Gr12 Mathematics: Term test 1

What is the definition of an arithmetic sequence?

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

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

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

How do you check if a sequence is arithmetic?

Verify if the ratio of consecutive terms is constant.
Verify if the sum of consecutive terms is constant.
Verify if the product of consecutive terms is constant.
Verify if the difference between consecutive terms is constant. (correct)

What is the arithmetic mean between two numbers?

The average of the two numbers (B) 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 happens to an arithmetic sequence if the common difference is positive?

The sequence increases (B) Signup and view all the answers

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

They form a linear pattern (D) Signup and view all the answers

How do you find the nth term of an arithmetic sequence?

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

What is the common ratio in a geometric sequence?

The ratio between consecutive terms (A) Signup and view all the answers

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

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

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

The square root of ab (C) Signup and view all the answers

What is the graph of a geometric sequence?

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

How do you test if a sequence is geometric?

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

What is the definition of a series?

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

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

Sn (D) Signup and view all the answers

What is the formula for an infinite series?

S∞ = T1 + T2 + ... (C) Signup and view all the answers

What is the purpose of sigma notation?

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

What is the index of summation in sigma notation?

The starting index of the sequence (B) Signup and view all the answers

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

Sn = a(1 - r^n) / (1 - r) (D) 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∞ = a / (1 - r) (A) Signup and view all the answers

What is the definition of an arithmetic sequence?

A sequence of numbers where each term is added by a constant value. (B) 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 method used by Karl Friedrich Gauss to find the sum of the first 100 integers?

Writing the numbers in ascending and descending order and adding corresponding pairs of terms. (A) 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 definition of a finite geometric series?

A series where a known number of terms in a geometric sequence are summed. (D) Signup and view all the answers

What is the definition of an infinite geometric series?

A series where an infinite number of terms in a geometric sequence are summed. (D) Signup and view all the answers

When does an infinite geometric series converge?

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

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

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

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

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

What is the restriction on the domain of y = ax^2 for its inverse to be a function?

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

What is the graph of f(x) = -x^2?

Falls rapidly (D) Signup and view all the answers

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

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

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

$S_n = \frac{n}{2} (a + l)$ (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 condition for a function to have an inverse that is also a function?

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

What is the shape of the graph of the logarithmic function y = logb(x)?

Increasing (A) Signup and view all the answers

What is the special logarithmic value of loga(a)?

1 (B) 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. (A) Signup and view all the answers

What is the definition of a relation?

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

What is the Quotient Rule of logarithms?

loga(x/y) = loga(x) - loga(y) (A) Signup and view all the answers

What is the formula for the sum of an arithmetic sequence if the last term is unknown?

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

What is the Power Rule of logarithms?

loga(x^b) = b loga(x) (A) Signup and view all the answers

What is the property of an inverse function?

It reverses the operation of a given function. (C) Signup and view all the answers

What is the notation for the inverse function?

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

What is the graphical representation of the inverse of a function?

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

What is the condition for a function to be many-to-one?

Multiple elements of the domain map to the same element of the range. (B) Signup and view all the answers

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

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

What is the formula to change the base of a logarithm?

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

What is the reflection of the graph of an exponential function about?

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

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

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

What is the formula for pH levels?

pH = -log10[H+] (B) Signup and view all the answers

What is the formula for the cosine of a difference?

cos( - ) = cos cos + sin sin (C) Signup and view all the answers

How are the compound angle formulas derived?

Using the distance formula and cosine rule (B) Signup and view all the answers

What is the application of logarithms in finance?

Calculating loan repayments and interest rates (A) Signup and view all the answers

What is the formula for population growth?

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

What is the formula to find the angles in the interval [0°, 360°] that satisfy the equation sin θ = x?

θ = sin⁻¹x + k × 360° (A), θ = 180° - sin⁻¹x + k × 360° (C) Signup and view all the answers

What is the formula for the sine of a difference?

$\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$ (B) Signup and view all the answers

When should you use the Area Rule to find the area of a triangle?

When no perpendicular height is given (B) Signup and view all the answers

What is the formula for the cosine of a sum?

$\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ (D) Signup and view all the answers

What is the formula to find the area of a triangle using the Sine Rule?

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

What is the formula for the sine of a double angle?

$\sin(2\alpha) = 2 \sin \alpha \cos \alpha$ (C) Signup and view all the answers

When should you use the Cosine Rule to find the length of a side in a triangle?

When three sides are given (D) Signup and view all the answers

What is the formula for the cosine of a double angle?

$\cos(2\alpha) = \cos^2 \alpha - \sin^2 \alpha$ (D) Signup and view all the answers

What is the formula to find the height of a pole using trigonometry?

h = (d sin α) / (sin β) (D) Signup and view all the answers

How many general solutions are there for a trigonometric equation?

Infinitely many (C) Signup and view all the answers

What is the general approach to solving problems in three dimensions?

Draw a sketch, consider the given information, apply appropriate rules, and calculate the desired quantities (B) Signup and view all the answers

What is the formula to find the length of a side in a triangle using the Cosine Rule?

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

What is the first step in solving a trigonometric equation?

Simplify the equation using algebraic methods and trigonometric identities (B) Signup and view all the answers

When should you use the Sine Rule to find the length of a side in a triangle?

When two angles and a side are given (A) Signup and view all the answers

What is the purpose of the CAST diagram in solving trigonometric equations?

To determine where the function is positive or negative (C) Signup and view all the answers

What is the formula to find the area of a triangle using the Area Rule?

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

What is the formula for the cosine of a difference?

$\cos(\alpha - \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ (A) Signup and view all the answers

What is the formula for the sine of a sum?

$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$ (D) Signup and view all the answers

What is the formula to find the height of a building using trigonometry?

h = (b sin α sin θ) / (sin(β + θ)) (A) Signup and view all the answers

What is an alternate form of the cosine of a double angle?

$\cos(2\alpha) = 1 - 2 \sin^2 \alpha$ (C) Signup and view all the answers

What is the common difference of an arithmetic sequence if the first term is 5 and the fifth term is 17?

3 (A) Signup and view all the answers

If an arithmetic sequence has a first term of 2 and a common difference of 4, what is the 10th term?

38 (B) Signup and view all the answers

Which of the following sequences is NOT an arithmetic sequence?

2, 4, 8, 16, 32 (D) Signup and view all the answers

What is the number of terms in an arithmetic sequence if the first term is 3, the common difference is 2, and the last term is 19?

9 (B) Signup and view all the answers

What is the arithmetic mean between 5 and 9?

7.5 (B) Signup and view all the answers

What is the graph of an arithmetic sequence when plotted against its position?

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

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

The terms form a linear pattern (C) Signup and view all the answers

If an arithmetic sequence has a first term of 10 and a common difference of -3, what is the direction of the sequence?

Decreasing (D) Signup and view all the answers

To find the nth term of a geometric sequence, what formula should be used?

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

What is the purpose of sigma notation?

To denote the sum of a finite sequence (D) Signup and view all the answers

What is the condition for a sequence to be geometric?

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

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

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

What is the graphical representation of a geometric sequence?

An exponential graph (A) Signup and view all the answers

What is the sum of a finite series denoted by?

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

What is the index of summation in sigma notation?

$i$ (C) Signup and view all the answers

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

A finite series has a specific number of terms, while an infinite series has infinitely many terms (D) Signup and view all the answers

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

To find the nth term of the sequence (B) Signup and view all the answers

What happens to a geometric sequence if the common ratio is negative?

The sequence alternates in sign (A) Signup and view all the answers

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

$S_n = \frac{a(1 - r^n)}{1 - r}$ (B) 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?

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

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

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

What is the method used by Karl Friedrich Gauss to find the sum of the first 100 integers?

Writing the numbers in ascending and descending order and adding corresponding pairs of terms (A) Signup and view all the answers

What is the sum of a finite arithmetic series?

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

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

A finite geometric series has a fixed number of terms, whereas an infinite geometric series has an infinite number of terms (C) Signup and view all the answers

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

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

When does an infinite geometric series converge?

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

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

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

What is the formula for the sum of a finite arithmetic series if the last term is unknown?

$S_n = \frac{n}{2} [2a + (n - 1) d]$ (A) 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 the condition for a function to have an inverse that is also a function?

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

What is the graphical representation of the inverse of a function?

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

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

f^{-1}(x) = \frac{x - q}{a} (B) Signup and view all the answers

What is the property of an inverse function?

An inverse function reverses the operation of a given function. (B) Signup and view all the answers

What is the notation for the inverse function?

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

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

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

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

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

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

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

What is the formula for the inverse of the exponential function f(x) = 10^x?

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

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

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

What is the formula for the cosine of a difference of two angles?

cos(α - β) = cos α cos β + sin α sin β (A) Signup and view all the answers

What is the formula for the population growth, where A is the population after n years, P is the initial population, and i is the growth rate?

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

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

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

What is the formula for the sine of a sum of two angles?

sin(α + β) = sin α cos β + cos α sin β (A) Signup and view all the answers

What is the formula for changing the base of a logarithm?

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

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

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

What is the exponential function that has an inverse of y = logb(x)?

y = b^x (D) Signup and view all the answers

What is the formula for the pH level of a solution?

pH = -log10[H+] (A) Signup and view all the answers

What is the logarithmic function that has an exponential function of y = b^x?

y = logb(x) (A) Signup and view all the answers

What is the reflection of the graph of an exponential function about the line y = x?

The graph of the logarithmic function (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 given value. (C) Signup and view all the answers

What is the shape of the graph of the logarithmic function y = logb(x)?

Increasing (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 Power Rule of logarithms?

loga(x^b) = b × loga(x) (D) Signup and view all the answers

What is the special logarithmic value of loga(a)?

1 (A) Signup and view all the answers

What is the Quotient Rule of logarithms?

loga(x/y) = loga(x) - loga(y) (D) Signup and view all the answers

What is the formula for the sine of a difference?

sin(α - β) = sin α cos β - cos α sin β (A) Signup and view all the answers

What is the formula for the cosine of a sum?

cos(α + β) = cos α cos β - sin α sin β (D) Signup and view all the answers

What is the formula for the sine of a double angle?

sin(2α) = 2 sin α cos α (D) Signup and view all the answers

What is the formula for the cosine of a double angle?

cos(2α) = cos^2 α - sin^2 α (A), cos(2α) = 2 cos^2 α - 1 (C), cos(2α) = 1 - 2 sin^2 α (D) Signup and view all the answers

What are the steps to solve a trigonometric equation?

<ol> <li>Simplify the equation, 2. Find the reference angle, 3. Use the CAST diagram, 4. Find angles within a specified interval (D)</li> </ol> Signup and view all the answers

What is the purpose of using the CAST diagram in solving trigonometric equations?

To determine where the function is positive or negative (D) Signup and view all the answers

What is the result of using the sum formula for sine with α = β?

sin(2α) = 2 sin α cos α (A) Signup and view all the answers

What is the result of using the sum formula for cosine with α = β?

cos(2α) = cos^2 α - sin^2 α (A), cos(2α) = 1 + 2 sin^2 α (B), cos(2α) = 2 cos^2 α - 1 (C) Signup and view all the answers

What is the general method to solve trigonometric equations?

Simplify, find the reference angle, use the CAST diagram, and find angles within a specified interval (A) Signup and view all the answers

What is the importance of finding the general solution of a trigonometric equation?

It helps to find all possible solutions to the equation (D) Signup and view all the answers

What is the formula to find the height of a pole in a three-dimensional problem?

h = FB tan β (B) Signup and view all the answers

What is the formula to calculate the area of a triangle using the sine rule?

Area = (1/2)ab sin C (A), Area = (1/2)bc sin A (C), Area = (1/2)ac sin B (D) Signup and view all the answers

What is the condition for using the cosine rule in a triangle?

When no right angle is given, and two sides and the included angle are given (D) Signup and view all the answers

What is the formula to find the length of a side in a triangle using the sine rule?

a = b sin A / sin B (C) Signup and view all the answers

What is the first step in solving a three-dimensional problem?

Draw a sketch of the problem (A) Signup and view all the answers

What is the formula to find the length of a side in a triangle using the cosine rule?

a^2 = b^2 + c^2 - 2bc cos A (B) Signup and view all the answers

What is the condition for using the area rule in a triangle?

When no perpendicular height is given (A) Signup and view all the answers

What is the formula to find the area of a triangle using the sine rule?

Area = (1/2)ab sin C (A) Signup and view all the answers

What is the formula to find the length of a side in a triangle using the tangent ratio?

tan A = opposite side / adjacent side (C) Signup and view all the answers

What is the purpose of the sine rule in trigonometry?

To find the area of a triangle (B), To find the length of a side in a triangle (D) Signup and view all the answers

What is the value of the 10th term in an arithmetic sequence with a common difference of 3 and a first term of 2?

31 (A) Signup and view all the answers

A sequence of numbers is defined by the formula Tn = 2n + 1. Is this sequence arithmetic?

No, because the common difference is not constant (A) Signup and view all the answers

What is the equation of the straight line obtained by plotting the terms of an arithmetic sequence against their positions?

y = dx + a (C) Signup and view all the answers

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

-7 (B) Signup and view all the answers

What is the sum of the first 10 terms of an arithmetic sequence with a first term of 2 and a common difference of 3?

150 (B) Signup and view all the answers

If the 5th term of an arithmetic sequence is 11 and the common difference is 2, what is the first term?

3 (C) Signup and view all the answers

What is the number of terms in an arithmetic sequence with a first term of 3, a common difference of 2, and a last term of 15?

8 (D) Signup and view all the answers

If the arithmetic mean between two numbers is 8, and one of the numbers is 6, what is the other number?

10 (D) Signup and view all the answers

What is the sum of the first 10 terms of the geometric series 2 + 6 + 18 + ...?

2366 (D) Signup and view all the answers

What is the common ratio of the geometric series 3 + 12 + 48 + ...?

4 (D) Signup and view all the answers

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

6 (C) Signup and view all the answers

What is the sum of the first 15 terms of the arithmetic series 5 + 7 + 9 + ...?

195 (C) Signup and view all the answers

What is the common difference of the arithmetic series 10 + 15 + 20 + ...?

5 (B) Signup and view all the answers

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

2 (A) Signup and view all the answers

What is the sum of the first 20 terms of the geometric series 3 + 9 + 27 + ...?

177147 (C) Signup and view all the answers

What is the common ratio of the geometric series 4 + 16 + 64 + ...?

4 (A) Signup and view all the answers

What is the sum of the infinite geometric series 2 + 4/3 + 16/9 + ...?

4 (C) Signup and view all the answers

What is the geometric sequence that forms when the geometric mean between 2 and 3 is calculated?

2, $-\sqrt{6}$, $-3\sqrt{6}$, ... (B), 2, $\sqrt{6}$, $3\sqrt{6}$, ... (C) Signup and view all the answers

What is the common ratio of a geometric sequence if the first term is 2 and the second term is 6?

3 (C) Signup and view all the answers

What is the sum of the first 12 terms of the arithmetic series 2 + 5 + 8 + ...?

84 (D) Signup and view all the answers

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

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

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

S_n (D) Signup and view all the answers

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

Sigma notation (B) Signup and view all the answers

What is the general form of the sum of terms in a sequence?

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

What is the characteristic of a geometric sequence if the common ratio is negative?

The sequence alternates in sign (C) Signup and view all the answers

What is the graphical representation of a geometric sequence?

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

What is the test to determine if a sequence is geometric?

Calculate the ratio between consecutive terms (C) Signup and view all the answers

What is the sum of an infinite geometric series if the first term is 2 and the common ratio is 1/2?

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

What is the general formula for the sum of a finite arithmetic series if the last term is unknown?

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

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

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

What is the graphical representation of the inverse of a function?

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

What is the formula for the sum of an arithmetic sequence if the first term and the common difference are known?

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

What is the property of an inverse function?

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

What is the definition of a relation?

A relation is a rule that associates each element of one set with at least one element of another set. (D) 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. (C) Signup and view all the answers

What is the notation for the inverse function?

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

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

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

What is the condition for a function to be many-to-one?

Multiple elements of the domain map to the same element of the range. (A) Signup and view all the answers

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

f^{-1}(x) = \sqrt{\frac{x}{a}} if x \geq 0 (A) Signup and view all the answers

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

y = log_2 x (A) Signup and view all the answers

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

x \geq 0 (D) Signup and view all the answers

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

Increasing (D) Signup and view all the answers

What is the special logarithmic value of log_a a?

1 (A) Signup and view all the answers

What is the property of an inverse function?

It is always one-to-one (B) Signup and view all the answers

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

It must be one-to-one (B) Signup and view all the answers

What is the notation for the inverse function?

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

What is the reflection of the graph of an exponential function about the line y = x?

A logarithmic function (C) Signup and view all the answers

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

a(1 - r^n) / (1 - r) (B) Signup and view all the answers

What is the reflection of the graph of a logarithmic function about?

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

What is the formula to change the base of a logarithm?

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

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

1/x (B) Signup and view all the answers

What is the formula for the population of a city after n years, given an initial population P and a constant growth rate i?

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

What is the formula for the cosine of a sum of two angles alpha and beta?

cos(alpha + beta) = cos alpha cos beta - sin alpha sin beta (B) Signup and view all the answers

What is the formula for the sine of a difference of two angles alpha and beta?

sin(alpha - beta) = sin alpha cos beta - cos alpha sin beta (B) Signup and view all the answers

What is the formula for the pH level of a solution given the hydrogen ion concentration [H+]?

pH = -log[H+] (D) Signup and view all the answers

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

A logarithmic curve (D) Signup and view all the answers

What is the general approach to solve three-dimensional problems?

Draw a sketch, consider the given information, apply trigonometric identities, and calculate the desired quantities (C) Signup and view all the answers

What is the formula for the height of a pole, given AB = d, ∠FBA = θ, ∠FAB = α, ∠FBT = β, and ∠TFB = 90°?

h = d sin α tan β / sin β (C) Signup and view all the answers

When is it appropriate to use the Cosine Rule?

When no right angle is given, and either two sides and the included angle or three sides are given (B) Signup and view all the answers

What is the formula for the area of a triangle ABC?

Area = (1/2)ac sin B (A), Area = (1/2)ab sin C (C), Area = (1/2)bc sin A (D) Signup and view all the answers

What is the formula for the height of a building, given BC = b, ∠DBA = α, ∠DBC = β, and ∠DCB = θ?

h = b sin α sin θ / sin(β + θ) (A) Signup and view all the answers

When is it appropriate to use the Sine Rule?

When no right angle is given, and either two sides and an angle (not the included angle) or two angles and a side are given (B) Signup and view all the answers

What is the formula for the area of a triangle, given the three sides?

Area = √(s(s-a)(s-b)(s-c)) (C) Signup and view all the answers

What is the general approach to solve trigonometric equations?

Find angles in the interval [0°, 360°] that satisfy the equation and add multiples of the period (D) Signup and view all the answers

What is the formula for the angle of elevation of a building, given the height and distance from the observer?

tan θ = h / d (C) Signup and view all the answers

What is the formula for the cosine of a sum?

$\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ (D) Signup and view all the answers

What is the formula for the sine of a double angle?

$\sin(2\alpha) = 2 \sin \alpha \cos \alpha$ (D) Signup and view all the answers

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

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

What is the formula for the cosine of a double angle?

$\cos(2\alpha) = \cos^2 \alpha - \sin^2 \alpha$ (C) Signup and view all the answers

What is the formula for the sine of a difference?

$\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$ (B) Signup and view all the answers

What is the method to derive the double angle formulas?

Using the sum and difference formulas (C) Signup and view all the answers

What is the purpose of the CAST diagram?

To determine where a trigonometric function is positive or negative (A) Signup and view all the answers

What is the step to find the general solution of a trigonometric equation?

Find angles within a specified interval by adding or subtracting multiples of the period (A) Signup and view all the answers

What is the formula for the cosine of a difference?

$\cos(\alpha - \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ (C) Signup and view all the answers

What is the formula for the sine of a sum?

$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$ (B) Signup and view all the answers

What is the purpose of using the sum and difference formulas?

To expand the trigonometric functions of sums and differences of angles (B) Signup and view all the answers

If an arithmetic sequence has a first term of 5 and a common difference of 3, what is the 10th term?

35 (D) Signup and view all the answers

What is the common difference of an arithmetic sequence if the 5th term is 11 and the 7th term is 15?

2 (B) Signup and view all the answers

An arithmetic sequence has a first term of 2 and a last term of 20. If the common difference is 2, how many terms are in the sequence?

10 (D) Signup and view all the answers

What is the arithmetic mean between 5 and 11?

8 (B) Signup and view all the answers

If an arithmetic sequence has a common difference of -3, which of the following is true?

The sequence is decreasing (D) Signup and view all the answers

What is the graphical representation of an arithmetic sequence?

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

An arithmetic sequence has a first term of 1 and a common difference of 2. Which of the following is true?

The sequence is linear (D) Signup and view all the answers

What is the purpose of checking the difference between consecutive terms in an arithmetic sequence?

To find the common difference (C) Signup and view all the answers

What is the sum of the first 10 terms of the geometric sequence 2, 6, 18, 54, ...?

4375 (D) Signup and view all the answers

What is the common ratio of the infinite geometric series 3, 12, 48, 192, ...?

4 (C) Signup and view all the answers

What is the sum of the first 5 terms of the arithmetic sequence 3, 7, 11, 15, ...?

40 (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 a finite geometric series with first term a and common ratio r?

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

What is the sum of the first 8 terms of the geometric sequence 4, 12, 36, 108, ...?

6536 (C) Signup and view all the answers

What is the common difference of the arithmetic sequence 2, 5, 8, 11, ...?

3 (A) 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) (B) Signup and view all the answers

What is the sum of the first 12 terms of the arithmetic sequence 1, 3, 5, 7, ...?

78 (A) Signup and view all the answers

What is the formula for the nth term of an arithmetic sequence with first term a and common difference d?

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

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

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

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

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

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

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

How do you test if a sequence is geometric?

Check if the ratio between consecutive terms is constant (C) Signup and view all the answers

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

$S_n = T_1 + T_2 + ... + T_n$ (A) Signup and view all the answers

What is the formula for an infinite series?

$S_\infty = T_1 + T_2 + T_3 + ...$ (D) Signup and view all the answers

What is the purpose of sigma notation?

To find the sum of a sequence (A) Signup and view all the answers

What is the index of summation in sigma notation?

The index of the term being summed (A) Signup and view all the answers

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

$S_n = a(r^n - 1) / (r - 1)$ (B) 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 formula to change the base of a logarithm?

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

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

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

What is the formula for the cosine of a difference?

$\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$ (B) Signup and view all the answers

What is the application of logarithms in calculating loan repayments?

To calculate the monthly repayment (D) Signup and view all the answers

What is the graph of the function y = log(x) like?

A curve that opens upwards (D) Signup and view all the answers

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

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

What is the formula for the population growth model?

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

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

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

What is the pH level formula?

pH = -log10[H+] (C) Signup and view all the answers

What is the sine of a sum formula?

$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$ (D) 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. (B) Signup and view all the answers

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

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

What is the property of an inverse function?

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

What is the notation for the inverse function?

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

What is the graphical representation of the inverse of a function?

The graph is reflected about the line y = x. (D) Signup and view all the answers

What is the correct inverse of the function y = ax^2, given that the domain of the original function is restricted to ensure the inverse is a function?

y = √(x/a) if x ≥ 0 (B) Signup and view all the answers

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

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

What is the condition for a function to be many-to-one?

Every horizontal line intersects the graph more than once. (D) Signup and view all the answers

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

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

What is the definition of a relation?

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

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

$S = rac{a}{1 - r}$ (C) Signup and view all the answers

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

y = log(x)/log(2) (A) Signup and view all the answers

What is the correct way to express the function y = x^2 in function notation?

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

What is the property of a function that is required for its inverse to be a function?

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

What is the formula to change the base of a logarithm from a to b?

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

What is the shape of the graph of the exponential function y = 2^x?

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

What is the definition of the logarithmic function?

The logarithmic function is the inverse of the exponential function (C) Signup and view all the answers

What is the restriction on the domain of y = ax^2 for its inverse to be a function?

x ≥ 0 if a > 0 (A) Signup and view all the answers

What is the formula for the sine of a difference of two angles?

sin(α - β) = sin(α)cos(β) - cos(α)sin(β) (B) Signup and view all the answers

What is the formula for the cosine of a sum of two angles?

cos(α + β) = cos(α)cos(β) + sin(α)sin(β) (A) Signup and view all the answers

What is the formula for the sine of a double angle?

sin(2α) = 2sin(α)cos(α) (C) Signup and view all the answers

What is the formula for the cosine of a double angle?

cos(2α) = cos^2(α) - sin^2(α) (C), cos(2α) = 1 - 2sin^2(α) (D) Signup and view all the answers

How do you simplify a trigonometric equation?

By using algebraic methods and trigonometric identities. (D) Signup and view all the answers

What is the purpose of the CAST diagram?

To determine where the function is positive or negative. (B) Signup and view all the answers

What is the formula for the sine of a sum of two angles?

sin(α + β) = sin(α)cos(β) + cos(α)sin(β) (C) Signup and view all the answers

What is the formula for the cosine of a difference of two angles?

cos(α - β) = cos(α)cos(β) + sin(α)sin(β) (B) Signup and view all the answers

What is the purpose of finding the reference angle?

To determine the restricted values within a specified interval. (C) Signup and view all the answers

If cosθ = x, what is the general solution for θ?

θ = cos^-1 x + k ⋅ 360° (C), θ = 180° - cos^-1 x + k ⋅ 360° (D) Signup and view all the answers

What is the importance of the periodic nature of trigonometric functions?

It implies that there are infinitely many angles that satisfy a given trigonometric equation. (A) Signup and view all the answers

What is the formula for the area of a triangle in terms of two sides and the included angle?

Area = (1/2)ac sin B (B), Area = (1/2)bc sin A (D) Signup and view all the answers

What is the formula for the height of a pole using the sine rule and tangent ratio?

h = (d sin α) / (sin β) tan β (D) Signup and view all the answers

What is the formula for the height of a building using the sine rule?

h = (b sin α sin θ) / (sin(β + θ)) (D) Signup and view all the answers

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

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

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

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

What is the general solution for θ if tanθ = x?

θ = tan^-1 x + k ⋅ 90° (D) Signup and view all the answers

What is the formula for the sine rule in trigonometry?

sin A / a = sin B / b = sin C / c (B) Signup and view all the answers

What is the formula for the cosine rule in trigonometry?

a^2 = b^2 + c^2 - 2bc cos A (D) Signup and view all the answers

What is the general approach to solving three-dimensional problems in trigonometry?

Draw a sketch, consider the given information, apply appropriate rules, and calculate desired quantities (B) Signup and view all the answers

Gr12 Mathematics: Term test 1

Choose a study mode

Podcast

Questions and Answers

What is the definition of an arithmetic sequence?

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

How do you check if a sequence is arithmetic?

What is the arithmetic mean between two numbers?

What is the graphical representation of an arithmetic sequence?

What happens to an arithmetic sequence if the common difference is positive?

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

How do you find the nth term of an arithmetic sequence?

What is the common ratio in a geometric sequence?

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

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

What is the graph of a geometric sequence?

How do you test if a sequence is geometric?

What is the definition of a series?

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

What is the formula for an infinite series?

What is the purpose of sigma notation?

What is the index of summation in sigma notation?

What is the general formula for the sum of 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 an arithmetic sequence?

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

What is the method used by Karl Friedrich Gauss to find the sum of the first 100 integers?

What is the sum of the first 100 integers?

What is the definition of a finite geometric series?

What is the definition of an infinite geometric series?

When does an infinite geometric series converge?

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

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

What is the restriction on the domain of y = ax^2 for its inverse to be a function?

What is the graph of f(x) = -x^2?

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

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

What is the definition of a logarithm?

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

What is the shape of the graph of the logarithmic function y = logb(x)?

What is the special logarithmic value of loga(a)?

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

What is the definition of a relation?

What is the Quotient Rule of logarithms?

What is the formula for the sum of an arithmetic sequence if the last term is unknown?

What is the Power Rule of logarithms?

What is the property of an inverse function?

What is the notation for the inverse function?

What is the graphical representation of the inverse of a function?

What is the condition for a function to be many-to-one?

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

What is the formula to change the base of a logarithm?

What is the reflection of the graph of an exponential function about?

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

What is the formula for pH levels?

What is the formula for the cosine of a difference?

How are the compound angle formulas derived?

What is the application of logarithms in finance?

What is the formula for population growth?

What is the formula to find the angles in the interval [0°, 360°] that satisfy the equation sin θ = x?

What is the formula for the sine of a difference?

When should you use the Area Rule to find the area of a triangle?

What is the formula for the cosine of a sum?

What is the formula to find the area of a triangle using the Sine Rule?

What is the formula for the sine of a double angle?

When should you use the Cosine Rule to find the length of a side in a triangle?

What is the formula for the cosine of a double angle?

What is the formula to find the height of a pole using trigonometry?

How many general solutions are there for a trigonometric equation?

What is the general approach to solving problems in three dimensions?

What is the formula to find the length of a side in a triangle using the Cosine Rule?

What is the first step in solving a trigonometric equation?

When should you use the Sine Rule to find the length of a side in a triangle?

What is the purpose of the CAST diagram in solving trigonometric equations?

What is the formula to find the area of a triangle using the Area Rule?

What is the formula for the cosine of a difference?

What is the formula for the sine of a sum?

What is the formula to find the height of a building using trigonometry?

What is an alternate form of the cosine of a double angle?