Gr 10 Math Nov P2 Hard

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In Quadrant II, which trigonometric ratios are positive?

sin θ and cosec θ (correct)

sin θ and tan θ

sec θ and tan θ

cos θ and cot θ

What is the value of tan 90°?

undefined (correct)

1

infinity

0

For an isosceles triangle, which statement is true?

It must be a right-angled triangle.

It has three equal angles.

It has two equal sides. (correct)

It has no equal sides.

According to the CAST diagram, which trigonometric functions are positive in Quadrant III?

tan θ and cot θ Signup and view all the answers

What is the relationship between the angle of elevation and angle of depression?

They are always equal. Signup and view all the answers

When using the sine ratio for a right triangle, how is it defined?

Opposite over hypotenuse Signup and view all the answers

If the interior angles of a triangle sum to 180°, what can be said about its exterior angles?

The sum of the two opposite interior angles equals the exterior angle. Signup and view all the answers

Which of the following statements is true regarding the Pythagorean theorem?

It is used only in right-angled triangles. Signup and view all the answers

In terms of trigonometric ratios, how is cotangent defined?

cos θ / sin θ Signup and view all the answers

What type of triangle has all interior angles less than 90°?

Acute triangle Signup and view all the answers

In a 30°-60°-90° triangle, what is the length of the hypotenuse if the length of the side opposite the 30° angle is 2?

4 Signup and view all the answers

What is the sine value of a 60° angle?

√3/2 Signup and view all the answers

When finding the angle in a triangle using the ratio tan A = 3/2, what will be the approximate measure of angle A?

56.3° Signup and view all the answers

For a point Q located at (-2, 3) on the Cartesian plane, what is the correct description of angle B formed between line OQ and the positive x-axis?

It is equal to 180° - 56.3°. Signup and view all the answers

In a 45°-45°-90° triangle, what is the ratio of the lengths of the sides to the hypotenuse?

1:√2 Signup and view all the answers

What is the tangent value of a 45° angle?

1 Signup and view all the answers

In solving a trigonometric equation, which step is NOT involved?

Draw a new triangle Signup and view all the answers

Considering the properties of sine and cosine, what is a valid rule regarding their values?

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

If the lengths of two sides of a right triangle are known, which function can be used to find the angle opposite one of the known sides?

Inverse trigonometric function Signup and view all the answers

What is the characteristic property of the sides in a 30°-60°-90° triangle related to the angles?

The side opposite 30° is the smallest side. Signup and view all the answers

Which ratio correctly represents the definition of cosecant?

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

What is the result of multiplying sine and cosecant for any angle $\theta$?

1 Signup and view all the answers

If the value of $\tan \theta$ is 3, what is the approximate value of $\cot \theta$?

0.33 Signup and view all the answers

To calculate secant on a scientific calculator, which adjustment must be made first?

Set the calculator to degrees mode Signup and view all the answers

What is the defining formula for tangent in a right-angled triangle?

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

Which of the following is NOT a reciprocal trigonometric ratio?

tan $\theta$ Signup and view all the answers

For which angle is $\sin \theta = $1?

90° Signup and view all the answers

What is the cosine of an angle if its adjacent side length equals 4 and the hypotenuse length equals 5?

$\frac{4}{5}$ Signup and view all the answers

What is a valid mnemonic for remembering the trigonometric ratios?

Soh Cah Toa Signup and view all the answers

Which relationship between the ratios is incorrect?

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

Which of the following is NOT a property of a rectangle?

All four sides must be equal in length. Signup and view all the answers

Which statement accurately describes the differences between a rhombus and a rectangle?

A rhombus has all sides equal, while a rectangle has right angles. Signup and view all the answers

What condition does NOT apply to two triangles being considered similar?

All three sides must be of equal length. Signup and view all the answers

Which relationship is true regarding the hierarchy of quadrilaterals?

A square is a type of rhombus. Signup and view all the answers

Which properties are shared by both a parallelogram and a kite?

Both have at least one pair of parallel sides. Signup and view all the answers

Which condition indicates that a triangle is right-angled based on the Pythagorean theorem?

The equality $b^2 = a^2 + c^2$ holds true. Signup and view all the answers

Which of the following accurately describes a trapezium?

A trapezium has only one pair of opposite sides parallel. Signup and view all the answers

Which type of quadrilateral has diagonals that bisect each other at 90°?

Square Signup and view all the answers

In the context of triangle congruence, AAS indicates which combination of equalities?

Two angles and the non-included side. Signup and view all the answers

Which of the following must be true for a quadrilateral to be classified as a parallelogram?

Both pairs of opposite angles must be equal. Signup and view all the answers

What specific property does the line segment joining the mid-points of two sides of a triangle possess according to the mid-point theorem?

It is parallel to the third side. Signup and view all the answers

In a parallelogram, if opposite sides are equal and parallel, which property can be used to conclude that the opposite angles are also equal?

Properties of Parallel Lines Signup and view all the answers

What conclusion can be made about a quadrilateral if both pairs of opposite angles are equal?

It is a parallelogram. Signup and view all the answers

If the mid-point theorem is applied to triangle ABC, and points D and E are the mid-points of sides AB and AC respectively, what can be inferred about line segment DE?

It is half the length of BC and parallel to it. Signup and view all the answers

Which of the following statements about kites is true?

A kite has two pairs of adjacent sides that are equal. Signup and view all the answers

In analytical geometry, when plotting quadrilateral ABCD on the Cartesian plane, what is the correct way to label it?

ABCD only Signup and view all the answers

What is the purpose of the mid-point theorem in coordinate geometry?

To calculate mid-points and distances. Signup and view all the answers

Which of the following correctly describes the characteristics of a trapezium?

A quadrilateral with one pair of parallel sides. Signup and view all the answers

What condition must be met for two triangles to be considered similar?

Their corresponding angles must be equal. Signup and view all the answers

In the context of distance between two points, which formula is used to calculate the distance between points A(x₁, y₁) and B(x₂, y₂)?

$ ext{Distance} = ext{the square root of }{((x_2-x_1)^2 + (y_2-y_1)^2)}$ Signup and view all the answers

For similar triangles, which statement about the ratios of their corresponding sides is true?

The ratios remain constant regardless of size. Signup and view all the answers

In similar triangles, if the ratio of one side in triangle A to the corresponding side in triangle B is 2:3, what can be said about another pair of corresponding sides?

They also have a ratio of 2:3. Signup and view all the answers

When investigating the mid-point theorem, what does shifting triangle ADE onto quadrilateral BDEC illustrate?

Variance in lengths DE and BC. Signup and view all the answers

How does the order of vertices affect the identification of similar triangles?

It indicates which angles correspond to one another. Signup and view all the answers

If triangle ABC is similar to triangle DEF, what can be concluded about angles A, B, and C?

They are equal to angles D, E, and F respectively. Signup and view all the answers

Which of the following ratios represents the relationship of the sides in two similar triangles?

$rac{AB}{DE} = rac{BC}{EF}$ Signup and view all the answers

What term is used to describe the value that occurs most frequently in a data set?

Mode Signup and view all the answers

When three similar triangles are compared, which of the following properties regarding their angles must hold true?

The sum of the angles must be equal to 180°. Signup and view all the answers

If two triangles have sides in the ratio 5:12 and the angle between them is 90°, what is true about their similarity?

They are similar as they maintain proportion despite different sizes. Signup and view all the answers

How is the median determined in a data set?

It's the value at the center after sorting. Signup and view all the answers

What does the mean of a data set represent?

The sum of the values divided by their count. Signup and view all the answers

If a data set has an even number of values, how is the median calculated?

It is the average of the two central values. Signup and view all the answers

An outlier in a data set is best described as which of the following?

A value that is significantly higher or lower than others. Signup and view all the answers

Which of the following statements is true regarding the mode?

A data set can have multiple modes or none at all. Signup and view all the answers

What characteristic distinguishes qualitative data from quantitative data?

Qualitative data are based on categories and cannot be measured numerically. Signup and view all the answers

When calculating the mean of a set of values, which variable denotes the number of values?

n Signup and view all the answers

In which situation is it appropriate to use the median rather than the mean?

When the data set contains extreme values. Signup and view all the answers

How do you determine the value of the percentile when the rank is not an integer?

Average the two values indicated by the rank. Signup and view all the answers

What defines the interquartile range (IQR)?

The difference between the first quartile (Q1) and third quartile (Q3). Signup and view all the answers

Which values are included in the five number summary?

0th, 25th, 50th, 75th, and 100th percentiles. Signup and view all the answers

What is a characteristic of a right prism?

Its side faces are perpendicular to the base. Signup and view all the answers

What represents the spread of data in a box-and-whisker plot?

The interquartile range. Signup and view all the answers

How is the area of a trapezium calculated?

Area = ½ × (a + b) × height. Signup and view all the answers

What is the purpose of calculating percentiles for grouped data?

To ascertain the range of values where the percentile lies. Signup and view all the answers

Which formula would you use to calculate the area of a rectangle?

Area = height × base. Signup and view all the answers

What does the term 'deciles' refer to in statistics?

The nine values that divide a dataset into ten equal parts. Signup and view all the answers

What is the correct formula to determine the gradient of a line connecting two points A(x₁, y₁) and B(x₂, y₂)?

$m = rac{y₂ - y₁}{x₂ - x₁}$ Signup and view all the answers

Which equation represents the standard form of a straight line?

$y = mx + c$ Signup and view all the answers

Which of the following statements about parallel lines is true?

They have the same slope but different y-intercepts. Signup and view all the answers

What is the mid-point M(x, y) between points A(x₁, y₁) and B(x₂, y₂)?

$M ext{(x, y)} = ig(rac{x₁ + x₂}{2}, rac{y₁ + y₂}{2}ig)$ Signup and view all the answers

Which of the following best describes an undefined gradient?

A line that runs parallel to the y-axis. Signup and view all the answers

For two perpendicular lines, what is true about their gradients?

The product of their gradients equals -1. Signup and view all the answers

Which type of data is classified as discrete?

The number of cars in a parking lot. Signup and view all the answers

In the context of distance formula, what does the expression $ ext{distance (d)} = ext{d} = ackslash sqrt{(x₁ - x₂)^2 + (y₁ - y₂)^2}$ represent?

The straight-line distance between two points in a coordinate system. Signup and view all the answers

How can you prove that two points are collinear?

By checking if the slopes between them are equal. Signup and view all the answers

What term describes the data that is observed and recorded from an experiment or survey?

Data Signup and view all the answers

What is true regarding the mode of a data set?

If all values occur with equal frequency, it is correct to say the data set has no mode. Signup and view all the answers

Which statement accurately defines an outlier in a data set?

An outlier is a value that is significantly different from other values. Signup and view all the answers

What is the primary purpose of grouping continuous data?

To convert it into discrete data for easier analysis. Signup and view all the answers

When calculating the mean of grouped data, what is assumed about the values within each class interval?

All values are located at the midpoint of the interval. Signup and view all the answers

What defines the modal group in a data set?

The class interval with the highest frequency of data values. Signup and view all the answers

Which statement about the range of a data set is correct?

The range is defined as the difference between the maximum and minimum values. Signup and view all the answers

How is the pth percentile of a data set calculated?

It is computed using the formula $r = \frac{p}{100} (n - 1) + 1$. Signup and view all the answers

Which statement best describes measures of dispersion?

They provide information about the distribution of values around the central tendency. Signup and view all the answers

In a data set with continuous values, what will happen to the mode calculation?

The mode is typically not computable due to unique values. Signup and view all the answers

What does the median group represent in grouped data?

The central group that contains the median value. Signup and view all the answers

What happens to the volume of a rectangular prism when all of its dimensions are multiplied by 2?

It is multiplied by 8 Signup and view all the answers

If the height of a cylinder is doubled while keeping the radius constant, how is the surface area affected?

It is multiplied by 2 Signup and view all the answers

When only one dimension of a rectangular prism is multiplied by 5, what factor is the surface area changed by?

More than 5, but less than 10 Signup and view all the answers

Which expression correctly represents the volume of a cylinder after its radius is multiplied by 3?

$9V$ Signup and view all the answers

If all dimensions of a rectangular prism are multiplied by 2 and then by 3, what is the total factor by which the volume is increased?

24 Signup and view all the answers

What factor does the surface area of a rectangular prism increase when two of its dimensions are multiplied by 2?

4 Signup and view all the answers

How does the volume of a rectangular prism change if one dimension is halved?

It is halved Signup and view all the answers

When multiplying all dimensions of a cylinder by a constant $k$, what is the new volume expressed in terms of the original volume?

$k^3V$ Signup and view all the answers

What defines the surface area of a triangular prism?

The sum of the areas of two triangles and three rectangles Signup and view all the answers

How is the volume of a right cone calculated?

$\frac{1}{3} \pi r^2 h$ Signup and view all the answers

Which statement correctly describes a cube?

It is a rectangular prism with all sides of equal length. Signup and view all the answers

What is the formula for the surface area of a sphere?

$4\pi r^2$ Signup and view all the answers

When a dimension of a prism is multiplied by a factor of $k$, how does this affect the surface area?

The surface area increases by a factor of $k^2$. Signup and view all the answers

What is the volume of a rectangular prism using its defined dimensions?

$\text{length} \times \text{width} \times \text{height}$ Signup and view all the answers

What distinguishes a right pyramid from a regular pyramid?

The apex of the right pyramid is perpendicular to the center of the base. Signup and view all the answers

Which formula is used to calculate the surface area of a right cone?

$\pi r (r + h)$ Signup and view all the answers

How is the volume of a triangular pyramid determined?

$\frac{1}{3} \times \text{Area of base} \times ext{Height}$ Signup and view all the answers

Which of the following statements correctly describes the relationship between the angles of similar triangles?

The angles of similar triangles are equal. Signup and view all the answers

If triangle ABC is similar to triangle DEF, which of the following must be true about their corresponding sides?

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

In a triangle where the angles are 30°, 60°, and 90°, what is the relationship between the lengths of the sides?

The side opposite the 30° angle is half the length of the hypotenuse. Signup and view all the answers

When labeling similar triangles, why is the order of the vertices important?

It establishes the correspondence between angles and sides. Signup and view all the answers

Which of the following sets of ratios is consistent with the property of similar triangles?

$rac{AB}{BC} = rac{DE}{EF} = rac{GH}{HK}$ Signup and view all the answers

Which of the following is NOT a property of similar triangles?

The perimeter of similar triangles is always the same. Signup and view all the answers

What does the notation $ riangle ABC ext{ similar to } riangle DEF$ imply?

The angles correspond: $A$ to $D$, $B$ to $E$, and $C$ to $F$. Signup and view all the answers

If triangles ABC and DEF are similar, which of the following ratio expressions is correct?

$rac{AB}{DE} = rac{BC}{EF}$ Signup and view all the answers

Which trigonometric ratio represents the relationship between the opposite side and the hypotenuse in a right triangle?

sin θ Signup and view all the answers

In which quadrant are both sine and cosine negative?

Quadrant III Signup and view all the answers

When calculating the cosecant of angle θ, which sides of the triangle are used?

Hypotenuse and opposite Signup and view all the answers

If the sine of an angle is $\frac{1}{2}$, what is the angle in degrees?

30° Signup and view all the answers

What is true about the angles of elevation and depression from the same point?

They are equal in measure. Signup and view all the answers

In terms of interior angles, which type of triangle has one angle greater than 90°?

Obtuse triangle Signup and view all the answers

Which statement correctly describes the relationship of the interior and exterior angles of a triangle?

An exterior angle is equal to the sum of the two opposite interior angles. Signup and view all the answers

In right-angled triangles, if the hypotenuse is denoted as 'r', which statement holds true about the adjacent side 'x' when θ is an acute angle?

x < r Signup and view all the answers

What can be concluded about the values of tan θ when θ is in Quadrant II?

tan θ is negative. Signup and view all the answers

If a triangle is defined as obtuse and isosceles, which property must it have?

Two sides are of equal length, and one angle is greater than 90°. Signup and view all the answers

In a 30°-60°-90° triangle, if the length of the hypotenuse is 10, what is the length of the side opposite the 60° angle?

5√3 Signup and view all the answers

What is the value of the cosine of 45°?

√2/2 Signup and view all the answers

If the tangent of an angle is 1.5, what is the ratio of the opposite to the adjacent sides in the corresponding right triangle?

3/2 Signup and view all the answers

What angle corresponds to a sine value of 0.866?

60° Signup and view all the answers

For the point P(4, 3) in the Cartesian plane, what is the tangent of the angle formed between line OP and the positive x-axis?

4/3 Signup and view all the answers

In which scenario might the equation ext{sine} θ = 2/1 have a solution in the context of trigonometric functions?

There is no solution Signup and view all the answers

If the opposite side of a 30°-60°-90° triangle measures 5, what is the length of the hypotenuse?

10 Signup and view all the answers

Which inverse trigonometric function would you use to find an angle θ if tan θ = 0.5?

tan^-1 Signup and view all the answers

What is the cosine of an angle if the length of the adjacent side is 6 and the hypotenuse is 10?

6/10 Signup and view all the answers

In triangle congruence, which condition is sufficient to declare two triangles congruent?

Three sides are equal. Signup and view all the answers

Which property is not true for a rhombus?

Diagonals are equal in length. Signup and view all the answers

Which trigonometric ratio corresponds to the ratio of the opposite side to the adjacent side in a right-angled triangle?

Tangent Signup and view all the answers

What does the cosecant of an angle represent in terms of the triangle's sides?

The ratio of the hypotenuse to the opposite side Signup and view all the answers

Which statement about a square is incorrect?

A square has opposite sides not parallel. Signup and view all the answers

Which of the following statements accurately reflects the conditions for triangle similarity?

All corresponding angles are equal, and sides are in proportion. Signup and view all the answers

The relationship $ an heta imes ext{cot } heta = 1$ signifies what about these two ratios?

They are reciprocals of each other. Signup and view all the answers

When calculating the value of secant on a scientific calculator, which operation needs to be converted first?

The secant function must be converted to cosine. Signup and view all the answers

Which quadrilateral has diagonals that are perpendicular to each other?

Rhombus Signup and view all the answers

What is the characteristic of a trapezium?

It has one pair of parallel sides. Signup and view all the answers

In the mnemonic 'Soh Cah Toa', what does 'Cah' refer to?

Cosine = adjacent/hypotenuse Signup and view all the answers

Which of the following statements is the correct representation of the sine function?

sin θ = opposite/hypotenuse Signup and view all the answers

What does the notation $ b^2 = a^2 + c^2 $ signify?

It represents the Pythagorean theorem for right-angled triangles. Signup and view all the answers

How do the special angles relate to calculating values of sine, cosine, and tangent?

They yield simple, known ratios. Signup and view all the answers

What condition is required for two triangles to be declared congruent using the SAS congruence rule?

Two sides and the included angle are equal. Signup and view all the answers

What is the value of $rac{1}{ an heta}$ in terms of sine and cosine?

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

Which property distinguishes a rectangle from a parallelogram?

All angles are equal to 90°. Signup and view all the answers

If the sides of a right-angled triangle are in the ratio 3:4:5, what can be said about this triangle?

It is a right-angled triangle. Signup and view all the answers

What does the relationship $ ext{sin}^2 heta + ext{cos}^2 heta = 1$ signify in the context of trigonometry?

It defines an identity in trigonometry. Signup and view all the answers

What can be concluded about the line segment connecting the mid-points of two sides of a triangle?

It is equal to the half-length of the third side and parallel. Signup and view all the answers

When applying the mid-point theorem, what does the converse state?

If a line through the mid-point is parallel to one side, it bisects the opposite side. Signup and view all the answers

In the context of quadrilaterals, which of the following statements is true regarding kites?

Kites have two pairs of adjacent sides that are equal. Signup and view all the answers

What is a consequence of the properties of parallelogram ABCD in proving that quadrilateral MNOP is also a parallelogram?

Both pairs of opposite angles are equal. Signup and view all the answers

Which application of the mid-point theorem is most commonly used in geometric proofs?

To prove segment proportions and parallelism in figures. Signup and view all the answers

Which property accurately describes trapeziums compared to other quadrilaterals?

They have one pair of parallel sides. Signup and view all the answers

When labeling figures in analytical geometry, which aspect is crucial for accuracy?

Maintaining alphabetical order of the vertices. Signup and view all the answers

For the coordinates A(1, 1), B(3, 1), C(3, 3), and D(1, 3), what type of quadrilateral is formed?

A rectangle. Signup and view all the answers

Which formula is derived from the Pythagorean theorem to find the distance between two points?

$d = ext{√}((x₁ - x₂)² + (y₁ - y₂)²)$ Signup and view all the answers

What distinguishes qualitative data from quantitative data?

Qualitative data cannot be written as numbers. Signup and view all the answers

How is the median calculated in a data set with an even number of values?

It is the average of the two middle values. Signup and view all the answers

What does the gradient of a line represent?

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

Which of the following equations represents a line with a gradient of zero?

y = 5 Signup and view all the answers

What defines an outlier in a data set?

It is a value that significantly differs from other values. Signup and view all the answers

Which statement accurately defines the mode of a data set?

The most frequently occurring value in the data set. Signup and view all the answers

If two lines are perpendicular, what can be concluded about their gradients?

The product of their gradients equals -1 Signup and view all the answers

Which of the following formulas represents the mean of a set of values?

$ar{x} = rac{x_1 + x_2 + ... + x_n}{n}$ Signup and view all the answers

To find the mid-point M(x, y) of points A(x₁, y₁) and B(x₂, y₂), which formula should be used?

M(x, y) = (rac{x₁ + x₂}{2}, rac{y₁ + y₂}{2}) Signup and view all the answers

If a line is vertical, what statement is true regarding its gradient?

It has an undefined gradient Signup and view all the answers

In what case can a data set have more than one mode?

When two or more values have the highest frequency. Signup and view all the answers

If a data set has values of {2, 3, 5, 7, 9}, what is the median?

5 Signup and view all the answers

Which statement correctly describes collinear points?

They lie on the same straight line Signup and view all the answers

What is required to prove that two points A(x₁, y₁) and B(x₂, y₂) are on the same line?

The slope calculations using both points must be identical Signup and view all the answers

What is a common characteristic of anecdotal data?

It originates from personal stories or interviews. Signup and view all the answers

Which term refers to the sum of a set of values divided by their count?

Mean Signup and view all the answers

In a coordinate plane, which form represents the standard equation of a straight line?

y = mx + c Signup and view all the answers

What type of data is represented by the number of objects or counts?

Discrete quantitative data Signup and view all the answers

If the coordinates of the points are A(1,3) and B(4,7), what is the gradient of the line joining these points?

2 Signup and view all the answers

What is indicated by having multiple modes in a data set?

At least two values appear with the same maximum frequency. Signup and view all the answers

Which statement best describes an outlier in a data set?

It is a value that significantly differs from other values. Signup and view all the answers

What happens to the calculation of the mean when data is grouped into intervals?

It is calculated using the midpoints of each interval. Signup and view all the answers

Which best describes the role of percentiles in a data set?

They divide the data set into two parts based on a fixed percentage. Signup and view all the answers

What is a critical characteristic of the range as a measure of dispersion?

It is sensitive to outliers. Signup and view all the answers

How is the median group defined in a set of grouped data?

It is the central group of values when ordered. Signup and view all the answers

What is the primary limitation when calculating statistics on grouped data?

Grouping results in loss of precision of measured values. Signup and view all the answers

In a frequency distribution, how is the modal group identified?

As the class interval with the highest frequency. Signup and view all the answers

Which is NOT a requirement for defining a class interval for grouping continuous data?

The total range of data must be ignored. Signup and view all the answers

When the height of a cylinder is multiplied by a factor of 3, how does this affect its volume?

The volume is multiplied by 3. Signup and view all the answers

What happens to the volume of a rectangular prism if only its width is multiplied by 5?

Volume becomes 5V. Signup and view all the answers

If the radius of a cylinder is increased by a factor of 2 and the height remains unchanged, what is the factor by which the volume increases?

4 Signup and view all the answers

If the surface area of a rectangular prism is initially A, what will it be if its height is multiplied by 3 and the other dimensions are unchanged?

3A Signup and view all the answers

How does multiplying the length and breadth of a rectangular prism by 2 while keeping height unchanged affect its volume?

Volume becomes 4V. Signup and view all the answers

For a cylinder, when both the radius and the height are multiplied by a factor of k, what is the new surface area expressed in terms of A?

k^2A Signup and view all the answers

In terms of volume, what is the effect of multiplying all dimensions of a rectangular prism by a factor of k?

Volume multiplies by k^3. Signup and view all the answers

What represents the second quartile in an ordered data set?

The median (50th percentile) Signup and view all the answers

When calculating the interquartile range, which values are used?

Q1 and Q3 Signup and view all the answers

How do you determine the range in which a percentile lies for grouped data?

Apply the percentile formula to determine the corresponding rank Signup and view all the answers

Which statement accurately describes the relationship between the five-number summary and the box-and-whisker plot?

They display the same information in different formats Signup and view all the answers

What term describes half of the interquartile range?

Semi-interquartile range Signup and view all the answers

Which percentile is also known as the first or lower quartile?

25th percentile Signup and view all the answers

In a box-and-whisker plot, what does the box represent?

The interquartile range Signup and view all the answers

Which information is NOT included in the five-number summary?

Mean value Signup and view all the answers

Which definition characterizes a right prism?

It has a polygon as its base with vertical faces perpendicular to the base Signup and view all the answers

What is the combined area formula for a trapezium given the bases and height?

Area = ½ × (a + b) × h Signup and view all the answers

What is the surface area formula for a triangular prism?

2 times area of triangle + perimeter of triangle times height Signup and view all the answers

Which of the following statements accurately describes the relationship between volume and dimensions of a prism when one dimension is multiplied by a factor of k?

Volume increases by $k^3$ Signup and view all the answers

What is the surface area of a right cone expressed as a formula?

πr^2 + πrh Signup and view all the answers

How is the volume of a square pyramid calculated?

Base area multiplied by height divided by 3 Signup and view all the answers

What defines a 'right pyramid'?

Base is any polygon and apex is perpendicular to the center of the base Signup and view all the answers

For a sphere, what is the formula for its surface area?

4πr^2 Signup and view all the answers

If the height of a cylinder is doubled, how does this affect its volume?

Increases by a factor of 2 Signup and view all the answers

In a triangular pyramid, which formula is used to calculate the volume?

$\frac{1}{3} \times \text{Area of base} \times \text{Height}$ Signup and view all the answers

Which geometric shape is defined by having a perfectly round surface?

Sphere Signup and view all the answers

How does the surface area change when each dimension of a cube is multiplied by a factor of k?

Surface area increases by $k^2$ Signup and view all the answers

In similar triangles, how do the ratios of corresponding sides behave?

They are equal regardless of the triangle sizes. Signup and view all the answers

Which of the following properties must be true for triangles to be considered similar?

Their ratios of side lengths must be equal. Signup and view all the answers

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

It defines which angles and sides correspond to each other. Signup and view all the answers

If triangles \Delta ABC and \Delta DEF are similar, what can be inferred about their angles?

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

What is the relationship demonstrated by the ratios \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}?

They illustrate a proportional relationship between corresponding sides. Signup and view all the answers

Which of the following statements accurately describes the ratios of the sides in a 30°-60°-90° triangle?

The ratio is consistent and equal among all triangles of this type. Signup and view all the answers

When drawing similar triangles, which measurements ensure their similarity?

Identical angles and proportional side lengths. Signup and view all the answers

Regarding the angles in a set of similar triangles, which statement is true?

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

What is the main property of the line segment joining the mid-points of two sides of a triangle according to the Mid-Point Theorem?

It is equal to half the length of the third side. Signup and view all the answers

In proving that a quadrilateral is a parallelogram, which property is NOT explicitly required?

Both pairs of opposite angles must be equal. Signup and view all the answers

What is the cosecant of an angle if the hypotenuse is 10 and the opposite side is 6?

10/6 Signup and view all the answers

Which application of the Mid-Point Theorem is essential for geometric proofs?

Proving similarity in geometric figures. Signup and view all the answers

In which quadrant is cosine negative and sine positive?

Quadrant II Signup and view all the answers

What is the value of $ an 30^ heta$ expressed in simplest form?

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

When labeling a quadrilateral in the Cartesian plane as ABCD, which order of coordinates represents a clockwise movement starting from point A?

(1, 1), (3, 1), (3, 3), (1, 3) Signup and view all the answers

Which of the following statements is true regarding the side lengths of a 45°-45°-90° triangle?

The hypotenuse is $\sqrt{2}$ times the length of each leg. Signup and view all the answers

Which of the following statements is NOT a property of kites?

Their diagonals are of equal length. Signup and view all the answers

What is the relationship between the angles of elevation and depression?

Angles of elevation and depression are supplementary. Signup and view all the answers

In the given conditions of parallelogram ABCD, which equation directly leads to proving that Mˆ2 = Oˆ2?

DCZˆ = BAXˆ and Mˆ1 = Mˆ2 Signup and view all the answers

Which triangle classification correctly describes a triangle with one angle measuring 120°?

Obtuse Signup and view all the answers

Which geometric figure is defined as having only one pair of parallel sides?

Trapezium Signup and view all the answers

What fundamental concept is used to derive the distance formula between two points in analytical geometry?

The theorem of Pythagoras Signup and view all the answers

If two triangles are congruent, which of the following properties must hold true?

Their corresponding angles and sides must be equal. Signup and view all the answers

In the context of triangle congruency, which condition must be proven to apply the Angle-Angle-Side (AAS) theorem?

Two angles and the non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle. Signup and view all the answers

What is the exterior angle of a triangle if its opposite interior angles are 40° and 60°?

100° Signup and view all the answers

If the sine of an angle is known to be $\frac{1}{2}$, which of the following angles could it represent?

30° Signup and view all the answers

Which of the following angles does NOT exist as a practical measurement in standard trigonometry?

180° Signup and view all the answers

What does the gradient of a line indicate in relation to two points?

The constant rate of change between the x-values and y-values of the points Signup and view all the answers

If two lines are perpendicular, what can be inferred about their gradients?

The gradients are negative reciprocals of each other. Signup and view all the answers

When calculating the mid-point of a line segment between points A(x₁, y₁) and B(x₂, y₂), which statement is true?

The mid-point coordinates are the averages of the x-values and y-values. Signup and view all the answers

Which of the following correctly represents the equation of a straight line in slope-intercept form?

y = mx + b Signup and view all the answers

In which scenario would the gradient of a line be undefined?

When the line is vertical Signup and view all the answers

What characteristic defines quantitative data?

Information that involves counting and can be expressed numerically. Signup and view all the answers

What is the main purpose of using the distance formula in coordinate geometry?

To calculate the straight-line distance between two points. Signup and view all the answers

Which statement is true regarding horizontal lines in terms of their gradient?

They have a gradient of zero. Signup and view all the answers

If point C is defined at coordinates (2, 5) and point D at (2, -3), what can be determined about the line segment CD?

It is a straight line with an undefined slope. Signup and view all the answers

When analyzing the concept of collinear points, what can be inferred?

They are connected by a straight line and share a constant gradient. Signup and view all the answers

Which of the following trigonometric ratios uses the adjacent and hypotenuse sides of a right-angled triangle?

Cosine Signup and view all the answers

If the sine of an angle $ heta$ is defined as $rac{3}{5}$, what is the value of cosecant $ heta$?

1.67 Signup and view all the answers

What relationship is true between sine and cosecant for any angle $ heta$?

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

What is the reciprocal of the tangent function?

Cotangent Signup and view all the answers

For a right-angled triangle where the adjacent side is 7 units and the hypotenuse is 25 units, what is the cosine of the angle $ heta$?

0.64 Signup and view all the answers

Which mnemonic helps in remembering the definitions of trigonometric ratios such as sine, cosine, and tangent?

SOHCAHTOA Signup and view all the answers

If $ an heta = rac{4}{3}$, what is the approximate value of $ heta$ in degrees?

53.13° Signup and view all the answers

What is the product of cosine and secant for any angle $ heta$?

1 Signup and view all the answers

Which statement is true concerning the trigonometric function cosecant?

Cosecant equals the hypotenuse over opposite. Signup and view all the answers

Which of the following statements correctly describes a property of a rhombus?

Diagonals intersect at 90° and bisect both pairs of opposite angles. Signup and view all the answers

Which triangle congruence condition does not require the angles of the triangles to be measured?

SSS (Side-Side-Side) Signup and view all the answers

What must be true for two triangles to be considered similar under the SSS similarity criterion?

The ratio of corresponding sides must be in proportion. Signup and view all the answers

In which quadrilateral do the diagonals always bisect each other but are not necessarily equal in length?

Kite Signup and view all the answers

Which property distinguishes a square from other quadrilaterals?

It has four right angles and equal sides. Signup and view all the answers

What characterizes a trapezium in relation to its sides?

Only one pair of opposite sides is parallel. Signup and view all the answers

In the context of triangle congruence, SAS requires which of the following conditions?

Two sides and the included angle must be equal. Signup and view all the answers

Which condition represents the converse of the Pythagorean theorem?

If $b^2 = a^2 + c^2$, then the triangle is right-angled. Signup and view all the answers

What is the total sum of the interior angles of a quadrilateral?

360° Signup and view all the answers

Which of the following is NOT a requirement for a shape to be classified as a parallelogram?

Diagonals must be of equal length. Signup and view all the answers

What is the value of the sine function for a 30° angle?

1/2 Signup and view all the answers

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

√2 Signup and view all the answers

When using the tangent ratio to find angle A for the point P(2, 3) in the Cartesian plane, what is the value of tan A?

3/2 Signup and view all the answers

For which angle is the cosine function equal to 1/2?

60° Signup and view all the answers

What is the approximate measure of angle B formed between line OQ and the positive x-axis for Q(-2, 3)?

123.7° Signup and view all the answers

What is the maximum value for the sine function of any angle?

1 Signup and view all the answers

Which of these represents the correct relationship for the tangent function concerning angle θ?

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

If the tangent of an angle is undefined, which of the following could be true regarding the angle?

It is 90° Signup and view all the answers

Which of the following statements about the cosine of 45° is true?

cos 45° = √2/2 Signup and view all the answers

What is the sine function value of an angle that corresponds to the opposite side measuring 3 and the hypotenuse measuring 5?

3/5 Signup and view all the answers

What is true about qualitative data?

It can consist of categorical and anecdotal data. Signup and view all the answers

What defines the median in a numerical data set with an even number of values?

It is the average of the two central values. Signup and view all the answers

If a data set has multiple modes, what does it imply?

More than one value has the highest frequency. Signup and view all the answers

Which statement best describes an outlier in a data set?

It significantly differs from other values in the data set. Signup and view all the answers

Which is the correct formula for calculating the mean of a data set?

$ar{x} = rac{x_1 + x_2 + ... + x_n}{n}$ Signup and view all the answers

Which characteristic is NOT true for the mode in a data set?

It is always the average of the dataset. Signup and view all the answers

In which scenario would the mean be an inappropriate measure of central tendency?

When the data set contains outliers. Signup and view all the answers

When calculating the mode, if all values in a data set occur with equal frequency, what can be concluded?

The mode is not defined. Signup and view all the answers

Why might one choose to use the median over the mean for a data set?

The median is less affected by outliers. Signup and view all the answers

What is the definition of an outlier in a data set?

A value that is much greater or much less than the rest Signup and view all the answers

How is the range of a data set calculated?

By subtracting the minimum value from the maximum value Signup and view all the answers

What does the modal group represent in a frequency distribution?

The group that has the highest frequency of values Signup and view all the answers

What happens to the mean calculation when dealing with grouped data?

It is based on the midpoints of class intervals Signup and view all the answers

What do percentiles indicate in a data set?

What percentage of values lie below a certain value Signup and view all the answers

How is the $p$th percentile determined in a dataset with $n$ values?

Applying the formula $r = \frac{p}{100} (n - 1) + 1$ Signup and view all the answers

Which of the following best describes measures of dispersion?

Statistics that determine how values are concentrated around the center Signup and view all the answers

Why is the mode typically not computable for continuous data sets?

Because each continuous value occurs only once Signup and view all the answers

When constructing a histogram, what do the height of the rectangles represent?

The frequency of values in each sub-range Signup and view all the answers

What describes a situation where a data set has no mode?

All data points occur with equal frequency Signup and view all the answers

Which of the following correctly describes the surface area of a triangular prism?

It consists of two triangles and three rectangles. Signup and view all the answers

What is the volume of a cube with a side length of $s$?

$s^3$ Signup and view all the answers

Which formula accurately expresses the volume of a right cone?

$rac{1}{3} imes ext{Area of base} imes ext{Height of cone}$ Signup and view all the answers

When the dimensions of a cylinder are multiplied by a factor of $k$, what is the effect on its volume?

The volume increases by a factor of $k^3$. Signup and view all the answers

The surface area of a sphere with radius $r$ is expressed as which of the following?

$4 ext{π}r^2$ Signup and view all the answers

Which statement about the formula for the volume of a triangular pyramid is correct?

It is given by $rac{1}{3} imes rac{1}{2}b imes h imes H$. Signup and view all the answers

For a right pyramid, how does its apex relate to the base?

The apex is perpendicular to the centroid of the base. Signup and view all the answers

What signifies that a shape is classified as a cylinder?

It has circular bases and curved sides. Signup and view all the answers

What is the surface area formula for a right cone?

$ ext{π} r^2 + ext{π} r h$ Signup and view all the answers

What is the primary purpose of the five number summary in data analysis?

To concisely represent the spread and distribution of data Signup and view all the answers

Which statement describes the semi-interquartile range?

It is half of the difference between the first quartile and the third quartile. Signup and view all the answers

How are percentiles determined in grouped data?

By locating the range within which the rank falls. Signup and view all the answers

What does the interquartile range represent?

The difference between the first and third quartiles. Signup and view all the answers

In the context of a box-and-whisker plot, what does the box represent?

The interquartile range. Signup and view all the answers

Which formula correctly represents the area for a trapezium?

Area = ½ × (side1 + side2) × height Signup and view all the answers

What characteristic defines a right prism?

It has a polygonal base with vertical faces perpendicular to the base. Signup and view all the answers

Which quartile corresponds to the median of the data set?

Q2 Signup and view all the answers

What does the 0th percentile represent in a data set?

The first value in an ordered data set. Signup and view all the answers

Which of the following is NOT a characteristic of deciles?

They can only be calculated for continuous data. Signup and view all the answers

What happens to the volume of a rectangular prism when all three dimensions are multiplied by a factor of 2?

It is multiplied by 8 Signup and view all the answers

How does multiplying the height of a cylinder by 3 affect its surface area?

Surface area is multiplied by 3 Signup and view all the answers

If the volume of a rectangular prism is defined as $V = l imes b imes h$, what would the new volume be if the length is doubled and the breadth is kept the same while height remains unchanged?

It becomes 2 times the original volume Signup and view all the answers

What is the impact on the surface area of a rectangular prism if its length is tripled?

It is multiplied by a factor involving 3 Signup and view all the answers

What is the new volume of a cylinder if its radius is halved and height is kept the same?

It is multiplied by 0.25 Signup and view all the answers

If a rectangular prism's dimensions are increased such that the height is multiplied by 2 and the length by 3, what is the overall change in volume?

It increases by a factor of 6 Signup and view all the answers

When all three dimensions of a rectangular prism are multiplied by a constant $k$, what is the relationship between the change in surface area and the change in volume?

Surface area is multiplied by $k^2$ and volume by $k^3$ Signup and view all the answers

What is the new surface area of a cylinder if both the radius is multiplied by 4 and the height is multiplied by 2?

It is multiplied by 16 Signup and view all the answers

What is the ratio defined as cosecant in terms of the sides of a right-angled triangle?

The ratio of the hypotenuse to the opposite side Signup and view all the answers

Which equation represents the relationship between sine and cosecant for any angle theta?

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

Which of the following statements correctly relates the definitions of tangent and cotangent?

tan θ = sine of angle / cosine of angle Signup and view all the answers

When calculating the secant of an angle using a calculator, what is the necessary initial step?

Ensure the calculator is in degrees mode Signup and view all the answers

Which of the following pairs of angles will yield the same values for their sine and cosine ratios?

45° and 135° Signup and view all the answers

If the sine value of an angle is represented as sin θ = 0.5, which of the following angles could this correspond to?

150° Signup and view all the answers

Which of the following relationships is accurate for the cosecant and sine functions?

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

For which angle will the value of cotangent equal 1?

45° Signup and view all the answers

Given that the cosine of an angle is 0, which of the following statements is true?

The angle must be 90° or 270° Signup and view all the answers

If the relationship sin² θ + cos² θ = 1 is true, which of the following represents this relationship for cosecant and secant?

sec² θ - tan² θ = 1 Signup and view all the answers

In which of the following scenarios can two triangles be considered congruent?

If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another. Signup and view all the answers

What is a requirement for two triangles to be classified as similar?

Their corresponding angles must be equal and the corresponding sides must be in proportion. Signup and view all the answers

Which property is unique to a square compared to other quadrilaterals?

All sides are equal in length. Signup and view all the answers

Which statement about the properties of a rhombus is true?

All sides are equal in length. Signup and view all the answers

Which description accurately defines a trapezium?

A quadrilateral with only one pair of opposite sides parallel. Signup and view all the answers

How can the interior angles of a quadrilateral be characterized mathematically?

They collectively equal 360 degrees. Signup and view all the answers

What is the relationship between a rectangle and a parallelogram?

All rectangles are parallelograms, but not all parallelograms are rectangles. Signup and view all the answers

Which property about the diagonals of a kite is true?

They bisect each other at right angles. Signup and view all the answers

Which theorem can be applied to determine if a triangle is right-angled?

Pythagorean theorem. Signup and view all the answers

How does a square differ from a rectangle?

All sides of a square must be equal, while a rectangle only requires equal opposite sides. Signup and view all the answers

What is the relationship between the angles of similar triangles?

They are equal to their corresponding angles. Signup and view all the answers

Which of the following statements regarding the ratios of similar triangles is true?

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

In which scenario would the property of similar triangles not hold?

If the corresponding angles of the triangles differ. Signup and view all the answers

If triangles $ riangle ABC $ and $ riangle DEF $ are similar, how are the sides related?

AB/DE = BC/EF = AC/DF. Signup and view all the answers

What must be true about the order of vertices when labeling similar triangles?

Each vertex must correspond directly to another in the same order. Signup and view all the answers

What holds true about the ratios of the sides in triangles $ riangle ABC $, $ riangle DEF $, and \( riangle GHK \$)

The ratios are equal regardless of triangle dimensions. Signup and view all the answers

For triangles to maintain similarity, what condition regarding their angles must be satisfied?

All corresponding angles must be equal. Signup and view all the answers

Which statement accurately reflects the significance of similar triangles in relation to trigonometric ratios?

They define the ratios used to establish the trigonometric functions. Signup and view all the answers

What is the relationship between the angles in a 30°-60°-90° triangle and the length of its sides?

The hypotenuse is twice the length of the side opposite the 30° angle. Signup and view all the answers

When solving the equation $ an A = rac{3}{2}$, what is the approximate value of angle A?

56.3° Signup and view all the answers

For the angle denoted as B in point Q(-2, 3), what is the relationship between angle B and angle α?

B is supplementary to α. Signup and view all the answers

What is the sine value of a 60° angle based on the trigonometric ratios of special angles?

√3/2 Signup and view all the answers

Which of the following statements about tangent in a right triangle is accurate?

Tangent is the ratio of the lengths of the opposite side to the adjacent side. Signup and view all the answers

Which of the following correctly describes the properties of angles formed by the radius of a circle in relation to the x-axis?

The angle can be measured directly from the positive x-axis. Signup and view all the answers

In the evaluation of the inverse trigonometric function $ an^{-1}(1)$, what is the resulting angle?

45° Signup and view all the answers

When given the lengths of two sides of a triangle, which rule is typically applied to find an unknown angle?

Law of Sines. Signup and view all the answers

Which statement about the ratios between sides in a 45°-45°-90° triangle is true?

Each leg is $rac{ ext{hypotenuse}}{ ext{sqrt}(2)}$. Signup and view all the answers

When evaluating the values of trigonometric functions for angles larger than 90°, which statement is accurate?

The functions can produce results outside their defined range. Signup and view all the answers

What does the Mid-Point Theorem state about the line segment joining the mid-points of two sides of a triangle?

It is parallel to the third side and equal to half its length. Signup and view all the answers

Which of the following properties is true for a kite?

It has two pairs of adjacent sides that are equal. Signup and view all the answers

If line segment DE is drawn through the mid-point of side AC of triangle ABC, parallel to side AB, what can be concluded about segment DE?

It will bisect side BC. Signup and view all the answers

What information can be derived when proving that MNOP is a parallelogram?

MN and OP must be equal in length. Signup and view all the answers

When observing a parallelogram, which condition does NOT ensure its classification?

The lengths of all sides are equal. Signup and view all the answers

In quadrilateral ABCD with vertices listed as A(1, 1), B(3, 1), C(3, 3), and D(1, 3), which characteristic accurately describes this shape?

It is a rectangle. Signup and view all the answers

Which geometric property can be concluded from the Mid-Point Theorem's converse?

A parallel segment through a midpoint divides the opposing side into two equal parts. Signup and view all the answers

When calculating the distance between two points A(x₁, y₁) and B(x₂, y₂), which formula is correctly derived from the Pythagorean theorem?

Distance = √((x₂ - x₁)² + (y₂ - y₁)²) Signup and view all the answers

Which of the following statements is a necessary condition for two triangles to be similar?

Corresponding angles must be congruent. Signup and view all the answers

What distinguishes qualitative data from quantitative data?

Qualitative data cannot be represented as numbers. Signup and view all the answers

In calculating the median of a data set with an even number of values, what is the correct process?

Find the average of the two middle values. Signup and view all the answers

Which statement about the mode of a data set is false?

If all values in a data set occur with equal frequency, it has a mode. Signup and view all the answers

What is the significance of an outlier in a data set?

It is a value that is markedly different from others. Signup and view all the answers

How is the mean of a data set calculated?

By summing all values and dividing by the count. Signup and view all the answers

Which scenario correctly describes when a data set has no mode?

All values appear with the same frequency. Signup and view all the answers

When identifying the median from a sorted list of values, how should the values be arranged?

From lowest to highest value. Signup and view all the answers

In what situation would the mean not accurately represent the central tendency of a data set?

When there is a high frequency of extreme values. Signup and view all the answers

Which of the following statements about the signs of trigonometric ratios in Quadrant II is true?

Sine and cosecant are positive, while cosine and tangent are negative. Signup and view all the answers

What is the primary difference between mean and median as measures of central tendency?

Mean takes all values into account, while median focuses on the middle. Signup and view all the answers

If the interior angles of a triangle measure 60°, 60°, and 60°, which classification is accurate?

Equilateral and acute Signup and view all the answers

What does an outlier in a dataset typically indicate?

It may suggest an error or exceptional condition in the data. Signup and view all the answers

What is the sine value of 30°?

$rac{ ext{1}}{2}$ Signup and view all the answers

In a right triangle, if the lengths of the adjacent side is 6 and the hypotenuse is 10, what is the cosine of the angle?

$rac{6}{10}$ Signup and view all the answers

If $ an heta = 4$, what is the value of $ an (90° - heta)$?

0.25 Signup and view all the answers

What does the gradient of a line represent in a coordinate system?

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

What is the relationship between the sine and cosine of complementary angles?

They are reciprocals of each other. Signup and view all the answers

Given a triangle where two sides are equal, which of the following classification accurately describes it?

Isosceles and right-angled Signup and view all the answers

What condition must hold true for two lines to be considered perpendicular?

The slopes of both lines are negative inverses of each other Signup and view all the answers

What can be inferred about a triangle if one of its exterior angles measures 120°?

At least one interior angle is obtuse. Signup and view all the answers

What does a horizontal line's gradient value indicate?

Gradient value of zero Signup and view all the answers

Which of the following represents the cosecant ratio?

$rac{r}{y}$ Signup and view all the answers

Which formulae accurately calculate the coordinates of the mid-point between points A(x₁, y₁) and B(x₂, y₂)?

M(x, y) = rac{x₁ + x₂}{2}, rac{y₁ + y₂}{2} Signup and view all the answers

When determining if three points are collinear, what method can be used?

Both the distance formula and the gradient method Signup and view all the answers

In trigonometric functions, which of the following is true for all angles?

Tangent is equal to 1 at 45°. Signup and view all the answers

How is quantitative data defined?

Data that can be written as numbers Signup and view all the answers

Which statement about the distance formula is true?

It determines the length of the hypotenuse of a right triangle Signup and view all the answers

What characteristic defines a straight line in terms of its gradient?

The gradient is constant between any two points Signup and view all the answers

In which scenario does a line have an undefined gradient?

When the line is vertical Signup and view all the answers

What does qualitative data refer to?

Categorical descriptions of characteristics Signup and view all the answers

What is the primary characteristic of the mode in a data set?

It is the value that appears most frequently. Signup and view all the answers

What happens to the mode if all values in a data set occur with equal frequency?

The data set is said to have no mode. Signup and view all the answers

In the context of grouped data, how is the mean approximated?

By assuming all values are at the midpoint of the intervals. Signup and view all the answers

How is the median group within a grouped data set defined?

It is the group that contains the middle value. Signup and view all the answers

Which of the following best describes the percentile in a data set?

It is a measure that divides the set into two parts based on rank. Signup and view all the answers

What is the impact of outliers on the range of a data set?

It inflates the range. Signup and view all the answers

When dividing continuous quantitative data into sub-ranges, what is essential?

Sub-ranges must completely cover the data range without overlap. Signup and view all the answers

What happens when continuous data is grouped into intervals?

Precision of individual measurements is lost. Signup and view all the answers

What is the formula for finding the rank of the pth percentile in an ordered data set?

$r = \frac{p}{100} (n - 1) + 1$ Signup and view all the answers

What is the correct method to determine the value of the 70th percentile in an ordered data set when the rank is not an integer?

Take the average of the two values indicated by the rank. Signup and view all the answers

Which calculation accurately defines the interquartile range?

The difference between the third quartile and the first quartile. Signup and view all the answers

Which statement correctly describes the five number summary of a data set?

It represents the minimum, lower quartile, median, upper quartile, and maximum. Signup and view all the answers

How does the semi-interquartile range differ from the interquartile range?

It is half of the value of the interquartile range. Signup and view all the answers

In the context of grouped data, how are percentiles determined?

Percentiles lie within a range derived from the calculated rank using specific formulas. Signup and view all the answers

What does the box-and-whisker plot visually represent?

The distribution of the data while indicating the interquartile range. Signup and view all the answers

When calculating ranges in data, what specifically defines the full data range?

The difference between the maximum and minimum values in the data set. Signup and view all the answers

Which property is true regarding right prisms based on their definition?

The bases are congruent shapes and the lateral faces are perpendicular to the base. Signup and view all the answers

How are deciles defined with respect to an ordered data set?

They split the dataset into ten equal parts with nine key values. Signup and view all the answers

What is the volume formula for a triangular prism?

Volume = rac{1}{2} imes base imes height of triangle imes height of prism Signup and view all the answers

Which of the following solids has both a polygon as its base and triangular faces converging at an apex?

Triangular pyramid Signup and view all the answers

What is the surface area formula for a right cone?

Surface Area = ext{Area of base} + ext{Area of walls} Signup and view all the answers

When one dimension of a cylinder is multiplied by a factor of k, what effect does this have on its volume?

The volume increases by a factor of k Signup and view all the answers

The surface area of a cube can also be expressed in terms of its side length. What is that expression?

Surface Area = 6s^2 Signup and view all the answers

Which of the following statements about the properties of a sphere is true?

A sphere's surface area formula is 4πr^2 Signup and view all the answers

The volume of a square pyramid depends on what key elements?

Base area and height Signup and view all the answers

Which solid's base area is used to determine its volume but not its surface area?

Cylinder Signup and view all the answers

What is the primary geometric feature of the base of a cylinder?

Circle Signup and view all the answers

In the formula for the surface area of a triangular pyramid, what is the term for the total area of its triangular sides?

h_s Signup and view all the answers

What happens to the volume of a rectangular prism if only one dimension is multiplied by a factor of 5?

The volume increases by a factor of 5. Signup and view all the answers

How does the surface area of a rectangular prism change when all three dimensions are multiplied by the same factor k?

The surface area increases by a factor of k^2. Signup and view all the answers

What is the new volume of a cylinder if both the radius and height are multiplied by a factor of k?

The volume is multiplied by k^3. Signup and view all the answers

If the height of a cylinder is multiplied by 5 and the radius remains constant, how does this affect the surface area?

The surface area increases by a factor of 5. Signup and view all the answers

When both the length and height of a rectangular prism are multiplied by 5, how is the new volume expressed in terms of the original volume V?

25V Signup and view all the answers

What is the change in surface area when only the radius of a cylinder is multiplied by 5?

The surface area increases by a factor of 25. Signup and view all the answers

If the volume of a rectangular prism changes from V to 125V, what conclusion can be drawn about the dimensions adjusted?

All three dimensions were multiplied by 5. Signup and view all the answers

What factor would you multiply the height of a cylinder by to only double its volume?

2 Signup and view all the answers

Gr 10 Math Nov P2 Hard

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Questions and Answers

In Quadrant II, which trigonometric ratios are positive?

What is the value of tan 90°?

For an isosceles triangle, which statement is true?

According to the CAST diagram, which trigonometric functions are positive in Quadrant III?

What is the relationship between the angle of elevation and angle of depression?

When using the sine ratio for a right triangle, how is it defined?

If the interior angles of a triangle sum to 180°, what can be said about its exterior angles?

Which of the following statements is true regarding the Pythagorean theorem?

In terms of trigonometric ratios, how is cotangent defined?

What type of triangle has all interior angles less than 90°?

In a 30°-60°-90° triangle, what is the length of the hypotenuse if the length of the side opposite the 30° angle is 2?

What is the sine value of a 60° angle?

When finding the angle in a triangle using the ratio tan A = 3/2, what will be the approximate measure of angle A?

For a point Q located at (-2, 3) on the Cartesian plane, what is the correct description of angle B formed between line OQ and the positive x-axis?

In a 45°-45°-90° triangle, what is the ratio of the lengths of the sides to the hypotenuse?

What is the tangent value of a 45° angle?

In solving a trigonometric equation, which step is NOT involved?

Considering the properties of sine and cosine, what is a valid rule regarding their values?

If the lengths of two sides of a right triangle are known, which function can be used to find the angle opposite one of the known sides?

What is the characteristic property of the sides in a 30°-60°-90° triangle related to the angles?

Which ratio correctly represents the definition of cosecant?

What is the result of multiplying sine and cosecant for any angle $\theta$?

If the value of $\tan \theta$ is 3, what is the approximate value of $\cot \theta$?

To calculate secant on a scientific calculator, which adjustment must be made first?

What is the defining formula for tangent in a right-angled triangle?

Which of the following is NOT a reciprocal trigonometric ratio?

For which angle is $\sin \theta = $1?

What is the cosine of an angle if its adjacent side length equals 4 and the hypotenuse length equals 5?

What is a valid mnemonic for remembering the trigonometric ratios?

Which relationship between the ratios is incorrect?

Which of the following is NOT a property of a rectangle?

Which statement accurately describes the differences between a rhombus and a rectangle?

What condition does NOT apply to two triangles being considered similar?

Which relationship is true regarding the hierarchy of quadrilaterals?

Which properties are shared by both a parallelogram and a kite?

Which condition indicates that a triangle is right-angled based on the Pythagorean theorem?

Which of the following accurately describes a trapezium?

Which type of quadrilateral has diagonals that bisect each other at 90°?

In the context of triangle congruence, AAS indicates which combination of equalities?

Which of the following must be true for a quadrilateral to be classified as a parallelogram?

What specific property does the line segment joining the mid-points of two sides of a triangle possess according to the mid-point theorem?

In a parallelogram, if opposite sides are equal and parallel, which property can be used to conclude that the opposite angles are also equal?

What conclusion can be made about a quadrilateral if both pairs of opposite angles are equal?

If the mid-point theorem is applied to triangle ABC, and points D and E are the mid-points of sides AB and AC respectively, what can be inferred about line segment DE?

Which of the following statements about kites is true?

In analytical geometry, when plotting quadrilateral ABCD on the Cartesian plane, what is the correct way to label it?

What is the purpose of the mid-point theorem in coordinate geometry?

Which of the following correctly describes the characteristics of a trapezium?

What condition must be met for two triangles to be considered similar?

In the context of distance between two points, which formula is used to calculate the distance between points A(x₁, y₁) and B(x₂, y₂)?

For similar triangles, which statement about the ratios of their corresponding sides is true?

In similar triangles, if the ratio of one side in triangle A to the corresponding side in triangle B is 2:3, what can be said about another pair of corresponding sides?

When investigating the mid-point theorem, what does shifting triangle ADE onto quadrilateral BDEC illustrate?

How does the order of vertices affect the identification of similar triangles?

If triangle ABC is similar to triangle DEF, what can be concluded about angles A, B, and C?

Which of the following ratios represents the relationship of the sides in two similar triangles?

What term is used to describe the value that occurs most frequently in a data set?

When three similar triangles are compared, which of the following properties regarding their angles must hold true?

If two triangles have sides in the ratio 5:12 and the angle between them is 90°, what is true about their similarity?

How is the median determined in a data set?

What does the mean of a data set represent?

If a data set has an even number of values, how is the median calculated?

An outlier in a data set is best described as which of the following?

Which of the following statements is true regarding the mode?

What characteristic distinguishes qualitative data from quantitative data?

When calculating the mean of a set of values, which variable denotes the number of values?

In which situation is it appropriate to use the median rather than the mean?

How do you determine the value of the percentile when the rank is not an integer?

What defines the interquartile range (IQR)?

Which values are included in the five number summary?

What is a characteristic of a right prism?

What represents the spread of data in a box-and-whisker plot?

How is the area of a trapezium calculated?

What is the purpose of calculating percentiles for grouped data?

Which formula would you use to calculate the area of a rectangle?

What does the term 'deciles' refer to in statistics?

In the context of distance formula, what does the expression $ ext{distance (d)} = ext{d} = ackslash sqrt{(x₁ - x₂)^2 + (y₁ - y₂)^2}$ represent?