Gr12 Mathematics: June Exam Hard P(1)

What is the primary concept that calculus is built upon?

Limits (correct)

Algebra

Optimization

Geometry

What is the name of the philosopher who introduced the concept of limits through his paradoxes?

Euclid

Achilles

Zeno (correct)

Pythagoras

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

-12

-14

-10

-8 (correct)

Why can we cancel the (x + 6) terms in the function y = ((x + 6)(x - 2))/(x + 6)?

Because x is never equal to -6 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 purpose of differential calculus in optimization problems?

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

What is the characteristic of an arithmetic sequence when plotted on a graph?

A straight line Signup and view all the answers

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

T_n = a + (n - 1)d 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 characteristic of a geometric sequence when plotted on a graph?

An exponential curve Signup and view all the answers

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

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

What is the purpose of testing for an arithmetic sequence?

To identify the sequence type Signup and view all the answers

What is the purpose of testing for a geometric sequence?

To identify the sequence type Signup and view all the answers

What is the definition of a series?

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

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

Sigma notation 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 value of the limit $\lim_{x \to -6} \frac{(x + 6)(x - 2)}{x + 6}$?

-8 Signup and view all the answers

What is the general rule for differentiating a function of the form $x^n$?

$\frac{d}{dx}[x^n] = nx^{n-1}$ Signup and view all the answers

Which expression correctly defines the derivative of a constant $k$?

$\frac{d}{dx}[k] = 0$ Signup and view all the answers

When should one specifically use differentiation from first principles?

When requested, or when defining a derivative. Signup and view all the answers

Which of the following notations is not equivalent to the derivative of a function $f(x)$?

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

What is the result of differentiating the sum of two functions $f(x)$ and $g(x)$?

$\frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$ Signup and view all the answers

What is one of the key reasons to use rules for differentiation instead of first principles?

It simplifies the process and reduces mistakes. Signup and view all the answers

Identify the incorrect statement regarding differential operators.

$D_xy$ is a common notation for integration. Signup and view all the answers

What relationship does the derivative have with the tangent to a curve?

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

What is the formula to find the sum of an arithmetic series from the first term to the last term?

$S_n = n(a + l)$ Signup and view all the answers

What is the characteristic of a one-to-one function?

Each element of the domain maps to exactly one element of the range Signup and view all the answers

What is the graphical representation of a many-to-one relation?

At least one vertical line intersects the graph more than once Signup and view all the answers

What is the definition of an inverse function?

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

What is the purpose of the horizontal line test?

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

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

$S_n = rac{n}{2} [2a + (n - 1) d]$ 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 graphical representation of an inverse function?

The graph is symmetrical about the line y = x Signup and view all the answers

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

To reverse the operation of the original function Signup and view all the answers

What is the formula to find the sum of an arithmetic series from the first term to the last term in terms of the common difference?

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

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

$m_{tangent} \times m_{normal} = -1$ Signup and view all the answers

A cubic function has a stationary point at ( x = 2 ). If the second derivative at ( x = 2 ) is positive, what can we conclude about the stationary point?

It is a local minimum. Signup and view all the answers

What is the correct notation for the second derivative of the function ( y = f(x) ) with respect to ( x )?

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

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

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

If the coefficient ( a ) in the cubic function ( y = ax^3 + bx^2 + cx + d ) is negative, what can we conclude about the graph of the function?

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

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

-1 Signup and view all the answers

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

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

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

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

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

The sign of the second derivative indicates whether the gradient of the original function is increasing, decreasing, or constant. Signup and view all the answers

What is the process for finding the stationary points of a cubic function?

Find the derivative of the function, set it equal to zero, and solve for ( x ). Signup and view all the answers

What is the significance of the second derivative ( f''(x) ) in relation to the concavity of a function ( f(x) )?

The second derivative ( f''(x) ) identifies the points where the concavity of ( f(x) ) changes. Signup and view all the answers

When finding the x-intercepts of a cubic polynomial, what equation must be solved?

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

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

Quadratic Formula Signup and view all the answers

If ( f(k) = 0 ), what can we conclude about the polynomial ( f(x) ) in relation to the factor ( x - k ) ?

The factor ( x - k ) is a factor of ( f(x) ). Signup and view all the answers

What is the purpose of using the opposite sign of the root of the divisor polynomial in synthetic division?

It facilitates the substitution of the root into the polynomial for the remainder theorem. Signup and view all the answers

What is the mathematical relationship between the remainder ( R ) of a polynomial ( p(x) ) divided by ( cx - d ) and the value of the polynomial at ( x = rac{d}{c} )?

They are equal. Signup and view all the answers

In the context of sketching the graph of a cubic polynomial, what information can be derived from analyzing the end behavior of the function?

The overall shape and direction of the graph as x approaches positive and negative infinity. Signup and view all the answers

In the division rule for polynomials, ( a(x) = b(x) \cdot Q(x) + R(x) ), what does ( Q(x) ) represent?

The quotient polynomial. Signup and view all the answers

Which of the following statements accurately describes the relationship between the derivative ( f'(x) ) and the turning points of a function ( f(x) ) ?

The derivative ( f'(x) ) changes sign at turning points. Signup and view all the answers

How does the sign of the leading coefficient ( a ) in a cubic polynomial ( f(x) = ax^3 + bx^2 + cx + d ) affect the shape of the graph?

It determines the overall direction of the graph as ( x ) approaches positive and negative infinity. Signup and view all the answers

What is the main condition for the convergence of an infinite geometric series?

The common ratio must satisfy $-1 < r < 1$. Signup and view all the answers

In the formula for the sum of a finite geometric series, what does the term $a(1 - r^n)$ represent?

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

What does the notation $ au_i = 1 + 2 + ... + i$ inherently represent?

The sum of the first $i$ natural numbers. Signup and view all the answers

Which of the following best describes the common ratio $r$ in a geometric sequence?

The factor by which each term is multiplied to obtain the next term. Signup and view all the answers

What must the common ratio $r$ be for the sum of an infinite geometric series to be calculable?

Between -1 and 1, exclusively. Signup and view all the answers

Which of the following equations derives the general formula for the sum of a finite geometric series?

Subtracting $rS_n$ from $S_n$. Signup and view all the answers

In the finite arithmetic series formula $S_n = \frac{n}{2}(T_1 + T_n)$, what is $T_n$?

The last term of the arithmetic sequence. Signup and view all the answers

What does the term $S_n = \frac{a(1 - r^n)}{1 - r}$ represent?

The sum of the first $n$ terms of a geometric series. Signup and view all the answers

Which statement about convergence of an infinite geometric series is true?

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

What is the result of applying the Product Rule to the expression $y = rac{x^2 + 2x}{3}$?

$rac{2x + 2}{3}$ Signup and view all the answers

Which statement is accurate regarding the range of the logarithmic function?

It includes all real numbers. Signup and view all the answers

In the change of base formula, which expression correctly represents $ ext{log}_2 8$ using base 10?

$rac{ ext{log}{10}8}{ ext{log}{10}2}$ Signup and view all the answers

Which of the following expressions describes the intercepts for the exponential function $f(x) = 10^x$?

(0, 1) and no x-intercept Signup and view all the answers

When using the logarithmic decay model for radioactive decay, how would you express the decay constant if the half-life is known?

$k = rac{ ext{log}2}{t_{1/2}}$ Signup and view all the answers

Which application is NOT typically associated with logarithms?

Determining the area of a triangle Signup and view all the answers

What is the first step to find the inverse of a linear function defined as $y = ax + q$?

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

What is the correct general form of the inverse function for $y = ax^2$ if $a > 0$?

$y = ext{sqrt}(rac{x}{a})$ for $x geq 0$ Signup and view all the answers

What condition must be met for $y = ax^2$ to have an inverse that is a function?

The domain must be restricted to allow one-to-one mapping. Signup and view all the answers

For the exponential function $y = b^x$, how is the inverse function expressed?

$y = ext{log}_b x$ 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

Which of the following equations correctly represents the conversion from exponential to logarithmic form?

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

If $0 < b < 1$, how will the graph of the exponential function $f(x) = b^x$ behave?

It will decrease slowly. Signup and view all the answers

What is the horizontal asymptote of the exponential function $y = b^x$?

$y = 0$ Signup and view all the answers

How are the domain and range of the inverse of a linear function related to those of the original function?

The domain and range switch places. Signup and view all the answers

A polynomial ( p(x) ) is divided by ( 2x - 3 ), resulting in a remainder of 5. What is the value of ( p \left( \frac{3}{2} \right) )?

5 Signup and view all the answers

Given that ( x - 2 ) is a factor of the polynomial ( p(x) = x^3 - 5x^2 + 8x - 4 ), what is the value of ( p(2) )?

0 Signup and view all the answers

Which of the following is a factor of the polynomial ( x^3 - 6x^2 + 11x - 6 )?

x - 2 Signup and view all the answers

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

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

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

35 Signup and view all the answers

If ( x - 3 ) is a factor of the polynomial ( p(x) = x^3 - 7x^2 + kx - 15 ), what is the value of ( k )?

12 Signup and view all the answers

Which of the following is a possible solution to the cubic equation ( 2x^3 - 9x^2 + 13x - 6 = 0 )?

2 Signup and view all the answers

What is the complete factorization of the polynomial ( x^3 - 7x^2 + 14x - 8 )?

(x - 1)(x - 2)(x - 4) Signup and view all the answers

What is the value of ( k ) if the polynomial ( x^3 - 5x^2 + kx - 6 ) has a factor of ( x - 3 )?

12 Signup and view all the answers

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

-19 Signup and view all the answers

What is the derivative of the function $f(x) = x^5$ using the general rule for differentiation?

$5x^4$ Signup and view all the answers

What is the derivative of the function $f(x) = 2x^2 + 3x - 4$ using the rules for differentiation?

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

What is the derivative of the function $f(x) = (x + 1)/(x - 1)$ using the definition of a derivative?

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

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

$y = 2x - 3$ Signup and view all the answers

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

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

What is the purpose of the rules for differentiation?

To find the derivative of a function. Signup and view all the answers

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

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

What is the differential operator used to denote differentiation?

$D$ Signup and view all the answers

What is the result of differentiating the sum of two functions $f(x)$ and $g(x)$?

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

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

The derivative is equal to the gradient of the tangent. Signup and view all the answers

What is the condition for a point of inflection on a curve?

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

What is the relationship between the coefficient a of a cubic polynomial and its shape?

If a < 0, the graph opens downwards Signup and view all the answers

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

To find the quotient and remainder of a polynomial division Signup and view all the answers

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

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

What behavioral pattern is exhibited by Achilles in Zeno's paradox when racing the tortoise?

He continuously catches up but never surpasses. Signup and view all the answers

What is the condition for a stationary point on a curve?

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

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

To plot the graph of the polynomial Signup and view all the answers

Why must the term $x + 6$ be excluded when determining the function's limit?

It becomes zero, making the function undefined at that point. Signup and view all the answers

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

All of the above Signup and view all the answers

What value does the function approach as $x$ approaches -6 from either side?

-8 Signup and view all the answers

What is the purpose of long division in factorizing cubic polynomials?

To find the quotient and remainder of a polynomial division Signup and view all the answers

In graphical representation, what characterizes the function $y = rac{x^2 + 4x - 12}{x + 6}$ at $x = -6$?

A hole indicating the limit exists but the function is undefined. Signup and view all the answers

What is the result of applying the Factor and Remainder Theorem to a cubic polynomial?

The polynomial can be factorized into the product of a linear and a quadratic factor Signup and view all the answers

Which statement best describes the main purpose of differential calculus as outlined in the content?

To analyze rates of change and solve optimization problems. Signup and view all the answers

What important mathematical concept does Zeno's paradox about Achilles and the tortoise illustrate?

The concept of limits and convergence. Signup and view all the answers

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

To find the turning points of the polynomial Signup and view all the answers

What is the necessary condition for a sequence to be an arithmetic sequence?

The differences between consecutive terms are 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 characteristic of a geometric sequence when plotted on a graph?

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

What is the purpose of testing for an arithmetic sequence?

To identify the first term and common difference Signup and view all the answers

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

Σ 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 formula to find the sum of an arithmetic series from the first term to the last term?

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

What is the characteristic of a one-to-one function?

Each input corresponds to exactly one output Signup and view all the answers

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

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

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

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

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

1 Signup and view all the answers

If logₐ(x) = 3, what is the value of x equal to?

a³ Signup and view all the answers

What is the change of base formula for logarithms?

logₐ(x) = logₓ(x)/logₓ(a) 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 the graph of the inverse of an exponential function?

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

What is the pH level of a solution with a hydrogen ion concentration of [H⁺] = 0.01?

pH = -log(0.01) = 2 Signup and view all the answers

What happens to the roles of dependent and independent variables when finding the inverse of a linear function?

They get switched. Signup and view all the answers

How is the domain affected when finding the inverse of the quadratic function $y = ax^2$?

It becomes the range of the inverse function. Signup and view all the answers

For which value of $a$ is the function $y = ax^2$ guaranteed to be one-to-one without any restrictions?

None of the above Signup and view all the answers

What is the correct expression for the inverse of the exponential function $y = b^x$?

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

What restriction is typically applied to the quadratic function $y = ax^2$ for its inverse to be a function?

Restrict $x$ to be non-negative for $a > 0$. Signup and view all the answers

What is the horizontal asymptote of the exponential function $y = b^x$ when $b > 1$?

y = 0 Signup and view all the answers

Which of the following statements is true regarding the logarithmic function $y = ext{log}_b x$?

It has a vertical asymptote at $x = 0$. Signup and view all the answers

When converting the exponential equation $2^7 = 128$ into logarithmic form, which expression is correct?

$ ext{log}_2 128 = 7$ Signup and view all the answers

What is the general shape of the graph of an exponential function $f(x) = b^x$ when $b < 1$?

Decreasing and concave up. Signup and view all the answers

Which of the following correctly describes the transformation of the range of the function and its inverse?

The range of the original function is the domain of the inverse function. Signup and view all the answers

What is the condition under which an infinite geometric series converges?

If the common ratio is between -1 and 1. Signup and view all the answers

Which of the following correctly describes the general form of a finite geometric series?

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

What is the primary definition of a series?

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

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

The variable representing the current term in the sum. Signup and view all the answers

For which of the following values of the common ratio does the infinite geometric series diverge?

r = 2 Signup and view all the answers

Which formula represents the sum of the first $n$ terms of an arithmetic series?

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

In the formula for the sum of an infinite geometric series, what does it mean if the series diverges?

The sum increases without bound. Signup and view all the answers

How would you express the n-th term of a geometric sequence?

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

What is the role of the common difference in an arithmetic series?

It is the consistent increment between terms. Signup and view all the answers

Which of the following statements about finite series is correct?

Finite series never contain infinite terms. Signup and view all the answers

Given the function ( f(x) = 2x^3 - 3x^2 + 4x - 1 ), what is the equation of the tangent line to the curve at the point (1, 2)?

y = 7x - 5 Signup and view all the answers

For the cubic function ( f(x) = -x^3 + 2x^2 - x + 3 ), determine the nature of the stationary point at ( x = 1 ).

Local Maximum Signup and view all the answers

Given the function ( y = 2x^3 - 5x^2 + 3x - 1 ), what is the value of the second derivative at ( x = 2 )?

10 Signup and view all the answers

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

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

For the cubic function ( y = 2x^3 - 3x^2 - 12x + 5 ), determine the x-coordinate(s) of the stationary point(s).

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

Determine the coefficient ( a ) for the cubic function ( f(x) = ax^3 + 2x^2 - 5x + 1 ) given that the function has a local maximum at ( x = 1 ).

-2 Signup and view all the answers

If a cubic function ( f(x) = ax^3 + bx^2 + cx + d ) has a stationary point at ( x = k ), what can we conclude about the factor ( (x - k) ) of the cubic polynomial?

It is a factor of the first derivative ( f'(x) ) but not the original function ( f(x) ). Signup and view all the answers

The gradient of the tangent to a curve at a given point is equal to the first derivative of the function at that point. What does the second derivative at that point represent?

The rate of change of the first derivative at the given point. Signup and view all the answers

A cubic function ( f(x) = ax^3 + bx^2 + cx + d ) has a local maximum at ( x = 2 ). What can we conclude about the sign of the second derivative ( f''(2) )?

It must be negative. Signup and view all the answers

Which of the following statements accurately describes the relationship between the first derivative and the second derivative of a function?

The first derivative represents the slope of the tangent line, while the second derivative represents the curvature of the function. Signup and view all the answers

If the sum of the first (n) terms of an arithmetic series is (S_n = rac{n}{2} (a + l)), what is the sum of the first (n) terms of an arithmetic series if the first term is doubled and the common difference is halved?

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

In the sum of an arithmetic series, (S_n = rac{n}{2} (a + l)), what does (l) represent?

The last term of the arithmetic sequence Signup and view all the answers

Which of the following statements is true about the inverse of a function?

The inverse of a function is only a function if the original function is one-to-one. Signup and view all the answers

A function (f(x)) is one-to-one. Which of the following statements must be true about its inverse function (f^{-1}(x))?

The domain of (f^{-1}(x)) is the range of (f(x)). Signup and view all the answers

Given a function (f(x)), what is the condition for its inverse function (f^{-1}(x)) to exist and be a function?

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

What is the inverse of the function (f(x) = 2x + 3)?

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

Which of the following functions has an inverse that is also a function?

(f(x) = x^3 ) Signup and view all the answers

If the graph of a function is symmetric about the line (y = x), what can be concluded about the function?

The function is its own inverse. Signup and view all the answers

Consider the function (f(x) = 3x - 2). What is the value of (f^{-1}(4))?

2 Signup and view all the answers

Which of the following functions does NOT have an inverse function?

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

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

( x - 1 ) Signup and view all the answers

What is the remainder when ( x^4 - 3x^3 + 2x^2 + 5x - 1 ) is divided by ( x - 2 )?

7 Signup and view all the answers

If ( x - 3 ) is a factor of ( 2x^3 + kx^2 - 11x + 6 ), what is the value of ( k )?

5 Signup and view all the answers

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

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

Given that ( x - 2 ) is a factor of the polynomial ( 2x^3 - 7x^2 + ax + 6 ), find the value of ( a ).

-1 Signup and view all the answers

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

3 Signup and view all the answers

If ( p(x) = x^3 + 2x^2 - 5x + 6 ) and ( p(2) = 0 ), which of the following is a factor of ( p(x) )?

( x - 2 ) Signup and view all the answers

Which of the following statements is NOT true about the Factor Theorem?

The Factor Theorem is only applicable to polynomials of degree 3 or higher. Signup and view all the answers

Given the polynomial ( p(x) = x^3 - 5x^2 + 8x - 4 ) and its factor ( x - 2 ), what is the quotient ( Q(x) )?

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

For what value of ( k ) will ( x + 2 ) be a factor of ( 2x^3 + kx^2 - 7x - 6 )?

-5 Signup and view all the answers

What is the fundamental concept that Zeno's paradoxes, including Achilles and the Tortoise, illustrates?

Limits Signup and view all the answers

What is the value of x that makes the function y = (x^2 + 4x - 12)/(x + 6) undefined?

-6 Signup and view all the answers

What happens to the value of y as x approaches -6 in the function y = (x^2 + 4x - 12)/(x + 6)?

y approaches -8 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

Why can we cancel the (x + 6) terms in the function y = ((x + 6)(x - 2))/(x + 6)?

Only when x is not equal to -6 Signup and view all the answers

What is the significance of the concept of limits in calculus?

It is the fundamental concept that calculus is built upon Signup and view all the answers

If $f(x) = x^2 + 3x - 4$, what is $f'(x)$ using the rules of differentiation?

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

What is the derivative of the function $f(x) = 2x^3 - 5x^2 + x - 1$ at $x = 1$?

3 Signup and view all the answers

If $y = f(x) = x^2 - 4x + 3$, what is the gradient of the tangent to the curve at $x = 2$?

-2 Signup and view all the answers

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

y = 3x - 2 Signup and view all the answers

If $f(x) = rac{x + 1}{x - 1}$, what is the derivative of $f(x)$?

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

Which of the following is a differential operator?

All of the above Signup and view all the answers

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

$rac{1}{2\sqrt{x}}$ Signup and view all the answers

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

$2x \sin x + x^2 \cos x$ Signup and view all the answers

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

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

What is the purpose of using differentiation from first principles?

To find the derivative of a function using the definition of a derivative. Signup and view all the answers

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

They are perpendicular Signup and view all the answers

What is the process for finding the stationary points of a cubic function?

Solving f'(x) = 0 for x and substituting back into f(x) Signup and view all the answers

What is the significance of the second derivative f''(x) in relation to the concavity of a function f(x)?

It indicates the concavity of the function Signup and view all the answers

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

y = 3x - 3 Signup and view all the answers

If f(k) = 0, what can we conclude about the polynomial f(x) in relation to the factor x - k?

x - k is a factor of f(x) Signup and view all the answers

What is the purpose of using the opposite sign of the root of the divisor polynomial in synthetic division?

To find the remainder Signup and view all the answers

What is the formula to find the sum of an arithmetic series from the first term to the last term?

S = n(a + l)/2 Signup and view all the answers

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

They are equal Signup and view all the answers

What is the correct notation for the second derivative of the function y = f(x) with respect to x?

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

What is the significance of the sign of the second derivative f''(x) in relation to the concavity of a function f(x)?

It indicates whether the function is concave up or concave down Signup and view all the answers

What is the relationship between the common difference and the gradient of the straight line formed by plotting an arithmetic sequence?

The common difference is equal to the gradient. Signup and view all the answers

If a sequence has a negative common ratio, what can be said about the sequence?

The sequence will oscillate between positive and negative values. Signup and view all the answers

What is the primary difference between an arithmetic sequence and a geometric sequence?

The arithmetic sequence has a constant difference, while the geometric sequence has a constant ratio. Signup and view all the answers

What is the purpose of finding the nth term of a sequence?

To calculate the nth term of a specific sequence. Signup and view all the answers

What is the relationship between the common ratio and the exponential graph of a geometric sequence?

The common ratio determines the rate of growth or decay of the exponential graph. Signup and view all the answers

What is the purpose of sigma notation?

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

What is the key 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 result of testing for an arithmetic sequence?

The common difference is 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 primary application of geometric sequences?

Modeling population growth and exponential decay. Signup and view all the answers

What is the formula to find the sum of an arithmetic series from the first term to the last term when the last term is unknown?

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

What is the key property of an inverse function?

All of the above Signup and view all the answers

What is the purpose of the horizontal line test?

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

What is the characteristic of a one-to-one function?

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

What is the graphical representation of a many-to-one relation?

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

What is the definition of an inverse function?

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

What is the formula to find the sum of an arithmetic series from the first term to the last term in terms of the common difference?

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

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

To solve for the input variable. Signup and view all the answers

What is the graphical representation of an inverse function?

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

What is the formula to find the sum of an arithmetic series from the first term to the last term?

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

What is the value of logₐ(a)?

1 Signup and view all the answers

If logₐ(x) = y, then what is a^y?

x Signup and view all the answers

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

x > 0 Signup and view all the answers

If logₐ(x) = 2, then what is logₐ(x^2)?

4 Signup and view all the answers

What is the range of the logarithmic function f(x) = logₐ(x)?

all real numbers Signup and view all the answers

What is the change of base formula for logarithms?

logₐ(x) = logᵦ(x) / logᵦ(a) Signup and view all the answers

What is the correct expression for the inverse of the linear function $f(x) = ax + q$?

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

Which of the following statements is true about the graph of an exponential function when the base $b$ is less than 1?

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

Why must the domain of the quadratic function $y = ax^2$ be restricted when finding its inverse?

To guarantee that the quadratic is one-to-one and passes the horizontal line test. Signup and view all the answers

When converting the expression $5^2 = 25$ into logarithmic form, what is the correct representation?

$ ext{log}_5 25 = 2$ 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

For the exponential function $f(x) = b^x$, which of the following is characteristic when $b$ is greater than 1?

The function increases without bound as $x$ approaches positive infinity. Signup and view all the answers

What is the inverse of the function $y = b^x$ expressed in terms of logarithms?

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

For the quadratic function $y = ax^2$, what must be the restriction on $x$ if $a > 0$ to ensure the inverse is a valid function?

$x ext{ must be restricted to only positive values}$ Signup and view all the answers

When does the function $f(x) = -x^2$ not cross the x-axis?

For all values of x. Signup and view all the answers

Given a cubic function ( f(x) = ax^3 + bx^2 + cx + d ), where ( a < 0 ), what can we conclude about the graph of the function?

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

If the second derivative of a function ( f(x) ) is positive at a stationary point, what can we conclude about the nature of that stationary point?

It is a local minimum. Signup and view all the answers

The tangent line to the curve ( y = x^3 - 2x^2 + 1 ) at the point ( (1, 0) ) has a gradient of:

-1 Signup and view all the answers

The relationship between the gradients of the tangent and the normal to a curve at a given point is expressed by:

(m_{\text{tangent}} \times m_{\text{normal}} = -1 ) Signup and view all the answers

Which of the following is NOT a valid notation for the second derivative of a function ( y = f(x) ) with respect to ( x ) ?

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

Given a cubic function ( f(x) = ax^3 + bx^2 + cx + d ), the y-intercept is determined by:

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

To find the stationary points of a cubic function ( f(x) = ax^3 + bx^2 + cx + d ), you need to:

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

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

Whether the gradient is increasing or decreasing Signup and view all the answers

Which of the following is NOT a valid method for factorizing cubic polynomials?

Quadratic formula Signup and view all the answers

If ( f(k) = 0 ), what can we conclude about the polynomial ( f(x) ) in relation to the factor ( x - k ) ?

(x - k) is a factor of ( f(x) ) Signup and view all the answers

What can be inferred if the remainder when dividing a polynomial by a linear divisor is non-zero?

The divisor does not divide the polynomial evenly. Signup and view all the answers

What result is achieved when using the Factor Theorem and finding that a polynomial equals zero at a certain value?

The polynomial has a root at that value. Signup and view all the answers

When solving for roots of a cubic equation using the Factor Theorem, which step is essential?

Dividing by the first linear factor found. Signup and view all the answers

If a cubic polynomial can be expressed as the product of a linear factor and a quadratic factor, what can be stated about its degree?

It must be of degree 3. Signup and view all the answers

In an arithmetic sequence, what would be the 10th term if the first term is 5 and the common difference is 3?

27 Signup and view all the answers

When applying the Quadratic Formula to find the roots of a quadratic polynomial, what condition must be satisfied for real roots to exist?

The discriminant must be positive. Signup and view all the answers

Which process is crucial after identifying a root of a cubic polynomial to continue solving for additional roots?

Perform polynomial long division. Signup and view all the answers

In the general expression for an arithmetic sequence, what does the variable 'd' represent?

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

How does the Factor Theorem relate to the solution of cubic equations?

It allows for successful polynomial division to find factors. Signup and view all the answers

What is the formula for the sum of the first n terms of a geometric sequence with the first term a and common ratio r, when r > 1?

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

If the sum of an infinite geometric series is finite, what must be true about the common ratio r?

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

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

4 Signup and view all the answers

Which of the following is NOT a characteristic of a finite geometric series?

The series always converges to a finite value. Signup and view all the answers

What is the sum of the first 5 terms of the geometric sequence 3, 6, 12, 24, ...?

93 Signup and view all the answers

If the 5th term of a geometric sequence is 16 and the common ratio is 2, what is the first term of the sequence?

1 Signup and view all the answers

Consider an arithmetic series with the first term a and common difference d. Which of the following expressions represents the sum of the first n terms of this series?

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

What is the sum of the first 10 terms of the arithmetic sequence 1, 4, 7, 10, ...?

155 Signup and view all the answers

If the sum of the first 8 terms of an arithmetic series is 100 and the common difference is 2, what is the first term of the series?

4 Signup and view all the answers

A sequence is defined by the recursive formula Tn = 2 * T(n - 1) + 1, where T1 = 3. What is the value of T4?

33 Signup and view all the answers

What is the primary concept behind Zeno's paradox of Achilles and the tortoise?

Limits Signup and view all the answers

Why can we cancel the (x + 6) terms in the function y = ((x + 6)(x - 2))/(x + 6)?

Because it is a common factor Signup and view all the answers

What happens to the value of y as x approaches -6 in the function y = (x^2 + 4x - 12)/(x + 6)?

y approaches -8 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 concept does the function y = (x^2 + 4x - 12)/(x + 6) illustrate?

Limits 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 Signup and view all the answers

What is the name of the mathematical operation that the symbols D and \frac{d}{dx} represent?

Differentiation Signup and view all the answers

What is the derivative of the function f(x) = x, evaluated at x = 2?

1 Signup and view all the answers

What is the notation for the derivative of a function f(x) with respect to x?

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

What is the general rule for differentiating a function of the form x^n?

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

What is the derivative of a constant k?

0 Signup and view all the answers

What is the result of differentiating the sum of two functions f(x) and g(x)?

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

What is the purpose of using rules for differentiation instead of first principles?

To simplify the process of differentiation Signup and view all the answers

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

The derivative is equal to the slope Signup and view all the answers

What is the equation of the tangent line to the curve y = f(x) at a given point (a, f(a))?

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

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

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

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

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

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

A polynomial of one degree less than p(x) 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 general form of a polynomial p(x) when cx - d is a factor?

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

What is the first step in solving a cubic equation?

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

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

Divide the polynomial by the factor Signup and view all the answers

What is the purpose of the Quadratic Formula?

To solve quadratic equations Signup and view all the answers

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

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

What is the key property of an arithmetic sequence?

Each term is obtained by adding a constant to the previous term Signup and view all the answers

What is the purpose of testing for an arithmetic sequence?

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

What is the primary characteristic of an arithmetic sequence?

The sequence has a constant difference between consecutive terms. 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 testing for an arithmetic sequence?

To determine if the sequence has a constant difference. Signup and view all the answers

What is the primary characteristic of a geometric sequence?

The sequence has a constant ratio between consecutive terms. Signup and view all the answers

What is the formula to find the sum of an arithmetic series?

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

What is the purpose of finding the common difference in an arithmetic sequence?

To determine the formula for the nth term. Signup and view all the answers

What is the characteristic of a geometric sequence when plotted on a graph?

The points form an exponential pattern. 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

What is the purpose of testing for a geometric sequence?

To determine if the sequence has a constant ratio. 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

Given a cubic polynomial ( f(x) = ax^3 + bx^2 + cx + d ) with a negative leading coefficient (a < 0), which of the following statements about the end behavior of the graph is true?

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

Which of these correctly identifies the relationship between a cubic polynomial and its derivative, in terms of the number of real roots?

The relationship between the number of real roots of a cubic polynomial and its derivative cannot be determined generally. Signup and view all the answers

A cubic polynomial has a stationary point at (x = 2). If the second derivative at (x = 2) is negative, what can we conclude about the stationary point?

The stationary point is a local maximum. Signup and view all the answers

Given a cubic polynomial ( f(x) = ax^3 + bx^2 + cx + d ), what is the degree of its derivative ( f'(x) )?

2 Signup and view all the answers

If the remainder ( R ) of a polynomial ( p(x) ) divided by ( cx - d ) is 0, what can we conclude about the relationship between ( p(x) ) and ( cx - d )?

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

Consider a cubic polynomial ( f(x) = ax^3 + bx^2 + cx + d ) where (a, b, c, d) are real numbers. Which of the following statements about its concavity is always true?

The graph changes concavity at least once. Signup and view all the answers

Which of these correctly describes the relationship between the x-intercepts of a cubic polynomial and its derivative?

There is no general relationship between the x-intercepts of a cubic polynomial and its derivative. Signup and view all the answers

If a cubic polynomial ( f(x) ) has a point of inflection at ( x = k ) , what can we conclude about the second derivative ( f''(x) ) at ( x = k )?

( f''(k) = 0 ) Signup and view all the answers

A cubic polynomial is divided by ( x - 2 ) , resulting in a remainder of 5. What is the value of ( f(2) )?

5 Signup and view all the answers

The derivative of a cubic polynomial ( f(x) ) is ( f'(x) = 3x^2 - 6x + 3 ). What is the value of the second derivative ( f''(x) ) at ( x = 1 )?

3 Signup and view all the answers

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

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

Which notation is NOT commonly used to represent the second derivative?

f'(x^2) Signup and view all the answers

For a cubic function defined as $f(x) = ax^3 + bx^2 + cx + d$, what is indicated if the coefficient $a$ is positive?

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

How is a local maximum of a cubic function characterized?

The derivative is zero and the function is decreasing afterwards. Signup and view all the answers

In finding the y-intercept of a cubic function $f(x) = ax^3 + bx^2 + cx + d$, what value of x should be used?

Zero. Signup and view all the answers

When finding the equation of a tangent line at a point on a curve, what must be calculated first?

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

What do stationary points indicate when examining a cubic function?

They can be local maxima or minima points. Signup and view all the answers

For a function (f(x)) and its inverse function (f^{-1}(x)), what is the relationship between their graphs in terms of the line (y = x)?

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

What is the role of the second derivative in relation to concavity of a function?

It indicates whether the function is concave up or concave down. Signup and view all the answers

Given a function (f(x)) and its inverse (f^{-1}(x)), which of the following statements is TRUE regarding their domains and ranges?

The domain of (f(x)) is equal to the range of (f^{-1}(x)), and vice versa. Signup and view all the answers

What is necessary to identify the x-intercepts of a cubic function $f(x) = ax^3 + bx^2 + cx + d$?

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

Consider a function (f(x)) that is not one-to-one. What can be concluded about its inverse function (f^{-1}(x))?

The inverse function (f^{-1}(x)) is not a function because it is not uniquely defined. Signup and view all the answers

Suppose (f(x)) is a function and (f^{-1}(x)) is its inverse. What is the value of (f(f^{-1}(5)))?

5 Signup and view all the answers

How do you determine the second derivative $f''(x)$ from the first derivative $f'(x)$?

By applying the rules of differentiation on $f'(x)$ once. Signup and view all the answers

If the graph of a function (f(x)) passes the horizontal line test, what can we conclude about its inverse function (f^{-1}(x))?

The inverse function (f^{-1}(x)) is a function and is also one-to-one. Signup and view all the answers

What is the sum of the first 100 odd positive integers?

10000 Signup and view all the answers

Consider a sequence where the first term is 3 and the common difference is 5. What is the sum of the first 20 terms of this arithmetic sequence?

1025 Signup and view all the answers

Which of the following is NOT a valid formula for the sum of an arithmetic series with 'n' terms, where 'a' is the first term, 'd' is the common difference, and 'l' is the last term?

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

What is the sum of the first 15 terms of the arithmetic sequence 2, 5, 8, 11, ...?

345 Signup and view all the answers

You are given the sum of the first 'n' terms of an arithmetic sequence and the common difference 'd'. What additional information is needed to determine the first term 'a'?

The last term 'l' Signup and view all the answers

What is the correct inverse of the linear function defined by the equation $f(x) = ax + q$?

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

Which restriction is typically applied when finding the inverse of the quadratic function $y = ax^2$?

The domain must be restricted to $x < 0$. Signup and view all the answers

For the exponential function $f(x) = b^x$, what is the behavior of the graph when $0 < b < 1$?

The function decreases and approaches a horizontal asymptote at $y = 0$. Signup and view all the answers

What is the correct expression for the inverse of the function $y = b^x$?

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

Which statement correctly describes the domain and range of the inverse of a linear function?

The domain and range are both $ ext{all real numbers}$. Signup and view all the answers

In order to find the inverse of the function $y = ax^2$, what is the first step required?

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

Which of the following accurately represents the properties of the logarithmic function $y = ext{log}_b(x)$?

It is only defined for values of $x > 0$. Signup and view all the answers

What does the negative sign indicate in the inverse function $f^{-1}(x) = ext{log}_{b}(x)$ when $b < 1$?

The function no longer represents an increasing function. Signup and view all the answers

Which characteristic is true of exponential functions where the base $b$ is greater than 1?

They show rapid growth as $x$ increases. Signup and view all the answers

Which of the following logarithmic properties accurately describes the relationship between logarithms of products?

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

What is the correct change of base formula for logarithms?

$ ext{log}_a x = rac{ ext{log}_b x}{ ext{log}_b a}$ Signup and view all the answers

For the function representing exponential growth, what is the range of the function $f(x) = 10^x$?

$y > 0$ Signup and view all the answers

In the context of logarithmic and exponential functions, which statement about their inverses is true?

The inverse function of $f(x) = ext{log}_b x$ is $f^{-1}(x) = b^x$ Signup and view all the answers

Which logarithmic property correctly applies when determining the logarithm of a quotient?

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

What is the condition for convergence of an infinite geometric series?

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

When analyzing the population growth formula $A = P(1 + i)^n$, what does 'n' represent?

The number of periods over which the growth occurs Signup and view all the answers

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

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

In the geometric series formula $S_n = \frac{a(1 - r^n)}{1 - r}$, what does $a$ represent?

The first term of the sequence Signup and view all the answers

If a geometric series diverges, what can be inferred about the common ratio $r$?

$r > 1$ or $r ext{ is irrational}$ Signup and view all the answers

What does the term $T_n = a \cdot r^{n-1}$ describe in a geometric sequence?

The nth term of the sequence Signup and view all the answers

How is the sum $S_n$ of a finite arithmetic series calculated?

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

In a finite arithmetic series, what does $d$ represent?

The common difference Signup and view all the answers

Which notation accurately represents the sum of a general sequence of terms from index $m$ to $n$?

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

What is the primary distinction between a finite and an infinite series?

Subset of terms; infinite takes all terms. Signup and view all the answers

Gr12 Mathematics: June Exam Hard P(1)

Questions and Answers

What is the primary concept that calculus is built upon?

What is the name of the philosopher who introduced the concept of limits through his paradoxes?

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

Why can we cancel the (x + 6) terms in the function y = ((x + 6)(x - 2))/(x + 6)?

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

What is the purpose of differential calculus in optimization problems?

What is the characteristic of an arithmetic sequence when plotted on a graph?

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

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

What is the characteristic of a geometric sequence when plotted on a graph?

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

What is the purpose of testing for an arithmetic sequence?

What is the purpose of testing for a geometric sequence?

What is the definition of a series?

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

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

What is the value of the limit $\lim_{x \to -6} \frac{(x + 6)(x - 2)}{x + 6}$?

What is the general rule for differentiating a function of the form $x^n$?

Which expression correctly defines the derivative of a constant $k$?

When should one specifically use differentiation from first principles?

Which of the following notations is not equivalent to the derivative of a function $f(x)$?

What is the result of differentiating the sum of two functions $f(x)$ and $g(x)$?

What is one of the key reasons to use rules for differentiation instead of first principles?

Identify the incorrect statement regarding differential operators.

What relationship does the derivative have with the tangent to a curve?

What is the formula to find the sum of an arithmetic series from the first term to the last term?

What is the characteristic of a one-to-one function?

What is the graphical representation of a many-to-one relation?

What is the definition of an inverse function?

What is the purpose of the horizontal line test?

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

What is the key property of an inverse function?

What is the graphical representation of an inverse function?

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

What is the formula to find the sum of an arithmetic series from the first term to the last term in terms of the common difference?

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

A cubic function has a stationary point at ( x = 2 ). If the second derivative at ( x = 2 ) is positive, what can we conclude about the stationary point?

What is the correct notation for the second derivative of the function ( y = f(x) ) with respect to ( x )?

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

If the coefficient ( a ) in the cubic function ( y = ax^3 + bx^2 + cx + d ) is negative, what can we conclude about the graph of the function?

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

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

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

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

What is the process for finding the stationary points of a cubic function?

What is the significance of the second derivative ( f''(x) ) in relation to the concavity of a function ( f(x) )?

When finding the x-intercepts of a cubic polynomial, what equation must be solved?

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

If ( f(k) = 0 ), what can we conclude about the polynomial ( f(x) ) in relation to the factor ( x - k ) ?

What is the purpose of using the opposite sign of the root of the divisor polynomial in synthetic division?

What is the mathematical relationship between the remainder ( R ) of a polynomial ( p(x) ) divided by ( cx - d ) and the value of the polynomial at ( x = rac{d}{c} )?

In the context of sketching the graph of a cubic polynomial, what information can be derived from analyzing the end behavior of the function?

In the division rule for polynomials, ( a(x) = b(x) \cdot Q(x) + R(x) ), what does ( Q(x) ) represent?

Which of the following statements accurately describes the relationship between the derivative ( f'(x) ) and the turning points of a function ( f(x) ) ?

How does the sign of the leading coefficient ( a ) in a cubic polynomial ( f(x) = ax^3 + bx^2 + cx + d ) affect the shape of the graph?

What is the main condition for the convergence of an infinite geometric series?

In the formula for the sum of a finite geometric series, what does the term $a(1 - r^n)$ represent?

What does the notation $ au_i = 1 + 2 + ... + i$ inherently represent?

Which of the following best describes the common ratio $r$ in a geometric sequence?

What must the common ratio $r$ be for the sum of an infinite geometric series to be calculable?

Which of the following equations derives the general formula for the sum of a finite geometric series?

In the finite arithmetic series formula $S_n = \frac{n}{2}(T_1 + T_n)$, what is $T_n$?

What does the term $S_n = \frac{a(1 - r^n)}{1 - r}$ represent?

Which statement about convergence of an infinite geometric series is true?

What is the result of applying the Product Rule to the expression $y = rac{x^2 + 2x}{3}$?

Which statement is accurate regarding the range of the logarithmic function?

In the change of base formula, which expression correctly represents $ ext{log}_2 8$ using base 10?

Which of the following expressions describes the intercepts for the exponential function $f(x) = 10^x$?

When using the logarithmic decay model for radioactive decay, how would you express the decay constant if the half-life is known?

Which application is NOT typically associated with logarithms?

What is the first step to find the inverse of a linear function defined as $y = ax + q$?

What is the correct general form of the inverse function for $y = ax^2$ if $a > 0$?

What condition must be met for $y = ax^2$ to have an inverse that is a function?

For the exponential function $y = b^x$, how is the inverse function expressed?

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

Which of the following equations correctly represents the conversion from exponential to logarithmic form?

If $0 < b < 1$, how will the graph of the exponential function $f(x) = b^x$ behave?

What is the horizontal asymptote of the exponential function $y = b^x$?