Podcast
Questions and Answers
What is the definition of a point of inflection?
What is the definition of a point of inflection?
What is the application of differential calculus in optimisation problems?
What is the application of differential calculus in optimisation problems?
What is the formula for synthetic division?
What is the formula for synthetic division?
What is the role of the remainder theorem in solving cubic equations?
What is the role of the remainder theorem in solving cubic equations?
Signup and view all the answers
What is the condition for a graph to be concave up?
What is the condition for a graph to be concave up?
Signup and view all the answers
What is the method to find the x-intercepts of a cubic polynomial?
What is the method to find the x-intercepts of a cubic polynomial?
Signup and view all the answers
What is the application of differential calculus in rates of change?
What is the application of differential calculus in rates of change?
Signup and view all the answers
What is the formula for long division of polynomials?
What is the formula for long division of polynomials?
Signup and view all the answers
What is the method to find the y-intercept of a cubic polynomial?
What is the method to find the y-intercept of a cubic polynomial?
Signup and view all the answers
What is the condition for a graph to be concave down?
What is the condition for a graph to be concave down?
Signup and view all the answers
What is the remainder when a polynomial p(x) is divided by cx - d?
What is the remainder when a polynomial p(x) is divided by cx - d?
Signup and view all the answers
What is the degree of the quotient Q(x) when a polynomial p(x) is divided by a linear polynomial cx - d?
What is the degree of the quotient Q(x) when a polynomial p(x) is divided by a linear polynomial cx - d?
Signup and view all the answers
What is the condition for cx - d to be a factor of a polynomial p(x)?
What is the condition for cx - d to be a factor of a polynomial p(x)?
Signup and view all the answers
What is the equation of a circle with centre at the origin and radius r?
What is the equation of a circle with centre at the origin and radius r?
Signup and view all the answers
How do we find a factor of a cubic polynomial using the factor theorem?
How do we find a factor of a cubic polynomial using the factor theorem?
Signup and view all the answers
What is the general form of a polynomial p(x) when divided by cx - d?
What is the general form of a polynomial p(x) when divided by cx - d?
Signup and view all the answers
What is the next step after finding a factor of a cubic polynomial using the factor theorem?
What is the next step after finding a factor of a cubic polynomial using the factor theorem?
Signup and view all the answers
What is the purpose of the quadratic formula in solving cubic equations?
What is the purpose of the quadratic formula in solving cubic equations?
Signup and view all the answers
What is the relationship between the roots of a polynomial and its factors?
What is the relationship between the roots of a polynomial and its factors?
Signup and view all the answers
Why is the factor theorem useful in solving cubic equations?
Why is the factor theorem useful in solving cubic equations?
Signup and view all the answers
What is the limit of the function (y = \frac{x^2 + 4x - 12}{x + 6}) as (x) approaches -6?
What is the limit of the function (y = \frac{x^2 + 4x - 12}{x + 6}) as (x) approaches -6?
Signup and view all the answers
Which of the following statements accurately describes the concept of limits in the context of Achilles and the tortoise paradox?
Which of the following statements accurately describes the concept of limits in the context of Achilles and the tortoise paradox?
Signup and view all the answers
Why is the function (y = \frac{x^2 + 4x - 12}{x + 6}) not defined at (x = -6)?
Why is the function (y = \frac{x^2 + 4x - 12}{x + 6}) not defined at (x = -6)?
Signup and view all the answers
What does the hole in the graph of (y = \frac{x^2 + 4x - 12}{x + 6}) at (x = -6) represent?
What does the hole in the graph of (y = \frac{x^2 + 4x - 12}{x + 6}) at (x = -6) represent?
Signup and view all the answers
Which of the following expressions represents the simplified form of the function (y = \frac{x^2 + 4x - 12}{x + 6}) for (x \neq -6)?
Which of the following expressions represents the simplified form of the function (y = \frac{x^2 + 4x - 12}{x + 6}) for (x \neq -6)?
Signup and view all the answers
How does the concept of limits help in understanding the behavior of a function near a point where it is not defined?
How does the concept of limits help in understanding the behavior of a function near a point where it is not defined?
Signup and view all the answers
What is the equation of a circle with center at the origin and radius r?
What is the equation of a circle with center at the origin and radius r?
Signup and view all the answers
A circle with center at the origin is symmetric about which of the following?
A circle with center at the origin is symmetric about which of the following?
Signup and view all the answers
What is the equation of a circle with center at (a, b) and radius r?
What is the equation of a circle with center at (a, b) and radius r?
Signup and view all the answers
What is the first step to find the equation of a tangent to a circle?
What is the first step to find the equation of a tangent to a circle?
Signup and view all the answers
What is the relationship between the gradients of the radius and tangent?
What is the relationship between the gradients of the radius and tangent?
Signup and view all the answers
What is the last step to find the equation of a tangent to a circle?
What is the last step to find the equation of a tangent to a circle?
Signup and view all the answers
What is the general form of a circle's equation?
What is the general form of a circle's equation?
Signup and view all the answers
What is a tangent to a circle?
What is a tangent to a circle?
Signup and view all the answers
What is a ratio?
What is a ratio?
Signup and view all the answers
What is the purpose of completing the square in finding the equation of a circle?
What is the purpose of completing the square in finding the equation of a circle?
Signup and view all the answers
What is the first step in determining the equation of a tangent to a curve?
What is the first step in determining the equation of a tangent to a curve?
Signup and view all the answers
What indicates the change in gradient of the original function?
What indicates the change in gradient of the original function?
Signup and view all the answers
Which of the following describes a local maximum of a function?
Which of the following describes a local maximum of a function?
Signup and view all the answers
If the coefficient 'a' in a cubic function is negative, which statement is true?
If the coefficient 'a' in a cubic function is negative, which statement is true?
Signup and view all the answers
How can one find the x-intercepts of a cubic function?
How can one find the x-intercepts of a cubic function?
Signup and view all the answers
What is the relationship between the gradient of the tangent and the normal to a curve?
What is the relationship between the gradient of the tangent and the normal to a curve?
Signup and view all the answers
What is the result of differentiating the function twice?
What is the result of differentiating the function twice?
Signup and view all the answers
To find stationary points on a graph, which condition must be satisfied?
To find stationary points on a graph, which condition must be satisfied?
Signup and view all the answers
Which notation indicates the second derivative of a function?
Which notation indicates the second derivative of a function?
Signup and view all the answers
What does the derivative allow us to determine about a function?
What does the derivative allow us to determine about a function?
Signup and view all the answers
What is the primary purpose of simplifying ratios?
What is the primary purpose of simplifying ratios?
Signup and view all the answers
Which of the following is a property of proportion?
Which of the following is a property of proportion?
Signup and view all the answers
What is the name of the theorem that states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally?
What is the name of the theorem that states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally?
Signup and view all the answers
What is the formula for the area of a triangle?
What is the formula for the area of a triangle?
Signup and view all the answers
What is the name of the polygon with four sides of equal length?
What is the name of the polygon with four sides of equal length?
Signup and view all the answers
Which of the following is not a type of polygon?
Which of the following is not a type of polygon?
Signup and view all the answers
What is the formula for the area of a trapezium?
What is the formula for the area of a trapezium?
Signup and view all the answers
What is the term used to describe the equality of ratios between corresponding sides or other measurements in polygons?
What is the term used to describe the equality of ratios between corresponding sides or other measurements in polygons?
Signup and view all the answers
Which of the following is a characteristic of similar polygons?
Which of the following is a characteristic of similar polygons?
Signup and view all the answers
What is the first step in solving proportional problems?
What is the first step in solving proportional problems?
Signup and view all the answers
What is the derivative of the function $f(x) = 3x^4$?
What is the derivative of the function $f(x) = 3x^4$?
Signup and view all the answers
Using differentiation from first principles, how is the gradient at a point $x = a$ calculated?
Using differentiation from first principles, how is the gradient at a point $x = a$ calculated?
Signup and view all the answers
What is the derivative of a constant function $f(x) = k$?
What is the derivative of a constant function $f(x) = k$?
Signup and view all the answers
What does the notation $rac{dy}{dx}$ represent?
What does the notation $rac{dy}{dx}$ represent?
Signup and view all the answers
Which of the following notations can be used to represent the derivative of a function?
Which of the following notations can be used to represent the derivative of a function?
Signup and view all the answers
When should the rules for differentiation be used instead of differentiation from first principles?
When should the rules for differentiation be used instead of differentiation from first principles?
Signup and view all the answers
What is the derivative of the function $f(x) = x^3 + 4x$?
What is the derivative of the function $f(x) = x^3 + 4x$?
Signup and view all the answers
What is the gradient at the point $x = -6$ for the function $y = rac{(x + 6)(x - 2)}{x + 6}$?
What is the gradient at the point $x = -6$ for the function $y = rac{(x + 6)(x - 2)}{x + 6}$?
Signup and view all the answers
Which operation does the differential operator $rac{d}{dx}$ indicate?
Which operation does the differential operator $rac{d}{dx}$ indicate?
Signup and view all the answers
What is the derivative of the function $f(x) = k imes f(x)$, where $k$ is a constant?
What is the derivative of the function $f(x) = k imes f(x)$, where $k$ is a constant?
Signup and view all the answers
What is the statement of the Pythagorean theorem?
What is the statement of the Pythagorean theorem?
Signup and view all the answers
What is the condition for two triangles to be similar?
What is the condition for two triangles to be similar?
Signup and view all the answers
What is the formula for the area of a triangle?
What is the formula for the area of a triangle?
Signup and view all the answers
What is the converse of the Pythagorean theorem?
What is the converse of the Pythagorean theorem?
Signup and view all the answers
What is the condition for triangles with equal bases between the same parallel lines?
What is the condition for triangles with equal bases between the same parallel lines?
Signup and view all the answers
What is the similarity condition for polygons?
What is the similarity condition for polygons?
Signup and view all the answers
What is the proof of the Pythagorean theorem based on?
What is the proof of the Pythagorean theorem based on?
Signup and view all the answers
What is the result of adding the two results from the similarity of triangles ABD and CAD?
What is the result of adding the two results from the similarity of triangles ABD and CAD?
Signup and view all the answers
If a line is drawn parallel to one side of a triangle, how does it affect the other two sides?
If a line is drawn parallel to one side of a triangle, how does it affect the other two sides?
Signup and view all the answers
What is the relationship between two triangles that are equiangular?
What is the relationship between two triangles that are equiangular?
Signup and view all the answers
In a right-angled triangle, how are the squares of the sides related?
In a right-angled triangle, how are the squares of the sides related?
Signup and view all the answers
What is the relationship between the areas of two triangles with the same height?
What is the relationship between the areas of two triangles with the same height?
Signup and view all the answers
If two triangles have equal bases and lie between the same parallel lines, what can you say about their areas?
If two triangles have equal bases and lie between the same parallel lines, what can you say about their areas?
Signup and view all the answers
If two triangles on the same base have equal areas, what can you conclude about their heights?
If two triangles on the same base have equal areas, what can you conclude about their heights?
Signup and view all the answers
According to the Proportion Theorem, what is the relationship between the segments created when a line parallel to one side of a triangle intersects the other two sides?
According to the Proportion Theorem, what is the relationship between the segments created when a line parallel to one side of a triangle intersects the other two sides?
Signup and view all the answers
What is the relationship between the line joining the midpoints of two sides of a triangle and the third side of the triangle?
What is the relationship between the line joining the midpoints of two sides of a triangle and the third side of the triangle?
Signup and view all the answers
If a line is drawn from the midpoint of one side of a triangle parallel to another side, what can you conclude about the third side?
If a line is drawn from the midpoint of one side of a triangle parallel to another side, what can you conclude about the third side?
Signup and view all the answers
What is the formula for the area of a triangle?
What is the formula for the area of a triangle?
Signup and view all the answers
If two triangles have all corresponding angles equal, what can we conclude about the triangles?
If two triangles have all corresponding angles equal, what can we conclude about the triangles?
Signup and view all the answers
In triangle ABC, D and E are midpoints of sides AB and AC respectively. What is the relationship between DE and BC?
In triangle ABC, D and E are midpoints of sides AB and AC respectively. What is the relationship between DE and BC?
Signup and view all the answers
Triangle ABC is similar to triangle DEF. If AB = 6, BC = 8, and DE = 3, what is the length of EF?
Triangle ABC is similar to triangle DEF. If AB = 6, BC = 8, and DE = 3, what is the length of EF?
Signup and view all the answers
Two polygons are similar if they have:
Two polygons are similar if they have:
Signup and view all the answers
If two triangles have corresponding sides in the same proportion, what can we conclude about the triangles?
If two triangles have corresponding sides in the same proportion, what can we conclude about the triangles?
Signup and view all the answers
In triangle ABC, D is the midpoint of side AB and E is the midpoint of side AC. If BC = 10 cm, what is the length of DE?
In triangle ABC, D is the midpoint of side AB and E is the midpoint of side AC. If BC = 10 cm, what is the length of DE?
Signup and view all the answers
Which of the following is NOT a condition for two polygons to be similar?
Which of the following is NOT a condition for two polygons to be similar?
Signup and view all the answers
Triangle ABC is similar to triangle DEF. If the ratio of the area of triangle ABC to the area of triangle DEF is 4:1, what is the ratio of the corresponding sides?
Triangle ABC is similar to triangle DEF. If the ratio of the area of triangle ABC to the area of triangle DEF is 4:1, what is the ratio of the corresponding sides?
Signup and view all the answers
If two triangles are similar, then which of the following must be true?
If two triangles are similar, then which of the following must be true?
Signup and view all the answers
In triangle ABC, D is a point on side AB such that AD = 1/3 AB. E is a point on side AC such that AE = 1/3 AC. What is the relationship between DE and BC?
In triangle ABC, D is a point on side AB such that AD = 1/3 AB. E is a point on side AC such that AE = 1/3 AC. What is the relationship between DE and BC?
Signup and view all the answers
What is the derivative of the function $f(x) = x^2$?
What is the derivative of the function $f(x) = x^2$?
Signup and view all the answers
What is the formula for the gradient of the tangent to a curve with equation $y = f(x)$ at $x = a$?
What is the formula for the gradient of the tangent to a curve with equation $y = f(x)$ at $x = a$?
Signup and view all the answers
What is the derivative of the function $f(x) = 3x^2 + 2x - 5$?
What is the derivative of the function $f(x) = 3x^2 + 2x - 5$?
Signup and view all the answers
What is the notation for the derivative of a function $f(x)$?
What is the notation for the derivative of a function $f(x)$?
Signup and view all the answers
What is the general rule for differentiating $x^n$?
What is the general rule for differentiating $x^n$?
Signup and view all the answers
What is the derivative of the function $f(x) = 2x + 1$?
What is the derivative of the function $f(x) = 2x + 1$?
Signup and view all the answers
What is the purpose of differentiating a function?
What is the purpose of differentiating a function?
Signup and view all the answers
What is the derivative of the function $f(x) = k$?
What is the derivative of the function $f(x) = k$?
Signup and view all the answers
What is the derivative of the function $f(x) = x^2 + 2x - 3$?
What is the derivative of the function $f(x) = x^2 + 2x - 3$?
Signup and view all the answers
What is the equation of the tangent to a curve with equation $y = f(x)$ at $x = a$?
What is the equation of the tangent to a curve with equation $y = f(x)$ at $x = a$?
Signup and view all the answers
Why is the function (y = \frac{x^2 + 4x - 12}{x + 6}) not defined at (x = -6)?
Why is the function (y = \frac{x^2 + 4x - 12}{x + 6}) not defined at (x = -6)?
Signup and view all the answers
What is the simplified form of the function (y = \frac{x^2 + 4x - 12}{x + 6}) for (x \neq -6)?
What is the simplified form of the function (y = \frac{x^2 + 4x - 12}{x + 6}) for (x \neq -6)?
Signup and view all the answers
What does the hole in the graph of (y = \frac{x^2 + 4x - 12}{x + 6}) at (x = -6) represent?
What does the hole in the graph of (y = \frac{x^2 + 4x - 12}{x + 6}) at (x = -6) represent?
Signup and view all the answers
How does the concept of limits help in understanding the behavior of a function near a point where it is not defined?
How does the concept of limits help in understanding the behavior of a function near a point where it is not defined?
Signup and view all the answers
Which of the following statements accurately describes the concept of limits in the context of Achilles and the tortoise paradox?
Which of the following statements accurately describes the concept of limits in the context of Achilles and the tortoise paradox?
Signup and view all the answers
What is the limit of the function (y = \frac{x^2 + 4x - 12}{x + 6}) as (x) approaches -6?
What is the limit of the function (y = \frac{x^2 + 4x - 12}{x + 6}) as (x) approaches -6?
Signup and view all the answers
What is the remainder when a polynomial p(x) is divided by cx - d?
What is the remainder when a polynomial p(x) is divided by cx - d?
Signup and view all the answers
What is the general form of a polynomial p(x) when divided by cx - d?
What is the general form of a polynomial p(x) when divided by cx - d?
Signup and view all the answers
What is the next step after finding a factor of a cubic polynomial using the Factor Theorem?
What is the next step after finding a factor of a cubic polynomial using the Factor Theorem?
Signup and view all the answers
What is the purpose of the Factor Theorem in solving cubic equations?
What is the purpose of the Factor Theorem in solving cubic equations?
Signup and view all the answers
What is the equation of a circle with center at (a, b) and radius r?
What is the equation of a circle with center at (a, b) and radius r?
Signup and view all the answers
What is the first step to find the equation of a tangent to a circle?
What is the first step to find the equation of a tangent to a circle?
Signup and view all the answers
What is the relationship between the gradients of the radius and tangent?
What is the relationship between the gradients of the radius and tangent?
Signup and view all the answers
What is the general form of a circle's equation?
What is the general form of a circle's equation?
Signup and view all the answers
What is the equation of a circle with center at the origin and radius r?
What is the equation of a circle with center at the origin and radius r?
Signup and view all the answers
What is a tangent to a circle?
What is a tangent to a circle?
Signup and view all the answers
What is the relationship between the gradients of the tangent and the normal to a curve at a given point?
What is the relationship between the gradients of the tangent and the normal to a curve at a given point?
Signup and view all the answers
If the coefficient 'a' in a cubic function is negative, which statement is true?
If the coefficient 'a' in a cubic function is negative, which statement is true?
Signup and view all the answers
What is the first step in determining the equation of a tangent to a curve?
What is the first step in determining the equation of a tangent to a curve?
Signup and view all the answers
What is the notation used to indicate the second derivative of a function y with respect to x?
What is the notation used to indicate the second derivative of a function y with respect to x?
Signup and view all the answers
Which of the following describes a local maximum of a function?
Which of the following describes a local maximum of a function?
Signup and view all the answers
What does the derivative allow us to determine about a function?
What does the derivative allow us to determine about a function?
Signup and view all the answers
To find stationary points on a graph, which condition must be satisfied?
To find stationary points on a graph, which condition must be satisfied?
Signup and view all the answers
How can one find the x-intercepts of a cubic function?
How can one find the x-intercepts of a cubic function?
Signup and view all the answers
What is the result of differentiating the function twice?
What is the result of differentiating the function twice?
Signup and view all the answers
What indicates the change in gradient of the original function?
What indicates the change in gradient of the original function?
Signup and view all the answers
What is the condition for two triangles to be similar?
What is the condition for two triangles to be similar?
Signup and view all the answers
If a line is drawn parallel to one side of a triangle, what is the effect on the other two sides?
If a line is drawn parallel to one side of a triangle, what is the effect on the other two sides?
Signup and view all the answers
What is the formula for the area of a triangle?
What is the formula for the area of a triangle?
Signup and view all the answers
What is the Mid-point Theorem?
What is the Mid-point Theorem?
Signup and view all the answers
What can be concluded about two triangles with equal bases and between the same parallel lines?
What can be concluded about two triangles with equal bases and between the same parallel lines?
Signup and view all the answers
What is the relationship between the areas of two triangles with the same height?
What is the relationship between the areas of two triangles with the same height?
Signup and view all the answers
What is the conclusion of the Proportion Theorem?
What is the conclusion of the Proportion Theorem?
Signup and view all the answers
What is the condition for two polygons to be similar?
What is the condition for two polygons to be similar?
Signup and view all the answers
What is the relationship between the heights of two triangles with equal areas?
What is the relationship between the heights of two triangles with equal areas?
Signup and view all the answers
Which theorem states that equiangular triangles are similar?
Which theorem states that equiangular triangles are similar?
Signup and view all the answers
What is the formula for the Proportion Theorem?
What is the formula for the Proportion Theorem?
Signup and view all the answers
What can be concluded about two triangles with equal heights and equal areas?
What can be concluded about two triangles with equal heights and equal areas?
Signup and view all the answers
What is the condition for two triangles to be similar?
What is the condition for two triangles to be similar?
Signup and view all the answers
What is the Mid-point Theorem used for?
What is the Mid-point Theorem used for?
Signup and view all the answers
If two triangles have corresponding sides in proportion, what can be concluded?
If two triangles have corresponding sides in proportion, what can be concluded?
Signup and view all the answers
What is the purpose of the Proportionality Theorem?
What is the purpose of the Proportionality Theorem?
Signup and view all the answers
What is the definition of similar polygons?
What is the definition of similar polygons?
Signup and view all the answers
What is the condition for a line to be parallel to one side of a triangle?
What is the condition for a line to be parallel to one side of a triangle?
Signup and view all the answers
What is the result of constructing a line parallel to one side of a triangle?
What is the result of constructing a line parallel to one side of a triangle?
Signup and view all the answers
What does it mean if a graph is concave up?
What does it mean if a graph is concave up?
Signup and view all the answers
Which method can be used to find turning points of a cubic function?
Which method can be used to find turning points of a cubic function?
Signup and view all the answers
In which situation would a point be classified as a point of inflection?
In which situation would a point be classified as a point of inflection?
Signup and view all the answers
What is the general procedure to sketch a cubic polynomial?
What is the general procedure to sketch a cubic polynomial?
Signup and view all the answers
What is the result when applying the Remainder Theorem to a polynomial p(x) divided by cx - d?
What is the result when applying the Remainder Theorem to a polynomial p(x) divided by cx - d?
Signup and view all the answers
How do you determine the y-intercept of a cubic function?
How do you determine the y-intercept of a cubic function?
Signup and view all the answers
Which statement is true regarding stationary points and turning points?
Which statement is true regarding stationary points and turning points?
Signup and view all the answers
Which of the following best describes concave down graphs?
Which of the following best describes concave down graphs?
Signup and view all the answers
What is the purpose of synthetic division in polynomial division?
What is the purpose of synthetic division in polynomial division?
Signup and view all the answers
When is a cubic polynomial guaranteed to have a real root?
When is a cubic polynomial guaranteed to have a real root?
Signup and view all the answers
What relationship is defined by the Pythagorean theorem in a right-angled triangle?
What relationship is defined by the Pythagorean theorem in a right-angled triangle?
Signup and view all the answers
What is the equation of a circle with center at ( (3, -2) ) and radius 5?
What is the equation of a circle with center at ( (3, -2) ) and radius 5?
Signup and view all the answers
What is the center of the circle represented by the equation ( x^2 + y^2 - 6x + 4y - 12 = 0 ) ?
What is the center of the circle represented by the equation ( x^2 + y^2 - 6x + 4y - 12 = 0 ) ?
Signup and view all the answers
Which condition confirms that two triangles are similar?
Which condition confirms that two triangles are similar?
Signup and view all the answers
Which statement is true regarding the areas of triangles with equal bases and heights?
Which statement is true regarding the areas of triangles with equal bases and heights?
Signup and view all the answers
A tangent to a circle intersects the circle at exactly:
A tangent to a circle intersects the circle at exactly:
Signup and view all the answers
What is the gradient of the tangent to the circle ( x^2 + y^2 = 25 ) at the point ( (3, 4) ) ?
What is the gradient of the tangent to the circle ( x^2 + y^2 = 25 ) at the point ( (3, 4) ) ?
Signup and view all the answers
In the context of triangle similarity, if two triangles are equiangular, what can be said about their sides?
In the context of triangle similarity, if two triangles are equiangular, what can be said about their sides?
Signup and view all the answers
Which of the following statements about the equation of a circle is TRUE?
Which of the following statements about the equation of a circle is TRUE?
Signup and view all the answers
What is the area formula for a triangle?
What is the area formula for a triangle?
Signup and view all the answers
What is the radius of the circle represented by the equation ( x^2 + y^2 + 8x - 10y + 16 = 0 ) ?
What is the radius of the circle represented by the equation ( x^2 + y^2 + 8x - 10y + 16 = 0 ) ?
Signup and view all the answers
Which of the following would make a triangle a right triangle according to the converse of the Pythagorean theorem?
Which of the following would make a triangle a right triangle according to the converse of the Pythagorean theorem?
Signup and view all the answers
Given the equation of a circle ( (x - 2)^2 + (y + 1)^2 = 9 ) , what is the gradient of the radius drawn to the point ( (5, 2) ) ?
Given the equation of a circle ( (x - 2)^2 + (y + 1)^2 = 9 ) , what is the gradient of the radius drawn to the point ( (5, 2) ) ?
Signup and view all the answers
How is proportionality in triangles determined if they share equal heights?
How is proportionality in triangles determined if they share equal heights?
Signup and view all the answers
What must be proven to establish that two triangles are similar under the conditions provided?
What must be proven to establish that two triangles are similar under the conditions provided?
Signup and view all the answers
A circle with center at the origin is symmetric about:
A circle with center at the origin is symmetric about:
Signup and view all the answers
What is the equation of the tangent to the circle ( x^2 + y^2 = 16 ) at the point ( (4, 0) ) ?
What is the equation of the tangent to the circle ( x^2 + y^2 = 16 ) at the point ( (4, 0) ) ?
Signup and view all the answers
Which of the following is NOT a step in finding the equation of a tangent to a circle?
Which of the following is NOT a step in finding the equation of a tangent to a circle?
Signup and view all the answers
What is the primary reason why ratios are considered unit-less?
What is the primary reason why ratios are considered unit-less?
Signup and view all the answers
Which property of proportion involves rearranging two equal ratios to form a new ratio?
Which property of proportion involves rearranging two equal ratios to form a new ratio?
Signup and view all the answers
In similar polygons, which statement is true about corresponding angles?
In similar polygons, which statement is true about corresponding angles?
Signup and view all the answers
What is the area formula for a rectangle defined by length and width?
What is the area formula for a rectangle defined by length and width?
Signup and view all the answers
What theorem states that a line parallel to one side of a triangle divides the other two sides proportionally?
What theorem states that a line parallel to one side of a triangle divides the other two sides proportionally?
Signup and view all the answers
When setting up proportional equations in problems, what is the first step?
When setting up proportional equations in problems, what is the first step?
Signup and view all the answers
Which formula correctly calculates the area of a trapezium?
Which formula correctly calculates the area of a trapezium?
Signup and view all the answers
Which of the following statements regarding ratios is NOT true?
Which of the following statements regarding ratios is NOT true?
Signup and view all the answers
In polygons, what defines a closed shape?
In polygons, what defines a closed shape?
Signup and view all the answers
What is the area of a rhombus calculated using its diagonals AC and BD?
What is the area of a rhombus calculated using its diagonals AC and BD?
Signup and view all the answers
What is the significance of the second derivative in determining the concavity of a curve?
What is the significance of the second derivative in determining the concavity of a curve?
Signup and view all the answers
What is the purpose of synthetic division in factorising cubic polynomials?
What is the purpose of synthetic division in factorising cubic polynomials?
Signup and view all the answers
What is the application of differential calculus in optimisation problems?
What is the application of differential calculus in optimisation problems?
Signup and view all the answers
What is the relationship between the gradient of the tangent and the normal to a curve?
What is the relationship between the gradient of the tangent and the normal to a curve?
Signup and view all the answers
What is the significance of the point of inflection in a curve?
What is the significance of the point of inflection in a curve?
Signup and view all the answers
What is the derivative of the function y = x^3?
What is the derivative of the function y = x^3?
Signup and view all the answers
What is the derivative of the function y = (x^2 + 3x - 4) / (x + 2)?
What is the derivative of the function y = (x^2 + 3x - 4) / (x + 2)?
Signup and view all the answers
What is the purpose of long division in polynomial division?
What is the purpose of long division in polynomial division?
Signup and view all the answers
What is the limit of the function y = (x - 2) / (x + 2) as x approaches -2?
What is the limit of the function y = (x - 2) / (x + 2) as x approaches -2?
Signup and view all the answers
What is the relationship between the roots of a polynomial and its factors?
What is the relationship between the roots of a polynomial and its factors?
Signup and view all the answers
What is the equation of the tangent line to the curve y = x^2 at the point (1, 1)?
What is the equation of the tangent line to the curve y = x^2 at the point (1, 1)?
Signup and view all the answers
What is the application of differential calculus in rates of change?
What is the application of differential calculus in rates of change?
Signup and view all the answers
What is the derivative of the function y = (2x + 1) / (x - 1)?
What is the derivative of the function y = (2x + 1) / (x - 1)?
Signup and view all the answers
What is the significance of the remainder theorem in solving cubic equations?
What is the significance of the remainder theorem in solving cubic equations?
Signup and view all the answers
What is the relationship between the coefficient 'a' and the graph of a cubic polynomial?
What is the relationship between the coefficient 'a' and the graph of a cubic polynomial?
Signup and view all the answers
What is the definition of the derivative of a function y = f(x)?
What is the definition of the derivative of a function y = f(x)?
Signup and view all the answers
What is the method used to find the derivative of a function y = f(x)?
What is the method used to find the derivative of a function y = f(x)?
Signup and view all the answers
What is the notation for the derivative of a function y = f(x)?
What is the notation for the derivative of a function y = f(x)?
Signup and view all the answers
What is the purpose of the differential operator D in differentiation?
What is the purpose of the differential operator D in differentiation?
Signup and view all the answers
What is the relationship between the gradient of the tangent and the gradient of the normal to a curve?
What is the relationship between the gradient of the tangent and the gradient of the normal to a curve?
Signup and view all the answers
If a polynomial p(x) is divided by cx - d, what is the degree of the quotient Q(x) if cx - d is a linear polynomial?
If a polynomial p(x) is divided by cx - d, what is the degree of the quotient Q(x) if cx - d is a linear polynomial?
Signup and view all the answers
What does the limit of the function $y = \frac{x^2 + 4x - 12}{x + 6}$ approach as $x$ approaches -6?
What does the limit of the function $y = \frac{x^2 + 4x - 12}{x + 6}$ approach as $x$ approaches -6?
Signup and view all the answers
Which of the following correctly represents the behavior of the function $y = \frac{x^2 + 4x - 12}{x + 6}$ near its point of discontinuity at $x = -6$?
Which of the following correctly represents the behavior of the function $y = \frac{x^2 + 4x - 12}{x + 6}$ near its point of discontinuity at $x = -6$?
Signup and view all the answers
What is the remainder when a polynomial p(x) is divided by cx - d?
What is the remainder when a polynomial p(x) is divided by cx - d?
Signup and view all the answers
In the context of the Achilles and the tortoise paradox, which fundamental principle of calculus is illustrated?
In the context of the Achilles and the tortoise paradox, which fundamental principle of calculus is illustrated?
Signup and view all the answers
If the remainder R is zero when a polynomial p(x) is divided by cx - d, what can be concluded?
If the remainder R is zero when a polynomial p(x) is divided by cx - d, what can be concluded?
Signup and view all the answers
What is the purpose of the Factor Theorem in solving cubic equations?
What is the purpose of the Factor Theorem in solving cubic equations?
Signup and view all the answers
What graphical feature is present in the function $y = \frac{x^2 + 4x - 12}{x + 6}$ at $x = -6$?
What graphical feature is present in the function $y = \frac{x^2 + 4x - 12}{x + 6}$ at $x = -6$?
Signup and view all the answers
What is the equation of a circle with center at (a, b) and radius r?
What is the equation of a circle with center at (a, b) and radius r?
Signup and view all the answers
What can be concluded about the limit of a function at a point where the function is not defined?
What can be concluded about the limit of a function at a point where the function is not defined?
Signup and view all the answers
What is the simplified form of the function $y = \frac{x^2 + 4x - 12}{x + 6}$ for $x \neq -6$?
What is the simplified form of the function $y = \frac{x^2 + 4x - 12}{x + 6}$ for $x \neq -6$?
Signup and view all the answers
If a polynomial p(x) has a root d/c, what can be concluded?
If a polynomial p(x) has a root d/c, what can be concluded?
Signup and view all the answers
What is the next step after finding a factor of a cubic polynomial using the Factor Theorem?
What is the next step after finding a factor of a cubic polynomial using the Factor Theorem?
Signup and view all the answers
What is the purpose of the Quadratic Formula in solving cubic equations?
What is the purpose of the Quadratic Formula in solving cubic equations?
Signup and view all the answers
What is the relationship between the roots of a polynomial and its factors?
What is the relationship between the roots of a polynomial and its factors?
Signup and view all the answers
Why is the Factor Theorem useful in solving cubic equations?
Why is the Factor Theorem useful in solving cubic equations?
Signup and view all the answers
If the equation of a circle is given by x^2 + y^2 + 4x - 6y + 9 = 0, what are the coordinates of its center?
If the equation of a circle is given by x^2 + y^2 + 4x - 6y + 9 = 0, what are the coordinates of its center?
Signup and view all the answers
What is the gradient of the tangent to the circle x^2 + y^2 = 25 at the point (3, 4)?
What is the gradient of the tangent to the circle x^2 + y^2 = 25 at the point (3, 4)?
Signup and view all the answers
What is the equation of the circle with center at (-1, 2) and radius 4?
What is the equation of the circle with center at (-1, 2) and radius 4?
Signup and view all the answers
What is the equation of the tangent to the circle x^2 + y^2 = 16 at the point (4, 0)?
What is the equation of the tangent to the circle x^2 + y^2 = 16 at the point (4, 0)?
Signup and view all the answers
What is the ratio of the circumference of a circle to its diameter?
What is the ratio of the circumference of a circle to its diameter?
Signup and view all the answers
What is the equation of the circle with center at (2, -3) and radius 5?
What is the equation of the circle with center at (2, -3) and radius 5?
Signup and view all the answers
What is the relationship between the gradients of the radius and tangent to a circle?
What is the relationship between the gradients of the radius and tangent to a circle?
Signup and view all the answers
What is the equation of the circle with center at the origin and radius 3?
What is the equation of the circle with center at the origin and radius 3?
Signup and view all the answers
What is the purpose of completing the square in finding the equation of a circle?
What is the purpose of completing the square in finding the equation of a circle?
Signup and view all the answers
What is a tangent to a circle?
What is a tangent to a circle?
Signup and view all the answers
If DE is parallel to BC in triangle ABC, what is the relationship between AD, DB, AE, and EC?
If DE is parallel to BC in triangle ABC, what is the relationship between AD, DB, AE, and EC?
Signup and view all the answers
What is the conclusion of the Triangle Proportionality Theorem?
What is the conclusion of the Triangle Proportionality Theorem?
Signup and view all the answers
What is the condition for triangles with the same height to have areas proportional to their bases?
What is the condition for triangles with the same height to have areas proportional to their bases?
Signup and view all the answers
What is the formula for the area of a triangle?
What is the formula for the area of a triangle?
Signup and view all the answers
What is the statement of the Mid-point Theorem?
What is the statement of the Mid-point Theorem?
Signup and view all the answers
What is the converse of the Mid-point Theorem?
What is the converse of the Mid-point Theorem?
Signup and view all the answers
What is the proportionality statement for triangles with equal bases and between the same parallel lines?
What is the proportionality statement for triangles with equal bases and between the same parallel lines?
Signup and view all the answers
What is the proportionality statement for triangles with the same height and same base?
What is the proportionality statement for triangles with the same height and same base?
Signup and view all the answers
What is the relationship between the heights of two triangles with equal areas and the same base?
What is the relationship between the heights of two triangles with equal areas and the same base?
Signup and view all the answers
What is the conclusion of the Proportion Theorem?
What is the conclusion of the Proportion Theorem?
Signup and view all the answers
What is the condition for two triangles to be similar?
What is the condition for two triangles to be similar?
Signup and view all the answers
What is the statement of the converse of the Pythagorean theorem?
What is the statement of the converse of the Pythagorean theorem?
Signup and view all the answers
What is the formula for the area of a triangle?
What is the formula for the area of a triangle?
Signup and view all the answers
What is the similarity condition for polygons?
What is the similarity condition for polygons?
Signup and view all the answers
What is the purpose of constructing a perpendicular line in the proof of the Pythagorean theorem?
What is the purpose of constructing a perpendicular line in the proof of the Pythagorean theorem?
Signup and view all the answers
What is the result of adding the two results from the similar triangles in the proof of the Pythagorean theorem?
What is the result of adding the two results from the similar triangles in the proof of the Pythagorean theorem?
Signup and view all the answers
What is the statement of the theorem that relates the lengths of the sides of a right-angled triangle?
What is the statement of the theorem that relates the lengths of the sides of a right-angled triangle?
Signup and view all the answers
What is the condition for two triangles to have areas proportional to their bases?
What is the condition for two triangles to have areas proportional to their bases?
Signup and view all the answers
Given a cubic function (f(x) = 2x^3 - 3x^2 + x - 1), what is the equation of the tangent line at the point where (x = 1)?
Given a cubic function (f(x) = 2x^3 - 3x^2 + x - 1), what is the equation of the tangent line at the point where (x = 1)?
Signup and view all the answers
The second derivative of a function (f(x)) is denoted by (f''(x)). If (f''(x) > 0) for all (x) in a particular interval, what can we conclude about the graph of (f(x)) within that interval?
The second derivative of a function (f(x)) is denoted by (f''(x)). If (f''(x) > 0) for all (x) in a particular interval, what can we conclude about the graph of (f(x)) within that interval?
Signup and view all the answers
A cubic function is defined as (f(x) = ax^3 + bx^2 + cx + d). If the coefficient (a) is negative, what can we say about the shape of the graph of (f(x)) as (x) approaches positive infinity?
A cubic function is defined as (f(x) = ax^3 + bx^2 + cx + d). If the coefficient (a) is negative, what can we say about the shape of the graph of (f(x)) as (x) approaches positive infinity?
Signup and view all the answers
If a function has a local maximum at a point, what must be true about the first derivative of the function at that point?
If a function has a local maximum at a point, what must be true about the first derivative of the function at that point?
Signup and view all the answers
What is the relationship between the gradients of the tangent and the normal to a curve at a given point?
What is the relationship between the gradients of the tangent and the normal to a curve at a given point?
Signup and view all the answers
A cubic function is defined as (f(x) = ax^3 + bx^2 + cx + d). What is the maximum number of x-intercepts this function can have?
A cubic function is defined as (f(x) = ax^3 + bx^2 + cx + d). What is the maximum number of x-intercepts this function can have?
Signup and view all the answers
To find the y-intercept of a cubic function (f(x) = ax^3 + bx^2 + cx + d), what value do we substitute for (x)?
To find the y-intercept of a cubic function (f(x) = ax^3 + bx^2 + cx + d), what value do we substitute for (x)?
Signup and view all the answers
What is the second derivative of the function (f(x) = x^4 - 3x^2 + 2)?
What is the second derivative of the function (f(x) = x^4 - 3x^2 + 2)?
Signup and view all the answers
Consider the cubic function (f(x) = x^3 - 3x^2 + 2x). If (f''(x) < 0) for all (x) in the interval (1, 2), what can we conclude about the graph of (f(x)) within this interval?
Consider the cubic function (f(x) = x^3 - 3x^2 + 2x). If (f''(x) < 0) for all (x) in the interval (1, 2), what can we conclude about the graph of (f(x)) within this interval?
Signup and view all the answers
Given a function (f(x) = x^3 - 4x^2 + 5x - 2), at which of the following values of (x) does the function have a stationary point?
Given a function (f(x) = x^3 - 4x^2 + 5x - 2), at which of the following values of (x) does the function have a stationary point?
Signup and view all the answers
In a triangle ABC, DE is parallel to BC and intersects sides AB and AC at D and E respectively. If AD = 4 cm, DB = 6 cm, and AE = 5 cm, what is the length of EC?
In a triangle ABC, DE is parallel to BC and intersects sides AB and AC at D and E respectively. If AD = 4 cm, DB = 6 cm, and AE = 5 cm, what is the length of EC?
Signup and view all the answers
A rectangle has a length of 12 cm and a width of 8 cm. A square has the same area as the rectangle. What is the side length of the square?
A rectangle has a length of 12 cm and a width of 8 cm. A square has the same area as the rectangle. What is the side length of the square?
Signup and view all the answers
A rhombus has diagonals of length 10 cm and 24 cm. What is the area of the rhombus?
A rhombus has diagonals of length 10 cm and 24 cm. What is the area of the rhombus?
Signup and view all the answers
Two similar triangles have corresponding sides in the ratio 3:5. If the perimeter of the smaller triangle is 24 cm, what is the perimeter of the larger triangle?
Two similar triangles have corresponding sides in the ratio 3:5. If the perimeter of the smaller triangle is 24 cm, what is the perimeter of the larger triangle?
Signup and view all the answers
A trapezium has bases of length 8 cm and 12 cm, and a height of 6 cm. What is the area of the trapezium?
A trapezium has bases of length 8 cm and 12 cm, and a height of 6 cm. What is the area of the trapezium?
Signup and view all the answers
A kite has diagonals of length 16 cm and 12 cm. What is the area of the kite?
A kite has diagonals of length 16 cm and 12 cm. What is the area of the kite?
Signup and view all the answers
Two similar polygons have corresponding sides in the ratio 2:3. If the area of the smaller polygon is 16 cm², what is the area of the larger polygon?
Two similar polygons have corresponding sides in the ratio 2:3. If the area of the smaller polygon is 16 cm², what is the area of the larger polygon?
Signup and view all the answers
In a triangle ABC, DE is parallel to BC, AD = 5 cm, DB = 8 cm, and AE = 7 cm. What is the length of EC?
In a triangle ABC, DE is parallel to BC, AD = 5 cm, DB = 8 cm, and AE = 7 cm. What is the length of EC?
Signup and view all the answers
A parallelogram has a base of 10 cm and a height of 6 cm. A triangle has the same base and height as the parallelogram. What is the ratio of the area of the triangle to the area of the parallelogram?
A parallelogram has a base of 10 cm and a height of 6 cm. A triangle has the same base and height as the parallelogram. What is the ratio of the area of the triangle to the area of the parallelogram?
Signup and view all the answers
Two similar triangles have corresponding sides in the ratio 4:7. If the area of the larger triangle is 196 cm², what is the area of the smaller triangle?
Two similar triangles have corresponding sides in the ratio 4:7. If the area of the larger triangle is 196 cm², what is the area of the smaller triangle?
Signup and view all the answers
Triangle ABC is similar to triangle DEF, with AB = 6 cm, DE = 9 cm, and AC = 8 cm. What is the length of DF?
Triangle ABC is similar to triangle DEF, with AB = 6 cm, DE = 9 cm, and AC = 8 cm. What is the length of DF?
Signup and view all the answers
Two triangles are similar if they have the same shape but differ in size. Which of the following statements is not a condition for similarity?
Two triangles are similar if they have the same shape but differ in size. Which of the following statements is not a condition for similarity?
Signup and view all the answers
If DE is the line joining the midpoints of sides AB and AC of triangle ABC, and BC = 10 cm, what is the length of DE?
If DE is the line joining the midpoints of sides AB and AC of triangle ABC, and BC = 10 cm, what is the length of DE?
Signup and view all the answers
In triangle ABC, D and E are the midpoints of sides AB and AC respectively. If DE = 5 cm, what is the length of BC?
In triangle ABC, D and E are the midpoints of sides AB and AC respectively. If DE = 5 cm, what is the length of BC?
Signup and view all the answers
Triangle ABC is similar to triangle DEF. If the ratio of their corresponding sides is 2:3, and the area of triangle ABC is 12 square cm, what is the area of triangle DEF?
Triangle ABC is similar to triangle DEF. If the ratio of their corresponding sides is 2:3, and the area of triangle ABC is 12 square cm, what is the area of triangle DEF?
Signup and view all the answers
Triangle ABC has sides AB = 8 cm, BC = 6 cm, and AC = 10 cm. Triangle DEF has sides DE = 12 cm, EF = 9 cm, and DF = 15 cm. Are the two triangles similar?
Triangle ABC has sides AB = 8 cm, BC = 6 cm, and AC = 10 cm. Triangle DEF has sides DE = 12 cm, EF = 9 cm, and DF = 15 cm. Are the two triangles similar?
Signup and view all the answers
Two triangles are equiangular. Which of the following statements is always true?
Two triangles are equiangular. Which of the following statements is always true?
Signup and view all the answers
In triangle ABC, D and E are points on sides AB and AC respectively, such that DE is parallel to BC. If AD = 4 cm, DB = 6 cm, and AE = 5 cm, what is the length of EC?
In triangle ABC, D and E are points on sides AB and AC respectively, such that DE is parallel to BC. If AD = 4 cm, DB = 6 cm, and AE = 5 cm, what is the length of EC?
Signup and view all the answers
Triangle ABC has a base BC of 10 cm and a height of 6 cm. Triangle DEF is similar to triangle ABC, and the ratio of their corresponding sides is 3:2. What is the area of triangle DEF?
Triangle ABC has a base BC of 10 cm and a height of 6 cm. Triangle DEF is similar to triangle ABC, and the ratio of their corresponding sides is 3:2. What is the area of triangle DEF?
Signup and view all the answers
Two polygons are similar if they have the same shape but differ in size. Which of the following is not a property of similar polygons?
Two polygons are similar if they have the same shape but differ in size. Which of the following is not a property of similar polygons?
Signup and view all the answers
What is the primary concept that the Achilles and the tortoise paradox illustrates?
What is the primary concept that the Achilles and the tortoise paradox illustrates?
Signup and view all the answers
What is the reason why the function y = (x^2 + 4x - 12)/(x + 6) is not defined at x = -6?
What is the reason why the function y = (x^2 + 4x - 12)/(x + 6) is not defined at x = -6?
Signup and view all the answers
What is the graphical representation of the function y = (x^2 + 4x - 12)/(x + 6)?
What is the graphical representation of the function y = (x^2 + 4x - 12)/(x + 6)?
Signup and view all the answers
What is the limiting value of the function y = (x^2 + 4x - 12)/(x + 6) as x approaches -6?
What is the limiting value of the function y = (x^2 + 4x - 12)/(x + 6) as x approaches -6?
Signup and view all the answers
What is the simplified form of the function y = (x^2 + 4x - 12)/(x + 6) for x ≠ -6?
What is the simplified form of the function y = (x^2 + 4x - 12)/(x + 6) for x ≠ -6?
Signup and view all the answers
What is the significance of the hole in the graph of the function y = (x^2 + 4x - 12)/(x + 6) at x = -6?
What is the significance of the hole in the graph of the function y = (x^2 + 4x - 12)/(x + 6) at x = -6?
Signup and view all the answers
If the coefficient a in a cubic function is positive, what is the shape of the graph?
If the coefficient a in a cubic function is positive, what is the shape of the graph?
Signup and view all the answers
What is the relationship between the gradients of the tangent and the normal to a curve?
What is the relationship between the gradients of the tangent and the normal to a curve?
Signup and view all the answers
What is the purpose of finding the second derivative of a function?
What is the purpose of finding the second derivative of a function?
Signup and view all the answers
What is the notation for the second derivative of a function y?
What is the notation for the second derivative of a function y?
Signup and view all the answers
What is the first step to find the equation of a tangent to a curve?
What is the first step to find the equation of a tangent to a curve?
Signup and view all the answers
What is the purpose of finding the stationary points of a function?
What is the purpose of finding the stationary points of a function?
Signup and view all the answers
What is the result of differentiating the function twice?
What is the result of differentiating the function twice?
Signup and view all the answers
What does the derivative allow us to determine about a function?
What does the derivative allow us to determine about a function?
Signup and view all the answers
What is the method to find the y-intercept of a cubic function?
What is the method to find the y-intercept of a cubic function?
Signup and view all the answers
What is the relationship between the gradient of the tangent and the curve at a point?
What is the relationship between the gradient of the tangent and the curve at a point?
Signup and view all the answers
If the remainder is zero when dividing a polynomial p(x) by cx - d, what can be concluded about cx - d?
If the remainder is zero when dividing a polynomial p(x) by cx - d, what can be concluded about cx - d?
Signup and view all the answers
If p(x) is divided by cx - d and the remainder is R, what is the general form of p(x)?
If p(x) is divided by cx - d and the remainder is R, what is the general form of p(x)?
Signup and view all the answers
What is the relationship between the degree of Q(x) and p(x) when p(x) is divided by a linear polynomial cx - d?
What is the relationship between the degree of Q(x) and p(x) when p(x) is divided by a linear polynomial cx - d?
Signup and view all the answers
What is the condition for a polynomial p(x) to have a root d/c, where c and d are constants?
What is the condition for a polynomial p(x) to have a root d/c, where c and d are constants?
Signup and view all the answers
What is the quadratic formula used for in solving cubic equations?
What is the quadratic formula used for in solving cubic equations?
Signup and view all the answers
If f(x) is a cubic polynomial and f(d/c) = 0, what can be concluded about cx - d?
If f(x) is a cubic polynomial and f(d/c) = 0, what can be concluded about cx - d?
Signup and view all the answers
What is the equation of a circle with center at the origin and radius r?
What is the equation of a circle with center at the origin and radius r?
Signup and view all the answers
What is the purpose of factorization in solving cubic equations?
What is the purpose of factorization in solving cubic equations?
Signup and view all the answers
What is the next step after finding a factor of a cubic polynomial using the Factor Theorem?
What is the next step after finding a factor of a cubic polynomial using the Factor Theorem?
Signup and view all the answers
Why is the Factor Theorem useful in solving cubic equations?
Why is the Factor Theorem useful in solving cubic equations?
Signup and view all the answers
Which condition indicates a point of inflection on a curve?
Which condition indicates a point of inflection on a curve?
Signup and view all the answers
What does it mean when a cubic polynomial is described as 'concave down'?
What does it mean when a cubic polynomial is described as 'concave down'?
Signup and view all the answers
How can the y-intercept of a cubic polynomial be determined?
How can the y-intercept of a cubic polynomial be determined?
Signup and view all the answers
Which of the following steps is essential to determine the turning points of a cubic polynomial?
Which of the following steps is essential to determine the turning points of a cubic polynomial?
Signup and view all the answers
What signifies the end behavior of a cubic polynomial?
What signifies the end behavior of a cubic polynomial?
Signup and view all the answers
What is the first step in the synthetic division method for polynomials?
What is the first step in the synthetic division method for polynomials?
Signup and view all the answers
What outcome occurs if the second derivative of a function is zero but does not change sign?
What outcome occurs if the second derivative of a function is zero but does not change sign?
Signup and view all the answers
In terms of cubic polynomial division, what does the Remainder Theorem state about the evaluation of a polynomial?
In terms of cubic polynomial division, what does the Remainder Theorem state about the evaluation of a polynomial?
Signup and view all the answers
What does the equation of the form $f(x) = ax^3 + bx^2 + cx + d$ represent?
What does the equation of the form $f(x) = ax^3 + bx^2 + cx + d$ represent?
Signup and view all the answers
What is the fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle?
What is the fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle?
Signup and view all the answers
If two triangles are equiangular, what can be said about their corresponding sides?
If two triangles are equiangular, what can be said about their corresponding sides?
Signup and view all the answers
What is the condition for a triangle to be a right-angled triangle according to the Converse of the Pythagorean Theorem?
What is the condition for a triangle to be a right-angled triangle according to the Converse of the Pythagorean Theorem?
Signup and view all the answers
What can be said about triangles with equal bases between the same parallel lines?
What can be said about triangles with equal bases between the same parallel lines?
Signup and view all the answers
What is the formula for the area of a triangle?
What is the formula for the area of a triangle?
Signup and view all the answers
In a triangle, a line joining the midpoints of two sides is:
In a triangle, a line joining the midpoints of two sides is:
Signup and view all the answers
What is the condition for polygons to be similar?
What is the condition for polygons to be similar?
Signup and view all the answers
If two polygons are similar, which of the following must be true?
If two polygons are similar, which of the following must be true?
Signup and view all the answers
In similar triangles, which of the following is NOT true?
In similar triangles, which of the following is NOT true?
Signup and view all the answers
What is the relationship between the areas of triangles with equal heights?
What is the relationship between the areas of triangles with equal heights?
Signup and view all the answers
If ∆ABC ∼ ∆DEF, which of the following is true?
If ∆ABC ∼ ∆DEF, which of the following is true?
Signup and view all the answers
What is the condition for two triangles to be similar?
What is the condition for two triangles to be similar?
Signup and view all the answers
What is the condition for two polygons to be similar?
What is the condition for two polygons to be similar?
Signup and view all the answers
If ∆ABC is similar to ∆DEF, which of the following is true?
If ∆ABC is similar to ∆DEF, which of the following is true?
Signup and view all the answers
What is the theorem that states that equiangular triangles are similar?
What is the theorem that states that equiangular triangles are similar?
Signup and view all the answers
What is the formula for the area of a triangle?
What is the formula for the area of a triangle?
Signup and view all the answers
If ∆ABC ∼ ∆DEF, what can be said about their areas?
If ∆ABC ∼ ∆DEF, what can be said about their areas?
Signup and view all the answers
What is the condition for triangles to be similar?
What is the condition for triangles to be similar?
Signup and view all the answers
What does it mean for two ratios to be in proportion?
What does it mean for two ratios to be in proportion?
Signup and view all the answers
Which property of proportion states that if you have ratios $\frac{w}{x} = \frac{y}{z}$, you can swap the numerator and denominator?
Which property of proportion states that if you have ratios $\frac{w}{x} = \frac{y}{z}$, you can swap the numerator and denominator?
Signup and view all the answers
What is the conclusion of the Basic Proportionality Theorem when a line is drawn parallel to one side of a triangle?
What is the conclusion of the Basic Proportionality Theorem when a line is drawn parallel to one side of a triangle?
Signup and view all the answers
What characteristic identifies polygons as being similar?
What characteristic identifies polygons as being similar?
Signup and view all the answers
In the context of finding area, what formula is used for the area of a parallelogram?
In the context of finding area, what formula is used for the area of a parallelogram?
Signup and view all the answers
If you have a rectangle with a length of 10 units and a width of 5 units, what is the area?
If you have a rectangle with a length of 10 units and a width of 5 units, what is the area?
Signup and view all the answers
What is a common property of all polygons?
What is a common property of all polygons?
Signup and view all the answers
Which formula represents the area of a kite?
Which formula represents the area of a kite?
Signup and view all the answers
When applying proportionality in geometric figures, which theorem is primarily referenced?
When applying proportionality in geometric figures, which theorem is primarily referenced?
Signup and view all the answers
What is the equation of the tangent line to the circle ( (x - 2)^2 + (y - 1)^2 = 9 ) at the point ( (4, 4) )?
What is the equation of the tangent line to the circle ( (x - 2)^2 + (y - 1)^2 = 9 ) at the point ( (4, 4) )?
Signup and view all the answers
What is the center and radius of the circle with equation ( x^2 + y^2 - 6x + 4y - 12 = 0 )?
What is the center and radius of the circle with equation ( x^2 + y^2 - 6x + 4y - 12 = 0 )?
Signup and view all the answers
The equation of a circle is ( x^2 + y^2 - 4x + 6y - 3 = 0 ). What is the radius of the circle?
The equation of a circle is ( x^2 + y^2 - 4x + 6y - 3 = 0 ). What is the radius of the circle?
Signup and view all the answers
If the point ( (3, -2) ) lies on the circle with equation ( (x - 1)^2 + (y + 3)^2 = r^2 ), what is the value of ( r )?
If the point ( (3, -2) ) lies on the circle with equation ( (x - 1)^2 + (y + 3)^2 = r^2 ), what is the value of ( r )?
Signup and view all the answers
What is the equation of the circle with center at ( (1, -2) ) and passing through the point ( (4, 1) )?
What is the equation of the circle with center at ( (1, -2) ) and passing through the point ( (4, 1) )?
Signup and view all the answers
What is the equation of the tangent line to the circle ( x^2 + y^2 = 25 ) at the point ( (3, 4) )?
What is the equation of the tangent line to the circle ( x^2 + y^2 = 25 ) at the point ( (3, 4) )?
Signup and view all the answers
A circle has a diameter with endpoints at ( (-2, 3) ) and ( (4, -1) ). What is the equation of this circle?
A circle has a diameter with endpoints at ( (-2, 3) ) and ( (4, -1) ). What is the equation of this circle?
Signup and view all the answers
What is the equation of the circle with center at ( (2, -5) ) and tangent to the line ( x + 2y = 1 )?
What is the equation of the circle with center at ( (2, -5) ) and tangent to the line ( x + 2y = 1 )?
Signup and view all the answers
A circle has a radius of 5 units and its center is at ( (3, -1) ). What is the equation of the tangent line to the circle at the point ( (8, -1) )?
A circle has a radius of 5 units and its center is at ( (3, -1) ). What is the equation of the tangent line to the circle at the point ( (8, -1) )?
Signup and view all the answers
What is the equation of the circle passing through the points ( (1, 2) ), ( (5, 2) ), and ( (1, 6) )?
What is the equation of the circle passing through the points ( (1, 2) ), ( (5, 2) ), and ( (1, 6) )?
Signup and view all the answers
In a triangle ABC, DE is drawn parallel to BC. If AD = 3, DB = 4, and EC = 5, what is the length of AE?
In a triangle ABC, DE is drawn parallel to BC. If AD = 3, DB = 4, and EC = 5, what is the length of AE?
Signup and view all the answers
If two triangles are similar, which of the following statements is true?
If two triangles are similar, which of the following statements is true?
Signup and view all the answers
In a triangle ABC, the area is 12 square units and the base is 4 units. What is the height of the triangle?
In a triangle ABC, the area is 12 square units and the base is 4 units. What is the height of the triangle?
Signup and view all the answers
If two triangles have equal heights and their areas are in the ratio 2:3, what is the ratio of their bases?
If two triangles have equal heights and their areas are in the ratio 2:3, what is the ratio of their bases?
Signup and view all the answers
In a triangle ABC, XYZ is a triangle with the same base and equal in area. What can be concluded about the heights of the triangles?
In a triangle ABC, XYZ is a triangle with the same base and equal in area. What can be concluded about the heights of the triangles?
Signup and view all the answers
If a line is drawn parallel to one side of a triangle, what is the effect on the other two sides?
If a line is drawn parallel to one side of a triangle, what is the effect on the other two sides?
Signup and view all the answers
What is the name of the theorem that states that triangles with equal heights have areas proportional to their bases?
What is the name of the theorem that states that triangles with equal heights have areas proportional to their bases?
Signup and view all the answers
If two triangles are equiangular, what is the relationship between their corresponding sides?
If two triangles are equiangular, what is the relationship between their corresponding sides?
Signup and view all the answers
What is the ratio of the areas of two triangles with equal heights and bases in the ratio 3:4?
What is the ratio of the areas of two triangles with equal heights and bases in the ratio 3:4?
Signup and view all the answers
What is the formula for the area of a triangle?
What is the formula for the area of a triangle?
Signup and view all the answers
What is the derivative of the function ( f(x) = 3x^2 + 2x - 1 ) using the general rule for differentiation?
What is the derivative of the function ( f(x) = 3x^2 + 2x - 1 ) using the general rule for differentiation?
Signup and view all the answers
What is the derivative of (f(x) = 5x^3 - 2x^2 + 7 ) using the rules for differentiation?
What is the derivative of (f(x) = 5x^3 - 2x^2 + 7 ) using the rules for differentiation?
Signup and view all the answers
What is the derivative of (f(x) = (x^2 + 3)(2x - 1)) using the product rule?
What is the derivative of (f(x) = (x^2 + 3)(2x - 1)) using the product rule?
Signup and view all the answers
What is the derivative of (f(x) = \frac{x^3 + 2x}{x^2}) using the quotient rule?
What is the derivative of (f(x) = \frac{x^3 + 2x}{x^2}) using the quotient rule?
Signup and view all the answers
What is the gradient of the tangent to the curve (y = x^2 - 3x + 2) at the point (x = 2)?
What is the gradient of the tangent to the curve (y = x^2 - 3x + 2) at the point (x = 2)?
Signup and view all the answers
What is the equation of the tangent to the curve (y = x^3 + 2x) at the point (x = 1)?
What is the equation of the tangent to the curve (y = x^3 + 2x) at the point (x = 1)?
Signup and view all the answers
What is the derivative of (f(x) = \sin(x)) using the first principles definition of the derivative?
What is the derivative of (f(x) = \sin(x)) using the first principles definition of the derivative?
Signup and view all the answers
What is the derivative of (f(x) = \cos(x)) using the first principles definition of the derivative?
What is the derivative of (f(x) = \cos(x)) using the first principles definition of the derivative?
Signup and view all the answers
What is the derivative of (f(x) = e^x) using the first principles definition of the derivative?
What is the derivative of (f(x) = e^x) using the first principles definition of the derivative?
Signup and view all the answers