Gr 10 Math June P1 Mix

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Which set includes all positive integers starting from 1?

Integers (Z)

Rational Numbers (Q)

Natural Numbers (N) (correct)

Whole Numbers (N0)

What is the definition of irrational numbers?

Numbers that can be expressed as a fraction

Whole numbers including negative integers

Numbers that have a negative square root

Numbers with non-repeating, non-terminating decimal expansions (correct)

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

e

√2

0.75 (correct)

π

What does the symbol Z represent in the real number system?

Integers Signup and view all the answers

Which of the following sets includes zero and all positive integers?

Whole Numbers (N0) Signup and view all the answers

Which statement accurately defines imaginary numbers?

Numbers that have a negative square root Signup and view all the answers

What is the result of applying the zero exponent law?

The result is 1 for any base not equal to 0. Signup and view all the answers

Which of the following accurately represents the multiplication of exponents with the same base?

a^{m+n} Signup and view all the answers

How do you simplify the expression (ab)^{2}?

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

Which operation will you use to solve the equation a^{x} = a^{3}?

Equating the exponents Signup and view all the answers

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

It represents the reciprocal of the base raised to the positive exponent. Signup and view all the answers

Which property applies if you have the expression rac{a^{1/2}}{a^{1/3}}?

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

How can you convert a square root into a rational exponent?

By taking the base to the exponent of 0.5. Signup and view all the answers

What is the correct way to factor the expression a^{2} - b^{2}?

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

What should be done first when faced with a complex fraction?

Change to prime bases and factorise. Signup and view all the answers

What is a binomial?

An expression with two terms Signup and view all the answers

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

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

How can a quadratic trinomial of the form ( ax^2 + bx + c ) be factorised?

By finding two binomials that produce the trinomial Signup and view all the answers

What is the result of multiplying a monomial by a binomial ( a(x + y) )?

( ax + ay ) Signup and view all the answers

Which of the following identities is used for factorising a difference of two squares?

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

When simplifying $rac{a}{b} imes rac{c}{d}$, which statement is correct?

( rac{ac}{bd}, b = 0, d = 0 ) Signup and view all the answers

What is the first step in simplifying algebraic fractions?

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

In the process of factorisation, what does factoring by grouping entail?

Identifying and extracting a common factor from grouped terms Signup and view all the answers

Which expression represents 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 solution to a system of simultaneous linear equations when solved graphically?

The coordinates of the intersection point Signup and view all the answers

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

Discard any irrelevant information Signup and view all the answers

What is a literal equation?

An equation that contains several letters or variables Signup and view all the answers

How does the value of 'm' affect the graph of the function $y = mx + c$?

It affects the slope of the graph Signup and view all the answers

What must be done to solve an inequality if both sides involve division by a negative number?

The inequality sign must be switched Signup and view all the answers

What happens to the graph of a linear function if the value of 'c' is less than 0?

The graph shifts vertically downwards Signup and view all the answers

In solving literal equations, what should be done if the unknown variable is in two or more terms?

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

When solving a linear inequality, what do you do if the inequality involves a multiplication by a negative number?

Switch the direction of the inequality sign Signup and view all the answers

What is needed to effectively tackle word problems involving simultaneous equations?

A clear set of algebraic equations Signup and view all the answers

If 'm' is greater than 0 in the function $y = mx + c$, how does the graph behave?

It slopes upwards from left to right 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

What is the purpose of checking for extraneous solutions?

To verify the solutions are valid in the original equation Signup and view all the answers

Which step comes first when solving linear equations?

Expand all brackets Signup and view all the answers

How many solutions can a linear equation have?

One Signup and view all the answers

What is a quadratic equation characterized by?

The exponent of the variable is at most 2 Signup and view all the answers

In the factorization method for quadratic equations, what form must the equation take?

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

How are simultaneous equations typically solved?

By balancing the coefficients of each variable Signup and view all the answers

What method can be used to reduce the number of equations in simultaneous equations?

Substitution Signup and view all the answers

What is a necessary step after solving for a variable in quadratic equations?

Verify the solutions by substitution Signup and view all the answers

What should always be performed on both sides of an equation?

The same operation Signup and view all the answers

Which of the following is true about irrational numbers?

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

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

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

Which of the following statements about rounding off is correct?

Rounding off can convert an irrational number into a rational approximation. Signup and view all the answers

How can you identify a surd?

It has the form \sqrt[n]{a} where a is a positive number. Signup and view all the answers

Which of the following correctly describes rational numbers?

Rational numbers can be written as \frac{a}{b} where b ≠ 0. Signup and view all the answers

What is the characteristic of terminating decimals?

They have a fixed number of decimal places. Signup and view all the answers

What is a key step in estimating surds?

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

Which of the following represents a rational number?

0.333... Signup and view all the answers

What happens when you round 9.999 to one decimal place?

It becomes 10.0. Signup and view all the answers

Which expression is a term in mathematics?

5x^2 Signup and view all the answers

What happens to the graph of an exponential function when the value of q is positive?

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

In the function of the form y = a sin θ + q, how does a affect the graph?

It alters the amplitude and may reflect the graph about the x-axis. Signup and view all the answers

Which of the following statements about the sine function is incorrect?

The maximum turning point occurs at 180°. Signup and view all the answers

For the function y = a cos θ + q, what does a < 0 indicate?

The graph is reflected about the x-axis. Signup and view all the answers

What defines the asymptote of an exponential function in the form y = ab^x + q?

It is defined as the line y = q. Signup and view all the answers

What is the domain of the tangent function y = tan θ?

{ θ : 0° ≤ θ ≤ 360°, θ ≠ 90°, 270° } Signup and view all the answers

When the value of b in the exponential function is between 0 and 1, what type of function does it represent?

Exponential decay. Signup and view all the answers

What is the range of the sine function y = a sin θ + q for a > 0?

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

What is the standard form of a straight-line graph?

y = mx + c Signup and view all the answers

How can the sine and cosine graphs be related?

The cosine graph can be shifted to the left by 90° to overlap with the sine graph. Signup and view all the answers

How can the y-intercept be calculated for the equation y = mx + c?

Let x = 0 Signup and view all the answers

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

The graph opens upwards and has a minimum turning point Signup and view all the answers

What happens to the graph of the cosine function with a positive q?

It shifts upwards. Signup and view all the answers

Which statement about the range of the function f(x) = mx + c is true?

The range is {f(x) : f(x) in ℝ} Signup and view all the answers

What effect does 'q' have on the graph of a parabola in the equation y = ax^2 + q?

It causes a vertical shift Signup and view all the answers

For which condition will a parabola have a maximum turning point?

a < 0 and q > 0 Signup and view all the answers

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

All real numbers Signup and view all the answers

If a parabola opens downwards, what can be inferred about the value of 'a'?

a < 0 Signup and view all the answers

When plotting a straight-line graph, which points are typically used?

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

What indicates the steepness of a line in the equation y = mx + c?

m Signup and view all the answers

What is the range of the function when $a < 0$?

(- text{infinity}, q] Signup and view all the answers

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

(0, q) Signup and view all the answers

Which of the following is the horizontal asymptote for hyperbolic functions of the form $y = \frac{a}{x} + q$?

$y = q$ Signup and view all the answers

What happens to a hyperbolic graph when $q > 0$?

It shifts vertically upwards. Signup and view all the answers

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

${x : x in ext{Real}, x eq 0}$ Signup and view all the answers

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

The line $x = 0$ Signup and view all the answers

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

Set $y = 0$. Signup and view all the answers

What effect does the parameter $a$ have in exponential functions $y = ab^x + q$?

It determines the direction of the graph. Signup and view all the answers

When sketching the graph of $y = b^x$, which characteristic is NOT considered?

The turning point. Signup and view all the answers

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

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

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

It is equal to the value of 'q'. Signup and view all the answers

How are asymptotes for a hyperbola determined?

By setting the denominator to zero. Signup and view all the answers

What determines the domain of a tangent function?

All values except where the function is undefined. Signup and view all the answers

When determining the equation of a hyperbola, what should you look for in the graph?

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

How is the vertical shift represented in the equation of a parabola?

By the constant 'q'. Signup and view all the answers

What is the significance of the value 'q' in the trigonometric functions?

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

Which method is used to find 'a' in the equation of a hyperbola?

By substituting known points into the equation. Signup and view all the answers

What can be derived from the range of a function like a hyperbola?

It determines possible y-values for all x-values. Signup and view all the answers

Which of the following statements is true about the intercepts of parabolic graphs?

The x-intercept can be found by setting y to zero. Signup and view all the answers

Which statement is true about whole numbers?

Whole numbers start from zero. Signup and view all the answers

What characterizes integers in the real number system?

They include both positive and negative whole numbers. Signup and view all the answers

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

$rac{22}{7}$ Signup and view all the answers

What distinguishes rational numbers from irrational numbers?

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

Which type of number includes negative values as well as zero?

Integers Signup and view all the answers

Which statement best describes real numbers?

Real numbers consist of both rational and irrational numbers. Signup and view all the answers

Which of the following decimal types is not considered a rational number?

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

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

Multiply by a power of 10 Signup and view all the answers

Which of the following statements about surds is true?

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

When rounding off a decimal number, what happens if the digit after the required place is 5 or greater?

The last digit increases by one. Signup and view all the answers

What represents the relationship between rational and irrational numbers in decimal form?

Rational numbers could be either terminating or repeating. Signup and view all the answers

Which of the following describes a key characteristic of irrational numbers?

Irrational numbers cannot be expressed as a simple fraction. Signup and view all the answers

How can you estimate the square root of a number that is not a perfect square?

By identifying and comparing perfect squares around that number. Signup and view all the answers

Which of the following must be true about the coefficients in a term of an expression?

Coefficients can be fractions. Signup and view all the answers

What is the characteristic of products in mathematical expressions?

A product is the result of multiplication. Signup and view all the answers

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

One Signup and view all the answers

Which method is typically used to solve simultaneous equations?

Substitution Signup and view all the answers

When solving quadratic equations, which of the following steps is NOT essential?

Check for extraneous solutions Signup and view all the answers

What does it mean to check for extraneous solutions?

Verify if solutions satisfy the original equation Signup and view all the answers

What is the first step in solving a quadratic equation?

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

How are the solutions of a quadratic equation generally found?

Solving each factor set to zero Signup and view all the answers

What is a critical step when performing operations on equations?

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

In order to solve simultaneous equations by elimination, what must be achieved first?

Equal coefficients of both variables Signup and view all the answers

Which of the following is true regarding the roots of a quadratic equation?

It may have one double root Signup and view all the answers

What is required to solve for two unknown variables using simultaneous equations?

Two independent equations Signup and view all the answers

What does the intersection of two linear equations represent when solved graphically?

The coordinates of the solutions Signup and view all the answers

Which of the following is the first step in the problem-solving strategy for word problems?

Read the whole question Signup and view all the answers

What is the main concept of a literal equation?

It is an equation with multiple variables or letters Signup and view all the answers

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

It switches direction Signup and view all the answers

What effect does a positive value for 'c' have on the graph of the function $y = mx + c$?

The graph shifts vertically upwards Signup and view all the answers

Which of the following steps is necessary when isolating an unknown variable in a literal equation?

Perform the same operation to both sides Signup and view all the answers

What must be taken into account when taking the square root of both sides of an equation?

Both positive and negative solutions must be considered Signup and view all the answers

Which statement describes the effect of increasing the value of 'm' in the equation $y = mx + c$?

The gradient of the graph increases Signup and view all the answers

When writing equations for word problems, what is the main goal?

To write equations that represent the problem mathematically Signup and view all the answers

In the equation $y = mx + c$, what property is represented by the variable 'm'?

The slope or gradient of the graph Signup and view all the answers

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

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

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

It will form a maximum turning point. Signup and view all the answers

What determines the steepness of the line in the equation $y = mx + c$?

The value of $m$ Signup and view all the answers

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

The graph shifts vertically upwards. Signup and view all the answers

What points are essential to sketch a straight-line graph using the dual intercept method?

One x-intercept and one y-intercept Signup and view all the answers

For the equation $y = mx + c$, which condition leads to a horizontal line?

$m = 0$ Signup and view all the answers

How does the value of $c$ contribute to the graph of a linear function?

It determines the y-intercept. Signup and view all the answers

When $a > 0$ in a parabola $y = ax^2 + q$, what is the behavior of the graph?

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

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

The slope of the line Signup and view all the answers

What happens to the graph of a parabola when $a$ approaches 0 from the positive side?

The graph becomes wider. Signup and view all the answers

What is the result of applying the law of exponents for dividing expressions with the same base?

The exponents are subtracted. Signup and view all the answers

Which expression correctly demonstrates the negative exponent law?

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

When simplifying $(ab)^{m/n}$, what is the correct application of the exponent laws?

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

Which statement about simplifying rational exponents is NOT true?

Rational exponents can only be expressed with prime bases. Signup and view all the answers

What is the first step in solving the exponential equation $a^x = a^2$?

Set the exponents equal to each other. Signup and view all the answers

Which property defines the zero exponent law?

a^0 = 1 for any a ≠ 0 Signup and view all the answers

What does the expression $a^{m/n}$ represent when $n$ is an integer?

The $n$th root of $a^m$. Signup and view all the answers

How should the expression $rac{a^{2/3}}{a^{1/3}}$ be simplified?

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

What strategy is recommended for simplifying complex fractions?

Change to prime bases before factoring. Signup and view all the answers

Which of the following correctly shows the application of the law of exponents for a product raised to a power?

(a imes b)^{n} = a^{n}b^{n} Signup and view all the answers

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

Horizontal asymptote at $y = q$ Signup and view all the answers

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

Shifts the graph upwards Signup and view all the answers

For the function $y = a an heta + q$, what happens when $a < 0$?

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

Which of the following describes the domain of the sine function $y = ext{sin}( heta)$?

$0° ext{ to } 360°$ Signup and view all the answers

What is the effect of the value of $b$ on the graph of an exponential function?

Determines the rate of growth or decay Signup and view all the answers

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

It determines the upward or downward curve Signup and view all the answers

What defines the range of a sine function given by $y = a ext{sin}( heta) + q$ for $a > 0$?

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

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

The graph shifts downwards by $|q|$ Signup and view all the answers

What characteristic do the sine and cosine functions share regarding their periodicity?

Both have a period of 360° Signup and view all the answers

How many x-intercepts does the sine function $y = ext{sin}( heta)$ have within its domain?

3 Signup and view all the answers

What is the range of the function if the parameter $a$ is less than 0?

From negative infinity to $q$ Signup and view all the answers

What can be concluded about the graph of the function $f(x) = ax^2 + q$ if $a > 0$?

It has a minimum turning point at $(0, q)$ Signup and view all the answers

What is the significance of the axes of symmetry for the function $f(x) = ax^2 + q$?

It is represented by the line $x = 0$ Signup and view all the answers

For the function $y = rac{a}{x} + q$, what is the behavior of the graph when $x = 0$?

The function is undefined Signup and view all the answers

What type of asymptote does the hyperbolic function $y = rac{a}{x} + q$ exhibit?

Both horizontal at $y = q$ and vertical at $x = 0$ Signup and view all the answers

How does the sign of parameter $a$ affect the quadrants in which the hyperbola $y = rac{a}{x} + q$ lies?

It lies in the second and fourth quadrants for $a < 0$ Signup and view all the answers

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

{x: x ∈ ℝ, x ≠ 0} Signup and view all the answers

What characteristic does the exponential function $y = ab^x + q$ exhibit for all real values of $x$?

The domain is all real numbers Signup and view all the answers

For the exponential function where $a > 0$, what is the behavior of its range?

The range is limited to $[q, ext{infinity})$ Signup and view all the answers

What shifting effect does the constant $q$ have on a hyperbola $y = rac{a}{x} + q$?

It shifts the graph vertically Signup and view all the answers

What is a monomial?

An expression with one term Signup and view all the answers

Which expression represents the product of a binomial and a trinomial?

(A + B)(C + D + E) 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 result of applying the difference of squares identity?

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

What is a trinomial?

An expression with three terms Signup and view all the answers

Which operation do you perform to simplify the expression ( \frac{a}{b} \div \frac{c}{d} )?

Multiply by the reciprocal of the second fraction Signup and view all the answers

Which of the following expressions is the correct identity for the sum of two cubes?

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

What does factorising by grouping involve?

Grouping terms with common factors Signup and view all the answers

In the multiplication of two linear binomials, what is the term that results from the product of the constants?

The constant term Signup and view all the answers

When simplifying algebraic fractions, what is the second step after factorizing the numerator and denominator?

Cancel common factors Signup and view all the answers

What is the effect of increasing the value of 'a' in the equation of a parabola?

It alters the steepness of the graph branches, making them steeper. Signup and view all the answers

How can the y-intercept of a trigonometric function in the form $y = a an heta + q$ be calculated?

By substituting $0$ for $ heta$ in the equation. Signup and view all the answers

Which statement accurately describes how to find the vertical shift 'q' for a parabola?

It can be found using the y-intercept point. Signup and view all the answers

What is the significance of the asymptotes in the graph of a hyperbola?

They indicate where the graph becomes undefined. Signup and view all the answers

What is required to determine the equation of a hyperbola using given points?

Substituting the given points to create equations for 'a' and 'q'. Signup and view all the answers

Which of the following best describes the domain of the tangent function?

All real numbers except $90°$ and $270°$. Signup and view all the answers

How does the vertical shift 'q' affect the graph of a trigonometric function?

It moves the graph up or down without changing its shape. Signup and view all the answers

Which method is used to find points of intersection between two graphs?

Equating the expressions of both graphs and solving for $x$ and $y$. Signup and view all the answers

What characterizes the range of the function represented by a hyperbola?

The output is infinite in both positive and negative directions. Signup and view all the answers

How is the steepness of the graph branches of a hyperbola influenced?

By the value of 'a' in the equation. Signup and view all the answers

Which of the following subsets of the real number system includes both positive and negative whole numbers along with zero?

Integers Signup and view all the answers

What distinguishes irrational numbers from rational numbers?

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

Which symbol represents the set of all numbers that can be expressed as a fraction of two integers where the denominator is not zero?

Q Signup and view all the answers

Which statement is true regarding the set of real numbers?

Real numbers encompass both rational and irrational numbers. Signup and view all the answers

What is the set of natural numbers represented by?

N Signup and view all the answers

Which of the following expressions is an irrational number?

$ ext{ } oot{2}$ 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 should come immediately after expanding all brackets when solving a linear equation?

Rearrange the terms Signup and view all the answers

In solving quadratic equations, what form should the equation be in before attempting factorization?

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

What is the primary purpose of checking your solution after solving an equation?

To verify that both sides of the equation are equal Signup and view all the answers

When solving simultaneous equations by elimination, what is the first step typically taken?

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

What is indicated when a quadratic equation has no real solutions?

The discriminant is negative Signup and view all the answers

Which of the following is true about the solution set of a linear equation?

It contains exactly one solution Signup and view all the answers

What is the consequence of not balancing an equation after performing an operation?

The solution will be incorrect Signup and view all the answers

When solving by substitution, what is the ultimate goal of the process?

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

What is the recommended approach if one of the factors in a quadratic equation is not easily factorable?

Use the quadratic formula Signup and view all the answers

Which of the following decimal representations is classified as a rational number?

1.555555... Signup and view all the answers

What type of number is represented by the expression ( \sqrt{5} )?

Irrational number Signup and view all the answers

Which statement correctly describes the rounding process of the number 4.786 to two decimal places?

It becomes 4.79 Signup and view all the answers

What is the primary challenge in estimating the value of the surd ( \sqrt{27} )?

Finding perfect squares surrounding it Signup and view all the answers

When converting the recurring decimal ( 0.6666... ) into a rational number, which of the following steps is necessary?

Multiply by 10 and subtract Signup and view all the answers

Which of these sets represents only irrational numbers?

{ ( \sqrt{2}, \sqrt{3}, \sqrt{5} ) } Signup and view all the answers

Which of the following choices accurately places the set of rational numbers (Q) in relation to natural numbers (N)?

N is a subset of Q Signup and view all the answers

What is the goal when solving a system of simultaneous equations graphically?

To identify the coordinates of the intersection point Signup and view all the answers

How can the process of rounding affect the representation of an irrational number?

It can convert it to a rational approximation Signup and view all the answers

What is a characteristic feature of a surd?

It cannot be expressed as a simple fraction Signup and view all the answers

What is the first step in the problem-solving strategy for word problems?

Read the whole question Signup and view all the answers

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

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

How is the value of 'm' in the linear function $y = mx + c$ represented?

As the gradient or slope of the graph Signup and view all the answers

When rearranging a literal equation, what should you do if the unknown variable is in the denominator?

Multiply both sides by the lowest common denominator Signup and view all the answers

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

The inequality sign is reversed Signup and view all the answers

Which of the following statements is true about word problems?

They require translating text into algebraic expressions Signup and view all the answers

Which expression correctly demonstrates the power of a power law?

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

What is the correct way to simplify the expression ( \frac{a^{1/2}}{a^{1/3}} )?

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

What is the effect of 'c' in the linear function $y = mx + c$ when 'c' is greater than 0?

It shifts the graph upwards Signup and view all the answers

Which statement correctly describes how to solve an exponential equation ( a^x = a^y )?

x must equal y when a is any positive number. Signup and view all the answers

What should be the result when solving a linear equation?

A unique value for the variable Signup and view all the answers

What are simultaneous equations best solved through when only two graphs are involved?

Graphical method Signup and view all the answers

How can you express the square root of a number in terms of a rational exponent?

a^{0.5} Signup and view all the answers

Which method can be used when the bases in the equation ( 2^x = 8 ) are not the same?

Use logarithms to isolate the exponent. Signup and view all the answers

What is the result when applying the law of negative exponents to the expression a^{-3}?

1/a^3 Signup and view all the answers

What is an essential step when simplifying an expression with complex fractions?

Factor out common factors in the numerator and denominator. Signup and view all the answers

Which of the following is a true statement about rational exponents?

Rational exponents can be simplified using the same laws as integer exponents. Signup and view all the answers

Which expression represents the multiplication of two exponents with the same base?

a^{m/n + p/q} Signup and view all the answers

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

(ab)^{m/n} = a^{m/n} b^{m/n} Signup and view all the answers

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

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

When simplifying the fraction (\frac{a}{b} \div \frac{c}{d}), which expression represents the correct operation?

(\frac{ad}{bc}) Signup and view all the answers

What is the correct formula for the difference of two cubes?

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

Which statement accurately describes factorizing by grouping?

Identify groups of terms with a common factor to factor them separately. Signup and view all the answers

To factor the quadratic trinomial (ax^2 + bx + c), what is typically the first step?

Identify any common factors. Signup and view all the answers

What does the expression ((A + B)(C + D + E)) expand to?

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

Which identity is used for factorizing the expression (a^2 - b^2)?

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

What is the purpose of factorization in simplifying expressions?

To rewrite expressions in terms of products of simpler expressions. Signup and view all the answers

Which method can be applied to simplify the expression (\frac{x^2 - 1}{x - 1})?

Factorization of the numerator and cancellation of common factors. Signup and view all the answers

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

The graph intersects the y-axis at a positive value. Signup and view all the answers

What happens in the multiplication of two fractions (\frac{a}{b} \times \frac{c}{d})?

(\frac{ac}{bd}) where both denominators must be added. Signup and view all the answers

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

It affects the direction in which the parabola opens. Signup and view all the answers

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

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

Which statement about the y-intercept of the line $y = mx + c$ is accurate?

It is calculated by the formula $y = mx + c$ when x = 0. Signup and view all the answers

When classifying the shape of a parabola, if 'a' is between 0 and 1, what happens to the graph?

The graph widens. Signup and view all the answers

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

It performs a vertical shift upwards or downwards. Signup and view all the answers

Which two points are primarily used to sketch the graph of a linear equation?

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

What defines the steepness of a line represented by the equation $y = mx + c$?

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

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

All real numbers. Signup and view all the answers

What does the sign of the parameter $a$ in the function $y = ab^x + q$ determine?

Whether the graph curves upwards or downwards Signup and view all the answers

Which characteristic is affected by the value of $q$ in the functions $y = a an heta + q$?

Vertical shift of the graph Signup and view all the answers

What is the period of the sine function represented by $y = ext{sin} heta$?

360° Signup and view all the answers

In the context of exponential functions, what does $b > 1$ indicate?

Exponential growth Signup and view all the answers

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

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

Which point represents the y-intercept of the function $y = a ext{cos} heta + q$?

$(0°, q)$ Signup and view all the answers

What are the x-intercepts of the cosine function $y = ext{cos} heta$ within one period?

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

Which of the following characteristics is NOT applicable to the tangent function $y = an heta$?

Period is 360° Signup and view all the answers

What effect does a negative value of $a$ have in the function $y = a ext{sin} heta + q$?

Reflection about the x-axis Signup and view all the answers

What is the range of the function if $a < 0$ in the form $f(x) = ax^2 + q$?

(-\infty; q] Signup and view all the answers

What determines whether the graph of $f(x) = ax^2 + q$ has a minimum or maximum turning point?

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

Which statement is true regarding hyperbolic functions of the form $y = \frac{a}{x} + q$?

The range is $, {f(x) : f(x) eq q}$ Signup and view all the answers

What do the vertical and horizontal asymptotes represent in hyperbolic functions?

Horizontal asymptote is $x = 0$, vertical is $y = q$ Signup and view all the answers

Given the function $y = ab^x + q$, which statement is true regarding the domain?

The domain is all real numbers Signup and view all the answers

What is the characteristic of the turning point in the function $y = ax^2 + q$?

It is always a maximum if $a < 0$ Signup and view all the answers

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

It shifts the graph vertically Signup and view all the answers

Which is true about the x-intercepts of the function $y = \frac{a}{x} + q$?

They can be calculated by setting $y = 0$ Signup and view all the answers

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

The line $x = 0$ Signup and view all the answers

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

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

What effect does increasing the value of 'a' have on the branches of the graph?

The branches become steeper. Signup and view all the answers

What determines the vertical shift of a parabola in the equation form $y = ax^2 + q$?

Value of 'q' Signup and view all the answers

Which points would help you determine the parameters 'a' and 'q' in the hyperbola equation $y = \frac{a}{x} + q$?

Given points $(x; y)$ Signup and view all the answers

What is the range of the function for the tangent graph represented by $y = a \tan \theta + q$?

All real numbers Signup and view all the answers

Which of the following is a characteristic of the asymptotes in a hyperbola?

They occur wherever the denominator equals zero. Signup and view all the answers

In the equation $y = a \sin \theta + q$, what does 'q' represent?

Vertical shift of the graph Signup and view all the answers

What identifies the smoothness of graph branches in a function based on the value of 'a'?

Magnitude of 'a' Signup and view all the answers

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

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

Which calculation is necessary to find the y-intercept for the equation of a parabola?

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

What determines the type of trigonometric graph when analyzing the equation $y = a \cos \theta + q$?

Both 'a' and 'q' Signup and view all the answers

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

One Signup and view all the answers

During the process of solving a linear equation, which step follows after expanding all brackets?

Group like terms together Signup and view all the answers

Why is it essential to check for extraneous solutions when solving equations?

To verify that all solutions are correct Signup and view all the answers

What unique characteristic distinguishes quadratic equations from linear equations?

Quadratic equations can have one or two solutions Signup and view all the answers

When solving simultaneous equations using the elimination method, what must be done first?

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

What form must a quadratic equation be in before applying the factorisation method?

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

How many independent equations are required to solve for 'n' unknown variables?

n Signup and view all the answers

Which method can effectively reduce the number of equations when solving simultaneous equations?

Substitution Signup and view all the answers

What must be performed consistently on both sides of an equation when solving?

Any operation performed Signup and view all the answers

What is the first action taken when solving quadratic equations?

Rewrite the equation in standard form Signup and view all the answers

Which subset of real numbers includes negative integers and zero?

Integers (Z) Signup and view all the answers

Which statement about irrational numbers is true?

Irrational numbers cannot be expressed as a simple fraction. Signup and view all the answers

Which of the following examples is not a rational number?

$rac{2}{0}$ Signup and view all the answers

Which of the following correctly represents the multiplication of rational exponents?

$a^{m/n} imes a^{p/q} = a^{m/n + p/q}$ Signup and view all the answers

What does the expression $a^{0}$ equal when $a eq 0$?

1 Signup and view all the answers

What type of numbers does the set of real numbers include?

Both rational and irrational numbers Signup and view all the answers

When simplifying the expression $rac{(a^2b^3)^3}{a^3b^4}$, what is the first step?

Apply the power of a power rule to simplify the numerator Signup and view all the answers

Which of the following statements best describes whole numbers?

Whole numbers include all natural numbers and zero. Signup and view all the answers

What differentiates rational numbers from irrational numbers?

Rational numbers can be expressed as fractions of integers. Signup and view all the answers

Which law of exponents would be used to simplify the expression $a^{-3}b^2$ before canceling?

Negative Exponent Rule Signup and view all the answers

How is the equation $4^{x} = 64$ solved when finding $x$?

By equating $64$ to $4^{3}$ and setting $x = 3$ Signup and view all the answers

What is the result of applying the power of a product rule to $(ab)^{2}$?

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

Which of the following expressions represents the simplification process for $rac{a^{m/n}}{a^{p/q}}$?

$a^{m/n - p/q}$ Signup and view all the answers

Which step should be taken first when faced with an exponential equation involving logarithms?

Logarithmically isolate the variable in the exponent Signup and view all the answers

What form does the expression $2^{x} = 8^{x-1}$ take when simplified?

$2^{x} = 2^{3(x-1)}$ Signup and view all the answers

When applying the difference of squares formula, which expression must be factored?

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

Which statement about irrational numbers is true?

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

What method can be used to convert a terminating decimal into a rational number?

Use the integer part and express the decimal part as a fraction. Signup and view all the answers

How can one identify a surd?

It cannot be simplified to a rational number and its root form is non-repeating. Signup and view all the answers

In rounding off 3.784 to two decimal places, what would be the result?

3.79 Signup and view all the answers

Given the surd $ ewline rac{ ext{1}}{4}$, which operation best describes its relationship in terms of simplicity?

It simplifies to a rational number. Signup and view all the answers

What distinguishes rational numbers from irrational numbers?

Rational numbers can be expressed as fractions whereas irrational cannot. Signup and view all the answers

What is a key characteristic of decimal numbers classified as rational?

They include both repeating and terminating decimals. Signup and view all the answers

When estimating the value of a surd such as $ ewline ext{between } ext{1}^2 ext{ and } ext{2}^2 ?$ Which is the best approximation?

1.5 Signup and view all the answers

Which of the following correctly describes the process of converting a recurring decimal into a fraction?

Align the decimal places and subtract one term from another. Signup and view all the answers

What is the correct expression for the product of a binomial and a trinomial?

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

Which operation is used to simplify the fraction ( \frac{a}{b} \div \frac{c}{d} )?

Multiply by ( \frac{d}{c} ) Signup and view all the answers

How is the difference of two cubes factorized?

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

What is the result of using the identity ( a^2 - b^2 = (a + b)(a - b) ) on the expression ( 16 - x^2 )?

( (4 - x)(4 + x) ) Signup and view all the answers

Which statement describes the role of the coefficient in the term ( 7x^2 )?

It serves as a multiplying factor for the variable's value. Signup and view all the answers

What do you obtain from the expansion of ( (ax + b)(cx + d) )?

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

What are the initial steps in simplifying an algebraic fraction?

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

Which process involves grouping terms to aid in factorization?

Factorizing by grouping pairs. Signup and view all the answers

Which expression correctly represents the multiplication of a monomial by a binomial?

( a(x + y) = ax + ay ) Signup and view all the answers

What is the solution to a system of simultaneous linear equations when solved graphically?

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

Which step is crucial when translating a word problem into mathematical equations?

Identifying all variables involved Signup and view all the answers

When solving a literal equation, how should an unknown variable that is in multiple terms be handled?

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

How does one manage the inequality sign when both sides of a linear inequality are divided by a negative number?

It is switched around Signup and view all the answers

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

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

Which general principle applies when solving a system of equations using substitution?

Substitute back into one of the original equations after isolation Signup and view all the answers

What happens to the graph of a linear function if 'm' is less than 0?

The graph slopes downwards from left to right Signup and view all the answers

What is indicated if the inequality $2x + 2 < 1$ is solved?

The variable 'x' is greater than a specific value Signup and view all the answers

When manipulating literal equations, what must be done if the unknown variable is in the denominator?

Multiply both sides by the lowest common denominator Signup and view all the answers

Which part of the problem-solving strategy emphasizes checking the solution for correctness?

Verifying the final answers Signup and view all the answers

What describes the y-intercept of the hyperbolic function defined as $y = \frac{a}{x} + q$?

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

What is the effect of increasing the value of $q$ in the hyperbolic function $y = \frac{a}{x} + q$ when $a > 0$?

The graph shifts vertically upwards. Signup and view all the answers

Which equations describe the asymptotes of the hyperbolic function $y = \frac{a}{x} + q$?

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

What can be determined about the graph of the function $f(x) = ax^2 + q$ if $a < 0$?

The maximum turning point is located at $(0, q)$. Signup and view all the answers

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

The range is always above $q$ if $a > 0$. Signup and view all the answers

Which statement accurately describes the domain of hyperbolic functions of the form $y = \frac{a}{x} + q$?

Domain excludes $x = 0$. Signup and view all the answers

What can be inferred about the graph of the function $y = ax^2 + q$ if $a > 0$?

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

Which of the following statements is true about the axes of symmetry for the hyperbolic function $y = \frac{a}{x} + q$?

The axes of symmetry are the lines $y = x + q$ and $y = -x + q$. Signup and view all the answers

What is the primary factor affecting the shape of the graph for the function $y = \frac{a}{x} + q$?

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

If the coefficient 'm' in the equation $y = mx + c$ is negative, which of the following must be true about the y-intercept?

It can be any real number. Signup and view all the answers

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

The graph will open upward. Signup and view all the answers

In the equation $y = mx + c$, which factor primarily affects the steepness of the line?

The absolute value of m. Signup and view all the answers

What determines whether a parabola is a 'smile' or a 'frown'?

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

If both 'm' and 'c' are zero in the equation $y = mx + c$, what is the nature of the resulting line?

It is the point at the origin only. Signup and view all the answers

For the parabola defined by $y = ax^2 + q$, what happens if 'q' is negative?

The parabola shifts downward. Signup and view all the answers

What is the range of the function defined by $y = ax^2 + q$ when 'a' is less than zero?

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

When using the dual intercept method to sketch a line, which two values are essential to determine?

y-intercept and x-intercept. Signup and view all the answers

What is the vertical shift of the graph when the value of q is -3?

Shift downwards by 3 units Signup and view all the answers

In relation to the function $f(x) = mx + c$, which statements about domain and range are accurate?

Domain and range are both all real numbers. Signup and view all the answers

Which of the following describes the effect of a negative value of a on the graph of the function y = a sin θ + q?

Reflection about the x-axis Signup and view all the answers

What is the range of the function y = 2 sin θ - 1?

[-3; 1] Signup and view all the answers

If a parabolic function has a positive 'q' and a negative 'a', what is the nature of its vertex?

It is at its maximum point. Signup and view all the answers

If the function y = -3b^x + 2 has an asymptote, what is the equation of the asymptote?

y = 2 Signup and view all the answers

For an exponential function to experience growth, which of the following conditions must hold true?

a > 0 and b > 1 Signup and view all the answers

What is the domain of the function y = tan θ?

θ ≠ 90° + k * 180° for k ∈ Z Signup and view all the answers

When comparing the sine and cosine functions, what is the maximum turning point for y = cos θ?

(0°, 1) Signup and view all the answers

In what scenario does the value of b affect the graph of a sine function?

It determines the period of the function. Signup and view all the answers

What are the x-intercepts for the function y = a cos θ + q?

(0°, 0), (180°, 0), (360°, 0) Signup and view all the answers

How does a positive value of 'a' affect the shape of a parabola?

The parabola opens upwards. Signup and view all the answers

What determines the vertical shift of a hyperbola?

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

What do the asymptotes of a hyperbola indicate?

Lines the graph approaches but never touches. Signup and view all the answers

In the function of a tangent graph, what happens when 'q' is negative?

The graph shifts down. Signup and view all the answers

To find the y-intercept of a trigonometric function like $y = a an heta + q$, which angle should you use?

$ heta = 0°$ Signup and view all the answers

What method is used to determine the equation of a hyperbola using given points?

Substitute the points into the hyperbola equation and solve simultaneously. Signup and view all the answers

What is the significance of setting the denominator to zero in analyzing the graph of functions?

It reveals the location of asymptotes. Signup and view all the answers

For trigonometric functions, which factor significantly modifies the amplitude of the graph?

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

What determines the domain of the tangent function?

All angles except at points where the cosine is zero. Signup and view all the answers

Gr 10 Math June P1 Mix

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Questions and Answers

Which set includes all positive integers starting from 1?

What is the definition of irrational numbers?

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

What does the symbol Z represent in the real number system?

Which of the following sets includes zero and all positive integers?

Which statement accurately defines imaginary numbers?

What is the result of applying the zero exponent law?

Which of the following accurately represents the multiplication of exponents with the same base?

How do you simplify the expression (ab)^{2}?

Which operation will you use to solve the equation a^{x} = a^{3}?

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

Which property applies if you have the expression rac{a^{1/2}}{a^{1/3}}?

How can you convert a square root into a rational exponent?

What is the correct way to factor the expression a^{2} - b^{2}?

What should be done first when faced with a complex fraction?

What is a binomial?

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

How can a quadratic trinomial of the form ( ax^2 + bx + c ) be factorised?

What is the result of multiplying a monomial by a binomial ( a(x + y) )?

Which of the following identities is used for factorising a difference of two squares?

When simplifying $ rac{a}{b} imes rac{c}{d}$, which statement is correct?

What is the first step in simplifying algebraic fractions?

In the process of factorisation, what does factoring by grouping entail?

Which expression represents the sum of two cubes?

What is the solution to a system of simultaneous linear equations when solved graphically?

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

What is a literal equation?

How does the value of 'm' affect the graph of the function $y = mx + c$?

What must be done to solve an inequality if both sides involve division by a negative number?

What happens to the graph of a linear function if the value of 'c' is less than 0?

In solving literal equations, what should be done if the unknown variable is in two or more terms?

When solving a linear inequality, what do you do if the inequality involves a multiplication by a negative number?

What is needed to effectively tackle word problems involving simultaneous equations?

If 'm' is greater than 0 in the function $y = mx + c$, how does the graph behave?

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

What is the purpose of checking for extraneous solutions?

Which step comes first when solving linear equations?

How many solutions can a linear equation have?

What is a quadratic equation characterized by?

In the factorization method for quadratic equations, what form must the equation take?

How are simultaneous equations typically solved?

What method can be used to reduce the number of equations in simultaneous equations?

What is a necessary step after solving for a variable in quadratic equations?

What should always be performed on both sides of an equation?

Which of the following is true about irrational numbers?

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

Which of the following statements about rounding off is correct?

How can you identify a surd?

Which of the following correctly describes rational numbers?

What is the characteristic of terminating decimals?

What is a key step in estimating surds?

Which of the following represents a rational number?

What happens when you round 9.999 to one decimal place?

Which expression is a term in mathematics?

What happens to the graph of an exponential function when the value of q is positive?

In the function of the form y = a sin θ + q, how does a affect the graph?

Which of the following statements about the sine function is incorrect?

For the function y = a cos θ + q, what does a < 0 indicate?

What defines the asymptote of an exponential function in the form y = ab^x + q?

What is the domain of the tangent function y = tan θ?

When the value of b in the exponential function is between 0 and 1, what type of function does it represent?

What is the range of the sine function y = a sin θ + q for a > 0?

What is the standard form of a straight-line graph?

How can the sine and cosine graphs be related?

How can the y-intercept be calculated for the equation y = mx + c?

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

What happens to the graph of the cosine function with a positive q?

Which statement about the range of the function f(x) = mx + c is true?

What effect does 'q' have on the graph of a parabola in the equation y = ax^2 + q?

For which condition will a parabola have a maximum turning point?

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

If a parabola opens downwards, what can be inferred about the value of 'a'?

When plotting a straight-line graph, which points are typically used?

What indicates the steepness of a line in the equation y = mx + c?

What is the range of the function when $a < 0$?

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

When simplifying $rac{a}{b} imes rac{c}{d}$, which statement is correct?