Gr 12 Mathematics: November Easy P(2)
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Questions and Answers

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

  • (x - a)^2 + (y - b)^2 = r^2
  • y = x^2
  • x^2 + y^2 = r^2
  • x^2 + Dx + y^2 + Ey + F = 0 (correct)

    • What is the radius of a circle with equation (x - 2)^2 + (y - 3)^2 = 25?

  • 10
  • 5 (correct)
  • 2
  • 25

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

  • (1, 3)
  • (-2, 3) (correct)
  • (2, 3)
  • (-2, -3)

    • What is a characteristic of a tangent line to a circle?

    <p>It touches the circle at exactly one point without crossing it.</p> Signup and view all the answers

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

    <p>They are perpendicular.</p> Signup and view all the answers

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

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

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

    <p>Group the x terms and the y terms.</p> Signup and view all the answers

    What is the purpose of completing the square?

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

    What is the relationship between the gradient of the radius and the gradient of the tangent?

    <p>m_radius × m_tangent = -1</p> Signup and view all the answers

    What is the first step in determining the equation of a tangent?

    <p>Write the equation of the circle in its standard form</p> Signup and view all the answers

    What is the purpose of a ratio?

    <p>To compare quantities with the same units</p> Signup and view all the answers

    What is the Cross Multiplication property of proportion?

    <p>w × z = x × y</p> Signup and view all the answers

    What is the Basic Proportionality Theorem also known as?

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

    What is a polygon?

    <p>A plane, closed shape with three or more line segments</p> Signup and view all the answers

    What is the formula for the area of a triangle?

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

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

    <p>To find the equation of a tangent</p> Signup and view all the answers

    What is the purpose of verifying the solution in proportional problems?

    <p>To check that the solution maintains the proportional relationships</p> Signup and view all the answers

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

    <p>The gradient of the radius is perpendicular to the point of tangency</p> Signup and view all the answers

    What is the formula for KL^2 in terms of α and β when considering two points on the unit circle?

    <p>(cos α - cos β)^2 + (sin α - sin β)^2</p> Signup and view all the answers

    What is the value of cos(-β) in terms of β?

    <p>cos β</p> Signup and view all the answers

    What is the formula for cos(α - β)?

    <p>cos α cos β + sin α sin β</p> Signup and view all the answers

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

    <p>cos α cos β - sin α sin β</p> Signup and view all the answers

    What is the value of sin(-β) in terms of β?

    <p>-sin β</p> Signup and view all the answers

    What is the purpose of the cosine rule in deriving trigonometric identities?

    <p>To find the distance between two points on the unit circle</p> Signup and view all the answers

    What is the formula for the sine of a difference?

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

    What is the formula for the cosine of a sum?

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

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

    <p>$\sin(2\alpha) = 2 \sin \alpha \cos \alpha$</p> Signup and view all the answers

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

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

    What is the first step in solving a trigonometric equation?

    <p>Simplify the equation using algebraic methods and trigonometric identities</p> Signup and view all the answers

    What is the purpose of the CAST diagram?

    <p>To determine where the function is positive or negative</p> Signup and view all the answers

    What is the formula for the cosine of a difference?

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

    What is the formula for the sine of a sum?

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

    What is the purpose of verifying the solutions in trigonometric equations?

    <p>To check if the solutions satisfy the equation</p> Signup and view all the answers

    What is the general solution method for trigonometric equations?

    <p>Simplify, reference angle, CAST diagram, restricted values, general solution, and check</p> Signup and view all the answers

    What does a correlation coefficient of 0 indicate?

    <p>No correlation</p> Signup and view all the answers

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

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

    What is the relationship between the standard deviation of the x-values (σx) and the standard deviation of the y-values (σy) in the formula for the Pearson’s product moment correlation coefficient?

    <p>They are directly proportional.</p> Signup and view all the answers

    What is the compound angle identity for cos(α - β)?

    <p>cos α cos β + sin α sin β</p> Signup and view all the answers

    What does a negative value of the linear correlation coefficient, r, indicate?

    <p>As x increases, y decreases.</p> Signup and view all the answers

    What is the range of values for the linear correlation coefficient, r?

    <p>[-1, 1]</p> Signup and view all the answers

    Which of the following statements is TRUE about the correlation coefficient, r?

    <p><em>r</em> measures the strength and direction of the relationship between two variables.</p> Signup and view all the answers

    What is the value of the correlation coefficient, r, for a perfect negative correlation?

    <p>-1</p> Signup and view all the answers

    What is the gradient (B) of the least squares regression line represented by the equation y = A + Bx?

    <p>B</p> Signup and view all the answers

    Which of the following represents the strongest positive correlation?

    <p>r = 0.95</p> Signup and view all the answers

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

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

    When are triangles considered similar?

    <p>If both their corresponding angles are equal and corresponding sides are in proportion.</p> Signup and view all the answers

    What formula represents the area of a triangle?

    <p>Area = rac{1}{2} imes base imes height</p> Signup and view all the answers

    What does the Proportionality Theorem assert regarding triangles?

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

    Which of the following conditions are necessary for two polygons to be similar?

    <p>All pairs of corresponding angles are equal.</p> Signup and view all the answers

    What does it mean for triangles to be equiangular?

    <p>All pairs of corresponding angles are equal.</p> Signup and view all the answers

    If two triangles have sides in proportion, what can be concluded?

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

    What is true about similar polygons?

    <p>They can differ in size but maintain the same shape.</p> Signup and view all the answers

    What relationship do the areas of triangles have if they share the same height?

    <p>Their areas are proportional to the lengths of their bases.</p> Signup and view all the answers

    What trigonometric rule should you use to find the length of a side when you know two sides and the included angle?

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

    Which of the following scenarios would require the use of the Sine Rule?

    <p>Finding the length of a side in a triangle given two angles and a side.</p> Signup and view all the answers

    When is the Area Rule most useful for calculating the area of a triangle?

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

    In a triangle ABC, what does the expression (rac{1}{2}ab \sin C) represent?

    <p>The area of the triangle.</p> Signup and view all the answers

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

    <p>Draw a sketch.</p> Signup and view all the answers

    If we know the height of a pole, the angle between the ground and the pole, and the distance from the base of the pole to a point on the ground, which trigonometric rule can be used to find the distance from the base of the pole to the point where the angle is measured?

    <p>Sine Rule</p> Signup and view all the answers

    What is the general solution for the equation (\sin heta = x) where x is a real number?

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

    Which of the following is NOT a step in solving a three-dimensional problem involving trigonometric ratios?

    <p>Use the Pythagorean theorem to find missing sides.</p> Signup and view all the answers

    Which condition must be satisfied for two triangles to be considered similar?

    <p>All pairs of corresponding angles are equal.</p> Signup and view all the answers

    What statement best describes the Pythagorean theorem?

    <p>It relates the lengths of the sides of a right-angled triangle.</p> Signup and view all the answers

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

    <p>The corresponding sides are in proportion.</p> Signup and view all the answers

    When using the formula for the area of a triangle, which elements are needed?

    <p>Base and height.</p> Signup and view all the answers

    What does a regression coefficient (r) of +1 indicate?

    <p>A strong positive correlation.</p> Signup and view all the answers

    For which type of triangle is the converse of the Pythagorean theorem applicable?

    <p>Right-angled triangle.</p> Signup and view all the answers

    In triangle similarity, the SSS condition applies under what circumstance?

    <p>When sides are in proportion.</p> Signup and view all the answers

    How is the linear regression line typically expressed?

    <p>y = A + Bx.</p> Signup and view all the answers

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

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

    What is the effect of a regression coefficient (r) close to -1?

    <p>It indicates a strong negative correlation.</p> Signup and view all the answers

    What is the formula for the area of a trapezium?

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

    Which polygon has vertices A, B, C, and D and has all sides and angles equal?

    <p>Square</p> Signup and view all the answers

    What does the basic proportionality theorem state?

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

    For a rhombus, how is the area calculated?

    <p>Area = 1/2 × diagonal AC × diagonal BD</p> Signup and view all the answers

    In triangles with the same height, how are their areas related?

    <p>The areas are proportional to the lengths of their bases.</p> Signup and view all the answers

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

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

    How is the area of a rectangle calculated?

    <p>Area = length × width</p> Signup and view all the answers

    What is the Mid-point Theorem about in triangles?

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

    What is the area formula for a square?

    <p>Area = side^2</p> Signup and view all the answers

    If two triangles are equiangular, how are their sides related?

    <p>The corresponding sides are in proportion.</p> Signup and view all the answers

    What is the formula to calculate the area of a rectangle?

    <p>length × width</p> Signup and view all the answers

    What is the condition for two polygons to be similar?

    <p>Proportional corresponding sides</p> Signup and view all the answers

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

    <p>diagonal AC × diagonal BD</p> Signup and view all the answers

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

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

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

    <p>0.5 × (base_1 + base_2) × height</p> Signup and view all the answers

    What is the condition for triangles to be equiangular?

    <p>Equal angles</p> Signup and view all the answers

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

    <p>0.5 × diagonal AC × diagonal BD</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 equal to half the length of the third side.</p> Signup and view all the answers

    What is the formula to calculate the area of a square?

    <p>side^2</p> Signup and view all the answers

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

    <p>Triangles with the Same Height Theorem</p> Signup and view all the answers

    What is the relationship between the gradient of the radius and the gradient of the tangent?

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

    What is the purpose of the ratio in geometry?

    <p>To compare quantities of the same kind</p> Signup and view all the answers

    What is the formula for the area of a triangle?

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

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

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

    What is the definition of a polygon?

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

    What is the step to determine the equation of a tangent?

    <p>Calculate the gradient of the radius</p> Signup and view all the answers

    What is the purpose of verifying the solution in proportional problems?

    <p>To check that the solution maintains the proportional relationships</p> Signup and view all the answers

    What is the formula for the equation of a tangent?

    <p>y - y1 = m(x - x1)</p> Signup and view all the answers

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

    <p>Alternating Proportion</p> Signup and view all the answers

    What does a correlation coefficient of -1 indicate?

    <p>Perfect negative correlation</p> Signup and view all the answers

    What is the formula for the Pearson’s product moment correlation coefficient?

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

    What does the cosine of a difference identity represent?

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

    What does a correlation coefficient of 0.8 indicate?

    <p>Strong positive correlation</p> Signup and view all the answers

    What is the purpose of calculating the correlation coefficient?

    <p>To identify the strength and direction of a relationship between two variables</p> Signup and view all the answers

    What is the sine of a sum identity?

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

    What is the range of values for the linear correlation coefficient, r?

    <p>-1 to 1</p> Signup and view all the answers

    What does a negative value of the linear correlation coefficient, r, indicate?

    <p>Negative correlation</p> Signup and view all the answers

    What is the purpose of using a calculator for linear regression?

    <p>To determine the equation of the regression line</p> Signup and view all the answers

    Using the cosine difference formula, what is the expanded form of (\cos(\alpha - (-\beta)))?

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

    Which of the following trigonometric identities is used to simplify the expression (\cos (-\beta)) in the derivation of (\cos(\alpha + \beta))?

    <p>Even-odd identity</p> Signup and view all the answers

    Which of the following expressions represents the simplified form of (\sin (-\beta)) used in the derivation of (\cos(\alpha + \beta))?

    <p>(-\sin \beta)</p> Signup and view all the answers

    What is the expanded form of (\cos(\alpha + \beta)) after applying the cosine difference formula and substituting the even-odd identities?

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

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

    <p>To express the sum of angles as a difference of angles.</p> Signup and view all the answers

    What is the general form of the cosine sum formula based on the derivation of (\cos(\alpha + \beta))?

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

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

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

    What is the center of the circle represented by the equation ( x^2 + 6x + y^2 - 8y = 0 )?

    <p>( (-3, 4) )</p> Signup and view all the answers

    What is the radius of the circle represented by the equation ( x^2 + y^2 - 10x + 4y + 20 = 0 )?

    <p>( 3 )</p> Signup and view all the answers

    Which of the following lines is a tangent to the circle ( (x - 2)^2 + (y - 1)^2 = 9 ) at the point ( (5, 1) )?

    <p>( y = 1 )</p> Signup and view all the answers

    What is the equation of the tangent line to the circle ( x^2 + y^2 = 16 ) at the point ( (4, 0) )?

    <p>( y = 0 )</p> Signup and view all the answers

    What is the equation of the circle with center ( (-2, 1) ) and passing through the point ( (1, 4) )?

    <p>( (x + 2)^2 + (y - 1)^2 = 18 )</p> Signup and view all the answers

    What is the equation of the tangent line to the circle ( (x + 1)^2 + (y - 3)^2 = 25 ) at the point ( (3, 7) )?

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

    What is the equation of the circle with center ( (0, 0) ) and passing through the point ( (5, -12) )?

    <p>( x^2 + y^2 = 169 )</p> Signup and view all the answers

    In the context of triangle similarity, what is the significance of equiangular triangles?

    <p>Equiangular triangles always have corresponding sides in proportion, leading to the conclusion that they are similar.</p> Signup and view all the answers

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

    <p>A line drawn from the midpoint of one side is parallel to another side.</p> Signup and view all the answers

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

    <p>All pairs of corresponding angles must be equal.</p> Signup and view all the answers

    Which statement accurately describes similar polygons?

    <p>They are enlarged versions of each other.</p> Signup and view all the answers

    According to the Proportionality Theorem, how does a line drawn parallel to one side of a triangle divide the other two sides?

    <p>The segments are proportional to the whole sides.</p> Signup and view all the answers

    Which formula represents the area of a triangle?

    <p>Area = rac{1}{2} imes base imes height</p> Signup and view all the answers

    What is the relationship between two equiangular triangles?

    <p>They are similar regardless of their side lengths.</p> Signup and view all the answers

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

    <p>All corresponding sides must be proportionate.</p> Signup and view all the answers

    How can one prove that triangles are similar using corresponding sides?

    <p>Prove that the ratios of their corresponding sides are equal.</p> Signup and view all the answers

    In the similarity proof of equiangular triangles, what is true about the corresponding sides?

    <p>They are in proportion to each other.</p> Signup and view all the answers

    What result follows if two triangles have their corresponding angles equal?

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

    Which rule is used to find the area of a triangle when no perpendicular height is given?

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

    When would you use the Sine Rule to solve a triangle?

    <p>When two angles and a side are known</p> Signup and view all the answers

    Which rule is used to find the length of a side when two sides and the included angle are known?

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

    In the formula for the height of a pole, what does the variable 'd' represent?

    <p>The distance from the base of the pole to the point where the angle of elevation is measured</p> Signup and view all the answers

    Which of the following is NOT a step in the general approach to solving three-dimensional problems involving trigonometric functions?

    <p>Use the Pythagoras Theorem</p> Signup and view all the answers

    What is the formula for the height of a building in a three-dimensional problem, given the information: BC = b, ∠DBA = α, ∠DBC = β, ∠DCB = θ?

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

    In the formula for the height of a pole, what does the variable 'h' represent?

    <p>The height of the pole</p> Signup and view all the answers

    What is the compound angle formula for the sine of a difference?

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

    Which identity can be used to rewrite $ ext{cos}(2 heta)$?

    <p>$2 ext{cos}^2 heta - 1$</p> Signup and view all the answers

    What is the general solution method's first step in solving trigonometric equations?

    <p>Simplify the equation</p> Signup and view all the answers

    What is the sine of a sum, according to the formulas?

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

    Which of the following forms is NOT equivalent to the cosine of double angle?

    <p>$2 ext{sin}^2 heta - 1$</p> Signup and view all the answers

    What is a method to determine where a trigonometric function is positive or negative?

    <p>Utilize the CAST diagram</p> Signup and view all the answers

    What is the cosine of a sum according to the compound angle formulas?

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

    Which of the following represents the sine of double angle?

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

    How can the reference angle be determined in the general solution method?

    <p>Using positive values only</p> Signup and view all the answers

    Which of the following indicates a strong negative correlation?

    <p>-0.9</p> Signup and view all the answers

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

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

    Which of the following values of the correlation coefficient indicates no correlation?

    <p>0</p> Signup and view all the answers

    What is the formula for the cosine of a difference?

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

    What is the formula for the linear correlation coefficient, r, in terms of the gradient of the least squares regression line (b), the standard deviation of the x-values (σx), and the standard deviation of the y-values (σy)?

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

    What is the range of values for the linear correlation coefficient, r?

    <p>[-1, 1]</p> Signup and view all the answers

    Which of the following describes a very weak correlation?

    <p>0 &lt; r &lt; 0.25</p> Signup and view all the answers

    What is the value of the correlation coefficient, r, for a perfect positive correlation?

    <p>1</p> Signup and view all the answers

    Which of the following is NOT a valid step in calculating the regression coefficients (A and B) using a calculator?

    <p>Calculate the mean of the x and y values.</p> Signup and view all the answers

    How is the gradient of the tangent line to the circle related to the gradient of the radius at the point of tangency?

    <p>Their product is -1.</p> Signup and view all the answers

    What is the first step in determining the equation of a tangent to a circle?

    <p>Write the equation of the circle in standard form.</p> Signup and view all the answers

    When two ratios are equal, how are they described?

    <p>Proportionate.</p> Signup and view all the answers

    What does the Basic Proportionality Theorem state about parallel lines in a triangle?

    <p>They divide the other two sides proportionally.</p> Signup and view all the answers

    In the context of ratios, what is an important characteristic of ratios?

    <p>Ratios should be simplified to their simplest form.</p> Signup and view all the answers

    What is the formula for the gradient of the radius from the center (a, b) to the point of tangency (x1, y1)?

    <p>m_{radius} = \frac{y_1 - b}{x_1 - a}</p> Signup and view all the answers

    What is the area formula for a triangle with base b and height h?

    <p>Area = \frac{1}{2} \times b \times h</p> Signup and view all the answers

    What shape is defined as a plane closed shape consisting of three or more line segments?

    <p>Polygon</p> Signup and view all the answers

    What does the inverse proportion property state?

    <p>If x increases, z decreases.</p> Signup and view all the answers

    When setting up proportional equations, what is the first step in solving these problems?

    <p>Identify the Given Ratios.</p> Signup and view all the answers

    What is the formula for calculating the area of a rhombus?

    <p>$rac{1}{2} imes ext{diagonal AC} imes ext{diagonal BD}$</p> Signup and view all the answers

    How does the area of triangles with the same height compare?

    <p>They have the same area if their bases are equal.</p> Signup and view all the answers

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

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

    What is the area formula for a trapezium?

    <p>$rac{1}{2} imes ( ext{base}_1 + ext{base}_2) imes ext{height}$</p> Signup and view all the answers

    When are two triangles considered similar?

    <p>When their corresponding sides are in proportion.</p> Signup and view all the answers

    What is the area formula for a rectangle?

    <p>$ ext{length} imes ext{width}$</p> Signup and view all the answers

    What does the Pythagorean Theorem state in a right triangle?

    <p>The sum of the squares of the two legs is equal to the square of the hypotenuse.</p> Signup and view all the answers

    For a square, what is the formula to find its area?

    <p>$ ext{side}^2$</p> Signup and view all the answers

    What is the characteristic of similar polygons?

    <p>Their corresponding angles are equal and corresponding sides are proportional.</p> Signup and view all the answers

    What does the Mid-point Theorem state?

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

    What is the negative angle identity applied in the derivation of (\cos(\alpha + \beta))?

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

    In the derivation of (\cos(\alpha + \beta)), which even-odd identity is used for (\cos(-\beta))?

    <p>(\cos(-\beta) = \cos(\beta))</p> Signup and view all the answers

    What is the final simplified expression for (\cos(\alpha + \beta)) after applying the cosine difference formula and even-odd identities?

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

    What is the formula for the distance KL^2 between two points on the unit circle, expressed using the distance formula?

    <p>(KL^2 = (\cos\alpha - \cos\beta)^2 + (\sin\alpha - \sin\beta)^2)</p> Signup and view all the answers

    What is the formula for the distance KL^2 between two points on the unit circle, expressed using the cosine rule?

    <p>(KL^2 = 2 - 2\cos(\alpha - \beta))</p> Signup and view all the answers

    What is the purpose of equating the two expressions for KL^2 derived using the distance formula and the cosine rule?

    <p>To prove the cosine difference formula</p> Signup and view all the answers

    What indicates that triangles AGH and ABC are similar?

    <p>All pairs of corresponding angles are equal.</p> Signup and view all the answers

    Which of the following conditions must be met for two triangles to be congruent?

    <p>Their corresponding sides are equal in length.</p> Signup and view all the answers

    What does the Pythagorean theorem specifically relate?

    <p>The squares of the lengths of the sides of a right-angled triangle.</p> Signup and view all the answers

    In the context of triangle areas, if two triangles share the same base, what can be concluded about their heights?

    <p>They must be equal to have equal area.</p> Signup and view all the answers

    What is the value range for the regression coefficient r?

    <p>-1 to 1.</p> Signup and view all the answers

    What does the Mid-point Theorem state?

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

    What does a correlation coefficient close to 0 indicate?

    <p>There is no correlation between the variables.</p> Signup and view all the answers

    Which of the following indicates that two polygons are similar?

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

    What is the significance of the least squares regression line?

    <p>It minimizes the sum of the squares of the vertical distances of the points from the line.</p> Signup and view all the answers

    What can be inferred if two triangles are described as equiangular?

    <p>Their corresponding angles are equal.</p> Signup and view all the answers

    When are two triangles considered similar?

    <p>If all pairs of corresponding angles are equal or all pairs of corresponding sides are in proportion.</p> Signup and view all the answers

    What does the Proportionality Theorem state?

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

    In the similarity conditions for polygons, which statement is true?

    <p>All pairs of corresponding angles must be equal and all sides be in proportion.</p> Signup and view all the answers

    How is the area of a triangle calculated?

    <p>Half the product of base and height.</p> Signup and view all the answers

    How can you determine that two triangles are similar using corresponding sides?

    <p>If the corresponding sides are in the same ratio.</p> Signup and view all the answers

    What is the formula for the area of a triangle?

    <p>Area = rac{1}{2} × base × height</p> Signup and view all the answers

    Which of the following triangles are guaranteed to be similar?

    <p>Equiangular triangles.</p> Signup and view all the answers

    If triangle ABC is similar to triangle DEF, what can be inferred about their angles?

    <p>Angle C equals angle F.</p> Signup and view all the answers

    What is the relationship between similar polygons?

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

    What does the statement 'BC = 2 × DE' imply when BC is parallel to DE?

    <p>BC is twice the length of DE.</p> Signup and view all the answers

    What does the sine of a difference formula express mathematically?

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

    Which of the following represents the cosine of a sum?

    <p>$ ext{cos}(eta + eta) = 2 ext{cos}^2 eta - 1$</p> Signup and view all the answers

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

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

    How can the cosine of a difference formula be summarized?

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

    What is derived from the sine and cosine of a double angle?

    <p>Two distinct identities for sine and cosine</p> Signup and view all the answers

    Which statement reflects a misunderstanding about the general solution of trigonometric equations?

    <p>Reference angles should always be negative.</p> Signup and view all the answers

    What is one of the formulas for the cosine of a double angle?

    <p>$ ext{cos}(2eta) = 2 ext{sin}^2 eta - 1$</p> Signup and view all the answers

    During the process of solving trigonometric equations, what is the role of the CAST diagram?

    <p>To define quadrants where functions are positive or negative.</p> Signup and view all the answers

    Which formula represents the sine of a sum?

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

    Which equation summarizes the reference angle concept in solving trigonometric equations?

    <p>Reference angles are always acute.</p> Signup and view all the answers

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

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

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

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

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

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

    Which of the following points lies on the circle represented by the equation ( (x - 1)^2 + (y + 2)^2 = 9 )?

    <p>( (2, -1) )</p> 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) )?

    <p>( 4x - 3y = 0 )</p> Signup and view all the answers

    What is the equation of the tangent to the circle ( (x + 2)^2 + (y - 1)^2 = 16 ) at the point ( (2, 5) )?

    <p>( 4x - 3y = -7 )</p> Signup and view all the answers

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

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

    When should the Sine Rule be used in triangle calculations?

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

    Which formula represents the Cosine Rule for side (b)?

    <p>(b^2 = a^2 + c^2 - 2ac \cos A)</p> Signup and view all the answers

    What is the formula to calculate the height of a pole using the Sine Rule?

    <p>(h = FB \tan \beta)</p> Signup and view all the answers

    Which equation represents the area of triangle ABC using side (b) and angle (A)?

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

    What should one do first when solving a problem involving three-dimensional triangles?

    <p>Draw a sketch to visualize the problem.</p> Signup and view all the answers

    How is the height of a building represented in terms of triangle BCD?

    <p>(h = BD \sin \alpha)</p> Signup and view all the answers

    What is the purpose of the Cosine Rule in triangles?

    <p>To find an angle when all three sides are known.</p> Signup and view all the answers

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

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

    Which of the following is a characteristic of a tangent line to a circle?

    <p>It touches the circle at exactly one point</p> Signup and view all the answers

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

    <p>To rewrite the equation in standard form</p> Signup and view all the answers

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

    <p>They are perpendicular</p> Signup and view all the answers

    What is the first step in finding the center and radius of a circle given the equation x^2 + y^2 + Dx + Ey + F = 0?

    <p>Group the x terms and the y terms</p> Signup and view all the answers

    What is the product of the gradients of the radius and tangent at the point of tangency?

    <p>-1</p> Signup and view all the answers

    What is the equation of a circle with center (-a, -b) and radius r, in standard form?

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

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

    <p>To make the equation easier to work with</p> Signup and view all the answers

    What is the first step in determining the equation of a tangent to a circle?

    <p>Write the equation of the circle in standard form</p> Signup and view all the answers

    What is the relationship between the center of a circle and the equation of the circle?

    <p>The equation of the circle is determined by the center</p> Signup and view all the answers

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

    <p>They are proportional</p> Signup and view all the answers

    What is the formula for the area of a triangle?

    <p>1/2 × base × height</p> Signup and view all the answers

    What is the purpose of the Basic Proportionality Theorem?

    <p>To compare different parts of geometric figures</p> Signup and view all the answers

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

    <p>They are reciprocals</p> Signup and view all the answers

    What is the formula for the equation of a tangent to a circle?

    <p>y - y1 = m(x - x1)</p> Signup and view all the answers

    What is the purpose of verifying the solution in proportional problems?

    <p>To check that the solution maintains the proportional relationships</p> Signup and view all the answers

    What is the Cross Multiplication property of proportion?

    <p>wx = yz</p> Signup and view all the answers

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

    <p>y - y1 = m(x - x1)</p> Signup and view all the answers

    What is the formula for the area of a parallelogram?

    <p>base × height</p> Signup and view all the answers

    What is the formula for the area of a rhombus?

    <p>diagonal AC × diagonal BD / 2</p> Signup and view all the answers

    What is the formula for the area of a trapezium?

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

    What is the definition of similar polygons?

    <p>Polygons with equal angles and proportional sides</p> Signup and view all the answers

    What is the Basic Proportionality Theorem also known as?

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

    What is the formula for the area of a triangle with the same height?

    <p>1/2 × base × height</p> Signup and view all the answers

    What is the conclusion of the Triangles with the Same Base theorem?

    <p>Triangles with equal bases and between the same parallel lines are equal in area</p> Signup and view all the answers

    What is the statement of the Mid-point Theorem?

    <p>The 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</p> Signup and view all the answers

    What is the definition of proportionality in polygons?

    <p>The equality of ratios between corresponding sides or other measurements in polygons</p> Signup and view all the answers

    What is the first step in deriving the formula for (\cos(\alpha + \beta)) from the formula for (\cos(\alpha - \beta))?

    <p>Rewrite (\alpha + \beta) as (\alpha - (-\beta))</p> Signup and view all the answers

    Which of the following identities is used to simplify the expression (\cos(\alpha - (-\beta))) during the derivation of the formula for (\cos(\alpha + \beta))?

    <p>The even-odd identities</p> Signup and view all the answers

    Which of the following is the formula for (\cos(\alpha + \beta))?

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

    What is the first step in deriving the formula for (\sin(\alpha - \beta)) and (\sin(\alpha + \beta))?

    <p>Rewrite the angles as sums or differences</p> Signup and view all the answers

    Which trigonometric identity is most useful in deriving the formula for (\sin(\alpha - \beta)) from the formula for (\cos(\alpha - \beta))?

    <p>The cosine difference formula</p> Signup and view all the answers

    The derivation of (\cos(\alpha + \beta)) and (\sin(\alpha + \beta)) relies on:

    <p>The angle addition formula</p> Signup and view all the answers

    Which of the following conditions must be met for two triangles to be similar?

    <p>They must have all corresponding angles equal.</p> Signup and view all the answers

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

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

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

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

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

    <p>The polygons must have the same area.</p> Signup and view all the answers

    What is the formula for the area of a triangle?

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

    What does the Pythagorean Theorem state?

    <p>The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.</p> Signup and view all the answers

    In the equation of the linear regression line, y = A + Bx, what does the variable 'B' represent?

    <p>The slope of the line</p> Signup and view all the answers

    What does a strong negative correlation between two sets of data imply?

    <p>As one variable increases, the other variable decreases.</p> Signup and view all the answers

    Which of the following values of the correlation coefficient, r, indicates the strongest positive correlation?

    <p>0.8</p> Signup and view all the answers

    What does the Converse of the Pythagorean Theorem state?

    <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 triangle is a right-angled triangle.</p> Signup and view all the answers

    What is the formula for the sine of a difference?

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

    What is the formula for the cosine of a sum?

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

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

    <p>2sin(α)cos(α)</p> Signup and view all the answers

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

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

    What is the general solution method for trigonometric equations?

    <p>All of the above</p> Signup and view all the answers

    What is the purpose of the CAST diagram?

    <p>To determine where the function is positive or negative</p> Signup and view all the answers

    What is the formula for the cosine of a difference?

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

    What is the formula for the sine of a sum?

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

    What is the purpose of verifying the solutions in trigonometric equations?

    <p>To check the solutions using a calculator</p> Signup and view all the answers

    What is the first step in solving a trigonometric equation?

    <p>Simplify the equation using algebraic methods and trigonometric identities</p> Signup and view all the answers

    What does a correlation coefficient of $r = 0$ indicate?

    <p>No correlation</p> Signup and view all the answers

    Which range of values indicates a strong positive correlation?

    <p>$0.8 &lt; r ext{ and } r ext{ less than } 1$</p> Signup and view all the answers

    In the equation of a circle $x^2 + y^2 = r^2$, what does $r$ represent?

    <p>The radius of the circle</p> Signup and view all the answers

    What is the formula for calculating the linear correlation coefficient $r$?

    <p>$r = b imes rac{ ext{std}(y)}{ ext{std}(x)}$</p> Signup and view all the answers

    What does a negative correlation value indicate about the relationship between two variables?

    <p>As $x$ increases, $y$ decreases</p> Signup and view all the answers

    Which of the following best describes a weak positive correlation?

    <p>$0 &lt; r ext{ and } r &lt; 0.4$</p> Signup and view all the answers

    In the context of correlation, what does the term 'gradient' refer to?

    <p>The slope of the least squares regression line</p> Signup and view all the answers

    How is the equation of a line derived from the regression coefficients calculated?

    <p>$y = A + Bx$</p> Signup and view all the answers

    What does the equation $x^2 + y^2 = r^2$ represent geometrically?

    <p>A circle centered at the origin</p> Signup and view all the answers

    The terms 'medium positive correlation' corresponds to which range of values?

    <p>$0.4 &lt; r ext{ and } r ext{ less than } 0.8$</p> Signup and view all the answers

    What happens when a line is drawn from the midpoint of one side of a triangle parallel to another side?

    <p>It bisects the third side of the triangle.</p> Signup and view all the answers

    What is required for two triangles to be similar based on sides?

    <p>The corresponding sides must be in proportion.</p> Signup and view all the answers

    Which of the following means the triangles are equiangular?

    <p>All corresponding angles are equal.</p> Signup and view all the answers

    What does the area of a triangle depend on?

    <p>The base and height of the triangle.</p> Signup and view all the answers

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

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

    When should the Cosine Rule be used?

    <p>When no right angle is given and either two sides and the included angle or three sides are known.</p> Signup and view all the answers

    What is the area of a triangle calculated using the area rule when given two sides and the included angle?

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

    In which scenario is the Sine Rule applicable?

    <p>When two sides and an angle (not the included angle) are known.</p> Signup and view all the answers

    For the equation \( ext{Area} = rac{1}{2}bc \sin A\), which components are necessary to compute the area?

    <p>Two sides and the angle between them.</p> Signup and view all the answers

    Which of the following defines the height of a pole using the given information in triangle FAB?

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

    What expression relates sides BD and height h in triangle ABD?

    <p>\(h = BD \sin \alpha\)</p> Signup and view all the answers

    What must be done as the first step in solving problems in three dimensions?

    <p>Draw a sketch to visualize the problem.</p> Signup and view all the answers

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