Gr12 Mathematics: Term test 2

What is the definition of a point of inflection?

A point where the graph opens downwards

A point where the gradient of the curve is zero

A point where the concavity of the graph changes (correct)

A point where the graph opens upwards

What is the application of differential calculus in optimisation problems?

To sketch the graph of a function

To find the derivative of a function

To find the maximum or minimum value of a function (correct)

To find the integral of a function

What is the formula for synthetic division?

a(x) = b(x) ⋅ Q(x) + R(x)

f'(x) = 0

a = b ⋅ q + r

q₂ = a₃, q₁ = a₂ + q₂ ⋅ d/c, q₀ = a₁ + q₁ ⋅ d/c, R = a₀ + q₀ ⋅ d/c (correct)

What is the role of the remainder theorem in solving cubic equations?

To find the remainder when divided by cx - d Signup and view all the answers

What is the condition for a graph to be concave up?

f''(x) > 0 Signup and view all the answers

What is the method to find the x-intercepts of a cubic polynomial?

Solve f(x) = 0 Signup and view all the answers

What is the application of differential calculus in rates of change?

To find the instantaneous rate of change Signup and view all the answers

What is the formula for long division of polynomials?

a(x) = b(x) ⋅ Q(x) + R(x) Signup and view all the answers

What is the method to find the y-intercept of a cubic polynomial?

Set x = 0 and evaluate f(x) Signup and view all the answers

What is the condition for a graph to be concave down?

f''(x) < 0 Signup and view all the answers

What is the remainder when a polynomial p(x) is divided by cx - d?

$p(d/c)$ 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?

One less than the degree of p(x) Signup and view all the answers

What is the condition for cx - d to be a factor of a polynomial p(x)?

$p(d/c) = 0$ Signup and view all the answers

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

$x^2 + y^2 = r^2$ Signup and view all the answers

How do we find a factor of a cubic polynomial using the factor theorem?

By trial and error Signup and view all the answers

What is the general form of a polynomial p(x) when divided by cx - d?

$p(x) = (cx - d)Q(x) + R$ Signup and view all the answers

What is the next step after finding a factor of a cubic polynomial using the factor theorem?

Factorize the polynomial by polynomial long division Signup and view all the answers

What is the purpose of the quadratic formula in solving cubic equations?

To find the roots of the quadratic polynomial Signup and view all the answers

What is the relationship between the roots of a polynomial and its factors?

The roots are the solutions of the polynomial when substituted into the factors Signup and view all the answers

Why is the factor theorem useful in solving cubic equations?

It helps in factorizing the polynomial 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?

-8 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?

The limit represents the point where the distance between Achilles and the tortoise becomes infinitely small. Signup and view all the answers

Why is the function (y = \frac{x^2 + 4x - 12}{x + 6}) not defined at (x = -6)?

Because the denominator becomes zero 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?

The point where the function is undefined. 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)?

(x - 2) 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?

Limits allow us to understand how the function behaves as it gets closer to the 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?

x^2 + y^2 = r^2 Signup and view all the answers

A circle with center at the origin is symmetric about which of the following?

The x-axis and the y-axis Signup and view all the answers

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

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

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

Write the equation of the circle in standard form Signup and view all the answers

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

m_radius = -1/m_tangent Signup and view all the answers

What is the last step to find the equation of a tangent to a circle?

Write down the gradient-point form of the straight line equation Signup and view all the answers

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

x^2 + y^2 + Dx + Ey + F = 0 Signup and view all the answers

What is a tangent to a circle?

A straight line that touches the circle at exactly one point Signup and view all the answers

What is a ratio?

A relationship between two quantities with the same units Signup and view all the answers

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

To convert the general form to the standard form Signup and view all the answers

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

Find the derivative using the rules of differentiation. Signup and view all the answers

What indicates the change in gradient of the original function?

The second derivative. Signup and view all the answers

Which of the following describes a local maximum of a function?

A point where the function changes from increasing to decreasing. Signup and view all the answers

If the coefficient 'a' in a cubic function is negative, which statement is true?

The graph will fall to the left and rise to the right. Signup and view all the answers

How can one find the x-intercepts of a cubic function?

Set y equal to zero and solve for x. Signup and view all the answers

What is the relationship between the gradient of the tangent and the normal to a curve?

Their slopes multiply to -1. Signup and view all the answers

What is the result of differentiating the function twice?

It gives the rate of change of the gradient. Signup and view all the answers

To find stationary points on a graph, which condition must be satisfied?

The first derivative must be zero. Signup and view all the answers

Which notation indicates the second derivative of a function?

f''(x) Signup and view all the answers

What does the derivative allow us to determine about a function?

The rate of change at any point. Signup and view all the answers

What is the primary purpose of simplifying ratios?

To express ratios in their simplest form Signup and view all the answers

Which of the following is a property of proportion?

Cross Multiplication 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?

Thales' Theorem Signup and view all the answers

What is the formula for the area of a triangle?

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

What is the name of the polygon with four sides of equal length?

Rhombus Signup and view all the answers

Which of the following is not a type of polygon?

Circle Signup and view all the answers

What is the formula for the area of a trapezium?

Area = 1/2 × (base1 + base2) × height 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?

Proportionality Signup and view all the answers

Which of the following is a characteristic of similar polygons?

Corresponding angles are equal Signup and view all the answers

What is the first step in solving proportional problems?

Identify the given ratios Signup and view all the answers

What is the derivative of the function $f(x) = 3x^4$?

$12x^3$ Signup and view all the answers

Using differentiation from first principles, how is the gradient at a point $x = a$ calculated?

$rac{f(a+h) - f(a)}{h}$ Signup and view all the answers

What is the derivative of a constant function $f(x) = k$?

$0$ Signup and view all the answers

What does the notation $rac{dy}{dx}$ represent?

The slope of the tangent to a curve at a point Signup and view all the answers

Which of the following notations can be used to represent the derivative of a function?

$D_xy$ Signup and view all the answers

When should the rules for differentiation be used instead of differentiation from first principles?

If the question does not specify how to determine the derivative Signup and view all the answers

What is the derivative of the function $f(x) = x^3 + 4x$?

$3x^2 + 4$ Signup and view all the answers

What is the gradient at the point $x = -6$ for the function $y = rac{(x + 6)(x - 2)}{x + 6}$?

$-8$ Signup and view all the answers

Which operation does the differential operator $rac{d}{dx}$ indicate?

Derivation Signup and view all the answers

What is the derivative of the function $f(x) = k imes f(x)$, where $k$ is a constant?

$k imes rac{d}{dx}[f(x)]$ Signup and view all the answers

What is the statement of the Pythagorean theorem?

The square on the hypotenuse is equal to the sum of the squares on the other two sides. Signup and view all the answers

What is the condition for two triangles to be similar?

All pairs of corresponding angles are equal and all pairs of corresponding sides are in the same proportion. Signup and view all the answers

What is the formula for the area of a triangle?

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

What is the converse of the Pythagorean theorem?

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

What is the condition for triangles with equal bases between the same parallel lines?

They have equal areas. Signup and view all the answers

What is the similarity condition for polygons?

All pairs of corresponding angles are equal, and all pairs of corresponding sides are in the same proportion. Signup and view all the answers

What is the proof of the Pythagorean theorem based on?

The similarity of triangles ABD and CBA. Signup and view all the answers

What is the result of adding the two results from the similarity of triangles ABD and CAD?

AB^2 + AC^2 = BC^2 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?

It divides the other two sides proportionally. Signup and view all the answers

What is the relationship between two triangles that are equiangular?

They are similar. Signup and view all the answers

In a right-angled triangle, how are the squares of the sides related?

The square of the hypotenuse is equal to the sum of the squares of the other two sides. Signup and view all the answers

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

The areas are proportional to their bases. 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?

They have the same area. Signup and view all the answers

If two triangles on the same base have equal areas, what can you conclude about their heights?

Their heights are equal. 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?

The segments are proportional to the corresponding sides of the triangle. 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?

The line is parallel to the third side and half its length. 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?

The third side is bisected by the line. Signup and view all the answers

What is the formula for the area of a triangle?

Area = 1/2 x base x height Signup and view all the answers

If two triangles have all corresponding angles equal, what can we conclude about the triangles?

They are always similar. 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?

DE is parallel to BC and DE = 1/2 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?

4 Signup and view all the answers

Two polygons are similar if they have:

All of the above. Signup and view all the answers

If two triangles have corresponding sides in the same proportion, what can we conclude about the triangles?

They are always similar. 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?

5 cm Signup and view all the answers

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

All corresponding sides are equal. 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?

2:1 Signup and view all the answers

If two triangles are similar, then which of the following must be true?

The triangles have the same shape. 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?

DE is parallel to BC and DE = 1/3 BC Signup and view all the answers

What is the derivative of the function $f(x) = x^2$?

$2x$ 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 $x = a$?

$\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$ Signup and view all the answers

What is the derivative of the function $f(x) = 3x^2 + 2x - 5$?

$6x + 2$ Signup and view all the answers

What is the notation for the derivative of a function $f(x)$?

All of the above Signup and view all the answers

What is the general rule for differentiating $x^n$?

$nx^{n-1}$ Signup and view all the answers

What is the derivative of the function $f(x) = 2x + 1$?

$2$ Signup and view all the answers

What is the purpose of differentiating a function?

To determine the gradient of the function Signup and view all the answers

What is the derivative of the function $f(x) = k$?

$0$ Signup and view all the answers

What is the derivative of the function $f(x) = x^2 + 2x - 3$?

$2x + 2$ Signup and view all the answers

What is the equation of the tangent to a curve with equation $y = f(x)$ at $x = a$?

$y = f'(a)(x - a) + f(a)$ Signup and view all the answers

Why is the function (y = \frac{x^2 + 4x - 12}{x + 6}) not defined at (x = -6)?

Because the denominator becomes zero, leading to an undefined result. 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)?

(y = x - 2) 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?

A point of discontinuity where the function is undefined. 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?

Limits help us understand the behavior of the function as it approaches the undefined point. 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?

The paradox demonstrates that the limit of the distance between Achilles and the tortoise as Achilles gets closer to the tortoise is zero. 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?

-8 Signup and view all the answers

What is the remainder when a polynomial p(x) is divided by cx - d?

p(d/c) Signup and view all the answers

What is the general form of a polynomial p(x) when divided by cx - d?

p(x) = (cx - d)Q(x) + R Signup and view all the answers

What is the next step after finding a factor of a cubic polynomial using the Factor Theorem?

Divide the cubic polynomial by the factor to get a quadratic polynomial. Signup and view all the answers

What is the purpose of the Factor Theorem in solving cubic equations?

To factorize the cubic polynomial into linear factors. Signup and view all the answers

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

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

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

Find the point of tangency. Signup and view all the answers

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

They are perpendicular. Signup and view all the answers

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

x^2 + y^2 = r^2 Signup and view all the answers

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

x^2 + y^2 = r^2 Signup and view all the answers

What is a tangent to a circle?

A line that intersects the circle at one 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?

The product of the gradients is equal to -1. Signup and view all the answers

If the coefficient 'a' in a cubic function is negative, which statement is true?

The graph falls to the right and rises to the left. Signup and view all the answers

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

Find the derivative of the function using the rules of differentiation. Signup and view all the answers

What is the notation used to indicate the second derivative of a function y with respect to x?

y'' Signup and view all the answers

Which of the following describes a local maximum of a function?

A point where the function changes from increasing to decreasing. Signup and view all the answers

What does the derivative allow us to determine about a function?

The rate of change of the function at a given point. Signup and view all the answers

To find stationary points on a graph, which condition must be satisfied?

The first derivative is equal to zero. Signup and view all the answers

How can one find the x-intercepts of a cubic function?

Set the function equal to zero and solve for x. Signup and view all the answers

What is the result of differentiating the function twice?

The second derivative of the function. Signup and view all the answers

What indicates the change in gradient of the original function?

The second derivative of the function. Signup and view all the answers

What is the condition for two triangles to be similar?

The corresponding sides are in proportion. 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?

They are divided proportionally. Signup and view all the answers

What is the formula for the area of a triangle?

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

What is the Mid-point Theorem?

The line joining the midpoints of two sides of a triangle is parallel to the third side and equal to half the length of the third side. Signup and view all the answers

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

They have equal areas. Signup and view all the answers

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

The areas are proportional to their bases. Signup and view all the answers

What is the conclusion of the Proportion Theorem?

The corresponding sides of two triangles are in proportion. Signup and view all the answers

What is the condition for two polygons to be similar?

All pairs of corresponding angles are equal and all pairs of corresponding sides are in the same proportion Signup and view all the answers

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

The heights are equal. Signup and view all the answers

Which theorem states that equiangular triangles are similar?

Equiangular triangle theorem Signup and view all the answers

What is the formula for the Proportion Theorem?

AD/DB = AE/EC Signup and view all the answers

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

They have equal bases. Signup and view all the answers

What is the condition for two triangles to be similar?

Both conditions b and c must be true Signup and view all the answers

What is the Mid-point Theorem used for?

To find the length of a line segment parallel to one side of a triangle and half its length Signup and view all the answers

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

They are similar Signup and view all the answers

What is the purpose of the Proportionality Theorem?

To show that corresponding sides of two triangles are in proportion Signup and view all the answers

What is the definition of similar polygons?

Polygons with the same shape but differ in size Signup and view all the answers

What is the condition for a line to be parallel to one side of a triangle?

The corresponding angles are equal Signup and view all the answers

What is the result of constructing a line parallel to one side of a triangle?

The line divides the other two sides proportionally Signup and view all the answers

What does it mean if a graph is concave up?

The second derivative is positive. Signup and view all the answers

Which method can be used to find turning points of a cubic function?

Set the derivative equal to zero. Signup and view all the answers

In which situation would a point be classified as a point of inflection?

The second derivative changes sign. Signup and view all the answers

What is the general procedure to sketch a cubic polynomial?

Consider the shape using the sign of a, find intercepts, turning points, and end behavior. Signup and view all the answers

What is the result when applying the Remainder Theorem to a polynomial p(x) divided by cx - d?

The remainder is the value of p at x = d/c. Signup and view all the answers

How do you determine the y-intercept of a cubic function?

By evaluating f(0). Signup and view all the answers

Which statement is true regarding stationary points and turning points?

A stationary point can indicate a local maximum or minimum. Signup and view all the answers

Which of the following best describes concave down graphs?

The second derivative is negative. Signup and view all the answers

What is the purpose of synthetic division in polynomial division?

To simplify the coefficients systematically. Signup and view all the answers

When is a cubic polynomial guaranteed to have a real root?

Always, due to the Intermediate Value Theorem. Signup and view all the answers

What relationship is defined by the Pythagorean theorem in a right-angled triangle?

The square of the hypotenuse equals the sum of the squares of the other two sides. Signup and view all the answers

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

( (x - 3)^2 + (y + 2)^2 = 25 ) 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 ) ?

( (3, -2) ) Signup and view all the answers

Which condition confirms that two triangles are similar?

All pairs of corresponding angles are equal. Signup and view all the answers

Which statement is true regarding the areas of triangles with equal bases and heights?

They can have different perimeters. Signup and view all the answers

A tangent to a circle intersects the circle at exactly:

One point 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) ) ?

( -\frac{4}{3} ) Signup and view all the answers

In the context of triangle similarity, if two triangles are equiangular, what can be said about their sides?

The corresponding sides are in proportion. Signup and view all the answers

Which of the following statements about the equation of a circle is TRUE?

The equation of a circle with center at the origin is always in the form of ( x^2 + y^2 = r^2 ) , where r is the radius. Signup and view all the answers

What is the area formula for a triangle?

Area = 1/2 × base × height. 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 ) ?

( 5 ) 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?

The square of one side equals the sum of the squares of the other two sides. 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) ) ?

( \frac{1}{3} ) Signup and view all the answers

How is proportionality in triangles determined if they share equal heights?

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

What must be proven to establish that two triangles are similar under the conditions provided?

Either equiangularity or proportionality of sides must be shown. Signup and view all the answers

A circle with center at the origin is symmetric about:

The x-axis, the y-axis, and the origin 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) ) ?

( y = 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?

Find the x-intercept of the tangent. Signup and view all the answers

What is the primary reason why ratios are considered unit-less?

They compare quantities of the same kind. Signup and view all the answers

Which property of proportion involves rearranging two equal ratios to form a new ratio?

Reciprocal Proportion Signup and view all the answers

In similar polygons, which statement is true about corresponding angles?

They are equal. Signup and view all the answers

What is the area formula for a rectangle defined by length and width?

Area = length × 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?

Basic Proportionality Theorem Signup and view all the answers

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

Identify the given ratios. Signup and view all the answers

Which formula correctly calculates the area of a trapezium?

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

Which of the following statements regarding ratios is NOT true?

Ratios provide actual measurements. Signup and view all the answers

In polygons, what defines a closed shape?

Each segment intersects exactly two others. Signup and view all the answers

What is the area of a rhombus calculated using its diagonals AC and BD?

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

What is the significance of the second derivative in determining the concavity of a curve?

It determines whether the curve is concave up or concave down Signup and view all the answers

What is the purpose of synthetic division in factorising cubic polynomials?

To find the factors of the polynomial Signup and view all the answers

What is the application of differential calculus in optimisation problems?

To find the maximum or minimum values of a function Signup and view all the answers

What is the relationship between the gradient of the tangent and the normal to a curve?

They are perpendicular to each other Signup and view all the answers

What is the significance of the point of inflection in a curve?

It is the point where the curve changes concavity Signup and view all the answers

What is the derivative of the function y = x^3?

3x^2 Signup and view all the answers

What is the derivative of the function y = (x^2 + 3x - 4) / (x + 2)?

((x + 1)(x - 2)) / (x + 2)^2 Signup and view all the answers

What is the purpose of long division in polynomial division?

To find the quotient and remainder Signup and view all the answers

What is the limit of the function y = (x - 2) / (x + 2) as x approaches -2?

1 Signup and view all the answers

What is the relationship between the roots of a polynomial and its factors?

The roots are the solutions to the equation p(x) = 0 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)?

y = 2x - 1 Signup and view all the answers

What is the application of differential calculus in rates of change?

To find the instantaneous rate of change of a function Signup and view all the answers

What is the derivative of the function y = (2x + 1) / (x - 1)?

(-x + 3) / (x - 1)^2 Signup and view all the answers

What is the significance of the remainder theorem in solving cubic equations?

It is used to find the remainder when dividing a polynomial by a linear polynomial Signup and view all the answers

What is the relationship between the coefficient 'a' and the graph of a cubic polynomial?

If a is positive, the graph opens upwards Signup and view all the answers

What is the definition of the derivative of a function y = f(x)?

The derivative of a function y = f(x) is the limit of the ratio of the change in y to the change in x as the change in x approaches zero. Signup and view all the answers

What is the method used to find the derivative of a function y = f(x)?

Differentiation from first principles Signup and view all the answers

What is the notation for the derivative of a function y = f(x)?

All of the above Signup and view all the answers

What is the purpose of the differential operator D in differentiation?

To indicate the operation of 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?

They are perpendicular 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?

One degree less than p(x) 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?

-8 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 $x = -6$?

The function has a removable discontinuity. Signup and view all the answers

What is the remainder when a polynomial p(x) is divided by cx - d?

p(d/c) Signup and view all the answers

In the context of the Achilles and the tortoise paradox, which fundamental principle of calculus is illustrated?

The concept of limits. 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?

cx - d is a factor of p(x) Signup and view all the answers

What is the purpose of the Factor Theorem in solving cubic equations?

To find one factor of the cubic polynomial Signup and view all the answers

What graphical feature is present in the function $y = \frac{x^2 + 4x - 12}{x + 6}$ at $x = -6$?

A hole in the graph. Signup and view all the answers

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

(x - a)^2 + (y - b)^2 = r^2 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?

The limit can exist even if the function is undefined at that point. Signup and view all the answers

What is the simplified form of the function $y = \frac{x^2 + 4x - 12}{x + 6}$ for $x \neq -6$?

$y = x - 2$ Signup and view all the answers

If a polynomial p(x) has a root d/c, what can be concluded?

cx - d is a factor of p(x) Signup and view all the answers

What is the next step after finding a factor of a cubic polynomial using the Factor Theorem?

Use polynomial division to find the remaining factors Signup and view all the answers

What is the purpose of the Quadratic Formula in solving cubic equations?

To find the roots of the quadratic polynomial Signup and view all the answers

What is the relationship between the roots of a polynomial and its factors?

The factors are the roots of the polynomial Signup and view all the answers

Why is the Factor Theorem useful in solving cubic equations?

It helps to find one factor of the cubic polynomial 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?

(-2, 3) 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)?

-3/4 Signup and view all the answers

What is the equation of the circle with center at (-1, 2) and radius 4?

(x + 1)^2 + (y - 2)^2 = 16 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)?

x = 4 Signup and view all the answers

What is the ratio of the circumference of a circle to its diameter?

pi Signup and view all the answers

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

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

What is the relationship between the gradients of the radius and tangent to a circle?

They are perpendicular Signup and view all the answers

What is the equation of the circle with center at the origin and radius 3?

x^2 + y^2 = 9 Signup and view all the answers

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

To find the equation of the circle in standard form Signup and view all the answers

What is a tangent to a circle?

A line that intersects the circle at one point 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?

AD/DB = AE/EC Signup and view all the answers

What is the conclusion of the Triangle Proportionality Theorem?

The corresponding sides are in proportion, and the triangles are similar. Signup and view all the answers

What is the condition for triangles with the same height to have areas proportional to their bases?

The heights must be equal. Signup and view all the answers

What is the formula for the area of a triangle?

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

What is the statement of the Mid-point Theorem?

The line joining the midpoints of two sides of a triangle is parallel to the third side and equal to half the length of the third side. Signup and view all the answers

What is the converse of the Mid-point Theorem?

The line drawn from the midpoint of one side of a triangle parallel to another side bisects the third side of the triangle. Signup and view all the answers

What is the proportionality statement for triangles with equal bases and between the same parallel lines?

The areas are equal. Signup and view all the answers

What is the proportionality statement for triangles with the same height and same base?

The areas are equal. Signup and view all the answers

What is the relationship between the heights of two triangles with equal areas and the same base?

The heights are equal. Signup and view all the answers

What is the conclusion of the Proportion Theorem?

The line divides the sides proportionally. Signup and view all the answers

What is the condition for two triangles to be similar?

They have equal corresponding angles and sides in the same proportion Signup and view all the answers

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

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

What is the formula for the area of a triangle?

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

What is the similarity condition for polygons?

All pairs of corresponding angles are equal, and all pairs of corresponding sides are in the same proportion. Signup and view all the answers

What is the purpose of constructing a perpendicular line in the proof of the Pythagorean theorem?

To show that the two triangles are similar 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?

AB^2 + AC^2 = BC^2 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?

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

What is the condition for two triangles to have areas proportional to their bases?

They have equal bases between the same parallel lines 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)?

y = 2x - 2 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 graph of (f(x)) is concave up. 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?

The graph falls to the right. 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?

The first derivative is zero. 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?

The product of the gradients is -1. 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?

3 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)?

0 Signup and view all the answers

What is the second derivative of the function (f(x) = x^4 - 3x^2 + 2)?

12x^2 - 6 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?

The graph is concave down. 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?

x = 3 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?

7.5 cm 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?

9.6 cm Signup and view all the answers

A rhombus has diagonals of length 10 cm and 24 cm. What is the area of the rhombus?

120 cm² 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?

40 cm 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?

60 cm² Signup and view all the answers

A kite has diagonals of length 16 cm and 12 cm. What is the area of the kite?

96 cm² 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?

36 cm² 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?

11.2 cm 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?

1:2 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?

64 cm² 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?

12 cm 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?

The triangles have the same area. 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?

5 cm 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?

10 cm 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?

27 square cm 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?

Yes, because all corresponding sides are in the same proportion. Signup and view all the answers

Two triangles are equiangular. Which of the following statements is always true?

The triangles are similar. 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?

7.5 cm 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?

45 square cm 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?

The polygons have the same area. Signup and view all the answers

What is the primary concept that the Achilles and the tortoise paradox illustrates?

The concept of limits 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?

The denominator becomes zero Signup and view all the answers

What is the graphical representation of the function y = (x^2 + 4x - 12)/(x + 6)?

A straight line with a hole at 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?

-8 Signup and view all the answers

What is the simplified form of the function y = (x^2 + 4x - 12)/(x + 6) for x ≠ -6?

x - 2 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?

It represents a limit of the function Signup and view all the answers

If the coefficient a in a cubic function is positive, what is the shape of the graph?

The graph rises to the right and falls to the left. Signup and view all the answers

What is the relationship between the gradients of the tangent and the normal to a curve?

m_tangent * m_normal = -1 Signup and view all the answers

What is the purpose of finding the second derivative of a function?

To determine the change in gradient of the original function. Signup and view all the answers

What is the notation for the second derivative of a function y?

y'' Signup and view all the answers

What is the first step to find the equation of a tangent to a curve?

Find the derivative of the function. Signup and view all the answers

What is the purpose of finding the stationary points of a function?

To find the maximum or minimum value of the function. Signup and view all the answers

What is the result of differentiating the function twice?

The second derivative of the function. Signup and view all the answers

What does the derivative allow us to determine about a function?

The gradient of the function. Signup and view all the answers

What is the method to find the y-intercept of a cubic function?

Set x = 0 and solve for y. Signup and view all the answers

What is the relationship between the gradient of the tangent and the curve at a point?

The gradient of the tangent is equal to the negative reciprocal of the gradient of the curve. 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?

It is a factor of p(x) 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)?

p(x) = (cx - d)Q(x) + R 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?

The degree of Q(x) is one less than the degree of p(x) 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?

p(d/c) = 0 Signup and view all the answers

What is the quadratic formula used for in solving cubic equations?

To solve the quadratic polynomial obtained after division 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?

It is a linear factor of f(x) Signup and view all the answers

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

x^2 + y^2 = r^2 Signup and view all the answers

What is the purpose of factorization in solving cubic equations?

To find the factors of the cubic polynomial Signup and view all the answers

What is the next step after finding a factor of a cubic polynomial using the Factor Theorem?

All of the above Signup and view all the answers

Why is the Factor Theorem useful in solving cubic equations?

It helps to find the factors of the polynomial Signup and view all the answers

Which condition indicates a point of inflection on a curve?

The concavity changes from concave up to concave down. Signup and view all the answers

What does it mean when a cubic polynomial is described as 'concave down'?

The second derivative is less than zero. Signup and view all the answers

How can the y-intercept of a cubic polynomial be determined?

By substituting $x=0$ into the polynomial function. Signup and view all the answers

Which of the following steps is essential to determine the turning points of a cubic polynomial?

Setting the derivative equal to zero and solving for $x$. Signup and view all the answers

What signifies the end behavior of a cubic polynomial?

The signs of the coefficients of the polynomial. Signup and view all the answers

What is the first step in the synthetic division method for polynomials?

Write the coefficients of the dividend polynomial. Signup and view all the answers

What outcome occurs if the second derivative of a function is zero but does not change sign?

No point of inflection exists. Signup and view all the answers

In terms of cubic polynomial division, what does the Remainder Theorem state about the evaluation of a polynomial?

The polynomial evaluated at $x=rac{d}{c}$ gives the remainder. Signup and view all the answers

What does the equation of the form $f(x) = ax^3 + bx^2 + cx + d$ represent?

A polynomial function of degree three. 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?

Pythagorean Theorem Signup and view all the answers

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

They are in proportion 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?

The square of one side is equal to the sum of the squares of the other two sides Signup and view all the answers

What can be said about triangles with equal bases between the same parallel lines?

They have equal areas Signup and view all the answers

What is the formula for the area of a triangle?

½ × base × height Signup and view all the answers

In a triangle, a line joining the midpoints of two sides is:

parallel to the third side Signup and view all the answers

What is the condition for polygons to be similar?

All pairs of corresponding angles are equal and all pairs of corresponding sides are in the same proportion Signup and view all the answers

If two polygons are similar, which of the following must be true?

All pairs of corresponding angles are equal and all pairs of corresponding sides are in the same proportion Signup and view all the answers

In similar triangles, which of the following is NOT true?

The areas of the triangles are equal Signup and view all the answers

What is the relationship between the areas of triangles with equal heights?

They are proportional to their bases Signup and view all the answers

If ∆ABC ∼ ∆DEF, which of the following is true?

AB/DE = BC/EF Signup and view all the answers

What is the condition for two triangles to be similar?

They are equiangular or have proportional sides Signup and view all the answers

What is the condition for two polygons to be similar?

All pairs of corresponding angles are equal and all pairs of corresponding sides are in the same proportion Signup and view all the answers

If ∆ABC is similar to ∆DEF, which of the following is true?

∠A = ∠D Signup and view all the answers

What is the theorem that states that equiangular triangles are similar?

Theorem of equiangular triangles Signup and view all the answers

What is the formula for the area of a triangle?

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

If ∆ABC ∼ ∆DEF, what can be said about their areas?

The areas are in the same proportion as the squares of their corresponding sides Signup and view all the answers

What is the condition for triangles to be similar?

All pairs of corresponding angles are equal or all pairs of corresponding sides are in the same proportion Signup and view all the answers

What does it mean for two ratios to be in proportion?

The cross products of the ratios are equal. 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?

Reciprocal Proportion 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?

The two other sides are divided proportionally. Signup and view all the answers

What characteristic identifies polygons as being similar?

Corresponding angles are equal and corresponding sides are in proportion. Signup and view all the answers

In the context of finding area, what formula is used for the area of a parallelogram?

Area = base × height 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?

50 square units Signup and view all the answers

What is a common property of all polygons?

They consist of at least three line segments. Signup and view all the answers

Which formula represents the area of a kite?

Area = \frac{1}{2} × diagonal AC × diagonal BD Signup and view all the answers

When applying proportionality in geometric figures, which theorem is primarily referenced?

Basic Proportionality Theorem 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) )?

( y = -3x + 16 ) 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 )?

Center: (3, -2), radius: 5 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?

( \sqrt{19} ) 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 )?

( \sqrt{29} ) 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) )?

( (x - 1)^2 + (y + 2)^2 = 18 ) 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) )?

( y = -\frac{4}{3}x + \frac{25}{3} ) 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?

( (x - 1)^2 + (y - 1)^2 = 13 ) 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 )?

( (x - 2)^2 + (y + 5)^2 = 5 ) 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) )?

( y = -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) )?

( (x - 3)^2 + (y - 4)^2 = 25 ) 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?

6.25 Signup and view all the answers

If two triangles are similar, which of the following statements is true?

The corresponding sides are in proportion 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?

3 units 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?

3:2 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?

The heights are equal 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?

They are divided proportionally 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?

The triangle proportionality theorem Signup and view all the answers

If two triangles are equiangular, what is the relationship between their corresponding sides?

They are in proportion 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?

3:4 Signup and view all the answers

What is the formula for the area of a triangle?

Area = 1/2 × base × height 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?

( 6x + 2 ) Signup and view all the answers

What is the derivative of (f(x) = 5x^3 - 2x^2 + 7 ) using the rules for differentiation?

( 15x^2 - 4x ) Signup and view all the answers

What is the derivative of (f(x) = (x^2 + 3)(2x - 1)) using the product rule?

( 4x^3 + 6x^2 - 3x ) Signup and view all the answers

What is the derivative of (f(x) = \frac{x^3 + 2x}{x^2}) using the quotient rule?

( 1 - \frac{4}{x^3} ) 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)?

(1) 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)?

(y = 5x - 3 ) Signup and view all the answers

What is the derivative of (f(x) = \sin(x)) using the first principles definition of the derivative?

( \cos(x) ) Signup and view all the answers

What is the derivative of (f(x) = \cos(x)) using the first principles definition of the derivative?

( -\sin(x) ) Signup and view all the answers

What is the derivative of (f(x) = e^x) using the first principles definition of the derivative?

( e^x ) Signup and view all the answers

Gr12 Mathematics: Term test 2

Questions and Answers

What is the definition of a point of inflection?

What is the application of differential calculus in optimisation problems?

What is the formula for synthetic division?

What is the role of the remainder theorem in solving cubic equations?

What is the condition for a graph to be concave up?

What is the method to find the x-intercepts of a cubic polynomial?

What is the application of differential calculus in rates of change?

What is the formula for long division of polynomials?

What is the method to find the y-intercept of a cubic polynomial?

What is the condition for a graph to be concave down?

What is the remainder when a polynomial p(x) is divided by cx - d?

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 condition for cx - d to be a factor of a polynomial p(x)?

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

How do we find a factor of a cubic polynomial using the factor theorem?

What is the general form of a polynomial p(x) when divided by cx - d?

What is the next step after finding a factor of a cubic polynomial using the factor theorem?

What is the purpose of the quadratic formula in solving cubic equations?

What is the relationship between the roots of a polynomial and its factors?

Why is the factor theorem useful in solving cubic equations?

What is the limit of the function (y = \frac{x^2 + 4x - 12}{x + 6}) as (x) approaches -6?

Which of the following statements accurately describes the concept of limits in the context of Achilles and the tortoise paradox?

Why is the function (y = \frac{x^2 + 4x - 12}{x + 6}) not defined at (x = -6)?

What does the hole in the graph of (y = \frac{x^2 + 4x - 12}{x + 6}) at (x = -6) represent?

Which of the following expressions represents the simplified form of the function (y = \frac{x^2 + 4x - 12}{x + 6}) for (x \neq -6)?

How does the concept of limits help in understanding the behavior of a function near a point where it is not defined?

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

A circle with center at the origin is symmetric about which of the following?

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

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

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

What is the last step to find the equation of a tangent to a circle?

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

What is a tangent to a circle?

What is a ratio?

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

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

What indicates the change in gradient of the original function?

Which of the following describes a local maximum of a function?

If the coefficient 'a' in a cubic function is negative, which statement is true?

How can one find the x-intercepts of a cubic function?

What is the relationship between the gradient of the tangent and the normal to a curve?

What is the result of differentiating the function twice?

To find stationary points on a graph, which condition must be satisfied?

Which notation indicates the second derivative of a function?

What does the derivative allow us to determine about a function?

What is the primary purpose of simplifying ratios?

Which of the following is a property of proportion?

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 formula for the area of a triangle?

What is the name of the polygon with four sides of equal length?

Which of the following is not a type of polygon?

What is the formula for the area of a trapezium?

What is the term used to describe the equality of ratios between corresponding sides or other measurements in polygons?

Which of the following is a characteristic of similar polygons?

What is the first step in solving proportional problems?

What is the derivative of the function $f(x) = 3x^4$?

Using differentiation from first principles, how is the gradient at a point $x = a$ calculated?

What is the derivative of a constant function $f(x) = k$?

What does the notation $rac{dy}{dx}$ represent?

Which of the following notations can be used to represent the derivative of a function?

When should the rules for differentiation be used instead of differentiation from first principles?

What is the derivative of the function $f(x) = x^3 + 4x$?

What is the gradient at the point $x = -6$ for the function $y = rac{(x + 6)(x - 2)}{x + 6}$?

Which operation does the differential operator $rac{d}{dx}$ indicate?

What is the derivative of the function $f(x) = k imes f(x)$, where $k$ is a constant?

What is the statement of the Pythagorean theorem?

What is the condition for two triangles to be similar?

What is the formula for the area of a triangle?

What is the converse of the Pythagorean theorem?

What is the condition for triangles with equal bases between the same parallel lines?

What is the similarity condition for polygons?

What is the proof of the Pythagorean theorem based on?

What is the result of adding the two results from the similarity of triangles ABD and CAD?

If a line is drawn parallel to one side of a triangle, how does it affect the other two sides?

What is the relationship between two triangles that are equiangular?

In a right-angled triangle, how are the squares of the sides related?

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