Gr12 Mathematics: June mix P(2)

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

x^2 + y^2 = r^2 (correct)
x^2 - y^2 = r^2
x^2 / y^2 = r^2
x^2 * y^2 = r^2

What is a property of a circle with center at the origin?

It is asymmetric about the origin.
It is asymmetric about the y-axis.
It is asymmetric about the x-axis.
It is symmetric about the x-axis and y-axis. (correct)

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

(x - a)^2 - (y - b)^2 = r^2
(x - a)^2 / (y - b)^2 = r^2
(x - a)^2 * (y - b)^2 = r^2
(x - a)^2 + (y - b)^2 = r^2 (correct)

What is the purpose of completing the square in the equation of a circle?

To rewrite the equation in standard form. (A) Signup and view all the answers

What is the first step in completing the square to find the center and radius of a circle?

Group the x terms and the y terms. (B) Signup and view all the answers

What is the general form of a circle's equation?

x^2 + y^2 + Dx + Ey + F = 0 (A) Signup and view all the answers

What is the advantage of completing the square?

It allows us to find the center and radius of the circle. (D) Signup and view all the answers

What is used to derive the equation of a circle with center at the origin?

Pythagorean theorem (A) Signup and view all the answers

What is the formula for the area of a triangle?

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

What is the Mid-point Theorem?

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

What is the condition for two polygons to be similar?

They have the same number of sides and all pairs of corresponding angles and sides are in the same proportion. (D) Signup and view all the answers

What is the Proportion Theorem?

A line drawn parallel to one side of a triangle divides the other two sides proportionally. (D) 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 parallel to another side bisects the third side. (D) Signup and view all the answers

What is the condition for two triangles to be similar?

They have the same number of sides and all pairs of corresponding angles and sides are in the same proportion. (B) Signup and view all the answers

What is the formula for the proportion theorem?

AD/DB = AE/EC (C) Signup and view all the answers

What is the theorem that states equiangular triangles are similar?

The Theorem of Equiangular Triangles (A) Signup and view all the answers

What is the condition for triangles on the same base and equal in area?

They lie between parallel lines. (A) Signup and view all the answers

What is the definition of similar polygons?

Polygons with the same shape but different sizes. (A) Signup and view all the answers

What is the gradient of the radius of a circle?

The slope of the line from the center of the circle to the point of tangency. (A) Signup and view all the answers

Which of the following is NOT a property of proportion?

Inverse proportion (B) Signup and view all the answers

What is the standard form of the equation of a circle?

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

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

They are negative reciprocals. (D) Signup and view all the answers

If a line is drawn parallel to one side of a triangle, what does it do to the other two sides?

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

What is the equation of a tangent line to a circle given the center of the circle and the point of tangency?

y - y1 = m(x - x1) (B) Signup and view all the answers

What is a tangent line to a circle?

A line that touches the circle at exactly one point without crossing it. (D) Signup and view all the answers

What is the gradient of the line from the center of a circle to the point of tangency called?

Slope of the radius (A) Signup and view all the answers

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

They are negative reciprocals. (A) Signup and view all the answers

What is the standard form of the equation of a circle?

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

What is the formula for the area of a parallelogram?

$Area = base imes height$ (A) Signup and view all the answers

What is the name of the theorem that states if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally?

Basic Proportionality Theorem (A) Signup and view all the answers

What is the relationship between the areas of two triangles with the same height?

The areas are proportional to their bases. (C) Signup and view all the answers

What is the formula for the area of a trapezoid?

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

Which of the following statements is true about similar polygons?

Corresponding angles are equal, and corresponding sides are in proportion. (D) Signup and view all the answers

Which of the following polygons has a formula for its area that involves multiplying its two diagonals?

Rhombus (D) Signup and view all the answers

What is the relationship between the areas of two triangles with equal bases and between the same parallel lines?

The areas are equal. (B) Signup and view all the answers

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

All four sides are equal in length. (A) Signup and view all the answers

Which theorem states that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides?

Pythagoras' Theorem (B) Signup and view all the answers

What is the formula for the area of a square?

$Area = side^2$ (A) Signup and view all the answers

Which of the following is the correct formula for the sine of a sum?

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

Which of the following is a correct form of the cosine of a double angle formula?

cos(2α) = 1 - 2sin²α (C) Signup and view all the answers

What is the first step in deriving the double angle formula for sine?

Use the sine of a sum formula. (A) Signup and view all the answers

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

cos(2α) = 1 + 2sin²α (B) Signup and view all the answers

What is the purpose of using a CAST diagram when solving trigonometric equations?

To determine the quadrant where the solution lies. (B) Signup and view all the answers

Which of the following is the correct formula for the sine of a difference?

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

What is the formula for the cosine of a difference?

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

Which of the following is NOT a valid step in the general solution method for trigonometric equations?

Find the solutions within a specified interval by adding or subtracting multiples of the period. (A) Signup and view all the answers

What is the double angle formula for cosine in terms of sine?

cos(2α) = 1 - 2sin²α (B) Signup and view all the answers

Which of the following is the formula for the cosine of a sum?

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

Which trigonometric rule is used when no perpendicular height is given in a triangle?

Area Rule (D) Signup and view all the answers

When is the Sine Rule used?

When two angles and a side are given (B) Signup and view all the answers

What is the formula for the area of triangle ABC?

All of the above (D) Signup and view all the answers

Which trigonometric rule is used to find the length of a side in a triangle when two sides and the included angle are given?

Cosine Rule (C) Signup and view all the answers

If (\sin \theta = x), which of the following represents the general solution for (\theta)?

$\theta = \sin^{-1} x + k \cdot 360^\circ$ (A), $\theta = 180^\circ - \sin^{-1} x + k \cdot 360^\circ$ (C) Signup and view all the answers

Which of the following is the general solution for (\theta) if (\cos \theta = x)?

$\theta = \cos^{-1} x + k \cdot 360^\circ$ (A), $\theta = 360^\circ - \cos^{-1} x + k \cdot 360^\circ$ (D) Signup and view all the answers

What is the general solution for (\theta) when (\tan \theta = x)?

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

Which of the following is the correct formula for the Cosine Rule?

$\a^2 = b^2 + c^2 - 2bc \cos A$ (C) Signup and view all the answers

What must be proven for triangles to establish similarity?

Equiangularity or proportionality of sides (D) Signup and view all the answers

Which formula represents the cosine of a difference?

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

Which condition is necessary for polygons to be similar?

All corresponding angles are equal and all corresponding sides are in the same proportion (A) Signup and view all the answers

What is the cosine of a sum formula?

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

What is the correct expression for sin(α - β)?

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

Which identity is used in the derivation of cos(α + β)?

Negative angle identity (B) Signup and view all the answers

How is cos(α - β) derived using the distance formula?

By using coordinates on the unit circle (B) Signup and view all the answers

Which of the following best describes sine addition?

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

Which of the following statements is true about triangles with sides in proportion?

They are always similar. (A) Signup and view all the answers

In the proof of the Pythagorean Theorem, why is (\triangle ABD) similar to (\triangle CBA)?

AAA similarity criterion (D) Signup and view all the answers

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

They have equal areas. (C) Signup and view all the answers

In the proof of the theorem 'Triangles with Sides in Proportion are Similar', why is (GH) parallel to (BC)?

Sides are in proportion. (B) Signup and view all the answers

What is the key concept used to prove the similarity of triangles in the text?

Angle-Angle-Angle (AAA) criterion. (D) Signup and view all the answers

Which of these is NOT a direct consequence of the proportionality theorem?

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

What is the purpose of constructing line segment (GH) in the proof of the theorem 'Triangles with Sides in Proportion are Similar'?

To prove that (\triangle AGH) is similar to (\triangle ABC). (D) Signup and view all the answers

What is the formula for the area of a triangle?

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

What is the name of the theorem that states if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally?

Basic Proportionality Theorem (C) Signup and view all the answers

What is the formula for the area of a parallelogram?

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

What is the relationship between the areas of two triangles with the same height?

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

What is the formula for the area of a rhombus?

Area = diagonal1 × diagonal2 / 2 (A) Signup and view all the answers

What is the formula for the area of a trapezium?

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

What is the relationship between the areas of two triangles with equal bases and between the same parallel lines?

They are equal (A) Signup and view all the answers

What is the formula for the area of a kite?

Area = diagonal1 × diagonal2 / 2 (A) Signup and view all the answers

Which of the following polygons has a formula for its area that involves multiplying its two diagonals?

Kite (C), Rhombus (D) Signup and view all the answers

What is the name of the theorem that states that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides?

Pythagoras' Theorem (B) Signup and view all the answers

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

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

Which of the following is NOT a symmetry property of a circle with center at the origin?

Symmetry about the line y = 2x (D) Signup and view all the answers

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

3 (C) Signup and view all the answers

What is the center of the circle represented by the equation x^2 + y^2 + 8x - 10y + 16 = 0?

(-4, 5) (B) Signup and view all the answers

Which of the following is the correct step to complete the square for the x terms in the equation x^2 - 10x + y^2 + 6y = 12?

Add 25 to the left side and 25 to the right side (B) Signup and view all the answers

What is the equation of the circle with center at the origin and passing through the point (4, 3)?

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

The equation of a circle is given as (x + 2)^2 + (y - 1)^2 = 9. What is the center of the circle?

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

Which of the following points lies on the circle with equation (x - 1)^2 + (y + 3)^2 = 16?

(5, -3) (D) Signup and view all the answers

If a line is drawn parallel to one side of a triangle, how does it divide the other two sides?

It divides the other two sides in the same proportion. (A) Signup and view all the answers

Two triangles are considered similar if they have:

The same shape but different sizes. (C) Signup and view all the answers

What is the relationship between the areas of two triangles with equal bases and between the same parallel lines?

The areas of the two triangles are equal. (C) Signup and view all the answers

What is the relationship between the areas of two triangles with equal heights?

The areas of the two triangles are proportional to their bases. (D) Signup and view all the answers

If two triangles are equiangular, what is the relationship between their corresponding sides?

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

The line joining the midpoints of two sides of a triangle is:

Parallel to the third side and equal to half the third side. (B) 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 shape. (B) Signup and view all the answers

Two triangles on the same side of the same base and equal in area lie:

Between two parallel lines. (A) Signup and view all the answers

If a line drawn from the midpoint of one side of a triangle is parallel to another side, what does it do to the third side?

It divides the third side into equal lengths. (A) Signup and view all the answers

What is the relationship between the gradients of the tangent line and the radius of a circle at the point of tangency?

They are perpendicular. (B) Signup and view all the answers

What is the compound angle identity for (\cos(\alpha - \beta))?

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

Which of the following is used to rewrite (\cos(\alpha + \beta)) as a difference of angles for deriving its compound angle identity?

Negative angle identity (C) Signup and view all the answers

Which compound angle identity can be used to directly derive (\cos(\alpha + \beta))?

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

The derivation of (\cos(\alpha - \beta)) involves equating two expressions. What are these expressions?

Distance formula and cosine rule (C) Signup and view all the answers

What is the compound angle identity for (\sin(\alpha - \beta))?

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

Which of the following identities is NOT used in the derivation of (\cos(\alpha + \beta))?

Double angle identity (D) Signup and view all the answers

What is the compound angle identity for (\sin(\alpha + \beta))?

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

In the derivation of (\cos(\alpha - \beta)), what is the relationship between the points K and L on the unit circle?

They are the endpoints of a chord (C) Signup and view all the answers

What is the formula for the radius of the circle in standard form?

r = rac{D^2 + E^2}{4} - F (A) Signup and view all the answers

Which formula correctly relates the gradients of the radius and the tangent line at the point of tangency?

m_{radius} imes m_{tangent} = -1 (C) Signup and view all the answers

In the equation of the circle, what does the term rac{E}{2} represent?

The y-coordinate of the center (A) Signup and view all the answers

Which of the following properties holds true if two ratios are in proportion?

The sum of the products of the extremes equals the sum of the products of the means. (C) Signup and view all the answers

Which statement is true about the properties of proportion?

If rac{w}{x} = rac{y}{z}, then w imes y = x imes z. (D) Signup and view all the answers

How is the equation of a tangent line to a circle at the point of tangency written?

y - y_1 = m_{tangent}(x - x_1) (D) Signup and view all the answers

What does the concept of ratio allow us to compare?

Two quantities with the same units (A) Signup and view all the answers

What does Thales' theorem state about a line drawn parallel to one side of a triangle?

It creates similar triangles with the original triangle. (A) Signup and view all the answers

What aspect of a circle is indicated by the center coordinates (a, b) in its standard form?

The location of the circle's center (A) Signup and view all the answers

What occurs if a line is drawn parallel to one side of a triangle in terms of the other two sides?

The other two sides are divided proportionally. (B) Signup and view all the answers

What is the compound angle formula for the sine of the difference of two angles?

$rac{ an(eta + eta)}{1 - an eta an eta}$ (B) Signup and view all the answers

How can the cosine of a sum be expressed using sine and cosine functions?

$ an(eta + eta) = rac{ an eta + an eta}{1 - an eta an eta}$ (C) Signup and view all the answers

Which of the following represents the correct formula for sine of double angle?

$ an(2 heta) = rac{2 an heta}{1 - an^2 heta}$ (D) Signup and view all the answers

What is the correct expression for the cosine of double angle in terms of sine?

$ an(2 heta) = rac{1 - 2rac{ an^2 heta}{ an^2 heta}}{1 + 2 an^2 heta}$ (B) Signup and view all the answers

What is the general method for solving trigonometric equations?

Identify exact values and their periodicities. (A) Signup and view all the answers

Which equation correctly represents the sine of a sum?

$ an( heta + heta) = rac{ an heta + an heta}{1 - an^2 heta}$ (A) Signup and view all the answers

When deriving the cosine of a double angle, which identity is essential?

$ ext{sec}^2 heta = 1 + an^2 heta$ (C) Signup and view all the answers

Which statement best describes the co-function identity for sine?

$ an(90^ ext{circ} - heta) = rac{1}{ an heta}$ (B) Signup and view all the answers

What does the CAST diagram help determine when solving trigonometric equations?

Quadrants where trigonometric functions are positive or negative. (A) Signup and view all the answers

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

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

Which statement accurately describes the proportionality theorem as stated in the content?

The ratios of the lengths of the sides of two triangles are equal. (D) Signup and view all the answers

How is the area of a triangle calculated?

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

What can be inferred if two triangles are equiangular?

They are similar triangles. (B) Signup and view all the answers

What does the Pythagorean theorem state?

The square on the hypotenuse equals the sum of the squares of the adjacent sides. (B) Signup and view all the answers

What does it mean when two triangles are similar?

Their corresponding angles are equal and their sides are in proportion. (B) Signup and view all the answers

How is similarity in polygons determined?

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

Which of the following accurately represents the converse of the Pythagorean theorem?

If the triangle's sides satisfy the Pythagorean theorem, then it is a right-angled triangle. (D) Signup and view all the answers

What is a necessary condition for a triangle to have equal areas when sharing a common base?

They must have equal altitudes. (B) Signup and view all the answers

When should the Cosine Rule be applied in triangle problems?

When two sides and the included angle are known. (A), When all three sides are known. (D) Signup and view all the answers

What is the correct general solution for the equation \tan \theta = x?

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

In the context of triangles, which rule allows the calculation of an area when no height is provided?

Area Rule (A) Signup and view all the answers

Which of the following equations represents the Sine Rule?

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} (D) Signup and view all the answers

To find the height of a pole using trigonometric ratios, which of the following is used?

Combining the Sine Rule and the Tangent Ratio. (C) Signup and view all the answers

What does the area of triangle ABC equal when using the Area Rule?

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

If \sin \theta = x, which relationships can determine \theta?

\theta = 180^\circ - \sin^{-1} x + k \cdot 360^\circ (B), \theta = \sin^{-1} x + k \cdot 360^\circ (C) Signup and view all the answers

Which condition indicates that the Sine Rule should be used?

Two angles and a side are known. (D) Signup and view all the answers

What is the formula for the area of a parallelogram?

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

What is the name of the theorem that states if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally?

Basic Proportionality Theorem (D) Signup and view all the answers

What is the formula for the area of a triangle?

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

What is the relationship between the areas of two triangles with the same height?

The areas are proportional to their bases (C) Signup and view all the answers

What is the formula for the area of a kite?

Area = (diagonal1 × diagonal2) / 2 (D) Signup and view all the answers

What is the relationship between the areas of two triangles with equal bases and between the same parallel lines?

The areas are equal (B) Signup and view all the answers

What is the name of the theorem that states the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides?

Pythagoras' Theorem (D) Signup and view all the answers

What is the formula for the area of a trapezoid?

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

What is the formula for the area of a rhombus?

Area = (diagonal1 × diagonal2) / 2 (D) Signup and view all the answers

What is the formula for the area of a square?

Area = side × side (B) Signup and view all the answers

Given the equation of a circle in standard form: (x - a)^2 + (y - b)^2 = r^2, what is the coordinate of the center of the circle?

(a, b) (C) Signup and view all the answers

What is the gradient of the tangent line to a circle at a point of tangency, if the gradient of the radius to that point is $m_{radius}$?

$-1/m_{radius}$ (A) Signup and view all the answers

If a line is drawn parallel to one side of a triangle, what relationship does it have with the other two sides?

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

If the ratio of two quantities is 3:4, what is the ratio of their reciprocals?

4:3 (A) Signup and view all the answers

Which of the following is NOT a property of proportion?

Adding corresponding terms (A) Signup and view all the answers

What is the simplified form of the equation: (x + D/2)^2 + (y + E/2)^2 = (D/2)^2 + (E/2)^2 - F?

(x - D/2)^2 + (y - E/2)^2 = F (D) Signup and view all the answers

What is the radius of the circle represented by the equation: (x - 2)^2 + (y + 3)^2 = 16?

4 (A) Signup and view all the answers

What is the gradient of the radius of the circle (x - 3)^2 + (y + 2)^2 = 25 at the point of tangency (5, 1)?

3/2 (B) Signup and view all the answers

If the equation of a circle is x^2 + y^2 - 6x + 4y - 12 = 0, what is the equation of the tangent line at the point (5, 3)?

y - 3 = (3/2)(x - 5) (C) Signup and view all the answers

Given a triangle ABC with a line DE parallel to BC and intersecting AB and AC at D and E respectively. If AD = 4, DB = 6, and AE = 5, what is the length of EC?

7.5 (D) Signup and view all the answers

Using the formula for (\cos(\alpha - \beta)), what is the simplified expression for (\cos(180^\circ - \theta))?

(-\cos(\theta)) (D) Signup and view all the answers

What is the formula for (\sin(\alpha + \beta)) in terms of sine and cosine of (\alpha) and (\beta)?

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

Which of the following correctly expresses (\cos(\alpha + \beta)) using the negative angle identity?

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

Given that triangles ABC and DEF have sides in proportion, what can we conclude about the triangles?

The triangles are similar. (D) Signup and view all the answers

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

They have three pairs of corresponding angles equal. (D) Signup and view all the answers

If two triangles have equal heights and their bases are in the ratio 2:3, what is the ratio of their areas?

2:3 (D) Signup and view all the answers

Which of the following statements is true about similar polygons?

Corresponding angles of similar polygons are equal. (D) Signup and view all the answers

What is the relationship between the areas of two triangles with the same base and between the same parallel lines?

The areas are equal. (C) Signup and view all the answers

In the proof of the Pythagorean Theorem, what is the purpose of drawing AD perpendicular to BC?

All of the above. (D) Signup and view all the answers

What is the converse of the Pythagorean Theorem?

If the square on one side of a triangle is equal to the sum of the squares on the other two sides, then the triangle is a right triangle. (A) Signup and view all the answers

What is the importance of the Proportionality Theorem in the context of similar triangles?

It states that if two triangles are similar, their sides are in proportion. (B) Signup and view all the answers

Which of the following conditions is NOT sufficient to prove that two triangles are similar?

Two pairs of corresponding angles are equal. (B) Signup and view all the answers

What is the significance of the Pythagorean Theorem in geometry?

It is a fundamental theorem that relates the sides of a right-angled triangle. (D) Signup and view all the answers

What can be concluded about triangles on the same base and equal in area?

They lie between parallel lines. (A) Signup and view all the answers

What does the Proportion Theorem state regarding a line drawn parallel to one side of a triangle?

It divides the other two sides in the same ratio. (A) 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. (D) Signup and view all the answers

What must be true for two polygons to be considered similar?

Their angles must be equal and sides must be proportional. (A) Signup and view all the answers

What does the equation of a circle with center at the origin state about any point on its circumference?

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

Which statement is correct regarding equiangular triangles?

Their sides are in proportion. (B) Signup and view all the answers

For a circle with center at point (a, b), how is the general equation expressed?

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

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

A line from the midpoint of one side parallel to another side bisects the third side. (A) Signup and view all the answers

What is a unique characteristic of a circle with respect to its symmetry?

It is symmetric about the origin. (B) Signup and view all the answers

How are similar polygons defined?

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

What is one method to prove that two triangles are similar?

Show that two pairs of corresponding angles are equal. (A) Signup and view all the answers

What is the purpose of completing the square in the equation of a circle?

To find the center and radius of the circle. (C) Signup and view all the answers

Which of the following steps is NOT part of completing the square for the circulation equation?

Finding roots of the resulting quadratic equations. (C) Signup and view all the answers

In the context of triangle areas, when triangles have the same height, what can be said about their bases?

Their areas are proportional to their bases. (C) Signup and view all the answers

Why is the expression $OP = r$ significant in deriving the equation of a circle?

It relates the radius to a point in Cartesian coordinates. (B) Signup and view all the answers

If two triangles are equiangular, what conclusion can be drawn?

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

Which equation corresponds to the standard form of a circle centered at (0, 0) with radius changes?

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

In which scenario would you use the equation $(x - a)^2 + (y - b)^2 = r^2$?

When the center of the circle is at a point (a, b). (C) Signup and view all the answers

Which of the following is a correct formula for the cosine of a double angle?

$\cos(2\alpha) = 2\cos^2\alpha - 1$ (A), $\cos(2\alpha) = \cos^2\alpha - \sin^2\alpha$ (B), $\cos(2\alpha) = 1 - 2\sin^2\alpha$ (C) Signup and view all the answers

What is the first step in deriving the double angle formula for sine?

Start with the sum formula for sine. (B) Signup and view all the answers

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

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

Which of the following is the correct formula for the sine of a sum?

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

Which trigonometric rule is used when no perpendicular height is given in a triangle?

The Cosine Rule (D) Signup and view all the answers

Which of the following is the correct formula for the sine of a difference?

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

Which of the following is the formula for the cosine of a difference?

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

What is the purpose of using a CAST diagram when solving trigonometric equations?

To determine where the function is positive or negative. (D) Signup and view all the answers

Which of the following is NOT a valid step in the general solution method for trigonometric equations?

Find angles in the interval ([0^\circ, 360^\circ]) that satisfy the equation and add multiples of the period, using the reference angle and the CAST diagram. (C) Signup and view all the answers

Which of the following is the formula for the cosine of a sum?

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

Which equation represents the correct general solution for $ an heta = x $?

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

When should the Sine Rule be applied in triangle problems?

When no right angle is given and two angles and a side are specified. (C) Signup and view all the answers

Which formula correctly calculates the area of triangle ABC using the Sine Rule?

$ \text{Area} = \frac{1}{2}bc \sin A $ (B) Signup and view all the answers

What conditions necessitate the use of the Cosine Rule?

When three sides are known, or two sides and the included angle are specified. (C) Signup and view all the answers

Which of the following relationships is true when using the Sine Rule?

$ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} $ (A) Signup and view all the answers

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

The height of a pole. (C) Signup and view all the answers

What is the correct generalization for the area of triangle ABC if only the lengths of two sides and the included angle are known?

$ \text{Area} = \frac{1}{2}ab \sin C $ (C) Signup and view all the answers

According to the Sine Rule, what is ( BD ) expressed in terms of length ( b ), angle ( \theta ), ( \alpha ), and ( \beta )?

$ BD = \frac{b \sin \theta}{\sin(\beta + \theta)} $ (A) Signup and view all the answers

If a circle has a center at (a, b) and radius r, what is the equation of the circle?

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

What is the significance of completing the square in the equation of a circle?

To find the center and radius of the circle. (B) Signup and view all the answers

A circle with center at the origin is symmetric about which of the following?

The x-axis, y-axis, and the origin (C) Signup and view all the answers

What is the first step in completing the square to find the center and radius of a circle?

Group the x terms and the y terms (B) 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 (B) Signup and view all the answers

What is the purpose of rewriting the equation of a circle in standard form?

To find the center and radius of the circle. (A) Signup and view all the answers

What is the significance of the distance formula in the derivation of the equation of a circle?

It helps to derive the equation of the circle. (D) Signup and view all the answers

What is the relationship between the equation of a circle with center at the origin and the equation of a circle with center at (a, b)?

They are different equations with the same form. (C) Signup and view all the answers

If two triangles are similar, which of the following statements is true?

The corresponding sides are in proportion and the corresponding angles are equal. (C) Signup and view all the answers

What is the formula for the area of a trapezoid?

Area = (base1 + base2) × height (A) Signup and view all the answers

If a line is drawn parallel to one side of a triangle, what does it do to the other two sides?

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

What is the formula for the area of a rhombus?

Area = (diagonal1 × diagonal2) / 2 (A) Signup and view all the answers

What is the name of the theorem that states if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally?

Thales' Theorem (A) Signup and view all the answers

What is the formula for the area of a kite?

Area = (diagonal1 × diagonal2) / 2 (B) Signup and view all the answers

What is the relationship between the areas of two triangles with the same height?

The areas are proportional to their bases. (C) Signup and view all the answers

What is the relationship between the areas of two triangles with equal bases and between the same parallel lines?

The areas are equal. (D) Signup and view all the answers

Which of the following is a property of similar polygons?

The corresponding angles are equal, and the corresponding sides are in proportion. (B) Signup and view all the answers

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

The diagonals are parallel. (B) Signup and view all the answers

If two triangles have equal heights and their bases are in the ratio 3:5, what is the ratio of their areas?

9:25 (A) Signup and view all the answers

In the figure below, if DE || BC and AD:DB = 2:3, what is the ratio of the area of triangle ADE to the area of triangle ABC?

4:9 (A) Signup and view all the answers

If two triangles are similar, which of the following statements is NOT true?

Their areas are equal. (A) Signup and view all the answers

In the figure below, if PQRST is similar to ABCDE, what is the ratio of PQ to AB?

2:3 (C) Signup and view all the answers

If a line is drawn parallel to one side of a triangle, what does it do to the other two sides?

Divides them proportionally. (A) Signup and view all the answers

What is the condition for two triangles to be similar?

They have equal corresponding angles and proportional corresponding sides. (C) Signup and view all the answers

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

They are proportional. (C) Signup and view all the answers

What is 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 its length. (D) Signup and view all the answers

What is the condition for two polygons to be similar?

They have the same shape but differ in size. (D) Signup and view all the answers

What is the Converse of the Mid-point Theorem?

A line drawn from the midpoint of one side of a triangle parallel to another side bisects the third side. (B) Signup and view all the answers

Which of the following is an alternate form of the cosine of a double angle formula?

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

If $\sin(\alpha - eta) = rac{1}{2}$, what is the value of $\cos(\alpha + eta)$?

$-rac{1}{2}$ (D) Signup and view all the answers

What is the derivation of $\sin(2\alpha)$ using the sum formula for sine?

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

What is the general solution method for trigonometric equations?

Simplify, Reference Angle, CAST Diagram, Restricted Values, General Solution, Check (A) Signup and view all the answers

What is the formula for $\cos(\alpha - eta)$?

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

What is the purpose of using a CAST diagram when solving trigonometric equations?

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

If $\cos(2\alpha) = rac{1}{2}$, what is the value of $\sin \alpha$?

$\pmrac{\sqrt{3}}{2}$ (C) Signup and view all the answers

What is the formula for $\sin(\alpha + eta)$?

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

What is the derivation of $\cos(2\alpha)$ using the sum formula for cosine?

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

What is the general solution for the equation $\sin( heta) = rac{1}{2}$?

$ heta = 30^\circ + 360^\circ k$ (B) Signup and view all the answers

What is the correct expression for the radius of a circle in standard form using parameters D, E, and F?

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

What is the gradient of the tangent line in relation to the gradient of the radius at the point of tangency?

$m_{tangent} = -m_{radius}$ (C) Signup and view all the answers

If the center of a circle is at (a, b), how would you express the gradient of the radius to a point of tangency (x1, y1)?

$m_{radius} = rac{y1 - b}{x1 - a}$ (D) Signup and view all the answers

What does the Basic Proportionality Theorem state about a triangle when a line is drawn parallel to one of its sides?

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

What is the relationship expressed by the reciprocal proportion of two ratios $\frac{w}{x} = \frac{y}{z}$?

$\frac{x}{w} = \frac{y}{z}$ (C) Signup and view all the answers

To determine the equation of a tangent, what is the correct point-slope form using the point of tangency (x1, y1) and the tangent's gradient?

$y - y1 = m_{tangent}(x - x1)$ (A) Signup and view all the answers

If a line is drawn parallel to one side of a triangle, what is the effect on the segments it creates?

It divides the other two sides in proportion. (D) Signup and view all the answers

What mathematical property does the statement $\frac{w}{x} = \frac{y}{z}$ exemplify?

Cross Multiplication. (C) Signup and view all the answers

What key characteristic defines a diagonal ratio in relation to areas of similar polygons?

It dictates the relationship between the areas of triangles. (A) Signup and view all the answers

If triangles ABC and DEF have corresponding sides in proportion, which angles must necessarily be equal?

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

Which of the following conditions guarantees the similarity of two triangles?

All corresponding angles are equal (B) Signup and view all the answers

For two triangles with equal height, how is their area related to their bases?

Areas are directly proportional to their bases (D) Signup and view all the answers

What conclusion can be made about the triangles AGH and ABC if AG is equal to DE and AH is equal to DF?

Triangle AGH is similar to triangle DEF (D) Signup and view all the answers

In proving the Pythagorean theorem, what relationship is established between the triangles ABD and CBA?

They are similar triangles (D) Signup and view all the answers

Which of the following correctly states the Pythagorean theorem?

$BC^2 = AB^2 + AC^2$ (B) Signup and view all the answers

If triangles ABC and DEF are equiangular, what can be said about their corresponding sides?

Their corresponding sides are in proportion (A) Signup and view all the answers

Which construction is essential for proving that \triangle AGH is similar to \triangle ABC?

Drawing a line parallel to BC from G (D) Signup and view all the answers

What does it imply if two triangles' sides are not in proportion?

The triangles are not similar (C) Signup and view all the answers

Using the distance formula and cosine rule, what is the first step in deriving the formula for (\cos(\alpha - \beta))?

Find the distance between two points on the unit circle, (K) and (L), using the distance formula. (A) Signup and view all the answers

What is the primary purpose of using the negative angle identity in the derivation of the formula for (\cos(\alpha + \beta))?

To express (\cos(\alpha + \beta)) as (\cos(\alpha - (-eta))) and apply the cosine difference formula. (C) Signup and view all the answers

During the construction of triangle AGH, GH is drawn parallel to which side, and why?

GH is parallel to BC to ensure corresponding angles are equal (B) Signup and view all the answers

Which of the following correctly expresses the cosine of a sum in terms of the cosine of a difference?

(\cos(\alpha + \beta) = \cos(\alpha - (-eta))) (A) Signup and view all the answers

Which trigonometric identity is directly used to derive the formula for (\sin(\alpha + \beta)) after applying the cosine difference formula?

Sine of a difference (B) Signup and view all the answers

Which of the following steps is NOT required in the derivation of (\sin(\alpha + \beta)) using the cosine difference formula and other trigonometric identities?

Applying the Pythagorean identity to the expression obtained in step (b). (C) Signup and view all the answers

What is the key difference between deriving (\cos(\alpha + \beta)) and deriving (\sin(\alpha + \beta))?

The derivation of (\cos(\alpha + \beta)) involves expressing the sum as a difference, while the derivation of (\sin(\alpha + \beta)) involves expressing the sum as a co-function. (D) Signup and view all the answers

Which of the following is NOT a direct consequence of the derivation of the cosine of a difference formula?

The tangent of a difference and tangent of a sum formulas can be derived directly from the sine of a difference and cosine of a difference formulas. (B) Signup and view all the answers

What is the essential role of the distance formula in the derivation of the cosine of a difference formula?

The distance formula is used to relate the distance between the two points on the unit circle to the cosine of the angle between their radii. (D) Signup and view all the answers

What is the correct application of the Area Rule for calculating the area of a triangle?

It is used when no perpendicular height is given. (B) Signup and view all the answers

Given $ an heta = x$, what is a general solution for $ heta$?

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

In which scenario would you use the Cosine Rule?

When three sides are known. (C) Signup and view all the answers

What formula is used to find the height of a pole given angle measurements and distance?

$ h = FB an eta $ (C) Signup and view all the answers

Which of the following correctly describes the Sine Rule's application?

It is used when two angles and a side are known. (A), It is used when one angle and two sides are known. (B) Signup and view all the answers

For the equation $ a^2 = b^2 + c^2 - 2bc \cos A \, $ what does it represent?

The Cosine Rule applied to any triangle. (A) Signup and view all the answers

What is the correct formulation to calculate the area of triangle ABC using the Sine Rule?

$ ext{Area} = rac{1}{2}bc \sin B $ (C) Signup and view all the answers

Gr12 Mathematics: June mix P(2)

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Questions and Answers

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

What is a property of a circle with center at the origin?

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

What is the purpose of completing the square in the equation of a circle?

What is the first step in completing the square to find the center and radius of a circle?

What is the general form of a circle's equation?

What is the advantage of completing the square?

What is used to derive the equation of a circle with center at the origin?

What is the formula for the area of a triangle?

What is the Mid-point Theorem?

What is the condition for two polygons to be similar?

What is the Proportion Theorem?

What is the Converse of the Mid-point Theorem?

What is the condition for two triangles to be similar?

What is the formula for the proportion theorem?

What is the theorem that states equiangular triangles are similar?

What is the condition for triangles on the same base and equal in area?

What is the definition of similar polygons?

What is the gradient of the radius of a circle?

Which of the following is NOT a property of proportion?

What is the standard form of the equation of a circle?

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

If a line is drawn parallel to one side of a triangle, what does it do to the other two sides?

What is the equation of a tangent line to a circle given the center of the circle and the point of tangency?

What is a tangent line to a circle?

What is the gradient of the line from the center of a circle to the point of tangency called?

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

What is the standard form of the equation of a circle?

What is the formula for the area of a parallelogram?

What is the name of the theorem that states if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally?

What is the relationship between the areas of two triangles with the same height?

What is the formula for the area of a trapezoid?

Which of the following statements is true about similar polygons?

Which of the following polygons has a formula for its area that involves multiplying its two diagonals?

What is the relationship between the areas of two triangles with equal bases and between the same parallel lines?

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

Which theorem states that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides?

What is the formula for the area of a square?

Which of the following is the correct formula for the sine of a sum?

Which of the following is a correct form of the cosine of a double angle formula?

What is the first step in deriving the double angle formula for sine?

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

What is the purpose of using a CAST diagram when solving trigonometric equations?

Which of the following is the correct formula for the sine of a difference?

What is the formula for the cosine of a difference?

Which of the following is NOT a valid step in the general solution method for trigonometric equations?

What is the double angle formula for cosine in terms of sine?

Which of the following is the formula for the cosine of a sum?

Which trigonometric rule is used when no perpendicular height is given in a triangle?

When is the Sine Rule used?

What is the formula for the area of triangle ABC?

Which trigonometric rule is used to find the length of a side in a triangle when two sides and the included angle are given?

If (\sin \theta = x), which of the following represents the general solution for (\theta)?

Which of the following is the general solution for (\theta) if (\cos \theta = x)?

What is the general solution for (\theta) when (\tan \theta = x)?

Which of the following is the correct formula for the Cosine Rule?

What must be proven for triangles to establish similarity?

Which formula represents the cosine of a difference?

Which condition is necessary for polygons to be similar?

What is the cosine of a sum formula?

What is the correct expression for sin(α - β)?

Which identity is used in the derivation of cos(α + β)?

How is cos(α - β) derived using the distance formula?

Which of the following best describes sine addition?

Which of the following statements is true about triangles with sides in proportion?

In the proof of the Pythagorean Theorem, why is (\triangle ABD) similar to (\triangle CBA)?

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

In the proof of the theorem 'Triangles with Sides in Proportion are Similar', why is (GH) parallel to (BC)?

What is the key concept used to prove the similarity of triangles in the text?

Which of these is NOT a direct consequence of the proportionality theorem?

What is the purpose of constructing line segment (GH) in the proof of the theorem 'Triangles with Sides in Proportion are Similar'?

What is the formula for the area of a triangle?

What is the name of the theorem that states if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally?

What is the formula for the area of a parallelogram?

What is the relationship between the areas of two triangles with the same height?

What is the formula for the area of a rhombus?