Gr12 Mathematics: June Easy P(2)
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Questions and Answers

What is the formula to find the area of a triangle?

  • $base - height$
  • $base \times height$
  • $\frac{1}{2} \times base \times height$ (correct)
  • $base + height$

    • 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?

  • Triangle Proportionality Theorem
  • Thales' Theorem (correct)
  • Pythagoras' Theorem
  • Basic Similarity Theorem

    • What is the formula to find the area of a parallelogram?

  • $\frac{1}{2} \times base \times height$
  • $base + height$
  • $base \times height$ (correct)
  • $base - height$

    • What is the name of the polygon with four sides?

    <p>Quadrilateral (B)</p> Signup and view all the answers

    What is the formula to find the area of a kite?

    <p>$\frac{1}{2} \times diagonal AC \times diagonal BD$ (D)</p> Signup and view all the answers

    What is the condition for two triangles to be similar?

    <p>The corresponding sides are in proportion and the corresponding angles are equal (D)</p> Signup and view all the answers

    What is the formula to find the area of a trapezium?

    <p>$\frac{1}{2} \times (base_1 + base_2) \times height$ (C)</p> Signup and view all the answers

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

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

    What is the name of the theorem that states the square on the hypotenuse is equal to the sum of the squares on the other two sides?

    <p>Pythagoras' Theorem (C)</p> Signup and view all the answers

    What is the formula to find the area of a rhombus?

    <p>$\frac{1}{2} \times diagonal AC \times diagonal BD$ (D)</p> 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?

    <p>m_radius * m_tangent = -1 (A)</p> Signup and view all the answers

    What is the condition for two polygons to be similar?

    <p>The corresponding sides are in proportion and the corresponding angles are equal (C)</p> Signup and view all the answers

    What is the property of proportion that states wz = xy?

    <p>Cross Multiplication (A)</p> Signup and view all the answers

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

    <p>Thales' Theorem (C)</p> Signup and view all the answers

    What is the definition of a tangent to a circle?

    <p>A straight line that touches the circle at exactly one point (D)</p> Signup and view all the answers

    What is the purpose of completing the square for the x terms and the y terms?

    <p>To put the equation of the circle in standard form (C)</p> Signup and view all the answers

    What is the relationship between the center of the circle and the point of tangency?

    <p>The radius is perpendicular to the tangent at the point of tangency (D)</p> Signup and view all the answers

    What is the formula for the gradient of the tangent?

    <p>m_tangent = -1 / m_radius (A)</p> Signup and view all the answers

    What is the definition of a ratio?

    <p>A comparison of two quantities with the same units (C)</p> Signup and view all the answers

    What is the purpose of simplifying ratios?

    <p>To compare the quantities more easily (D)</p> Signup and view all the answers

    What is the formula for the area of a triangle?

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

    What can be concluded about two triangles with equal heights?

    <p>Their areas are proportional to their bases. (A)</p> Signup and view all the answers

    What is the statement of the Proportion Theorem?

    <p>A line drawn parallel to one side of a triangle divides the other two sides proportionally. (D)</p> Signup and view all the answers

    What is the Mid-point Theorem?

    <p>The line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. (D)</p> Signup and view all the answers

    What is true about similar polygons?

    <p>They have the same shape but differ in size and orientation. (B)</p> Signup and view all the answers

    What are the conditions for similarity of two polygons?

    <p>All pairs of corresponding angles are equal, and all pairs of corresponding sides are in the same proportion. (C)</p> Signup and view all the answers

    What is the converse of the Mid-point Theorem?

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

    What is the formula for the areas of two triangles with the same height?

    <p>Area1 : Area2 = Base1 : Base2 (B)</p> Signup and view all the answers

    What is true about two triangles with equal bases and between the same parallel lines?

    <p>They have equal areas. (B)</p> Signup and view all the answers

    What can be concluded about two triangles with proportional sides?

    <p>They are similar and have proportional areas. (A)</p> Signup and view all the answers

    What is the condition required to prove that two triangles are similar?

    <p>Both b and c (D)</p> Signup and view all the answers

    What is the statement of the theorem that equiangular triangles are similar?

    <p>If two triangles have equal corresponding angles, they are similar. (B)</p> Signup and view all the answers

    What is the conclusion of the proof that equiangular triangles are similar?

    <p>$\frac{AB}{DE} = \frac{AC}{DF} = \frac{BC}{EF}$ (B)</p> Signup and view all the answers

    What is the method used in the proof that triangles with sides in proportion are similar?

    <p>Proportionality theorem (B)</p> Signup and view all the answers

    What is the result of the theorem that triangles with sides in proportion are similar?

    <p>The two triangles are similar. (B)</p> Signup and view all the answers

    What is the name of the theorem that states the square on the hypotenuse is equal to the sum of the squares on the other two sides?

    <p>Pythagorean theorem (A)</p> Signup and view all the answers

    What is the diagram used in the proof of the Pythagorean theorem?

    <p>Right-angled triangle with a square drawn on each side (B)</p> Signup and view all the answers

    What is the formula for the area of a triangle?

    <p>Area = 0.5 * base * height (A)</p> Signup and view all the answers

    What is the concept of proportionality in triangles?

    <p>Triangles with equal heights have areas proportional to their bases. (B)</p> Signup and view all the answers

    What is the property of similar triangles?

    <p>Corresponding sides are in proportion. (B)</p> Signup and view all the answers

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

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

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

    <p>line y = -x + 1 (B)</p> Signup and view all the answers

    What is the center and radius of the circle with the equation (x + 2)^2 + (y - 3)^2 = 16?

    <p>center: (-2, 3), radius = 4 (C)</p> Signup and view all the answers

    Which of the following steps is NOT involved in completing the square to find the center and radius of a circle given in general form?

    <p>Divide both sides of the equation by the coefficient of x^2 and y^2 (C)</p> Signup and view all the answers

    What is the equation of a circle with center (3, -2) and radius 7?

    <p>(x - 3)^2 + (y + 2)^2 = 49 (D)</p> Signup and view all the answers

    What is the equation of a circle with center (0, 0) and passing through the point (4, 3)?

    <p>x^2 + y^2 = 25 (E)</p> Signup and view all the answers

    Which of the following equations represents a circle with center at (-1, 2) and a radius of 3?

    <p>(x + 1)^2 + (y - 2)^2 = 9 (C)</p> Signup and view all the answers

    What is the radius of the circle with the equation x^2 + y^2 - 6x + 4y - 3 = 0?

    <p>√13 (B)</p> Signup and view all the answers

    What is the formula to find the general solution of sin(θ) = x?

    <p>θ = sin^(-1) x + k ⋅ 360° (D)</p> Signup and view all the answers

    What is the formula to find the area of a triangle using the sine rule?

    <p>Area = (1/2)bc sin A (C)</p> Signup and view all the answers

    When should the cosine rule be used to find the length of a side of a triangle?

    <p>When no right angle is given, and three sides are given (A)</p> Signup and view all the answers

    What is the formula to find the height of a pole using the sine rule and tangent ratio?

    <p>h = (d sin α) / (sin β) ⋅ tan β (A)</p> Signup and view all the answers

    When should the area rule be used to find the area of a triangle?

    <p>When no perpendicular height is given (D)</p> Signup and view all the answers

    What is the formula to find the general solution of tan(θ) = x?

    <p>θ = tan^(-1) x + k ⋅ 180° (B)</p> Signup and view all the answers

    What is the formula to find the area of a triangle using the cosine rule?

    <p>Area = √(s(s-a)(s-b)(s-c)) (A)</p> Signup and view all the answers

    When should the sine rule be used to find the length of a side of a triangle?

    <p>When no right angle is given, and two sides and an angle (not the included angle) are given (D)</p> Signup and view all the answers

    What is the formula to find the height of a building using the sine rule?

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

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

    <p>Draw a sketch, consider the given information, apply appropriate rules, and calculate the desired quantities (C)</p> Signup and view all the answers

    In the proof of the Pythagorean theorem, what is the key relationship established between the triangles ΔABD and ΔCBA?

    <p>They are similar. (D)</p> Signup and view all the answers

    What is the primary trigonometric identity used in the derivation of the compound angle formulas?

    <p>The Unit Circle Definition (A)</p> Signup and view all the answers

    Which of the following correctly expresses the cosine of a difference using the compound angle identity?

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

    What is the main concept used to derive the formula for cos(α - β) in the provided explanation?

    <p>The Law of Cosines (A)</p> Signup and view all the answers

    What is the primary reason for proving the similarity of triangles in the proof of the Pythagorean theorem?

    <p>To establish proportions between corresponding sides. (E)</p> Signup and view all the answers

    What is the relationship between angles α1 and α2 in the proof of the Pythagorean Theorem?

    <p>They are complementary angles. (D)</p> Signup and view all the answers

    Which of the following is NOT a similarity condition for triangles?

    <p>Two sides of one triangle are congruent to two sides of the other triangle. (E)</p> Signup and view all the answers

    In the derivation of cos(α + β) from cos(α - β), what is the key identity used?

    <p>The Negative Angle Identity (D)</p> Signup and view all the answers

    What is the main idea behind the proof of the Pythagorean Theorem, as explained in the provided content?

    <p>Using the area of similar triangles to derive a relationship between the sides. (A)</p> Signup and view all the answers

    What are the compound angle identities used to express the sine of a sum and a difference?

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

    Which formula correctly expresses the sine of a difference?

    <p>$ an(eta - eta)$ (C), $ an eta an eta$ (D)</p> Signup and view all the answers

    What is the cosine of a sum represented as?

    <p>$ an(eta + eta)$ (A)</p> Signup and view all the answers

    Which of the following describes the sine of double angle?

    <p>$rac{ an 2 heta}{2}$ (A)</p> Signup and view all the answers

    How can the cosine of a sum be derived using known identities?

    <p>Deriving from the sine of a sum (D)</p> Signup and view all the answers

    What does the cosine of a double angle NOT equal?

    <p>$$rac{ an^2}{ an^2}$$ (D)</p> Signup and view all the answers

    Which of the following is true regarding the sine of double angle?

    <p>$2 an eta$ (B)</p> Signup and view all the answers

    What are the steps involved in finding general solutions for trigonometric equations?

    <p>Using a CAST diagram and reference angles (B)</p> Signup and view all the answers

    Which of the following correctly represents the cosine of a difference?

    <p>$ an(eta - eta)$ (C)</p> Signup and view all the answers

    When deriving the sine of double angle from the sum formula, which substitutions are made?

    <p>$eta = heta$ (A)</p> Signup and view all the answers

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

    <p>x^2 + y^2 = r^2 (B)</p> Signup and view all the answers

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

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

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

    <p>It is symmetric about the x-axis, y-axis, and origin. (B)</p> Signup and view all the answers

    What is the purpose of completing the square for the x terms and the y terms?

    <p>To find the center and radius of a circle. (B)</p> Signup and view all the answers

    What is the equation of a circle with center (3, -2) and radius 7?

    <p>(x - 3)^2 + (y + 2)^2 = 49 (C)</p> Signup and view all the answers

    What is the symmetry of a circle with center at the origin about?

    <p>The x-axis, y-axis, origin, and lines y = x and y = -x. (A)</p> Signup and view all the answers

    What is the center and radius of the circle with the equation (x + 2)^2 + (y - 3)^2 = 16?

    <p>Center (-2, 3) and radius 4. (B)</p> Signup and view all the answers

    What is NOT a symmetry of a circle with center at the origin?

    <p>The line y = 2x. (B)</p> Signup and view all the answers

    What is the purpose of completing the square for the x terms and the y terms?

    <p>To determine the equation of a circle in standard form (B)</p> 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?

    <p>Their product is -1 (A)</p> Signup and view all the answers

    What is the definition of a ratio?

    <p>A comparison of two quantities with the same units (B)</p> Signup and view all the answers

    What is the property of proportion that states wz = xy?

    <p>Cross Multiplication (C)</p> 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?

    <p>Thales' theorem (B), Basic Proportionality Theorem (C)</p> Signup and view all the answers

    What is the equation of a circle in standard form?

    <p>(x - a)^2 + (y - b)^2 = r^2 (D)</p> Signup and view all the answers

    What is the definition of a tangent to a circle?

    <p>A straight line that touches the circle at exactly one point (D)</p> Signup and view all the answers

    What is the formula for the gradient of the tangent?

    <p>-1 / m_radius (A)</p> Signup and view all the answers

    What is the condition for two triangles to be similar?

    <p>They have proportional sides (C)</p> Signup and view all the answers

    What is the purpose of the gradient-point form of the straight line equation?

    <p>To find the equation of a tangent to a circle (B)</p> Signup and view all the answers

    Under what condition will two triangles have areas proportional to their bases?

    <p>If they have the same height. (A)</p> Signup and view all the answers

    What is the ratio of the areas of two triangles with the same height?

    <p>The ratio of their bases. (C)</p> Signup and view all the answers

    What can be concluded about two triangles with equal bases and between the same parallel lines?

    <p>They have equal areas. (D)</p> Signup and view all the answers

    What is the statement of the Proportion Theorem?

    <p>If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. (A)</p> Signup and view all the answers

    What is the Mid-point Theorem?

    <p>The line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. (B)</p> Signup and view all the answers

    What is the converse of the Mid-point Theorem?

    <p>If a line is drawn from the midpoint of one side of a triangle parallel to another side, it bisects the third side. (D)</p> Signup and view all the answers

    What is the definition of similar polygons?

    <p>Polygons with the same shape but different sizes. (C)</p> Signup and view all the answers

    What is the condition for two polygons to be similar?

    <p>All pairs of corresponding angles are equal and all pairs of corresponding sides are in the same proportion. (D)</p> Signup and view all the answers

    What is true about equiangular triangles?

    <p>They are similar. (B)</p> Signup and view all the answers

    What is the property of similar triangles?

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

    What is the condition required to prove that two triangles are similar?

    <p>All pairs of corresponding sides are in the same proportion (B), All pairs of corresponding angles are equal (C)</p> Signup and view all the answers

    What is the result of the theorem that equiangular triangles are similar?

    <p>The corresponding sides are in the same proportion (C)</p> Signup and view all the answers

    What is the statement of the Pythagorean theorem?

    <p>The square on the hypotenuse is equal to the sum of the squares on the other two sides (A)</p> Signup and view all the answers

    What is the concept of proportionality in triangles?

    <p>Triangles with equal heights have areas proportional to their bases (A)</p> Signup and view all the answers

    What is the method used in the proof that triangles with sides in proportion are similar?

    <p>Construction of a triangle (D)</p> Signup and view all the answers

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

    <p>Their areas are proportional to their bases (C)</p> Signup and view all the answers

    What is the statement of the theorem that triangles with sides in proportion are similar?

    <p>If the corresponding sides of two triangles are in proportion, then the two triangles are similar (A)</p> Signup and view all the answers

    What is true about two triangles with proportional sides?

    <p>They are similar (C)</p> Signup and view all the answers

    What is the name of the theorem that equiangular triangles are similar?

    <p>Equiangular triangles theorem (B)</p> Signup and view all the answers

    What is the condition for two triangles to be similar?

    <p>All pairs of corresponding angles are equal or all pairs of corresponding sides are in the same proportion (A)</p> Signup and view all the answers

    What defines a polygon?

    <p>A plane, closed shape consisting of three or more line segments. (D)</p> Signup and view all the answers

    What is the formula for the area of a trapezium?

    <p>Area = rac{1}{2} imes (base1 + base2) imes height (C)</p> Signup and view all the answers

    In which type of triangle are all corresponding angles equal?

    <p>Equiangular triangle (A)</p> Signup and view all the answers

    What do similar polygons have in common?

    <p>Their corresponding angles are equal, and corresponding sides are in proportion. (B)</p> Signup and view all the answers

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

    <p>Basic Proportionality Theorem (Thales' Theorem) (A)</p> Signup and view all the answers

    What is a common characteristic of a kite?

    <p>Its diagonals are perpendicular. (B)</p> Signup and view all the answers

    Which of the following is NOT a type of polygon?

    <p>Ellipse (D)</p> Signup and view all the answers

    What is the area of a rhombus based on its diagonals?

    <p>Area = rac{1}{2} imes diagonal AC imes diagonal BD (B)</p> Signup and view all the answers

    Which property is true for a right-angled triangle according to Pythagoras' theorem?

    <p>The hypotenuse is the longest side. (C)</p> Signup and view all the answers

    Which of the following options identifies the base in the context of the area of a triangle?

    <p>Any one of the sides of the triangle. (D)</p> Signup and view all the answers

    Which angles are equal in the triangles ( riangle ABD) and ( riangle CBA)?

    <p>(\angle A_1 = \angle C), (\angle B = \angle A_2), (\angle D_1 = \angle C) (A)</p> Signup and view all the answers

    Which angles are equal in the triangles ( riangle CAD) and ( riangle CBA)?

    <p>(\angle A_2 = \angle B), (\angle C = \angle A_1), (\angle D_2 = \angle C) (A)</p> Signup and view all the answers

    Based on the similarity of ( riangle ABD) and ( riangle CBA), what is the proportion that can be written using the corresponding sides?

    <p>(rac{AB}{BC} = rac{BD}{AB}) (B)</p> Signup and view all the answers

    Based on the similarity of ( riangle CAD) and ( riangle CBA), what is the proportion that can be written using the corresponding sides?

    <p>(rac{AC}{CB} = rac{DC}{AC}) (A)</p> Signup and view all the answers

    What is the final equation that is derived from the similarity of the triangles ( riangle ABD), ( riangle CAD), and ( riangle CBA)?

    <p>(BC^2 = AB^2 + AC^2) (B)</p> Signup and view all the answers

    What is the statement of the converse of the Pythagorean theorem?

    <p>If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the angle included by these two sides is a right angle. (D)</p> Signup and view all the answers

    What is the area of a triangle given its base and height?

    <p>(rac{1}{2} \cdot b \cdot h) (C)</p> Signup and view all the answers

    What is the condition required for two triangles to be similar?

    <p>All corresponding angles are equal and all corresponding sides are in the same proportion. (A)</p> Signup and view all the answers

    What is the formula for the cosine of the difference between two angles, (\alpha) and (eta)?

    <p>(\cos(\alpha - eta) = \cos \alpha \cos eta + \sin \alpha \sin eta) (A)</p> Signup and view all the answers

    What is the formula for the sine of the sum of two angles, (\alpha) and (eta)?

    <p>(\sin(\alpha + eta) = \sin \alpha \cos eta + \cos \alpha \sin eta) (B)</p> Signup and view all the answers

    Which of the following is a correct double angle formula for (\cos(2\alpha))?

    <p>(\cos^2 \alpha - \sin^2 \alpha) (A), (2\cos^2 \alpha - 1) (D)</p> Signup and view all the answers

    What is the first step in deriving the double angle formula for sine, (\sin(2\alpha))?

    <p>Use the sum formula for sine: (\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta) (B)</p> Signup and view all the answers

    Which of the following is NOT a correct form of the double angle formula for cosine, (\cos(2\alpha))?

    <p>(\sin^2 \alpha - \cos^2 \alpha) (D)</p> Signup and view all the answers

    Which of the following formulas is used to derive the double angle formula for cosine, (\cos(2\alpha))?

    <p>The sum formula for cosine: (\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta) (A)</p> Signup and view all the answers

    Which of the following trigonometric identities is NOT used in deriving the alternate forms of the double angle formula for cosine?

    <p>(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta) (B)</p> Signup and view all the answers

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

    <p>(\frac{24}{25}) (B)</p> Signup and view all the answers

    Which of the following is NOT a correct step in the derivation of the double angle formula for sine, (\sin(2\alpha))?

    <p>Use the Pythagorean Identity: (\sin^2 \alpha + \cos^2 \alpha = 1) to simplify the expression further (A)</p> Signup and view all the answers

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

    <p>(\frac{7}{9}) (B)</p> Signup and view all the answers

    What is the double angle formula for sine in terms of tangent, given that (\tan \alpha = \frac{\sin \alpha}{\cos \alpha})?

    <p>(\frac{2\tan \alpha}{1 - \tan^2 \alpha}) (A)</p> Signup and view all the answers

    What is the value of (\sin(2\alpha)) if (\tan \alpha = \frac{1}{2})?

    <p>(\frac{4}{5}) (B)</p> Signup and view all the answers

    What is the general solution for the equation \( an heta = x\)?

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

    Which rule should be applied when given two sides and an included angle of a triangle?

    <p>Cosine Rule (B)</p> Signup and view all the answers

    What equation represents the area of triangle ABC using side lengths b and c and angle A?

    <p>(\text{Area} = \frac{1}{2}bc\sin A) (D)</p> Signup and view all the answers

    When is it appropriate to use the Sine Rule for solving triangle problems?

    <p>When no right angle is present, and two sides along with a non-included angle are given (B)</p> Signup and view all the answers

    What is the formula for determining the height of a pole using triangle FAB?

    <p>(h = \frac{d \sin \alpha}{\sin \beta}) (C)</p> Signup and view all the answers

    What does the area of triangle ABC equal when using the Side-Angle-Side method?

    <p>(\frac{1}{2}bc\sin A) (C)</p> Signup and view all the answers

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

    <p>(\theta = \sin^{-1} x + k \cdot 360^\circ) or (\theta = 180^\circ - \sin^{-1} x + k \cdot 360^\circ) (D)</p> Signup and view all the answers

    Which condition must be satisfied to effectively use the Cosine Rule?

    <p>Three sides of the triangle are given (D)</p> Signup and view all the answers

    In triangle ABC, what is the relationship expressed by the Sine Rule?

    <p>(\frac{AB}{\sin C} = \frac{BC}{\sin A}) (B)</p> Signup and view all the answers

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

    <p>x^2 + y^2 = r^2 (C)</p> Signup and view all the answers

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

    <p>(x - a)^2 + (y - b)^2 = r^2 (D)</p> Signup and view all the answers

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

    <p>The x-axis, the y-axis, the origin, and the lines y = x and y = -x (C)</p> Signup and view all the answers

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

    <p>To find the center and radius of the circle (C)</p> Signup and view all the answers

    What is the equation of a circle with center (a, b) and passing through the point (x, y)?

    <p>(x - a)^2 + (y - b)^2 = r^2 (C)</p> Signup and view all the answers

    What is the radius of the circle with the equation x^2 + y^2 - 4x + 6y - 3 = 0?

    <p>5 (A)</p> Signup and view all the answers

    What is the center of the circle with the equation (x + 2)^2 + (y - 3)^2 = 16?

    <p>(-2, 3) (A)</p> Signup and view all the answers

    Which of the following equations represents a circle with center at (-1, 2) and a radius of 3?

    <p>(x + 1)^2 + (y - 2)^2 = 9 (B)</p> Signup and view all the answers

    What is true about two triangles with equal heights?

    <p>Their areas are proportional to their bases. (D)</p> Signup and view all the answers

    What is the conclusion of the Mid-point Theorem?

    <p>The line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. (C)</p> Signup and view all the answers

    What is the condition required to prove that two triangles are similar?

    <p>Their corresponding sides are proportional. (C)</p> Signup and view all the answers

    What is true about two triangles with equal bases and between the same parallel lines?

    <p>Their areas are equal. (D)</p> Signup and view all the answers

    What is the statement of the Proportion Theorem?

    <p>A line drawn parallel to one side of a triangle divides the other two sides proportionally. (A)</p> Signup and view all the answers

    What is true about similar polygons?

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

    What are the conditions for similarity of two polygons?

    <p>All pairs of corresponding angles are equal and all pairs of corresponding sides are in the same proportion. (B)</p> Signup and view all the answers

    What is the converse of the Mid-point Theorem?

    <p>The line drawn from the midpoint of one side of a triangle parallel to another side bisects the third side of the triangle. (D)</p> Signup and view all the answers

    What is true about two triangles with proportional sides?

    <p>They are similar. (A)</p> Signup and view all the answers

    What is the formula for the areas of two triangles with the same height?

    <p>Area1 / Area2 = base1 / base2 (A)</p> Signup and view all the answers

    What is the formula for the area of a triangle?

    <p>$\frac{1}{2} \times \text{base} \times \text{height}$ (B)</p> Signup and view all the answers

    Which of the following is a property of similar polygons?

    <p>Corresponding angles are equal. (D)</p> 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?

    <p>Basic Proportionality Theorem (D)</p> Signup and view all the answers

    What is the formula for the area of a parallelogram?

    <p>$\text{base} \times \text{height}$ (C)</p> Signup and view all the answers

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

    <p>Diagonals are equal. (C)</p> Signup and view all the answers

    What is the formula for the area of a trapezium?

    <p>$\frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height}$ (B)</p> Signup and view all the answers

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

    <p>Corresponding angles are congruent. (C)</p> Signup and view all the answers

    What is the name of the theorem that states the square on the hypotenuse is equal to the sum of the squares on the other two sides?

    <p>Pythagorean Theorem (B)</p> Signup and view all the answers

    What is the formula for the area of a kite?

    <p>$\frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2$ (A)</p> Signup and view all the answers

    What is the name of the polygon with four sides?

    <p>Quadrilateral (A)</p> Signup and view all the answers

    What is the general solution for the equation (\sin \theta = x)?

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

    When is the Sine Rule used?

    <p>When two sides and an angle (not the included angle) are given. (A), When two angles and a side are given. (C)</p> Signup and view all the answers

    Which of these rules is used to calculate the area of a triangle when no perpendicular height is given?

    <p>Area Rule (A)</p> Signup and view all the answers

    What is the formula for the height of a pole, given that (AB = d), (\angle FBA = \theta), (\angle FAB = \alpha), (\angle FBT = \beta), and (\angle TFB = 90^\circ)?

    <p>(h = \frac{d \sin \alpha}{\sin \beta} \tan \beta) (C)</p> Signup and view all the answers

    What is the first step in solving a three-dimensional problem involving trigonometric functions?

    <p>Draw a sketch (A)</p> Signup and view all the answers

    Which of the following is NOT a condition for using the Cosine Rule?

    <p>Two angles and a side are given. (D)</p> Signup and view all the answers

    What is the formula for calculating the area of a triangle using the Sine Rule?

    <p>All of the above. (D)</p> Signup and view all the answers

    What is the general solution for the equation (\cos \theta = x)?

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

    Which trigonometric rule is used when two sides and the included angle are given in a triangle?

    <p>Cosine Rule (A)</p> Signup and view all the answers

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

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

    What is the purpose of completing the square for the x terms and the y terms?

    <p>To simplify the equation of a circle in general form (A)</p> 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?

    <p>Their product is -1 (A)</p> Signup and view all the answers

    What is the definition of a ratio?

    <p>A relationship between two quantities with the same units (C)</p> Signup and view all the answers

    What is the property of proportion that states wz = xy?

    <p>Cross multiplication (A)</p> 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?

    <p>Thales' theorem (C)</p> Signup and view all the answers

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

    <p>(x - a)^2 + (y - b)^2 = r^2 (D)</p> Signup and view all the answers

    What is the definition of a tangent to a circle?

    <p>A straight line that touches the circle at exactly one point (C)</p> Signup and view all the answers

    What is the purpose of simplifying ratios?

    <p>To make them easier to compare (C)</p> Signup and view all the answers

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

    <p>x^2 + y^2 = 25 (A)</p> Signup and view all the answers

    What can be concluded about two triangles with proportional sides?

    <p>They are similar (B)</p> Signup and view all the answers

    What must be true for two triangles to be classified as similar?

    <p>All pairs of corresponding angles are equal, or all pairs of corresponding sides are in proportion. (A)</p> Signup and view all the answers

    In the proof for equiangular triangles, what can be concluded once corresponding angles are found to be equal?

    <p>The triangles are similar. (A)</p> Signup and view all the answers

    Which statement is true about triangles when it is known that their sides are in proportion?

    <p>The corresponding angles of the triangles are equal. (B)</p> Signup and view all the answers

    What is a necessary step when proving that two triangles are similar by showing their sides are proportional?

    <p>Establishing that a line parallel to one side creates corresponding angles. (D)</p> Signup and view all the answers

    Which condition is NOT required to prove similarity using the theorem for equiangular triangles?

    <p>All sides must be congruent. (A)</p> Signup and view all the answers

    What aspect of triangle similarity allows for the area comparison to be drawn from their proportional sides?

    <p>The sides being proportional. (B)</p> Signup and view all the answers

    Why is the Pythagorean theorem important in the context of triangle properties?

    <p>It relates the lengths of sides of right-angled triangles. (B)</p> Signup and view all the answers

    If triangle ABC has a right angle at A, what does the Pythagorean theorem state?

    <p>The length of side BC squared equals the sum of the squares of sides AB and AC. (C)</p> Signup and view all the answers

    What can be concluded about two triangles with equal bases and equal heights?

    <p>They have equal areas. (B)</p> Signup and view all the answers

    In the construction for proving two triangles are similar from their equal angles, which step is relevant?

    <p>Drawing parallels to establish angle connections. (D)</p> Signup and view all the answers

    What is the sine of the difference of two angles according to the sine difference formula?

    <p>$egin{align*} ext{sin} heta ext{cos} eta - ext{cos} heta ext{sin} eta ewline ext{or } ext{sin} ( heta - eta) ext{ } ext{if } heta,eta= ext{c}. ext{} ext{} ext{} ext{ } heta,eta ext{ angles} ext{ } ext{ using complex angle formulas. } ext{ } ext{} ext{ } ext{} ext{ } ext{ } ext{ } ext{ } ext{ } ext{ angles} ext{ } ext{ } ext{ and arcs are cancels.} ext{ } ext{ } ext{ } ext{} ext{ } ext{ } ext{ } ext{ } ext{ } ewline ext{origin are many such complex numbers included.} ext{ } ext{ } ext{ } ewline ext{} ext{} ext{ } ewline ext{ angles } ext{ } ext{ and their hyperbolic counterparts.} ext{ } ext{ } ext{ } ewline} ext{ angle.} ext{ }$ (A)</p> Signup and view all the answers

    Which of the following is the correct formula for the cosine of the sum of two angles?

    <p>$ ext{cos} heta ext{cos} eta - ext{sin} heta ext{sin} eta$ (A)</p> Signup and view all the answers

    What is the correct expression for the sine of a double angle?

    <p>$2 ext{sin} heta ext{cos} heta$ (B)</p> Signup and view all the answers

    Which formula correctly represents the cosine of a double angle?

    <p>$ ext{cos}^2 heta - ext{sin}^2 heta$ (A)</p> Signup and view all the answers

    What is a key step in finding the general solution of a trigonometric equation?

    <p>Simplify the equation using algebraic methods (B)</p> Signup and view all the answers

    What does the CAST diagram help to determine?

    <p>Where the function is positive or negative (B)</p> Signup and view all the answers

    Which of the following represents the sine of a sum formula?

    <p>$ ext{sin} heta ext{cos} eta + ext{cos} heta ext{sin} eta$ (D)</p> Signup and view all the answers

    In the context of trigonometric identities, what is the cosine of a difference formula?

    <p>$ ext{cos}( heta - eta) = ext{cos} heta ext{cos} eta + ext{sin} heta ext{sin} eta$ (A)</p> Signup and view all the answers

    Which step is NOT part of the method for finding general solutions of trigonometric equations?

    <p>Finding the derivative of the function (D)</p> Signup and view all the answers

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