Gr 10 Math Term 1 Test

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Which of the following sets includes the number -3?

Whole Numbers

Natural Numbers

Integers (correct)

Rational Numbers (correct)

Which number can be classified as an irrational number?

0.5

π (correct)

-4

2

What is the definition of Whole Numbers?

Numbers that can be expressed as a fraction

Only positive integers

All integers including negatives

All natural numbers and zero (correct)

What is the main characteristic of Rational Numbers?

They can be expressed as a fraction of two integers Signup and view all the answers

Which of the following is NOT a subset of the real number system?

Imaginary Numbers Signup and view all the answers

Which statement accurately describes the Real Numbers?

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

Which of the following is a characteristic of rational numbers?

They can be expressed as a fraction. Signup and view all the answers

What is a feature of irrational numbers?

They have non-repeating decimal forms. Signup and view all the answers

Which of the following scenarios correctly describes the rounding of the number 5.678 to one decimal place?

It becomes 5.7. Signup and view all the answers

Identify the incorrect statement about surds:

Surds can be expressed as perfect squares. Signup and view all the answers

Which of these is a correct method to convert a recurring decimal into a rational number?

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

What defines an exponential term in mathematics?

It includes a base and a power. Signup and view all the answers

When estimating the value of a surd like $, \sqrt{50}$, what would be the key initial step?

Identify perfect squares around $50$. Signup and view all the answers

Which example represents a terminating decimal?

0.75 Signup and view all the answers

Which choice correctly describes the relationship between rational and irrational numbers?

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

What kind of number is represented by the expression $\sqrt{-5}$?

Imaginary number Signup and view all the answers

What is a binomial?

An expression with two terms Signup and view all the answers

What is the result of multiplying the binomials (2x + 3)(4x + 5)?

8x^2 + 22x + 15 Signup and view all the answers

How do you factor the expression $x^2 - 9$?

$(x - 3)(x + 3)$ Signup and view all the answers

What does a trinomial consist of?

Three terms Signup and view all the answers

What is the correct formula to multiply a monomial by a binomial?

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

To simplify the fraction ( \frac{2x^2 + 6x}{2x} ), what is the first step?

Factorise the numerator and then cancel Signup and view all the answers

What is the formula for 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 final result of multiplying $3(x + 4) + 2(x + 5)$?

$5x + 23$ Signup and view all the answers

In the operation ( \frac{a}{b} \div \frac{c}{d} ), what is the equivalent multiplication?

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

What is the first step when solving an exponential equation using logarithms?

Isolate the exponent Signup and view all the answers

Which method is NOT typically used when solving quadratic equations?

Using logarithms 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 step is NOT part of solving simultaneous equations?

Isolate variables separately in each equation Signup and view all the answers

What is the procedure to verify a solution for a linear equation?

Substitute the solution back into the original equation Signup and view all the answers

When solving quadratic equations, what is the form they should be rewritten into?

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

What should you do if an equation is not balanced?

Balance by performing the same operation on both sides Signup and view all the answers

What is a possible outcome for a quadratic equation aside from two solutions?

No solutions Signup and view all the answers

How can you simplify the process of solving simultaneous equations?

Substitute values from one equation into the other Signup and view all the answers

Which step comes after expanding all brackets in a linear equation?

Group like terms together Signup and view all the answers

What is the initial step in solving word problems mathematically?

Read the whole question. Signup and view all the answers

When using substitution to solve simultaneous equations, what is the first action taken?

Substitute the value of one variable into the second equation. Signup and view all the answers

What must be done when rearranging literal equations to solve for one variable?

Isolate the unknown variable by performing opposite operations. Signup and view all the answers

How can one determine the solution of a system of simultaneous equations graphically?

By finding the coordinates where the lines intersect. Signup and view all the answers

What effect does dividing or multiplying both sides of an inequality by a negative number have?

The inequality sign switches direction. Signup and view all the answers

Which of the following represents a linear inequality?

3x + 5 > 12 Signup and view all the answers

What is the primary objective when solving equations derived from word problems?

To identify the unknown quantities clearly. Signup and view all the answers

In the context of similar triangles, what is true about their corresponding sides?

Their sides are proportionate regardless of their size. Signup and view all the answers

What process should be followed after finding the value of one variable in simultaneous equations?

Substitute the value back to find the other variable. Signup and view all the answers

What is a necessary step when solving linear inequalities?

Combine all like terms before isolating the variable. Signup and view all the answers

What is the result of simplifying the expression $rac{a^5 imes a^3}{a^4}$?

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

Which of the following expressions represents $(3x^2y^3)^4$ when simplified using the exponent laws?

$3^4 x^8 y^{12}$ Signup and view all the answers

How can the expression $a^{-3}$ be rewritten using positive exponents?

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

When simplifying the complex fraction $rac{rac{a^2}{b}}{rac{b^2}{a}}$, what is the result?

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

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

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

To solve the equation $5^x = 125$, which is the correct method?

$x = 3$ Signup and view all the answers

What is the result of the expression $(a^{1/2}b^{-3})^2$?

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

Which expression correctly shows the factorization of $x^2 - 9$?

$(x - 3)(x + 3)$ Signup and view all the answers

When simplifying $(2^3 imes 2^4)/(2^5)$, what is the final result?

$2^{6}$ Signup and view all the answers

How do you express $rac{rac{a^4}{b^2}}{rac{a^2}{b}}$ in simplified form?

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

What is the value of angle $B$ formed between line $OQ$ and the positive x-axis if $\alpha$ is $56.3^ heta$?

123.7° Signup and view all the answers

In which quadrant are all trigonometric ratios positive according to the CAST diagram?

Quadrant I Signup and view all the answers

What is the tangent value for an angle of $30^ heta$?

$\frac{1}{\sqrt{3}}$ Signup and view all the answers

What is the value of $\sin 90^ heta$?

1 Signup and view all the answers

What is the cosecant value for an angle where $\sin \theta = \frac{1}{2}$?

2 Signup and view all the answers

Which of the following ratios represents cosine?

$\frac{x}{r}$ Signup and view all the answers

For which angle is the tangent undefined?

$90^ heta$ Signup and view all the answers

What is the radius $r$ in relation to angle $ heta$ if the opposite side is $3$ and the adjacent side is $2$?

$\sqrt{13}$ Signup and view all the answers

What is the value of sin 30°?

1/2 Signup and view all the answers

What is the tangent of 45°?

1 Signup and view all the answers

When using a scientific calculator, why is it important to set it to 'degrees' mode?

To ensure accuracy in angle measurements. Signup and view all the answers

For the 45°-45°-90° triangle, what is the ratio of the lengths of the sides?

1 : 1 : √2 Signup and view all the answers

What values must the sine function be limited to for valid angles?

|sin θ| ≤ 1 Signup and view all the answers

Which of the following ratios does not represent a reciprocal relationship among the trigonometric functions?

cos θ = 1/sin θ Signup and view all the answers

What is the cosine of 60°?

1/2 Signup and view all the answers

What value corresponds to tan 30°?

√3/3 Signup and view all the answers

What can be used to find unknown angles in right-angled triangles?

Trigonometric ratios Signup and view all the answers

If you plot the point P(2, 3) on the Cartesian plane, what is one way to determine angles relative to the x-axis?

By using the coordinates to determine sine and cosine ratios. Signup and view all the answers

What remains constant in similar triangles regardless of size?

The ratio of their corresponding sides Signup and view all the answers

Which relationship is true for trigonometric ratios in a right triangle?

The hypotenuse is always the longest side. Signup and view all the answers

What is the correct mnemonic for remembering the trigonometric ratios?

Soh Cah Toe Signup and view all the answers

How is the cosecant ratio defined in terms of sine?

cosec $ heta = rac{1}{ ext{sin} heta}$ Signup and view all the answers

Which of the following corresponds angles in triangles $\Delta ABC$ and $\Delta DEF$?

$\angle B$ corresponds to $\angle E$ Signup and view all the answers

Which statement about the ratios of similar triangles is false?

The ratios can vary based on the angles. Signup and view all the answers

What defines the tangent of an angle in a right triangle?

$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ Signup and view all the answers

Which equation correctly represents a reciprocal trigonometric identity?

$\tan \theta \times \text{cot } \theta = 1$ Signup and view all the answers

What is the value of $\sin 90°$?

1 Signup and view all the answers

If $\sin \theta = 0.6$, what is the value of $\text{cosec } \theta$?

1.67 Signup and view all the answers

Which of the following sets includes the number 0?

Whole Numbers (N0) Signup and view all the answers

Irrational numbers are best described as which of the following?

Numbers that have non-repeating and non-terminating decimal expansions Signup and view all the answers

What differentiates whole numbers from natural numbers?

Whole numbers include zero, whereas natural numbers do not. Signup and view all the answers

Which of the following correctly states a property of integers?

Integers can be both positive and negative including zero. Signup and view all the answers

Which statement accurately describes rational numbers?

They can be expressed as a fraction where the denominator is not zero. Signup and view all the answers

What set of numbers includes all possible real numbers?

Real Numbers (R) Signup and view all the answers

Which of the following best describes the term 'irrational numbers'?

They are numbers with non-repeating, non-terminating decimal expansions. Signup and view all the answers

What is a necessary consideration when converting a terminating decimal into a rational number?

Each decimal position is represented by a power of 10. Signup and view all the answers

When estimating surds, what is the first essential step?

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

Which of these numbers is a valid example of a rational number?

$0.666...$ Signup and view all the answers

What defines the process of rounding a number?

The digit following the last significant digit determines whether to round up or down. Signup and view all the answers

Which of the following is true regarding non-terminating decimals?

They can sometimes represent surds if they do not repeat. Signup and view all the answers

What is a key characteristic of integers when compared to rational numbers?

Rational numbers include integers and fractions. Signup and view all the answers

When assessing the value of $rac{3}{8}$, which of the following best describes this number?

A terminating decimal and a rational number. Signup and view all the answers

Which method is effective in converting a recurring decimal to a rational number?

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

What is the first step in solving an exponential equation using the method of equating exponents?

Ensure both sides of the equation have the same base. Signup and view all the answers

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

They can have at most one solution. Signup and view all the answers

When solving quadratic equations, what is the method typically used?

Factorization of the expression into two binomials. Signup and view all the answers

Which step is NOT necessary when checking for extraneous solutions in exponential equations?

Rewriting the original equation. Signup and view all the answers

What is the result of multiplying the binomials $(3x + 4)(2x + 5)$?

$6x^2 + 22x + 20$ 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 of the following statements best describes the method for factorising a quadratic trinomial of the form $ax^2 + bx + c$?

Find two binomials whose product matches the original trinomial. Signup and view all the answers

What is the first action to take when solving simultaneous equations by substitution?

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

What must be ensured when manipulating both sides of an equation?

The equation must always be balanced. Signup and view all the answers

When multiplying a monomial $a$ with a binomial $(x + y)$, what is the resulting expression?

$ax + ay$ Signup and view all the answers

Which expression correctly represents the difference of two cubes $x^3 - y^3$ when factored?

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

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

Not all solutions derived algebraically are valid. Signup and view all the answers

What is the step following the factorisation of both the numerator and denominator in simplifying algebraic fractions?

Cancel any common factors. Signup and view all the answers

In what order should the steps of solving linear equations be followed?

Expand, rearrange, group, factorize, then solve. Signup and view all the answers

Which of the following describes a trinomial?

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

How do you factor the expression $x^2 - 16$ using the difference of squares?

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

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

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

Which statement describes the product of multiplying two binomials $(ax + b)(cx + d)$?

It results in a quadratic expression. Signup and view all the answers

To simplify $rac{a}{b} imes rac{c}{d}$, which of the following represents the correct result?

$rac{ac}{bd}$ where $b = 0$ and $d = 0$ Signup and view all the answers

What is the relationship between the sides of similar triangles?

The ratios of corresponding sides are equal regardless of triangle size. Signup and view all the answers

Which angles correspond to each other in the similar triangles ΔABC and ΔDEF?

A corresponds to D. Signup and view all the answers

Which of the following correctly defines the sine of an angle in a right-angled triangle?

$rac{ ext{opposite}}{ ext{hypotenuse}}$ Signup and view all the answers

What is the reciprocal of the cosine ratio?

secant Signup and view all the answers

Which of the following pairs of trigonometric ratios correctly represents a fundamental relationship?

sin θ × cosec θ = 1 Signup and view all the answers

In the mnemonic 'Soh Cah Toa', what does 'Cah' represent?

cos θ = adjacent/hypotenuse Signup and view all the answers

What is the tangent of an angle defined as?

$rac{ ext{opposite}}{ ext{adjacent}}$ Signup and view all the answers

Which of the following correctly states the relationship between sine and cosecant?

cosec θ = 1/sin θ Signup and view all the answers

When calculating $ ext{sec } 34°$, what type of ratio is expected?

Ratio of hypotenuse to adjacent Signup and view all the answers

Which trigonometric ratio is equal to 1 for a 45° angle?

tan 45° Signup and view all the answers

What is the hypotenuse length of a 30°-60°-90° triangle?

2 Signup and view all the answers

If sin θ is 1/2, which angle could θ represent?

30° Signup and view all the answers

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

Use Pythagorean theorem Signup and view all the answers

In what mode should a scientific calculator be set to calculate trigonometric ratios accurately?

Degrees Signup and view all the answers

What is the value of cos 60°?

1/2 Signup and view all the answers

To find an unknown angle in a right triangle when two sides are known, which ratio can be used?

Trigonometric ratio Signup and view all the answers

Which triangle has side lengths of 1, 1, and √2?

45°-45°-90° triangle Signup and view all the answers

What is sin² θ equal to in terms of sin θ?

(sin θ)² Signup and view all the answers

What must be true for the sine function in terms of its values?

|sin θ| ≤ 1 Signup and view all the answers

What is the first step in solving a word problem using equations?

Read the whole question. Signup and view all the answers

When using the elimination method to solve simultaneous equations, what is done after making the coefficients the same?

Subtract the equations to eliminate the variable. Signup and view all the answers

How can the solution of simultaneous equations be visually represented?

By plotting the equations and finding their intersection point. Signup and view all the answers

What is an essential principle when rearranging literal equations to solve for one variable?

Performing the opposite operation to isolate the unknown variable. Signup and view all the answers

What should you do after finding the value of one variable in simultaneous equations?

Substitute it back into one of the original equations. Signup and view all the answers

Which of the following is a characteristic of linear inequalities compared to linear equations?

They involve operations that change the inequality sign. Signup and view all the answers

In the context of similar triangles, what relationship exists between corresponding sides?

They are proportional to each other based on angle similarity. Signup and view all the answers

What is involved when using substitution to solve simultaneous equations for the first time?

Replacing one variable with an expression derived from the other equation. Signup and view all the answers

What is the role of checking the solution of an equation?

To verify that the found solution satisfies all original equations. Signup and view all the answers

What type of equations have terms that contain two or more variables?

Literal equations. Signup and view all the answers

What is the result of applying the law of exponents for multiplication, specifically $a^3 \times a^4$?

$a^{7}$ Signup and view all the answers

Which of the following is the correct application of the zero exponent rule?

$a^0 = 1$ for any $a \neq 0$ Signup and view all the answers

Which expression correctly represents the division of the same base according to the laws of exponents?

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

Which step is NOT part of simplifying a complex fraction?

Convert all bases to their prime factors Signup and view all the answers

When simplifying the expression $\frac{(ab)^{2}}{b^{2}}$, what is the simplified form?

$a^2$ Signup and view all the answers

Which of the following expressions represents $(\frac{2}{3})^{3/2}$ when simplified?

$\frac{8}{9}$ Signup and view all the answers

What is the result of simplifying the expression $\frac{a^{5} \times a^{3}}{a^{4}}$?

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

Which expression correctly illustrates the raising of a quotient to a power?

$\left(\frac{a}{b}\right)^{n} = \frac{a^n}{b^n}$ Signup and view all the answers

To simplify the expression $a^{-3}$, what is the final result using positive exponents?

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

In solving the exponential equation $2^x = 8$, which statement is true?

$x = 3$ Signup and view all the answers

What is the angle B formed between line OQ and the positive x-axis if α is 56.3°?

123.7° Signup and view all the answers

Which quadrant has only cosine and secant as positive trigonometric ratios?

Quadrant IV Signup and view all the answers

What is the value of tan 0°?

0 Signup and view all the answers

If r is the radius of a circle centered at the origin, which formula represents sin θ?

sin θ = y / r Signup and view all the answers

At what angle does tan θ equal 1?

45° Signup and view all the answers

Which trigonometric function is undefined at 90°?

tan θ Signup and view all the answers

How many trigonometric ratios are positive in Quadrant III?

2 Signup and view all the answers

What is the sine value of 60°?

√3/2 Signup and view all the answers

Which statement accurately describes natural numbers?

Natural numbers are always greater than or equal to one. Signup and view all the answers

Which of the following sets contains only integers?

The set of whole numbers. Signup and view all the answers

What distinguishes irrational numbers from rational numbers?

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

Which of the following is a characteristic of the set of rational numbers?

They can be expressed as a ratio of two integers. Signup and view all the answers

Which of the following examples represents a real number?

The number π. Signup and view all the answers

Which statement about imaginary numbers is correct?

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

Which statement accurately describes a characteristic of irrational numbers?

Irrational numbers can be expressed as surds. Signup and view all the answers

What is the main step involved in converting a repeating decimal to a rational number?

Align the decimals by multiplying by an appropriate power of 10. Signup and view all the answers

When rounding the number 7.845 to two decimal places, what will be the result?

7.85 Signup and view all the answers

Which example represents a non-terminating, non-repeating decimal?

0.1010010001... Signup and view all the answers

Which of the following statements about surds is true?

A surd is the root of a whole number that cannot be simplified to a rational number. Signup and view all the answers

What is the correct relationship between rational and irrational numbers regarding their decimal representations?

All rational numbers are represented by cyclic decimals. Signup and view all the answers

When estimating the value of a surd such as $rac{3}{2} imes rac{4}{5}$, which is the most appropriate method to find its bounds using perfect powers?

Identify and calculate the perfect square roots surrounding the decimal. Signup and view all the answers

Which of the following transformations is necessary when rounding 9.999 to the nearest whole number?

Round up to 10. Signup and view all the answers

When expressing a rational number as a decimal, which characteristic must it fulfill?

It will either terminate or repeat in its decimal form. Signup and view all the answers

What method allows you to solve an exponential equation when the bases can be made identical?

Equating exponents Signup and view all the answers

Which equation represents a quadratic equation?

x^2 - 4x + 4 = 0 Signup and view all the answers

Which step must be completed before factoring a quadratic equation?

Express in standard form Signup and view all the answers

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

Two Signup and view all the answers

Which method can be used to check the validity of solutions found for an equation?

Substituting back into the original equation Signup and view all the answers

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

Isolate one variable in one equation Signup and view all the answers

How many independent equations are needed to solve for three variables?

Three Signup and view all the answers

What is the result of applying the zero exponent law to the expression $a^0$?

1 Signup and view all the answers

What could be a possible result when solving a quadratic equation?

One real solution or no solutions Signup and view all the answers

When simplifying the expression $rac{a^{5/2}}{a^{1/2}}$, what is the resulting exponent?

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

What does the expression $(3x^2y^3)^4$ simplify to using the laws of exponents?

$81x^8y^{12}$ Signup and view all the answers

What must be done to ensure an equation remains balanced after an operation?

Apply the operation to both sides Signup and view all the answers

Which of the following correctly represents the result of $(a^{-3})$ when rewritten using positive exponents?

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

In solving a linear equation, what is the purpose of grouping like terms?

To simplify the equation for easier solving Signup and view all the answers

When simplifying the expression $\frac{(x^2y)^{3}}{xy^{2}}$, what is the final result?

$x^{5}y^{1}$ Signup and view all the answers

What is the first step in simplifying the complex fraction $\frac{\frac{1}{x}}{\frac{2}{x^2}}$?

Rewrite as $\frac{1}{x} \cdot \frac{x^2}{2}$ Signup and view all the answers

When factoring the expression $a^4 - b^4$, which formula is applicable?

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

What principles apply when solving an exponential equation $a^x = a^y$?

If $a > 0$ and $a \neq 1$, then $x = y$ Signup and view all the answers

Which expression is equivalent to $(\frac{a}{b})^{m/n}$?

$\frac{a^{m/n}}{b^{m/n}}$ Signup and view all the answers

Which expression represents the result of multiplying a monomial by a binomial?

$a(x + y)$ Signup and view all the answers

What is the correct result of expanding the binomials $(3x + 2)(5x + 4)$?

$15x^2 + 22x + 8$ Signup and view all the answers

Which of the following identities is used to factor the difference of two squares?

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

Which describes the result of the expression $(x + 3)(x^2 - 2x + 1)$ when fully expanded?

$x^3 + 3x^2 - 2x^2 + 3x + 3$ Signup and view all the answers

To factor the trinomial $2x^2 + 3x - 2$, which pair of factors is considered?

$(2, -1)$ and $(1, -2)$ Signup and view all the answers

What is the result of simplifying the expression $\frac{6x^2 + 12x}{6x}$?

$x + 2$ Signup and view all the answers

Which property holds when dividing fractions expressed as $\frac{a}{b} ÷ \frac{c}{d}$?

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

What is the correct procedure for factorizing the expression $x^2 + 5x + 6$?

List factors of 6 and find pairs that add to 5: $(x + 2)(x + 3)$ Signup and view all the answers

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

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

What is the primary method used when solving linear inequalities?

Apply the same operations as in linear equations, adjusting for negative multipliers. Signup and view all the answers

What is essential to establish when solving word problems mathematically?

Translate the problem into a set of equations representing the situation. Signup and view all the answers

Which of the following describes the approach to solve literal equations?

Rearrange the equation to isolate the unknown variable. Signup and view all the answers

When solving simultaneous equations using elimination, what is the correct initial step?

Make the coefficients of one variable the same in both equations. Signup and view all the answers

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

The coordinates of the point at which the two graphs intersect. Signup and view all the answers

What conclusion can be drawn from the principle of similar triangles?

Their corresponding sides are proportional irrespective of triangle size. Signup and view all the answers

Which operation must be performed carefully when dealing with inequalities?

Dividing both sides by a negative number. Signup and view all the answers

What is the role of substitution in solving simultaneous equations?

Replace one variable in the equation with an expression involving the other variable. Signup and view all the answers

When solving an equation involving the square root, what is an important factor to remember?

The solution may yield both positive and negative results. Signup and view all the answers

What is often the first step taken when applying the problem-solving strategy to a word problem?

Read the whole question thoroughly. Signup and view all the answers

What is the value of angle B when the tangent ratio is given as $\tan A = \frac{3}{2}$?

123.7° Signup and view all the answers

Which trigonometric function is undefined at $90^ullet$?

tan 90° Signup and view all the answers

In which quadrant are both tangent and cotangent positive according to the CAST diagram?

Quadrant III Signup and view all the answers

If point P has coordinates (2, 3), what is the radius r when determining the angle formed by OP?

√13 Signup and view all the answers

What is the sine value for an angle measuring 30°?

0.5 Signup and view all the answers

Which ratio is represented by $\tan \theta$ in terms of x and y?

$\frac{y}{x}$ Signup and view all the answers

Which trigonometric function is positive only in Quadrant IV?

cos θ Signup and view all the answers

Given the values for special angles, what is $\cos 60°$?

$\frac{1}{2}$ Signup and view all the answers

What is the relationship between the sides of two similar triangles, such as $\Delta ABC$ and $\Delta DEF$?

The ratios of their corresponding sides are always equal. Signup and view all the answers

Which of the following correctly defines the sine of an angle $\theta$ in a right-angled triangle?

$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ Signup and view all the answers

Which of the following statements about trigonometric ratios is accurate?

Trigonometric ratios relate specific relationships of the triangle's sides. Signup and view all the answers

What is the cosecant of an angle $\theta$ defined as?

$\text{cosec } \theta = \frac{\text{hypotenuse}}{\text{opposite}}$ Signup and view all the answers

If two triangles are similar and one has a side length of 10 units with its corresponding side in the other triangle being 5 units, what is the ratio of their sides?

2:1 Signup and view all the answers

What does the mnemonic 'Soh Cah Toa' help in remembering?

The basic trigonometric ratios. Signup and view all the answers

Which equation correctly expresses the relationship between tangent and cotangent?

$\cot \theta = \frac{1}{\tan \theta}$ Signup and view all the answers

Which of the following statements about angles in similar triangles is true?

Corresponding angles remain equal in similar triangles. Signup and view all the answers

What must be true for triangles $\Delta ABC$ and $\Delta DEF$ to be considered similar?

They must have equal corresponding angles. Signup and view all the answers

Which relationship is correct based on the reciprocal ratios?

$\sin \theta \times \csc \theta = 1$ Signup and view all the answers

What is the value of $ an 30^ ext{o}$?

$rac{ oot{3}}{3}$ Signup and view all the answers

In a 45°-45°-90° triangle, what is the length of the hypotenuse if each leg length is 1?

$√2$ Signup and view all the answers

Which of the following statements is NOT true regarding trigonometric ratios?

Inverse trigonometric functions can return angles outside of defined ranges Signup and view all the answers

For which angle does the relationship $ an heta = rac{ ext{opposite}}{ ext{adjacent}}$ fail when calculating using side lengths?

90 Signup and view all the answers

What is the angle corresponding to the trigonometric ratio $rac{ oot{3}}{2}$ for cosine?

60 Signup and view all the answers

Which trigonometric ratio is correctly represented by the information given for the special angle triangles?

cos 30° = $rac{ oot{3}}{2}$ Signup and view all the answers

If the length of the opposite side in a right triangle is 3 and the hypotenuse is 5, what is $ heta$?

$ ext{sin}^{-1}(3/5)$ Signup and view all the answers

In the Cartesian plane, what is true about the sine function for an angle plotted in the first quadrant?

Its maximum value is 1 Signup and view all the answers

If a calculator is set to radians, how should the angle be adjusted when calculating the sine value for 30°?

Convert it to $rac{ ext{pi}}{6}$ Signup and view all the answers

To find the length of a side using trigonometric ratios, which method is typically used?

Isolate the desired ratio and rearrange the equation Signup and view all the answers

Which statement accurately differentiates between rational and irrational numbers?

Rational numbers can be expressed as fractions of integers, while irrational numbers cannot. Signup and view all the answers

What is a distinguishing feature of integers within the real number system?

They encompass whole numbers and their negative counterparts. Signup and view all the answers

Identify the number that is not classified as a natural number.

-2 Signup and view all the answers

How do imaginary numbers differ from real numbers?

Imaginary numbers have a negative square root. Signup and view all the answers

Which of the following sets contains only whole numbers?

Whole numbers Signup and view all the answers

Which characteristic correctly defines irrational numbers?

They cannot be expressed as simple fractions of two integers. Signup and view all the answers

Which of the following decimal representations corresponds to a rational number?

0.200200200... Signup and view all the answers

What is the outcome when rounding the number 2.4567 to two decimal places?

2.46 Signup and view all the answers

In estimating the surd $, ext{sqrt}(20)$, which perfect squares are closest to 20?

16 and 25 Signup and view all the answers

Which of the following statements accurately describes irrational numbers?

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

When converting the recurring decimal 0.666... to a rational number, what is the resulting fraction?

$\frac{2}{3}$ Signup and view all the answers

Which operation is performed first when rounding off the decimal 8.934 to one decimal place?

Look at the second decimal digit. Signup and view all the answers

What is the correct method to simplify the term $rac{12x^2y}{4xy^2}$?

$3x^{1}y^{-1}$ Signup and view all the answers

Which number is classified as a surd?

$ ext{sqrt}(3)$ Signup and view all the answers

Which of the following best describes the term 'coefficient' in mathematics?

The numeral factor in a term. Signup and view all the answers

Which expression represents the form of an irrational number?

$ ext{sqrt}(7)$ Signup and view all the answers

What result do you obtain when applying the law of exponents to simplify the expression $a^{5/2} imes a^{3/4}$?

$a^{11/4}$ Signup and view all the answers

How do you express $\sqrt{b^5}$ using rational exponents?

$b^{5/2}$ Signup and view all the answers

If $\frac{a^{6}}{a^{-2}}$ is simplified, what would the result be?

$a^{8}$ Signup and view all the answers

Which expression correctly represents the power of a power rule applied to $(x^2)^{3}$?

$x^6$ Signup and view all the answers

When simplifying the expression $\left(\frac{3}{4}\right)^{-2}$, what is the resulting value?

$\frac{16}{9}$ Signup and view all the answers

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

At most two solutions Signup and view all the answers

Which of the following denotes the correct simplification of the expression $\frac{(ab)^{2}}{a^{4}b^{3}}$?

$\frac{1}{a^{2}b}$ Signup and view all the answers

What simplification results from the expression $\left(\frac{m^2}{n^3}\right)^{1/2}$?

$\frac{m^{1/2}}{n^{3/2}}$ Signup and view all the answers

Which of the following steps is not typically involved in solving a quadratic equation?

Set the exponents equal Signup and view all the answers

If $a^3 = b^3$, what conclusion can we draw about the relationship between $a$ and $b$ given the condition that $a > 0$ and $b > 0$?

$a = b$ Signup and view all the answers

When solving a linear equation, what must be done after expanding all brackets?

Rearrange the terms Signup and view all the answers

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

Using logarithms Signup and view all the answers

What is the result when simplifying the expression $a^{2/3}b^{-4/3} \cdot a^{-1/3}b^{1/3}$?

$a^{1/3}b^{-5/3}$ Signup and view all the answers

How many independent equations are needed to solve for three unknown variables in simultaneous equations?

Three equations Signup and view all the answers

What happens to the solutions of a simultaneous equation if one variable is expressed in terms of another?

One variable is eliminated Signup and view all the answers

Which of the following statements about exponents in exponential equations is incorrect?

Using logarithms is unnecessary if bases match Signup and view all the answers

What should be done after obtaining the solutions from factored quadratic expressions?

Check the solutions Signup and view all the answers

In what situation will a quadratic equation have only one solution?

When the discriminant is zero Signup and view all the answers

What is the importance of verifying solutions in linear equations?

To confirm that both sides balance Signup and view all the answers

What conclusion can be drawn about the ratios of the sides of similar triangles?

They remain constant regardless of triangle size. Signup and view all the answers

Which of the following is NOT one of the trigonometric ratios defined?

Quotient Signup and view all the answers

What is the significance of the order of the vertices in similar triangles?

It shows which sides are corresponding to each other. Signup and view all the answers

Which of the following accurately represents the relationship defined by trigonometric ratios?

The cosine ratio equals the sine ratio when the angle is 45°. Signup and view all the answers

What is a key property of the angles in similar triangles?

The corresponding angles are equal. Signup and view all the answers

Which reciprocal ratio corresponds to $ an heta$?

Cotangent Signup and view all the answers

In the context of right-angled triangles, which side is always opposite to the right angle?

Hypotenuse Signup and view all the answers

What mnemonic helps to remember the definitions of the trigonometric ratios?

Soh Cah Toa Signup and view all the answers

What is established by the relationship $ an heta imes ext{cot } heta = 1 $?

It defines the reciprocal relationship between these functions. Signup and view all the answers

What does the sine ratio depend on in a right-angled triangle?

The length of the opposite side and the hypotenuse. Signup and view all the answers

What is the product of the binomials $(2x + 5)(3x - 2)$?

$6x^2 + 7x - 10$ Signup and view all the answers

Which equation represents the factorization of the expression $x^2 - 16$?

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

When multiplying the monomial $3x^2$ by the binomial $(x + 4)$, what is the result?

$3x^3 + 12$ Signup and view all the answers

Which of the following expresses the sum of two cubes $x^3 + 27$ in factored form?

$(x + 3)(x^2 + 3x + 9)$ Signup and view all the answers

What is the outcome of simplifying the expression $rac{x^2 - 4}{x - 2}$?

$x + 2$ for $x eq 2$ Signup and view all the answers

Which expression correctly factors the quadratic trinomial $x^2 + 5x + 6$?

$(x + 2)(x + 3)$ Signup and view all the answers

What is the result of multiplying two binomials $(x + 2)(x + 3)$?

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

Which operation correctly summarizes the process of simplifying the fraction $rac{2x^2 + 10x}{2x}$?

Factor out $2x$ to simplify to $x + 5$ Signup and view all the answers

If $x^3 - 8$ is factored using the difference of cubes formula, what is the result?

$(x - 2)(x^2 + 2x + 4)$ Signup and view all the answers

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

Substitute one variable from one equation into the other. Signup and view all the answers

When using elimination to solve a system of equations, what is the purpose of making the coefficients of one variable the same?

To eliminate that variable and solve for the other. Signup and view all the answers

What characteristic is true for the sides of similar triangles?

They are proportional to each other. Signup and view all the answers

In solving literal equations, what action is taken when the unknown variable is part of two terms?

You take it out as a common factor. Signup and view all the answers

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

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

What is the last step to ensure the correctness of your solution in word problems?

Substitute the solution back into the equations. Signup and view all the answers

Which method is a common graphical approach to solve simultaneous equations?

By graphing both equations and identifying their intersection point. Signup and view all the answers

What is the primary objective when rearranging a literal equation?

To express one variable in terms of others. Signup and view all the answers

What common mistake should be avoided when solving linear inequalities?

Assuming the inequality sign doesn’t change. Signup and view all the answers

When writing a set of equations to solve a word problem, what initial step is crucial?

Assign letters to unknown values in the problem. Signup and view all the answers

What is the sine value for a 45° angle in a right-angled triangle?

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

Which equation can be solved using the cosine ratio for finding an angle in a right triangle?

$rac{adjacent}{hypotenuse}$ Signup and view all the answers

In the context of a 30°-60°-90° triangle, what is the length of the side opposite the 60° angle?

$ ext{√3}$ Signup and view all the answers

When calculating the tangent of an angle, which of the following ratios is used?

$rac{opposite}{adjacent}$ Signup and view all the answers

If the cosine of an angle is 0.5, which of the following angles could it represent?

30° Signup and view all the answers

Which of the following cannot be a value for the sine function of an angle?

$2$ Signup and view all the answers

For a triangle with known lengths adjacent to an angle, which step is involved in finding the angle using trigonometric ratios?

Using inverse trigonometric functions Signup and view all the answers

What is the hypotenuse length in a 30°-60°-90° triangle?

2 Signup and view all the answers

What is the value of angle B if tan α = 3/2?

123.7° Signup and view all the answers

Which point in the Cartesian plane would correspond to an angle of approximately 56.31°?

(2, 3) Signup and view all the answers

Which trigonometric ratio is not positive in Quadrant II?

cos θ Signup and view all the answers

What is the rule that must be followed when finding the unknown lengths in right-angled triangles?

Pythagorean Theorem: Hypotenuse² = Opposite² + Adjacent² Signup and view all the answers

What is the sine value for 60° according to the special angle table?

√3/2 Signup and view all the answers

How is tan θ defined geometrically?

opposite length divided by adjacent length Signup and view all the answers

For the point Q(-2, 3), what is the calculated length from the origin O(0, 0) to point Q?

√13 Signup and view all the answers

What is the cosine value for 30° according to the special angle definitions?

√3/2 Signup and view all the answers

In the context of the CAST diagram, which quadrant has positive tan θ?

Quadrant III Signup and view all the answers

If α = 56.3°, what is the corresponding cotangent value based on its definition?

√3 Signup and view all the answers

Gr 10 Math Term 1 Test

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Questions and Answers

Which of the following sets includes the number -3?

Which number can be classified as an irrational number?

What is the definition of Whole Numbers?

What is the main characteristic of Rational Numbers?

Which of the following is NOT a subset of the real number system?

Which statement accurately describes the Real Numbers?

Which of the following is a characteristic of rational numbers?

What is a feature of irrational numbers?

Which of the following scenarios correctly describes the rounding of the number 5.678 to one decimal place?

Identify the incorrect statement about surds:

Which of these is a correct method to convert a recurring decimal into a rational number?

What defines an exponential term in mathematics?

When estimating the value of a surd like $, \sqrt{50}$, what would be the key initial step?

Which example represents a terminating decimal?

Which choice correctly describes the relationship between rational and irrational numbers?

What kind of number is represented by the expression $\sqrt{-5}$?

What is a binomial?

What is the result of multiplying the binomials (2x + 3)(4x + 5)?

How do you factor the expression $x^2 - 9$?

What does a trinomial consist of?

What is the correct formula to multiply a monomial by a binomial?

To simplify the fraction ( \frac{2x^2 + 6x}{2x} ), what is the first step?

What is the formula for the sum of two cubes?

What is the final result of multiplying $3(x + 4) + 2(x + 5)$?

In the operation ( \frac{a}{b} \div \frac{c}{d} ), what is the equivalent multiplication?

What is the first step when solving an exponential equation using logarithms?

Which method is NOT typically used when solving quadratic equations?

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

Which step is NOT part of solving simultaneous equations?

What is the procedure to verify a solution for a linear equation?

When solving quadratic equations, what is the form they should be rewritten into?

What should you do if an equation is not balanced?

What is a possible outcome for a quadratic equation aside from two solutions?

How can you simplify the process of solving simultaneous equations?

Which step comes after expanding all brackets in a linear equation?

What is the initial step in solving word problems mathematically?

When using substitution to solve simultaneous equations, what is the first action taken?

What must be done when rearranging literal equations to solve for one variable?

How can one determine the solution of a system of simultaneous equations graphically?

What effect does dividing or multiplying both sides of an inequality by a negative number have?

Which of the following represents a linear inequality?

What is the primary objective when solving equations derived from word problems?

In the context of similar triangles, what is true about their corresponding sides?

What process should be followed after finding the value of one variable in simultaneous equations?

What is a necessary step when solving linear inequalities?

What is the result of simplifying the expression $ rac{a^5 imes a^3}{a^4}$?

Which of the following expressions represents $(3x^2y^3)^4$ when simplified using the exponent laws?

How can the expression $a^{-3}$ be rewritten using positive exponents?

When simplifying the complex fraction $ rac{ rac{a^2}{b}}{ rac{b^2}{a}}$, what is the result?

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

To solve the equation $5^x = 125$, which is the correct method?

What is the result of the expression $(a^{1/2}b^{-3})^2$?

Which expression correctly shows the factorization of $x^2 - 9$?

When simplifying $(2^3 imes 2^4)/(2^5)$, what is the final result?

How do you express $ rac{ rac{a^4}{b^2}}{ rac{a^2}{b}}$ in simplified form?

What is the value of angle $B$ formed between line $OQ$ and the positive x-axis if $\alpha$ is $56.3^ heta$?

In which quadrant are all trigonometric ratios positive according to the CAST diagram?

What is the tangent value for an angle of $30^ heta$?

What is the value of $\sin 90^ heta$?

What is the cosecant value for an angle where $\sin \theta = \frac{1}{2}$?

Which of the following ratios represents cosine?

For which angle is the tangent undefined?

What is the radius $r$ in relation to angle $ heta$ if the opposite side is $3$ and the adjacent side is $2$?

What is the value of sin 30°?

What is the tangent of 45°?

When using a scientific calculator, why is it important to set it to 'degrees' mode?

For the 45°-45°-90° triangle, what is the ratio of the lengths of the sides?

What values must the sine function be limited to for valid angles?

Which of the following ratios does not represent a reciprocal relationship among the trigonometric functions?

What is the cosine of 60°?

What value corresponds to tan 30°?

What can be used to find unknown angles in right-angled triangles?

If you plot the point P(2, 3) on the Cartesian plane, what is one way to determine angles relative to the x-axis?

What remains constant in similar triangles regardless of size?

Which relationship is true for trigonometric ratios in a right triangle?

What is the correct mnemonic for remembering the trigonometric ratios?

What is the result of simplifying the expression $rac{a^5 imes a^3}{a^4}$?

When simplifying the complex fraction $rac{rac{a^2}{b}}{rac{b^2}{a}}$, what is the result?

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

How do you express $rac{rac{a^4}{b^2}}{rac{a^2}{b}}$ in simplified form?

When assessing the value of $rac{3}{8}$, which of the following best describes this number?

To simplify $rac{a}{b} imes rac{c}{d}$, which of the following represents the correct result?