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 (B)</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 (D)</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 (D)</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 (D)</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 (A)</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. (B)</p> Signup and view all the answers

What is the gradient of a vertical line?

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

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

<p>They are always equal. (B)</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) (A)</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. (D)</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. (D)</p> Signup and view all the answers

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

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

What is the gradient of a horizontal line?

<p>0 (A)</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. (B)</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. (A)</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 (C)</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 (C)</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 (B)</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 (C)</p> Signup and view all the answers

What is the Pythagorean identity in trigonometry?

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

What does the square identity simplify to?

<p>sin^2 θ + cos^2 θ = 1 (D)</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 (D)</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 (B)</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) (A)</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) (A)</p> Signup and view all the answers

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

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

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

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

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

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

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

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

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

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

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

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

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

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

What is the value of sin(180° + θ)?

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

What is the value of cos(360° - θ)?

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

What is the value of tan(180° + θ)?

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

What is the value of sin(-θ)?

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

What is the value of cos(90° + θ)?

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

What is the value of tan(90° - θ)?

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

What is the value of sin(360° + θ)?

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

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

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

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

<p>1/2bhH (B)</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 (D)</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 (C)</p> Signup and view all the answers

What is the formula for the area of a circle?

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

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

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

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

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

What is the formula for the volume of a sphere?

<p>4/3πr^3 (C)</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 (D)</p> Signup and view all the answers

What is the formula for the area of a trapezium?

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

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

<p>360°/k (B)</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. (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>It shifts the graph vertically. (B)</p> Signup and view all the answers

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

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

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

<p>[-1, 1] (C)</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. (B)</p> Signup and view all the answers

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

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

What is the formula for the area of a trapezium?

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

What is the formula for the area of a circle?

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

What is the formula for the circumference of a circle?

<p>C = 2π r (C)</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. (C)</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 (C)</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) (D)</p> Signup and view all the answers

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

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

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

<p>SA = 4r^2 (A)</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 (A)</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] (C)</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 (C)</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) (D)</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) (A)</p> Signup and view all the answers

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

<p>180° (C)</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. (C)</p> Signup and view all the answers

What is the range of y = tan(θ)?

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

What is the domain of y = cos(kθ)?

<p>{θ : θ ∈ ℝ} (D)</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. (B)</p> Signup and view all the answers

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

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

What is the domain of y = tan(kθ)?

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

What is the asymptote of y = tan(kθ)?

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

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

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

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

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

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

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

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

<p>−tan(θ) (A)</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°. (B)</p> Signup and view all the answers

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

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

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

<p>1 (D)</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. (B)</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 (C)</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 (A)</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 (B)</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 (C)</p> Signup and view all the answers

What is the condition for two lines to be parallel?

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

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

<p>It is undefined (B)</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) (B)</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 (D)</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° (D)</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 (C)</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 (C)</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 (A)</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) (B)</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 (A)</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) (D)</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 (C)</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 (A)</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 (B)</p> Signup and view all the answers

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

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

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

<p>θ = 90° (C)</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 (D)</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 (D)</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) (B)</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) (A)</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 (A)</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 (A)</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) (C)</p> Signup and view all the answers

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

What is the formula for the area of a trapezium?

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

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

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

What is the formula for the area of a circle?

<p>πr² (A)</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. (A)</p> Signup and view all the answers

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

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

What is the formula for the volume of a cylinder?

<p>πr²h (B)</p> Signup and view all the answers

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

<p>4πr² (C)</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 (A)</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 (C)</p> Signup and view all the answers

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

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

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

<p>b²h/3 (D)</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. (A)</p> Signup and view all the answers

What is the formula for the area of a trapezium?

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

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

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

What is the formula for the area of a circle?

<p>πr² (B)</p> Signup and view all the answers

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

<p>bhH/2 (D)</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. (C)</p> Signup and view all the answers

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<p>πr^2 + πrL (D)</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 (B)</p> Signup and view all the answers

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

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

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

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

What is the formula for the volume of a cylinder?

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

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

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

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

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

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

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

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

<p>180°/|k| (C)</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 (A)</p> Signup and view all the answers

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

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

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

<p>(-∞, ∞) (D)</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} (A)</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 (C)</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 (C)</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 (A)</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 (D)</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 (D)</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 (A)</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) (C)</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 (A), The gradient of the line is known (D)</p> Signup and view all the answers

What is the condition for tan θ to be undefined?

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

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

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

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

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

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

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

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

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

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

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

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

<p>cos θ (C)</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 (A)</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 (A)</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 (D)</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 (D)</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 (B)</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 (B)</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) (B)</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 (C)</p> Signup and view all the answers

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

<p>It is negative (A)</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 (B)</p> Signup and view all the answers

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

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

What is the purpose of the square identity in trigonometry?

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

What is the condition for tanθ to be undefined?

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

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

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

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

<p>-1/2 (A)</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 (D)</p> Signup and view all the answers

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

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

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

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

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

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

What is the formula for the volume of a cylinder?

<p>πr²h (C)</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 (C)</p> Signup and view all the answers

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

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

What is the formula for the volume of a sphere?

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

What is the formula for the area of a trapezium?

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

What is the formula for the volume of a sphere?

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

What is the formula for the area of a circle?

<p>πr² (A)</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 (B)</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 (A)</p> Signup and view all the answers

What is the formula for the area of a trapezium?

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

What is the formula for the area of a parallelogram?

<p>bh (B)</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 (D)</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 (C)</p> Signup and view all the answers

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

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

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

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

What is the formula for the area of a triangle?

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

What is the formula for the area of a circle?

<p>A = πr^2 (C)</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 (B)</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 (C)</p> Signup and view all the answers

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

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

What is the formula for the circumference of a circle?

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

What is the formula for the area of a triangle?

<p>A = ½ b × h (C)</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. (B)</p> Signup and view all the answers

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

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

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

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

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

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

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

<p>[-1, 1] (B)</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. (C)</p> Signup and view all the answers

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

<p>(90°, 0) (A), (360°, 0) (B), (180°, 0) (C), (0°, 0) (D)</p> Signup and view all the answers

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

<p>(0°, 1) (B), (360°, 1) (C)</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) (A)</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) (A)</p> Signup and view all the answers

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<p>360°/|k| (C)</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. (B)</p> Signup and view all the answers

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

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

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

<p>180°/|k| (A)</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. (C)</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} (A)</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. (B)</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. (D)</p> Signup and view all the answers

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

<p>(-∞, ∞) (A)</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. (D)</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. (A)</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. (A)</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. (C)</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. (C)</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. (C)</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. (A)</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. (D)</p> Signup and view all the answers

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<p>sin θ (A)</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 (C)</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)° (D)</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 (B)</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 (B)</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) (C)</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 (D)</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 (D)</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 (A)</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 (A)</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 (C)</p> Signup and view all the answers

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

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

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

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

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

<p>1 (C)</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. (B)</p> Signup and view all the answers

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

<p>They are negative reciprocals. (A)</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. (B)</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. (A)</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) (C)</p> Signup and view all the answers

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

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

What is the formula for the area of a trapezium?

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

What is the formula for the volume of a cylinder?

<p>πr^2 × h (D)</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. (C)</p> Signup and view all the answers

What is the formula for the area of a circle?

<p>πr^2 (B)</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 (D)</p> Signup and view all the answers

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

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

What is the formula for the circumference of a circle?

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

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

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

What is the surface area of a rectangular prism?

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

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

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

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

<p>1/3 × b^2 × H (C)</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 (D)</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 (B)</p> Signup and view all the answers

What is the formula for the volume of a sphere?

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

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

<p>360°/|k| (A)</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 (D)</p> Signup and view all the answers

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

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

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

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

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

<p>(180°, 0) (B), (0°, 0) (C), (360°, 0) (D)</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 (A)</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° (B)</p> Signup and view all the answers

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

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

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

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

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

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

What is the formula for the volume of a sphere?

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

What is the formula for the area of a trapezium?

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

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

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

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

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

What is the formula for the area of a circle?

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

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

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

What is the range of y = tan(θ)?

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

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

<p>(0, 0) (C)</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 (A)</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} (A)</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 (A)</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 (C)</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 (A)</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 (C)</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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