Gr 10 Math June P1 Medium

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

$ ext{e}$ (correct)

$0.5$

$-3$

$rac{4}{5}$

Which category do the numbers $0$ and $1$ belong to?

Whole Numbers (correct)

Irrational Numbers

Natural Numbers

Imaginary Numbers

What is the main feature of rational numbers?

They have infinite decimal expansions.

They include positive and negative values only.

They can be expressed as the fraction of two integers. (correct)

They are always integers.

Which of the following sets does not include the number $-5$?

Natural Numbers Signup and view all the answers

Which of the following represents the set of integers?

$ ext{Z}$ Signup and view all the answers

What differentiates real numbers from imaginary numbers?

Real numbers have a defined place on the number line, while imaginary numbers do not. Signup and view all the answers

What is the highest exponent of the variable in a linear equation?

1 Signup and view all the answers

How many solutions can a linear equation have at most?

One solution Signup and view all the answers

What is the first step in solving a linear equation?

Expand all brackets Signup and view all the answers

Which of the following methods is NOT used to solve quadratic equations?

Substitution Signup and view all the answers

When solving quadratic equations, what form must the equation be in?

ax^2 + bx + c = 0 Signup and view all the answers

What is the method of elimination primarily used for in simultaneous equations?

To eliminate one variable Signup and view all the answers

If a quadratic equation has no solutions, what can be said about its graph?

It does not intersect the x-axis. Signup and view all the answers

What should be done after solving for one variable in simultaneous equations by substitution?

Substitute back into the first equation Signup and view all the answers

What operation must be performed on both sides of an equation to maintain balance?

Any operation Signup and view all the answers

Which step is NOT part of the general method for solving linear equations?

Rewrite the equation Signup and view all the answers

Which statement accurately describes rational numbers?

They can be expressed as fractions with a non-zero denominator. Signup and view all the answers

What characterizes an irrational number?

It has a non-repeating, non-terminating decimal expansion. Signup and view all the answers

Which of the following is a correct way to estimate the value of the surd $ ext{√5}$?

By determining that $ ext{√5}$ is greater than 2 but less than 3. Signup and view all the answers

How can a terminating decimal be converted into a rational number?

By writing it as a fraction where the denominator is a power of 10. Signup and view all the answers

Which of the following best illustrates the difference between rational and irrational decimal forms?

0.333… is rational, while 0.101001001… is irrational. Signup and view all the answers

Which of the following statements about rounding off decimal numbers is correct?

If the next digit is less than 5, you leave the rounding digit unchanged. Signup and view all the answers

In the expression 5x^2 + 3x - 2, which term is the coefficient?

5 Signup and view all the answers

What defines a surd?

It is the n-th root of a number that cannot be simplified to a rational number. Signup and view all the answers

Which of the following represents a method to convert recurring decimals to rational numbers?

Identify and align the recurring part by multiplying with powers of 10. Signup and view all the answers

What does the gradient "m" represent in the equation of a straight-line graph?

The steepness of the line Signup and view all the answers

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

The value of y when x equals zero Signup and view all the answers

What effect does a negative value of "a" have on a parabolic graph?

It opens downwards with a maximum point Signup and view all the answers

What is the domain of the function defined by the equation $y = ax^2 + q$?

All real numbers Signup and view all the answers

Which statement is true about the effect of "q" on the parabolic graph?

It causes a vertical shift of the graph Signup and view all the answers

Which statement correctly describes the effect of increasing the absolute value of "a" on the parabolic graph?

The graph becomes narrower Signup and view all the answers

In the expression $m = \frac{\text{change in } y}{\text{change in } x}$, what does "change in y" represent?

The vertical rise between two points on the line Signup and view all the answers

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

[q, \infty) Signup and view all the answers

Which characteristic must be calculated to draw a straight line using the gradient and y-intercept method?

The y-intercept and one more point Signup and view all the answers

If a quadratic equation has a value of $q < 0$, where is the turning point located?

Below the x-axis Signup and view all the answers

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

Read the whole question. Signup and view all the answers

What can be concluded when two graphs of linear equations intersect?

The coordinates of the intersection point represent the solution to the system. Signup and view all the answers

What effect does increasing the value of $m$ have on the graph of the function $y = mx + c$?

The slope of the graph increases. Signup and view all the answers

When solving a linear inequality, what happens to the inequality sign if both sides are divided by a negative number?

The inequality sign is reversed. Signup and view all the answers

In the equation of a circle $A = \pi r^2$, what variable is represented by $A$?

The area of the circle. Signup and view all the answers

What does the value of $c$ affect in the linear equation $y = mx + c$?

The y-intercept of the graph. Signup and view all the answers

What is meant by a literal equation?

An equation with several letters or variables. Signup and view all the answers

Which of the following conditions would indicate that a system of equations has no solutions?

The lines are parallel. Signup and view all the answers

Which action is inappropriate when solving literal equations?

Distributing terms improperly. Signup and view all the answers

In the inequality $2x + 2 \leq 1$, how would you solve for $x$?

Subtract 2 from both sides and divide by 2. Signup and view all the answers

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

It influences the direction and width of the parabola. Signup and view all the answers

How can the vertical shift of a trigonometric graph be determined?

By evaluating the value of 'q'. Signup and view all the answers

Where are the asymptotes for the tangent function located?

At $θ = 90°$ and $θ = 270°$. Signup and view all the answers

What does the domain of a function represent?

The set of all possible x values. Signup and view all the answers

In the equation of a hyperbola, what does the term 'q' signify?

The vertical shift of the graph. Signup and view all the answers

When analyzing the graph of a function, how can x-intercepts be found?

By setting y equal to zero. Signup and view all the answers

Which equation represents the general form of a sine function?

y = a ext{sin} heta + q Signup and view all the answers

What is necessary to solve for 'a' in a hyperbola's equation?

Substituting given points into the equation. Signup and view all the answers

What indicates a vertical shift in the graph of a tangent function?

The value of 'q'. Signup and view all the answers

To determine the range of a parabolic function, which method is applicable?

Analyzing the direction of opening of the parabola. Signup and view all the answers

What is the range of the function if the coefficient $a$ is less than zero?

(-∞; q] Signup and view all the answers

What do the x-intercepts of the function $y = ax^2 + q$ represent?

Values of $x$ for which $y$ equals zero. Signup and view all the answers

Which characteristic is true if the graph of $f(x) = ax^2 + q$ is a 'smile'?

The axis of symmetry is $x = 0$. Signup and view all the answers

In the function $y = rac{a}{x} + q$, what is true about the vertical asymptote?

It occurs at the y-axis, or $x = 0$. Signup and view all the answers

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

It results in vertical shifts of the graph. Signup and view all the answers

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

It does not exist since the function is undefined at $x = 0$. Signup and view all the answers

Which of the following correctly describes the domain of the hyperbolic function $y = rac{a}{x} + q$?

{x: $x eq 0$, $x ext{ is a real number}$} Signup and view all the answers

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

A line defined by $y = q$. Signup and view all the answers

How does the sign of $a$ affect the graph of the exponential function $y = ab^x + q$?

It influences whether the graph increases or decreases. Signup and view all the answers

Which statement is true regarding the range of the exponential function $y = ab^x + q$ when $a > 0$?

The range is $(q; ext{∞})$. Signup and view all the answers

What happens to the horizontal asymptote of the function when $q$ is decreased?

It decreases vertically. Signup and view all the answers

Which statement about the effects of the coefficient $a$ in the function $y = a an heta + q$ is true?

It causes a reflection over the y-axis if $a < 0$. Signup and view all the answers

What is the effect of a coefficient $b$ where $0 < b < 1$ in an exponential function?

The function represents exponential decay. Signup and view all the answers

Which of the following correctly describes the domain of the sine function?

$[0°, 360°]$ Signup and view all the answers

Where do the x-intercepts of the cosine function occur?

At $(180°, 0)$ and $(270°, 0)$ Signup and view all the answers

What distinguishes the range of the function $y = a an heta + q$ from the range of the sine function?

It can take on all real values. Signup and view all the answers

For the sine function, which point signifies the maximum turning point?

$(90°, 1)$ Signup and view all the answers

What is the vertical shift effect when $q$ is greater than zero in a cosine function?

The graph shifts upward by $q$ units. Signup and view all the answers

What does the value of $|a|$ represent in the context of the sine function?

The amplitude of the function. Signup and view all the answers

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

A(C + D + E) + B(C + D + E) Signup and view all the answers

Which expression correctly represents the difference of two squares?

(a + b)(a - b) Signup and view all the answers

What is the first step in factorizing a trinomial in the form of extbf{ax^2 + bx + c}?

Identify the common factors. Signup and view all the answers

When simplifying the algebraic fraction extbf{(ax + b)/(cx + d)}, what must you do first?

Factorize the numerator and denominator. Signup and view all the answers

What is the outcome of multiplying the monomial extbf{a} by the binomial extbf{(x + y)}?

ax + ay Signup and view all the answers

Which operation represents the simplification of the algebraic fraction extbf{(3x^2)/(6x)}?

x/2 Signup and view all the answers

In multiplying two binomials extbf{(ax + b)(cx + d)}, what term would be calculated to represent the coefficient of $x^2$?

ac Signup and view all the answers

Which of the following correctly identifies a trinomial?

x^2 + 2x + 1 Signup and view all the answers

What is the result of the operation extbf{(x + 2)(x + 3)} upon expansion?

x^2 + 5x + 6 Signup and view all the answers

When performing multiplication of fractions, what expression shows the result of $rac{2}{3} imes rac{4}{5}$?

$rac{8}{15}$ Signup and view all the answers

What is the result of applying the zero exponent law to the base 7?

1 Signup and view all the answers

If you have the expression rac{a^5}{a^2}, what is the simplified form?

a^3 Signup and view all the answers

Which of the following represents the application of raising a product to a power?

(xy)^n = x^n y^n Signup and view all the answers

How would you simplify the expression a^{3/2} imes a^{1/2}?

a^4 Signup and view all the answers

What is the first step in solving an exponential equation like 2^x = 16?

Convert 16 to a power of 2 Signup and view all the answers

When applying the negative exponent rule to a^n, what is the result?

1/a^n Signup and view all the answers

If 3^x = 81, what is the value of x?

4 Signup and view all the answers

Which method can be used when the bases of an exponential equation are difficult to match?

Use logarithms Signup and view all the answers

How can rational exponents be expressed in terms of roots?

a^{m/n} = rac{ oot{n}{a}^m} Signup and view all the answers

What is the simplified form of rac{(2^3)(2^2)}{2^5}?

2^0 Signup and view all the answers

Which statement about integer representation is accurate?

All integers are rational numbers but not all rational numbers are integers. Signup and view all the answers

What is the outcome when rounding the number 2.675 to two decimal places?

2.68 Signup and view all the answers

Which decimal representation indicates a rational number?

0.333... Signup and view all the answers

Which method can be used to convert the recurring decimal 0.666... into a rational number?

Write as 2/3 by defining it as x. Signup and view all the answers

Which of the following correctly describes surds?

Surds are always irrational numbers. Signup and view all the answers

In estimating a surd like $ ext{√12}$, which perfect squares are nearest?

3 and 4 Signup and view all the answers

Which of the following is a characteristic of rational numbers?

They can always be expressed in the form ${rac{a}{b}}$. Signup and view all the answers

If the number $x = 4.7825$ is rounded to three decimal places, what will be the result?

4.79 Signup and view all the answers

Which property distinguishes irrational numbers from rational numbers?

Irrational numbers have non-repeating, non-terminating decimals. Signup and view all the answers

Which of the following correctly explains how to find the coefficient in an expression?

The coefficient is the numerical factor that multiplies the variable. Signup and view all the answers

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

Two solutions Signup and view all the answers

Which step is NOT included in the method for solving linear equations?

Group unlike terms together Signup and view all the answers

What must be true for an equation to be valid after performing operations?

Both sides must remain equal Signup and view all the answers

When solving quadratic equations, what must the equation be in the form of?

$ax^2 + bx + c = 0$ Signup and view all the answers

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

Identify the simpler equation Signup and view all the answers

What happens after isolating a variable when using the substitution method?

The other variable is plugged back into the original equation Signup and view all the answers

In which of the following methods do you eliminate a variable by making coefficients the same?

Elimination Signup and view all the answers

What indicates a quadratic equation can have no solutions?

The discriminant is negative Signup and view all the answers

Which algebraic method is specifically included in solving quadratic equations?

Factorisation Signup and view all the answers

What is true about the solutions of a simultaneous equation system with two equations and two variables?

They represent the intersection point on a graph Signup and view all the answers

What is a monomial?

An expression with one term Signup and view all the answers

Which of the following correctly represents the product of a binomial and a trinomial?

(A + B)(C + D + E) = A(C + D + E) + B(C + D + E) Signup and view all the answers

What is the purpose of factorization?

To break down an expression into simpler expressions Signup and view all the answers

Which identity is used for factoring a difference of two squares?

a^2 - b^2 = (a + b)(a - b) Signup and view all the answers

What does the term 'coefficient' refer to in an expression?

The numerical factor in a term Signup and view all the answers

Which method can be used to simplify a complex fraction?

Factorise all terms in the numerator and denominator Signup and view all the answers

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

Identify any common factors Signup and view all the answers

What is the product of two binomials described by the formula (ax + b)(cx + d)?

acx^2 + bcx + adx + bd Signup and view all the answers

What is the result when simplifying the fraction ( \frac{a}{b} \div \frac{c}{d} )?

( \frac{a}{b} \times \frac{d}{c} ) Signup and view all the answers

What does the sum of cubes identity express?

x^3 + y^3 = (x + y)(x^2 - xy + y^2) Signup and view all the answers

What happens to the value of an exponent when a number is raised to the power of zero?

It equals one Signup and view all the answers

Which of the following correctly simplifies the expression $\frac{a^{3}}{a^{5}}$?

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

When combining the expressions $a^{2} \times a^{3}$, what is the resulting exponent?

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

How do you express the square root of a number using exponents?

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

Which property of exponents allows you to write $(ab)^{n}$ as $a^{n}b^{n}$?

Power of a Product Property Signup and view all the answers

What does a negative exponent indicate?

It indicates reciprocal in the denominator Signup and view all the answers

What is the result of simplifying the expression $\left( \frac{a}{b} \right)^{3}$?

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

Which of the following is required to solve exponential equations when the bases are not the same?

Use logarithms Signup and view all the answers

What is the first step in simplifying an expression with rational exponents?

Convert roots to fractional exponents Signup and view all the answers

Which strategy can be used to solve exponential equations of the form $a^x = a^y$?

Equate the exponents Signup and view all the answers

What is the first step in solving a set of word problems mathematically?

Read the whole question. Signup and view all the answers

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

The inequality sign is flipped. Signup and view all the answers

What does the value of 'c' in the linear equation $y = mx + c$ determine?

The y-intercept of the graph. Signup and view all the answers

What is a literal equation primarily used for?

To represent a relationship among various variables. Signup and view all the answers

What graphical representation is used to solve simultaneous equations?

The intersection of the graphs. Signup and view all the answers

In the equation $v = \frac{D}{t}$, what does 'v' represent?

Speed. Signup and view all the answers

How should an unknown variable in a literal equation be isolated?

By performing the opposite operation of what it is joined by. Signup and view all the answers

If a linear function has a positive gradient, what can be inferred about its graph's direction?

It slopes upwards from left to right. Signup and view all the answers

When rearranging the formula for the area of a circle, what is a necessary consideration?

Understanding how the variables are related. Signup and view all the answers

In the inequality $\frac{3x + 1}{2} \geq 4$, what must be done first to isolate 'x'?

Multiply both sides by 2. Signup and view all the answers

How does the sign of the coefficient 'a' in the equation of a parabola affect its graph?

It indicates whether the graph opens upwards or downwards. Signup and view all the answers

What is the value of the y-intercept in the equation of a straight line represented by the function $y = mx + c$?

The value of 'c' when $x = 0$. Signup and view all the answers

If the value of 'c' in the equation $y = mx + c$ is less than zero, what can be stated about the y-intercept?

It lies below the x-axis. Signup and view all the answers

What impact does increasing the value of 'q' have on the parabola defined by $y = ax^2 + q$?

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

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

[−∞; q) Signup and view all the answers

If a linear graph has a positive gradient 'm' while having a y-intercept 'c' that is equal to zero, what does this imply about the graph?

The line passes through the origin and rises as it moves from left to right. Signup and view all the answers

What happens to the shape of a parabola when the absolute value of 'a' becomes larger than 1?

It becomes narrower. Signup and view all the answers

In the gradient formula $m = \frac{\text{change in } y}{\text{change in } x}$, what does 'change in x' refer to?

The horizontal distance between two points on the graph. Signup and view all the answers

If a parabola described by $y = ax^2 + q$ has a minimum turning point, what can be inferred about the values of 'a' and 'q'?

a must be greater than 0 and q can be any real number. Signup and view all the answers

Which set includes all whole numbers?

Whole Numbers Signup and view all the answers

What defines a rational number?

Any number that can be expressed as a fraction with a non-zero denominator Signup and view all the answers

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

$rac{5}{ ext{√2}}$ Signup and view all the answers

Which statement correctly identifies the set of integers?

Includes all whole numbers and their negative counterparts Signup and view all the answers

What distinguishes real numbers from imaginary numbers?

Real numbers cannot have negative square roots, whereas imaginary numbers do Signup and view all the answers

Which set does not include the number zero?

Natural Numbers Signup and view all the answers

What is the effect of the coefficient 'a' in the equation of a trigonometric function?

It affects the amplitude and reflection of the graph. Signup and view all the answers

How can the parameter 'q' in the equation of a parabola be determined?

Using the y-intercept point to solve for it. Signup and view all the answers

Where are the asymptotes of the tangent function located?

At $ heta = 90°$ and $ heta = 270°$. Signup and view all the answers

What is the behavior of the graph when the coefficient 'a' is negative for an exponential function with base 'b' greater than 1?

The graph will curve downwards. Signup and view all the answers

For the function defined as $y = a an \theta + q$, what effect does a positive 'q' have on the graph?

It shifts the graph vertically upwards by 'q' units. Signup and view all the answers

What determines the width and direction of a parabola?

The absolute value of 'a' only. Signup and view all the answers

To determine the equation of a hyperbola, what initial step should be taken?

Examine the quadrants where the curves lie. Signup and view all the answers

What forms the horizontal asymptote for exponential functions such as $y = ab^x + q$?

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

What is the domain of the function defined by the equation $y = a an heta + q$?

$ heta$ cannot equal 90° or 270°. Signup and view all the answers

How does the value of 'b' affect the behavior of the function $y = ab^x$?

It defines whether the function represents growth or decay. Signup and view all the answers

What is the range of the sine function $y = a ext{ sin } \theta + q$ when $a > 0$?

[q - a, q + a] Signup and view all the answers

What method is used to determine the value of 'a' in the equation of a parabola?

Substituting another given point into the equation. Signup and view all the answers

In the functions of the form $y = a ext{ cos } \theta + q$, what is the effect of applying $a < 0$?

It causes a reflection about the x-axis. Signup and view all the answers

What do the y-intercepts of graph functions allow you to find?

The vertical shift and value of 'q'. Signup and view all the answers

In the context of interpreting graphs, what is calculated to find distances between points?

The distance formula or simple subtraction of coordinates. Signup and view all the answers

What distinguishes exponential decay from growth in the function $y = ab^x$?

The base 'b' must be less than 1. Signup and view all the answers

For the sine function, what is the maximum obtainable value of $y = a ext{ sin } \theta + q$ when $a > 0$?

q + a Signup and view all the answers

What is the significance of identifying the vertical shifts in trigonometric graphs?

It affects the vertical position of the graph described by 'q'. Signup and view all the answers

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

180° Signup and view all the answers

What is the vertical shift of the cosine function $y = a ext{ cos } \theta + q$ when $q < 0$?

It shifts down by 'q' units. Signup and view all the answers

What can be concluded about the range of the function when $a < 0$ in the equation $y = ax^2 + q$?

The range is $(- ext{infinity}; q]$ Signup and view all the answers

Which statement correctly describes the y-intercept of the hyperbolic function $y = \frac{a}{x} + q$?

There is no y-intercept Signup and view all the answers

What is the effect of a positive value for $a$ in the equation $y = rac{a}{x} + q$?

The graph lies in the first and third quadrants Signup and view all the answers

What type of turning point occurs when $a > 0$ in the quadratic function $f(x) = ax^2 + q$?

Minimum turning point Signup and view all the answers

Where is the axis of symmetry for any quadratic function of the form $f(x) = ax^2 + q$?

The y-axis, or the line $x = 0$ Signup and view all the answers

Which statement is true regarding the range of an exponential function $y = ab^x + q$ when $a < 0$?

The range is $( ext{infinity}; q)$ Signup and view all the answers

What is the primary characteristic of the vertical asymptote in the hyperbolic function $y = \frac{a}{x} + q$?

Defined at $x = 0$ Signup and view all the answers

What happens to the graph of $f(x) = ax^2 + q$ if $q < 0$?

The turning point will be below the x-axis Signup and view all the answers

In the graph of a function of the form $y = ab^x + q$, what does the value of $q$ control?

Vertical shift Signup and view all the answers

What is true about the x-intercept for functions of the form $y = \frac{a}{x} + q$?

It is determined by setting $y = 0$ Signup and view all the answers

Which of the following best defines natural numbers?

The set of positive integers starting from 1. Signup and view all the answers

What is included in the set of whole numbers?

All natural numbers and zero. Signup and view all the answers

Which statement is true regarding irrational numbers?

Irrational numbers cannot be expressed as a fraction of two integers. Signup and view all the answers

What do real numbers encompass?

All rational numbers and irrational numbers. Signup and view all the answers

Which of the following best describes rational numbers?

Numbers that can be expressed as a fraction of two integers. Signup and view all the answers

Which of the following statements about imaginary numbers is accurate?

Imaginary numbers are those having a negative square root. Signup and view all the answers

Which of the following correctly describes a rational number?

A number that can be written as a repeating decimal. Signup and view all the answers

What is an example of an irrational number?

$ ext{√7}$ Signup and view all the answers

What happens to the digit being rounded if it is 9 and is in the decimal place to be rounded?

It becomes 0, and the preceding digit increases by 1. Signup and view all the answers

If a decimal is non-terminating and non-repeating, what type of number is it classified as?

Irrational Number Signup and view all the answers

How can a recurring decimal be represented as a rational number?

By aligning the decimal and subtracting from a multiple of ten. Signup and view all the answers

Which statement accurately describes surds?

Surds are square roots of numbers that cannot be simplified to rational. Signup and view all the answers

What is the first step in estimating a surd like $ ext{√5}$?

Finding nearby perfect squares. Signup and view all the answers

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

Leave the last digit unchanged. Signup and view all the answers

What is a characteristic feature of rational numbers?

They can be expressed in the form $rac{a}{b}$ where $b eq 0$. Signup and view all the answers

What is the process of breaking down an expression into simpler expressions known as?

Factorization Signup and view all the answers

When multiplying the binomial $(A + B)$ by the trinomial $(C + D + E)$, which of the following correctly represents the product?

$A(C + D + E) + B(C + D + E)$ Signup and view all the answers

Which identity is used for factorizing the difference of two squares?

$a^2 - b^2 = (a + b)(a - b)$ Signup and view all the answers

What is the first step in the general procedure for factorizing a trinomial of the form $ax^2 + bx + c$?

Identify any common factors. Signup and view all the answers

Which formula represents the general product of two linear binomials?

$(ax + b)(cx + d) = acx^2 + adx + bcx + bd$ Signup and view all the answers

What must be true for the denominator of a fraction when performing operations like multiplication or division?

It must not be zero. Signup and view all the answers

In the expression $a(x + y)$, what does the entire expression represent following multiplication?

$ax + ay$ Signup and view all the answers

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

$x^3 + y^3 = (x + y)(x^2 - xy + y^2)$ Signup and view all the answers

What is the outcome when cancelling common factors in a fraction during simplification?

The fraction is simplified to its lowest terms. Signup and view all the answers

When performing the operation $rac{a}{b} imes rac{c}{d}$, what is the resulting expression?

$rac{ac}{bd}$ Signup and view all the answers

What is the primary method to determine the solution of a system of simultaneous equations graphically?

Finding the intersection point of the graphs Signup and view all the answers

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

Ignore the final solution Signup and view all the answers

What does isolating the unknown variable often involve when solving literal equations?

Multiplying through by the lowest common denominator Signup and view all the answers

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

The sign is reversed Signup and view all the answers

What effect does increasing the value of the constant 'm' in the equation of a straight line have on the graph?

It increases the slope of the line Signup and view all the answers

When given the area formula of a circle, what does the variable 'r' represent?

The radius Signup and view all the answers

What characterizes a linear inequality compared to a linear equation?

It involves an inequality sign Signup and view all the answers

Which of the following describes the y-intercept in the equation of a straight line accurately?

The point where the line intersects the y-axis Signup and view all the answers

What is the initial step in solving a set of simultaneous equations using substitution?

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

What is the effect on the graph of the function as the value of 'c' becomes negative in the linear equation?

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

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

It shifts the graph vertically upwards by q units. Signup and view all the answers

Which of the following describes the range of the function $f(x) = ax^2 + q$ when $a < 0$?

The range is $(- ext{infinity}; q)$. Signup and view all the answers

How is the y-intercept of a straight-line graph calculated?

By setting $x = 0$ in the equation. Signup and view all the answers

What characterizes the graph of the function $y = mx + c$ if 'm' is negative?

The graph decreases as x increases. Signup and view all the answers

Which of the following statements is true regarding the gradient 'm' in the equation of a straight-line graph?

It represents the steepness of the line. Signup and view all the answers

What happens to the graph of the function $y = ax^2 + q$ as 'a' approaches zero but remains positive?

The graph becomes wider. Signup and view all the answers

Which of the following best describes the domain of a function defined by the equation $y = mx + c$?

All real numbers. Signup and view all the answers

What characteristic must be determined to plot a straight line using the gradient and y-intercept method?

The gradient and at least one intercept are needed. Signup and view all the answers

If the gradient 'm' is zero in the equation $y = mx + c$, what is true about the graph?

The graph is a horizontal line. Signup and view all the answers

What determines the direction in which the graph of the quadratic function $y = ax^2 + q$ opens?

The sign of 'a'. Signup and view all the answers

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

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

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

All real numbers except zero. Signup and view all the answers

Which of the following statements is true regarding the x-intercept of the function $y = ax^2 + q$?

It is calculated by setting $y = 0$. Signup and view all the answers

What does the range of the function $y = ax^2 + q$ depend on?

The sign of 'a' and the value of 'q'. Signup and view all the answers

Where is the vertical asymptote located for the hyperbolic function $y = \frac{a}{x} + q$?

At $x = 0$. Signup and view all the answers

How does the variable 'q' affect the graph of the function $y = \frac{a}{x} + q$?

It introduces a vertical shift. Signup and view all the answers

What best describes a characteristic of exponential functions of the form $y = ab^x + q$ when $a > 0$?

The range consists of values greater than 'q'. Signup and view all the answers

Which statement correctly describes the axis of symmetry for parabolic functions of the form $f(x) = ax^2 + q$?

The axis of symmetry is along $x = 0$. Signup and view all the answers

What does the turning point of the parabolic function $y = ax^2 + q$ indicate when $a > 0$?

It is a minimum point of the graph. Signup and view all the answers

What happens to the graph of an exponential function if 'a' is set to a negative value?

The graph flips downwards. Signup and view all the answers

What describes the range of an exponential function when $a < 0$?

$f(x) < q$ Signup and view all the answers

Which statement is true concerning the effect of the parameter $q$ on the graph of an exponential function?

It causes a vertical shift. Signup and view all the answers

What is the y-intercept of the function $y = a an heta + q$ when evaluated at $ heta = 0°$?

$q$ Signup and view all the answers

For the function $y = a heta + q$ ($ heta$ in degrees), how does the value of $a$ affect the graph?

It adjusts the amplitude of the wave. Signup and view all the answers

Which characteristic distinguishes the cosine function from the sine function within their respective shares?

The cosine function starts at its maximum value. Signup and view all the answers

What is the period of the sine and cosine functions?

360° Signup and view all the answers

What defines the behavior of the function $y = a an heta + q$ concerning its vertical shift?

$q$ affects the asymptotes of the tangent function. Signup and view all the answers

In the function $y = a an heta + q$, what defines the intercepts?

Both the x and y intercepts are at $(0, 0)$. Signup and view all the answers

What effect does a coefficient $a < 0$ have on the graph of the sine function?

The graph reflects across the x-axis. Signup and view all the answers

Which of the following characteristics is solely based on the parameter $q$ in the sine and cosine functions?

Establishing the vertical shift. Signup and view all the answers

What is the result of applying the rule for division of exponents when simplifying the expression ( \frac{a^5}{a^2} )?

a^{3} Signup and view all the answers

Which expression can be simplified using the power of a power rule?

(x^2)^3 Signup and view all the answers

Which of the following statements about rational exponents is true?

Rational exponents can represent both roots and powers. Signup and view all the answers

What happens when simplifying the expression ( (3x^2y^3)^2 )?

3^2x^{22}y^{32} Signup and view all the answers

To solve the equation ( 4^x = 16 ), what is the first step?

Convert 4 and 16 to the same base. Signup and view all the answers

What is the result of ( a^{-3} ) when expressed in terms of positive exponents?

\frac{1}{a^3} Signup and view all the answers

How would you simplify the expression ( \left(\frac{2}{3}\right)^{-2} )?

\frac{9}{4} Signup and view all the answers

Which of the following is the correct process for solving ( a^x = a^5 )?

Set x = 5. Signup and view all the answers

Which expression represents a correct simplification using the laws of exponents?

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

What effect does an increase in the value of 'a' have on the graph of a parabola?

Increases the curvature steepness Signup and view all the answers

What are the y-intercepts of the function defined by the equation $y = a an heta + q$?

At $y = q$ Signup and view all the answers

What do the vertical shifts in trigonometric graphs determine?

The value of 'q' Signup and view all the answers

Where are the asymptotes of the hyperbola defined by the equation $y = \frac{a}{x} + q$ located?

At $x = 0$ and $y = 0$ Signup and view all the answers

How can the domain of the function $y = a an heta + q$ be defined?

${ \theta : 0° \leq \theta \leq 360°, \theta \neq 90°, 270° }$ Signup and view all the answers

What is the correct way to determine the vertical shift of a hyperbola?

By determining the value of 'q' Signup and view all the answers

What information can be gathered from the x-intercepts of a graph?

They show where the function outputs a value of zero Signup and view all the answers

What is the shape of a parabola characterized by when 'a' is negative?

A 'frown' shape Signup and view all the answers

What is a characteristic of the tangent function graph?

It intersects the x-axis periodically Signup and view all the answers

Which of the following describes the process of finding the equation of a parabola?

Using the y-intercept to determine 'q' and another point for 'a' Signup and view all the answers

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

Two Signup and view all the answers

Which of the following steps is NOT part of solving a linear equation?

Add constants to both sides Signup and view all the answers

When solving simultaneous equations by substitution, what is the primary goal?

To reduce the equations to a single variable Signup and view all the answers

What form must a quadratic equation take before it can be solved?

ax^2 + bx + c = 0 Signup and view all the answers

Which operation maintains balance when manipulating an equation?

Performing the same operation on both sides Signup and view all the answers

How can one check if the solution of an equation is valid?

By substituting the solution back into the original equation Signup and view all the answers

What is the purpose of factorising a quadratic equation during the solving process?

To create two linear equations Signup and view all the answers

What is unique about the solutions to a linear equation compared to a quadratic equation?

Linear equations can have at most one solution Signup and view all the answers

What step comes after 'group like terms together' in solving a linear equation?

Factorise if necessary Signup and view all the answers

In the elimination method for solving simultaneous equations, what is the purpose of making coefficients the same?

To easily add or subtract the equations Signup and view all the answers

Which set of numbers is defined as including all positive integers starting from 1?

Natural Numbers Signup and view all the answers

What is the primary distinction of irrational numbers compared to rational numbers?

Their decimal expansions are non-repeating and non-terminating. Signup and view all the answers

Which of the following correctly represents the set of integers?

{..., -2, -1, 0, 1, 2, ...} Signup and view all the answers

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

Integers Signup and view all the answers

What symbol represents rational numbers in the real number system?

Q Signup and view all the answers

Which of the following best defines real numbers?

All rational and irrational numbers Signup and view all the answers

Which statement accurately characterizes irrational numbers?

Irrational numbers have non-repeating, non-terminating decimal expansions. Signup and view all the answers

What is the correct way to convert a terminating decimal into a rational number?

Identify the integer part and convert each decimal place to a fraction. Signup and view all the answers

Which of the following methods is used to estimate the value of a surd?

By identifying the nearest perfect squares or cubes surrounding the surd. Signup and view all the answers

Which process is essential when rounding off a decimal number?

Identify the next digit and apply rules based on its value. Signup and view all the answers

What distinguishes a rational number from an irrational number in terms of decimal representation?

Rational numbers can be expressed as decimals with periodic digits. Signup and view all the answers

What happens if an irrational number is rounded off?

It can become a rational approximation. Signup and view all the answers

When converting recurring decimals to rational numbers, what is the first step?

Multiply by a power of 10 to align the recurring parts. Signup and view all the answers

Which term refers to the root of a number that cannot be simplified to produce a rational result?

Surd Signup and view all the answers

What defines a rational number in relation to its representation?

It can be expressed in the form of a fraction where the denominator is non-zero. Signup and view all the answers

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

Two solutions Signup and view all the answers

What step should be taken immediately after expanding all brackets when solving a linear equation?

Rearrange the terms Signup and view all the answers

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

It distributes the monomial to each term of the binomial. Signup and view all the answers

In solving simultaneous equations by elimination, what is the first step that must be taken?

Make coefficients of one variable the same Signup and view all the answers

Which of the following represents the correct expansion of the expression $(ax + b)(cx + d)$?

$acx^2 + adx + bcx + bd$ Signup and view all the answers

How can a quadratic equation be identified as having no solutions?

When the discriminant is negative Signup and view all the answers

How is a trinomial defined in terms of its terms?

An expression with three terms. Signup and view all the answers

What is the main purpose of factorization in algebra?

To simplify complex expressions into simpler products. Signup and view all the answers

What is the last step in solving both linear and quadratic equations?

Check the answer Signup and view all the answers

Which method involves expressing one variable in terms of another to solve simultaneous equations?

Substitution Signup and view all the answers

Which identity is used for factoring a difference of two squares?

$a^2 - b^2 = (a + b)(a - b)$ Signup and view all the answers

What is a common approach to simplifying algebraic fractions?

Factorize both the numerator and denominator, then cancel common factors. Signup and view all the answers

When solving a quadratic equation, what is the form that it should be rewritten into?

ax^2 + bx + c = 0 Signup and view all the answers

Which configuration correctly represents the multiplication of fractions?

$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$ Signup and view all the answers

What is a crucial point to remember when manipulating equations?

Operations on one side must be performed on the other side Signup and view all the answers

Which formula is used to factor the sum of two cubes?

$x^3 + y^3 = (x + y)(x^2 + xy + y^2)$ Signup and view all the answers

What is the primary focus when checking for extraneous solutions?

To confirm whether the solution satisfies the original equation Signup and view all the answers

What does the process of factorizing a quadratic trinomial typically involve?

Finding two binomials that multiply to give the original trinomial. Signup and view all the answers

Which method reduces the number of variables by substituting one variable into another equation?

Substitution Signup and view all the answers

Which law of exponents states that when multiplying two expressions with the same base, you add the exponents?

Multiplication of Exponents Signup and view all the answers

What happens when you raise a quotient to a power?

Both the numerator and denominator are raised to the power. Signup and view all the answers

Which of the following correctly represents the expression with a negative exponent rewritten as a fraction?

$a^{-n} = \frac{1}{a^n}$ Signup and view all the answers

Which method is recommended when the bases of an exponential equation are not the same?

Using Logarithms Signup and view all the answers

What is the simplified form of the expression $\frac{a^{3/4}}{a^{1/4}}$?

$a^{2/4}$ Signup and view all the answers

How would you express the square root of 'a' as a rational exponent?

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

In the equation $a^x = a^5$, what conclusion can be drawn if $a > 0$ and $a \neq 1$?

x must equal 5. Signup and view all the answers

What is the first step in solving an exponential equation involving different bases?

Change the bases to be the same. Signup and view all the answers

What is the primary method for finding the solution of simultaneous equations graphically?

Identifying the intersection point of the lines Signup and view all the answers

Which statement is true regarding the expression $(3x^2y)^3$?

It simplifies to $3^3x^{2*3}y^3$. Signup and view all the answers

When given the expression $a^{m/n}$, what does it represent?

It is the n-th root of $a^m$. Signup and view all the answers

In solving a word problem, which of the following is the initial step according to the problem-solving strategy?

Read the whole question Signup and view all the answers

How should the unknown variable be treated if it appears in two or more terms in a literal equation?

Take it out as a common factor Signup and view all the answers

Which statement correctly describes what happens to an inequality sign during division by a negative number?

The inequality is reversed Signup and view all the answers

What effect does the value of 'c' have in the linear function equation $y = mx + c$?

It indicates the vertical position of the line Signup and view all the answers

When solving inequalities, what happens after multiplying by -1?

The inequality sign is inverted Signup and view all the answers

In the expression for speed given by $v = \frac{D}{t}$, what does 'D' represent?

Distance traveled Signup and view all the answers

To solve a literal equation in terms of one variable, what is the first thing to do?

Rearrange it to isolate the variable Signup and view all the answers

What characteristic must be evaluated to accurately plot a straight line using the gradient and y-intercept method?

Both the gradient and the y-intercept Signup and view all the answers

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

It shifts the graph vertically upwards by 'q' units. Signup and view all the answers

For an exponential function where 'a' is negative and 'b' is greater than 1, what can be said about the shape of the graph?

The graph will curve downwards exhibiting exponential decay. Signup and view all the answers

How does the amplitude of a sine function change if the value of 'a' is negative?

The graph will reflect about the x-axis. Signup and view all the answers

What is the period of a sine or cosine function of the form $y = a ext{sin} heta + q$?

360° Signup and view all the answers

What does the range of the function $y = a ext{sin} heta + q$ become when 'a' is equal to 1 and 'q' is equal to 2?

[1, 3] Signup and view all the answers

In the context of trigonometric functions, what defines the vertical shift produced by the parameter 'q'?

It moves the graph vertically by 'q' units. Signup and view all the answers

How are the x-intercepts of the sine function positioned within the domain [0°, 360°]?

At 0°, 180°, and 360°. Signup and view all the answers

What is true about the asymptotes of the tangent function?

They occur at 90° and 270°. Signup and view all the answers

For an exponential function with parameters a < 0 and b < 1, which statement is true?

The function will experience rapid decay and curve upwards. Signup and view all the answers

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

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

How does increasing 'm' in the equation $y = mx + c$ affect the slope of the line?

The line becomes steeper. Signup and view all the answers

What characterizes the y-intercept in the equation $y = mx + c$?

It is determined by setting $x = 0$. Signup and view all the answers

What does the sign of 'q' influence in the parabolic function $y = ax^2 + q$?

The y-intercept of the graph. Signup and view all the answers

Which of the following conditions provides the range of the function $f(x) = mx + c$?

The range is $ ext{R}$ because 'y' can take any real value. Signup and view all the answers

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

By setting $y = 0$ and solving for $x$. Signup and view all the answers

What effect does a negative value of 'a' have on the graph of the parabola $y = ax^2 + q$?

The graph is flipped upside down. Signup and view all the answers

In the standard form $y = ax^2 + q$, what happens to the graph when 'q' is positive?

The graph shifts vertically upwards by $q$ units. Signup and view all the answers

When using the gradient and y-intercept method, what is the first step to plot the graph of $y = mx + c$?

Calculate the y-intercept by setting $x = 0$. Signup and view all the answers

Which of the following accurately defines the term 'gradient' in the context of linear functions?

The ratio of the vertical change to horizontal change. Signup and view all the answers

What determines the steepness of the branches in the graph of a tangent function?

The value of a Signup and view all the answers

Where are the asymptotes located for the function defined as $y = a \tan \theta + q$?

$\theta = 90°$ and $\theta = 270°$ Signup and view all the answers

In the equation of a parabolic graph $y = ax^2 + q$, what does the term q affect?

The vertical shift Signup and view all the answers

What is the range of the function defined by $y = ax^2 + q$ where $a > 0$?

${ f(x) : f(x) \geq q }$ Signup and view all the answers

Which of the following is essential to determine the equation of a hyperbola?

Substituting given points into the equation Signup and view all the answers

What does the amplitude of a trigonometric graph depend on?

The value of a Signup and view all the answers

When an asymptote is present in the function $y = \frac{a}{x} + q$, what does this indicate?

The graph will have no limits in that direction Signup and view all the answers

To find the output values of a function with respect to different inputs, which characteristics must be examined?

Intercepts and shifts Signup and view all the answers

What is the correct form of the equation of a trigonometric function that includes both sine and a vertical shift?

$y = a \sin \theta + q$ Signup and view all the answers

What must be true about the angle ( \theta ) in the domain of the tangent function?

It cannot equal 90° or 270° Signup and view all the answers

What is the y-intercept of the function represented by the equation $y = ax^2 + q$?

(0, q) Signup and view all the answers

For the function $y = \frac{a}{x} + q$, which of the following statements is true regarding its intercepts?

It has no y-intercept. Signup and view all the answers

What happens to the range of the function $f(x) = ax^2 + q$ when $a < 0$?

The range is (-\infty, q]. Signup and view all the answers

Which line represents the vertical asymptote for the hyperbolic function $y = \frac{a}{x} + q$?

x = 0 Signup and view all the answers

How does the value of $q$ affect the graph of the hyperbolic function $y = \frac{a}{x} + q$?

It determines the horizontal asymptote. Signup and view all the answers

For the function $f(x) = ab^x + q$, what is the correct domain?

All real numbers Signup and view all the answers

Which characteristic is used to determine the direction of the parabola represented by $y = ax^2 + q$?

The sign of $a$ Signup and view all the answers

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

It is a maximum point at (0, q). Signup and view all the answers

Gr 10 Math June P1 Medium

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Questions and Answers

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

Which category do the numbers $0$ and $1$ belong to?

What is the main feature of rational numbers?

Which of the following sets does not include the number $-5$?

Which of the following represents the set of integers?

What differentiates real numbers from imaginary numbers?

What is the highest exponent of the variable in a linear equation?

How many solutions can a linear equation have at most?

What is the first step in solving a linear equation?

Which of the following methods is NOT used to solve quadratic equations?

When solving quadratic equations, what form must the equation be in?

What is the method of elimination primarily used for in simultaneous equations?

If a quadratic equation has no solutions, what can be said about its graph?

What should be done after solving for one variable in simultaneous equations by substitution?

What operation must be performed on both sides of an equation to maintain balance?

Which step is NOT part of the general method for solving linear equations?

Which statement accurately describes rational numbers?

What characterizes an irrational number?

Which of the following is a correct way to estimate the value of the surd \( ext{√5}\)?

How can a terminating decimal be converted into a rational number?

Which of the following best illustrates the difference between rational and irrational decimal forms?

Which of the following statements about rounding off decimal numbers is correct?

In the expression 5x^2 + 3x - 2, which term is the coefficient?

What defines a surd?

Which of the following represents a method to convert recurring decimals to rational numbers?

What does the gradient "m" represent in the equation of a straight-line graph?

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

What effect does a negative value of "a" have on a parabolic graph?

What is the domain of the function defined by the equation $y = ax^2 + q$?

Which statement is true about the effect of "q" on the parabolic graph?

Which statement correctly describes the effect of increasing the absolute value of "a" on the parabolic graph?

In the expression $m = \frac{\text{change in } y}{\text{change in } x}$, what does "change in y" represent?

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

Which characteristic must be calculated to draw a straight line using the gradient and y-intercept method?

If a quadratic equation has a value of $q < 0$, where is the turning point located?

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

What can be concluded when two graphs of linear equations intersect?

What effect does increasing the value of $m$ have on the graph of the function $y = mx + c$?

When solving a linear inequality, what happens to the inequality sign if both sides are divided by a negative number?

In the equation of a circle $A = \pi r^2$, what variable is represented by $A$?

What does the value of $c$ affect in the linear equation $y = mx + c$?

What is meant by a literal equation?

Which of the following conditions would indicate that a system of equations has no solutions?

Which action is inappropriate when solving literal equations?

In the inequality $2x + 2 \leq 1$, how would you solve for $x$?

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

How can the vertical shift of a trigonometric graph be determined?

Where are the asymptotes for the tangent function located?

What does the domain of a function represent?

In the equation of a hyperbola, what does the term 'q' signify?

When analyzing the graph of a function, how can x-intercepts be found?

Which equation represents the general form of a sine function?

What is necessary to solve for 'a' in a hyperbola's equation?

What indicates a vertical shift in the graph of a tangent function?

To determine the range of a parabolic function, which method is applicable?

What is the range of the function if the coefficient $a$ is less than zero?

What do the x-intercepts of the function $y = ax^2 + q$ represent?

Which characteristic is true if the graph of $f(x) = ax^2 + q$ is a 'smile'?

In the function $y = rac{a}{x} + q$, what is true about the vertical asymptote?

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

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

Which of the following correctly describes the domain of the hyperbolic function $y = rac{a}{x} + q$?

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

How does the sign of $a$ affect the graph of the exponential function $y = ab^x + q$?

Which statement is true regarding the range of the exponential function $y = ab^x + q$ when $a > 0$?

What happens to the horizontal asymptote of the function when $q$ is decreased?

Which statement about the effects of the coefficient $a$ in the function $y = a an heta + q$ is true?

What is the effect of a coefficient $b$ where $0 < b < 1$ in an exponential function?

Which of the following correctly describes the domain of the sine function?

Where do the x-intercepts of the cosine function occur?

What distinguishes the range of the function $y = a an heta + q$ from the range of the sine function?

For the sine function, which point signifies the maximum turning point?

What is the vertical shift effect when $q$ is greater than zero in a cosine function?

What does the value of $|a|$ represent in the context of the sine function?

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

Which expression correctly represents the difference of two squares?

When performing multiplication of fractions, what expression shows the result of $rac{2}{3} imes rac{4}{5}$?