Gr 11 Math June P2 (Hard)
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Questions and Answers

What is the name of the form of the straight line equation that is used when the gradient and one point on the line are known?

  • Slope-Intercept Form
  • Gradient–Point Form (correct)
  • Gradient–Intercept Form
  • Two-Point Form

    • What is the value of y when x is 0 in the gradient–intercept form of a straight line equation?

  • mx + c
  • c (correct)
  • mx
  • mx - c

    • What is the definition of the constant c in the gradient–intercept form of a straight line equation?

  • c = y_1 + mx_1
  • c = x_1 + my_1
  • c = y_1 - mx_1 (correct)
  • c = x_1 - my_1

    • What form of the straight line equation is derived from the two-point form?

    <p>Gradient–Point Form</p> Signup and view all the answers

    What are the two pieces of information required to use the gradient–point form of a straight line equation?

    <p>Gradient and one point</p> Signup and view all the answers

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

    <p>Determining the equation of a line given two points</p> Signup and view all the answers

    What is the step to derive the gradient–intercept form from the gradient–point form?

    <p>Make y the subject of the formula</p> Signup and view all the answers

    What is the name of the form of the straight line equation that is used when two points on the line are known?

    <p>Two-Point Form</p> Signup and view all the answers

    What is the relationship between the gradient of a line and its inclination?

    <p>The gradient is the tangent of the inclination.</p> Signup and view all the answers

    What is the gradient of a vertical line?

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

    If two lines are parallel, what can be said about their gradients?

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

    What is the formula to find the inclination of a line with a negative gradient?

    <p>θ = 180° + tan⁻¹(m)</p> Signup and view all the answers

    What is the purpose of the point-slope form of a line?

    <p>To find the equation of a line given the gradient and a point.</p> Signup and view all the answers

    Why is it important to ensure the equation of a line is in gradient-intercept form?

    <p>To directly read off the gradient.</p> Signup and view all the answers

    What can be said about the tangent of an acute angle?

    <p>It is always positive.</p> Signup and view all the answers

    What is the gradient of a horizontal line?

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

    What is the relationship between the inclination of a line and the positive x-axis?

    <p>The inclination is the angle formed between the line and the positive x-axis.</p> Signup and view all the answers

    What is the purpose of using the CAST diagram in trigonometry?

    <p>To determine the quadrant of an angle.</p> Signup and view all the answers

    What is the condition for two lines to be parallel?

    <p>They have the same gradient but different y-intercepts</p> Signup and view all the answers

    If two lines have gradients of 2 and -1/2, what can be concluded about the lines?

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

    What is the equation of a line perpendicular to y = 3x + 2 and passes through the point (1, 4)?

    <p>y = -1/3x + 10/3</p> Signup and view all the answers

    What is the trigonometric identity for tangent in terms of sine and cosine?

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

    What is the Pythagorean identity in trigonometry?

    <p>sin^2 θ + cos^2 θ = 1</p> Signup and view all the answers

    What is a useful tip for proving trigonometric identities?

    <p>Change all trigonometric ratios to sine and cosine</p> Signup and view all the answers

    What does the square identity simplify to?

    <p>sin^2 θ + cos^2 θ = 1</p> Signup and view all the answers

    What is a condition for tan θ to be undefined?

    <p>θ = k × 90°, where k is an odd integer</p> Signup and view all the answers

    What is the first step in finding the equation of a line perpendicular to a given line?

    <p>Identify the gradient of the given line</p> Signup and view all the answers

    What is the point-slope form of the equation of a line?

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

    What is the value of tan(θ) in terms of sine and cosine functions?

    <p>tan(θ) = sin(θ) / cos(θ)</p> Signup and view all the answers

    What is the result of the expression sin²(θ) + cos²(θ)?

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

    What is the value of sin(180° - θ) in terms of sine of θ?

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

    What is the value of cos(180° + θ) in terms of cosine of θ?

    <p>-cos(θ)</p> Signup and view all the answers

    What is the value of tan(180° + θ) in terms of tangent of θ?

    <p>-tan(θ)</p> Signup and view all the answers

    What is the expression for sin²(θ) in terms of cosine of θ?

    <p>1 - cos²(θ)</p> Signup and view all the answers

    What is the expression for cos²(θ) in terms of sine of θ?

    <p>1 - sin²(θ)</p> Signup and view all the answers

    What is the characteristic of the reduction formula for trigonometric functions?

    <p>It can be used to simplify trigonometric expressions involving angles of the form 90° ± θ, 180° ± θ, and 360° ± θ.</p> Signup and view all the answers

    If point P is rotated through 90°, what are the coordinates of P'?

    <p>(-y, x)</p> Signup and view all the answers

    What is the value of sin(180° + θ)?

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

    If points P and P' are symmetrical about the x-axis, what are the coordinates of P'?

    <p>(x, -y)</p> Signup and view all the answers

    What is the value of cos(360° - θ)?

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

    What is the value of tan(180° + θ)?

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

    If points P and P' are symmetrical about the origin, what are the coordinates of P'?

    <p>(-x, -y)</p> Signup and view all the answers

    What is the value of sin(-θ)?

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

    What is the value of cos(90° + θ)?

    <p>-cos(θ)</p> Signup and view all the answers

    What is the value of tan(90° - θ)?

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

    What is the value of sin(360° + θ)?

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

    What is the formula for the surface area of a square pyramid?

    <p>b(b + 2h_s)</p> Signup and view all the answers

    What is the formula for the volume of a triangular prism?

    <p>1/2bhH</p> Signup and view all the answers

    If the dimensions of a solid are multiplied by a factor k, how does the volume scale?

    <p>k^3</p> Signup and view all the answers

    What is the name of the theorem that states that the angle at the center of a circle is twice the angle at the circumference?

    <p>Angle at the Center Theorem</p> Signup and view all the answers

    What is the formula for the area of a circle?

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

    What is the name of the line that touches a circle at only one point?

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

    What is the formula for the surface area of a right cylinder?

    <p>2πr(h + r)</p> Signup and view all the answers

    What is the formula for the volume of a sphere?

    <p>4/3πr^3</p> Signup and view all the answers

    What is the name of the theorem that states that the perpendicular bisector of a chord passes through the center of the circle?

    <p>Perpendicular Bisector of Chord Passes Through Circle Center</p> Signup and view all the answers

    What is the formula for the area of a trapezium?

    <p>1/2(a + b)h</p> Signup and view all the answers

    A line segment from the center of a circle is perpendicular to a chord. What is the effect on the chord?

    <p>The chord is bisected.</p> Signup and view all the answers

    What is the relationship between the angles subtended by the same arc in a circle?

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

    What is the period of the function y = sin(kθ)?

    <p>360°/k</p> Signup and view all the answers

    What is the effect of a on the shape of the function y = a sin(θ)?

    <p>It increases the amplitude.</p> Signup and view all the answers

    What is the effect of q on the graph of the function y = a sin(θ) + q?

    <p>It shifts the graph vertically.</p> Signup and view all the answers

    What are the x-intercepts of the function y = cos(θ)?

    <p>(90°, 0), (270°, 0)</p> Signup and view all the answers

    What is the effect of k on the period of the function y = sin(kθ)?

    <p>It decreases the period.</p> Signup and view all the answers

    What is the range of the function y = sin(θ)?

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

    What is the effect of a on the amplitude of the function y = a cos(θ)?

    <p>It increases the amplitude.</p> Signup and view all the answers

    What is the y-intercept of the function y = cos(θ)?

    <p>(0°, 1)</p> Signup and view all the answers

    What is the primary purpose of using reduction formulae and co-function rules in trigonometry?

    <p>To simplify trigonometric expressions and facilitate evaluation.</p> Signup and view all the answers

    What is the first step in solving a trigonometric equation?

    <p>Use a calculator to find the reference angle.</p> Signup and view all the answers

    What is the formula for the area of a trapezium?

    <p>A = (1/2) (a + b) × h</p> Signup and view all the answers

    What is the surface area of a prism or cylinder?

    <p>The total area of the exposed or outer surfaces.</p> Signup and view all the answers

    What is the name of the geometric solid that has a polygon as its base and vertical sides perpendicular to the base?

    <p>Right prism</p> Signup and view all the answers

    What is the primary purpose of unfolding a prism or cylinder into a net?

    <p>To calculate the surface area.</p> Signup and view all the answers

    What is the formula for the area of a circle?

    <p>A = π r^2</p> Signup and view all the answers

    What is the general approach to solving quadratic trigonometric equations?

    <p>Substitute suitable values to find the solutions within the given interval.</p> Signup and view all the answers

    What is the formula for the circumference of a circle?

    <p>C = 2π r</p> Signup and view all the answers

    What is the purpose of using the CAST diagram in trigonometry?

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

    What is the formula to calculate the volume of a triangular prism?

    <p>V = (1/2)b x h x H</p> Signup and view all the answers

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

    <p>SA = r(r + h)</p> Signup and view all the answers

    What is the formula to calculate the volume of a cylinder?

    <p>V = r^2 x h</p> Signup and view all the answers

    If the dimensions of a rectangular prism are multiplied by a constant factor k, what is the formula to calculate the new volume?

    <p>V_1 = kV</p> Signup and view all the answers

    What is the formula to calculate the surface area of a sphere?

    <p>SA = 4r^2</p> Signup and view all the answers

    What is the formula to calculate the volume of a right pyramid?

    <p>V = (1/3)bh x H</p> Signup and view all the answers

    If the dimensions of a rectangular prism are multiplied by a constant factor k, what is the formula to calculate the new surface area?

    <p>A_1 = 2[klh + lb + kbh]</p> Signup and view all the answers

    What is the formula to calculate the volume of a square pyramid?

    <p>V = (1/3)b^2 x H</p> Signup and view all the answers

    What is the formula to calculate the surface area of a triangular pyramid?

    <p>SA = (1/2)b(h_b + 3h_s)</p> Signup and view all the answers

    If the dimensions of a rectangular prism are multiplied by a constant factor k, what is the formula to calculate the new surface area when two dimensions are multiplied by k?

    <p>A_2 = k^2 * 2(lh + lb + bh)</p> Signup and view all the answers

    What is the period of the function y = cos(2θ)?

    <p>180°</p> Signup and view all the answers

    What is the effect of k on the period of y = tan(kθ)?

    <p>The period decreases as k increases.</p> Signup and view all the answers

    What is the range of y = tan(θ)?

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

    What is the effect of p on the graph of y = tan(θ + p)?

    <p>The graph shifts to the left by p units.</p> Signup and view all the answers

    What is the domain of y = cos(kθ)?

    <p>{θ : θ ∈ ℝ}</p> Signup and view all the answers

    What is the effect of a on the graph of y = a tan(θ)?

    <p>The branches of the graph are steeper.</p> Signup and view all the answers

    What is the y-intercept of y = cos(kθ)?

    <p>(0, 1)</p> Signup and view all the answers

    What is the effect of k on the graph of y = cos(kθ)?

    <p>The period decreases as k increases.</p> Signup and view all the answers

    What is the domain of y = tan(kθ)?

    <p>{θ : θ ∈ ℝ, θ ≠ 90°/k, 270°/k}</p> Signup and view all the answers

    What is the asymptote of y = tan(kθ)?

    <p>θ = 90°/k</p> Signup and view all the answers

    What is the value of sin(180° - θ) in terms of sine of θ?

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

    What is the condition for tan(θ) to be undefined?

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

    What is the expression for cos²(θ) in terms of sine of θ?

    <p>1 - sin²(θ)</p> Signup and view all the answers

    What is the value of tan(180° + θ) in terms of tangent of θ?

    <p>−tan(θ)</p> Signup and view all the answers

    What is the characteristic of the reduction formula for trigonometric functions?

    <p>It is used to find the values of trigonometric functions for angles greater than 90°.</p> Signup and view all the answers

    What is the value of cos(360° - θ) in terms of cosine of θ?

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

    What is the proof of the quotient identity tan(θ) = sin(θ) / cos(θ)?

    <p>Geometry of the unit circle</p> Signup and view all the answers

    What is the result of the expression sin²(θ) + cos²(θ)?

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

    What is the main difference between the two-point form and the gradient-point form of the straight line equation?

    <p>The two-point form is used when two points on the line are known, while the gradient-point form is used when the gradient and one point on the line are known.</p> Signup and view all the answers

    What is the purpose of rearranging the gradient-point form to make y the subject of the formula?

    <p>To derive the gradient-intercept form</p> Signup and view all the answers

    What is the relationship between the gradient of a line and its inclination?

    <p>The gradient of a line is tan(θ) where θ is the inclination of the line</p> Signup and view all the answers

    What can be said about the y-intercept of a line in the gradient-intercept form?

    <p>It is the point where the line intersects the y-axis</p> Signup and view all the answers

    What is the advantage of using the gradient-intercept form of a straight line equation?

    <p>It is easier to find the y-intercept of the line</p> Signup and view all the answers

    What is the condition for two lines to be parallel?

    <p>They have the same gradient</p> Signup and view all the answers

    What is the purpose of using the two-point form of a straight line equation?

    <p>To find the equation of a line when two points on the line are known</p> Signup and view all the answers

    What can be said about the gradient of a vertical line?

    <p>It is undefined</p> Signup and view all the answers

    What is the relationship between the gradient of a line and its inclination in the second quadrant?

    <p>m = -tan(180° - θ)</p> Signup and view all the answers

    What is the equation of a line that passes through the point (2, 3) and is parallel to the line y = 2x + 1?

    <p>y = 2x + 5</p> Signup and view all the answers

    What is the value of θ when the gradient of a line is -3/4?

    <p>tan⁻¹(-3/4) + 180°</p> Signup and view all the answers

    What is the condition for two lines to be parallel in terms of their inclination?

    <p>θ₁ = θ₂</p> Signup and view all the answers

    What is the equation of a line that passes through the point (1, 2) and has a gradient of 3/2?

    <p>y = 3/2x + 1/2</p> Signup and view all the answers

    What is the gradient of a line that is perpendicular to the line y = 2x + 1 and passes through the origin?

    <p>-1/2</p> Signup and view all the answers

    What is the value of θ when the gradient of a line is 2/3?

    <p>tan⁻¹(2/3)</p> Signup and view all the answers

    What is the equation of a line that is parallel to the line y = 3x - 2 and passes through the point (4, 5)?

    <p>y = 3x + 1</p> Signup and view all the answers

    What is the relationship between the gradient of a line and its inclination in the first quadrant?

    <p>m = tan(θ)</p> Signup and view all the answers

    What is the equation of a line that passes through the point (3, 4) and is parallel to the line y = 2x + 3?

    <p>y = 2x + 5</p> Signup and view all the answers

    If two lines are perpendicular, what is the relationship between their gradients?

    <p>The product of their gradients is -1</p> Signup and view all the answers

    What is the purpose of the point-slope form of a line?

    <p>To find the equation of a line given its slope and one point on the line</p> Signup and view all the answers

    What is the trigonometric identity for sin²(θ) + cos²(θ)

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

    What is the condition for tan(θ) to be undefined?

    <p>θ = 90°</p> Signup and view all the answers

    If two lines have gradients of 3 and -1/3, what can be concluded about the lines?

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

    What is the first step in finding the equation of a line perpendicular to a given line?

    <p>Identify the gradient of the given line</p> Signup and view all the answers

    What is the result of the expression sin(θ)cos(θ) + cos(θ)sin(θ)?

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

    What is the value of tan(θ) in terms of sine and cosine functions?

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

    What is the useful tip for proving trigonometric identities?

    <p>Change all trigonometric ratios to sine and cosine</p> Signup and view all the answers

    What is the equation of a line perpendicular to y = 2x - 3 and passing through the point (1, 5)?

    <p>y = -1/2x + 7</p> Signup and view all the answers

    If points P and P' are symmetrical about the line y = x, what are the coordinates of P'?

    <p>(y, x)</p> Signup and view all the answers

    What is the value of sin(360° - θ) in terms of sine of θ?

    <p>sin θ</p> Signup and view all the answers

    If P is rotated through 90°, what are the coordinates of P'?

    <p>(-y, x)</p> Signup and view all the answers

    What is the value of cos(180° + θ) in terms of cosine of θ?

    <p>-cos θ</p> Signup and view all the answers

    What is the value of tan(90° + θ) in terms of tangent of θ?

    <p>-tan θ</p> Signup and view all the answers

    What is the value of sin(90° - θ) in terms of cosine of θ?

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

    If points P and P' are symmetrical about the origin, what are the coordinates of P'?

    <p>(-x, -y)</p> Signup and view all the answers

    What is the value of cos(360° + θ) in terms of cosine of θ?

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

    What is the value of tan(180° - θ) in terms of tangent of θ?

    <p>-tan θ</p> Signup and view all the answers

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

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

    What is the purpose of using the CAST diagram in trigonometry?

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

    What is the reduction formula for tan(180° + θ)?

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

    What is the general solution of the equation sin(θ) = 1/2?

    <p>θ = 30° + 360°n</p> Signup and view all the answers

    What is the surface area of a rectangular prism with a length of 6 cm, a width of 4 cm, and a height of 3 cm?

    <p>78 cm²</p> Signup and view all the answers

    What is the formula for the area of a trapezium?

    <p>1/2(a + b)h</p> Signup and view all the answers

    What is the value of sin(90° + θ) in terms of cosine of θ?

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

    What is the characteristic of the reduction formula for trigonometric functions?

    <p>It allows us to express trigonometric functions in terms of acute angles.</p> Signup and view all the answers

    What is the formula for the area of a circle?

    <p>πr²</p> Signup and view all the answers

    What is the purpose of unfolding a prism into a net?

    <p>To find the surface area of the prism.</p> Signup and view all the answers

    What is the value of cos(180° - θ) in terms of cosine of θ?

    <p>-cos(θ)</p> Signup and view all the answers

    What is the formula for the volume of a cylinder?

    <p>πr²h</p> Signup and view all the answers

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

    <p>4πr²</p> Signup and view all the answers

    What is the effect of scaling the dimensions of a solid by a factor k on its volume?

    <p>It scales by k³</p> Signup and view all the answers

    What is the name of the theorem that states that the angle at the center of a circle is twice the angle at the circumference?

    <p>Angle at the Center Theorem</p> Signup and view all the answers

    What is the formula for the surface area of a triangular pyramid?

    <p>bh + 3hs</p> Signup and view all the answers

    What is the formula for the volume of a square pyramid?

    <p>b²h/3</p> Signup and view all the answers

    What is the effect of a on the shape of the function y = a sin θ?

    <p>The amplitude increases but the graph is not reflected.</p> Signup and view all the answers

    What is the formula for the area of a trapezium?

    <p>h(b + a)/2</p> Signup and view all the answers

    What is the period of the function y = sin kθ?

    <p>360°/|k|</p> Signup and view all the answers

    What is the formula for the area of a circle?

    <p>πr²</p> Signup and view all the answers

    What is the formula for the volume of a triangular prism?

    <p>bhH/2</p> Signup and view all the answers

    What is the effect of p on the graph of the function y = sin(θ + p)?

    <p>The graph shifts to the left by p units.</p> Signup and view all the answers

    What is the y-intercept of the function y = cos θ?

    <p>(0, 1)</p> Signup and view all the answers

    What is the effect of scaling the dimensions of a solid by a factor k on its surface area?

    <p>It scales by k²</p> Signup and view all the answers

    What is the amplitude of the function y = a sin θ + q?

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

    What is the effect of q on the graph of the function y = a sin θ + q?

    <p>The graph shifts vertically upwards by q units.</p> Signup and view all the answers

    What is the formula for the volume of a right cone?

    <p>1/3πr^2h</p> Signup and view all the answers

    What is the x-intercept of the function y = sin θ?

    <p>(180, 0)</p> Signup and view all the answers

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

    <p>4πr^2</p> Signup and view all the answers

    What is the range of the function y = sin kθ?

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

    What is the domain of the function y = cos θ?

    <p>{θ: θ ∈ ℝ}</p> Signup and view all the answers

    If the dimensions of a solid are multiplied by a factor k, how does the surface area scale?

    <p>It scales by k^2</p> Signup and view all the answers

    What is the minimum turning point of the function y = cos θ?

    <p>(180, -1)</p> Signup and view all the answers

    What is the formula for the volume of a triangular pyramid?

    <p>1/3bhH</p> Signup and view all the answers

    What is the formula for the surface area of a right cone?

    <p>πr^2 + πrL</p> Signup and view all the answers

    If the dimensions of a solid are multiplied by a factor k, how does the volume scale?

    <p>It scales by k^3</p> Signup and view all the answers

    What is the formula for the volume of a rectangular prism?

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

    What is the formula for the surface area of a square pyramid?

    <p>b^2 + 2bh</p> Signup and view all the answers

    What is the formula for the volume of a cylinder?

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

    What is the formula for the surface area of a triangular prism?

    <p>bh + 2l(b + h)</p> Signup and view all the answers

    What is the period of the function y = cos kθ?

    <p>360°/|k|</p> Signup and view all the answers

    What is the effect of k on the graph of y = cos kθ when k > 1?

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

    What is the y-intercept of the function y = cos kθ?

    <p>(0, 1)</p> Signup and view all the answers

    What is the effect of p on the graph of y = cos(θ + p) when p > 0?

    <p>The graph is shifted to the left by p units</p> Signup and view all the answers

    What is the period of the function y = tan kθ?

    <p>180°/|k|</p> Signup and view all the answers

    What is the effect of k on the graph of y = tan kθ when k > 1?

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

    What is the y-intercept of the function y = tan kθ?

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

    What is the effect of p on the graph of y = tan(θ + p) when p > 0?

    <p>The graph is shifted to the left by p units</p> Signup and view all the answers

    What is the range of the function y = tan kθ?

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

    What is the domain of the function y = tan kθ?

    <p>{θ: θ ∈ ℝ, θ ≠ 90°/k, 270°/k}</p> Signup and view all the answers

    What is the condition required to use the two-point form of the straight line equation?

    <p>Two points on the line are known</p> Signup and view all the answers

    What is the formula derived from the gradient–point form of the straight line equation?

    <p>y = mx - mx_1 + y_1</p> Signup and view all the answers

    What is the value of y_1 - mx_1 in the gradient–intercept form of the straight line equation?

    <p>The y-intercept of the line</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 line when the gradient and one point are known</p> Signup and view all the answers

    What is the relationship between the gradient of a line and the constant c in the gradient–intercept form?

    <p>The gradient is not related to the constant c</p> Signup and view all the answers

    What is the advantage of using the gradient–intercept form of the straight line equation?

    <p>It provides a standard form for the equation of a line</p> Signup and view all the answers

    What is the formula derived from the two-point form of the straight line equation?

    <p>y - y_1 = m(x - x_1)</p> Signup and view all the answers

    What is the condition required to use the gradient–intercept form of the straight line equation?

    <p>The y-intercept of the line is known</p> Signup and view all the answers

    What is the condition for tan θ to be undefined?

    <p>cos θ = 0</p> Signup and view all the answers

    What is the result of the expression sin²(θ) + cos²(θ)?

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

    What is the value of sin(180° - θ) in terms of sine of θ?

    <p>sin θ</p> Signup and view all the answers

    What is the expression for sin²(θ) in terms of cosine of θ?

    <p>1 - cos²(θ)</p> Signup and view all the answers

    What is the characteristic of the reduction formula for trigonometric functions?

    <p>It can be used to simplify trigonometric functions</p> Signup and view all the answers

    What is the value of tan(180° + θ) in terms of tangent of θ?

    <p>-tan θ</p> Signup and view all the answers

    What is the relationship between the gradient of a line and its inclination when the angle is acute?

    <p>m = tan(θ)</p> Signup and view all the answers

    What is the value of cos(180° - θ) in terms of cosine of θ?

    <p>-cos θ</p> Signup and view all the answers

    What is the value of sin(90° + θ) in terms of cosine of θ?

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

    Which form of the equation of a line is most useful when finding the equation of a parallel line?

    <p>Gradient-intercept form</p> Signup and view all the answers

    If a line has a negative gradient, what can be said about the angle of inclination?

    <p>It is an obtuse angle</p> Signup and view all the answers

    What is the relationship between the gradient of a line and its y-intercept in the gradient-intercept form?

    <p>m is independent of c</p> Signup and view all the answers

    What is the purpose of using the CAST diagram in trigonometry when dealing with negative gradients?

    <p>To determine the quadrants of the tangent function</p> Signup and view all the answers

    What is the condition for two lines to be parallel in terms of their gradients?

    <p>m1 = m2</p> Signup and view all the answers

    What is the step to derive the equation of a parallel line from a given line?

    <p>Identify the gradient of the given line</p> Signup and view all the answers

    What is the trigonometric relationship between the gradient of a line and its inclination when the angle is obtuse?

    <p>m = tan(180° + θ)</p> Signup and view all the answers

    What is the importance of ensuring the equation of a line is in gradient-intercept form when finding the equation of a parallel line?

    <p>To identify the gradient</p> Signup and view all the answers

    What is the characteristic of the tangent function in the second quadrant?

    <p>It is negative</p> Signup and view all the answers

    What is the condition for two lines to be perpendicular in terms of their gradients?

    <p>m1 × m2 = -1</p> Signup and view all the answers

    What is the trigonometric identity for sine in terms of cosine?

    <p>sinθ = sqrt(1 - cos²θ)</p> Signup and view all the answers

    What is the first step in finding the equation of a line perpendicular to a given line?

    <p>Identify the gradient of the given line</p> Signup and view all the answers

    What is the purpose of the square identity in trigonometry?

    <p>To simplify trigonometric expressions</p> Signup and view all the answers

    What is the condition for tanθ to be undefined?

    <p>θ = 90°</p> Signup and view all the answers

    What is the result of the expression sin²(θ) + cos²(θ)?

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

    What is the gradient of a line perpendicular to y = 2x + 3?

    <p>-1/2</p> Signup and view all the answers

    What is the equation of a line perpendicular to y = 3x + 2 and passes through the point (2, 5)?

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

    What is the trigonometric identity for cosine in terms of sine?

    <p>cosθ = sqrt(1 - sin²θ)</p> Signup and view all the answers

    What is the purpose of using the quotient identity in trigonometry?

    <p>To simplify trigonometric expressions</p> Signup and view all the answers

    What is the formula for the surface area of a right cone?

    <p>2πr(r + h)</p> Signup and view all the answers

    What is the formula for the volume of a cylinder?

    <p>πr²h</p> Signup and view all the answers

    If the dimensions of a solid are multiplied by a factor k, how does the surface area scale?

    <p>k²</p> Signup and view all the answers

    What is the formula for the surface area of a right cone?

    <p>πr(r + h)</p> Signup and view all the answers

    What is the formula for the volume of a sphere?

    <p>4/3πr³</p> Signup and view all the answers

    What is the formula for the area of a trapezium?

    <p>(a + b)h/2</p> Signup and view all the answers

    Which of the following trigonometric identities is used to express trigonometric functions in terms of acute angles?

    <p>Reduction formulae</p> Signup and view all the answers

    If the dimensions of a rectangular prism are multiplied by a factor k, how does the surface area change?

    <p>It increases by k^2</p> Signup and view all the answers

    What is the first step in solving a trigonometric equation?

    <p>Use a calculator to find the reference angle</p> Signup and view all the answers

    What is the formula for the volume of a sphere?

    <p>4/3πr^3</p> Signup and view all the answers

    What is the formula for the area of a circle?

    <p>πr²</p> Signup and view all the answers

    If the length of a rectangle is multiplied by a factor k, how does the volume of the rectangular prism change?

    <p>It increases by k</p> Signup and view all the answers

    What is the theorem that states that the angle at the center of a circle is twice the angle at the circumference?

    <p>Angle at the Center is Twice the Angle at the Circumference</p> Signup and view all the answers

    What is the formula for the area of a trapezium?

    <p>A = ½ (a + b) × h</p> Signup and view all the answers

    What is the formula for the surface area of a triangular pyramid?

    <p>1/2b(h_b + 3h_s)</p> Signup and view all the answers

    What is the name of the geometric solid that has a polygon as its base and vertical sides perpendicular to the base?

    <p>Right prism</p> Signup and view all the answers

    What is the formula for the area of a parallelogram?

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

    What is the purpose of unfolding a prism or cylinder into a net?

    <p>To find the surface area of the prism or cylinder</p> Signup and view all the answers

    If the height of a cylinder is multiplied by a factor k, how does the volume change?

    <p>It increases by k</p> Signup and view all the answers

    What is the formula for the volume of a square pyramid?

    <p>b²h/3</p> Signup and view all the answers

    What is the formula for the volume of a triangular prism?

    <p>1/2bhH</p> Signup and view all the answers

    What is the formula for the area of a triangle?

    <p>bh/2</p> Signup and view all the answers

    What is the formula for the area of a circle?

    <p>A = πr^2</p> Signup and view all the answers

    If the radius of a sphere is multiplied by a factor k, how does the volume change?

    <p>It increases by k^3</p> Signup and view all the answers

    What is the characteristic of the reduction formula for trigonometric functions?

    <p>It is used to express trigonometric functions in terms of acute angles</p> Signup and view all the answers

    What is the formula for the surface area of a square pyramid?

    <p>b(b + 2h_s)</p> Signup and view all the answers

    What is the purpose of using the CAST diagram in trigonometry?

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

    If the base of a right pyramid is multiplied by a factor k, how does the surface area change?

    <p>It increases by k^2</p> Signup and view all the answers

    What is the formula for the circumference of a circle?

    <p>C = 2πr</p> Signup and view all the answers

    What is the formula for the area of a triangle?

    <p>A = ½ b × h</p> Signup and view all the answers

    What is the effect of a > 1 on the graph of y = a sin θ?

    <p>The amplitude increases.</p> Signup and view all the answers

    What is the period of the graph of y = sin kθ?

    <p>360°/k</p> Signup and view all the answers

    What is the minimum turning point of the graph of y = cos θ?

    <p>(180°, -1)</p> Signup and view all the answers

    What is the effect of q on the graph of y = a sin θ + q?

    <p>The graph shifts vertically upwards by q units.</p> Signup and view all the answers

    What is the domain of the function y = sin kθ?

    <p>{θ : θ ∈ ℝ}</p> Signup and view all the answers

    What is the range of the function y = a sin θ?

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

    What is the effect of p on the graph of y = sin(θ + p)?

    <p>The graph shifts horizontally to the left by p units.</p> Signup and view all the answers

    What is the x-intercept of the graph of y = sin θ?

    <p>(90°, 0)</p> Signup and view all the answers

    What is the maximum turning point of the graph of y = cos θ?

    <p>(0°, 1)</p> Signup and view all the answers

    What is the y-intercept of the graph of y = a cos θ + q?

    <p>(0°, a + q)</p> Signup and view all the answers

    If points P and P' are symmetrical about the y-axis, what are the coordinates of P'?

    <p>(-x, y)</p> Signup and view all the answers

    What is the value of sin(360° - θ) in terms of sine of θ?

    <p>sin θ</p> Signup and view all the answers

    If points P and P' are symmetrical about the line y = x, what are the coordinates of P'?

    <p>(y, x)</p> Signup and view all the answers

    What is the value of cos(90° + θ) in terms of sine of θ?

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

    What is the value of tan(90° - θ) in terms of tangent of θ?

    <p>tan θ</p> Signup and view all the answers

    What is the value of sin(180° + θ) in terms of sine of θ?

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

    What is the value of cos(360° + θ) in terms of cosine of θ?

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

    What is the value of tan(180° - θ) in terms of tangent of θ?

    <p>-tan θ</p> Signup and view all the answers

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

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

    What is the value of cos(90° - θ) in terms of cosine of θ?

    <p>sin θ</p> Signup and view all the answers

    What is the period of the function y = cos kθ?

    <p>360°/|k|</p> Signup and view all the answers

    What is the effect of k on the period of y = cos kθ?

    <p>For k &gt; 1, the period decreases, and for 0 &lt; k &lt; 1, the period increases.</p> Signup and view all the answers

    What is the range of the function y = cos kθ?

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

    What is the effect of p on the graph of y = cos(θ + p)?

    <p>For p &gt; 0, the graph shifts to the left, and for p &lt; 0, the graph shifts to the right.</p> Signup and view all the answers

    What is the period of the function y = tan kθ?

    <p>180°/|k|</p> Signup and view all the answers

    What is the effect of k on the period of y = tan kθ?

    <p>For k &gt; 1, the period decreases, and for 0 &lt; k &lt; 1, the period increases.</p> Signup and view all the answers

    What is the domain of the function y = tan kθ?

    <p>{θ: θ ∈ ℝ, θ ≠ 90°/k, 270°/k}</p> Signup and view all the answers

    What is the effect of p on the graph of y = tan(θ + p)?

    <p>For p &gt; 0, the graph shifts to the left, and for p &lt; 0, the graph shifts to the right.</p> Signup and view all the answers

    What is the effect of a on the shape of the graph of y = a tan θ + q?

    <p>For a &gt; 1, the branches of the graph are steeper, and for 0 &lt; a &lt; 1, the branches are less steep.</p> Signup and view all the answers

    What is the range of the function y = tan kθ?

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

    Which of the following is true about the gradient–point form of a straight line equation?

    <p>It is used when the gradient and one point on the line are known.</p> Signup and view all the answers

    What is the advantage of expressing a straight line equation in the gradient–intercept form?

    <p>It is easier to find the y-intercept of the line.</p> Signup and view all the answers

    Given a straight line equation in the form y = mx + c, what can be said about the value of c?

    <p>It is the y-intercept of the line.</p> Signup and view all the answers

    Which of the following is a step in deriving the gradient–intercept form from the gradient–point form?

    <p>Expanding the brackets and making y the subject of the formula.</p> Signup and view all the answers

    What is the significance of the two-point form of a straight line equation?

    <p>It is used to find the equation of a line given two points on the line.</p> Signup and view all the answers

    Which of the following is a characteristic of the gradient–point form of a straight line equation?

    <p>It is used when the gradient and one point on the line are known.</p> Signup and view all the answers

    What is the relationship between the two-point form and the gradient–point form of a straight line equation?

    <p>The gradient–point form is derived from the two-point form.</p> Signup and view all the answers

    What is the advantage of expressing a straight line equation in the two-point form?

    <p>It is easier to find the equation of a line given two points on the line.</p> Signup and view all the answers

    What is the reduction formula for ∇(180° - θ) in terms of tan(θ)?

    <p>-tan(θ)</p> Signup and view all the answers

    If points P and P' are symmetrical about the y-axis, what are the coordinates of P' if P is (√3, 1)?

    <p>(-√3, 1)</p> Signup and view all the answers

    What is the result of the expression sin²(θ) + cos²(θ)?

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

    What is the reduction formula for cos(180° + θ) in terms of cos(θ)?

    <p>-cos(θ)</p> Signup and view all the answers

    What is the value of tan(180° - θ) in terms of tan(θ)?

    <p>-tan(θ)</p> Signup and view all the answers

    What is the expression for cos²(θ) in terms of sine of θ?

    <p>1 - sin²(θ)</p> Signup and view all the answers

    What is the reduction formula for sin(180° + θ) in terms of sin(θ)?

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

    What is the characteristic of the reduction formula for trigonometric functions?

    <p>It can be used to rewrite trigonometric functions in terms of θ.</p> Signup and view all the answers

    If point P is rotated through 90°, what are the coordinates of P'?

    <p>(-y, x)</p> Signup and view all the answers

    What is the value of cos(360° - θ) in terms of cosine of θ?

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

    If points P and P' are symmetrical about the x-axis, what are the coordinates of P'?

    <p>(x, -y)</p> Signup and view all the answers

    What is the value of sin(180° + θ) in terms of sine of θ?

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

    What is the value of tan(180° + θ) in terms of tangent of θ?

    <p>tan θ</p> Signup and view all the answers

    If points P and P' are symmetrical about the origin, what are the coordinates of P'?

    <p>(-x, -y)</p> Signup and view all the answers

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

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

    What is the value of cos(90° + θ) in terms of sine of θ?

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

    What is the value of tan(90° - θ) in terms of tangent of θ?

    <p>cot θ</p> Signup and view all the answers

    What is the value of sin(360° + θ) in terms of sine of θ?

    <p>sin θ</p> Signup and view all the answers

    What is the relationship between the gradient of a line and its inclination in the first quadrant?

    <p>m = tan θ</p> Signup and view all the answers

    If a line has a gradient of 3, what is the angle of inclination in degrees?

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

    What is the condition for two lines to be parallel in terms of their gradients?

    <p>m₁ = m₂</p> Signup and view all the answers

    What is the equation of a line parallel to y = 2x + 1 and passing through the point (3, 4)?

    <p>y = 2x + 5</p> Signup and view all the answers

    If a line has a negative gradient, what is the relationship between its inclination and the positive x-axis?

    <p>θ = 180° + tan⁻¹(m)</p> Signup and view all the answers

    What is the trigonometric identity for the tangent of an angle in terms of sine and cosine?

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

    What is the purpose of the CAST diagram in trigonometry?

    <p>To identify the quadrants of the coordinate plane</p> Signup and view all the answers

    What is the relationship between the gradient of a line and its inclination in the second quadrant?

    <p>m = -tan θ</p> Signup and view all the answers

    What is the equation of a line parallel to y = -3x + 2 and passing through the point (2, 5)?

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

    What is the characteristic of the reduction formula for trigonometric functions?

    <p>It is used to simplify trigonometric identities</p> Signup and view all the answers

    What is the main condition for two lines to be perpendicular?

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

    What is the formula to derive the equation of a line perpendicular to a given line and passing through a specific point?

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

    What is the purpose of the quotient identity in trigonometry?

    <p>To find the tangent of an angle in terms of sine and cosine.</p> Signup and view all the answers

    What is the condition for tan(θ) to be undefined?

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

    What is the result of the expression sin²(θ) + cos²(θ) in trigonometry?

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

    What is the first step in finding the equation of a line perpendicular to a given line?

    <p>Identify the gradient of the given line.</p> Signup and view all the answers

    What is the relationship between the gradients of two perpendicular lines?

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

    What is the purpose of using the square identity in trigonometry?

    <p>To simplify expressions involving sine and cosine.</p> Signup and view all the answers

    What is the main difference between the point-slope form and the standard form of a line equation?

    <p>The gradient is explicit in the point-slope form.</p> Signup and view all the answers

    What is the trigonometric identity that relates the tangent of an angle to the sine and cosine of the same angle?

    <p>tan(θ) = sin(θ) / cos(θ)</p> Signup and view all the answers

    What is the value of sin(68°) in terms of cosine?

    <p>cos(22°)</p> Signup and view all the answers

    To solve a trigonometric equation, what is the first step?

    <p>Use a calculator to find the reference angle.</p> Signup and view all the answers

    What is the formula for the area of a trapezium?

    <p>1/2(a + b) × h</p> Signup and view all the answers

    What is the surface area of a prism or cylinder?

    <p>The total area of the exposed or outer surfaces.</p> Signup and view all the answers

    What is the characteristic of the reduction formula for trigonometric functions?

    <p>It allows you to express trigonometric functions in terms of acute angles.</p> Signup and view all the answers

    What is the purpose of using the CAST diagram in trigonometry?

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

    What is the formula for the volume of a cylinder?

    <p>πr^2 × h</p> Signup and view all the answers

    If no interval is given, what is the general solution of a trigonometric equation?

    <p>The solution within the interval [0°; 360°] and adding multiples of the period to each answer.</p> Signup and view all the answers

    What is the formula for the area of a circle?

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

    If the dimensions of a rectangular prism are multiplied by a factor k, how does the surface area change?

    <p>It increases by a factor of k^2</p> Signup and view all the answers

    What is the formula for the surface area of a right cone?

    <p>πr^2 + πr × h</p> Signup and view all the answers

    What is the formula for the circumference of a circle?

    <p>2πr</p> Signup and view all the answers

    What is the formula for the volume of a triangular pyramid?

    <p>1/3 × b × h × H</p> Signup and view all the answers

    What is the surface area of a rectangular prism?

    <p>The total area of the six rectangles.</p> Signup and view all the answers

    If the dimensions of a rectangular prism are multiplied by a factor k, how does the volume change?

    <p>It increases by a factor of k^3</p> Signup and view all the answers

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

    <p>4πr^2</p> Signup and view all the answers

    What is the formula for the volume of a right pyramid?

    <p>1/3 × b^2 × H</p> Signup and view all the answers

    If the dimensions of a cylinder are multiplied by a factor k, how does the volume change?

    <p>It increases by a factor of k^3</p> Signup and view all the answers

    What is the formula for the surface area of a triangular pyramid?

    <p>1/2 × b × h_b + 3 × 1/2 × b × h_s</p> Signup and view all the answers

    What is the formula for the volume of a sphere?

    <p>4/3 × πr^3</p> Signup and view all the answers

    If a line segment from the center of a circle is perpendicular to a chord, what will it do to the chord?

    <p>Bisect the chord</p> Signup and view all the answers

    What is the period of the function y = sin(kθ)?

    <p>360°/|k|</p> Signup and view all the answers

    What is the effect of q on the graph of y = a sin(θ) + q?

    <p>Vertical shift upwards</p> Signup and view all the answers

    What is the amplitude of the function y = a sin(θ)?

    <p>|a|</p> Signup and view all the answers

    What is the range of the function y = sin(θ)?

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

    What is the x-intercept of the function y = sin(θ)?

    <p>(180°, 0)</p> Signup and view all the answers

    What is the effect of a on the graph of y = a sin(θ)?

    <p>Change in amplitude</p> Signup and view all the answers

    What is the equation of the axis of symmetry of the function y = sin(θ)?

    <p>θ = 90°</p> Signup and view all the answers

    What is the domain of the function y = sin(θ)?

    <p>All real numbers</p> Signup and view all the answers

    What is the y-intercept of the function y = cos(θ)?

    <p>(0°, 1)</p> Signup and view all the answers

    What is the formula for the surface area of a right cone?

    <p>$\pi r (r + h_s)$</p> Signup and view all the answers

    What is the effect of multiplying the dimensions of a rectangular prism by a factor of 2?

    <p>The surface area is quadrupled, and the volume is octupled.</p> Signup and view all the answers

    What is the formula for the volume of a sphere?

    <p>$rac{4}{3} \pi r^3$</p> Signup and view all the answers

    What is the name of the theorem that states that the angle at the center of a circle is twice the angle at the circumference?

    <p>Angle at the Center Theorem</p> Signup and view all the answers

    What is the formula for the area of a trapezium?

    <p>$rac{1}{2} (a + b) * h$</p> Signup and view all the answers

    What is the name of the line that touches a circle at only one point on the circumference?

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

    What is the formula for the surface area of a triangular pyramid?

    <p>$rac{1}{2} b (h_b + 3h_s)$</p> Signup and view all the answers

    What is the effect of multiplying the dimensions of a cylinder by a factor of 3?

    <p>The surface area is tripled, and the volume is sextupled.</p> Signup and view all the answers

    What is the formula for the volume of a rectangular prism?

    <p>$l * b * h$</p> Signup and view all the answers

    What is the formula for the area of a circle?

    <p>$\pi r^2$</p> Signup and view all the answers

    What is the period of y = cos(kθ) if k = 2?

    <p>90°/2</p> Signup and view all the answers

    What is the range of y = tan(θ)?

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

    What is the effect of k on the period of y = tan(kθ)?

    <p>The period decreases as k increases</p> Signup and view all the answers

    What is the y-intercept of y = tan(kθ)?

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

    What is the effect of q on the graph of y = a tan(θ) + q?

    <p>The graph shifts vertically by q units</p> Signup and view all the answers

    What is the domain of one branch of y = tan(kθ)?

    <p>{θ : -90°/k &lt; θ &lt; 90°/k, θ ∈ ℝ}</p> Signup and view all the answers

    What is the effect of a on the graph of y = a tan(θ) + q?

    <p>The branches of the graph are steeper for a &gt; 1</p> Signup and view all the answers

    What is the effect of p on the graph of y = tan(θ + p)?

    <p>The graph shifts horizontally by p units</p> Signup and view all the answers

    What is the period of y = cos(θ) if the period of y = cos(kθ) is 120°?

    <p>360°/k</p> Signup and view all the answers

    What is the asymptote of y = tan(kθ)?

    <p>The lines θ = 90°/k and θ = 270°/k</p> Signup and view all the answers

    Study Notes

    Here are the study notes for the provided text:

    • Equation of a Line*

    Two-Point Form

    • The equation of a straight line can be derived from two points (x1,y1)(x_1, y_1)(x1​,y1​) and (x2,y2)(x_2, y_2)(x2​,y2​)
    • Formula: y−y1x−x1=y2−y1x2−x1\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}x−x1​y−y1​​=x2​−x1​y2​−y1​​

    Gradient-Point Form

    • Derived from the definition of gradient and the two-point form
    • Formula: y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​)
    • Requires the gradient of the line and the coordinates of one point on the line

    Gradient-Intercept Form

    • Derived from the gradient-point form
    • Formula: y=mx+cy = mx + cy=mx+c
    • ccc is the y-intercept of the straight line
    • mmm is the gradient of the line
    • Inclination of a Line*

    Relationship between Gradient and Inclination

    • Gradient mmm of a line is equal to the tangent of the angle θ\thetaθ it makes with the positive x-axis
    • Formula: m=tan⁡θm = \tan \thetam=tanθ for 0∘≤θ<180∘0^\circ \leq \theta < 180^\circ0∘≤θ<180∘

    Special Cases

    • Vertical lines: θ=90∘\theta = 90^\circθ=90∘, mmm is undefined
    • Horizontal lines: θ=0∘\theta = 0^\circθ=0∘, m=0m = 0m=0
    • Lines with negative gradients: m<0m < 0m<0, tan⁡θ<0\tan \theta < 0tanθ<0
    • Parallel Lines*

    Gradient Relationship

    • Two lines are parallel if and only if their gradients are equal
    • Formula: m1=m2m_1 = m_2m1​=m2​

    Finding the Equation of a Parallel Line

    • Identify the gradient of the given line
    • Use the point-slope form of the equation of a line
    • Simplify the equation to the standard form y=mx+cy = mx + cy=mx+c
    • Perpendicular Lines*

    Gradient Relationship

    • Two lines are perpendicular if and only if the product of their gradients is equal to -1
    • Formula: m1×m2=−1m_1 \times m_2 = -1m1​×m2​=−1

    Finding the Equation of a Perpendicular Line

    • Identify the gradient of the given line
    • Calculate the perpendicular gradient
    • Use the point-slope form of the equation of a line
    • Simplify the equation to the standard form y=mx+cy = mx + cy=mx+c
    • Trigonometric Identities*

    Quotient Identity

    • Formula: tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}tanθ=cosθsinθ​
    • Defined for all values of θ\thetaθ except where cos⁡θ=0\cos \theta = 0cosθ=0

    Square Identity

    • Formula: sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1sin2θ+cos2θ=1
    • Alternative forms: sin⁡2θ=1−cos⁡2θ\sin^2 \theta = 1 - \cos^2 \thetasin2θ=1−cos2θ, cos⁡2θ=1−sin⁡2θ\cos^2 \theta = 1 - \sin^2 \thetacos2θ=1−sin2θ
    • Reduction Formulae*

    Angle Sum and Difference Formulae

    • Formulae for sine, cosine, and tangent of 90∘±θ90^\circ \pm \theta90∘±θ, 180∘±θ180^\circ \pm \theta180∘±θ, and 360∘±θ360^\circ \pm \theta360∘±θ
    • Reduction formulae allow for the expression of trigonometric functions in terms of acute angles, facilitating simplification and evaluation
    • Trigonometric Equations*

    Solving Trigonometric Equations

    • Use a calculator to find the reference angle
    • Use the CAST diagram to determine the quadrants
    • Use reduction formulae to find the values of the angle
    • Check the solution using a calculator
    • Area of a Polygon*

    Types of Polygons

    • Square: Area=s2Area = s^2Area=s2
    • Rectangle: Area=b×hArea = b \times hArea=b×h
    • Triangle: Area=12b×hArea = \frac{1}{2} b \times hArea=21​b×h
    • Trapezium: Area=12(a+b)×hArea = \frac{1}{2} (a + b) \times hArea=21​(a+b)×h
    • Parallelogram: Area=b×hArea = b \times hArea=b×h
    • Circle: Area=πr2Area = \pi r^2Area=πr2, Circumference = 2πr2\pi r2πr
    • Right Prisms and Cylinders*

    Surface Area

    • Calculate the area of each face and add them together
    • Formulas for surface area of rectangular prism, cube, triangular prism, and cylinder

    Volume

    • Formula: Volume=Area×HeightVolume = Area \times HeightVolume=Area×Height
    • Formulas for volume of rectangular prism, cube, triangular prism, and cylinder### Rectangular Prism, Triangular Prism, and Cylinder
    • Volume of a rectangular prism: l × b × h
    • Volume of a triangular prism: (1/2)b × h × H
    • Volume of a cylinder: πr^2 × h

    Right Pyramids, Right Cones, and Spheres

    • Definition of a pyramid: a geometric solid with a polygon base and sides that converge at a point (apex)
    • Definition of a right pyramid: a pyramid with a line from the apex to the center of the base perpendicular to the base
    • Definition of a cone: a geometric solid with a circular base and sides that converge at a point (apex)
    • Definition of a sphere: a perfectly round solid, looking the same from any direction

    Surface Area and Volume of Pyramids, Cones, and Spheres

    • Surface area of a square pyramid: b(b + 2hs)
    • Surface area of a triangular pyramid: (1/2)b(hb + 3hs)
    • Surface area of a right cone: πr(r + h)
    • Surface area of a sphere: 4πr^2
    • Volume of a square pyramid: (1/3)b^2 × H
    • Volume of a triangular pyramid: (1/3) × (1/2)bh × H
    • Volume of a right cone: (1/3)πr^2 × H
    • Volume of a sphere: (4/3)πr^3

    Effects of Scaling

    • If one or more dimensions of a prism or cylinder are multiplied by a constant factor k, the surface area and volume will change
    • Volume scales by k^3
    • Surface area scales by k^2

    Circle Geometry

    • Definitions:
      • Arc: a portion of the circumference of a circle
      • Chord: a straight line joining two points on a circle
      • Circumference: the perimeter or boundary line of a circle
      • Radius: a line from the center of a circle to a point on the circumference
      • Diameter: a special chord that passes through the center of the circle
      • Segment: a part of the circle cut off by a chord
      • Tangent: a straight line that touches the circle at only one point
    • Theorem of Pythagoras: in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides
    • Theorems:
      • A tangent to a circle is perpendicular to the radius drawn to the point of contact
      • Perpendicular line from circle center bisects chord
      • Perpendicular bisector of chord passes through circle center
      • Angle at the center is twice the angle at the circumference
      • Angles subtended by the same arc are equal
      • Opposite angles of a cyclic quadrilateral are supplementary
      • Exterior angle of a cyclic quadrilateral is equal to the interior opposite angle
      • Tangent-chord theorem: the angle between a tangent to a circle and a chord drawn from the point of contact is equal to the angle which the chord subtends in the alternate segment

    The Sine Function

    • Period: 360°
    • Amplitude: 1
    • Domain: [0°, 360°]
    • Range: [-1, 1]
    • x-intercepts: (0°, 0), (180°, 0), (360°, 0)
    • y-intercept: (0°, 0)
    • Maximum turning point: (90°, 1)
    • Minimum turning point: (270°, -1)
    • Effects of a and q on the graph of y = a sin(θ) + q
    • Effects of k on the graph of y = sin(kθ)

    The Cosine Function

    • Period: 360°
    • Amplitude: 1
    • Domain: [0°, 360°]
    • Range: [-1, 1]
    • x-intercepts: (90°, 0), (270°, 0)
    • y-intercept: (0°, 1)
    • Maximum turning points: (0°, 1), (360°, 1)
    • Minimum turning point: (180°, -1)
    • Effects of a and q on the graph of y = a cos(θ) + q
    • Effects of k on the graph of y = cos(kθ)

    The Tangent Function

    • Period: 180°
    • Domain: {θ: 0° ≤ θ ≤ 360°, θ ≠ 90°, 270°}
    • Range: {f(θ): f(θ) ∈ ℝ}
    • x-intercepts: (0°, 0), (180°, 0), (360°, 0)
    • y-intercept: (0°, 0)
    • Asymptotes: θ = 90°, θ = 270°
    • Effects of a and q on the graph of y = a tan(θ) + q
    • Effects of k on the graph of y = tan(kθ)

    Here are the study notes for the provided text:

    • Equation of a Line*

    Two-Point Form

    • The equation of a straight line can be derived from two points (x1,y1)(x_1, y_1)(x1​,y1​) and (x2,y2)(x_2, y_2)(x2​,y2​)
    • Formula: y−y1x−x1=y2−y1x2−x1\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}x−x1​y−y1​​=x2​−x1​y2​−y1​​

    Gradient-Point Form

    • Derived from the definition of gradient and the two-point form
    • Formula: y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​)
    • Requires the gradient of the line and the coordinates of one point on the line

    Gradient-Intercept Form

    • Derived from the gradient-point form
    • Formula: y=mx+cy = mx + cy=mx+c
    • ccc is the y-intercept of the straight line
    • mmm is the gradient of the line
    • Inclination of a Line*

    Relationship between Gradient and Inclination

    • Gradient mmm of a line is equal to the tangent of the angle θ\thetaθ it makes with the positive x-axis
    • Formula: m=tan⁡θm = \tan \thetam=tanθ for 0∘≤θ<180∘0^\circ \leq \theta < 180^\circ0∘≤θ<180∘

    Special Cases

    • Vertical lines: θ=90∘\theta = 90^\circθ=90∘, mmm is undefined
    • Horizontal lines: θ=0∘\theta = 0^\circθ=0∘, m=0m = 0m=0
    • Lines with negative gradients: m<0m < 0m<0, tan⁡θ<0\tan \theta < 0tanθ<0
    • Parallel Lines*

    Gradient Relationship

    • Two lines are parallel if and only if their gradients are equal
    • Formula: m1=m2m_1 = m_2m1​=m2​

    Finding the Equation of a Parallel Line

    • Identify the gradient of the given line
    • Use the point-slope form of the equation of a line
    • Simplify the equation to the standard form y=mx+cy = mx + cy=mx+c
    • Perpendicular Lines*

    Gradient Relationship

    • Two lines are perpendicular if and only if the product of their gradients is equal to -1
    • Formula: m1×m2=−1m_1 \times m_2 = -1m1​×m2​=−1

    Finding the Equation of a Perpendicular Line

    • Identify the gradient of the given line
    • Calculate the perpendicular gradient
    • Use the point-slope form of the equation of a line
    • Simplify the equation to the standard form y=mx+cy = mx + cy=mx+c
    • Trigonometric Identities*

    Quotient Identity

    • Formula: tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}tanθ=cosθsinθ​
    • Defined for all values of θ\thetaθ except where cos⁡θ=0\cos \theta = 0cosθ=0

    Square Identity

    • Formula: sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1sin2θ+cos2θ=1
    • Alternative forms: sin⁡2θ=1−cos⁡2θ\sin^2 \theta = 1 - \cos^2 \thetasin2θ=1−cos2θ, cos⁡2θ=1−sin⁡2θ\cos^2 \theta = 1 - \sin^2 \thetacos2θ=1−sin2θ
    • Reduction Formulae*

    Angle Sum and Difference Formulae

    • Formulae for sine, cosine, and tangent of 90∘±θ90^\circ \pm \theta90∘±θ, 180∘±θ180^\circ \pm \theta180∘±θ, and 360∘±θ360^\circ \pm \theta360∘±θ
    • Reduction formulae allow for the expression of trigonometric functions in terms of acute angles, facilitating simplification and evaluation
    • Trigonometric Equations*

    Solving Trigonometric Equations

    • Use a calculator to find the reference angle
    • Use the CAST diagram to determine the quadrants
    • Use reduction formulae to find the values of the angle
    • Check the solution using a calculator
    • Area of a Polygon*

    Types of Polygons

    • Square: Area=s2Area = s^2Area=s2
    • Rectangle: Area=b×hArea = b \times hArea=b×h
    • Triangle: Area=12b×hArea = \frac{1}{2} b \times hArea=21​b×h
    • Trapezium: Area=12(a+b)×hArea = \frac{1}{2} (a + b) \times hArea=21​(a+b)×h
    • Parallelogram: Area=b×hArea = b \times hArea=b×h
    • Circle: Area=πr2Area = \pi r^2Area=πr2, Circumference = 2πr2\pi r2πr
    • Right Prisms and Cylinders*

    Surface Area

    • Calculate the area of each face and add them together
    • Formulas for surface area of rectangular prism, cube, triangular prism, and cylinder

    Volume

    • Formula: Volume=Area×HeightVolume = Area \times HeightVolume=Area×Height
    • Formulas for volume of rectangular prism, cube, triangular prism, and cylinder### Rectangular Prism, Triangular Prism, and Cylinder
    • Volume of a rectangular prism: l × b × h
    • Volume of a triangular prism: (1/2)b × h × H
    • Volume of a cylinder: πr^2 × h

    Right Pyramids, Right Cones, and Spheres

    • Definition of a pyramid: a geometric solid with a polygon base and sides that converge at a point (apex)
    • Definition of a right pyramid: a pyramid with a line from the apex to the center of the base perpendicular to the base
    • Definition of a cone: a geometric solid with a circular base and sides that converge at a point (apex)
    • Definition of a sphere: a perfectly round solid, looking the same from any direction

    Surface Area and Volume of Pyramids, Cones, and Spheres

    • Surface area of a square pyramid: b(b + 2hs)
    • Surface area of a triangular pyramid: (1/2)b(hb + 3hs)
    • Surface area of a right cone: πr(r + h)
    • Surface area of a sphere: 4πr^2
    • Volume of a square pyramid: (1/3)b^2 × H
    • Volume of a triangular pyramid: (1/3) × (1/2)bh × H
    • Volume of a right cone: (1/3)πr^2 × H
    • Volume of a sphere: (4/3)πr^3

    Effects of Scaling

    • If one or more dimensions of a prism or cylinder are multiplied by a constant factor k, the surface area and volume will change
    • Volume scales by k^3
    • Surface area scales by k^2

    Circle Geometry

    • Definitions:
      • Arc: a portion of the circumference of a circle
      • Chord: a straight line joining two points on a circle
      • Circumference: the perimeter or boundary line of a circle
      • Radius: a line from the center of a circle to a point on the circumference
      • Diameter: a special chord that passes through the center of the circle
      • Segment: a part of the circle cut off by a chord
      • Tangent: a straight line that touches the circle at only one point
    • Theorem of Pythagoras: in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides
    • Theorems:
      • A tangent to a circle is perpendicular to the radius drawn to the point of contact
      • Perpendicular line from circle center bisects chord
      • Perpendicular bisector of chord passes through circle center
      • Angle at the center is twice the angle at the circumference
      • Angles subtended by the same arc are equal
      • Opposite angles of a cyclic quadrilateral are supplementary
      • Exterior angle of a cyclic quadrilateral is equal to the interior opposite angle
      • Tangent-chord theorem: the angle between a tangent to a circle and a chord drawn from the point of contact is equal to the angle which the chord subtends in the alternate segment

    The Sine Function

    • Period: 360°
    • Amplitude: 1
    • Domain: [0°, 360°]
    • Range: [-1, 1]
    • x-intercepts: (0°, 0), (180°, 0), (360°, 0)
    • y-intercept: (0°, 0)
    • Maximum turning point: (90°, 1)
    • Minimum turning point: (270°, -1)
    • Effects of a and q on the graph of y = a sin(θ) + q
    • Effects of k on the graph of y = sin(kθ)

    The Cosine Function

    • Period: 360°
    • Amplitude: 1
    • Domain: [0°, 360°]
    • Range: [-1, 1]
    • x-intercepts: (90°, 0), (270°, 0)
    • y-intercept: (0°, 1)
    • Maximum turning points: (0°, 1), (360°, 1)
    • Minimum turning point: (180°, -1)
    • Effects of a and q on the graph of y = a cos(θ) + q
    • Effects of k on the graph of y = cos(kθ)

    The Tangent Function

    • Period: 180°
    • Domain: {θ: 0° ≤ θ ≤ 360°, θ ≠ 90°, 270°}
    • Range: {f(θ): f(θ) ∈ ℝ}
    • x-intercepts: (0°, 0), (180°, 0), (360°, 0)
    • y-intercept: (0°, 0)
    • Asymptotes: θ = 90°, θ = 270°
    • Effects of a and q on the graph of y = a tan(θ) + q
    • Effects of k on the graph of y = tan(kθ)

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    Derive and use different forms of the straight line equation, including two-point, gradient-point, and gradient-intercept forms, depending on the problem information.

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