Gr 12 Mathematics: November Medium P(2)

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If the gradient of the radius of a circle is (\frac{1}{2}), what is the gradient of the tangent at the point of tangency?

-$\frac{1}{2}$

2

-2 (correct)

$\frac{1}{2}$

In a circle with center (3, 4) and radius 5, what is the gradient of the radius drawn to the point (7, 1)?

-$\frac{4}{3}$

$\frac{3}{4}$

$\frac{4}{3}$

-$\frac{3}{4}$ (correct)

A line parallel to one side of a triangle divides the other two sides proportionally. This theorem is known as:

Midpoint Theorem

Pythagorean Theorem

Angle Bisector Theorem

Basic Proportionality Theorem (correct)

In a triangle ABC, a line DE is drawn parallel to BC, dividing AB in the ratio 2:3. If AD = 8 cm, what is the length of DB?

12 cm Signup and view all the answers

What is the area of a triangle with a base of 10 cm and a height of 6 cm?

30 cm Signup and view all the answers

Given the equation of a circle: (x - 2) + (y + 1) = 9, what is the center of the circle?

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

A ratio compares two quantities with the same units. Which of the following is NOT a ratio?

10% Signup and view all the answers

If the equation of a tangent to a circle is y = 3x - 2, and the point of tangency is (1, 1), what is the gradient of the radius drawn to that point?

-(\frac{1}{3}) Signup and view all the answers

Which of these properties is NOT a property of proportions?

Direct Proportion Signup and view all the answers

Given a triangle ABC, a line DE is drawn parallel to BC, dividing AB in the ratio 2:1. If AD = 6 cm, what is the length of AB?

12 cm Signup and view all the answers

What is the condition for two triangles to be similar?

They have proportional sides and equal corresponding angles Signup and view all the answers

What is the formula for the area of a triangle?

Area = 1/2 × base × height Signup and view all the answers

What is the Pythagorean theorem used for?

To find the sum of squares of two sides of a right-angled triangle Signup and view all the answers

What is the regression coefficient r used for?

To measure the strength of correlation between two sets of data Signup and view all the answers

What is the linear regression line used for?

To find the equation of the line of best fit Signup and view all the answers

What is the converse of the Pythagorean theorem used for?

To prove that an angle is a right angle Signup and view all the answers

What is the condition for two triangles to have areas proportional to their bases?

They have equal heights Signup and view all the answers

What is the advantage of using the linear regression line?

It helps in finding the pattern in a set of data Signup and view all the answers

What is the value of the regression coefficient r?

Always between -1 and +1 Signup and view all the answers

What is the purpose of the Pythagorean theorem proof?

To prove that the square on the hypotenuse is equal to the sum of the squares on the other two sides Signup and view all the answers

What is the formula for the area of a triangle?

$rac{1}{2} \times \text{base} \times \text{height}$ Signup and view all the answers

What is the condition for two polygons to be similar?

All pairs of corresponding angles are equal and all pairs of corresponding sides are in the same proportion. Signup and view all the answers

What is the statement of the Mid-point Theorem?

The line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. Signup and view all the answers

What is the converse of the Mid-point Theorem?

The line drawn from the midpoint of one side of a triangle bisects the third side of the triangle. Signup and view all the answers

What is the Proportion Theorem?

If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. Signup and view all the answers

If two triangles are equiangular, what can be concluded about their corresponding sides?

The corresponding sides are in the same proportion. Signup and view all the answers

What is the definition of similar polygons?

Polygons that have the same shape but differ in size. Signup and view all the answers

To prove two triangles are similar, what condition must be shown?

All pairs of corresponding angles are equal or all pairs of corresponding sides are in the same proportion. Signup and view all the answers

What does a correlation value of 0 indicate?

No correlation Signup and view all the answers

What is the statement of the theorem: Triangles with Sides in Proportion are Similar?

If the corresponding sides of two triangles are in proportion, then the two triangles are similar. Signup and view all the answers

What is the condition for two triangles to be equiangular?

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

What is the formula to calculate the linear correlation coefficient r?

r = b * (σx / σy) Signup and view all the answers

What is the equation of a circle with center at the origin and radius r?

x^2 + y^2 = r^2 Signup and view all the answers

What is the cosine of a difference formula in trigonometry?

cos(α - β) = cos α cos β - sin α sin β Signup and view all the answers

What does a correlation value of -1 indicate?

A perfect negative correlation Signup and view all the answers

What is the interpretation of a strong negative correlation?

As x increases, y decreases Signup and view all the answers

What is the formula to calculate the gradient of the least squares regression line?

B = r * (σy / σx) Signup and view all the answers

What is the equation of the regression line?

y = A + Bx Signup and view all the answers

What is the sine of a sum formula in trigonometry?

sin(α + β) = sin α cos β + cos α sin β Signup and view all the answers

What is the interpretation of a very strong correlation?

0.9 < r < 1 Signup and view all the answers

What is the formula for the area of a trapezium (trapezoid)?

$rac{1}{2} imes (base_1 + base_2) imes height$ Signup and view all the answers

Which statement accurately describes a property of similar triangles?

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

How do you calculate the area of a rhombus using its diagonals?

$rac{1}{2} imes diagonal_{AC} imes diagonal_{BD}$ Signup and view all the answers

In the context of triangle proportionality, if a line is drawn parallel to one side of a triangle, what happens to the other sides?

They are divided in the same proportion. Signup and view all the answers

What is the area of a square if the length of one side is 5 units?

25 square units Signup and view all the answers

Which theorem states that a line joining the midpoints of two sides of a triangle is parallel to the third side?

Mid-point Theorem Signup and view all the answers

Which equation represents Pythagoras' Theorem in a right-angled triangle?

$a^2 + b^2 = c^2$ Signup and view all the answers

If a triangle has a base of 8 units and a height of 5 units, what is its area?

20 square units Signup and view all the answers

What is true regarding polygons that are similar?

Their areas are proportional to the square of the ratio of their corresponding sides. Signup and view all the answers

What is the formula for the area of a rectangle?

$length imes width$ Signup and view all the answers

What is the standard form of the equation for a circle with center at (3, -2) and radius 5?

(x - 3)^2 + (y + 2)^2 = 25 Signup and view all the answers

What does the term 'perpendicularity' refer to in the context of a tangent to a circle?

The radius at the point of tangency is perpendicular to the tangent line. Signup and view all the answers

Which step is NOT part of completing the square for the equation of a circle?

Square both sides of the radius equation. Signup and view all the answers

If a circle's equation is given as (x + 2)^2 + (y - 5)^2 = 36, what is the radius of the circle?

6 Signup and view all the answers

In the standard form of a circle's equation, what do the variables a and b represent?

The center coordinates (x, y) of the circle. Signup and view all the answers

What is the first step in rearranging the general form of a circle's equation into standard form?

Group the x terms and y terms. Signup and view all the answers

For a circle represented by the equation (x - 1)^2 + (y + 4)^2 = 16, what are the coordinates of the center?

(1, -4) Signup and view all the answers

What is the relationship between the radius of a circle and the coordinates of a point on its circumference?

The radius squared equals the sum of the differences squared. Signup and view all the answers

What is the compound angle formula for sin(α - β)?

sin α cos β - cos α sin β Signup and view all the answers

What is the double angle formula for sin(2α)?

2sin α cos α Signup and view all the answers

What is the general method for solving trigonometric equations?

Simplifying, using reference angles, and adding multiples of the period Signup and view all the answers

What is the compound angle formula for cos(α + β)?

cos α cos β - sin α sin β Signup and view all the answers

What is the double angle formula for cos(2α)?

cos²α - sin²α Signup and view all the answers

What is the purpose of the CAST diagram?

To determine where the function is positive or negative Signup and view all the answers

What is the co-function identity for cos(90° - α)?

sin α Signup and view all the answers

What is the step in the general method for solving trigonometric equations that involves adding multiples of the period?

Finding restricted values Signup and view all the answers

What is the compound angle formula for sin(α + β)?

sin α cos β + cos α sin β Signup and view all the answers

What is the purpose of the reference angle in the general method for solving trigonometric equations?

To determine the restricted values Signup and view all the answers

What is the formula used to calculate the area of triangle ABC if two sides and an included angle are known?

Area = $rac{1}{2}bc imes an A$ Signup and view all the answers

Which rule should be used if two sides and the included angle of a triangle are known?

Cosine Rule Signup and view all the answers

Under what condition would you use the Sine Rule?

When two angles and a side are known Signup and view all the answers

What is the correct expression to find $ heta$ if $ an heta = x$?

$ heta = an^{-1} x + k imes 180^ heta$ Signup and view all the answers

In triangle ABC, which ratio represents the Cosine Rule?

$a^2 = b^2 + c^2 - 2bc imes an A$ Signup and view all the answers

What is the formula to calculate the height of a pole given distance AB and angle FBA?

$h = rac{d imes an heta}{ an eta}$ Signup and view all the answers

When using the Cosine Rule, which variables are essential?

Two sides and one angle Signup and view all the answers

To find the height of a building using the Sine Rule, which angles must be known?

Both angles DBC and DCB Signup and view all the answers

What is the relationship between the distance formula and the cosine rule for points on the unit circle?

Both express the distance between two points as a function of the cosine of their angle difference. Signup and view all the answers

Using the cosine rule, how is $KL^2$ expressed in relation to angles $eta$ and $eta$?

$KL^2 = 2 - 2 ext{cos}(eta - eta)$ Signup and view all the answers

What simplification leads to the final expression for $ ext{cos}( heta)$ in terms of $ ext{cos}(eta)$ and $ ext{sin}(eta)$?

The angle difference formula results in $ ext{cos}( heta) = ext{cos}(eta) ext{cos}(eta) + ext{sin}(eta) ext{sin}(eta)$ Signup and view all the answers

How is $ ext{cos}(eta)$ used in relation to the negative angle identity when deriving $ ext{cos}( heta)$?

It is equal to $ ext{cos}(eta)$ regardless of the angle's sign due to even-odd identities. Signup and view all the answers

What simplification is made when deriving $ ext{cos}( heta)$ from the cosine difference formula?

The addition is simplified to fit the angle addition theorem. Signup and view all the answers

What does the expression $ ext{cos}( heta) = ext{cos}(eta) ext{cos}(eta) - ext{sin}(eta) ext{sin}(eta)$ represent?

It represents the cosine of the sum of two angles. Signup and view all the answers

What is the formula for the area of a kite?

1/2 × diagonal AC × diagonal BD Signup and view all the answers

What is the condition for two polygons to be similar?

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

What is the formula for the area of a parallelogram?

base × height Signup and view all the answers

What is the statement of the Triangle Proportionality Theorem?

If two triangles are equiangular, then the corresponding sides are in proportion Signup and view all the answers

What is the formula for the area of a trapezium?

1/2 × (base1 + base2) × height Signup and view all the answers

If two triangles have equal heights, what can be concluded about their areas?

They are proportional to their bases Signup and view all the answers

What is the statement of the Basic Proportionality Theorem?

If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally Signup and view all the answers

What is the formula for the area of a rhombus?

1/2 × diagonal AC × diagonal BD Signup and view all the answers

What is the condition for two triangles to have areas proportional to their bases?

They have equal heights Signup and view all the answers

What is the formula for the area of a rectangle?

length × width Signup and view all the answers

What is the equation of a circle with center at ( (2, -3) ) and radius ( 5 )?

( (x - 2)^2 + (y + 3)^2 = 25 ) Signup and view all the answers

What is the center and radius of the circle represented by the equation ( x^2 + y^2 - 6x + 4y - 12 = 0 )?

Center ( (3, -2) ), radius ( \sqrt{37} ) Signup and view all the answers

Given a circle with equation ( (x - 4)^2 + (y + 1)^2 = 9 ), what is the equation of the tangent line at the point ( (7, -1) )?

( y = -rac{4}{3}x + rac{25}{3} ) Signup and view all the answers

Which of the following statements is true about the tangent line to a circle?

The tangent line is perpendicular to the radius drawn to the point of tangency. Signup and view all the answers

What is the equation of the tangent to the circle ( x^2 + y^2 = 25 ) at the point ( (3, 4) )?

( 3x + 4y = 25 ) Signup and view all the answers

What is the distance between the center of the circle ( (x - 3)^2 + (y + 2)^2 = 16 ) and the point ( (7, 1) )?

( \sqrt{65} ) Signup and view all the answers

What is the equation of the circle with center ( (-1, 2) ) that passes through the point ( (3, 4) )?

( (x + 1)^2 + (y - 2)^2 = 20 ) Signup and view all the answers

If a circle has a diameter of 10 units, what is its radius?

( 5 ) Signup and view all the answers

What does the relationship between the gradients of the radius and the tangent in a circle indicate?

Their product is -1. Signup and view all the answers

In order to find the gradient of the tangent at a point on a circle, what must be calculated first?

The gradient of the radius. Signup and view all the answers

Which statement correctly defines a proportion?

A statement that two ratios are equal. Signup and view all the answers

What is the condition for two triangles to be similar?

They have corresponding sides in proportion Signup and view all the answers

What property of proportions states that if two ratios are equal, the cross-products are also equal?

Cross multiplication. Signup and view all the answers

In which form can a ratio be expressed?

As a fraction. Signup and view all the answers

What is the purpose of the Pythagorean theorem?

To find the length of the hypotenuse of a right-angled triangle Signup and view all the answers

If a line is drawn parallel to one side of a triangle, which effect does it have on the other sides?

It divides the other sides proportionally. Signup and view all the answers

What is the equation of a linear regression line?

y = A + Bx Signup and view all the answers

What is the standard form of a circle's equation that allows the identification of its center?

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

What is the condition for two polygons to be similar?

All pairs of corresponding angles are equal, and all pairs of corresponding sides are in the same proportion Signup and view all the answers

How is the area of a triangle calculated using its base and height?

Area = (1/2) × base × height Signup and view all the answers

What is the formula for the area of a triangle?

Area = 1/2 × base × height Signup and view all the answers

What characterizes a polygon?

It is constructed from at least three line segments. Signup and view all the answers

What is the regression coefficient (r) used for?

To measure the strength of correlation between two sets of data Signup and view all the answers

What is the converse of the Pythagorean theorem used for?

To prove a triangle is right-angled Signup and view all the answers

What is the condition for two triangles to have areas proportional to their bases?

They have equal heights Signup and view all the answers

What is the advantage of using the linear regression line?

It is used to make predictions based on a set of data Signup and view all the answers

If two triangles are equiangular, what can be concluded about their corresponding sides?

They are in proportion Signup and view all the answers

What is the condition that must be met for two polygons to be determined as similar?

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

According to the Mid-point Theorem, what is true about the line joining the midpoints of two sides of a triangle?

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

Which of the following statements is true about equiangular triangles?

Their corresponding sides are in proportion. Signup and view all the answers

If a triangle has a base of 4 cm and a height of 10 cm, what is its area?

20 cm² Signup and view all the answers

In the statement of the Proportion Theorem, what must be true if a line is drawn parallel to one side of a triangle?

It divides the other two sides proportionally. Signup and view all the answers

What can be concluded about two triangles if their corresponding sides are in proportion?

They must be similar. Signup and view all the answers

Which condition is NOT necessary to prove that two triangles are similar?

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

What does the converse of the Mid-point Theorem state?

If a line bisects a side, it is parallel to another side. Signup and view all the answers

In which situation would the two triangles riangle ABC and riangle DEF be considered similar?

If all pairs of corresponding sides are in the same proportion. Signup and view all the answers

Which of the following is a characteristic of similar triangles?

They have the same shape. Signup and view all the answers

What is the simplified form of the double angle formula for sine?

$\sin(2\alpha) = 2\sin(\alpha)\cos(\alpha)$ Signup and view all the answers

Which of the following is NOT a valid form of the double angle formula for cosine?

$\cos(2\alpha) = \sin^2(\alpha) + \cos^2(\alpha)$ Signup and view all the answers

What is the value of (\sin(2\alpha)) if (\sin(\alpha) = \frac{3}{5}) and (\alpha) is in the first quadrant?

$\frac{24}{25}$ Signup and view all the answers

What is the value of (\cos(2\alpha)) if (\cos(\alpha) = \frac{1}{3}) and (\alpha) is in the first quadrant?

$\frac{7}{9}$ Signup and view all the answers

Which of the following steps is NOT involved in finding the general solution of a trigonometric equation?

Solve the equation for the unknown variable and then find its value in the given interval Signup and view all the answers

Which of the following trigonometric identities can be used to prove the double angle formula for cosine, (\cos(2\alpha) = \cos^2(\alpha) - \sin^2(\alpha))?

$\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)$ Signup and view all the answers

What is the general solution of the equation (\sin(x) = \frac{1}{2})?

$\x = 30^\circ + 360^\circ k$ or $\x = 150^\circ + 360^\circ k$, where (k) is any integer Signup and view all the answers

Which of the following is NOT a valid step in deriving the double angle formula for sine?

Use the Pythagorean identity (\sin^2(\alpha) + \cos^2(\alpha) = 1) to simplify the expression Signup and view all the answers

Which of the following trigonometric identities can be used to derive the alternate form of the double angle formula for cosine, (\cos(2\alpha) = 2\cos^2(\alpha) - 1)?

$\sin^2(\alpha) + \cos^2(\alpha) = 1$ Signup and view all the answers

What is the value of (\cos(2\alpha)) if (\sin(\alpha) = \frac{4}{5}) and (\alpha) is in the first quadrant?

$\frac{7}{25}$ Signup and view all the answers

What does a correlation coefficient of -0.5 indicate in terms of the relationship between two variables?

There is a moderate negative correlation. Signup and view all the answers

Which of the following correctly represents the equation of a circle centered at the origin with a radius of 4?

x^2 + y^2 = 16 Signup and view all the answers

In the equation of a regression line, what does the variable 'A' represent?

The y-intercept. Signup and view all the answers

Which correlation coefficient value indicates a perfect positive correlation?

r = 1 Signup and view all the answers

What is the interpretation of a correlation value of 0.3?

A weak positive correlation. Signup and view all the answers

How is the linear correlation coefficient, r, calculated from the gradient of the least squares regression line?

r = b (σ_x / σ_y) Signup and view all the answers

Which of the following structures is indicative of a strong negative correlation?

-0.8 to -1 Signup and view all the answers

What does the equation y = A + Bx describe?

A linear relationship between two variables. Signup and view all the answers

What is the value of r when there is a perfect negative correlation?

-1 Signup and view all the answers

What is the outcome of equating the distance formula with the cosine rule for points on the unit circle?

It leads to the expression for $ heta$ in sine and cosine. Signup and view all the answers

Which identity correctly defines $ ext{cos}( heta + eta)$ using the negative angle identity?

$ ext{cos}( heta + eta) = ext{cos}( heta) ext{cos}(eta) - ext{sin}( heta) ext{sin}(eta)$ Signup and view all the answers

Which of the following transformations represents the cosine difference formula?

$ ext{cos}( heta - eta) = ext{cos}( heta) ext{cos}(eta) + ext{sin}( heta) ext{sin}(eta)$ Signup and view all the answers

What does the equation $ ext{cos}( heta - eta) = ext{cos}( heta) ext{cos}(eta) + ext{sin}( heta) ext{sin}(eta)$ signify about the relationship between cosine and sine?

It reveals the coexistence of sine and cosine in representing angle differences. Signup and view all the answers

Which identity correctly showcases the even-odd properties of trigonometric functions applied to cosine and sine?

$ ext{cos}(-eta) = ext{cos}(eta)$ and $ ext{sin}(-eta) = - ext{sin}(eta)$ Signup and view all the answers

What is the result of substituting the even-odd identities for sine and cosine into the expression for $ ext{cos}( heta + eta)$?

$ ext{cos}( heta + eta) = ext{cos}( heta) ext{cos}(eta) - ext{sin}( heta) ext{sin}(eta)$ Signup and view all the answers

If sinsθ = x, what is the general solution for θ?

θ = sins^-1 x + k ⋅ 360° Signup and view all the answers

In a triangle ABC, which rule would be used to find the side a if the sides b and c and angle A are given?

Cosine Rule Signup and view all the answers

What is the general approach for solving problems in three dimensions?

Draw a sketch, consider the given information, apply appropriate rules, and calculate the desired quantities Signup and view all the answers

In a triangle ABC, if the area of the triangle is given by (1/2)bc sin A, what is the other way to express the area?

(1/2)ab sin C Signup and view all the answers

What is the formula for the height of a pole in three dimensions?

h = d sin α / sin β Signup and view all the answers

What is the formula for the height of a building in three dimensions?

h = b sin α sin θ / sin (β + θ) Signup and view all the answers

If cos θ = x, what is the general solution for θ?

θ = cos^-1 x + k ⋅ 360° Signup and view all the answers

If tan θ = x, what is the general solution for θ?

θ = tan^-1 x + k ⋅ 90° Signup and view all the answers

What is the equation of a circle with center at the origin and radius 5?

x^2 + y^2 = 25 Signup and view all the answers

What does a correlation value of -0.75 indicate?

Moderate negative correlation Signup and view all the answers

What is the formula to calculate the cosine of a difference in trigonometry?

cos(α - β) = cos α cos β + sin α sin β Signup and view all the answers

What is the formula for calculating the gradient (slope) of the least squares regression line?

b = (Σxy - (Σx)(Σy) / n) / (Σx^2 - (Σx)^2 / n) Signup and view all the answers

Which of the following values represents a strong positive correlation?

0.85 Signup and view all the answers

What is the interpretation of a very strong negative correlation?

As x increases, y decreases significantly. Signup and view all the answers

What does a correlation value of 0 indicate?

No correlation Signup and view all the answers

What is the formula for the sine of a sum in trigonometry?

sin(α + β) = sin α cos β + cos α sin β Signup and view all the answers

Which of the following represents the equation of the regression line?

y = A + Bx Signup and view all the answers

What is the value of the linear correlation coefficient 'r' when there is a perfect negative correlation?

-1 Signup and view all the answers

If two triangles are similar, which of the following must be true?

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

What is the converse of the Mid-point Theorem?

The line drawn from the midpoint of one side of a triangle bisects the third side of the triangle. Signup and view all the answers

What is the condition for two polygons to be similar?

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

What is the formula for the area of a triangle?

Area = (base × height) / 2 Signup and view all the answers

What is the statement of the Proportion Theorem?

If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. Signup and view all the answers

What is true about equiangular triangles?

They are similar. Signup and view all the answers

What is the condition for two triangles to have areas proportional to their bases?

They are similar. Signup and view all the answers

What is the theorem that states that triangles with equal heights have areas proportional to their bases?

None of the above Signup and view all the answers

What is the expression for the cosine of a difference in trigonometry?

$\cos \alpha \cos eta + \sin \alpha \sin eta$ Signup and view all the answers

What is the expression for the cosine of a sum in trigonometry?

$\cos \alpha \cos eta - \sin \alpha \sin eta$ Signup and view all the answers

What is the definition of similar polygons?

Polygons with the same shape but not necessarily the same size. Signup and view all the answers

What is the relationship between the gradient of the radius and the gradient of the tangent at the point of tangency?

m_radius × m_tangent = -1 Signup and view all the answers

How can the cosine of a sum be derived?

Using the negative angle identity and the even-odd identities Signup and view all the answers

In a triangle, if a line is drawn parallel to one side, what happens to the other two sides?

They are divided in the same ratio Signup and view all the answers

How do you prove two triangles are similar?

By showing all corresponding angles are equal and all corresponding sides are in proportion. Signup and view all the answers

What is the advantage of using the cosine of a difference formula?

It can be used to find the cosine of a difference Signup and view all the answers

What is the formula for the area of a triangle?

Area = 1/2 × base × height Signup and view all the answers

What is a polygon?

A plane, closed shape consisting of three or more line segments Signup and view all the answers

What is the cosine rule used for?

Finding the distance between two points Signup and view all the answers

What is the relationship between the cosine of a difference and the cosine of a sum?

They are equal Signup and view all the answers

What is the property of proportion that states w × z = x × y?

Cross Multiplication Signup and view all the answers

If two ratios are equal, what can be concluded about the quantities involved?

They are in proportion Signup and view all the answers

What is the condition for two triangles to be similar?

They have proportional corresponding sides Signup and view all the answers

What is the formula for the equation of a circle with center (a, b) and radius r?

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

What is the purpose of the Basic Proportionality Theorem?

To divide the other two sides of a triangle proportionally Signup and view all the answers

What is the result of cross multiplying two ratios?

An equation is formed Signup and view all the answers

If (\sin \theta = x), what is (\theta) equal to?

(\theta = \sin^{-1} x + k \cdot 360^\circ) Signup and view all the answers

What is the formula for the area of a triangle:

(\frac{1}{2}ac \sin B) Signup and view all the answers

When would you use the cosine rule?

When no right angle is given, and either two sides and the included angle or three sides are given. Signup and view all the answers

What is the formula for the height of a pole?

(h = \frac{d \sin \alpha}{\sin \beta} \tan \beta) Signup and view all the answers

What is the general approach to solving three-dimensional problems?

Draw a sketch, consider the given information, apply appropriate rules, and calculate the desired quantities. Signup and view all the answers

What is the formula for the height of a building?

(h = \frac{b \sin \alpha \sin \theta}{\sin(\beta + \theta)}) Signup and view all the answers

When would you use the sine rule?

When no right angle is given, and either two sides and an angle (not the included angle) or two angles and a side are given. Signup and view all the answers

What is the formula for the area of a triangle in terms of its sides?

(\sqrt{s(s-a)(s-b)(s-c)}) Signup and view all the answers

What is the area of a parallelogram with a base of 10 cm and a height of 4 cm?

40 cm² Signup and view all the answers

Which of the following formulas correctly represents the area of a trapezium?

Area = (1/2) × (base₁ + base₂) × height Signup and view all the answers

What can be concluded if two triangles are equiangular?

Their corresponding sides are proportional. Signup and view all the answers

If a triangle has bases of 6 cm and 8 cm, and a height of 5 cm, what is the area of the triangle?

25 cm² Signup and view all the answers

What distinguishes a kite from a rhombus?

Kites do not need to have equal angles, while rhombuses do. Signup and view all the answers

What theorem states that triangles with the same base and height have equal areas?

Triangle Area Theorem Signup and view all the answers

How is the area of a rectangle calculated?

Area = length × width Signup and view all the answers

What is Thales' Theorem related to triangle proportions?

A line parallel to a side divides the other two sides proportionally. Signup and view all the answers

If the diagonals of a rhombus are 10 cm and 24 cm, what is its area?

120 cm² Signup and view all the answers

Which statement is true about the bases of a trapezium?

The height is the distance between the bases. Signup and view all the answers

What is the standard equation of a circle with center at the point (a, b) and radius r?

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

What step is performed first when completing the square for the equation of a circle in general form?

Group the x and y terms Signup and view all the answers

If the equation of a circle is written as $(x - 3)^2 + (y + 2)^2 = 25$, what is the center of the circle?

(3, -2) Signup and view all the answers

What does the tangent line to a circle represent?

A line that touches the circle at exactly one point Signup and view all the answers

Given the general equation of a circle $x^2 + y^2 + 4x - 8y + 4 = 0$, what is the value of the radius?

2 Signup and view all the answers

Which statement correctly describes the relationship between the radius and the tangent at the point of tangency on a circle?

The radius forms a right angle with the tangent Signup and view all the answers

From the equation $x^2 + y^2 - 6x + 8y + 9 = 0$, what is the center of the circle?

(3, 4) Signup and view all the answers

What formula is used to derive the radius when converting the general form of a circle's equation?

$r = ext{sqrt}igg{igg(}rac{D}{2}igg)^2 + igg(rac{E}{2}igg)^2 - Figg{igg)}$ Signup and view all the answers

Which of the following correctly represents the sine of a difference formula?

$ ext{sine}( heta - eta) = ext{sine} heta ext{cos} eta - ext{cos} heta ext{sine} eta$ Signup and view all the answers

What is the cosine of a sum formula?

$ ext{cos}(eta + heta) = ext{cos} eta ext{cos} heta - ext{sin} eta ext{sin} heta$ Signup and view all the answers

Which identity is correctly derived for the sine of double angle?

$ ext{sine}(2 heta) = 2 ext{cos} heta ext{sine} heta$ Signup and view all the answers

What is one form of the cosine of double angle?

$ ext{cos}(2 heta) = ext{cos}^2 heta - ext{sin}^2 heta$ Signup and view all the answers

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

Determining the values of the function for negative angles. Signup and view all the answers

The sine of a sum formula states:

$ ext{sine}( heta + eta) = ext{sine} heta ext{cos} eta + ext{cos} heta ext{sine} eta$ Signup and view all the answers

To find the cosine of a difference, which formula is utilized?

$ ext{cos}( heta - eta) = ext{cos} heta ext{cos} eta - ext{sin} heta ext{sin} eta$ Signup and view all the answers

What does the term reference angle refer to in trigonometric solutions?

The angle in the first quadrant corresponding to the given angle. Signup and view all the answers

Which of the following represents a correct form of the sine double angle identity?

$ ext{sine}(2 heta) = 2 ext{sine} heta ext{cos} heta$ Signup and view all the answers

In the context of similar triangles, what is the significance of the proportionality of corresponding sides?

It allows us to determine the ratio of the areas of the triangles. Signup and view all the answers

Which of the following is NOT a condition for two polygons to be similar?

The polygons have the same area. Signup and view all the answers

In the proof of the Pythagorean theorem, why are triangles ABD and CBA similar?

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

What is the relationship between the area of two triangles with equal bases but different heights?

The areas are proportional to their heights. Signup and view all the answers

What is the purpose of the linear regression line in statistics?

To predict the value of one variable based on the value of another variable. Signup and view all the answers

What is the meaning of a regression coefficient 'r' value close to +1?

A strong positive correlation between the variables. Signup and view all the answers

Which of the following statements accurately describes the relationship between the Pythagorean theorem and right-angled triangles?

The theorem relates the lengths of the sides of a right-angled triangle to each other. Signup and view all the answers

In the context of triangle proportionality, if a line is drawn parallel to one side of a triangle, what happens to the other two sides?

The other two sides are divided in the same ratio. Signup and view all the answers

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

They have the same area. Signup and view all the answers

If two triangles are equiangular, what can be concluded about their corresponding sides?

The corresponding sides are proportional. Signup and view all the answers

What is the equation of a circle with center (2, -3) and radius 4?

(x - 2)^2 + (y + 3)^2 = 16 Signup and view all the answers

What are the coordinates of the center of the circle defined by the equation x^2 + y^2 - 6x + 4y - 12 = 0?

(3, -2) Signup and view all the answers

What is the radius of the circle defined by the equation x^2 + y^2 + 8x - 10y + 25 = 0?

3 Signup and view all the answers

A tangent line to a circle is always perpendicular to which of the following?

The radius at the point of tangency Signup and view all the answers

Which of the following statements accurately describes the relationship between a tangent line and a radius at the point of tangency?

They are perpendicular lines. Signup and view all the answers

What is the equation of the line that is tangent to the circle with the equation (x - 1)^2 + (y + 2)^2 = 9 at the point (4, 1)?

y = -2x + 9 Signup and view all the answers

What is the gradient of the tangent line to the circle with equation (x + 3)^2 + (y - 2)^2 = 16 at the point (-1, 6)?

-2 Signup and view all the answers

What is the equation of the tangent to the circle (x - 2)^2 + (y + 1)^2 = 25 at the point (7, -4)?

y = -3x + 17 Signup and view all the answers

What is the formula for the area of a parallelogram?

base × height Signup and view all the answers

If two triangles are similar, what can be concluded about their corresponding angles?

They are equal Signup and view all the answers

What is the condition for two triangles to have areas proportional to their bases?

They have equal heights Signup and view all the answers

What is the formula for the area of a kite?

(diagonal AC × diagonal BD) / 2 Signup and view all the answers

What is the statement of the Basic Proportionality Theorem?

If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. Signup and view all the answers

What is the formula for the area of a trapezium (trapezoid)?

(base1 + base2) × height / 2 Signup and view all the answers

What is the statement of the Mid-point Theorem?

A line joining the midpoints of two sides of a triangle is parallel to the third side and equal to half the length of the third side. Signup and view all the answers

What is the formula for the area of a rectangle?

length × width Signup and view all the answers

What is the statement of the Proportion Theorem?

A line drawn parallel to one side of a triangle divides the other two sides proportionally. Signup and view all the answers

What is the condition for two triangles to be similar?

They are equiangular Signup and view all the answers

Given the equation of a circle: ( (x - 2)^2 + (y + 1)^2 = 9 ), what is the gradient of the tangent at the point (5, 1)?

-3/2 Signup and view all the answers

In a triangle ABC, a line DE is drawn parallel to BC, dividing AB in the ratio 2:3. If AD = 8 cm, what is the length of DB?

12 cm Signup and view all the answers

If the gradient of the radius of a circle is (\frac{1}{2}), what is the gradient of the tangent at the point of tangency?

-2 Signup and view all the answers

Which of the following properties is NOT a property of proportions?

Sum of Proportions: ( \frac{w + x}{y + z} = \frac{w}{y} = \frac{x}{z} ) Signup and view all the answers

What is the area of a triangle with a base of 10 cm and a height of 6 cm?

30 cm Signup and view all the answers

If the equation of a tangent to a circle is ( y = 3x - 2 ), and the point of tangency is (1, 1), what is the gradient of the radius drawn to that point?

-1/3 Signup and view all the answers

Given the equation of a circle: ( (x - 2)^2 + (y + 1)^2 = 9 ), what is the center of the circle?

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

In a triangle ABC, a line DE is drawn parallel to BC, dividing AB in the ratio 2:1. If AD = 6 cm, what is the length of AB?

9 cm Signup and view all the answers

Which of these is NOT a valid form of a ratio?

2 + 3 Signup and view all the answers

Which of the following properties does NOT apply to similar polygons?

Areas are equal. Signup and view all the answers

Which of the following is NOT a condition for two triangles to be similar?

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

In the context of the Pythagorean Theorem, what is the relationship between the lengths of the sides of a right-angled triangle?

The sum of the squares of the two shorter sides equals the square of the longest side. Signup and view all the answers

What does a regression coefficient (r) value close to -1 indicate?

A strong negative correlation between the two sets of data. Signup and view all the answers

Which of the following statements accurately describes a property of similar triangles?

They have the same shape but different sizes. Signup and view all the answers

In the proof of the Pythagorean Theorem, why are triangles ABD and CBA similar?

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

Given two triangles with equal heights, how are their areas related?

Their areas are in the ratio of their bases. Signup and view all the answers

What is the primary purpose of the linear regression line?

To predict the value of one variable based on the value of another variable. Signup and view all the answers

What is the converse of the Pythagorean Theorem used for?

To prove that a triangle is a right-angled triangle. Signup and view all the answers

Which of the following is a condition for polygons to be similar?

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

If two triangles are equiangular, what can be concluded about their corresponding sides?

Their corresponding sides are in proportion. Signup and view all the answers

What is the formula for the sine of a difference of two angles?

$\sin(\alpha) \cos(\beta) - \cos(\alpha) \sin(\beta)$ Signup and view all the answers

What is the formula for the cosine of a sum of two angles?

$\cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta)$ Signup and view all the answers

What is the formula for the sine of a double angle?

$\sin(2\alpha) = 2 \sin(\alpha) \cos(\alpha)$ Signup and view all the answers

What is the formula for the cosine of a double angle?

$\cos(2\alpha) = 2 \cos^2(\alpha) - 1$ Signup and view all the answers

What is the general approach to solving trigonometric equations?

Use algebraic methods and trigonometric identities, then find reference angles and use the CAST diagram Signup and view all the answers

What is the purpose of the CAST diagram in solving trigonometric equations?

To determine the sign of the trigonometric function Signup and view all the answers

What is the formula for the cosine of a difference of two angles?

$\cos(\alpha - \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta)$ Signup and view all the answers

What is the formula for the sine of a sum of two angles?

$\sin(\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta)$ Signup and view all the answers

What is the purpose of the general solution method in solving trigonometric equations?

To find all possible solutions to the equation Signup and view all the answers

What is the advantage of using trigonometric identities in solving trigonometric equations?

They simplify the equation and make it easier to solve Signup and view all the answers

What is the correct expression derived from equating the distance formula and the cosine rule for two points on the unit circle?

KL^2 = 2 - 2 ig(rac{1}{2}ig) Signup and view all the answers

Which identity allows the conversion of ext{cos}(eta - eta) during the derivation of ext{cos}(eta + eta)?

Negative angle identities Signup and view all the answers

What formula is obtained from the cosine of a sum identity for angles ext{alpha} and ext{beta}?

<pre><code>ext{cos}( ext{alpha} + ext{beta}) = ext{cos} ext{alpha} ext{cos} ext{beta} + ext{sin} ext{alpha} ext{sin} ext{beta} </code></pre> Signup and view all the answers

Using the distance formula, how can you express the distance KL between two points on the unit circle?

KL^2 = ( ext{cos}( ext{alpha}) - ext{cos}( ext{beta}))^2 + ( ext{sin}( ext{alpha}) - ext{sin}( ext{beta}))^2 Signup and view all the answers

What is the main purpose of using the negative angle identity in the derivation process?

To express ext{cos}(-eta) and ext{sin}(-eta) Signup and view all the answers

Which equation correctly reflects the relationship derived from the cosine rule related to angle differences on the unit circle?

2 - 2 ext{cos}( ext{alpha} - ext{beta}) = 2 - 2 ( ext{cos} ext{alpha} ext{cos} ext{beta} + ext{sin} ext{alpha} ext{sin} ext{beta}) Signup and view all the answers

What is the formula to find the area of triangle ABC using the sides and an angle?

Area = $\frac{1}{2}ac \sin B$ Signup and view all the answers

When is it appropriate to use the Sine Rule instead of the Cosine Rule?

When two angles and the included side are known Signup and view all the answers

Which equation correctly represents the Cosine Rule for side b?

$b^2 = a^2 + c^2 - 2ac \cos B$ Signup and view all the answers

What type of problem would likely require the use of the Area Rule for triangles?

When no perpendicular height is provided Signup and view all the answers

If $ an \theta = x$, what is the expression for (\theta)?

(\theta = \tan^{-1} x + k \cdot 180^\circ) Signup and view all the answers

Which of the following accurately describes how to determine which trigonometric rule to apply to a triangle?

Use the Area Rule when height is not provided. Signup and view all the answers

When calculating the height of a pole using triangles, which trigonometric function do you primarily apply?

Tangent to find the height Signup and view all the answers

What does the formula (h = \frac{d \sin \alpha}{\sin \beta} \tan \beta) calculate?

The height of a pole Signup and view all the answers

What does a correlation value of 0 indicate?

There is no relationship between the variables. Signup and view all the answers

What is the equation of a circle with center at the origin and radius r?

x^2 + y^2 = r^2 Signup and view all the answers

Which of the following statements is true for a strong negative correlation?

As x increases, y decreases significantly. Signup and view all the answers

What does the linear correlation coefficient r range from?

-1 to 1 Signup and view all the answers

How is the linear correlation coefficient r calculated?

Using the formula: r = b(σ_x/σ_y) Signup and view all the answers

In terms of correlation strength, what does an r value of 0.3 indicate?

Weak positive correlation Signup and view all the answers

What does the cosine of a sum formula describe?

The relationship between cosine and sine of angles. Signup and view all the answers

Which of the following defines a perfect positive correlation?

r = 1 Signup and view all the answers

What geometric shape's equation is given by x^2 + y^2 = r^2?

Circle Signup and view all the answers

What is the interpretation of a weak positive correlation?

As one variable increases, the other variable increases slightly. Signup and view all the answers

Two triangles are similar if they have the same shape but differ in size. This statement refers to which concept?

Similar triangles Signup and view all the answers

A line drawn parallel to one side of a triangle divides the other two sides proportionally. This statement is the core of which theorem?

Proportion theorem Signup and view all the answers

In a triangle ABC, a line DE is drawn parallel to BC, dividing AB in the ratio 2:3. If AD = 8 cm, what is the length of DB?

6 cm Signup and view all the answers

If two triangles are equiangular, what can be concluded about their corresponding sides?

They are proportional. Signup and view all the answers

Which of the following is NOT a condition for two polygons to be similar?

All pairs of corresponding sides are equal. Signup and view all the answers

What is the formula for the area of a triangle?

1/2 * base * height Signup and view all the answers

Which theorem states that a line joining the midpoints of two sides of a triangle is parallel to the third side and half its length?

Mid-point Theorem Signup and view all the answers

If two triangles have equal bases between the same parallel lines, what can be concluded about their areas?

Their areas are equal. Signup and view all the answers

What is the statement of the Converse of the Mid-point Theorem?

If a line is drawn from the midpoint of one side of a triangle parallel to another side, it bisects the third side of the triangle. Signup and view all the answers

Gr 12 Mathematics: November Medium P(2)

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Questions and Answers

If the gradient of the radius of a circle is (\frac{1}{2}), what is the gradient of the tangent at the point of tangency?

In a circle with center (3, 4) and radius 5, what is the gradient of the radius drawn to the point (7, 1)?

A line parallel to one side of a triangle divides the other two sides proportionally. This theorem is known as:

In a triangle ABC, a line DE is drawn parallel to BC, dividing AB in the ratio 2:3. If AD = 8 cm, what is the length of DB?

What is the area of a triangle with a base of 10 cm and a height of 6 cm?

Given the equation of a circle: (x - 2) + (y + 1) = 9, what is the center of the circle?

A ratio compares two quantities with the same units. Which of the following is NOT a ratio?

If the equation of a tangent to a circle is y = 3x - 2, and the point of tangency is (1, 1), what is the gradient of the radius drawn to that point?

Which of these properties is NOT a property of proportions?

Given a triangle ABC, a line DE is drawn parallel to BC, dividing AB in the ratio 2:1. If AD = 6 cm, what is the length of AB?

What is the condition for two triangles to be similar?

What is the formula for the area of a triangle?

What is the Pythagorean theorem used for?

What is the regression coefficient r used for?

What is the linear regression line used for?

What is the converse of the Pythagorean theorem used for?

What is the condition for two triangles to have areas proportional to their bases?

What is the advantage of using the linear regression line?

What is the value of the regression coefficient r?

What is the purpose of the Pythagorean theorem proof?

What is the formula for the area of a triangle?

What is the condition for two polygons to be similar?

What is the statement of the Mid-point Theorem?

What is the converse of the Mid-point Theorem?

What is the Proportion Theorem?

If two triangles are equiangular, what can be concluded about their corresponding sides?

What is the definition of similar polygons?

To prove two triangles are similar, what condition must be shown?

What does a correlation value of 0 indicate?

What is the statement of the theorem: Triangles with Sides in Proportion are Similar?

What is the condition for two triangles to be equiangular?

What is the formula to calculate the linear correlation coefficient r?

What is the equation of a circle with center at the origin and radius r?

What is the cosine of a difference formula in trigonometry?

What does a correlation value of -1 indicate?

What is the interpretation of a strong negative correlation?

What is the formula to calculate the gradient of the least squares regression line?

What is the equation of the regression line?

What is the sine of a sum formula in trigonometry?

What is the interpretation of a very strong correlation?

What is the formula for the area of a trapezium (trapezoid)?

Which statement accurately describes a property of similar triangles?

How do you calculate the area of a rhombus using its diagonals?

In the context of triangle proportionality, if a line is drawn parallel to one side of a triangle, what happens to the other sides?

What is the area of a square if the length of one side is 5 units?

Which theorem states that a line joining the midpoints of two sides of a triangle is parallel to the third side?

Which equation represents Pythagoras' Theorem in a right-angled triangle?

If a triangle has a base of 8 units and a height of 5 units, what is its area?

What is true regarding polygons that are similar?

What is the formula for the area of a rectangle?

What is the standard form of the equation for a circle with center at (3, -2) and radius 5?

What does the term 'perpendicularity' refer to in the context of a tangent to a circle?

Which step is NOT part of completing the square for the equation of a circle?

If a circle's equation is given as (x + 2)^2 + (y - 5)^2 = 36, what is the radius of the circle?

In the standard form of a circle's equation, what do the variables a and b represent?

What is the first step in rearranging the general form of a circle's equation into standard form?

For a circle represented by the equation (x - 1)^2 + (y + 4)^2 = 16, what are the coordinates of the center?

What is the relationship between the radius of a circle and the coordinates of a point on its circumference?

What is the compound angle formula for sin(α - β)?

What is the double angle formula for sin(2α)?

What is the general method for solving trigonometric equations?

What is the compound angle formula for cos(α + β)?

What is the double angle formula for cos(2α)?

What is the purpose of the CAST diagram?

What is the co-function identity for cos(90° - α)?

What is the step in the general method for solving trigonometric equations that involves adding multiples of the period?

What is the compound angle formula for sin(α + β)?

What is the purpose of the reference angle in the general method for solving trigonometric equations?

What is the formula used to calculate the area of triangle ABC if two sides and an included angle are known?

Which rule should be used if two sides and the included angle of a triangle are known?

Under what condition would you use the Sine Rule?

What is the correct expression to find $ heta$ if $ an heta = x$?

In triangle ABC, which ratio represents the Cosine Rule?

What is the formula to calculate the height of a pole given distance AB and angle FBA?

When using the Cosine Rule, which variables are essential?

To find the height of a building using the Sine Rule, which angles must be known?