Gr12 Mathematics: June Exam Mix P(1)

What is the process of determining the derivative of a given function called?

Differentiation from first principles (correct)

First principle of differentiation

Derivative definition

Gradient calculation

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

lim(h → 0) [f(x) + f(x + h)]/h

lim(h → 0) [f(x + h) - f(x)]/h (correct)

lim(h → 0) [f(x) - f(x + h)]/h

lim(h → 0) [f(x + h) + f(x)]/h

What is the general rule for differentiation?

d/dx [x^n] = nx^(n+1), where n ∈ ℝ and n ≠ 0

d/dx [x^n] = nx^(n-1), where n ∈ ℝ and n ≠ 0 (correct)

d/dx [x^n] = nx^(2n-1), where n ∈ ℝ and n ≠ 0

d/dx [x^n] = nx^n, where n ∈ ℝ and n ≠ 0

What is the derivative of a constant?

0 Signup and view all the answers

What is the derivative of a constant multiplied by a function?

k ∙ d/dx [f(x)] Signup and view all the answers

What is the derivative of a sum?

d/dx [f(x)] + d/dx [g(x)] Signup and view all the answers

What is the derivative of a difference?

d/dx [f(x)] - d/dx [g(x)] Signup and view all the answers

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

f'(x) = y' = dy/dx = df/dx = d/dx[f(x)] = Df(x) = D_xy Signup and view all the answers

What does the symbol d/dx mean?

y differentiated with respect to x Signup and view all the answers

Why is the equation of a tangent to a curve important?

It helps to find the gradient of a tangent to a curve at a point Signup and view all the answers

What is the limit 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 concept illustrated by Achilles and the Tortoise paradox?

The concept of limits 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 value of `y = (x^2 + 4x - 12)/(x + 6)` when x = -6?

undefined Signup and view all the answers

What is the main focus of differential calculus?

Solving optimization problems Signup and view all the answers

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

To determine the concavity of a function 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?

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

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

To identify the local maximum and local minimum of a function Signup and view all the answers

How do you find the y-intercept of a cubic function f(x) = ax^3 + bx^2 + cx + d?

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

What is the effect of a > 0 on the shape of a cubic graph y = ax^3 + bx^2 + cx + d?

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

What is the derivative of a function used for?

To find the gradient of the equation of a tangent line Signup and view all the answers

What is concavity, in the context of graphs?

The rate of change of the gradient of a curve Signup and view all the answers

What is the purpose of finding the normal to a curve?

To find the line perpendicular to the tangent line Signup and view all the answers

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

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

What is the purpose of finding the equation of a tangent line?

To find the gradient of a curve at a given point Signup and view all the answers

What is the formula for the nth term of an arithmetic sequence?

Tn = a + (n - 1)d Signup and view all the answers

What is the common characteristic of an arithmetic sequence?

The sequence forms a linear pattern when plotted on a graph Signup and view all the answers

How do you find the geometric mean between two numbers a and b?

Geometric Mean = √(ab) Signup and view all the answers

What is the formula for the nth term of a geometric sequence?

Tn = ar^(n-1) Signup and view all the answers

What is a series in mathematics?

The sum of the terms of a sequence Signup and view all the answers

What is the notation for the sum of the first n terms of a sequence?

S(n) = T1 + T2 + ... + Tn Signup and view all the answers

What is an infinite series?

The sum of infinitely many terms of a sequence Signup and view all the answers

How do you test if a sequence is geometric?

Verify if the ratio between consecutive terms is constant Signup and view all the answers

What is the general form of sigma notation?

∑(i=m to n) Ti = Tm + Tm+1 + ... + Tn-1 + Tn Signup and view all the answers

What happens when you plot the terms of a geometric sequence?

The points form an exponential curve Signup and view all the answers

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

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

What is the formula for the n-th term of an arithmetic sequence?

Tn = a + (n - 1)d Signup and view all the answers

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

To find the quadratic polynomial Signup and view all the answers

What is the common difference in an arithmetic sequence?

The difference between consecutive terms Signup and view all the answers

What is the graphical representation of an arithmetic sequence?

A straight line Signup and view all the answers

How do you test if a sequence is arithmetic?

By verifying if the difference between consecutive terms is constant Signup and view all the answers

What is the first step in solving a cubic equation using the Factor Theorem?

Find a factor by trial and error Signup and view all the answers

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

A factor is a root of the polynomial Signup and view all the answers

What is the arithmetic mean of two numbers?

The average of the two numbers Signup and view all the answers

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

To factorize the polynomial Signup and view all the answers

What is the notation used to denote the sum of terms in a sequence?

$\sum$ (sigma) Signup and view all the answers

What is the formula for the sum of a finite geometric series?

$S_n = rac{a(1 - r^n)}{1 - r}$ Signup and view all the answers

What is the condition for an infinite geometric series to converge?

$-1 < r < 1$ Signup and view all the answers

What is the formula for the sum of an infinite geometric series?

$S_\infty = a/(1 - r)$ Signup and view all the answers

What is the definition of an arithmetic sequence?

A sequence where each term is added by a constant value Signup and view all the answers

What is the formula for the nth term of an arithmetic sequence?

$T_n = a + (n - 1)d$ Signup and view all the answers

Who is credited with a method for finding the sum of an arithmetic series?

Karl Friedrich Gauss Signup and view all the answers

What is the sum of the first n terms of a sequence denoted as?

$S_n$ Signup and view all the answers

What is the term used to describe the constant value in an arithmetic sequence?

Common difference Signup and view all the answers

What is the term used to describe the constant value in a geometric sequence?

Common ratio Signup and view all the answers

What is the general formula for a finite arithmetic series?

$S_n = \frac{n}{2} (2a + (n - 1) d)$ Signup and view all the answers

What is the inverse of the function y = ax + q?

y = 1/ax - q/a Signup and view all the answers

What is the general form of the inverse of y = ax^2?

y = ±√(x/a) Signup and view all the answers

What is the definition of an exponent?

The number of times a base number is multiplied by itself Signup and view all the answers

What is the inverse of the function y = b^x?

y = log_b(x) Signup and view all the answers

What is the definition of a logarithm?

The exponent to which a base must be raised to yield a given value Signup and view all the answers

What is the shape of the graph of the exponential function y = b^x?

Increasing or decreasing Signup and view all the answers

What is the domain of the logarithmic function y = log_b(x)?

All positive real numbers Signup and view all the answers

What is the result of applying the product rule of logarithms to log_a(xy)?

log_a(x) + log_a(y) Signup and view all the answers

What is the result of applying the power rule of logarithms to log_a(x^b)?

b × log_a(x) Signup and view all the answers

What is the change of base formula for logarithms?

log_a(x) = log_b(x) / log_b(a) Signup and view all the answers

What is the graph of the inverse of an exponential function?

A reflection of the original function about the line y = x Signup and view all the answers

What is the domain of the logarithmic function f(x) = log x?

x > 0 Signup and view all the answers

What is an application of logarithms in real-life scenarios?

Calculating the pH levels of a solution Signup and view all the answers

What is the asymptote of the exponential function f(x) = 10^x?

y = 0 Signup and view all the answers

What is the formula to calculate the population growth, given a constant rate?

A = P(1 + i)^n Signup and view all the answers

What is a point of inflection?

The point where the graph changes concavity. Signup and view all the answers

Which method is NOT typically used for dividing polynomials?

Polynomial Interpolation Signup and view all the answers

Which equation represents the division of a polynomial by another polynomial?

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

To find the x-intercepts of a cubic polynomial, which equation should be solved?

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

What does the Remainder Theorem state regarding a polynomial and a linear divisor?

The remainder is found by evaluating the polynomial at the root of the divisor. Signup and view all the answers

Which of the following measures the instantaneous rate of change of a function?

The derivative Signup and view all the answers

In cubic polynomial functions, the nature of the leading coefficient 'a' determines which characteristic?

The end behavior of the graph Signup and view all the answers

What is the role of stationary points in the context of graph sketching?

They help determine local maxima and minima. Signup and view all the answers

Which formula can be used for finding the coefficient of a cubic polynomial through synthetic division?

R = a_0 + q_0 imes d/c Signup and view all the answers

Why is the concept of limits essential in calculus?

To understand the behavior of functions as the input values approach a specific point Signup and view all the answers

What does the graph of the function y = (x^2 + 4x - 12)/(x + 6) represent?

A straight line with a hole at x = -6 Signup and view all the answers

What is the significance of Zeno's paradox involving Achilles and the tortoise?

It highlights the concept of limits in calculus Signup and view all the answers

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

Because the denominator is zero at x = -6 Signup and view all the answers

What is the main focus of differential calculus?

To optimize functions and find rates of change Signup and view all the answers

What do we examine when finding the limit of a function?

The behavior of the function as the input values approach a specific point Signup and view all the answers

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

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

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

To find the roots of a quadratic equation Signup and view all the answers

What is the general form of an arithmetic sequence?

Tn = a + (n - 1)d Signup and view all the answers

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

If a polynomial has a root, then it must have a corresponding factor Signup and view all the answers

What is the first step in solving a cubic equation using the Factor Theorem?

Identify a factor by trial and error Signup and view all the answers

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

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

What is the graphical representation of an arithmetic sequence?

A straight line Signup and view all the answers

What is the formula to find the nth term of an arithmetic sequence?

Tn = a + (n - 1)d Signup and view all the answers

How do you test if a sequence is arithmetic?

By finding the common difference Signup and view all the answers

What is the common characteristic of a geometric sequence?

Each term is multiplied by a constant value Signup and view all the answers

What is the common characteristic of an arithmetic sequence?

Each term is the sum of the previous term and a constant Signup and view all the answers

What is the formula to find the nth term of a geometric sequence?

Tn = ar^(n-1) Signup and view all the answers

What is the purpose of sigma notation?

To represent the sum of terms in a sequence Signup and view all the answers

What is the arithmetic mean of two numbers?

The average of the two numbers Signup and view all the answers

What is the general form of sigma notation?

∑(i=m to n) Ti = Tm + T(m+1) + ... + T(n-1) + Tn Signup and view all the answers

What is the difference between a finite series and an infinite series?

A finite series has a fixed number of terms, while an infinite series has an infinite number of terms Signup and view all the answers

What is the graphical representation of a geometric sequence?

An exponential graph Signup and view all the answers

How do you test if a sequence is geometric?

By checking if the ratio between consecutive terms is constant Signup and view all the answers

What is the geometric mean between two numbers a and b?

√(ab) Signup and view all the answers

What is the importance of finding the common ratio in a geometric sequence?

To find the nth term of the sequence 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 of the tangent and the normal is -1. Signup and view all the answers

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

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

Which of the following steps is NOT involved in drawing the graph of a cubic polynomial?

Find the points of inflection by solving (f'(x) = 0) Signup and view all the answers

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

To find the x-intercepts of the polynomial. Signup and view all the answers

What does the term "point of inflection" represent in a cubic graph?

The point where the graph changes concavity. Signup and view all the answers

Which of the following methods is NOT used for factorising cubic polynomials?

Quadratic formula Signup and view all the answers

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

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

What does the sign of (a) in the cubic polynomial (f(x) = ax^3 + bx^2 + cx + d) tell us about the graph?

The end behavior of the graph. Signup and view all the answers

If the first derivative of a cubic function is zero at (x = 2), what can you conclude?

The function has a turning point at (x = 2). Signup and view all the answers

What is the general form of the formula for synthetic division of the polynomial (a(x) = a_3x^3 + a_2x^2 + a_1x + a_0) by (cx - d)?

(q_2 = a_3, q_1 = a_2 + q_2 \cdot (d/c), q_0 = a_1 + q_1 \cdot (d/c), R = a_0 + q_0 \cdot (d/c)) Signup and view all the answers

Which of the following is NOT a method for solving cubic equations?

Quadratic Formula Signup and view all the answers

What is the inverse of the function y = ax^2?

y = ±√(x/a) Signup and view all the answers

What is the definition of the logarithm of a number x with base b?

The exponent to which b must be raised to yield x Signup and view all the answers

What is the shape of the graph of the logarithmic function y = log_b x?

Increasing and concave down Signup and view all the answers

What is the product rule of logarithms?

log_a(xy) = log_a x + log_a y Signup and view all the answers

What is the inverse of the function y = b^x?

y = log_b x Signup and view all the answers

What is the domain of the inverse of the function y = ax^2?

x ≥ 0 if a > 0, x ≤ 0 if a < 0 Signup and view all the answers

What is the range of the exponential function y = b^x?

y > 0 Signup and view all the answers

What is the change of base formula for logarithms?

log_a x = log_b x / log_b a Signup and view all the answers

What is the value of log_a 1?

0 Signup and view all the answers

What is the graph of the exponential function y = b^x like?

A curve that increases rapidly Signup and view all the answers

What is the general formula for the sum of the first (n) terms of a geometric series?

(S_n = rac{a(r^n - 1)}{r - 1}) Signup and view all the answers

Which of the following is NOT a requirement for the sum of an infinite geometric series to exist (i.e., to converge)?

The common ratio (r) must be greater than 0. Signup and view all the answers

Which of the following sequences is NOT an arithmetic sequence?

4, 8, 12, 16, 20, ... Signup and view all the answers

What is the sum of the first 5 terms of the geometric series (2 + 6 + 18 + 54 + \cdots)?

242 Signup and view all the answers

In the general form of sigma notation, what does the index of summation represent?

The position of each term in the sequence. Signup and view all the answers

What is the formula for the nth term of the arithmetic sequence (1, 4, 7, 10, 13, ...)?

(T_n = 1 + 3(n - 1)) Signup and view all the answers

If an infinite geometric series converges, what can we say about its common ratio (r)?

(|r| < 1) Signup and view all the answers

Which of the following is the correct formula for the sum of the first (n) terms of an arithmetic series?

(S_n = rac{n}{2}(a + l)) Signup and view all the answers

Which of these options best describes the use of Karl Friedrich Gauss's method for calculating the sum of the first 100 positive integers?

It involves pairing the integers and finding their average. Signup and view all the answers

What is the formula for the sum of an infinite geometric series that converges?

(S_\infty = rac{a}{1 - r}) Signup and view all the answers

What does the derivative of a function indicate about the function at a specific point?

The rate of change of the function's values. Signup and view all the answers

Which of the following notations represents the derivative of a function f(x)?

D_xf Signup and view all the answers

Which statement is true regarding the differentiation rules?

The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. Signup and view all the answers

What is the limit expression that defines the derivative of a function?

lim (f(a + h) - f(a)) / h as h approaches 0 Signup and view all the answers

When should one use differentiation from first principles?

When derivatives are explicitly requested. Signup and view all the answers

Which of the following correctly applies the derivative of a constant multiplied by a function?

If k is a constant and f(x) is a function, then d(kf(x))/dx = kf'(x). Signup and view all the answers

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

The slope of a curve at a specific point. Signup and view all the answers

What is represented by the notation $rac{dy}{dx}$?

The derivative of y with respect to x. Signup and view all the answers

Which of the following is NOT a concept related to differentiation?

Calculating the area under a curve. Signup and view all the answers

What establishes the relationship between the gradient of a curve and the gradient of its tangent?

The gradient of the tangent equals the derivative of the curve at that point. Signup and view all the answers

If a population doubles in size after 5 years, what is the annual growth rate (rounded to the nearest percent)?

14% Signup and view all the answers

What is the pH level of a solution with a hydrogen ion concentration of ( 10^{-7} ) moles per liter?

7 Signup and view all the answers

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

( f^{-1}(x) = \log_2 x ) Signup and view all the answers

Which of the following statements is NOT true about the graph of ( f(x) = \log x )?

The graph is symmetric about the y-axis. Signup and view all the answers

A radioactive substance decays to half its original amount in 10 years. What is the decay rate (rounded to the nearest percent)?

6.9% Signup and view all the answers

What is the formula for the sum of the first n terms of an arithmetic series when the last term is known?

$S_n = \frac{n}{2}(a + l)$ Signup and view all the answers

Which statement is true about the properties of functions and relations?

Functions strictly require one output for each input. Signup and view all the answers

What is necessary for a function to have an inverse that is also a function?

It must be one-to-one. Signup and view all the answers

To find the inverse of a linear function defined by $y = ax + q$, which of the following steps is first?

Interchange x and y. Signup and view all the answers

In the context of inverse functions, what does the notation $f^{-1}(x)$ represent?

The inverse function of f(x). Signup and view all the answers

If a function fails the horizontal line test, what can be concluded about its inverse?

The inverse cannot be uniquely defined. Signup and view all the answers

What does the derived formula for the sum of the first n terms of an arithmetic series, $S_n = \frac{n}{2}(2a + (n - 1)d)$, represent?

The sum of terms in an arithmetic sequence. Signup and view all the answers

What is the significance of interchanging x and y when finding the inverse of a function?

It allows the output to become the new input. Signup and view all the answers

How is a one-to-one function graphically represented?

Every horizontal line intersects the graph at most once. Signup and view all the answers

Which of the following statements is true concerning many-to-one functions?

Every element in the range is connected to at least one element of the domain. Signup and view all the answers

What happens to the function $y = \frac{x^2 + 4x - 12}{x + 6}$ as $x$ approaches -6?

The function approaches -8. Signup and view all the answers

What concept does the Achilles and the Tortoise paradox illustrate?

The concept of limits in calculus. Signup and view all the answers

Which statement is true regarding the function $y = \frac{x^2 + 4x - 12}{x + 6}$?

It has a hole at $x = -6$. Signup and view all the answers

What is the simplified form of the function when $x \neq -6$?

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

What is the graphical representation of $y = \frac{x^2 + 4x - 12}{x + 6}$?

A straight line with a hole at $x = -6$. Signup and view all the answers

What is the key focus of differential calculus as introduced?

The optimization of functions and rates of change. 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?

They are multiplicative inverses. Signup and view all the answers

Which of the following is NOT a notation for the second derivative of a function f(x)?

f'(x) Signup and view all the answers

If a cubic function has a positive leading coefficient (a > 0), what is the general shape of its graph?

Rises to the right and falls to the left. Signup and view all the answers

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

To identify points where the function changes from increasing to decreasing or vice versa. Signup and view all the answers

How do you find the y-intercept of a cubic function f(x) = ax^3 + bx^2 + cx + d?

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

What is the relationship between the derivative of a function and the tangent line to its graph at a given point?

The derivative is the slope of the tangent line. Signup and view all the answers

If the second derivative of a function is positive, what does this indicate about the graph of the original function?

The graph is concave up. Signup and view all the answers

Which of the following is a step involved in finding the equation of a tangent line to a curve at a given point?

Calculate the y-value at the given point. Signup and view all the answers

What is the purpose of finding the normal to a curve?

To find a line perpendicular to the tangent at a given point. Signup and view all the answers

Which of the following is NOT a use of the derivative?

Determining the area under a curve. Signup and view all the answers

What indicates a point of inflection in a function's graph?

Where the second derivative changes from positive to negative or vice versa Signup and view all the answers

Which method is NOT typically used for factorizing cubic polynomials?

Completing the square Signup and view all the answers

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

By solving the equation for x = 0 Signup and view all the answers

What is the remainder when dividing a polynomial p(x) by a linear polynomial cx - d using the Remainder Theorem?

The value of p at $x = rac{d}{c}$ Signup and view all the answers

Which aspect is NOT determined when sketching a cubic graph?

The absolute maximum value Signup and view all the answers

What do the stationary points of a function indicate?

Points where the derivative is zero Signup and view all the answers

Which property is essential for a polynomial's quotient when dividing by a linear polynomial?

It is a polynomial of one degree less than the original polynomial Signup and view all the answers

Which of the following methods can be used to determine points of inflection?

Setting the second derivative to zero and checking for sign change Signup and view all the answers

What is indicated by the shape of the cubic graph when the coefficient a is negative?

The graph opens downwards Signup and view all the answers

Which statement about the average and instantaneous rates of change is true?

Instantaneous rate of change is the derivative at a point Signup and view all the answers

What confirms that a polynomial ( p(x) ) has a factor ( cx - d )?

( p\left( \frac{d}{c} \right) = 0 ) Signup and view all the answers

Which of the following describes the first step to solve cubic equations using the Factor Theorem?

Substitute potential roots into the polynomial. Signup and view all the answers

How can a polynomial ( p(x) ) be expressed once a factor ( cx - d ) is found?

( p(x) = (cx - d) \cdot Q(x) ) Signup and view all the answers

What is the general formula for finding the n-th term ( T_n ) of an arithmetic sequence?

( T_n = a + (n - 1)d ) Signup and view all the answers

What indicates that a sequence is arithmetic?

The difference between consecutive terms is constant. Signup and view all the answers

What is necessary to use the Quadratic Formula on a cubic polynomial after the factor is found?

Obtain a quadratic polynomial by dividing the cubic by the factor. Signup and view all the answers

Which formula is used to calculate the arithmetic mean of two numbers?

( ext{Arithmetic Mean} = \frac{ ext{First Term} + ext{Second Term}}{2} ) Signup and view all the answers

What is the outcome when you substitute ( x = \frac{d}{c} ) into a polynomial ( p(x) ) where ( cx - d ) is a factor?

The result is zero. Signup and view all the answers

What does the gradient of the line representing an arithmetic sequence indicate?

It represents the common difference of the sequence. Signup and view all the answers

What is the range of the exponential function defined by $f(x) = 10^x$?

y > 0 Signup and view all the answers

Which of the following describes the domain of the logarithmic function $f^{-1}(x) = ext{log } x$?

x > 0 Signup and view all the answers

In the formula for population growth, what does the variable 'n' represent?

Number of growth periods Signup and view all the answers

What is the y-intercept of the exponential function $f(x) = 10^x$?

(0, 1) Signup and view all the answers

For a logarithmic function, what is the asymptote?

x = 0 Signup and view all the answers

How is the common ratio in a geometric sequence determined?

By calculating the ratio between any two consecutive terms. Signup and view all the answers

What characteristic does a geometric sequence exhibit when the common ratio is greater than 1?

The sequence grows exponentially. Signup and view all the answers

What is the formula for finding the $n$-th term of an arithmetic sequence?

$T_n = a + (n - 1)d$ Signup and view all the answers

Which of the following sums describes a finite series?

$S_n = T_1 + T_2 + ext{up to } T_n$ Signup and view all the answers

When plotted on a graph, what does a geometric sequence represent?

An exponential curve. Signup and view all the answers

What happens to the terms of a geometric sequence when the common ratio is negative?

The terms alternate in sign. Signup and view all the answers

Which expression represents the geometric mean of two numbers $a$ and $b$?

$ ext{Geometric Mean} = ext{sqrt}(ab)$ Signup and view all the answers

In sigma notation, what does the symbol $ ext{Σ}$ represent?

The summation of a sequence of terms. Signup and view all the answers

To verify if a sequence is arithmetic, what should be checked?

If the difference between all consecutive terms is constant. Signup and view all the answers

What is depicted when plotting the terms of an arithmetic sequence?

A straight line with a slope. 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 derivative of the function `f(x) = 5`?

0 Signup and view all the answers

What is the derivative of the function `f(x) = 4x^3 - 2x^2 + 7x`?

12x^2 - 4x + 7 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 = 1`?

5 Signup and view all the answers

What is the derivative of the function `f(x) = (x^2 + 1)(x - 2)`?

3x^2 - 4x + 1 Signup and view all the answers

What is the derivative of the function `f(x) = \frac{1}{x^2}`?

-2/x^3 Signup and view all the answers

Which of the following is NOT a valid notation for the derivative of a function `y = f(x)`?

d/dx[f(x)] Signup and view all the answers

What is the equation of the tangent to the curve `y = x^3 - 2x + 1` at the point `(1, 0)`?

y = x - 1 Signup and view all the answers

Which of the following statements is TRUE about the derivative of a function at a point?

The derivative represents the instantaneous rate of change of the function at that point. Signup and view all the answers

What is the derivative of the function `f(x) = \sqrt{x}`?

1/(2\sqrt{x}) Signup and view all the answers

What is the correct formula for the sum of the first n terms of an arithmetic series?

$S_n = rac{n}{2}(2a + (n - 1)d)$ Signup and view all the answers

In a one-to-one function, which of the following statements is true?

Each element in the domain maps to a unique element in the range. Signup and view all the answers

Which of the following is necessary for a function to have an inverse that is also a function?

The function must be one-to-one. Signup and view all the answers

How do you find the inverse of a linear function given as $y = ax + q$?

Interchange x and y, then solve for y: $x = ay + q$. Signup and view all the answers

What is true about the graph of an inverse function?

It is the reflection of the original function's graph across the line $y = x$. Signup and view all the answers

For the equation $f(x) = 2x + 3$, what is its inverse function?

$f^{-1}(x) = rac{x - 3}{2}$ Signup and view all the answers

Which of the following statements about arithmetic series is true?

The sum can be found using $S_n = rac{n}{2}(a + l)$. Signup and view all the answers

What is represented by the notation $f^{-1}(x)$?

The inverse function of f(x). Signup and view all the answers

In which scenario does a horizontal line test indicate that a function is one-to-one?

If the horizontal line intersects the graph at most once. Signup and view all the answers

What is the correct expression for the inverse of the linear function given by the equation $y = ax + q$?

$f^{-1}(x) = rac{1}{a}x - rac{q}{a}$ Signup and view all the answers

What is the restriction on the domain when finding the inverse of the quadratic function $y = ax^2$?

$x < 0$ if $a > 0$ Signup and view all the answers

Which statement is true regarding the graph of the exponential function $f(x) = b^x$ when $b > 1$?

The graph is increasing and always remains above the x-axis. Signup and view all the answers

If $y = b^x$, what is the correct form for expressing its inverse?

$y = ext{log}_b x$ Signup and view all the answers

For the logarithmic function $y = ext{log}_b x$, which of the following is true regarding its domain?

$x ext{ must be greater than zero}$ Signup and view all the answers

What does the product rule of logarithms state?

$ ext{log}_a(xy) = ext{log}_a x + ext{log}_a y$ Signup and view all the answers

What is the shape of the graph of a logarithmic function $y = ext{log}_b x$?

It is increasing and approaches negative infinity as x approaches zero. Signup and view all the answers

If $x = b^y$, which of the following describes the relationship of $b$, $x$, and $y$?

$x$ is the output of raising $b$ to the power of $y$. Signup and view all the answers

What is the range of the exponential function $f(x) = b^x$ when $b > 0$?

$(0, ext{inf})$ Signup and view all the answers

What is the formula for the sum of the first n terms of a finite geometric series where the common ratio (r) is greater than 1?

$S_n = \frac{a(r^n - 1)}{r - 1}$ Signup and view all the answers

What is the condition for an infinite geometric series to converge?

The common ratio (r) must be between -1 and 1 (exclusive). Signup and view all the answers

What is the formula for the sum of an infinite geometric series that converges?

$S_\infty = \frac{a}{1 - r}$ Signup and view all the answers

What is the general formula for a finite arithmetic series, where a is the first term, d is the common difference, and n is the number of terms?

$S_n = \frac{n}{2} [a + (n - 1)d]$ Signup and view all the answers

What is the formula for the nth term of a geometric sequence with the first term 'a' and the common ratio 'r'?

$T_n = a * r^{n-1}$ Signup and view all the answers

What is the sum of the first 100 natural numbers, using the method of Gauss?

5050 Signup and view all the answers

What does the symbol 'Σ' represent in sigma notation?

The sum of terms in a sequence Signup and view all the answers

What is the general form of the sigma notation for the sum of the first n terms of a sequence?

$\sum_{i=1}^{n} T_i$ Signup and view all the answers

What is the common ratio (r) in a geometric sequence where the second term is 6 and the fifth term is 48?

2 Signup and view all the answers

Which of the following statements correctly describes the convergence of an infinite geometric series?

An infinite geometric series converges if the absolute value of the common ratio (r) is less than 1. Signup and view all the answers

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

$2x$ Signup and view all the answers

If y = f(x) and f'(x) = nx^(n-1), what is the value of n?

any real number Signup and view all the answers

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

$0$ Signup and view all the answers

What is the equation of the tangent to the curve y = x^2 at the point (1, 1)?

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

If f'(x) = 2x, what is the function f(x)?

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

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

$kf'(x)$ Signup and view all the answers

What is the derivative of the function f(x) = f(x) + g(x)?

$f'(x) + g'(x)$ Signup and view all the answers

What is the derivative of the function f(x) = f(x) - g(x)?

$f'(x) - g'(x)$ 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 process of finding the derivative of a function from first principles?

differentiation from first principles Signup and view all the answers

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

To determine the concavity of the function 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?

Their product is equal to -1 Signup and view all the answers

What is the effect of a > 0 on the shape of a cubic graph y = ax^3 + bx^2 + cx + d?

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

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

To identify local maxima and minima Signup and view all the answers

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

d^2y/dx^2 Signup and view all the answers

What is concavity, in the context of graphs?

The rate of change of the gradient Signup and view all the answers

What is the purpose of finding the equation of a tangent line?

To describe the rate of change at a specific point Signup and view all the answers

What is the process of finding the equation of a tangent line to a curve at a point?

Differentiate the function, substitute the x-coordinate, and use the point-slope form Signup and view all the answers

What is the purpose of finding the second derivative of a function in the context of sketching cubic graphs?

To determine the concavity of the graph Signup and view all the answers

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

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

What is the relationship between the second derivative and the concavity of a function?

If f''(x) > 0, the graph is concave up Signup and view all the answers

What is the purpose of finding the normal to a curve?

To find the perpendicular line to the curve at a specific point Signup and view all the answers

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

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

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

To find the remainder of the division Signup and view all the answers

What is the general form of a cubic polynomial?

ax^3 + bx^2 + cx + d Signup and view all the answers

What is the concept of optimisation problems in the context of differential calculus?

Finding the maximum or minimum of a function Signup and view all the answers

What is the purpose of finding the turning points of a cubic graph?

To determine the shape of the graph Signup and view all the answers

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

They are perpendicular Signup and view all the answers

What is the purpose of finding the point of inflection of a cubic graph?

To determine the concavity of the graph Signup and view all the answers

What is the application of differential calculus in solving optimisation problems?

Finding the maximum or minimum of a function Signup and view all the answers

If (p(x) = x^3 - 6x^2 + 11x - 6) and (cx - d = x - 1), what can be concluded about (p(x))?

(x - 1) is a factor of (p(x)) Signup and view all the answers

What is the relationship between the factor (cx - d) and the root of the polynomial (p(x))?

(cx - d) is a factor of (p(x)) if and only if (p(d/c) = 0) Signup and view all the answers

If a polynomial (p(x)) has a root at (x = rac{d}{c}), what can be concluded about the factorization of (p(x))?

(p(x) = (cx - d) \cdot Q(x)) where (Q(x)) is a quadratic polynomial Signup and view all the answers

What is the first step in solving a cubic equation using the Factor Theorem?

Identify a factor of the polynomial by trial and error Signup and view all the answers

What is the graphical representation of an arithmetic sequence?

A straight line Signup and view all the answers

What is the common characteristic of an arithmetic sequence?

The difference between consecutive terms is constant Signup and view all the answers

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

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

If a polynomial (p(x)) has a factor (cx - d), what can be concluded about the root of the polynomial?

The root is (x = d/c) Signup and view all the answers

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

A polynomial has a root at (x = a) if and only if (x - a) is a factor Signup and view all the answers

What is the common difference in an arithmetic sequence?

The difference between consecutive terms Signup and view all the answers

What is the formula for the sum of the first n terms of an arithmetic sequence?

$S_n = rac{n}{2}(a + l)$ Signup and view all the answers

What is a necessary condition for a function to have an inverse function?

The function must be one-to-one. Signup and view all the answers

What is the graph of the inverse function like, compared to the graph of the original function?

It is a reflection across the line y = x. Signup and view all the answers

What is the process of finding the inverse of a linear function?

Interchange x and y, solve for x, and then substitute back into the original function. Signup and view all the answers

What is the formula for the inverse of a linear function f(x) = ax + q?

$f^{-1}(x) = rac{1}{a}x - rac{q}{a}$ Signup and view all the answers

What is the purpose of the horizontal line test in determining if a function has an inverse?

To determine if the function is one-to-one. Signup and view all the answers

What is the key property of an inverse function?

It undoes the operation of the original function. Signup and view all the answers

What is the general formula for the sum of the first n terms of a finite geometric series?

$S_n = \frac{a(r^n - 1)}{r - 1}$ Signup and view all the answers

What is the condition for an infinite geometric series to converge?

-1 < r < 1 Signup and view all the answers

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

$f^{-1}(x)$ Signup and view all the answers

What is the relationship between the graphs of a function and its inverse?

They are symmetrical about the line y = x. Signup and view all the answers

What is the general formula for the sum of an infinite geometric series?

$S_\infty = \frac{a}{1 - r}$ Signup and view all the answers

What is the definition of an infinite geometric series?

A series with an infinite number of terms Signup and view all the answers

What is the formula for the nth term of a geometric sequence?

$T_n = a \cdot r^{n-1}$ Signup and view all the answers

What is the formula for the sum of a finite arithmetic series?

$S_n = \frac{n}{2}(a + l)$ Signup and view all the answers

What is the definition of a finite geometric series?

A series with a finite number of terms Signup and view all the answers

What is the inverse function of y = ax^2?

y = ±√(x/a) Signup and view all the answers

What is the notation used to denote the sum of terms in a sequence?

$\sum_{i=m}^{n} T_i$ Signup and view all the answers

What is the formula for the nth term of an arithmetic sequence?

$T_n = a + (n - 1)d$ Signup and view all the answers

What is the domain and range of the inverse function of y = ax^2?

Domain: ℝ, Range: ℝ+ Signup and view all the answers

What is the purpose of Karl Friedrich Gauss's method for finding the sum of an arithmetic series?

To find the sum of an arithmetic series Signup and view all the answers

What is the logarithmic function equivalent to the exponential function y = 2^x?

y = log₂x Signup and view all the answers

What is the property of logarithms that states loga(xy) = loga(x) + loga(y)?

Product Rule Signup and view all the answers

What is the graph of the exponential function y = b^x like?

It rises rapidly if b > 1 and falls rapidly if 0 Signup and view all the answers

What is the inverse function of y = ax + q?

y = x/a - q/a Signup and view all the answers

What is the logarithmic value of loga(a) equal to?

1 Signup and view all the answers

What is the purpose of finding the inverse of a function?

To solve equations involving the original function Signup and view all the answers

What is the general form of the inverse function of y = ax^2?

y = ±√(x/a) Signup and view all the answers

What is the special logarithmic value of loga(1) equal to?

0 Signup and view all the answers

A population of bacteria doubles every 3 hours. Assuming exponential growth, how long will it take for the population to increase 16 times its initial size?

48 hours Signup and view all the answers

A radioactive substance has a half-life of 10 days. After 30 days, what fraction of the original substance remains?

1/8 Signup and view all the answers

The pH of a solution is 4. What is the concentration of hydrogen ions, [H+], in the solution?

10^-4 M Signup and view all the answers

A loan of $10,000 is taken out at an annual interest rate of 5%, compounded monthly. If the loan is to be repaid in 5 years, what is the monthly payment amount (rounded to the nearest dollar)?

$189 Signup and view all the answers

If the graph of an exponential function (f(x) = a^x) passes through the point ((2, 9)), what is the value of (a)?

3 Signup and view all the answers

What happens to the sequence when the common ratio, r, in a geometric sequence is less than 1 but greater than 0?

The sequence decays exponentially. Signup and view all the answers

Which of the following describes the graphical representation of a geometric sequence?

An exponential curve. Signup and view all the answers

How can you verify that a sequence is arithmetic?

The differences between consecutive terms are constant. Signup and view all the answers

What is the purpose of sigma notation in mathematics?

To denote the sum of a series of terms in a compact form. Signup and view all the answers

What characterizes the growth behavior of a geometric sequence when the common ratio is negative?

The terms alternate in sign. Signup and view all the answers

In the context of finite series, what does the notation S_n specifically represent?

The sum of the first n terms of a sequence. Signup and view all the answers

Given the formula for the nth term of a geometric sequence, T_n = ar^{n-1}, how would you identify the first term?

It is represented by the symbol a. Signup and view all the answers

What is the geometric mean of two numbers a and b, as defined in mathematics?

The value that forms a geometric sequence between a and b. Signup and view all the answers

How would you define an infinite series in mathematical terms?

A series that sums all terms indefinitely. Signup and view all the answers

What condition must the common ratio r satisfy for a geometric series to converge?

r must be less than 1 in absolute value. Signup and view all the answers

The function `y = (x^2 + 4x - 12)/(x + 6)` has a discontinuity at x = -6. What is the nature of this discontinuity?

Removable discontinuity, because the function can be simplified to a continuous function for x ≠ -6 Signup and view all the answers

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

The paradox shows that even in situations where one object is constantly gaining on another, the limit of their distance can still be zero. Signup and view all the answers

The graph of the function `y = (x^2 + 4x - 12)/(x + 6)` is a straight line with a hole at x = -6. What does the existence of this hole indicate about the limit of the function as x approaches -6?

The limit exists and is equal to -8, even though the function is not defined at x = -6. Signup and view all the answers

Which of the following statements about the limit of the function `y = (x^2 + 4x - 12)/(x + 6)` as x approaches -6 is TRUE?

The limit exists and is equal to -8, because as x approaches -6, the function approaches -8. Signup and view all the answers

If the tortoise gets a head start in the race, Achilles appears to never overtake it. This is because:

The paradox highlights the concept of infinite divisibility and the fact that Achilles must repeatedly cover smaller and smaller distances to catch up. Signup and view all the answers

Why is the concept of limits important in calculus?

Limits are used to find the derivative of a function. Signup and view all the answers

Gr12 Mathematics: June Exam Mix P(1)

Questions and Answers

What is the process of determining the derivative of a given function called?

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

What is the general rule for differentiation?

What is the derivative of a constant?

What is the derivative of a constant multiplied by a function?

What is the derivative of a sum?

What is the derivative of a difference?

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

What does the symbol d/dx mean?

Why is the equation of a tangent to a curve important?

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

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

What is the concept illustrated by Achilles and the Tortoise paradox?

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

What is the value of y = (x^2 + 4x - 12)/(x + 6) when x = -6?

What is the main focus of differential calculus?

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

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

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

How do you find the y-intercept of a cubic function f(x) = ax^3 + bx^2 + cx + d?

What is the effect of a > 0 on the shape of a cubic graph y = ax^3 + bx^2 + cx + d?

What is the derivative of a function used for?

What is concavity, in the context of graphs?

What is the purpose of finding the normal to a curve?

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

What is the purpose of finding the equation of a tangent line?

What is the formula for the nth term of an arithmetic sequence?

What is the common characteristic of an arithmetic sequence?

How do you find the geometric mean between two numbers a and b?

What is the formula for the nth term of a geometric sequence?

What is a series in mathematics?

What is the notation for the sum of the first n terms of a sequence?

What is an infinite series?

How do you test if a sequence is geometric?

What is the general form of sigma notation?

What happens when you plot the terms of a geometric sequence?

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

What is the formula for the n-th term of an arithmetic sequence?

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

What is the common difference in an arithmetic sequence?

What is the graphical representation of an arithmetic sequence?

How do you test if a sequence is arithmetic?

What is the first step in solving a cubic equation using the Factor Theorem?

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

What is the arithmetic mean of two numbers?

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

What is the notation used to denote the sum of terms in a sequence?

What is the formula for the sum of a finite geometric series?

What is the condition for an infinite geometric series to converge?

What is the formula for the sum of an infinite geometric series?

What is the definition of an arithmetic sequence?

What is the formula for the nth term of an arithmetic sequence?

Who is credited with a method for finding the sum of an arithmetic series?

What is the sum of the first n terms of a sequence denoted as?

What is the term used to describe the constant value in an arithmetic sequence?

What is the term used to describe the constant value in a geometric sequence?

What is the general formula for a finite arithmetic series?

What is the inverse of the function y = ax + q?

What is the general form of the inverse of y = ax^2?

What is the definition of an exponent?

What is the inverse of the function y = b^x?

What is the definition of a logarithm?

What is the shape of the graph of the exponential function y = b^x?

What is the domain of the logarithmic function y = log_b(x)?

What is the result of applying the product rule of logarithms to log_a(xy)?

What is the result of applying the power rule of logarithms to log_a(x^b)?

What is the change of base formula for logarithms?

What is the graph of the inverse of an exponential function?

What is the domain of the logarithmic function f(x) = log x?

What is an application of logarithms in real-life scenarios?

What is the asymptote of the exponential function f(x) = 10^x?

What is the formula to calculate the population growth, given a constant rate?

What is a point of inflection?

Which method is NOT typically used for dividing polynomials?

Which equation represents the division of a polynomial by another polynomial?

To find the x-intercepts of a cubic polynomial, which equation should be solved?

What does the Remainder Theorem state regarding a polynomial and a linear divisor?

Which of the following measures the instantaneous rate of change of a function?

What is the limit of the function `y = (x^2 + 4x - 12)/(x + 6)` as x approaches -6?

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

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

What is the value of `y = (x^2 + 4x - 12)/(x + 6)` when x = -6?