Gr 10 Math Nov P1 Medium

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Questions and Answers

Which of the following defines natural numbers?

  • All whole numbers and their negatives
  • The set of positive integers starting from 1 (correct)
  • All positive integers including zero
  • Numbers that can be expressed as a fraction

What is the main difference between rational and irrational numbers?

  • Irrational numbers can be expressed as fractions
  • Rational numbers include negative numbers
  • Irrational numbers are always positive
  • Rational numbers have decimal expansions that are repeating or terminating (correct)

Which of the following symbols represents integers?

  • Q
  • N
  • Z (correct)
  • N_0

Which of these is an example of an irrational number?

<p>√3 (A)</p> Signup and view all the answers

Whole numbers are defined as:

<p>All natural numbers including zero (C)</p> Signup and view all the answers

What is true about imaginary numbers?

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

Which subset includes both positive and negative whole numbers, as well as zero?

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

Which of the following statements about real numbers is correct?

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

What is the result of multiplying the binomial (A + B) by the trinomial (C + D + E)?

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

When factorising a quadratic trinomial of the form ax^2 + bx + c, which step is critical to finding the correct factors?

<p>Finding pairs of factors of a and c (B)</p> Signup and view all the answers

Which of the following represents the difference of two squares?

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

What is the general process for simplifying algebraic fractions?

<p>Factorise both numerator and denominator to cancel common factors (D)</p> Signup and view all the answers

Which of the following is NOT an exponent law?

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

In factorising by grouping, what is the first step?

<p>Grouping suitable pairs of terms together (D)</p> Signup and view all the answers

Which expression represents the sum of two cubes?

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

What is the first step in the general procedure for factorising a trinomial?

<p>Identify any common factors (B)</p> Signup and view all the answers

Which operation allows you to divide fractions correctly?

<p>Multiply by the reciprocal of the second fraction (D)</p> Signup and view all the answers

What is the expanded form of the expression (ax + b)(cx + d)?

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

Which of the following is true about irrational numbers?

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

How do you convert a terminating decimal like 0.75 into a rational number?

<p>By expressing it as a fraction of integers. (B)</p> Signup and view all the answers

What is the first step in rounding a decimal number to the nearest hundredth?

<p>Marking the digit after the hundredth place. (A)</p> Signup and view all the answers

Which of the following is NOT an example of a rational number?

<p>√3 (B)</p> Signup and view all the answers

Which statement regarding surds is correct?

<p>Surds are typically expressed in the form $\sqrt[n]{a}$ where $a$ is not a perfect power. (A)</p> Signup and view all the answers

What is the result of multiplying two binomials $(ax + b)(cx + d)$?

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

What characterizes a surd?

<p>It has a non-recurring decimal form. (D)</p> Signup and view all the answers

When rounding the number 2.367 to one decimal place, what is the result?

<p>2.4 (A)</p> Signup and view all the answers

How would you estimate the value of $ oot{3}{28}$?

<p>Between 3 and 4. (C)</p> Signup and view all the answers

What is the result of rac{a^m}{a^n} when simplified?

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

Which expression illustrates the power of a power law?

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

Which condition must be met to equate the exponents in exponential equations where a^x = a^y?

<p>a &gt; 0 and a != 1 (B)</p> Signup and view all the answers

When simplifying the expression (ab)^{m/n}, which of the following is true?

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

What is the effect of a negative exponent on a base?

<p>It inverts the base as rac{1}{a^n}. (A)</p> Signup and view all the answers

To solve the equation a^x = a^5 by equating the exponents, what conclusion can be drawn?

<p>x = 5 (A)</p> Signup and view all the answers

How can complex fractions be simplified with exponents?

<p>By converting to prime bases and dividing. (A)</p> Signup and view all the answers

What is the value of a^0 when a != 0?

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

Which method is used to solve exponential equations when bases differ?

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

Which of the following is part of the method for simplifying rational exponents?

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

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

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

Which of the following correctly describes the first step in solving a word problem?

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

In solving literal equations, what is the first principle to remember?

<p>Always apply the same operation to both sides (C)</p> Signup and view all the answers

What happens to the inequality sign when both sides are divided by a negative number?

<p>It is reversed (A)</p> Signup and view all the answers

What is the general formula for a linear sequence commonly represented as?

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

How is the common difference 'd' in a sequence calculated?

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

Which of the following is an example of a linear inequality?

<p>$ rac{4}{3}x - 6 &lt; 7x + 2$ (B)</p> Signup and view all the answers

Which variable represents the constant in the general formula for a linear sequence?

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

What does isolating the unknown variable in a literal equation involve?

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

Which term is used to describe individual items in a sequence?

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

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

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

Which step is NOT part of solving a linear equation?

<p>Divide by common factors (A)</p> Signup and view all the answers

What is the first step in solving a quadratic equation?

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

When solving simultaneous equations by elimination, what is done to the coefficients?

<p>They are made equal (C)</p> Signup and view all the answers

Which method would you most likely use to solve the equation $y = 3x + 5$ and $y = -2x + 1$?

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

Which reasoning supports the requirement of performing the same operation on both sides of an equation?

<p>To maintain balance (B)</p> Signup and view all the answers

What can be concluded about the roots of a quadratic equation with a discriminant of zero?

<p>A single repeated solution (D)</p> Signup and view all the answers

In terms of the number of variables, what is required to solve simultaneous equations?

<p>One equation for each variable (D)</p> Signup and view all the answers

What does the factorization of a quadratic equation allow you to do?

<p>Set each factor to zero (C)</p> Signup and view all the answers

What should you always do after finding the solution to an equation?

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

What is the effect of increasing the value of 'a' when 'a' is greater than 0 on the graph of the function?

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

For which value of 'a' does the graph of the function have a maximum turning point?

<p>When a &lt; 0 (B)</p> Signup and view all the answers

What is the axis of symmetry for the parabolic function in standard form?

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

What is the key characteristic of hyperbolic functions regarding intercepts?

<p>They have no y-intercept due to being undefined at x = 0. (C)</p> Signup and view all the answers

If 'q' is less than 0 for a parabolic function, what can be concluded about its range?

<p>Its range is (negative infinity, q]. (B)</p> Signup and view all the answers

What does the value of 'q' determine in the function of a hyperbola?

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

What distinguishes the graphs of hyperbolic functions when 'a' is positive from when 'a' is negative?

<p>The positions of the graphs in the quadrants. (C)</p> Signup and view all the answers

How do you find the x-intercept of a function of the form y = ax^2 + q?

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

What happens to the graph of a parabolic function as 'a' approaches 0 for '0 < a < 1'?

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

What defines the range of the function y = a/x + q?

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

What determines whether the graph of an exponential function curves upwards or downwards?

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

What is the y-intercept of the exponential function of the form y = ab^x + q when x = 0?

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

What is the characteristic of the range for an exponential function with a < 0?

<p>It is less than q (D)</p> Signup and view all the answers

For the sine function y = a sin θ + q, what effect does a < 0 have on the graph?

<p>Reflection about the x-axis (B)</p> Signup and view all the answers

Which of the following is true about the x-intercepts of the sine function y = sin θ?

<p>They occur at odd multiples of 90° (A)</p> Signup and view all the answers

What is the effect of q > 0 on the graph of the cosine function y = a cos θ + q?

<p>Shifts the graph vertically up by q units (C)</p> Signup and view all the answers

What is the maximum turning point of the sine function y = sin θ?

<p>(90°, 1) (D)</p> Signup and view all the answers

Which characteristic is correct for a function that shows exponential growth?

<p>b &gt; 1 (D)</p> Signup and view all the answers

What is the period of the sine and cosine functions defined in degrees?

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

What is the range of values for a probability?

<p>Between 0 and 1 (D)</p> Signup and view all the answers

Which of the following describes relative frequency?

<p>Empirical measure from experimental trials (A)</p> Signup and view all the answers

What does the symbol $ P(E) $ represent in probability?

<p>The ratio of outcomes in the event set to total outcomes (D)</p> Signup and view all the answers

How is the union of two sets denoted?

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

What does a probability of 0 indicate about an event?

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

What does it mean if two sets have complete containment?

<p>One set is fully contained within the other (B)</p> Signup and view all the answers

According to the Venn diagram definition, what represents elements that belong to a set?

<p>The region inside the closed curve (D)</p> Signup and view all the answers

Which statement about theoretical probability is true?

<p>It applies to events with equal likelihood (A)</p> Signup and view all the answers

What is the formula for calculating theoretical probability $ P(E) $?

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

What is the result of union operation $ A ∪ B $ when sets A and B have no elements in common?

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

What does the variable $q$ do in the equation for trigonometric functions?

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

In calculating the x-intercept of a function, which value do you set to zero?

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

What is the formula for calculating simple interest?

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

Which of the following statements is true about compound interest?

<p>Interest is calculated on the accumulated interest as well. (A)</p> Signup and view all the answers

How does inflation relate to the compound interest formula?

<p>Inflation affects the future price calculated using the formula. (B)</p> Signup and view all the answers

What is the main advantage of compound interest for investments?

<p>It causes exponential growth of the investment. (D)</p> Signup and view all the answers

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

<p>Simple interest is based on principal only, compound interest includes accrued interest. (D)</p> Signup and view all the answers

In a hire purchase agreement, how is interest calculated?

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

What does the term 'currency strength' refer to?

<p>The value of a currency compared to another currency. (D)</p> Signup and view all the answers

How is population growth calculated?

<p>Using the compound interest formula. (D)</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|] (C)</p> Signup and view all the answers

What is the period of the sine and cosine functions?

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

Which values of theta are excluded from the domain of the tangent function defined as $y = \tan \theta$?

<p>90° and 270° (D)</p> Signup and view all the answers

In the equation of a parabola $y = ax^2 + q$, what does the parameter $q$ represent?

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

How does the value of 'a' affect the graph of the tangent function $y = a \tan \theta + q$?

<p>It modifies the steepness of the graph branches (A)</p> Signup and view all the answers

What is the x-intercept for the tangent function $y = \tan \theta$ within the specified domain?

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

Which characteristic is true about the equation of a hyperbola in the form $y = \frac{a}{x} + q$?

<p>The curves lie in quadrants based on the sign of 'a' (D)</p> Signup and view all the answers

What is the effect of a positive value of 'q' on the graph of the function $y = a \tan \theta + q$?

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

When determining the equation of a parabola, what is the first step to take?

<p>Examine the sketch and direction of the parabola (D)</p> Signup and view all the answers

What happens to the cosine graph when it is shifted to the right by 90°?

<p>It becomes a sine graph (B)</p> Signup and view all the answers

What is the correct interpretation of the formula $P(A ext{ union } B) = P(A) + P(B) - P(A ext{ intersection } B)$?

<p>It accurately calculates the probability of at least one of the events occurring. (D)</p> Signup and view all the answers

Which statement is true regarding mutually exclusive events?

<p>The probability of their union is $P(A) + P(B)$. (A)</p> Signup and view all the answers

What is the complement of an event $A$ denoted as $A'$?

<p>The event that includes all outcomes in the sample space except those in $A$. (B)</p> Signup and view all the answers

Which of the following accurately describes the relationship between complementary events?

<p>Their union always equals the sample space $S$. (B)</p> Signup and view all the answers

For the probabilities of complementary events $P(A)$ and $P(A')$, which statement is accurate?

<p>The sum of their probabilities always equals 1. (A)</p> Signup and view all the answers

What does it mean when two events are described as mutually exclusive?

<p>They cannot happen together at all. (B)</p> Signup and view all the answers

If $P(A) = 0.3$ and $P(B) = 0.5$ for two mutually exclusive events $A$ and $B$, what is $P(A ext{ union } B)$?

<p>0.8 (D)</p> Signup and view all the answers

When visualizing probabilities with Venn diagrams, what is the purpose of subtracting $P(A ext{ intersection } B)$ from $P(A) + P(B)$?

<p>To account for the overlap where the events both occur. (B)</p> Signup and view all the answers

In the context of complementary events, what is expressed by the relationship $A ext{ union } A' = S$?

<p>It shows that all outcomes are accounted for in the sample space. (B)</p> Signup and view all the answers

What is represented in a Venn diagram for two mutually exclusive events?

<p>The events $A$ and $B$ are depicted in separate circles. (C)</p> Signup and view all the answers

What characteristic of a linear function is affected by the value of $m$ in the equation $y = mx + c$?

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

Which of the following describes the effect of a negative value of $c$ in the equation $y = mx + c$?

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

What is the general form of a quadratic function?

<p>$y = ax^2 + q$ (A)</p> Signup and view all the answers

How does the value of $a$ in the quadratic function $y = ax^2 + q$ affect its graph?

<p>It changes the shape and direction of the parabola. (D)</p> Signup and view all the answers

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

<p>$ orall x ext{ in } f{R}$ (B)</p> Signup and view all the answers

Which method can be used to sketch a graph of the form $y = mx + c$?

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

What happens to the graph of a linear function if $m = 0$?

<p>The graph is a horizontal line. (D)</p> Signup and view all the answers

Which point on the graph of a linear function is found by setting $x = 0$?

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

What characteristic is determined by both the sign and value of $q$ in the quadratic function $y = ax^2 + q$?

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

In the equation $y = mx + c$, what does $c$ represent?

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

What operation is used to multiply a binomial by a trinomial?

<p>Multiply each term of the binomial by the trinomial (B)</p> Signup and view all the answers

Which of the following describes a coefficient?

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

Which expression is an example of the difference of two squares?

<p>$4 - y^2$ (A), $x^2 - 9$ (C)</p> Signup and view all the answers

What is the first step in simplifying a fraction?

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

In rewriting the operation $ rac{a}{b} imes rac{c}{d}$ in a different form, what is true?

<p>$ rac{a imes c}{b imes d}$ (C)</p> Signup and view all the answers

What is the method used for factorising by grouping?

<p>Group terms with common factors (C)</p> Signup and view all the answers

Which of the following represents the sum of two cubes?

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

Which step is essential when factorising a quadratic trinomial?

<p>Find two binomials whose product matches the expression (A)</p> Signup and view all the answers

What is an exponent law that describes the product of exponents?

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

Which of the following correctly describes the set of natural numbers?

<p>The set of positive integers starting from 1 (C)</p> Signup and view all the answers

Which of the following decimal representations is classified as irrational?

<p>2.7182818284 (B)</p> Signup and view all the answers

Which number belongs to the set of rational numbers?

<p>0.333... (D)</p> Signup and view all the answers

What is the result of rounding the number 5.678 to the nearest tenth?

<p>5.7 (C)</p> Signup and view all the answers

Which of the following correctly illustrates the relationship between whole numbers and integers?

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

What is a characteristic of irrational numbers?

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

When converting the recurring decimal 0.666... into a rational number, what fraction do you get?

<p>2/3 (A)</p> Signup and view all the answers

What is one method to estimate the value of the surd \sqrt{30}?

<p>Identify the nearest perfect square less than 30, which is 25 (C)</p> Signup and view all the answers

What symbol represents the set of all real numbers?

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

Which of these is included in the set of integers?

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

Which statement about irrational numbers is accurate?

<p>They cannot be expressed in fraction form. (A)</p> Signup and view all the answers

Which of the following statements about imaginary numbers is accurate?

<p>Imaginary numbers arise from the square root of negative numbers. (D)</p> Signup and view all the answers

In the expression (2x + 1)(3x + 4), what is the coefficient of x after expanding?

<p>11 (A)</p> Signup and view all the answers

What is the appropriate decimal representation for the rational number 3/8?

<p>0.375 (D)</p> Signup and view all the answers

Which of these numbers represents an irrational number?

<p>√3 (D)</p> Signup and view all the answers

Which of the following best describes a surd?

<p>The n-th root of a non-integer that cannot be simplified. (C)</p> Signup and view all the answers

What is the first step when multiplying a monomial by a binomial?

<p>Distribute the binomial over the monomial. (A)</p> Signup and view all the answers

What is the rounded value of the irrational number π when rounded to two decimal places?

<p>3.14 (A)</p> Signup and view all the answers

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

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

Which of the following is a step in solving quadratic equations using factorisation?

<p>Divide by a common factor (A)</p> Signup and view all the answers

What must be true about the operations performed on both sides of an equation?

<p>They must balance and be equal after simplification (D)</p> Signup and view all the answers

Which method can be used to solve simultaneous equations involving two variables?

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

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

<p>ax^2 + bx + c = 0 (D)</p> Signup and view all the answers

Which scenario can lead to a quadratic equation having one solution?

<p>The discriminant equals zero (C)</p> Signup and view all the answers

What is often the first step in solving linear equations?

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

What happens when writing a quadratic equation in the standard form?

<p>All terms are moved to one side (D)</p> Signup and view all the answers

In which method do you eliminate a variable by adjusting coefficients before adding or subtracting equations?

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

Which of the following conditions applies to quadratic equations compared to linear equations?

<p>Quadratics can have one, two, or no solutions (B)</p> Signup and view all the answers

What is the result of raising a product to a power, according to the laws of exponents?

<p>Both factors are raised to the power individually (C)</p> Signup and view all the answers

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

<p>The coordinates of the point at which the equations intersect (A)</p> Signup and view all the answers

Which expression correctly exemplifies the law of multiplying exponents with the same base?

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

Which step comes first when solving a word problem?

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

What is the primary goal when rewriting a literal equation in terms of one variable?

<p>To isolate the variable of interest (D)</p> Signup and view all the answers

What can be concluded when both sides of an equation can be expressed with the same base?

<p>The exponents must be equated (B)</p> Signup and view all the answers

What occurs when both sides of a linear inequality are multiplied by a negative number?

<p>The inequality sign must be reversed (A)</p> Signup and view all the answers

If $a^{-n}$ appears in an expression, what is its equivalent form?

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

In the context of number sequences, what does the variable 'd' represent in the general formula?

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

What does the expression $(a^{m/n})^2$ simplify to using the power of a power rule?

<p>$a^{2m/n}$ (D)</p> Signup and view all the answers

How should you proceed when simplifying an expression containing rational exponents?

<p>Convert roots into fractional exponents (B)</p> Signup and view all the answers

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

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

When simplifying the expression $a^0$, what is the resulting value if $a$ is not zero?

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

When solving a literal equation, what should you consider about the unknown variable?

<p>It is often in multiple terms, which should be factored out (C)</p> Signup and view all the answers

What method is appropriate when bases differ in exponential equations?

<p>Use logarithms to isolate the exponent (A)</p> Signup and view all the answers

Which of the following is an essential aspect to remember when dealing with sequences?

<p>The nth term can be expressed using a general formula (C)</p> Signup and view all the answers

In solving word problems, what does assigning a variable help achieve?

<p>It represents an unknown quantity for better understanding (D)</p> Signup and view all the answers

To simplify a complex fraction involving exponents, which of the following steps is essential?

<p>Factorize and cancel common terms (A)</p> Signup and view all the answers

What principle must be kept in mind for the equation $a^x = a^y$ to hold true?

<p>The base must be positive and not equal to one (A)</p> Signup and view all the answers

What does the coefficient 'm' in the equation of a straight line affect?

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

Which statement best describes the effect of shifting the graph of a function vertically using 'q'?

<p>It moves the entire graph up or down without changing its shape. (A)</p> Signup and view all the answers

What is the domain of the function defined by the equation $y = mx + c$?

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

In the equation of a parabola $y = ax^2 + q$, what does the sign of 'a' indicate?

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

How is the y-intercept of the graph expressed for the function $y = mx + c$?

<p>As the value of $c$ when $x$ is zero (B)</p> Signup and view all the answers

What is the result of increasing the value of 'm' in a linear function?

<p>The gradient of the graph increases, becoming steeper (A)</p> Signup and view all the answers

How can you calculate the x-intercept of a straight line graph?

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

Which of the following characteristics does not describe a linear function?

<p>It can have multiple x-intercepts. (A)</p> Signup and view all the answers

What determines if the graph of a quadratic function is concave up or concave down?

<p>The sign of 'a' (B)</p> Signup and view all the answers

What is the significance of the sign of $a$ in the function form $y = ab^x + q$?

<p>It indicates whether the graph will curve upwards or downwards. (D)</p> Signup and view all the answers

For an exponential function of the form $y = ab^x + q$, what is the range when $a < 0$?

<p>{f(x) : f(x) &lt; q} (A)</p> Signup and view all the answers

What happens to the graph of a parabola when the value of $a$ is greater than 1?

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

Which of the following describes the range of a parabola when $a < 0$?

<p>The range is $(- ext{infinity}; q]$ (C)</p> Signup and view all the answers

What are the x-intercepts of the sine function defined as $y = ext{sin} heta$?

<p>(0°, 0), (180°, 0), and (360°, 0) (B)</p> Signup and view all the answers

What is the effect of $q$ on the graph of a sine function of the form $y = a ext{sin} heta + q$?

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

What is the y-intercept of the function $y = rac{a}{x} + q$?

<p>It does not exist. (C)</p> Signup and view all the answers

How does the value of $q$ affect the graph of a hyperbola?

<p>It leads to a vertical shift. (A)</p> Signup and view all the answers

When sketching the graph of $y = rac{a}{x} + q$, which characteristic is NOT needed?

<p>Maximum turning point (A)</p> Signup and view all the answers

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

<p>It has a minimum at $(0, q)$. (C)</p> Signup and view all the answers

What is the maximum turning point of the sine function defined as $y = a ext{sin} heta + q$ when $a > 0$?

<p>(90°, 1) (C)</p> Signup and view all the answers

Which statement accurately describes the behavior of the function $y = ab^x + q$ when $b < 1$?

<p>It exhibits exponential decay. (D)</p> Signup and view all the answers

In the context of the parabola, what does the axis of symmetry refer to?

<p>A vertical line that divides the parabola into two equal halves. (C)</p> Signup and view all the answers

What effect does a negative value of $a$ have on the shape of the parabolic graph?

<p>It creates a frown shape. (D)</p> Signup and view all the answers

What defines the vertical asymptote of a function in the form $y = rac{a}{x} + q$?

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

What role does the variable $q$ play in trigonometric functions?

<p>It affects the vertical shift. (D)</p> Signup and view all the answers

Which of the following formulas is used to calculate accumulated amount with simple interest?

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

For the cosine function $y = a ext{cos} heta + q$, which range is correct when $a < 0$?

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

Which statement is true about the x-intercepts of a function of the form $y = rac{a}{x} + q$?

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

How is compound interest most beneficial in financial terms?

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

In the graph of the parabola $y = ax^2 + q$, if $a$ approaches zero from the negative side, how does the graph behave?

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

What is the domain of the hyperbolic function $y = rac{a}{x} + q$?

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

When calculating the x-intercept of a function, which equation is set to zero?

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

What does inflation measure in economic terms?

<p>The average annual increase in prices. (B)</p> Signup and view all the answers

In the context of hire purchase agreements, what is charged interest on?

<p>The remaining loan amount. (C)</p> Signup and view all the answers

Which variable represents the principal amount in the simple interest formula?

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

What is the effect of compound interest on investments over time?

<p>It causes exponential growth. (A)</p> Signup and view all the answers

Which of the following factors influences currency strength?

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

In the context of the domain of a function, what does it represent?

<p>All possible $x$ values. (D)</p> Signup and view all the answers

Which of the following statements accurately describes probabilities?

<p>A probability of 0 indicates that the event cannot occur. (B)</p> Signup and view all the answers

What is the correct formula to calculate relative frequency?

<p>Relative frequency = positive outcomes divided by total trials (A)</p> Signup and view all the answers

In Venn diagrams, what does the intersection of two sets represent?

<p>Elements common to both sets (C)</p> Signup and view all the answers

What is the relationship between theoretical probability and relative frequency as trials increase?

<p>Relative frequency gets closer to theoretical probability (C)</p> Signup and view all the answers

Which of the following describes the union of two sets?

<p>The set containing elements from one or both sets (A)</p> Signup and view all the answers

If a sample space contains all possible outcomes of an experiment, what is the probability of observing an outcome from this sample space?

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

Which statement correctly describes local purchasing?

<p>It strengthens the local currency by keeping money in the country. (A)</p> Signup and view all the answers

What format can probabilities NOT be expressed in?

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

In the context of exchanging currencies, what is the correct equation for converting an amount from one currency to another?

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

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

<p>It approaches theoretical probability (C)</p> Signup and view all the answers

What must be subtracted from the sum of the probabilities of two events to calculate their union correctly?

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

What characterizes two events as mutually exclusive?

<p>They have no outcomes in common. (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

What is the complement of an event A denoted as?

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

What can be said about the union of complementary events A and A'?

<p>It equals the sample space. (D)</p> Signup and view all the answers

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

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

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

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

If events A and B are mutually exclusive, which of the following must be true?

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

Which statement correctly represents the relationship between complementary events?

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

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

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

Which of the following describes the period of the tangent function $y = a \tan \theta + q$?

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

To determine the equation of a parabola, what does the variable $q$ signify?

<p>2.5 (C)</p> Signup and view all the answers

What is the effect of a positive value for $q$ in the function $y = a \tan \theta + q$?

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

When interpreting the graph of $y = a \sin \theta + q$, which step is NOT necessary?

<p>3.2 (D)</p> Signup and view all the answers

For the equation of the hyperbola $y = \frac{a}{x} + q$, what does $a$ affect?

<p>4 (A)</p> Signup and view all the answers

What identifies the x-intercepts of the tangent function $y = \tan \theta$?

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

How does a negative value for $a$ in the function $y = ax^2 + q$ affect the parabola?

<p>It reflects the parabola downwards. (C)</p> Signup and view all the answers

What do the asymptotes of the function $y = \tan \theta$ represent?

<p>Values where the function is undefined. (C)</p> Signup and view all the answers

What determines the direction of a parabola defined by $y = ax^2 + q$?

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

Which set of numbers includes only positive integers?

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

Which of the following classes of numbers includes both whole numbers and negative integers?

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

Which statement accurately characterizes rational numbers?

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

Which subset of numbers cannot be written as a fraction of two integers?

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

Which of the following includes both rational and irrational numbers?

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

Which symbol designates the set of rational numbers?

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

What type of numbers are represented by examples such as $ rac{1}{2}$ and $0.75$?

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

Which characteristic is true for imaginary numbers?

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

Which statement correctly describes the nature of irrational numbers?

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

What process should be followed to convert a recurring decimal into a rational number?

<p>Multiply the decimal by a power of 10 so the repeating part aligns, then subtract. (D)</p> Signup and view all the answers

What is the first step in rounding off a decimal number?

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

Which method is incorrect when estimating the value of a surd?

<p>Using the fraction of the surd directly in a calculation. (B)</p> Signup and view all the answers

When converting the decimal 0.875 into a rational number, what is the result?

<p>7/8 (A)</p> Signup and view all the answers

Which of the following correctly identifies a characteristic of surds?

<p>Surds are roots of numbers that cannot simplify to rational numbers. (D)</p> Signup and view all the answers

What does the coefficient in a mathematical expression represent?

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

What is the simplified form of the expression (3x + 2)(2x + 1)?

<p>6x^2 + 7x + 2 (B)</p> Signup and view all the answers

Which statement is true regarding rounding off to one decimal place?

<p>Round down if the next digit is 0 to 4. (D)</p> Signup and view all the answers

What is the process to multiply a binomial by a trinomial?

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

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

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

Which step is essential when solving a quadratic equation after rewriting it in the standard form?

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

When solving simultaneous equations by substitution, what is the first step?

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

What is the first action taken in the method to solve linear equations?

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

Which of the following statements is true concerning solving quadratic equations?

<p>A quadratic equation can have no solutions or up to two solutions. (B)</p> Signup and view all the answers

What is the result of applying the law of exponents for division when simplifying $ rac{a^5}{a^2}$?

<p>a^3 (D)</p> Signup and view all the answers

In solving simultaneous equations by elimination, what is a key requirement?

<p>One variable's coefficients must be the same. (B)</p> Signup and view all the answers

What does $a^{m/n}$ represent in terms of roots?

<p>The nth root of a raised to m (D)</p> Signup and view all the answers

When checking the solution to an equation, what should be done?

<p>Substitute the solution back into the original equation (A)</p> Signup and view all the answers

Which method is commonly used to solve a quadratic equation?

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

How can you simplify the expression $a^{-3}$?

<p>1/a^3 (A)</p> Signup and view all the answers

What is a defining characteristic of simultaneous equations?

<p>Require two or more independent equations (D)</p> Signup and view all the answers

What is the first step when solving the equation $2^x = 2^3$?

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

What is the result of raising a quotient to a power, $( rac{b}{c})^3$?

<p>$ rac{b^3}{c^3}$ (D)</p> Signup and view all the answers

What principle allows the conclusion that if $a^x = a^y$, then $x = y$?

<p>Base equality principle (D)</p> Signup and view all the answers

To simplify $a^{1/2} imes a^{2/3}$, what is the correct application of exponent laws?

<p>a^{7/6} (A)</p> Signup and view all the answers

What does $a^0$ equal when $a$ is not equal to zero?

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

Which method can be used when the bases in an exponential equation cannot be easily made the same?

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

What is the result of multiplying the binomial (x + 2) by the trinomial (3 + 4x + x^2)?

<p>x^3 + 8x^2 + 6 (C)</p> Signup and view all the answers

What does factorising a quadratic trinomial ax^2 + bx + c typically involve?

<p>Identifying two binomials that produce the original trinomial (D)</p> Signup and view all the answers

Which identity represents the difference of two squares?

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

What method is used for simplifying algebraic fractions?

<p>Factoring both numerator and denominator before cancelling common factors (D)</p> Signup and view all the answers

When applying the law of exponents, what is the simplified form of a^m / a^n?

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

Which of the following correctly describes the multiplication of fractions?

<p>(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}) (C)</p> Signup and view all the answers

What is the first step in factorising by grouping?

<p>Split terms into pairs (B)</p> Signup and view all the answers

Which expression accurately 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

When combining like terms in polynomial expressions, what is required?

<p>Terms must have the same base and exponent to combine (A)</p> Signup and view all the answers

What does the variable 'm' represent in the equation of a straight-line function?

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

If the value of 'c' in the equation of a linear function is negative, what effect does it have on the graph?

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

How can the x-intercept of the graph of a linear function be determined?

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

For a parabolic function in the form of $y = ax^2 + q$, what does a positive value of 'a' indicate?

<p>The graph is a 'smile' shape. (C)</p> Signup and view all the answers

What happens to the graph of $y = ax^2 + q$ when 'q' is greater than zero?

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

Which of the following is true about the domain and range of the function $f(x) = mx + c$?

<p>Both domain and range include all real numbers. (C)</p> Signup and view all the answers

In the context of linear equations, what is meant by the term 'gradient'?

<p>The slope of the line. (C)</p> Signup and view all the answers

Which characteristics are necessary to sketch a graph of the form $y = mx + c$?

<p>The sign of 'm', y-intercept, and x-intercept are needed. (B)</p> Signup and view all the answers

What type of graph is formed when 'a' is less than zero in the function $y = ax^2 + q$?

<p>A parabolic curve opening downwards. (D)</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 stretches vertically upwards and becomes narrower. (B)</p> Signup and view all the answers

What is the range of the function if 'a' is negative and 'q' is zero?

<p>Less than zero or equal to zero. (B)</p> Signup and view all the answers

Which statement accurately describes the characteristics of the graph of the function 'y = ax^2 + q'?

<p>The domain includes all real numbers. (B)</p> Signup and view all the answers

If 'q' is negative in the hyperbolic function 'y = \frac{a}{x} + q', what does this imply about the vertical shift?

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

What is the correct expression for the horizontal asymptote of the function 'y = \frac{a}{x} + q'?

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

For a hyperbola defined by 'y = \frac{a}{x} + q', what happens to the function when 'x' approaches 0?

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

If the value of 'a' is negative, which of the following describes the shape and direction of the parabolic graph?

<p>The graph is a frown and opens downwards. (C)</p> Signup and view all the answers

How would the graph of the function change if 'a' is set to a value between -1 and 0?

<p>The graph will become wider as 'a' approaches 0. (A)</p> Signup and view all the answers

Which characteristic is true for the x-intercepts of the function 'y = \frac{a}{x} + q'?

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

In a parabolic function where 'a' is positive, what type of turning point is found at (0; q)?

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

What describes the x-intercepts of the tangent function in the interval $0° \leq \theta \leq 360°$?

<p>(0°, 0), (180°, 0), (360°, 0) (A)</p> Signup and view all the answers

What effect does the parameter 'a' have on the graph of the function $y = a \tan \theta + q$?

<p>It changes the steepness of the graph branches. (C)</p> Signup and view all the answers

What is the range of the function $y = a \cos \theta + q$?

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

What type of shift occurs if the parameter 'q' is negative in the function $y = a \tan \theta + q$?

<p>Shift down by |q| units. (A)</p> Signup and view all the answers

Which of the following statements regarding the cosine and sine functions is correct?

<p>The cosine graph can be shifted to overlap the sine graph by 90°. (B)</p> Signup and view all the answers

How can the y-intercept of a tangent function be expressed in the form $y = a \tan \theta + q$?

<p>It equals $q$. (B)</p> Signup and view all the answers

When analyzing a parabola represented by $y = ax^2 + q$, how is the sign of 'a' determined?

<p>By examining the graph's direction (smile or frown). (A)</p> Signup and view all the answers

What is the period of the tangent function represented in the form $y = \tan \theta$?

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

In which quadrant does a hyperbola represented by $y = \frac{a}{x} + q$ have curves when 'a' is positive?

<p>Quadrants I and IV (D)</p> Signup and view all the answers

What represents the solution to a system of simultaneous equations?

<p>The coordinates of the point at which the two graphs intersect (C)</p> Signup and view all the answers

What is the first step in solving a word problem mathematically?

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

What does 'changing the subject of the formula' imply when solving literal equations?

<p>To isolate the unknown variable in the equation (D)</p> Signup and view all the answers

Which is a critical aspect when solving linear inequalities?

<p>The inequality sign must switch if divided by a negative number (A)</p> Signup and view all the answers

Which of the following terms refers to the individual items in a sequence?

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

How can the common difference 'd' in a linear sequence be calculated?

<p>By subtracting the preceding term from the current term (B)</p> Signup and view all the answers

What is the general formula for a linear sequence represented as?

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

Which of the following describes a literal equation?

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

What is necessary to check after solving an equation analytically?

<p>Verify the solution by substituting back into the original equation (B)</p> Signup and view all the answers

What characterizes a linear inequality compared to a linear equation?

<p>It can have more than one solution (A)</p> Signup and view all the answers

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

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

How is the y-intercept calculated for functions?

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

Which component does not represent how compound interest differs from simple interest?

<p>Interest is calculated on the initial investment only. (A)</p> Signup and view all the answers

What is the formula used to calculate compound interest?

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

In the hire purchase agreement, what does the principal amount refer to?

<p>Cash price minus the deposit (C)</p> Signup and view all the answers

What does inflation typically indicate?

<p>An increase in the price of goods and services (C)</p> Signup and view all the answers

When determining the domain of a function, what is primarily considered?

<p>All possible $x$ values (D)</p> Signup and view all the answers

In the context of population growth, what does $P$ represent in the formula $A = P(1 + i)^n$?

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

Which best describes the relationship between currency strength and investment?

<p>More investments often strengthen the currency. (B)</p> Signup and view all the answers

In calculating the accumulated amount for simple interest, which variable represents the time period?

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

What characterizes the y-intercept of the function in the form $y = \frac{a}{x} + q$?

<p>It does not exist for this function. (D)</p> Signup and view all the answers

For what condition does the graph of the function $y = ab^x + q$ represent exponential growth?

<p>When $a &gt; 0$ and $b &gt; 1$. (D)</p> Signup and view all the answers

What is the vertical shift associated with the parameter $q$ in the function $y = a \sin \theta + q$?

<p>It shifts the graph upward by $q$ units. (A)</p> Signup and view all the answers

Which effect does a negative sign in the parameter $a$ have on the function $y = a \cos \theta + q$?

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

What condition describes the range of the function $y = ab^x + q$ when $a < 0$?

<p>The range is $\{f(x) : f(x) &lt; q\}$. (D)</p> Signup and view all the answers

How can the horizontal asymptote of the exponential function $y = ab^x + q$ be defined?

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

In the sine function $y = a \sin \theta + q$, what does $|a| > 1$ imply?

<p>The graph will experience vertical stretching. (D)</p> Signup and view all the answers

How is relative frequency calculated?

<p>Dividing the number of occurrences of the event by total trials. (B)</p> Signup and view all the answers

Which of the following defines the term 'union' in set theory?

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

Which x-intercepts are characteristic of the function $y = \sin \theta$?

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

In a Venn diagram, what does the intersection of two sets represent?

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

What characteristic of the cosine function $y = \cos \theta$ is reflected in its y-intercept?

<p>The y-intercept is $(0°, 1)$. (D)</p> Signup and view all the answers

What can a probability of 1 be interpreted as?

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

What is the purpose of a Venn diagram in probability?

<p>To visually represent the relationships between sets. (C)</p> Signup and view all the answers

What conclusion can be drawn from the sample space of an experiment?

<p>It represents all possible outcomes of the experiment. (A)</p> Signup and view all the answers

What does a probability expressed as a fraction indicate?

<p>The likelihood of an event occurring among all possibilities. (A)</p> Signup and view all the answers

What is a characteristic of theoretical probability?

<p>It is computed when all outcomes have equal chances. (A)</p> Signup and view all the answers

How does increasing the number of trials affect relative frequency?

<p>It causes relative frequency to converge toward theoretical probability. (A)</p> Signup and view all the answers

What is the probability of the union of two events that are mutually exclusive?

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

What does the notation A' represent?

<p>The complement of set A (B)</p> Signup and view all the answers

What relationship holds for the probabilities of complementary events?

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

Which of the following is a correct property of mutually exclusive events?

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

How is the probability of the union of two events calculated?

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

What does the intersection of two events A and B imply?

<p>Events A and B occur at the same time (A)</p> Signup and view all the answers

Which of the following statements is NOT true regarding complementary events?

<p>P(A ∩ A') &gt; 0. (D)</p> Signup and view all the answers

Which identity correctly represents the probability of the union of two events A and B?

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

If events A and B are mutually exclusive, what can be said about P(A ∩ B)?

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

What is the relationship between Venn diagrams and probabilities?

<p>They visually represent probability identities (A)</p> Signup and view all the answers

Which statement about natural numbers is true?

<p>They start from 1 and include all positive integers. (C)</p> Signup and view all the answers

What distinguishes rational numbers from irrational numbers?

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

Which set does not include zero?

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

Which of the following is an example of an irrational number?

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

Which statement about imaginary numbers is correct?

<p>They consist of a negative square root. (D)</p> Signup and view all the answers

How are whole numbers defined in the number system?

<p>All natural numbers and zero. (D)</p> Signup and view all the answers

Which subset of numbers contains both negative and positive integers?

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

Which statement correctly describes real numbers?

<p>They encompass all rational and irrational numbers. (C)</p> Signup and view all the answers

What is the result when multiplying a binomial by a trinomial?

<p>The sum of the products of each term in the binomial with the trinomial. (C)</p> Signup and view all the answers

Which of the following is a common step in factorising a quadratic trinomial?

<p>Identifying two numbers that multiply to 'c' and add to 'b'. (D)</p> Signup and view all the answers

What is one method to factorise a quadratic trinomial expression?

<p>Grouping terms and extracting common factors. (D)</p> Signup and view all the answers

When cancelling common factors in an algebraic fraction, what is required?

<p>Both the numerator and the denominator must be factored. (D)</p> Signup and view all the answers

What does the identity $a^2 - b^2 = (a + b)(a - b)$ represent?

<p>Difference of two squares. (C)</p> Signup and view all the answers

Which of the following correctly represents the sum of two cubes?

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

What is the primary factorisation technique used to simplify a rational expression?

<p>Identifying and factoring common factors. (D)</p> Signup and view all the answers

In the simplification of fractions, what does rewriting division as multiplication by the reciprocal accomplish?

<p>It makes it easier to simplify complex fractions. (D)</p> Signup and view all the answers

In the context of exponent laws, what is the result of applying the law $a^m imes a^n$?

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

What is the effect of applying the law $(ab)^n = a^n b^n$?

<p>It allows for distributing the exponent across multiplied bases. (B)</p> Signup and view all the answers

What characterizes irrational numbers?

<p>They are non-repeating, non-terminating decimals. (A)</p> Signup and view all the answers

Which of the following methods is used for converting a recurring decimal into a rational number?

<p>Identifying the recurring pattern and isolating it. (B)</p> Signup and view all the answers

When rounding a number, if the digit after the rounding position is 9, what happens to the rounding digit?

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

What is the correct definition of a surd?

<p>The nth root of a non-perfect power that cannot be simplified. (B)</p> Signup and view all the answers

Which decimal form indicates a rational number?

<p>Both B and C (D)</p> Signup and view all the answers

How do you express a terminating decimal like 0.625 as a rational number?

<p>By using the formula for converting decimals to fractions. (D)</p> Signup and view all the answers

When estimating a surd, what is the first step?

<p>Identify adjacent perfect powers. (B)</p> Signup and view all the answers

What is a binomial?

<p>A mathematical expression with two terms. (C)</p> Signup and view all the answers

Which component of a mathematical expression is always a number that multiplies the variable?

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

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

<p>acx^2 + bd + (ad + bc)x (B)</p> Signup and view all the answers

What does the common difference in a linear sequence represent?

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

What is the role of the constant 'c' in the equation of a linear function?

<p>It is the y-intercept of the line (D)</p> Signup and view all the answers

Which statement correctly describes the effect of a negative gradient 'm' on a graph?

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

In the function $y = ax^2 + q$, what effect does a positive value of 'a' have on the parabola?

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

How is the y-intercept of the function $y = mx + c$ calculated?

<p>By letting 'x' equal zero (D)</p> Signup and view all the answers

Which of the following represents the general formula for a linear sequence?

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

What does the vertical shift in a parabola signify in terms of the constant 'q'?

<p>The turning point changes location vertically (D)</p> Signup and view all the answers

What information can be derived from the gradient 'm' of a linear function?

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

Which of the following describes a characteristic of the range for linear functions?

<p>It can take any real value (D)</p> Signup and view all the answers

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

<p>The coordinates where the two graphs intersect (D)</p> Signup and view all the answers

What is the first step in solving a word problem mathematically?

<p>Determine what we are asked to solve for (C)</p> Signup and view all the answers

What happens to the inequality sign when both sides of an inequality are divided by a negative number?

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

How is the common difference 'd' in a sequence calculated?

<p>By taking the difference between any term and the previous term (C)</p> Signup and view all the answers

What should you do first when solving a literal equation?

<p>Identify what is unknown and what is known (D)</p> Signup and view all the answers

What is the general formula for a linear sequence expressed as?

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

What is the correct approach when the unknown variable is in the denominator of an equation?

<p>Multiply both sides by the lowest common denominator (C)</p> Signup and view all the answers

What is a characteristic of a linear inequality?

<p>The largest exponent of a variable is 1 (B)</p> Signup and view all the answers

What is involved in describing the terms of a sequence?

<p>Denoting each term with a consistent label (B)</p> Signup and view all the answers

How do we check the solution of an equation after solving it?

<p>Substituting the solution back into the original equation (A)</p> Signup and view all the answers

How does the value of 'a' affect the shape of the graph of the function $y = ax^2 + q$?

<p>A smaller $a$ (between 0 and 1) makes the graph wider. (B)</p> Signup and view all the answers

What is the range of the function $y = ax^2 + q$ when $a < 0$?

<p>(-\infty; q] (A)</p> Signup and view all the answers

What indicates a minimum turning point in the graph of the function $y = ax^2 + q$?

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

What is true about the y-intercept of the function $y = ax^2 + q$?

<p>It is $q$ when $x = 0$. (B)</p> Signup and view all the answers

Identify the characteristic of the domain of the hyperbolic function $y = \frac{a}{x} + q$.

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

What determines the horizontal asymptote of the hyperbolic function $y = \frac{a}{x} + q$?

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

Which of the following statements about axes of symmetry is true for parabolic graphs of the form $y = ax^2 + q$?

<p>The axis of symmetry is the line $x = 0$. (D)</p> Signup and view all the answers

How does the graph of a function change when $q$ is positive in the hyperbolic function $y = \frac{a}{x} + q$?

<p>The graph shifts upwards by $q$ units. (B)</p> Signup and view all the answers

What happens to the turning point of the graph when $q$ is less than zero in the function $y = ax^2 + q$?

<p>The turning point shifts downwards to $(0; q)$. (D)</p> Signup and view all the answers

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

<p>$f(x) &gt; q$ (D)</p> Signup and view all the answers

How do you find the x-intercept of a function of the form $y = ab^x + q$?

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

What type of asymptote does a function of the form $y = ab^x + q$ possess?

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

What is the effect of the constant $q$ on the graph of $y = a an heta + q$?

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

How is the amplitude of the sine function $y = a an heta + q$ affected by the constant $a$?

<p>It can cause vertical compression if $|a| &lt; 1$ (D)</p> Signup and view all the answers

What determines whether the graph of an exponential function curves upwards or downwards?

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

What is the period of the sine function $y = a an heta + q$?

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

In a trigonometric function of the form $y = a an heta + q$, how is the y-intercept determined?

<p>By evaluating at $θ = 0$ (C)</p> Signup and view all the answers

What determines exponential decay in a function of the form $y = ab^x + q$?

<p>When $b &lt; 1$ and $a &gt; 0$ (B)</p> Signup and view all the answers

What determines the vertical shift in trigonometric functions?

<p>The parameter q (D)</p> Signup and view all the answers

What is the effect of the parameter a in trigonometric functions?

<p>It affects the amplitude and reflection (C)</p> Signup and view all the answers

How is the x-intercept of a function calculated?

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

What is the main advantage of compound interest compared to simple interest?

<p>It allows interest to earn more interest (C)</p> Signup and view all the answers

In the formula for simple interest, what does A represent?

<p>The accumulated amount (A)</p> Signup and view all the answers

How is the future price calculated using inflation?

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

What do the terms principal, interest rate, and time period represent in the context of simple interest?

<p>Key variables in calculating interest earned (B)</p> Signup and view all the answers

How does a hire purchase agreement calculate interest on the remaining amount?

<p>Using a simple interest rate on the balance after the deposit (D)</p> Signup and view all the answers

What happens to a currency when it strengthens?

<p>More money is invested in the country (B)</p> Signup and view all the answers

Which of the following correctly identifies a function that uses asymptotes?

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

What is the simplified form of $(x^2 y^3)^2$?

<p>x^4 y^6 (D)</p> Signup and view all the answers

When simplifying the exponent expression $a^{3/4} imes a^{1/2}$, what is the result?

<p>a^{5/4} (C)</p> Signup and view all the answers

What is the value of $5^0$?

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

If $a^2 = b^2$, which of the following is always correct?

<p>a = b ext{ or } a = -b (D)</p> Signup and view all the answers

Which of the following correctly describes how to handle a negative exponent?

<p>Reciprocate the base and change the exponent to positive (A)</p> Signup and view all the answers

Which expression correctly illustrates the power of a power law?

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

How would you simplify the expression ${ rac{x^{5}}{x^{3}}}$?

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

When raising a fraction to an exponent, which expression represents the correct application of the law?

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

What is the correct interpretation of the expression $a^{-3}$?

<p> rac{1}{a^3} (A)</p> Signup and view all the answers

When solving the equation $3^x = 9$, what is the first step to take?

<p>Convert 9 to an exponent of 3 (B)</p> Signup and view all the answers

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

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

Which of the following is NOT a step in solving linear equations?

<p>Divide by Common Factors (B)</p> Signup and view all the answers

In which situation can a quadratic equation have only one solution?

<p>When the two roots are equal (B)</p> Signup and view all the answers

What do you need to do before applying the factorisation method to a quadratic equation?

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

When solving simultaneous equations by elimination, you should aim to:

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

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

<p>To confirm that both sides of the equation are equal (C)</p> Signup and view all the answers

Which method is commonly used to solve quadratic equations?

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

How can you reduce the number of variables in simultaneous equations when using substitution?

<p>Isolate one variable in one of the equations (B)</p> Signup and view all the answers

What is a common characteristic of quadratic equations compared to linear equations?

<p>Quadratics have a higher degree than one (D)</p> Signup and view all the answers

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

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

What is the correct formula to calculate the probability of the union of two non-mutually exclusive events?

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

Which statement accurately describes mutually exclusive events?

<p>They cannot have any outcomes in common. (B)</p> Signup and view all the answers

What does the complement of an event A, denoted as A', include?

<p>Elements in the sample space that are not in A. (D)</p> Signup and view all the answers

For mutually exclusive events A and B, what is the probability of their intersection, P(A ∩ B)?

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

How does the probability of complementary events relate to the sample space?

<p>Their union covers the entire sample space. (C)</p> Signup and view all the answers

If events A and B are not mutually exclusive, which formula correctly describes their union?

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

What identity is true for complementary events A and A'?

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

When two events A and B overlap, how should their probability be calculated?

<p>By subtracting the intersection from the sum of their probabilities. (A)</p> Signup and view all the answers

Which statement correctly defines the union of two events A and B?

<p>The area representing outcomes in both A and B. (B)</p> Signup and view all the answers

In a Venn diagram, how is the probability of the intersection of events A and B visually represented?

<p>By the overlapping area of both circles. (A)</p> Signup and view all the answers

What is the period of the cosine function?

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

Given the equation of a tangent function, what is the domain of y = tan(θ)?

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

What happens to the tangent function y = a tan(θ) + q when a > 1?

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

What does the parameter 'q' represent in the equation of a parabola y = ax^2 + q?

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

Which statement is true regarding the x-intercepts of the tangent function?

<p>They occur at 0°, 180°, and 360°. (A)</p> Signup and view all the answers

How does the steepness of branches in a hyperbola behave as the value of 'a' changes?

<p>The branches become steeper as 'a' increases. (C)</p> Signup and view all the answers

What effect does increasing 'q' have on the graph of the tangent function y = a tan(θ) + q?

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

What characterizes the sine and cosine functions in terms of their graphs?

<p>They can be made to overlap with simple horizontal shifts. (A)</p> Signup and view all the answers

To analyze the equation of a hyperbola, what is the first step?

<p>Determine the sign of 'a'. (D)</p> Signup and view all the answers

How do you determine the equation of a trigonometric function from a graph?

<p>Substitute given points into the function to solve for 'a' and 'q'. (A)</p> Signup and view all the answers

What does a probability of 0.5 signify?

<p>The event occurs half the time. (C)</p> Signup and view all the answers

In probability, the sample space refers to what?

<p>All possible outcomes of an experiment. (B)</p> Signup and view all the answers

What is the relationship between theoretical probability and relative frequency?

<p>Theoretical probability can be calculated, while relative frequency is experimental. (D)</p> Signup and view all the answers

Which of the following symbols represents the union of two sets?

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

How is the theoretical probability of an event calculated?

<p>By finding the ratio of favorable outcomes to total outcomes. (A)</p> Signup and view all the answers

If an event has a relative frequency of 0.8 in 50 trials, how many times did the event occur?

<p>40 (B)</p> Signup and view all the answers

What does a probability of 1 indicate?

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

Which of the following would represent an event that never occurs in probability?

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

What is meant by the intersection of two sets?

<p>Elements that are in both sets. (D)</p> Signup and view all the answers

In Venn diagrams, what does the area outside the curves represent?

<p>Elements excluded from both sets. (C)</p> Signup and view all the answers

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