Gr 12 Mathematics: November Mix P(1)

What is the definition of an arithmetic sequence?

A sequence of numbers in which each consecutive term is calculated by dividing a constant value

A sequence of numbers in which each consecutive term is calculated by adding a constant value (correct)

A sequence of numbers in which each consecutive term is calculated by subtracting a constant value

A sequence of numbers in which each consecutive term is calculated by multiplying a constant value

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

Tn = a + (n - 1)d (correct)

Tn = a + (n + 1)d

Tn = a × (n - 1)d

Tn = a - (n - 1)d

How do you find the common difference of an arithmetic sequence?

By subtracting consecutive terms (correct)

By dividing consecutive terms

By adding the first and last terms

By multiplying consecutive terms

What is the arithmetic mean between two numbers?

The average of the two numbers 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 effect of a positive common difference on an arithmetic sequence?

The sequence increases Signup and view all the answers

What is the pattern formed by the terms of an arithmetic sequence when plotted on a graph?

Linear Signup and view all the answers

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

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

What is the general formula for the n-th term of a geometric sequence?

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

What is the general form for a finite arithmetic series?

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

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

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

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

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

What does the common ratio, r, represent in a geometric sequence?

The constant value multiplied to each term to obtain the next term Signup and view all the answers

Which of the following is an example of a geometric sequence?

1, 2, 4, 8, 16, ... Signup and view all the answers

What is the common difference, d, in an arithmetic sequence?

The constant value added to each term to obtain the next term Signup and view all the answers

Which of the following is an example of an arithmetic sequence?

2, 4, 6, 8, 10, ... Signup and view all the answers

What is the formula for the sum of an arithmetic series?

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

What is a one-to-one function?

A function where each element in the domain maps to exactly one element in the range Signup and view all the answers

What is the graphical representation of a one-to-one function?

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

What is the inverse of a function?

A function that reverses the operation of a given function Signup and view all the answers

What is the formula for the sum of an arithmetic series when the last term is unknown?

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

What is a many-to-one function?

A function where multiple elements of the domain map to the same element of the range Signup and view all the answers

What is the graphical symmetry of the inverse function?

The graph of the inverse function is the reflection of the original function's graph across the line y = x Signup and view all the answers

What is the horizontal line test used for?

To determine if a function has an inverse that is also a function Signup and view all the answers

What is the formula for the inverse of a linear function?

$y = \frac{1}{a}x - \frac{q}{a}$ Signup and view all the answers

What is the notation $f^{-1}(x)$ used for?

To represent the inverse of a function Signup and view all the answers

What does the variable 'i' represent in the future value of an annuity formula?

The interest rate per period Signup and view all the answers

What is the formula for calculating the present value of an annuity?

PV = P \frac{1 - (1 + i)^{-n}}{i} Signup and view all the answers

Which of the following is NOT a characteristic of a future value annuity?

Payments are made at the beginning of each period Signup and view all the answers

What does the variable 'n' represent in the present value of an annuity formula?

The number of periods Signup and view all the answers

Which of the following is NOT a characteristic of a present value annuity?

The payments are made at the end of each period Signup and view all the answers

Which formula would you use to calculate the future value of an annuity where the payment amount is $500, the interest rate is 5%, and the number of periods is 10?

FV = 500 \frac{(1 + 0.05)^{10} - 1}{0.05} Signup and view all the answers

What is the formula for the present value of an annuity?

$P = x \left[ \frac{1 - (1 + i)^{-n}}{i} \right]$ Signup and view all the answers

What is the formula for calculating the period of an investment using compound interest?

$n = \frac{\log(\frac{A}{P})}{\log(1 + i)}$ Signup and view all the answers

Which of the following is the formula for simple interest?

$A = P(1 + in)$ Signup and view all the answers

What is the formula for calculating the total interest paid on a loan?

$I = T - P$ Signup and view all the answers

What is the formula for the future value of a series of payments?

$F = x \left[ \frac{(1 + i)^n - 1}{i} \right]$ Signup and view all the answers

What is the formula for calculating the effective annual rate (EAR)?

$EAR = \left(1 + \frac{i_{nominal}}{m} \right)^m - 1$ Signup and view all the answers

What is the formula for calculating the outstanding loan balance?

$P_{balance} = x \left[ \frac{1 - (1 + i)^{-n_{remaining}}}{i} \right]$ Signup and view all the answers

Which formula is used for calculating the payment amount for a future value annuity?

$x = \frac{F \cdot i}{(1 + i)^n - 1}$ Signup and view all the answers

What is the formula for calculating the total amount paid on a loan?

$T = n \times x$ Signup and view all the answers

What is the formula for compound depreciation?

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

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

(0, 1) Signup and view all the answers

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

All positive real numbers. Signup and view all the answers

Which formula is used to calculate the period of an investment in compound interest?

$n = rac{ ext{log }(A/P)}{ ext{log }(1 + i)}$ Signup and view all the answers

What type of interest is calculated on the principal and includes interest from previous periods?

Compound interest Signup and view all the answers

In financial calculations involving logarithms, which application involves determining the decay rate?

Radioactive decay Signup and view all the answers

What is the future value annuity concept primarily used for?

Accumulating a sum of money Signup and view all the answers

What is implied by the value of the pH level calculated using the formula $\text{pH} = -\log_{10}[\text{H}^+]$?

It increases as the concentration of hydrogen ions decreases. Signup and view all the answers

Which equation represents the formula for compounded depreciation?

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

How is the nominal interest rate defined in relation to the effective rate?

It indicates the rate compounded a specific number of times per year. Signup and view all the answers

In the context of logarithms, what does the term 'pH' measure?

Concentration of hydrogen ions 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

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

Find the value of f(0) Signup and view all the answers

What does the sign of the coefficient 'a' tell you about the shape of a cubic polynomial?

The sign of 'a' determines the end behavior of the graph Signup and view all the answers

What is the formula used in synthetic division for finding the quotient and remainder?

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

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

If f(k) = 0, then x - k is a factor of the polynomial Signup and view all the answers

What is the derivative of the function f(x) = ax^3 + bx^2 + cx + d?

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

What does it mean for a function to be concave up?

The graph opens upwards Signup and view all the answers

How do you find the x-intercepts of a cubic polynomial?

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

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

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

What does it mean for a function to be decreasing?

The derivative is negative Signup and view all the answers

What does the Remainder Theorem state about the remainder when a polynomial is divided by a linear polynomial?

The remainder is the value of the polynomial at the divisor's root. Signup and view all the answers

Which of these statements is not true regarding the Factor Theorem?

The Factor Theorem applies only to quadratic polynomials. Signup and view all the answers

When dividing a polynomial by a linear polynomial, what form does the polynomial take?

The polynomial can be expressed as a product of the divisor and a quotient plus a remainder. Signup and view all the answers

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

Identifying and substituting a root into the polynomial. Signup and view all the answers

For two mutually exclusive events, how is the addition rule simplified?

It eliminates the need to consider their intersection. Signup and view all the answers

Which of the following equations represents the Remainder Theorem?

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

Which polynomial expression represents the relationship between a polynomial, its divisor, and remainder?

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

What does the Quadratic Formula solve for in the context of cubic equations?

It helps in solving the quadratic part after factorization. Signup and view all the answers

What happens when you substitute $x = \frac{d}{c}$ into a polynomial and the result is zero?

The divisor is a factor of the polynomial. Signup and view all the answers

Which of the following accurately describes cubic equations?

They necessitate factoring and quadratic solutions. Signup and view all the answers

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

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

What is the common ratio of the geometric sequence 2, 6, 18, 54...?

3 Signup and view all the answers

What is the geometric mean between 4 and 9?

6 Signup and view all the answers

In a geometric sequence, if the common ratio is greater than 1, what happens to the terms as the sequence progresses?

The terms increase exponentially Signup and view all the answers

What is the sum of the first 5 terms of the geometric sequence 1, 2, 4, 8...?

31 Signup and view all the answers

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

∑_(i=1)^n T_i Signup and view all the answers

Which of the following statements about infinite series is true?

An infinite geometric series always converges if the common ratio is less than 1. Signup and view all the answers

What is the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ...?

2 Signup and view all the answers

Which of the following is an example of a geometric sequence?

1, 2, 4, 8, ... Signup and view all the answers

Which of the following statements accurately describes the graph of a geometric sequence?

An exponential graph with discrete points Signup and view all the answers

What is the probability of either event A or event B occurring if they are mutually exclusive?

P(A) + P(B) Signup and view all the answers

What is the complementary rule used to calculate the probability of event A not occurring?

P(not A) = 1 - P(A) Signup and view all the answers

How are two-way contingency tables helpful in probability problems?

They help determine if events are dependent or independent. Signup and view all the answers

According to the Fundamental Counting Principle, if there are n outcomes in event A and m outcomes in event B, how many total outcomes exist for the combination of both events?

n × m Signup and view all the answers

Which notation represents the factorial of an integer n?

n! Signup and view all the answers

If event A has a probability of 0.3 and event B has a probability of 0.4, what is the probability that either event A or B occurs if they are not mutually exclusive?

0.3 + 0.4 Signup and view all the answers

What is the outcome of 0! in factorial notation?

1 Signup and view all the answers

How is the probability of independent events A and B calculated?

P(A) × P(B) Signup and view all the answers

In probability, what does the symbol A ∩ B represent?

The intersection of events A and B Signup and view all the answers

What is the inverse function of the linear function defined by $y = ax + q$?

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

In the function $y = ax^2$, what must be done to find its inverse?

Interchange x and y, then solve for y Signup and view all the answers

What restriction must be applied to the function $y = ax^2$ to ensure its inverse is a function?

Range must be positive Signup and view all the answers

In which case is the function $y = b^x$ described as increasing?

$b > 1$ Signup and view all the answers

What does the inverse of the exponential function $y = b^x$ express?

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

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

$x > 0$ Signup and view all the answers

What is the vertical asymptote for the graph of the logarithmic function?

$x = 0$ Signup and view all the answers

What is a property of logarithms that states $ ext{log}_a (xy) = ext{log}_a x + ext{log}_a y$?

Product Rule Signup and view all the answers

Which of the following is true for the exponential function $y = b^x$ if $0 < b < 1$?

The graph falls rapidly Signup and view all the answers

Under which condition can the inverse of a quadratic function be expressed as a standard function?

If the domain is restricted to make it one-to-one Signup and view all the answers

What does the complementary rule state about event A?

P(not A) = 1 - P(A) Signup and view all the answers

Which condition indicates that events A and B are mutually exclusive?

P(A and B) = 0 Signup and view all the answers

How is the addition rule for probabilities expressed?

P(A or B) = P(A) + P(B) - P(A and B) Signup and view all the answers

Which statement about independent events is correct?

P(A and B) = P(A) * P(B) Signup and view all the answers

What is the result of P(A or not A)?

1 Signup and view all the answers

Which rule applies to the union of two mutually exclusive events?

P(A or B) = P(A) + P(B) Signup and view all the answers

What does a tree diagram help to visualize?

The outcomes of a sequence of events Signup and view all the answers

Which of the following describes the relationship between independent events A and B?

P(A and B) = P(A) * P(B) Signup and view all the answers

What is true about mutually exclusive events?

They do not overlap Signup and view all the answers

Which is a characteristic of complementary events?

P(A) + P(A') = 1 Signup and view all the answers

What does the notation $f'(x)$ represent?

The first derivative of the function $f(x)$ Signup and view all the answers

When differentiating, what does $rac{dy}{dx}$ signify?

The change in $y$ over the change in $x$ Signup and view all the answers

Which equation is used to find the gradient of the tangent to a curve?

Finding the derivative of the function at a specific point Signup and view all the answers

What do the signs of the second derivative indicate?

The direction of the curve's gradient changes Signup and view all the answers

How do we 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

The relationship between the gradients of the tangent and the normal can be expressed as what?

$m_{ ext{tangent}} imes m_{ ext{normal}} = -1$ Signup and view all the answers

Which operation do differential operators perform?

Differentiation of functions Signup and view all the answers

To find the equation of the tangent line, what is the last step after determining the gradient?

Use the point-slope form of the line equation Signup and view all the answers

What is denoted by $y''$?

The second derivative of $y$ Signup and view all the answers

What can the derivative of a function be used to identify?

The rate of change and stationary points of a graph 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 gradient of the tangent to the curve ( y = f(x) ) at the point ( x = a )?

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

What is the derivative of ( f(x) )?

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

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

( \frac{df}{dx} ) Signup and view all the answers

What is the derivative of ( x^n ) with respect to ( x )?

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

What is the derivative of a constant ( k ) with respect to ( x )?

( 0 ) Signup and view all the answers

What is the derivative of ( k \cdot f(x) ) with respect to ( x )?

( k \cdot f'(x) ) Signup and view all the answers

What is the derivative of ( f(x) + g(x) ) with respect to ( x )?

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

What is the derivative of ( f(x) - g(x) ) with respect to ( x )?

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

What is the formula used to determine the period (n) of an investment when using the compound interest formula?

$n = \frac{\log(\frac{A}{P})}{\log(1 + i)}$ Signup and view all the answers

What is the primary purpose of a present value annuity?

To pay off a loan or debt over time with regular payments Signup and view all the answers

Which of the following formulas is used to calculate simple depreciation?

$A = P(1 - in)$ Signup and view all the answers

Which of the following is a characteristic of the exponential function (f(x) = 10^x)?

The function has a horizontal asymptote at y = 0 Signup and view all the answers

Which of the following scenarios would NOT be a suitable application of logarithms?

Calculating the amount of interest earned on a savings account Signup and view all the answers

What does the variable 'i' represent in the future value of an annuity formula?

The interest rate per period Signup and view all the answers

Which of the following formulas represents the relationship between nominal and effective interest rates?

$1 + i = \left(1 + \frac{i_{(m)}}{m}\right)^m$ Signup and view all the answers

Which of the following is a key difference between simple interest and compound interest?

Simple interest is calculated on the principal only, while compound interest is calculated on the principal and accumulated interest. Signup and view all the answers

Which of the following is NOT a common application of annuities?

Simple interest calculations Signup and view all the answers

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

The graphs are reflections of each other across the line y = x. Signup and view all the answers

What is the primary characteristic of a geometric sequence?

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

Which formula represents the sum of the first n terms of a finite series?

$S_n = T_1 + T_2 + T_3 + ext{...} + T_n$ Signup and view all the answers

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

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

What is the formula for calculating the total interest paid on a loan?

$T - P$ Signup and view all the answers

When is a geometric series considered to be convergent?

When the common ratio is less than one in absolute value. Signup and view all the answers

In sigma notation, what does the symbol $orall$ represent?

The index of summation Signup and view all the answers

Which formula would you use to calculate the period of an investment using compound interest?

$n = rac{\log\left(rac{A}{P} ight)}{\log(1 + i)}$ Signup and view all the answers

What is the formula for calculating the outstanding loan balance?

$P_{ ext{balance}} = x \left[rac{1 - (1 + i)^{-n_{ ext{remaining}}}}{i} ight]$ Signup and view all the answers

What represents the common ratio in a geometric sequence?

The ratio of any term to its preceding term. Signup and view all the answers

What does the expression $T_n = ar^{n-1}$ calculate in a geometric sequence?

The nth term of the series. Signup and view all the answers

What is the formula for the future value of a series of payments?

$F = x \left[rac{(1 + i)^n - 1}{i} ight]$ Signup and view all the answers

If the common ratio r of a geometric sequence is 0.5, what can be said about the sequence?

The sequence decays exponentially. Signup and view all the answers

What is the formula for calculating the payment amount for a future value annuity?

$x = rac{F \cdot i}{(1 + i)^n - 1}$ Signup and view all the answers

How do you verify if a sequence is geometric?

Verify if ratios between consecutive terms remain equal. Signup and view all the answers

What is the formula for calculating the total amount paid on a loan?

$T = n imes x$ Signup and view all the answers

Which of the following is the formula for simple interest?

$A = P(1 + in)$ Signup and view all the answers

What type of graph is formed by plotting the terms of a geometric sequence?

Exponential graph Signup and view all the answers

Which formula is used to calculate the period of an investment in compound interest?

$n = rac{\log\left(rac{A}{P} ight)}{\log(1 + i)}$ Signup and view all the answers

Which formula is used for calculating the effective annual rate (EAR)?

$ ext{EAR} = \left(1 + rac{i_{ ext{nominal}}}{m} ight)^m - 1$ Signup and view all the answers

What type of interest is calculated on the principal and includes interest from previous periods?

Compound Interest Signup and view all the answers

What is the formula used to calculate the sum of the first n terms of a finite geometric series?

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

Under what condition does an infinite geometric series converge?

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

What is the common ratio in a geometric sequence defined as $T_n = a \cdot r^{n-1}$?

The constant factor between terms Signup and view all the answers

Using Karl Friedrich Gauss's method, what is the sum of the first 100 positive integers?

5050 Signup and view all the answers

In a finite arithmetic series, what do the variables 'a' and 'd' represent?

The first term and common difference, respectively Signup and view all the answers

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

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

What happens to the series if the common ratio r is greater than or equal to 1 in an infinite geometric series?

The series diverges Signup and view all the answers

What does the variable ( P ) represent in both the future value and present value annuity formulas?

Payment amount per period Signup and view all the answers

What simplifies the calculation of the sum of the first n terms of an arithmetic series?

The average of the first and last terms multiplied by n Signup and view all the answers

What does the index of summation i indicate in the general form of a series?

The variable term that adjusts the summation Signup and view all the answers

How is the total value of the future value annuity calculated at the end of the investment period?

Sum of all payments plus interest earned Signup and view all the answers

What is the term used to describe the difference between consecutive terms in an arithmetic sequence?

Common difference Signup and view all the answers

What is the correct application of the present value annuity formula?

To assess loan repayment amounts Signup and view all the answers

Which of the following statements about the future value of an annuity is accurate?

It accounts for future payments and the interest earned Signup and view all the answers

In the present value annuity formula, what does the term ( i ) signify?

The interest rate per payment period Signup and view all the answers

Which formula would you use to calculate the future value of an annuity where the payment amount is ( x )?

( F = x \left[rac{(1 + i)^n - 1}{i} ight] ) Signup and view all the answers

What is the simplified value of $S_{100}$ calculated from the equation $2S_{100} = 10100$?

5050 Signup and view all the answers

Which formula represents the sum of a finite arithmetic series when the last term is unknown?

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

What characteristic must a relation possess for it to be classified as a function?

Each element in set A must map to exactly one element in set B. Signup and view all the answers

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

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

Which of the following describes the graphical representation of inverse functions?

The graphs are symmetrical about the line $y = x$. Signup and view all the answers

What does the horizontal line test determine?

If a function is one-to-one. Signup and view all the answers

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

Interchange $x$ and $y$, and solve for $y$. Signup and view all the answers

What is the geometric representation of a many-to-one function?

Every vertical line can intersect the graph at the same $y$-value. Signup and view all the answers

For a function to be considered a one-to-one function, what must be true concerning vertical lines?

Each vertical line intersects the graph at most once. Signup and view all the answers

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

The inverse function, not the reciprocal. Signup and view all the answers

Given the arithmetic sequence: 2, 5, 8, 11, ..., what is the 15th term?

44 Signup and view all the answers

In an arithmetic sequence, if the common difference is -5, what can be said about the sequence?

The sequence will always be decreasing. Signup and view all the answers

Determine the arithmetic mean of the numbers 10 and 22.

16 Signup and view all the answers

Which of the following sequences is NOT an arithmetic sequence?

2, 4, 8, 16, ... Signup and view all the answers

If the 7th term of an arithmetic sequence is 23 and the common difference is 4, what is the first term?

3 Signup and view all the answers

Consider the arithmetic sequence: -1, 2, 5, 8, ... . What is the value of the 20th term?

57 Signup and view all the answers

The first term of an arithmetic sequence is 12, and the common difference is -3. How many terms are needed to reach a value of -39?

17 Signup and view all the answers

Which of the following statements is TRUE about the graphical representation of an arithmetic sequence?

The graph is always a straight line. Signup and view all the answers

What is the expression for the inverse of the linear function defined as $y = ax + q$?

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

Which of the following represents the inverse of the quadratic function $y = ax^2$ after appropriate domain restriction?

$y = rac{1}{ ext{sqrt}(a)} x$ Signup and view all the answers

Which statement is true regarding the domain and range of the inverse function of a quadratic function?

The domain of the inverse function may be restricted to remain a function. Signup and view all the answers

What is the correct method to find the inverse of the exponential function $y = b^x$?

Interchange $x$ and $y$, then solve for $y$ to get $y = ext{log}_b x$. Signup and view all the answers

What is the vertical asymptote of the logarithmic function $y = ext{log}_b x$?

$x = 0$ Signup and view all the answers

Which of the following statements is true for the exponential function when $b > 1$?

The function increases and rises rapidly. Signup and view all the answers

Which of the following represents a law of logarithms?

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

How can you convert the equation $5^2 = 25$ into logarithmic form?

$ ext{log}_5 25 = 2$ Signup and view all the answers

In the context of logarithmic functions, what does $ ext{log}_a 1 = 0$ signify?

Any non-zero number raised to the power of 0 is 1. Signup and view all the answers

What does the derivative of a function represent?

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

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

f''(x) 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

How do you find the equation of a tangent to a curve at a given point?

Find the derivative using the rules of differentiation, substitute the x-coordinate of the given point into the derivative, and substitute the gradient of the tangent and the coordinates of the given point into an appropriate form of the straight line equation Signup and view all the answers

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

It affects the shape and orientation of the graph Signup and view all the answers

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

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

What is the purpose of the second derivative?

To determine the concavity of a curve Signup and view all the answers

What is the equation of the tangent line to f(x) at x = a?

y - y1 = m(x - x1) Signup and view all the answers

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

The product of the gradients of the tangent and the normal is -1 Signup and view all the answers

What is the use of the derivative in finding the equation of a tangent to a curve?

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

For a cubic function, what can be said about the concavity at a point of inflection?

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

What is the y-intercept of the cubic polynomial ( f(x) = 2x^3 - 5x^2 + 3x - 1 )?

-1 Signup and view all the answers

What is the general form of the quotient (Q(x)) when dividing the polynomial (a(x) = 3x^3 - 2x^2 + 5x - 1) by (b(x) = x - 2) using long division?

3x^2 + 4x + 13 + \frac{25}{x - 2} Signup and view all the answers

What is the remainder (R(x)) when dividing the polynomial (a(x) = 2x^3 - 3x^2 + 4x - 1) by (b(x) = x + 1) using synthetic division?

0 Signup and view all the answers

Which of the following is a stationary point of the cubic function (f(x) = x^3 - 3x^2 + 2x + 1)?

(1, 1) Signup and view all the answers

If a cubic function has a local maximum at (x = 2) and a local minimum at (x = -1), what is the sign of the coefficient of the (x^3) term?

Negative Signup and view all the answers

If a cubic polynomial has a root at (x = 3), which of the following is a factor of the polynomial?

x - 3 Signup and view all the answers

Given that the cubic polynomial (f(x) = 2x^3 + 5x^2 - 4x - 3) has a root at (x = -1), what is the quotient (Q(x)) after dividing by (x + 1)?

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

Which of the following is a valid method for solving cubic equations?

All of the above. Signup and view all the answers

What is the end behavior of the cubic function (f(x) = -2x^3 + 4x^2 - 3x + 1)?

As (x) approaches positive infinity, (f(x)) approaches negative infinity. Signup and view all the answers

What is the probability of the complement of event A?

$1 - P(A)$ Signup and view all the answers

What is the limit of the function as x approaches -6?

-8 Signup and view all the answers

If two events A and B are independent, what is the probability of their intersection?

$P(A) * P(B)$ Signup and view all the answers

What is the condition for two events to be mutually exclusive?

$P(A ext{ and } B) = 0$ Signup and view all the answers

Why is the function not defined at x = -6?

-8 Signup and view all the answers

What is the derivative of a constant function?

0 Signup and view all the answers

What is the probability of a sequence of outcomes in a tree diagram?

The product of the probabilities along the branches of the sequence Signup and view all the answers

What is the purpose of a Venn diagram in probability?

To visualize the relationships between events Signup and view all the answers

What is the general rule for differentiating a power function?

Signup and view all the answers

What is the formula for the probability of the union of two events A and B?

$P(A) + P(B) - P(A ext{ and } B)$ Signup and view all the answers

What is the formula for the probability of mutually exclusive events A and B?

P(A or B) = P(A) + P(B) Signup and view all the answers

Which statement is true regarding the notation for derivatives?

1 Signup and view all the answers

What is the definition of complementary events?

Events that are mutually exclusive and make up the whole sample space Signup and view all the answers

What is the notation for the sample space in probability?

S Signup and view all the answers

What is the purpose of a tree diagram in probability?

To organize and visualize the different possible outcomes of a sequence of events Signup and view all the answers

When should first principles be used to determine a derivative?

0 Signup and view all the answers

What is the formula for the number of outcomes in event A and event B combined?

n(A) × n(B) Signup and view all the answers

What is the symbol for the sample space in probability?

$S$ Signup and view all the answers

What happens to the function's graph at x = -6?

-8 Signup and view all the answers

What is the formula for the total number of possibilities in a fundamental counting principle problem?

n_1 × n_2 × ... × n_k Signup and view all the answers

What does the notation dy/dx represent?

0 Signup and view all the answers

What is the notation for the factorial of a number n?

n! Signup and view all the answers

What is the formula for the probability of the intersection of two independent events A and B?

$P(A) * P(B)$ Signup and view all the answers

What is the value of 0!?

1 Signup and view all the answers

What is the definition of mutually exclusive events?

Events that cannot happen at the same time Signup and view all the answers

Which of the following is NOT a rule for differentiation?

Derivative of a product Signup and view all the answers

What is the formula for the probability of complementary events?

P(not A) = 1 - P(A) Signup and view all the answers

What is the formula for the union of sets A and B?

A ∪ B Signup and view all the answers

What is the formula for the intersection of sets A and B?

A ∩ B Signup and view all the answers

How is the remainder calculated when dividing a polynomial by a linear divisor?

By evaluating the polynomial at the point $x = rac{d}{c}$. Signup and view all the answers

What indicates that a linear divisor is a factor of a polynomial according to the Factor Theorem?

If $pigg(rac{d}{c}igg) = 0$. Signup and view all the answers

Which of the following describes one of the steps in solving cubic equations?

Identify a potential factor using the Factor Theorem. Signup and view all the answers

What is the form of the polynomial after it has been divided by a linear divisor?

The polynomial is expressed as $p(x) = (cx - d) imes Q(x) + R$. Signup and view all the answers

Which of the following statements about the Remainder Theorem is true?

The remainder can be determined by substituting $x = rac{d}{c}$ into the polynomial. Signup and view all the answers

What condition must be met for two events to be classified as mutually exclusive?

They cannot happen at the same time. Signup and view all the answers

When using the Quadratic Formula to solve a quadratic expression, what is the first step?

Identify the coefficients $a$, $b$, and $c$. Signup and view all the answers

In the addition rule for probability, what is required to calculate the probability of the union of two events?

The probabilities of each event and their intersection must be known. Signup and view all the answers

What role does the remainder function play when a polynomial is divided by a linear factor?

It indicates whether the factor is a root of the polynomial. Signup and view all the answers

What is the effect of a negative common difference on an arithmetic sequence?

The sequence decreases Signup and view all the answers

If a sequence has a common difference of 3, what is the difference between the 5th and 3rd terms?

6 Signup and view all the answers

What is the 10th term of an arithmetic sequence with a first term of 2 and a common difference of 4?

38 Signup and view all the answers

How many terms are in an arithmetic sequence with a first term of 5, a common difference of 2, and a last term of 17?

6 Signup and view all the answers

What is the first term of an arithmetic sequence with a common difference of -2 and a 5th term of -6?

6 Signup and view all the answers

What can be said about the graph of an arithmetic sequence?

It is a straight line Signup and view all the answers

Why is it important to verify if the difference between consecutive terms is constant?

To determine if the sequence is arithmetic Signup and view all the answers

What is the common difference of an arithmetic sequence with a 2nd term of 7 and a 1st term of 3?

4 Signup and view all the answers

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

Sn = (n/2)(2a + (n-1)d) Signup and view all the answers

What is the 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 graphical representation of a one-to-one function?

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

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

f^(-1)(x) = (1/a)x - (q/a) Signup and view all the answers

What is the horizontal line test used for?

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

What is the notation f^(-1)(x) used for?

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

What is the graphical symmetry of the inverse function?

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

What is the formula for the sum of an arithmetic series when the last term is unknown?

Sn = (n/2)(2a + (n-1)d) Signup and view all the answers

What is the definition of an inverse function?

A function that is the reverse of another function. Signup and view all the answers

What is the general formula for a finite arithmetic series?

Sn = a + (a + d) + (a + 2d) + ... + (l - 2d) + (l - d) + l Signup and view all the answers

What determines whether a sequence is classified as geometric?

The ratio between consecutive terms remains constant. Signup and view all the answers

Given the first term of a geometric sequence as 3 and a common ratio of 2, what is the 4th term?

24 Signup and view all the answers

Which of the following statements about the common ratio of a geometric sequence is true?

If the common ratio is 1, all terms are the same. Signup and view all the answers

What is the formula for the geometric mean of two numbers, 4 and 9?

6 Signup and view all the answers

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

The sum of the first n terms in the series. Signup and view all the answers

What is the sum of the first 5 terms of a geometric sequence where the first term is 2 and the common ratio is 3?

364 Signup and view all the answers

What is the defining characteristic of an infinite series?

It has no defined endpoint to its summation. Signup and view all the answers

Which condition must be met for an infinite geometric series to converge?

The common ratio must be between 0 and 1. Signup and view all the answers

Which of the following represents a proper way to test if a sequence is geometric?

Verifying that the ratios of consecutive terms are equal. Signup and view all the answers

Which graphical representation is expected for a geometric sequence?

An exponential graph that displays discrete points. Signup and view all the answers

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

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

What is the formula for calculating the future value of an annuity?

$FV = PMT * \frac{[(1 + i)^n - 1]}{i(1 + i)^n}$ Signup and view all the answers

When finding the inverse of the function $y = ax^2$, what is the first step?

Interchange $x$ and $y$ Signup and view all the answers

For the function $y = ax^2$, when $a < 0$, which domain restriction is typically applied?

$x ext{ must be negative}$ Signup and view all the answers

Which of the following is a key concept in financial mathematics that deals with the reduction in value of an asset over time using a linear method?

Simple Depreciation Signup and view all the answers

What is the general form of the inverse function for $y = b^x$?

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

What is the formula used to determine the period (n) of an investment in compound interest calculations?

$n = \frac{\log(A/P)}{\log(1 + i)}$ Signup and view all the answers

What does the present value of an annuity formula calculate?

The current value of a series of future payments Signup and view all the answers

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

$y ext{ can be any real number}$ Signup and view all the answers

Which type of annuity is commonly used for loan repayments and mortgage bonds, where regular payments are made to pay off a debt over time?

Present Value Annuity Signup and view all the answers

In the future value of an annuity formula, which variable represents the total future value accrued from regular payments?

F Signup and view all the answers

The product rule of logarithms states which of the following?

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

What is the main purpose of a future value annuity (FVA)?

To calculate the total value of an investment after all payments have been made and compounded with interest over the investment period. Signup and view all the answers

What happens to the graph of the exponential function $f(x)=b^x$ when $b > 1$?

The graph increases rapidly Signup and view all the answers

What is the formula for simple depreciation?

$A = P(1 - in)$ Signup and view all the answers

When calculating the future value of an annuity, which aspect does NOT contribute to increasing the total value at the end of the investment period?

The present value of the annuity, $PV$ Signup and view all the answers

Which formula reflects the difference between the stated interest rate and the actual rate received or paid?

$1 + i = \left(1 + \frac{i_{(m)}}{m}\right)^m$ Signup and view all the answers

If $x = b^y$, what is the relationship expressed in logarithmic terms?

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

Which component is NOT necessary for calculating the present value of an annuity?

Future value of the annuity, $F$ Signup and view all the answers

Which statement correctly defines the asymptote of the exponential function?

Horizontal asymptote at $y = 0$ Signup and view all the answers

Which of the following applications of logarithms is used to determine the decay rate in radioactive decay?

Radioactive Decay Signup and view all the answers

What does standardize payment amount per period, denoted as $x$, signify in the future value annuity formula?

The amount to be invested over the periods Signup and view all the answers

Which of the following represents the correct interpretation of the variable $i$ in both annuity formulas?

The interest rate per period Signup and view all the answers

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

At $(1, 0)$ Signup and view all the answers

What is the formula for calculating the effective annual interest rate (EAR) based on the nominal interest rate (i_(m)) compounded m times per year?

$EAR = \left(1 + \frac{i_{(m)}}{m}\right)^m - 1$ Signup and view all the answers

What is the formula used to calculate the sum of the first n terms of a finite geometric series when the common ratio r is greater than 1?

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

Under what condition does an infinite geometric series converge?

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

What is the common difference in an arithmetic sequence defined by the term $T_n = a + (n - 1)d$?

$d$ Signup and view all the answers

What is the formula for the sum of the first n integers, which form an arithmetic series?

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

In the context of finite geometric series, what does the variable 'r' represent?

The common ratio between consecutive terms Signup and view all the answers

Which of the following represents the general formula for a term in an arithmetic sequence?

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

What is the formula for the sum of an infinite geometric series if the common ratio r is between -1 and 1?

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

In the derivation of the formula for a finite geometric series, what does subtracting $rS_n$ from $S_n$ help to establish?

A simplified expression to isolate $S_n$ Signup and view all the answers

What is the purpose of Karl Friedrich Gauss's method when summing an arithmetic series?

To simplify the calculation of larger series Signup and view all the answers

What does the variable 'i' represent within the context of the summation formula?

The index of summation Signup and view all the answers

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

To solve optimisation problems Signup and view all the answers

What does the derivative of a function represent?

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

What is the condition for a point of inflection to occur on a cubic graph?

The second derivative is zero Signup and view all the answers

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

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

What is the formula for the second derivative of a function?

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

What is the purpose of finding the concavity of a cubic graph?

To sketch the graph of the function Signup and view all the answers

What is the formula for the quotient in synthetic division?

All of the above Signup and view all the answers

What is the purpose of finding the y-intercept of a cubic polynomial?

To sketch the graph of the function Signup and view all the answers

What is the condition for a cubic polynomial to have a local maximum or local minimum?

The first derivative changes sign Signup and view all the answers

What is the purpose of using the Remainder Theorem in solving cubic equations?

To factorise the polynomial Signup and view all the answers

What does the variable 'P' represent in the present value of an annuity formula?

Present value of the loan or annuity Signup and view all the answers

Which of the following formulas is used to calculate the future value of annuities?

F = x[(1 + i)^n - 1] / i Signup and view all the answers

Which formula would you use to determine the accumulated amount using simple interest?

A = P(1 + in) Signup and view all the answers

In the effective annual rate formula, what does the variable 'm' denote?

The number of compounding periods per year Signup and view all the answers

Which statement correctly describes the principle of the Achilles and the Tortoise paradox?

The paradox demonstrates how limits can lead to unexpected conclusions. Signup and view all the answers

What does 'x' represent when calculating the payment amount for future value annuities?

Regular payment amount per period Signup and view all the answers

What is the purpose of the formula for calculating total interest paid on a loan?

To establish the difference between total amount paid and principal Signup and view all the answers

In the context of compound interest, what does the variable 'n' usually represent?

Total number of compounding periods Signup and view all the answers

Which formula correctly represents the calculation of remaining loan balance?

P_balance = x[(1 - (1 + i)^{-n_remaining})/i] Signup and view all the answers

What is the limit of the function as $x$ approaches -6?

-8 Signup and view all the answers

Why can the term $x + 6$ be canceled in the function $y = \frac{(x + 6)(x - 2)}{x + 6}$?

Cancellation is valid when $x eq -6$. Signup and view all the answers

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

0 Signup and view all the answers

In the context of limits, when is it necessary to state that the limit is not defined?

$x$ is equal to the point of discontinuity. Signup and view all the answers

Which of the following correctly describes the behavior of the graph at $x = -6$?

It has a hole. Signup and view all the answers

If event A has 3 possible outcomes and event B has 4 possible outcomes, how many total possible outcomes are there for both events combined?

12 Signup and view all the answers

What does the notation "n!" represent?

The product of all positive integers up to n Signup and view all the answers

What do the differential operators in differentiation denote?

Differentiation Signup and view all the answers

Which of the following events are mutually exclusive?

Flipping a coin and getting heads and flipping a coin and getting tails Signup and view all the answers

When is it appropriate to use differentiation from first principles?

When specifically requested. Signup and view all the answers

What is the remainder when the polynomial ( p(x) = x^3 - 2x^2 + 5x - 3 ) is divided by ( x - 2 )?

11 Signup and view all the answers

Which of the following is a factor of the polynomial ( p(x) = 2x^3 + 5x^2 - 4x - 3 )?

2x + 1 Signup and view all the answers

What is the probability of an event not happening, given the probability of the event happening is 0.6?

0.4 Signup and view all the answers

Which of the following formulas correctly calculates the probability of events A and B happening, assuming they are independent?

$P(A ext{ and } B) = P(A) imes P(B)$ Signup and view all the answers

What is the quotient when the polynomial ( p(x) = x^4 - 3x^3 + 2x^2 + x - 1 ) is divided by ( x - 1 )?

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

What are the solutions to the cubic equation ( x^3 - 6x^2 + 11x - 6 = 0 )?

1, 2, 3 Signup and view all the answers

What is the probability of drawing a red card or a king from a standard deck of cards?

7/13 Signup and view all the answers

What is the complement of the event "choosing a red ball" from a bag containing 5 red balls and 3 blue balls?

Choosing a blue ball Signup and view all the answers

If ( P(A) = 0.4 ), ( P(B) = 0.3 ), and ( P(A ext{ and } B) = 0.1 ), what is ( P(A ext{ or } B) )?

0.6 Signup and view all the answers

If ( P(A) = 0.5 ), ( P(B) = 0.2 ), and ( A ) and ( B ) are mutually exclusive, what is ( P(A ext{ or } B) )?

0.7 Signup and view all the answers

A restaurant offers 3 appetizers, 4 main courses, and 2 desserts. How many different three-course meals can be ordered?

24 Signup and view all the answers

If a password can be made up of any combination of 4 letters, how many different passwords are possible?

456,976 Signup and view all the answers

What is the remainder when the polynomial ( p(x) = 3x^3 - 5x^2 + 2x - 1 ) is divided by ( 2x - 1 )?

1/2 Signup and view all the answers

Which of the following is NOT a factor of the polynomial ( p(x) = x^3 - 7x^2 + 14x - 8 )?

x - 6 Signup and view all the answers

What is the value of 5!

120 Signup and view all the answers

What is the quotient when the polynomial ( p(x) = 2x^4 - 5x^3 + 3x^2 + x - 2 ) is divided by ( x + 1 )?

2x^3 - 7x^2 + 10x - 8 Signup and view all the answers

What is the remainder when the polynomial ( p(x) = x^4 + 2x^3 - 3x^2 + 4x - 5 ) is divided by ( 2x - 3 )?

-1/2 Signup and view all the answers

What does the complementary rule state about the probability of event A?

P(A') = 1 - P(A) Signup and view all the answers

Which of the following statements is true regarding mutually exclusive events?

P(A or B) = P(A) + P(B) Signup and view all the answers

What is the correct application of the product rule for independent events?

P(A and B) = P(A) imes P(B) Signup and view all the answers

How does the addition rule simplify for mutually exclusive events?

P(A or B) = P(A) + P(B) - P(A and B) Signup and view all the answers

What distinguishes independent events from mutually exclusive events?

Mutually exclusive events cannot occur at the same time. Signup and view all the answers

Which equation reflects the relationship of probabilities for complementary events?

P(A') + P(A) = 1 Signup and view all the answers

In a probability tree diagram, what do the branches represent?

The outcomes of an event and their associated probabilities. Signup and view all the answers

What happens to the probability when two events are dependent?

P(A and B) ≠ P(A) × P(B) Signup and view all the answers

What is represented by the symbol $A'$ in probability?

The complement of event A. Signup and view all the answers

Which of the following correctly represents the second derivative of the function ( y ) with respect to ( x )?

( \frac{d^2 y}{dx^2} ) Signup and view all the answers

If the gradient of a tangent to a curve at a point is 2, what is the gradient of the normal to the curve at the same point?

-1/2 Signup and view all the answers

What does the second derivative of a function tell us about the original function?

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

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

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

What is the effect of the coefficient ( a ) on the shape of the cubic function ( y = ax^3 + bx^2 + cx + d ) when ( a < 0 )?

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

Which of the following steps is NOT required to find the equation of the tangent line to ( f(x) ) at ( x = a )?

Find the second derivative ( f''(x) ). 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 should be substituted for ( x )?

0 Signup and view all the answers

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

Find the area under the curve of a function. Signup and view all the answers

If the derivative of a function is positive at a particular point, what can be concluded about the function at that point?

The function is increasing. 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 -1. Signup and view all the answers

Given a geometric sequence with the first term (a) and common ratio (r), what is the value of the (n)-th term if the common ratio is negative and (n) is odd?

Positive Signup and view all the answers

Suppose the geometric mean of two numbers (a) and (b) is (m). What is the value of (ab)?

(m^2) Signup and view all the answers

If an arithmetic sequence has a first term of 2 and a common difference of 3, what is the 5th term of the sequence?

14 Signup and view all the answers

What is the common difference of an arithmetic sequence with a first term of 5 and a 4th term of 11?

2 Signup and view all the answers

What is the 10th term of an arithmetic sequence with a first term of 7 and a common difference of -2?

-13 Signup and view all the answers

What is the arithmetic mean of 3 and 9?

6 Signup and view all the answers

What is the gradient of the line formed by plotting an arithmetic sequence with a common difference of 4?

4 Signup and view all the answers

What is the number of terms in an arithmetic sequence with a first term of 2, a common difference of 3, and a last term of 20?

7 Signup and view all the answers

If an arithmetic sequence has a first term of 10 and a common difference of -5, what is the sequence decreasing or increasing?

Decreasing Signup and view all the answers

What is the effect of a negative common difference on an arithmetic sequence?

The sequence decreases Signup and view all the answers

What is the primary difference between the formulas for the future value and present value of an annuity?

The future value formula uses the number of periods as the exponent, while the present value formula uses the interest rate as the exponent. Signup and view all the answers

If an investment of $10,000 earns an interest rate of 8% per year for 5 years, what is the future value of the investment?

$15,559.26 Signup and view all the answers

A loan of $20,000 has an interest rate of 6% per year and is repaid in equal annual installments over 10 years. What is the present value of the loan?

$13,464.12 Signup and view all the answers

Which of the following statements is true about the relationship between the interest rate and the number of periods in an annuity?

The interest rate and number of periods are inversely proportional. Signup and view all the answers

What is the effect of an increase in the interest rate on the future value of an annuity?

The future value increases at a faster rate. Signup and view all the answers

A company invests $5,000 per year for 8 years, earning an interest rate of 7% per year. What is the present value of the investment?

$26,515.17 Signup and view all the answers

What is the sum of the first n terms of a geometric series with a first term of 4 and a common ratio of 2?

$4(1 - 2^n) / (1 - 2)$ Signup and view all the answers

Which of the following statements is true regarding the convergence of an infinite geometric series?

It converges only if -1 < r < 1. Signup and view all the answers

If $T_n = 3 + (n-1)(5)$ defines an arithmetic sequence, what is the 10th term, $T_{10}$?

51 Signup and view all the answers

In an infinite geometric series where the first term is 8 and the common ratio is 0.5, what is the sum of the series?

16 Signup and view all the answers

Using Gauss's method, what is the total sum from 1 to 50?

5050 Signup and view all the answers

What is the result of the expression $rac{a(1 - r^n)}{1 - r}$ when $a = 5$, $r = 3$, and $n = 4$?

5(40)/2 Signup and view all the answers

How does the convergence condition of an infinite geometric series affect the value of the sum?

The sum remains finite only when the absolute value of $r$ is less than 1. Signup and view all the answers

What will be the sum of a finite arithmetic series if the first term is 10 and the common difference is 2, summing 20 terms?

400 Signup and view all the answers

What term $T_n$ in geometric sequence has a first term 6 and a common ratio 4 if $n$ is 3?

144 Signup and view all the answers

What is the value of $S_{100}$ when using the formula $S_n = \frac{n}{2} (2a + (n - 1) d)$ with $a = 1$ and $d = 1$?

5050 Signup and view all the answers

For a function to have an inverse that is also a function, which property must it satisfy?

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

Which statement best describes the relationship between a function and its inverse?

They undo each other's operations. Signup and view all the answers

What is the significance of the expression $2S_n = n \times (a + l)$ in the context of an arithmetic series?

It represents the average of the first and last terms. Signup and view all the answers

Which of the following statements about functions is false?

A function can map one input to several outputs. Signup and view all the answers

When finding the inverse of a linear function $y = ax + q$, what is the first step you take?

Interchange $x$ and $y$. Signup and view all the answers

What does the horizontal line test determine about a function?

If the function is a one-to-one function. Signup and view all the answers

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

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

What represents the range of the logarithmic function defined as $f^{-1}(x) = \log x$?

y , \in , \mathbb{R} Signup and view all the answers

Which of the following correctly describes the concept of compound interest?

Interest is calculated on the principal plus any earned interest. Signup and view all the answers

Which of the following correctly summarizes the graphical representation of a one-to-one function?

Every vertical line intersects the graph only once. Signup and view all the answers

When solving for the time period $n$ in compound interest, which formula is applied?

$ n = \frac{\log(A) - \log(P)}{\log(1 + i)} $ Signup and view all the answers

Which characteristic is unique to the future value annuity compared to the present value annuity?

It is designed to accumulate funds over time. Signup and view all the answers

What is the formula for calculating the accumulated amount using simple interest?

$ A = P(1 + in) $ Signup and view all the answers

Which of the following illustrates a common misconception about nominal and effective interest rates?

Nominal rates represent the actual interest earned. Signup and view all the answers

What does the intercept of the exponential function $f(x) = 10^x$ represent?

(0, 1) Signup and view all the answers

In the context of exponential growth, what does the formula $3P = P(1 + i)^n$ solve for?

The total periods required for tripling the population. Signup and view all the answers

A loan is being repaid with monthly payments of $500 over a period of 5 years. The interest rate on the loan is 6% per annum, compounded monthly. What is the outstanding loan balance after 3 years?

$10,000 Signup and view all the answers

Which statement is true regarding the characteristics of the logarithmic function?

It is defined only for positive inputs. Signup and view all the answers

Which application correctly uses logarithms in financial contexts?

Determining the decay rate of a radioactive material. Signup and view all the answers

An investor is considering two investment options. Option A offers a 10% annual return compounded quarterly, while Option B offers a 9.5% annual return compounded monthly. Which investment option yields a higher effective annual rate?

Option A Signup and view all the answers

You wish to accumulate $50,000 in 10 years. You plan to make regular monthly payments into an account that earns an annual interest rate of 4% compounded monthly. What is the required monthly payment amount?

$400 Signup and view all the answers

A loan of $20,000 is taken out with an annual interest rate of 8% compounded monthly. The loan is to be repaid over 10 years with equal monthly payments. What is the total amount of interest paid over the life of the loan?

$13,000 Signup and view all the answers

You are offered a loan with an annual interest rate of 7% compounded quarterly. What is the equivalent effective annual rate (EAR)?

7.23% Signup and view all the answers

A company purchases a piece of machinery for $100,000. The machinery depreciates at a rate of 10% per year compounded annually. What will the value of the machinery be after 5 years?

$59,049 Signup and view all the answers

You are considering investing in a bond that pays a 6% annual coupon rate, compounded semi-annually. The bond has a face value of $1,000 and matures in 5 years. What is the present value of the bond if the current market interest rate is 5% compounded semi-annually?

$1,052 Signup and view all the answers

You want to purchase a car that costs $25,000. The dealer offers you a loan with an annual interest rate of 4.5% compounded monthly. If you plan to repay the loan over 5 years with equal monthly payments, what is the total amount of interest you will pay on the loan?

$3,500 Signup and view all the answers

A company invests $50,000 at the beginning of each year for the next 10 years. The investment earns an annual return of 7% compounded annually. What is the total value of the investment at the end of the 10th year?

$850,000 Signup and view all the answers

Which statement is true regarding the inverse of a linear function?

The inverse function's domain and range are always different from the original function. Signup and view all the answers

What must be done to express the inverse of the quadratic function $y = ax^2$?

First interchange $x$ and $y$ then restrict the domain. Signup and view all the answers

Which condition must be satisfied for the inverse of the function $y = b^x$ to be expressed correctly?

The value of $b$ must be greater than 0. Signup and view all the answers

Which of the following statements accurately describes logarithmic functions?

They have a vertical asymptote at $x = 0$. Signup and view all the answers

What happens to the graph of an exponential function when $0 < b < 1$?

The graph decreases rapidly and approaches zero. Signup and view all the answers

Which of these describes the process to convert an exponential equation to logarithmic form?

Interchange the base and the exponent with the result. Signup and view all the answers

For the function (y = \frac{x^2 + 4x - 12}{x + 6}), what is the value of (\lim_{x \to -6} \frac{(x + 6)(x - 2)}{x + 6})?

-8 Signup and view all the answers

If $y = ax^2$ with $a < 0$, how is the domain for the inverse function typically restricted?

Restrict to $x ext{ less than or equal to } 0$. Signup and view all the answers

What does the notation (\frac{dy}{dx}) represent?

All of the above Signup and view all the answers

What is the significance of the intercept for the function defined as $y = b^x$ where $b > 1$?

The y-intercept is always 1. Signup and view all the answers

Which property of logarithms allows the expression $rac{x}{y}$ to be separated?

Quotient Rule Signup and view all the answers

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

(\frac{dy}{dx^2}) Signup and view all the answers

The derivative of a function (f(x)) is defined as:

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

What is the derivative of (x^3) using the general rule for differentiation?

(3x^2) Signup and view all the answers

Which rule of differentiation would you use to find the derivative of (5x^2 + 3x - 2)?

Derivative of a sum Signup and view all the answers

What is the derivative of (7)?

(0) Signup and view all the answers

What is the derivative of (4x^2) using the rule for the derivative of a constant multiplied by a function?

(8x) Signup and view all the answers

Which of the following is a valid way to find the derivative of a function?

All of the above Signup and view all the answers

What is the derivative of (f(x) = x^2 + 3x - 5) at (x = 2) using first principles?

(7) Signup and view all the answers

What condition indicates that a polynomial divisor is indeed a factor of the polynomial?

The polynomial evaluated at the divisor's root equals zero. Signup and view all the answers

How can the remainder of a polynomial divided by a linear polynomial be expressed?

As the evaluation of the polynomial at a specific point. Signup and view all the answers

When solving a cubic equation, what is typically the first step to find the roots?

Identify a potential factor using the Factor Theorem. Signup and view all the answers

Which of the following correctly describes the relationship between polynomial division and the structure of the polynomial?

The quotient is derived from the division process and is of lower degree than the original polynomial. Signup and view all the answers

For mutually exclusive events, what simplification applies to the addition rule of probabilities?

The formula reduces to the simple sum of probabilities. Signup and view all the answers

In the context of polynomial division, what role does the term 'remainder' play?

It represents the excess after dividing the polynomial. Signup and view all the answers

What must be true about a polynomial if its remainder is zero when divided by a linear polynomial?

The linear polynomial is considered a factor of the polynomial. Signup and view all the answers

Which formula is essential for finding the roots of a quadratic polynomial after factorization?

The Quadratic Formula. Signup and view all the answers

What concept directly confirms whether two events can occur simultaneously according to the addition rule?

Whether they are mutually exclusive. Signup and view all the answers

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

To identify the local maxima and minima for graphing. Signup and view all the answers

Which of the following statements correctly defines a point of inflection?

It occurs where the second derivative changes sign. Signup and view all the answers

In the context of cubic functions, what does it imply if a function is described as concave up?

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

What is the result of solving the equation $f'(x) = 0$ in the context of finding stationary points?

It reveals the turning points of the graph. Signup and view all the answers

Which method is typically NOT used for factorising cubic polynomials?

Root Finding Techniques Signup and view all the answers

What does a local maximum indicate in the behavior of a cubic function?

The function is increasing before and decreasing after the point. Signup and view all the answers

How is the y-intercept of a cubic polynomial found?

By setting $x = 0$ in the equation $f(x) = ax^3 + bx^2 + cx + d$. Signup and view all the answers

What does it mean for a function's graph to display end behavior where $f(x)$ approaches positive infinity as $x$ approaches negative infinity?

The leading coefficient is positive. Signup and view all the answers

When performing synthetic division, which of the following coefficients is used in the division process?

The root of the divisor polynomial, used with the opposite sign. Signup and view all the answers

What aspect does the Rational Root Theorem primarily assist in determining for polynomial equations?

The potential rational roots of the polynomial. Signup and view all the answers

What is the probability of a sequence of outcomes in a tree diagram?

The product of the probabilities along the branches of the sequence Signup and view all the answers

What is the purpose of a two-way contingency table?

To determine whether events are dependent or independent Signup and view all the answers

What is the formula for the probability of two mutually exclusive events?

P(A or B) = P(A) + P(B) Signup and view all the answers

What is the formula for the probability of the complement of an event?

P(not A) = 1 - P(A) Signup and view all the answers

What is the formula for the total number of outcomes in a fundamental counting principle problem?

n(A) × n(B) × ... × n(k) Signup and view all the answers

What is the formula for the total number of possibilities when there are n objects to choose from and you choose from them r times?

n × n × ... × n (r times) Signup and view all the answers

What is the formula for the total number of arrangements of n different objects?

n! Signup and view all the answers

What is the value of 0!?

1 Signup and view all the answers

What is the formula for the addition rule for any two events A and B?

P(A or B) = P(A) + P(B) - P(A and B) Signup and view all the answers

What is the product rule for independent events A and B?

P(A and B) = P(A) × P(B) Signup and view all the answers

Two events, ( A ) and ( B ), are mutually exclusive. If ( P(A) = 0.3 ) and ( P(B) = 0.5 ), what is ( P(A \text{ or } B) )?

0.8 Signup and view all the answers

If events ( A ) and ( B ) are independent, and ( P(A) = 0.4 ) and ( P(B) = 0.6 ), what is ( P(A \text{ and } B) )?

0.24 Signup and view all the answers

Suppose ( P(A) = 0.7 ). Using the complementary rule, what is ( P(\text{not } A) )?

0.3 Signup and view all the answers

Which of the following statements about mutually exclusive events is TRUE?

Mutually exclusive events cannot occur at the same time. Signup and view all the answers

If events ( A ) and ( B ) are independent, then ( P(A|B) ) is equal to:

( P(A) ) Signup and view all the answers

Consider the events ( A ) and ( B ). Which of the following is NOT a valid expression in probability?

( P(A + B) ) Signup and view all the answers

Which of the following statements is TRUE about complementary events?

The sum of probabilities of complementary events is always 1. Signup and view all the answers

If ( P(A) = 0.2 ) and ( P(B) = 0.6 ), and ( A ) and ( B ) are mutually exclusive, what is ( P(A \text{ and } B) )?

0 Signup and view all the answers

Which of the following pairs of events are NOT mutually exclusive?

Drawing a red card and drawing a queen from a standard deck of cards. Signup and view all the answers

If events ( A ) and ( B ) are independent, and ( P(A) = 0.3 ) and ( P(B) = 0.7 ), what is ( P(A | B) )?

0.3 Signup and view all the answers

Given the function ( f(x) = 2x^3 - 3x^2 + 5x - 1 ), what is the equation of the tangent line at the point where ( x = 1 )?

( y = 6x - 2 ) Signup and view all the answers

A cubic function has the form ( f(x) = ax^3 + bx^2 + cx + d ). What is the y-intercept of the graph of the function ( f(x) = 2x^3 - 4x^2 + 3x - 1 )?

( -1 ) Signup and view all the answers

Which of the following is a correct way to express the second derivative of ( y = f(x) ) with respect to ( x )?

( \frac{d}{dx} (f'(x)) ) Signup and view all the answers

Which of the following statements accurately describes the relationship between the gradients of a tangent and its corresponding normal at a point on a curve?

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

If the second derivative of a function ( f(x) ) is positive at a point ( x = a ), what does this tell us about the original function ( f(x) ) at ( x = a )?

The function ( f(x) ) is concave up at ( x = a ). Signup and view all the answers

What is the effect of the coefficient ( a ) on the shape of the graph of a cubic function ( f(x) = ax^3 + bx^2 + cx + d ) when ( a < 0 )?

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

Given the cubic function ( f(x) = 3x^3 - 2x^2 + x + 5 ), what is the x-intercept of the graph of this function?

The x-intercept cannot be determined without further calculations. Signup and view all the answers

Which of the following statements is TRUE about the derivative of a function ( f(x) ) at a point ( x = a )?

The derivative at ( x = a ) represents the rate of change of the function at ( x = a ). Signup and view all the answers

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

To determine the concavity of the function. Signup and view all the answers

If the derivative of a function is equal to zero at a point, what can we conclude about the function at that point?

The function has a stationary point (either a maximum, minimum, or inflection point) at that point. Signup and view all the answers

Gr 12 Mathematics: November Mix P(1)

Questions and Answers

What is the definition of an arithmetic sequence?

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

How do you find the common difference of an arithmetic sequence?

What is the arithmetic mean between two numbers?

What is the graphical representation of an arithmetic sequence?

How do you test if a sequence is arithmetic?

What is the effect of a positive common difference on an arithmetic sequence?

What is the pattern formed by the terms of an arithmetic sequence when plotted on a graph?

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

What is the general formula for the n-th term of a geometric sequence?

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

What is the general form for a finite arithmetic series?

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

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

What does the common ratio, r, represent in a geometric sequence?

Which of the following is an example of a geometric sequence?

What is the common difference, d, in an arithmetic sequence?

Which of the following is an example of an arithmetic sequence?

What is the formula for the sum of an arithmetic series?

What is a one-to-one function?

What is the graphical representation of a one-to-one function?

What is the inverse of a function?

What is the formula for the sum of an arithmetic series when the last term is unknown?

What is a many-to-one function?

What is the graphical symmetry of the inverse function?

What is the horizontal line test used for?

What is the formula for the inverse of a linear function?

What is the notation $f^{-1}(x)$ used for?

What does the variable 'i' represent in the future value of an annuity formula?

What is the formula for calculating the present value of an annuity?

Which of the following is NOT a characteristic of a future value annuity?

What does the variable 'n' represent in the present value of an annuity formula?

Which of the following is NOT a characteristic of a present value annuity?

Which formula would you use to calculate the future value of an annuity where the payment amount is $500, the interest rate is 5%, and the number of periods is 10?

What is the formula for the present value of an annuity?

What is the formula for calculating the period of an investment using compound interest?

Which of the following is the formula for simple interest?

What is the formula for calculating the total interest paid on a loan?

What is the formula for the future value of a series of payments?

What is the formula for calculating the effective annual rate (EAR)?

What is the formula for calculating the outstanding loan balance?

Which formula is used for calculating the payment amount for a future value annuity?

What is the formula for calculating the total amount paid on a loan?

What is the formula for compound depreciation?

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

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

Which formula is used to calculate the period of an investment in compound interest?

What type of interest is calculated on the principal and includes interest from previous periods?

In financial calculations involving logarithms, which application involves determining the decay rate?

What is the future value annuity concept primarily used for?

What is implied by the value of the pH level calculated using the formula $\text{pH} = -\log_{10}[\text{H}^+]$?

Which equation represents the formula for compounded depreciation?

How is the nominal interest rate defined in relation to the effective rate?

In the context of logarithms, what does the term 'pH' measure?

What is the general form of a cubic polynomial?

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

What does the sign of the coefficient 'a' tell you about the shape of a cubic polynomial?

What is the formula used in synthetic division for finding the quotient and remainder?

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

What is the derivative of the function f(x) = ax^3 + bx^2 + cx + d?

What does it mean for a function to be concave up?

How do you find the x-intercepts of a cubic polynomial?

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

What does it mean for a function to be decreasing?

What does the Remainder Theorem state about the remainder when a polynomial is divided by a linear polynomial?

Which of these statements is not true regarding the Factor Theorem?

When dividing a polynomial by a linear polynomial, what form does the polynomial take?

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

For two mutually exclusive events, how is the addition rule simplified?

Which of the following equations represents the Remainder Theorem?

Which polynomial expression represents the relationship between a polynomial, its divisor, and remainder?

What does the Quadratic Formula solve for in the context of cubic equations?

What happens when you substitute $x = \frac{d}{c}$ into a polynomial and the result is zero?

Which of the following accurately describes cubic equations?

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

What is the common ratio of the geometric sequence 2, 6, 18, 54...?

What is the geometric mean between 4 and 9?

In a geometric sequence, if the common ratio is greater than 1, what happens to the terms as the sequence progresses?