Gr12 Mathematics: June Exam Easy P(1)

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What is the definition of a limit in the context of calculus?

The value of a function at a specific point.
The maximum value of a function.
The derivative of a function at a point.
The value a function approaches as the input approaches a certain point. (correct)

In the function $y = \frac{x^2 + 4x - 12}{x + 6}$, what happens when $x$ equals -6?

The function is undefined. (correct)
The function has a maximum value.
The function value is 0.
The function approaches positive infinity.

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

$y = \frac{x^2 + 4x - 12}{x - 6}$
$y = x - 2$ (correct)
$y = x + 2$
$y = x^2 - 2$

According to the Achilles and the Tortoise paradox, what does this illustrate about limits?

A competitor can approach a goal without ever overtaking it. (C) Signup and view all the answers

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

There is a hole in the graph at that point. (B) Signup and view all the answers

As $x$ approaches -6 in the function $y = \frac{x^2 + 4x - 12}{x + 6}$, what does $y$ approach?

-8 (B) Signup and view all the answers

What is the derivative of $x^4$?

$4x^3$ (D) Signup and view all the answers

What is the derivative of $5x^2$?

$10x$ (A) Signup and view all the answers

What is the derivative of $3x^3 + 2x$?

$9x^2 + 2$ (B) Signup and view all the answers

What is the derivative of $2x^2 - 3x$?

$4x - 3$ (C) Signup and view all the answers

What is the derivative of $7$?

$0$ (D) Signup and view all the answers

Which of the following is a correct notation for the derivative of a function $f(x)$?

All of the above (D) Signup and view all the answers

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

$2x$ (B) Signup and view all the answers

What is the gradient of the tangent to the curve $y = x^3$ at the point $x = 2$?

$12$ (A) Signup and view all the answers

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

$y = 2x + 1$ (C) Signup and view all the answers

What is the derivative of $f(x) = \frac{1}{x}$?

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

What does the second derivative indicate about a function?

The change in gradient of the original function. (B) Signup and view all the answers

If the coefficient 'a' of a cubic function is negative, what can be inferred about the graph?

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

Which of the following denotes the second derivative of a function?

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

What is the first step in finding the equation of a tangent line to a function?

Find the derivative of the function. (C) Signup and view all the answers

What relationship exists between the gradients of the tangent and the normal at a point on a curve?

Their product equals negative one. (D) Signup and view all the answers

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

Set x equal to 0. (A) Signup and view all the answers

What does a stationary point signify on a graph?

The function stops changing. (B) Signup and view all the answers

What is required to determine the normal line to a curve at a given point?

The derivative of the function. (B) Signup and view all the answers

What type of stationary point occurs when a function changes from increasing to decreasing?

Local maximum. (A) Signup and view all the answers

To find the x-coordinates of stationary points for a function, what must be solved?

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

What does the symbol (\Sigma) represent?

The sum of a sequence (B) Signup and view all the answers

What is the general formula for a finite geometric series?

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

What is the condition for the sum of an infinite geometric series to exist?

-1 < r < 1 (B) Signup and view all the answers

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

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

What is the definition of a finite series?

The sum of a specific number of terms of a sequence (D) Signup and view all the answers

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

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

What is the definition of a geometric sequence?

A sequence of numbers with a constant ratio (B) Signup and view all the answers

What is the general form of the summation notation?

(\sum_{i=m}^{n} T_i = T_m + T_{m+1} + \cdots + T_{n-1} + T_n) (C) 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 (A) Signup and view all the answers

What happens to an infinite geometric series when r = 1?

The series diverges (A) Signup and view all the answers

Which of the following is the correct expression for the remainder (R) when a polynomial (p(x)) is divided by (cx - d)?

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

According to the Factor Theorem, if (cx - d) is a factor of polynomial (p(x)), what is the value of (p(d/c)) ?

0 (A) Signup and view all the answers

What is the degree of the quotient (Q(x)) when a polynomial (p(x)) of degree (n) is divided by a linear polynomial (cx - d)?

n - 1 (B) Signup and view all the answers

Which of the following is NOT a key formula used in solving cubic equations?

Remainder Theorem (D) Signup and view all the answers

In the process of solving a cubic equation, after finding a factor, how do you obtain the remaining polynomial factors?

Polynomial Long Division (D) Signup and view all the answers

What is the common difference in an arithmetic sequence if the first term is 5 and the fourth term is 17?

4 (D) Signup and view all the answers

Which of the following is the correct formula for the (n)-th term of an arithmetic sequence?

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

Given an arithmetic sequence with the first term 3 and a common difference of 2, what is the 10th term?

21 (D) Signup and view all the answers

If the 5th term of an arithmetic sequence is 12 and the common difference is 3, what is the first term?

3 (C) Signup and view all the answers

Which of the following sequences is NOT an arithmetic sequence?

1, 3, 6, 10, 15 (D) Signup and view all the answers

What is the steps to find the inverse of a linear function?

Interchange x and y, Solve for y, Express in function notation (B) Signup and view all the answers

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

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

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

Interchange x and y, Solve for y, Express in function notation (B) Signup and view all the answers

What is the domain and range of the inverse function f^-1(x) = 1/a x - q/a?

Domain: R, Range: R, assuming a ≠ 0 (A) Signup and view all the answers

What is the definition of the logarithm?

If x = b^y, then y = log_b(x) (D) Signup and view all the answers

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

Increasing or decreasing (D) Signup and view all the answers

What is the intercept of the logarithmic function y = log_b x?

(1, 0) (B) Signup and view all the answers

What is the asymptote of the logarithmic function y = log_b x?

Vertical asymptote at x = 0 (D) Signup and view all the answers

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

x > 0 (C) Signup and view all the answers

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

y > 0 (D) Signup and view all the answers

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

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

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

3 (A) Signup and view all the answers

Which of the following sequences is an arithmetic sequence?

1, 3, 5, 7, ... (A) Signup and view all the answers

What is the sum of the first 10 terms of the arithmetic sequence 3, 7, 11, 15, ...?

210 (D) Signup and view all the answers

Which of the following is a characteristic of an arithmetic sequence?

The terms form a linear pattern when plotted on a graph. (B) Signup and view all the answers

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

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

What does the gradient (slope) of the line representing an arithmetic sequence represent?

The common difference (A) Signup and view all the answers

What is the sum of the first 5 terms of the geometric sequence 2, 4, 8, 16, ...?

62 (A) Signup and view all the answers

Which of the following is true about the graphical representation of a geometric sequence?

It is an exponential curve. (B) Signup and view all the answers

What does it indicate when a graph is concave up?

The gradient is increasing. (C) Signup and view all the answers

Which equation indicates a point of inflection?

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

When performing synthetic division, what should you use as the divisor's root?

The opposite sign of the divisor's root (A) Signup and view all the answers

What does the Remainder Theorem state about polynomial division?

The remainder can be found by evaluating at the divisor's root. (A) Signup and view all the answers

Which method is not used for factorizing cubic polynomials?

Polynomial Interpolation (A) Signup and view all the answers

Which statement correctly describes a stationary point?

It occurs where f'(x) = 0. (C) Signup and view all the answers

What is meant by the term 'optimisation problem' in calculus?

Determining maximum or minimum values of functions. (B) Signup and view all the answers

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

By setting x to zero in the polynomial equation. (B) Signup and view all the answers

What do the signs of the coefficients in a cubic polynomial indicate?

The shape of the graph. (D) Signup and view all the answers

Which of the following correctly defines a cubic polynomial?

A polynomial of the form $ax^3 + bx^2 + cx + d$. (D) Signup and view all the answers

What is the sum of the first 100 positive integers?

5050 (A) Signup and view all the answers

Which of the following is NOT a property of an inverse function?

The inverse function is always a linear function. (B) Signup and view all the answers

What is the value of ( S_{100} ) in the context of Gauss's method for finding the sum of the first 100 integers?

5050 (C) Signup and view all the answers

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

( S_n = rac{n}{2} (a + (n - 1) d) ) (A), ( S_n = rac{n}{2} (a + l) ) (B), ( S_n = rac{n}{2} (2a + (n - 1) d) ) (F) Signup and view all the answers

Which type of function has an inverse that is also a function?

One-to-one function (A) Signup and view all the answers

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

Interchange ( x ) and ( y ) in the equation ( y = f(x) ) and then solve for ( y ). (A) Signup and view all the answers

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

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

Which of the following is NOT a valid method for determining if a function has an inverse that is also a function?

Checking if the function is continuous (B) Signup and view all the answers

In a one-to-one function, how many elements in the range are associated with each element in the domain?

One (D) Signup and view all the answers

What is the difference between a relation and a function?

A relation can have multiple outputs for a single input, while a function can only have one output for a single input. (D) Signup and view all the answers

What is the value of $log_a 1$?

0 (C) Signup and view all the answers

What does the Product Rule for logarithms state?

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

What is the domain of the logarithmic function $f^{-1}(x) = log x$?

$x > 0$ (C) Signup and view all the answers

What is the correct formula for calculating pH levels?

$pH = -log_{10}[ ext{H}^+]$ (A) Signup and view all the answers

Which of the following correctly describes the characteristics of the exponential function $f(x) = 10^x$?

Intercept is $(0, 1)$, asymptote is $y = 0$ (D) Signup and view all the answers

Using the Change of Base formula, how can $log_a x$ be expressed?

$rac{log x}{log a}$ (C) Signup and view all the answers

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

The derivative of the function at x = a (C) Signup and view all the answers

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

All of the above (D) Signup and view all the answers

What is the rule for differentiating a constant?

The derivative of a constant is 0 (B) Signup and view all the answers

What is the general rule for differentiating x^n?

The derivative of x^n is nx^(n-1) (B) Signup and view all the answers

What is the derivative of a sum of two functions?

The sum of the derivatives of the two functions (C) Signup and view all the answers

What is the derivative of a difference of two functions?

The difference of the derivatives of the two functions (C) Signup and view all the answers

When should you use the rules for differentiation?

When the question does not specify how to determine the derivative (B) Signup and view all the answers

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

To find the equation of the tangent to the function (A) Signup and view all the answers

What is the notation for the differential operator?

All of the above (D) Signup and view all the answers

What does the derivative of a function describe?

The rate of change of the function with respect to x (A) Signup and view all the answers

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

To indicate the change in gradient of the original function (D) Signup and view all the answers

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

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

What is the first step in finding the equation of a tangent line to a function?

Find the derivative using the rules of differentiation (C) Signup and view all the answers

What is the effect of a negative coefficient 'a' on a cubic graph?

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

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

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

What does a stationary point signify on a graph?

A point where the function changes from increasing to decreasing or vice versa (B) Signup and view all the answers

What is required to determine the normal line to a curve at a given point?

All of the above (D) Signup and view all the answers

What type of stationary point occurs when a function changes from increasing to decreasing?

Local maximum (A) Signup and view all the answers

To find the x-coordinates of stationary points for a function, what must be solved?

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

What notation is used to denote the second derivative of a function?

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

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

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

What is the degree of the quotient Q(x) when a polynomial p(x) of degree n is divided by a linear polynomial cx - d?

n - 1 (C) Signup and view all the answers

According to the Factor Theorem, what is the value of p(d/c) if cx - d is a factor of polynomial p(x)?

0 (D) Signup and view all the answers

What is the general form of an arithmetic sequence?

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

What is the formula to solve a quadratic equation?

x = -b ± √(b^2 - 4ac) / 2a (A) Signup and view all the answers

What is the first step to solve a cubic equation?

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

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

They are perpendicular (B) Signup and view all the answers

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

a / (1 - r) (C) Signup and view all the answers

What is the condition for the sum of an infinite geometric series to exist?

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

What is the definition of an arithmetic sequence?

A sequence of numbers in which each consecutive term is calculated by adding a constant value (B) Signup and view all the answers

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

The second derivative of the function changes sign. (C), The second derivative of the function is equal to zero. (D) Signup and view all the answers

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

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

Which of the following methods can be used to factorize a cubic polynomial?

Long Division (A), Synthetic Division (B) Signup and view all the answers

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

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

In the division rule for polynomials, what does the term (R(x)) represent?

The remainder polynomial (B) Signup and view all the answers

What is the first step in sketching the graph of a cubic polynomial?

Finding the y-intercept (D) Signup and view all the answers

Which of the following is a correct notation for the second derivative of a function f(x)?

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

What does the sign of the second derivative of a function tell us about the concavity of the graph?

Positive second derivative indicates concave up, negative second derivative indicates concave down. (C) Signup and view all the answers

Which of the following is NOT a method for finding the x-intercepts of a cubic polynomial?

Using the quadratic formula (B) Signup and view all the answers

What does a stationary point indicate on the graph of a function?

A point where the graph has a maximum or minimum value. (B), A point where the graph changes direction. (D) Signup and view all the answers

What happens to the function when the variable $x$ approaches -6?

The function approaches a value of -8 (C) Signup and view all the answers

How can the numerator of the function $y = rac{x^2 + 4x - 12}{x + 6}$ be simplified?

It can be simplified to $(x + 6)(x - 2)$ (D) Signup and view all the answers

In the context of Zeno's paradox, what does Achilles' perceived failure to overtake the tortoise demonstrate?

The idea of limits in mathematics (A) Signup and view all the answers

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

It shows a discontinuity with a hole at the point (C) Signup and view all the answers

What condition must be met for the terms of the function $y$ to be canceled?

It is valid only when $x eq -6$ (A) Signup and view all the answers

Which of the following statements correctly describes the limit of the function as it approaches -6?

The limit approaches -8 from both sides (A) Signup and view all the answers

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

To switch the roles of the dependent and independent variables (C) Signup and view all the answers

What is the correct inverse of the function y = 2x + 3?

y = x/2 - 3/2 (C) Signup and view all the answers

What is the domain and range of the inverse function f^-1(x) = 1/a x - q/a, assuming a ≠ 0?

Domain: ℝ, Range: ℝ (B) Signup and view all the answers

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

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

What is the purpose of restricting the domain of a quadratic function when finding its inverse?

To ensure the function is one-to-one (C) Signup and view all the answers

What is the definition of a logarithm?

The exponent to which a base must be raised to yield a number (D) Signup and view all the answers

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

A curve that rises or falls rapidly (A) Signup and view all the answers

What is the relationship between the graphs of exponential and logarithmic functions?

They are reflections about the line y = x (B) Signup and view all the answers

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

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

What is the property of an exponential function f(x) = b^x when b > 1?

The function is increasing (B) Signup and view all the answers

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

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

What is the formula for the geometric mean between two numbers, 'a' and 'b'?

$\sqrt{ab}$ (A) Signup and view all the answers

If the common ratio 'r' of a geometric sequence is greater than 1, what happens to the sequence?

The sequence increases exponentially. (D) Signup and view all the answers

Which of the following represents the sum of the first 'n' terms of a sequence?

$S_n$ (A) Signup and view all the answers

What is the formula for the arithmetic mean between two numbers?

$rac{a + b}{2}$ (A) Signup and view all the answers

What does the gradient of a line represent when plotting the terms of an arithmetic sequence against their positions?

The common difference (D) Signup and view all the answers

What is the condition for a sequence to be considered geometric?

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

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

$T_n = ar^{n-1}$ (D) Signup and view all the answers

What happens to the terms of a geometric sequence if the common ratio 'r' is negative?

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

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

S = a / (1 - r) (C) Signup and view all the answers

Which of the following statements about the logarithmic function is true?

The log function approaches negative infinity as x approaches 0. (D) Signup and view all the answers

What is the product rule for logarithms?

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

What can be inferred about the graph of an exponential function?

The domain includes all real numbers. (B) Signup and view all the answers

What does the change of base formula for logarithms express?

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

Which application of logarithms could be used to model population growth?

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

Which of the following is the correct logarithmic identity?

log_a(a) = 1 (B) Signup and view all the answers

What is the sum of the first 100 positive integers?

5050 (A) Signup and view all the answers

Which of the following is NOT a characteristic of a function?

A single element in the domain can map to multiple elements in the range. (B) Signup and view all the answers

What is the key property required for a function to have an inverse function that is also a function?

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

Which of the following correctly describes the graphical relationship between a function and its inverse?

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

What is the first step in finding the inverse of a function?

Interchange the independent and dependent variables (x and y). (B) Signup and view all the answers

What does the notation (f^{-1}(x)) represent?

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

Which of the following is the general formula for the sum of a finite arithmetic series?

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

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

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

What is the condition for the sum of an infinite geometric series to exist?

The common ratio (r) must be less than 1. (A) Signup and view all the answers

What is the general form of the summation notation?

(\sum_{i=1}^{n} a_i) (A) Signup and view all the answers

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

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

What is the general form of the summation notation?

$\sum_{i=m}^{n} T_i = T_m + T_{m+1} + \cdots + T_{n-1} + T_n$ (A) Signup and view all the answers

What is the condition for the sum of an infinite geometric series to exist?

-1 < r < 1 (A) Signup and view all the answers

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

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

What is the definition of a finite series?

A series that sums a specific number of terms of a sequence. (C) Signup and view all the answers

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

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

What is the definition of a geometric sequence?

A sequence where each term is found by multiplying the previous term by a constant value. (C) 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 unlimited number of terms. (A) Signup and view all the answers

What happens to an infinite geometric series when r = 1?

The series diverges to infinity. (B) Signup and view all the answers

Which of the following is NOT a key formula used in solving cubic equations?

The discriminant. (C) Signup and view all the answers

What is the derivative of x^n?

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

What is the derivative of a constant?

0 (C) Signup and view all the answers

What is the derivative of kf(x)?

k*f'(x) (C) Signup and view all the answers

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

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

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

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

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

f'(x), y', Dy/Dx, Df(x) and D_xy (D) Signup and view all the answers

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

The gradient of the tangent to the curve at a point. (C) Signup and view all the answers

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

All of the above. (D) Signup and view all the answers

What does concavity indicate about a curve?

Whether the gradient of a curve is increasing or decreasing (B) Signup and view all the answers

When should you use the rules for differentiation?

When the question does not specify how to determine the derivative. (A) Signup and view all the answers

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

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

What is the purpose of finding the equation of a tangent to a curve?

To analyze the behavior of the function at a point. (B) Signup and view all the answers

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

To find the quotient and remainder (D) Signup and view all the answers

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

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

What does the first derivative of a function indicate?

The rate of change of the function (C) Signup and view all the answers

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

To find the equation of a tangent line (A) Signup and view all the answers

What is the general form of a cubic polynomial?

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

What does the second derivative of a function indicate?

The rate of change of the gradient (C) Signup and view all the answers

What is the purpose of the Remainder Theorem?

To find the remainder when a polynomial is divided by a linear polynomial (B) Signup and view all the answers

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

The product is -1 (D) Signup and view all the answers

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

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

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

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

What is the significance of a stationary point on a graph?

It indicates where the gradient of the curve is zero (D) Signup and view all the answers

What does a stationary point on a graph signify?

A point where the derivative is zero (D) Signup and view all the answers

How do you find the x-coordinates of stationary points for a function?

Find the derivative of the function and set it to 0 (D) Signup and view all the answers

What is the last step in finding the equation of a tangent line to a function?

Write the equation in the form y = mx + c (A) Signup and view all the answers

What type of stationary point occurs when a function changes from increasing to decreasing?

Local maximum (C) Signup and view all the answers

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

f''(x) (A), d^2y/dx^2 (D) Signup and view all the answers

What is the effect of the coefficient 'a' on a cubic function?

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

What is the first step in finding the equation of a tangent line to a function?

Find the derivative of the function (D) Signup and view all the answers

In the function (y = \frac{x^2 + 4x - 12}{x + 6}), what happens to the function when (x = -6)?

The function is undefined because the denominator becomes zero. (D) Signup and view all the answers

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

A hole. (C) Signup and view all the answers

Which of the following is a correct notation for the limit of a function (f(x)) as (x) approaches (a)?

(lim_{x \to a} f(x)) (A) Signup and view all the answers

In the context of the Achilles and the Tortoise paradox, what does this illustrate about limits?

That the distance between Achilles and the tortoise will eventually become infinitely small. (B) 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 (A) Signup and view all the answers

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

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

What is the graphical representation of an arithmetic sequence?

A straight line (D) Signup and view all the answers

What is the formula to find the arithmetic mean between two numbers?

(a + b) / 2 (C) Signup and view all the answers

What is the condition for the sum of an infinite geometric series to exist?

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

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

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

What is the graphical representation of a geometric sequence?

An exponential curve (D) Signup and view all the answers

What is the formula to find the geometric mean between two numbers?

sqrt(ab) (B) Signup and view all the answers

What is the definition of a series?

A sum of the terms of a sequence (B) Signup and view all the answers

What is the symbol used to represent the sum of a series?

Σ (A) 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 (C) Signup and view all the answers

What is the symbol Σ used to denote in mathematics?

A sum of terms in a sequence (A) Signup and view all the answers

What is the general form of the summation notation?

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

What is the definition of a geometric sequence?

A sequence of numbers where each term is found by multiplying the previous term by a constant value (C) Signup and view all the answers

What is the general formula for a finite geometric series?

Sn = a(1 - rn) / (1 - r) (B), Sn = a(rn - 1) / (r - 1) (D) Signup and view all the answers

What is the condition for the sum of an infinite geometric series to exist?

-1 < r < 1 (A) Signup and view all the answers

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

S∞ = a / (1 - r) (C) Signup and view all the answers

What is the definition of a finite series?

The sum of the terms of a sequence where the number of terms is finite (D) 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 (D) Signup and view all the answers

What happens to an infinite geometric series when r = 1?

The series diverges to infinity (A) Signup and view all the answers

What is the definition of an arithmetic sequence?

A sequence of numbers where each term is found by adding a constant value to the previous term (B) Signup and view all the answers

What is the value of loga 1?

0 (A) Signup and view all the answers

What is the product rule of logarithms?

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

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

Reflection of the original function about the line y = x (D) Signup and view all the answers

What is the application of logarithms in pH levels?

To use the formula pH = -log10[H+] to calculate pH levels (D) Signup and view all the answers

What is the range of the logarithmic function?

y ∈ ℝ (D) Signup and view all the answers

What is the formula for population growth?

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

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

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

What is the inverse of the function (y = ax^2) when (a > 0)?

(y = \sqrt{rac{x}{a}}) (C) Signup and view all the answers

If a quadratic function is not naturally one-to-one, what needs to be done to ensure its inverse is a function?

Restrict the domain of the original function. (A) Signup and view all the answers

Which of the following is the correct definition of a logarithm?

If (x = b^y), then (y = \log_b(x)). (B) Signup and view all the answers

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

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

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

(x > 0) (B) Signup and view all the answers

Which of the following functions is increasing?

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

What is the shape of the graph of the exponential function (y = b^x) where (b > 1)?

Increasing and concave up (A) Signup and view all the answers

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

(y = 0) (C) Signup and view all the answers

Which of the following is a correct conversion from exponential to logarithmic form?

(4^3 = 64) to (\log_4 64 = 3) (B) Signup and view all the answers

Which of the following is a correct formula for the sum of an arithmetic series?

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

Which of the following describes a function where each element of the domain maps to a unique element of the range?

One-to-one function (A) Signup and view all the answers

Which of the following is a key property of inverse functions?

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

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

The inverse function is a reflection of the original function across the line (y = x). (A) Signup and view all the answers

What is the sum of the first 100 positive integers?

5050 (C) Signup and view all the answers

Which of the following is NOT a correct statement about relations and functions?

All relations are functions. (C), A function can have multiple outputs for a single input. (D) Signup and view all the answers

Which of the following graphical representations represents a function?

A graph where every vertical line intersects the graph at most once. (D) Signup and view all the answers

Which of the following best describes the horizontal line test for determining if a function has an inverse that is also a function?

If a horizontal line intersects the graph at most once, then the function has an inverse that is also a function. (A) Signup and view all the answers

What is the first step in finding the inverse of a function?

Interchange (x) and (y) in the equation (y = f(x)). (D) Signup and view all the answers

Which of the following is NOT a characteristic of a one-to-one function?

It can have multiple outputs for a single input. (C) Signup and view all the answers

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

11 (B) Signup and view all the answers

According to the Factor Theorem, which of the following is a factor of (p(x) = x^3 - 6x^2 + 11x - 6)?

x - 1 (A), x - 2 (C) Signup and view all the answers

Which of the following is the correct general form of the polynomial (p(x)) after division by a linear factor (cx - d)?

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

What is the degree of the quotient polynomial (Q(x)) when a polynomial (p(x)) of degree (n) is divided by a linear polynomial (cx - d)?

(n - 1) (A) Signup and view all the answers

What is the first step in solving a cubic equation using factorization methods?

Identify a factor using the Factor Theorem. (C) Signup and view all the answers

Which of the following is the correct formula for the (n)-th term of an arithmetic sequence?

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

If the first term of an arithmetic sequence is 5 and the fourth term is 17, what is the common difference?

4 (B) Signup and view all the answers

Which of the following is a factor of the polynomial (p(x) = x^3 - 3x^2 - 4x + 12)?

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

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

(2) (D) Signup and view all the answers

Given an arithmetic sequence with the first term 3 and a common difference of 2, what is the 10th term?

21 (B) Signup and view all the answers

What happens to the function when $x$ approaches -6?

The function is undefined but has a limit. (B) Signup and view all the answers

How does the Achilles and the Tortoise paradox relate to limits?

It illustrates that limits can exist even if a function is undefined. (C) Signup and view all the answers

What graphical feature does the function $y = rac{x^2 + 4x - 12}{x + 6}$ exhibit at $x = -6$?

A hole at $y = -8$. (D) Signup and view all the answers

What does the factorization of the function $y = rac{x^2 + 4x - 12}{x + 6}$ reveal?

When factored, it reveals a removable discontinuity. (A) Signup and view all the answers

What is the role of limits in calculus?

To analyze the behavior of functions as they approach specific points. (C) Signup and view all the answers

In the context of the function $y = rac{x^2 + 4x - 12}{x + 6}$, what limit does $y$ approach as $x$ nears -6?

-8 (B) Signup and view all the answers

What is the correct definition of the derivative using first principles?

$f'(x) = rac{f(x + h) - f(x)}{h}$ (C) Signup and view all the answers

Which of the following notations represents the derivative of a function correctly?

$rac{df}{dx}$ (A), $D_x f$ (B) Signup and view all the answers

What does the derivative of a constant equal?

0 (C) Signup and view all the answers

Which rule allows you to differentiate a sum of two functions?

Sum Rule (C) Signup and view all the answers

What is the general formula for differentiating a power function?

$rac{d}{dx}[x^n] = nx^{n-1}$ (D) Signup and view all the answers

In the context of calculus, what does the symbol $D$ represent?

The derivative operator (D) Signup and view all the answers

When would you choose to apply differentiation from first principles instead of rules of differentiation?

When asked specifically to do so (B) Signup and view all the answers

What is the meaning of $f'(x)$?

The derivative of $f(x)$ with respect to $x$ (B) Signup and view all the answers

What happens to the gradient of the tangent to a curve at a point?

It changes with the function. (D) Signup and view all the answers

Which rule is applied when differentiating a product of two functions?

Product Rule (B) Signup and view all the answers

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

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

If a polynomial p(x) is divided by cx - d and the remainder is zero, what can be concluded?

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

What is the degree of the quotient Q(x) when a polynomial p(x) of degree n is divided by a linear polynomial cx - d?

n-1 (C) Signup and view all the answers

What is the first step in solving a cubic equation of the form ax^3 + bx^2 + cx + d = 0?

Find a factor using the factor theorem (D) Signup and view all the answers

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

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

What is the common difference in an arithmetic sequence if the first term is 5 and the fourth term is 17?

4 (A) Signup and view all the answers

Which of the following is NOT a step in solving a cubic equation?

Taking the derivative of the polynomial (D) Signup and view all the answers

What is the degree of the polynomial Q(x) when a polynomial p(x) of degree n is divided by a linear polynomial cx - d?

n-1 (D) Signup and view all the answers

What is the formula for the sum of an infinite geometric series with first term a and common ratio r?

a / (1 - r) (A) Signup and view all the answers

What is the condition for the sum of an infinite geometric series to exist?

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

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

The second derivative of the function changes sign. (B) Signup and view all the answers

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

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

Which of the following is a correct step in determining the concavity of a cubic function?

Calculate the second derivative and determine its sign. (D) Signup and view all the answers

When dividing a polynomial (a(x)) by (b(x)), what does (R(x)) represent?

The remainder of the division. (B) Signup and view all the answers

What is the first step in finding the equation of a tangent line to a function?

Determine the slope of the tangent line. (C) Signup and view all the answers

Which of the following is NOT a key formula used in solving cubic equations?

Quadratic Formula (D) Signup and view all the answers

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

a_n = a_1 + (n - 1)d (A) Signup and view all the answers

Which of the following is a correct step in finding the x-intercepts of a cubic function?

Solve the equation f(x) = 0 to find the roots of the function. (D) Signup and view all the answers

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

Quadratic Formula (A) Signup and view all the answers

What does the symbol (\Sigma) represent?

The sum of a sequence (C) Signup and view all the answers

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

The product of their gradients is -1. (A) Signup and view all the answers

What is the second derivative of a function, and what does it indicate?

The second derivative is the derivative of the first derivative and indicates the change in gradient of the original function. (C) 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)?

The coefficient 'a' controls the direction of the graph's rise and fall. (C) Signup and view all the answers

What are the steps involved in finding the equation of a tangent line to a function (f(x)) at a point (x = a)?

Find (f'(x)), calculate (f'(a)), use the point-slope form. (D) Signup and view all the answers

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

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

What does the summation symbol (\Sigma) represent?

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

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

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

What is the first step in finding the equation of a tangent line to a function?

Find the derivative of the function. (A) Signup and view all the answers

What does a stationary point signify on a graph of a function?

A point where the function changes from increasing to decreasing or vice versa. (A), A point where the tangent line is horizontal. (C) Signup and view all the answers

What is the condition for the sum of an infinite geometric series to exist?

-1 < r < 1 (B) Signup and view all the answers

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

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

What type of stationary point occurs when a function changes from increasing to decreasing?

Local Maximum (D) Signup and view all the answers

What are the steps involved in finding the normal line to a curve at a given point?

Find the derivative, find the negative reciprocal of the slope, use the point-slope form. (B) Signup and view all the answers

What is the definition of a finite series?

The sum of the first n terms of a sequence (A) Signup and view all the answers

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

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

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

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

What is the definition of a geometric sequence?

A sequence with a common ratio (A) Signup and view all the answers

What is the general form of the summation notation?

( \sum_{i=m}^{n} T_i = T_m + T_{m+1} + \cdots + T_{n-1} + T_n ) (B) Signup and view all the answers

What happens to an infinite geometric series when r = 1?

The series diverges to infinity (B) Signup and view all the answers

What is the sum of the first 100 integers?

5050 (A) Signup and view all the answers

What is the general formula for a finite arithmetic series?

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

What is a relation?

A rule that associates each element of one set with at least one element of another set (B) Signup and view all the answers

What is a function?

A relation where each element in set A maps to exactly one element in set B (C) Signup and view all the answers

What is an inverse function?

A function that reverses the operation of another function (A) 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 (D) Signup and view all the answers

What is the horizontal line test?

A test to determine if a function has an inverse that is also a function (A) Signup and view all the answers

What is the formula to find the inverse of a function?

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

What is the notation for the inverse function?

f^-1(x) (D) Signup and view all the answers

What is the graph of the inverse function symmetrical to?

The line y = x (D) Signup and view all the answers

What is the common difference in an arithmetic sequence if the first term is 4 and the second term is 10?

5 (B) Signup and view all the answers

Which formula correctly defines the arithmetic mean of two numbers?

Mean = $\frac{a + b}{2}$ (A) Signup and view all the answers

How can you identify if a sequence is geometric?

By calculating the ratios between consecutive terms. (D) Signup and view all the answers

What occurs when the common ratio in a geometric sequence is between 0 and 1?

The sequence decays exponentially. (C) Signup and view all the answers

What type of graph is formed when plotting the terms of an arithmetic sequence against their positions?

Straight line (C) Signup and view all the answers

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

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

What defines a series in the context of sequences?

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

What will the sum of the first n terms of a geometric sequence be in sigma notation?

$S_n = \sum_{i=1}^n ar^{i-1}$ (A) Signup and view all the answers

What happens to an infinite geometric series when the common ratio is greater than 1?

It diverges to infinity. (B) Signup and view all the answers

What does the geometric mean between two numbers a and b equal?

$\sqrt{ab}$ (B) Signup and view all the answers

What is the value of (\log_a 1)?

0 (B) Signup and view all the answers

Which of the following is NOT a law of logarithms?

Inverse Rule (C) Signup and view all the answers

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

All positive real numbers (A) Signup and view all the answers

Which of the following is the formula for calculating the pH level of a solution?

(\text{pH} = -\log_{10}[\text{H}^+]) (D) Signup and view all the answers

If a population triples in size, what is the value of (n) in the formula (3P = P(1 + i)^n)?

Cannot be determined without knowing the value of (i) (A) Signup and view all the answers

Which of the following is the correct formula for the change of base rule for logarithms?

(\log_a x = \frac{\log_b x}{\log_b a}) (B) Signup and view all the answers

What is the first step in finding the inverse of the function $y = ax + q$?

Interchange $x$ and $y$. (A) Signup and view all the answers

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

The range of the original function becomes the domain of the inverse function. (A) Signup and view all the answers

For the quadratic function $y = ax^2$, what is necessary for its inverse to also be a function?

The function must be restricted to its vertex. (B) Signup and view all the answers

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

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

What occurs to the graph of an exponential function when the base $b$ is between 0 and 1?

The function decreases rapidly. (A) Signup and view all the answers

Which of the following statements about logarithms is correct?

The logarithm is undefined when the input is less than or equal to zero. (B) Signup and view all the answers

In the function $y = b^x$, what does the horizontal asymptote represent?

The value it approaches at infinity. (D) Signup and view all the answers

Which of the following represents the general form for the inverse of the function $y = ax^2$?

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

For which condition is the function $y = b^x$ not defined?

$b < 0$ (D) Signup and view all the answers

Gr12 Mathematics: June Exam Easy P(1)

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Questions and Answers

What is the definition of a limit in the context of calculus?

In the function $y = \frac{x^2 + 4x - 12}{x + 6}$, what happens when $x$ equals -6?

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

According to the Achilles and the Tortoise paradox, what does this illustrate about limits?

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

As $x$ approaches -6 in the function $y = \frac{x^2 + 4x - 12}{x + 6}$, what does $y$ approach?

What is the derivative of $x^4$?

What is the derivative of $5x^2$?

What is the derivative of $3x^3 + 2x$?

What is the derivative of $2x^2 - 3x$?

What is the derivative of $7$?

Which of the following is a correct notation for the derivative of a function $f(x)$?

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

What is the gradient of the tangent to the curve $y = x^3$ at the point $x = 2$?

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

What is the derivative of $f(x) = \frac{1}{x}$?

What does the second derivative indicate about a function?

If the coefficient 'a' of a cubic function is negative, what can be inferred about the graph?

Which of the following denotes the second derivative of a function?

What is the first step in finding the equation of a tangent line to a function?

What relationship exists between the gradients of the tangent and the normal at a point on a curve?

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

What does a stationary point signify on a graph?

What is required to determine the normal line to a curve at a given point?

What type of stationary point occurs when a function changes from increasing to decreasing?

To find the x-coordinates of stationary points for a function, what must be solved?

What does the symbol (\Sigma) represent?

What is the general formula for a finite geometric series?

What is the condition for the sum of an infinite geometric series to exist?

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

What is the definition of a finite series?

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

What is the definition of a geometric sequence?

What is the general form of the summation notation?

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

What happens to an infinite geometric series when r = 1?

Which of the following is the correct expression for the remainder (R) when a polynomial (p(x)) is divided by (cx - d)?

According to the Factor Theorem, if (cx - d) is a factor of polynomial (p(x)), what is the value of (p(d/c)) ?

What is the degree of the quotient (Q(x)) when a polynomial (p(x)) of degree (n) is divided by a linear polynomial (cx - d)?

Which of the following is NOT a key formula used in solving cubic equations?

In the process of solving a cubic equation, after finding a factor, how do you obtain the remaining polynomial factors?

What is the common difference in an arithmetic sequence if the first term is 5 and the fourth term is 17?

Which of the following is the correct formula for the (n)-th term of an arithmetic sequence?

Given an arithmetic sequence with the first term 3 and a common difference of 2, what is the 10th term?

If the 5th term of an arithmetic sequence is 12 and the common difference is 3, what is the first term?

Which of the following sequences is NOT an arithmetic sequence?

What is the steps to find the inverse of a linear function?

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

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

What is the domain and range of the inverse function f^-1(x) = 1/a x - q/a?

What is the definition of the logarithm?

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

What is the intercept of the logarithmic function y = log_b x?

What is the asymptote of the logarithmic function y = log_b x?

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

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

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

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

Which of the following sequences is an arithmetic sequence?

What is the sum of the first 10 terms of the arithmetic sequence 3, 7, 11, 15, ...?

Which of the following is a characteristic of an arithmetic sequence?

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

What does the gradient (slope) of the line representing an arithmetic sequence represent?

What is the sum of the first 5 terms of the geometric sequence 2, 4, 8, 16, ...?

Which of the following is true about the graphical representation of a geometric sequence?

What does it indicate when a graph is concave up?

Which equation indicates a point of inflection?

When performing synthetic division, what should you use as the divisor's root?

What does the Remainder Theorem state about polynomial division?

Which method is not used for factorizing cubic polynomials?

Which statement correctly describes a stationary point?

What is meant by the term 'optimisation problem' in calculus?

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

What do the signs of the coefficients in a cubic polynomial indicate?

Which of the following correctly defines a cubic polynomial?

What is the sum of the first 100 positive integers?