D)</p> Signup and view all the answers

What is the maximum number of solutions a quadratic equation can have?

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

What is the first step in solving linear equations?

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

When using substitution to solve simultaneous equations, what is typically done first?

<p>Express one variable in terms of the other (D)</p> Signup and view all the answers

What method is commonly used to solve quadratic equations when they can be factored?

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

What is a necessary condition for solving a quadratic equation effectively?

<p>It must be balanced (A)</p> Signup and view all the answers

In solving simultaneous equations by elimination, what should be done to the coefficients of one variable?

<p>They should be made the same (B)</p> Signup and view all the answers

What is the last step in solving a quadratic equation?

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

Which process is used to reduce the number of unknown variables in simultaneous equations?

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

When rearranging an equation, what principle must always be followed?

<p>Both sides must remain equal (A)</p> Signup and view all the answers

What distinguishes a linear equation from a quadratic equation?

<p>Linear equations have one solution; quadratic can have two. (A)</p> Signup and view all the answers

What is the general formula for a linear sequence?

<p>T_n = dn + c (B)</p> Signup and view all the answers

What effect does a positive value of 'm' have on the graph of a linear function?

<p>The graph slopes upwards from left to right. (C)</p> Signup and view all the answers

What happens to a graph when the value of 'c' is negative?

<p>The graph shifts vertically downward. (C)</p> Signup and view all the answers

How is the y-intercept of a linear function calculated?

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

What is the shape of the graph when 'a' in the equation y = ax^2 + q is greater than zero?

<p>The graph is a parabola that opens upward. (A)</p> Signup and view all the answers

What defines the vertical shift of a parabola in the function y = ax^2 + q?

<p>The value of 'q'. (B)</p> Signup and view all the answers

Which of these points is essential for sketching a straight-line graph using the gradient and y-intercept method?

<p>Y-intercept. (A)</p> Signup and view all the answers

What would be the result of a negative value of 'a' in the equation y = ax^2 + q?

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

What is the effect of varying the value of 'm' in a linear function's equation y = mx + c?

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

What does the variable $q$ affect in the trigonometric functions?

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

What is the primary difference between simple interest and compound interest?

<p>Simple interest is calculated only on the principal amount. (C)</p> Signup and view all the answers

How is the y-intercept of a function calculated?

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

In the context of hire purchase agreements, how is interest calculated?

<p>On the remaining balance after the deposit (B)</p> Signup and view all the answers

What does the interest rate represent in the formula for simple interest?

<p>The percentage earned or paid per year (B)</p> Signup and view all the answers

How can the domain of a function be determined?

<p>By identifying all possible $x$ values (B)</p> Signup and view all the answers

Which statement is true regarding inflation calculations?

<p>It is calculated using the compound interest formula. (D)</p> Signup and view all the answers

What does a strong currency indicate about a country's economy?

<p>Increased foreign investment (C)</p> Signup and view all the answers

Which formula is used for calculating the accumulated amount in simple interest?

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

What happens to interest in the context of compound interest over multiple periods?

<p>It grows exponentially due to interest on interest. (B)</p> Signup and view all the answers

What is the theoretical probability of an event that occurs once in a total of 4 possible outcomes?

<p>0.25 (C)</p> Signup and view all the answers

Given two events A and B that have partial overlap, the intersection of A and B represents what?

<p>Elements in both A and B (B)</p> Signup and view all the answers

What happens to the relative frequency as the number of trials increases?

<p>It tends to converge to theoretical probability (B)</p> Signup and view all the answers

Which formula would you use to convert an amount from dollars to euros if the exchange rate is 0.85 euros per dollar?

<p>Amount in new currency = Amount in original currency / Exchange rate (C)</p> Signup and view all the answers

When is the probability of an event equal to 0?

<p>When it is impossible for the event to occur (D)</p> Signup and view all the answers

Which statement about the union of two sets A and B is correct?

<p>It includes all elements that are in either A or B (C)</p> Signup and view all the answers

If the probability of an event occurring is 0.5, what can be inferred about its outcomes?

<p>It will occur half the time in a series of trials (A)</p> Signup and view all the answers

What is represented by a Venn diagram?

<p>Relationships between different sets (A)</p> Signup and view all the answers

How is the relative frequency of an event defined in an experimental scenario?

<p>The number of times the event occurs divided by the total trials (D)</p> Signup and view all the answers

What is the probability identity used to calculate the union of two events?

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

Which of the following statements is true about mutually exclusive events?

<p>Their union equals the sum of their individual probabilities. (D)</p> Signup and view all the answers

What is the probability relationship for two mutually exclusive events A and B?

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

Which of the following best defines the complement of a set A?

<p>All elements that are not in A. (B)</p> Signup and view all the answers

Which statement is true about complementary events?

<p>They are mutually exclusive events. (A)</p> Signup and view all the answers

What is the sum of the probabilities of an event A and its complement A'?

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

Which of the following correctly represents the intersection of event A and its complement A'?

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

When combining events A and B, what happens if they are not mutually exclusive?

<p>You subtract the intersection of A and B. (D)</p> Signup and view all the answers

In the context of Venn diagrams, what does the area where two events overlap represent?

<p>The probability of both events occurring. (B)</p> Signup and view all the answers

Which of the following describes the relationship between the sample space S and the complement of event A?

<p>A ∪ A' = S (C)</p> Signup and view all the answers

What happens to the graph of a parabola as the value of 'a' increases while 'a' is greater than 0?

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

What happens to the graph of a function when the value of 'a' is greater than 0?

<p>The graph becomes narrower as 'a' approaches infinity. (A), The graph is vertically stretched and has a minimum turning point. (C)</p> Signup and view all the answers

For the parabola defined by the equation $y = ax^2 + q$, what does the variable 'q' represent?

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

How is the range of the function $y = ax^2 + q$ determined when 'a' is less than 0?

<p>The range is from negative infinity to 'q'. (C)</p> Signup and view all the answers

What are the axes of symmetry for the parabolas in the form of $y = ax^2 + q$?

<p>The line $x = 0$. (C)</p> Signup and view all the answers

What is true about the behavior of the graph of a hyperbolic function when 'a' is less than 0?

<p>It lies in the second and fourth quadrants. (C)</p> Signup and view all the answers

What is true regarding the domain of the function $y = \frac{a}{x} + q$?

<p>It includes all real numbers except x = 0. (D)</p> Signup and view all the answers

How do changes in the coefficient 'q' affect the hyperbolic function $y = \frac{a}{x} + q$?

<p>They shift the graph vertically by 'q' units. (D)</p> Signup and view all the answers

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

<p>All real numbers except for 'q'. (D)</p> Signup and view all the answers

When finding the x-intercept of the function $y = ax^2 + q$, what condition must be satisfied?

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

How do the vertical asymptotes of the hyperbola behave?

<p>The vertical asymptote is the line x = 0. (B)</p> Signup and view all the answers

What is the range of the function defined by the equation $y = a \cos \theta + q$ for $a > 0$?

<p>[q - |a|, q + |a|] (A)</p> Signup and view all the answers

What is the period of the tangent function $y = a \tan \theta + q$?

<p>180 (D)</p> Signup and view all the answers

What is the maximum number of solutions that a quadratic equation can have?

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

Which step is NOT part of solving linear equations?

<p>Remember the Pythagorean theorem (B)</p> Signup and view all the answers

When solving simultaneous equations by substitution, what is the first action typically taken?

<p>Express one variable in terms of the other (B)</p> Signup and view all the answers

What must always be true about an equation when performing operations on it?

<p>Both sides must balance (B)</p> Signup and view all the answers

Which method can be used to solve quadratic equations?

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

In solving quadratic equations, when you factor into the form ((rx + s)(ux + v) = 0), what is the next step?

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

Which statement best describes a linear equation?

<p>It includes constants and can only have one solution (B)</p> Signup and view all the answers

What method can simplify solving simultaneous equations?

<p>Elimination of a variable by equalizing coefficients (A)</p> Signup and view all the answers

What is the importance of checking the solution after solving an equation?

<p>It ensures the values satisfy the original equation (D)</p> Signup and view all the answers

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

<p>x^6 (A)</p> Signup and view all the answers

What does the negative exponent rule state for the expression $a^{-m}$?

<p>$ rac{1}{a^m}$ (C)</p> Signup and view all the answers

In simplifying the expression $ rac{a^{3/4}}{a^{1/2}}$, what is the simplified result?

<p>a^{1/4} (D)</p> Signup and view all the answers

When raising a product to a power, what is the correct form of $(xy)^m$?

<p>$x^m imes y^m$ (A)</p> Signup and view all the answers

What happens when solving the exponential equation $3^x = 3^4$?

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

How can the expression $ rac{(x^2y)^3}{x^4y^2}$ be simplified?

<p>x^{-2}y^{3} (A)</p> Signup and view all the answers

What is the value of $a^0$ for any nonzero $a$?

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

What can be concluded when $x^2 = 16$?

<p>$x = 4$ (C), $x = -4$ (D)</p> Signup and view all the answers

Which method can be used when the bases are different in an exponential equation?

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

Which of the following sets includes all positive integers starting from 1?

<p>Natural Numbers (N) (D)</p> Signup and view all the answers

What is the primary distinction between whole numbers and natural numbers?

<p>Whole numbers include zero. (B)</p> Signup and view all the answers

Which of the following is true about rational numbers?

<p>They can be expressed as a fraction of two integers. (B)</p> Signup and view all the answers

Which numbers fall under the category of integers?

<p>Whole numbers and their negative counterparts. (D)</p> Signup and view all the answers

What characterizes irrational numbers compared to rational numbers?

<p>They cannot be expressed as a simple fraction. (C)</p> Signup and view all the answers

What is the correct set symbol for irrational numbers?

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

Which of the following statements is true about real numbers?

<p>Real numbers can be either positive or negative. (D)</p> Signup and view all the answers

What defines imaginary numbers?

<p>They have a negative square root. (B)</p> Signup and view all the answers

What happens to the graph of a parabola as the value of $a$ increases beyond 1?

<p>The graph gets narrower and stretches upward. (A)</p> Signup and view all the answers

For the standard form of a hyperbola $y = \frac{a}{x} + q$, which of the following describes the domain?

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

What is the x-intercept of the function represented by the equation $y = \frac{a}{x} + q$?

<p>It can be found by setting $y = 0$. (D)</p> Signup and view all the answers

If $a < 0$ in the equation $y = ax^2 + q$, what type of turning point does the graph have?

<p>A maximum turning point. (C)</p> Signup and view all the answers

What characterizes the graph of a parabola when $q < 0$?

<p>The turning point is below the x-axis. (C)</p> Signup and view all the answers

How does the value of $q$ affect a hyperbolic function's graph?

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

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

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

What type of shape does the graph exhibit when $a < 0$?

<p>It resembles a parabola opening downwards. (B)</p> Signup and view all the answers

What is the axis of symmetry for the functions of the form $f(x) = ax^2 + q$?

<p>The y-axis (line $x = 0$). (A)</p> Signup and view all the answers

What does the term 'variable' represent in algebra?

<p>A symbol representing an unknown quantity. (D)</p> Signup and view all the answers

Which identity is used to factorise the sum of two cubes?

<p>$x^3 + y^3 = (x + y)(x^2 + xy + y^2)$ (C)</p> Signup and view all the answers

Which of the following statements correctly describes a coefficient?

<p>The numerical factor in a term. (A)</p> Signup and view all the answers

What is the first step when simplifying an algebraic fraction?

<p>Factorise the numerator and the denominator. (D)</p> Signup and view all the answers

In the equation $a^m imes a^n$, what is the result according to the exponent laws?

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

What is the correct formulation for multiplying a monomial and a binomial?

<p>$ax + ay$ (A)</p> Signup and view all the answers

When factorising a quadratic trinomial $ax^2 + bx + c$, what is the general first step?

<p>Identify the values of a, b, and c. (B)</p> Signup and view all the answers

What is the formula for finding the difference of two squares?

<p>$a^2 - b^2 = (a - b)(a + b)$ (D)</p> Signup and view all the answers

Which of these is a valid step in the process of factorisation by grouping?

<p>Group terms that share a common factor. (B)</p> Signup and view all the answers

What distinguishes irrational numbers from rational numbers?

<p>Irrational numbers have non-terminating, non-repeating decimal expansions. (D)</p> Signup and view all the answers

Which of the following correctly converts a terminating decimal into a rational number?

<p>Using the formula: integer part + $ rac{ ext{decimal digits}}{10^n}$ where $n$ is the position of the last digit. (D)</p> Signup and view all the answers

When rounding a decimal number, what should be done if the next digit is exactly 5?

<p>Always round up the last digit in the required position. (C)</p> Signup and view all the answers

What is a key characteristic of surds?

<p>Surds cannot be expressed as a simple fraction. (A)</p> Signup and view all the answers

Which of the following statements about the method of converting recurring decimals to rational numbers is correct?

<p>You should multiply by a sufficiently high power of 10 to line up the repeating digits. (C)</p> Signup and view all the answers

Which expression best describes how to compare the sizes of two surds, $ oot{n}{a}$ and $ oot{n}{b}$?

<p>$ oot{n}{a} &gt; oot{n}{b}$ if $a &gt; b$. (D)</p> Signup and view all the answers

What method is typically used to multiply a monomial by a binomial?

<p>Distribute the monomial across each term in the binomial. (A)</p> Signup and view all the answers

What is the primary purpose of estimating surds?

<p>To determine bounds using perfect squares or cubes. (C)</p> Signup and view all the answers

In which scenario would a number like $ rac{1}{3}$ be considered irrational?

<p>When approximated to two decimal places. (C)</p> Signup and view all the answers

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

<p>It is given by the equation $y = q$. (B)</p> Signup and view all the answers

For the function $y = rac{a}{x} + q$, how is the x-intercept calculated?

<p>By letting $y = 0$ and solving for $x$. (C)</p> Signup and view all the answers

What determines whether a graph of $y = ab^x + q$ curves upwards or downwards?

<p>The sign of $a$. (A)</p> Signup and view all the answers

What is the range of an exponential function $y = ab^x + q$ when $a > 0$?

<p>$ orall y : y &gt; q$. (B)</p> Signup and view all the answers

Which of the following statements about the domain of the function $y = a an heta + q$ is accurate?

<p>The domain includes all real numbers except where the function is undefined. (A)</p> Signup and view all the answers

What does the parameter $q$ affect in the function $y = a an heta + q$?

<p>The vertical shift of the graph. (C)</p> Signup and view all the answers

Which statement is true about the characteristics of the sine function $y = rac{1}{2} an heta + q$?

<p>Its y-intercept varies with $q$. (B)</p> Signup and view all the answers

In the context of trigonometric functions, what is typically the y-intercept of the cosine function $y = rac{3}{2} an heta + q$?

<p>At $(0°, 1)$. (D)</p> Signup and view all the answers

What does the sign of $b$ signify in an exponential function of the form $y = ab^x + q$?

<p>The direction of exponential growth or decay. (A)</p> Signup and view all the answers

What effect does having $|a| > 1$ in the function $y = a an heta + q$ have on the graph?

<p>It stretches the graph vertically. (C)</p> Signup and view all the answers

What effect does the parameter $q$ have in trigonometric functions such as sine, cosine, and tangent?

<p>It determines the vertical shift of the function. (C)</p> Signup and view all the answers

Which formula is used to calculate accumulated amount in simple interest?

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

How does compound interest differ from simple interest?

<p>Compound interest is earned on both the principal and accumulated interest. (D)</p> Signup and view all the answers

What is the main characteristic of a hire purchase agreement?

<p>It involves paying for a product in installments with simple interest on the remaining amount. (A)</p> Signup and view all the answers

Which factors contribute to the strength of a currency?

<p>Increasing investments in the country. (B)</p> Signup and view all the answers

What does the range of a function represent?

<p>The set of all outputs that the function can produce. (A)</p> Signup and view all the answers

How is inflation measured in terms of price increases over time?

<p>Using the compound interest formula. (B)</p> Signup and view all the answers

Which of the following formulas is applied in population growth calculations?

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

What defines the domain of a function?

<p>The set of all values that can be inputted into the function. (D)</p> Signup and view all the answers

In the context of finance, what does the term 'principal' refer to?

<p>The initial amount of money invested or borrowed. (B)</p> Signup and view all the answers

What does the variable $c$ represent in the equation of a linear function $y = mx + c$?

<p>The y-intercept of the graph (A)</p> Signup and view all the answers

What is the effect of a negative value of $m$ in the equation $y = mx + c$?

<p>The graph slopes downwards from left to right (A)</p> Signup and view all the answers

In the context of a linear sequence, what is a common difference?

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

How does an increase in the value of $c$ affect the graph of a linear function?

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

What characteristic of the graph is determined by the value of $a$ in the equation of the form $y = ax^2 + q$?

<p>The direction in which the parabola opens (B)</p> Signup and view all the answers

Which of the following methods is used to determine the graph of $y = mx + c$?

<p>Calculating and plotting both the x-intercept and y-intercept. (A)</p> Signup and view all the answers

What is the domain and range of a linear function $f(x) = mx + c$?

<p>Domain: all real numbers, Range: all real numbers (D)</p> Signup and view all the answers

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

<p>It shifts the graph vertically up or down. (A)</p> Signup and view all the answers

What do you calculate to find the x-intercept of a linear function?

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

In which scenario will a parabola open upwards?

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

What is the formula for converting an amount from one currency to another?

<p>Amount in new currency = Amount in original currency / Exchange rate (A)</p> Signup and view all the answers

If an event has a probability of 0, what does it indicate about the event?

<p>The event will never occur (D)</p> Signup and view all the answers

In the context of probability, what does the ratio $P(E) = \frac{n(E)}{n(S)}$ represent?

<p>The likelihood of event E occurring (C)</p> Signup and view all the answers

What does a probability of 0.5 indicate about an event?

<p>The event will occur half the time (B)</p> Signup and view all the answers

What is a Venn diagram primarily used for in probability?

<p>To illustrate the relationships between different sets (B)</p> Signup and view all the answers

What does the term 'union' denote in set theory?

<p>Elements in at least one of the sets (A)</p> Signup and view all the answers

Relative frequency provides what type of probability?

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

If two events A and B have no overlap, what can be said about their intersection?

<p>A ∩ B is empty (C)</p> Signup and view all the answers

As the number of trials increases, relative frequency tends to approach which form of probability?

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

What does a probability of 1 indicate about an event?

<p>The event will always occur (A)</p> Signup and view all the answers

What is the range of the function defined by the equation $y = a \cos \theta + q$ for $a > 0$?

<p>[q - |a|, q + |a|] (B)</p> Signup and view all the answers

What is the period of the cosine and sine functions?

<p>360° (D)</p> Signup and view all the answers

What is the range of the tangent function $y = \tan \theta$?

<p>[f(\theta) : f(\theta) \in \mathbb{R}] (A)</p> Signup and view all the answers

How does changing the value of $q$ in the equation $y = a \tan \theta + q$ affect the graph?

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

To find the equation of a parabola, which point is specifically used to calculate the value of $q$?

<p>The y-intercept. (A)</p> Signup and view all the answers

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

<p>It stretches or compresses the branches. (D)</p> Signup and view all the answers

What are the x-intercepts of the tangent function $y = \tan \theta$?

<p>(0°, 0), (180°, 0), (360°, 0) (C)</p> Signup and view all the answers

What are the asymptotes of the tangent function defined by $y = \tan \theta$?

<p>At $\theta = 90°$ and $\theta = 270°$ (C)</p> Signup and view all the answers

What defines the direction in which a parabola opens?

<p>The sign of $a$. (C)</p> Signup and view all the answers

Which statement accurately reflects the relationship between sine and cosine functions?

<p>The cosine graph can be shifted left by 90° to overlap with the sine graph. (A)</p> Signup and view all the answers

What is the correct expression to calculate the probability of the union of two events?

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

What defines mutually exclusive events?

<p>Events that cannot occur at the same time. (B)</p> Signup and view all the answers

Which statement is true regarding the probabilities of complementary events?

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

In a Venn diagram, which representation indicates mutually exclusive events?

<p>Two circles that do not touch. (D)</p> Signup and view all the answers

What is the intersection of two mutually exclusive events?

<p>The empty set. (C)</p> Signup and view all the answers

How can the probability of the union of two mutually exclusive events be calculated?

<p>By adding their individual probabilities. (C)</p> Signup and view all the answers

What is the complement of event A as represented?

<p>All outcomes not in event A. (D)</p> Signup and view all the answers

What does the identity P(A) + P(A') = 1 signify?

<p>The combined probability of complementary events. (B)</p> Signup and view all the answers

What happens if you add the probabilities of events A and B without recognizing their intersection?

<p>You double-count the intersection area. (D)</p> Signup and view all the answers

What is indicated by the notation P(A ∩ B) = ∅?

<p>Events A and B are mutually exclusive. (D)</p> Signup and view all the answers

What do the coordinates of the intersection point represent in a system of linear equations?

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

Which of the following is a key step in solving word problems mathematically?

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

What is the result when multiplying both sides of an inequality by a negative number?

<p>The inequality must be switched (A)</p> Signup and view all the answers

In the formula for a linear sequence, what does the term $c$ represent?

<p>The constant term (D)</p> Signup and view all the answers

Which principle is essential for isolating an unknown variable in a literal equation?

<p>Performing the same operation to both sides (C)</p> Signup and view all the answers

What is the common difference in a linear sequence denoted by?

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

How is the common difference $d$ calculated in a sequence?

<p>By subtracting $T_{n-1}$ from $T_n$ (B)</p> Signup and view all the answers

When changing a literal equation to isolate a variable in the denominator, what operation is usually performed?

<p>Multiply both sides by the lowest common denominator (C)</p> Signup and view all the answers

Which of the following sequences is classified as a linear sequence?

<p>2, 4, 6, 8 (C)</p> Signup and view all the answers

What characterizes surds?

<p>They are the n-th root of numbers that cannot simplify to rational numbers. (B)</p> Signup and view all the answers

When converting a recurring decimal to a rational number, what is the first step?

<p>Multiply by a power of 10 before defining the recurring pattern. (D)</p> Signup and view all the answers

How can rounding off an irrational number be interpreted?

<p>As obtaining a rational approximation. (D)</p> Signup and view all the answers

Which of the following describes the nature of rational numbers?

<p>They encompass variations of both terminating and repeating decimals. (A)</p> Signup and view all the answers

In the context of estimating surds, what is the role of identifying perfect powers?

<p>It provides boundaries for comparison to better estimate the value. (B)</p> Signup and view all the answers

What type of decimal numbers qualify as rational numbers?

<p>Only repeating and terminating decimals. (B)</p> Signup and view all the answers

What happens if the digit to be rounded in a decimal is 9?

<p>It converts to 0 and increases the previous digit by one. (D)</p> Signup and view all the answers

Which statement best describes the procedure for multiplying a monomial by a binomial?

<p>You distribute the monomial across both terms of the binomial. (D)</p> Signup and view all the answers

What does the expression $(ax + b)(cx + d)$ expand to?

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

Which statement accurately describes the set of rational numbers?

<p>They can be expressed as a quotient of two integers. (A)</p> Signup and view all the answers

What distinguishes irrational numbers from rational numbers?

<p>Irrational numbers have non-terminating decimal expansions. (C)</p> Signup and view all the answers

Which of the following numbers is classified as a natural number?

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

In the context of the real number system, which of the following subsets contains negative values?

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

Which example best represents an irrational number?

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

What is the relationship between rational numbers and real numbers?

<p>All rational numbers are also real numbers. (B)</p> Signup and view all the answers

Which of the following statements about imaginary numbers is true?

<p>They are defined as numbers with negative square roots. (A)</p> Signup and view all the answers

What is the complete set of real numbers represented by?

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

What is the result of applying the power of a power rule to the expression $(x^3)^4$?

<p>$x^{12}$ (C)</p> Signup and view all the answers

If $a^m = 2$ and $a^n = 8$ for positive values of $a$, what is the relationship between $m$ and $n$?

<p>$m = n/3$ (D)</p> Signup and view all the answers

How would you simplify the expression $ rac{a^{5/2}}{a^{3/2}}$ using the laws of exponents?

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

What does the expression $( rac{2}{3})^{2/3}$ simplify to?

<p>$ rac{2^{2/3}}{3^{2/3}}$ (A)</p> Signup and view all the answers

If $a > 0$ and $a^{-2}$ is present in an expression, how is it equivalently expressed?

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

In solving the equation $3^x = 27$, which method would you use to isolate $x$?

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

What is the equivalent expression for $(ab)^{3/4}$?

<p>$a^{3/4}b^{3/4}$ (C)</p> Signup and view all the answers

Which of the following is NOT a correct application of the zero exponent rule?

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

When simplifying the expression $(x^4y^2)^3$, which step is first?

<p>Use the power of a power rule (D)</p> Signup and view all the answers

Which statement best describes how to handle a negative exponent in the expression $a^{-3}$?

<p>It becomes $ rac{1}{a^3}$ (D)</p> Signup and view all the answers

What do the coordinates of the intersection point of two linear equations represent?

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

In the equation $v = \frac{D}{t}$, what does $D$ represent?

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

What is the common difference in the linear sequence defined by $T_n = 3n + 5$?

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

When converting the equation $4x - 2 > 10$ into an inequality solution, what is the first step?

<p>Add 2 to both sides (B)</p> Signup and view all the answers

In the context of solving literal equations, what does 'changing the subject of the formula' mean?

<p>Isolating any variable in the formula (D)</p> Signup and view all the answers

When solving the inequality $2 - x \geq 3$, what is the resulting inequality after isolation of $x$?

<p>x \leq -1 (B)</p> Signup and view all the answers

In a linear sequence, if $d$ is not positive, what can we conclude about the sequence?

<p>The terms are decreasing (D)</p> Signup and view all the answers

How do you find the value of $d$ in a linear sequence defined by the general term $T_n = 7n - 2$?

<p>By identifying the coefficient of $n$ (D)</p> Signup and view all the answers

If the linear equation $y = 3x + 2$ intersects the x-axis at a certain point, what is the x-coordinate of that point?

<p>-2/3 (D)</p> Signup and view all the answers

What is the maximum number of solutions a quadratic equation can possess?

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

Which step is essential to verify the accuracy of a solution in both linear and quadratic equations?

<p>Substituting the solution back into the original equation (C)</p> Signup and view all the answers

What process is employed to eliminate a variable when solving simultaneous equations?

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

When rearranging terms in a linear equation, what is primarily moved to one side?

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

Which of the following statements is true regarding the factorisation of a quadratic equation?

<p>It can sometimes yield a single repeated factor. (D)</p> Signup and view all the answers

In the method for solving linear equations, what is the purpose of expanding all brackets?

<p>To simplify both sides of the equation (B)</p> Signup and view all the answers

What is the first step when using the substitution method for solving simultaneous equations?

<p>Isolating one variable in equation (C)</p> Signup and view all the answers

What must be true about the balance of an equation when performing operations?

<p>Both sides must be altered equally (D)</p> Signup and view all the answers

In what scenario can a quadratic equation have no real solutions?

<p>When the discriminant is negative (C)</p> Signup and view all the answers

What does the term 'factorisation' refer to in algebra?

<p>Breaking down an expression into simpler expressions that multiply to give the original. (A)</p> Signup and view all the answers

Which of the following represents the correct expansion of the product of a binomial and a trinomial?

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

When simplifying the fraction $ rac{a}{b} + rac{c}{b}$, what is the simplest form of the expression?

<p>$ rac{a + c}{b}$ (A)</p> Signup and view all the answers

Which equation correctly represents the factorisation of a difference of squares?

<p>$a^2 - b^2 = (a + b)(a - b)$ (B)</p> Signup and view all the answers

What is the first step in the general procedure for factorising a trinomial of the form $ax^2 + bx + c$?

<p>Identify any common factors. (D)</p> Signup and view all the answers

Which identity is used for factorising the sum of two cubes?

<p>$x^3 + y^3 = (x + y)(x^2 - xy + y^2)$ (A)</p> Signup and view all the answers

When multiplying two binomials, which form does the result take?

<p>$acx^2 + (ad + bc)x + bd$ (A)</p> Signup and view all the answers

What is the resulting form of the difference of two cubes represented by the expression $x^3 - y^3$?

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

In the operation of multiplying a binomial by a trinomial, what does each term of the binomial get multiplied by?

<p>Each term of the trinomial. (A)</p> Signup and view all the answers

What represents the y-intercept in a linear equation of the form $y = mx + c$?

<p>It is the point where the graph crosses the y-axis. (D)</p> Signup and view all the answers

How does the value of $m$ affect the graph of a linear function?

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

In the quadratic function $y = ax^2 + q$, what aspect of the graph is significantly influenced by the value of $q$?

<p>The vertical position of the turning point. (D)</p> Signup and view all the answers

What does a negative value for $a$ in the quadratic function $y = ax^2 + q$ indicate about the graph?

<p>The graph opens downwards, producing a maximum turning point. (A)</p> Signup and view all the answers

What is the domain of linear functions represented by $f(x) = mx + c$?

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

What role does the y-intercept ($c$) play when $c < 0$ in the graph of a linear function?

<p>It shifts the entire graph downwards. (C)</p> Signup and view all the answers

What are the three characteristics needed to sketch the graph of a linear function $f(x) = mx + c$?

<p>The sign of $m$, the y-intercept, and the x-intercept. (B)</p> Signup and view all the answers

What is the defining characteristic of a linear sequence?

<p>The elements change at a constant rate. (D)</p> Signup and view all the answers

In the equation $m = \frac{\text{change in } y}{\text{change in } x}$, what does $m$ represent?

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

What happens to the graph of a parabola when the value of $a$ is decreased while being less than zero?

<p>The graph becomes wider as it approaches zero. (B), The graph narrows and remains a frown. (D)</p> Signup and view all the answers

How does the function $y = ax^2 + q$ behave when $a$ is positive?

<p>The graph is a parabola that opens upwards with a minimum turning point. (A)</p> Signup and view all the answers

Which statement accurately describes the domain and range for the hyperbolic function $y = \frac{a}{x} + q$?

<p>Domain: all real numbers except for $0$; Range: all values except for $q$. (C)</p> Signup and view all the answers

What is the effect of choosing $q$ to be less than zero in the hyperbola $y = \frac{a}{x} + q$?

<p>The graph shifts vertically downwards by $|q|$ units. (A)</p> Signup and view all the answers

What characteristic of the parabola is defined by the value of $q$?

<p>It sets the vertical shift of the graph. (B)</p> Signup and view all the answers

Which statement is true about the turning points of the graph $y = ax^2 + q$ when $a < 0$?

<p>The maximum turning point occurs at $(0, q)$. (C)</p> Signup and view all the answers

What happens to the hyperbolic graph's shape when $a$ is negative?

<p>The graph lies in the second and fourth quadrants. (B)</p> Signup and view all the answers

What is the result of evaluating the x-intercept for the function $y = \frac{a}{x} + q$?

<p>It can be found by setting $y$ equal to $0$. (A), It does not exist because of the vertical asymptote. (B)</p> Signup and view all the answers

Which statement about the axes of symmetry of the parabolic function $y = ax^2 + q$ is correct?

<p>The axis of symmetry is always at the line $x = 0$. (D)</p> Signup and view all the answers

What is the range of the cosine function expressed in terms of parameters a and q?

<p>[q - |a|, q + |a|] (B)</p> Signup and view all the answers

For the tangent function, what are the specified asymptotes within the range of 0° to 360°?

<p>theta = 90° and theta = 270° (C)</p> Signup and view all the answers

In the context of the parabola, what does a negative value of a indicate?

<p>The parabola opens downward. (D)</p> Signup and view all the answers

When determining the equation of a hyperbola, what is the first step in the process?

<p>Examine the sketch to determine the sign of a. (A)</p> Signup and view all the answers

If q in the tangent function is greater than 0, what effect does this have on the graph?

<p>The graph shifts upward by q units. (A)</p> Signup and view all the answers

What does the process of calculating intercepts involve for parabolas and lines?

<p>Setting y to 0 to find x-intercepts and x to 0 for the y-intercept. (D)</p> Signup and view all the answers

How do the graphs of sine and cosine functions relate to each other?

<p>The cosine graph can be achieved by shifting the sine graph to the left by 90°. (A)</p> Signup and view all the answers

What primary factor influences the steepness of the branches in the tangent function?

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

What is the period of the tangent function?

<p>180° (D)</p> Signup and view all the answers

What does a probability of 0 indicate about an event?

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

When describing probability as a fraction, which represents a probability of 0.5?

<p>$ rac{1}{2}$ (D)</p> Signup and view all the answers

Which formula correctly defines the theoretical probability of an event?

<p>$P(E) = rac{n(E)}{n(S)}$ (B)</p> Signup and view all the answers

How does relative frequency differ from theoretical probability?

<p>It is determined through experimental trials. (D)</p> Signup and view all the answers

In relation to Venn diagrams, which of the following correctly represents 'A and B'?

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

What is represented by a probability that approaches 1 as more trials are conducted?

<p>The relative frequency is becoming constant. (D)</p> Signup and view all the answers

Which of the following statements about union and intersection is incorrect?

<p>Union is always a subset of both sets. (D)</p> Signup and view all the answers

Which of the following situations describes 'complete containment' in the context of Venn diagrams?

<p>Set B contains all elements of A. (D)</p> Signup and view all the answers

What is the range of values for probability?

<p>From 0 to 1 (A)</p> Signup and view all the answers

When calculating the amount in a new currency, which component clearly indicates the conversion involved?

<p>Divide the amount in the original currency by the exchange rate. (C)</p> Signup and view all the answers

How does the variable $q$ affect the sine function $y = a an \theta + q$?

<p>It determines the vertical shift of the graph. (D)</p> Signup and view all the answers

What characterizes the difference between simple interest and compound interest?

<p>Compound interest is calculated on both the principal and the accumulated interest. (B)</p> Signup and view all the answers

In the context of hyperbolas, how can asymptotes be identified?

<p>By finding the values that make the denominator equal to zero. (B)</p> Signup and view all the answers

Which formula accurately represents the accumulated amount with simple interest?

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

What is the defining characteristic of the population growth formula compared to simple interest?

<p>It uses the same compound interest formula as for financial contexts. (C)</p> Signup and view all the answers

What component is crucial in determining the vertical shift of the cosine function?

<p>The constant $q$. (B)</p> Signup and view all the answers

When calculating the accumulated amount from inflation, which variable plays the same role as the principal in other financial contexts?

<p>$P$, signifying the current price. (B)</p> Signup and view all the answers

What is a key factor that distinguishes a hire purchase agreement from other loan types?

<p>Simple interest is applied on the remaining balance. (B)</p> Signup and view all the answers

What effect does an increase in the interest rate have on the accumulated amount for both simple and compound interest?

<p>It increases the final amount for both simple and compound interest. (B)</p> Signup and view all the answers

Which statement best describes the impact of foreign exchange rates on international trade?

<p>They can either inflate or deflate international prices. (C)</p> Signup and view all the answers

What is the probability relationship for mutually exclusive events?

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

In a Venn diagram representing two events A and B, which area corresponds to the probability of their intersection?

<p>Only the overlapping area of A and B (C)</p> Signup and view all the answers

If event A has a probability of 0.4, what must the probability of its complement A' be?

<p>0.6 (D)</p> Signup and view all the answers

Which identity correctly expresses the relationship between the probabilities of complementary events?

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

Which statement correctly describes mutually exclusive events?

<p>They cannot have any outcomes in common. (B)</p> Signup and view all the answers

Which of the following statements about the probability of unions of two events is true?

<p>P(A ∪ B) requires knowing P(A ∩ B) to avoid double counting. (D)</p> Signup and view all the answers

In the context of unions and intersections, what does the notation P(A ∩ B) represent?

<p>The probability that both A and B occur simultaneously (B)</p> Signup and view all the answers

What does the symbol P(S) signify in probability theory?

<p>The probability of the sample space (C)</p> Signup and view all the answers

What is one consequence of events A and B being mutually exclusive?

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

What does the sign of the constant $a$ determine in the graph of the function $y = ab^x + q$?

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

In the function $y = ab^x + q$, what is the impact of $b$ when $b > 1$?

<p>The function represents exponential growth (D)</p> Signup and view all the answers

What is the range of the function $y = a ext{sin}( heta) + q$ when $a < 0$ and $q = 2$?

<p>[2 - |a|, 2] (D)</p> Signup and view all the answers

For the sine and cosine functions, how do their y-intercepts differ at $x = 0$?

<p>Sine has a y-intercept of 0, while cosine has 1 (A)</p> Signup and view all the answers

What defines the period of the sine and cosine functions?

<p>The interval over which the function repeats (A)</p> Signup and view all the answers

Which characteristic is specific to the x-intercepts of the sine function?

<p>They are located at specific multiples of $180°$ (B)</p> Signup and view all the answers

What effect does a negative value of $a$ have on a function of the form $y = a ext{cos}( heta) + q$?

<p>It reflects the graph about the x-axis (C)</p> Signup and view all the answers

What is the domain restriction for the sine function described?

<p>$[0°, 360°]$ (D)</p> Signup and view all the answers

How is the y-intercept of the exponential function defined mathematically?

<p>$y = ab^0 + q$ (D)</p> Signup and view all the answers

In the context of exponential functions, what happens to the graph when $q < 0$?

<p>The graph shifts down by $q$ units (B)</p> Signup and view all the answers

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