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Questions and Answers
Given matrices A and B of order $3 \times m$ and $3 \times n$ respectively, and assuming $m = n$, what is the order of the resultant matrix $(5A - 2B)$?
Given matrices A and B of order $3 \times m$ and $3 \times n$ respectively, and assuming $m = n$, what is the order of the resultant matrix $(5A - 2B)$?
- $3 \times n$ (correct)
- $m \times 3$
- $m \times n$
- $3 \times 3$
Given $\begin{bmatrix} 2p + q & p - 2q \ 5r - s & 4r + 3s \end{bmatrix} = \begin{bmatrix} 4 & -3 \ 11 & 24 \end{bmatrix}$, determine the value of $p + q + 2s$.
Given $\begin{bmatrix} 2p + q & p - 2q \ 5r - s & 4r + 3s \end{bmatrix} = \begin{bmatrix} 4 & -3 \ 11 & 24 \end{bmatrix}$, determine the value of $p + q + 2s$.
- 4
- -8
- 8 (correct)
- 10
The matrix $\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}$ is best described as which type of matrix?
The matrix $\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}$ is best described as which type of matrix?
- Symmetric matrix
- Skew-symmetric matrix
- Identity matrix (correct)
- Nilpotent matrix
The expression $\frac{2x + 1}{(x + 1)(x - 1)}$ is best described as:
The expression $\frac{2x + 1}{(x + 1)(x - 1)}$ is best described as:
For what values of $x$ is the identity $(5x + 4)^2 = 25x^2 + 40x + 16$ true?
For what values of $x$ is the identity $(5x + 4)^2 = 25x^2 + 40x + 16$ true?
The partial fraction decomposition of $\frac{x - 2}{(x - 1)(x + 2)}$ will have the form:
The partial fraction decomposition of $\frac{x - 2}{(x - 1)(x + 2)}$ will have the form:
Determine the nature of the equation $(x + 3)^2 = x^2 + 6x + 9$.
Determine the nature of the equation $(x + 3)^2 = x^2 + 6x + 9$.
Given the inductive hypothesis $2^k \geq k^2$ for $k \geq 4$, what must be proven in the inductive step to demonstrate that $2^{k+1} \geq (k+1)^2$?
Given the inductive hypothesis $2^k \geq k^2$ for $k \geq 4$, what must be proven in the inductive step to demonstrate that $2^{k+1} \geq (k+1)^2$?
Consider a recursively defined sequence where $a_1 = 3$ and $a_{n+1} = \frac{a_n^2 + 5}{2a_n}$. Determine the limit of the sequence as $n$ approaches infinity, assuming it converges, and justify the convergence using advanced mathematical principles.
Consider a recursively defined sequence where $a_1 = 3$ and $a_{n+1} = \frac{a_n^2 + 5}{2a_n}$. Determine the limit of the sequence as $n$ approaches infinity, assuming it converges, and justify the convergence using advanced mathematical principles.
Let $S(n)$ be the statement that $n^3 + 5n$ is divisible by 6 for all positive integers $n$. In the inductive step, we assume $S(k)$ is true. Which of the following congruences must be proven to hold true to complete the induction, demonstrating $S(k+1)$?
Let $S(n)$ be the statement that $n^3 + 5n$ is divisible by 6 for all positive integers $n$. In the inductive step, we assume $S(k)$ is true. Which of the following congruences must be proven to hold true to complete the induction, demonstrating $S(k+1)$?
Given the Dedekind cut $(\mathbb{Q} \cap (-\infty, \sqrt{2}) , \mathbb{Q} \cap (\sqrt{2}, \infty))$, which of the following operations, when applied to this cut, results in another valid Dedekind cut representing a real number?
Given the Dedekind cut $(\mathbb{Q} \cap (-\infty, \sqrt{2}) , \mathbb{Q} \cap (\sqrt{2}, \infty))$, which of the following operations, when applied to this cut, results in another valid Dedekind cut representing a real number?
Consider the statement: $n^2 - n + 41$ is prime for all positive integers $n$. What is the smallest positive integer $n$ for which this statement fails, and what implications does this have for mathematical induction?
Consider the statement: $n^2 - n + 41$ is prime for all positive integers $n$. What is the smallest positive integer $n$ for which this statement fails, and what implications does this have for mathematical induction?
Given a sequence defined by $a_n = \frac{1}{n^2} + \frac{1}{(n+1)^2} + \frac{1}{(n+2)^2} + ...$, analyze its convergence properties using advanced techniques from real analysis. Which statement accurately describes its behavior as $n\rightarrow \infty$?
Given a sequence defined by $a_n = \frac{1}{n^2} + \frac{1}{(n+1)^2} + \frac{1}{(n+2)^2} + ...$, analyze its convergence properties using advanced techniques from real analysis. Which statement accurately describes its behavior as $n\rightarrow \infty$?
Let $x$ be a real number represented by a Cauchy sequence of rational numbers ${x_n}$. If we define $x > 0$ if there exists an $N$ such that $x_n > 0$ for all $n > N$, how can we rigorously prove that this definition is independent of the choice of the Cauchy sequence representing $x$?
Let $x$ be a real number represented by a Cauchy sequence of rational numbers ${x_n}$. If we define $x > 0$ if there exists an $N$ such that $x_n > 0$ for all $n > N$, how can we rigorously prove that this definition is independent of the choice of the Cauchy sequence representing $x$?
Consider two irrational numbers, $a$ and $b$, such that $a^b$ is rational. This demonstrates the existence of such numbers without explicitly constructing them. Now, suppose both $a$ and $b$ are transcendental. How can we demonstrate, without constructing specific examples, that there exist transcendental numbers $x$ and $y$ such that $x^y$ is rational?
Consider two irrational numbers, $a$ and $b$, such that $a^b$ is rational. This demonstrates the existence of such numbers without explicitly constructing them. Now, suppose both $a$ and $b$ are transcendental. How can we demonstrate, without constructing specific examples, that there exist transcendental numbers $x$ and $y$ such that $x^y$ is rational?
Define a 'super-composite' number as a positive integer $n$ such that the sum of its digits in base 10 is a prime number, and the product of its digits is a perfect square. Which of the following statements regarding the distribution and properties of super-composite numbers is most accurate?
Define a 'super-composite' number as a positive integer $n$ such that the sum of its digits in base 10 is a prime number, and the product of its digits is a perfect square. Which of the following statements regarding the distribution and properties of super-composite numbers is most accurate?
Consider the expression $ \lim_{x \to \infty} (\sqrt{x^2 + ax} - \sqrt{x^2 + bx}) $, where $a$ and $b$ are real numbers. For what condition on $a$ and $b$ does this limit converge to a finite non-zero value?
Consider the expression $ \lim_{x \to \infty} (\sqrt{x^2 + ax} - \sqrt{x^2 + bx}) $, where $a$ and $b$ are real numbers. For what condition on $a$ and $b$ does this limit converge to a finite non-zero value?
Given the recurrence relation $a_{n+2} = 5a_{n+1} - 6a_n$ with initial conditions $a_0 = 1$ and $a_1 = 0$, determine a closed-form expression for $a_n$.
Given the recurrence relation $a_{n+2} = 5a_{n+1} - 6a_n$ with initial conditions $a_0 = 1$ and $a_1 = 0$, determine a closed-form expression for $a_n$.
Suppose a relation $R$ is defined on the set of integers $\mathbb{Z}$ such that $(a, b) \in R$ if and only if $a^2 \equiv b^2 \pmod{5}$. Which of the following statements accurately characterizes the properties of $R$?
Suppose a relation $R$ is defined on the set of integers $\mathbb{Z}$ such that $(a, b) \in R$ if and only if $a^2 \equiv b^2 \pmod{5}$. Which of the following statements accurately characterizes the properties of $R$?
Consider the following predicate logic statement: $ \forall x \in D, \exists y \in E : P(x, y) \rightarrow Q(x, y) $. Under what conditions is this statement false, assuming non-empty domains $D$ and $E$?
Consider the following predicate logic statement: $ \forall x \in D, \exists y \in E : P(x, y) \rightarrow Q(x, y) $. Under what conditions is this statement false, assuming non-empty domains $D$ and $E$?
Let $A$ and $B$ be two finite sets. Given that $|A| = m$ and $|B| = n$, determine the cardinality of the set of all relations from $A$ to $B$.
Let $A$ and $B$ be two finite sets. Given that $|A| = m$ and $|B| = n$, determine the cardinality of the set of all relations from $A$ to $B$.
In the context of formal language theory, consider a language $L$ defined over the alphabet $ \Sigma = {0, 1} $ that consists of all strings in which the number of 0s is a multiple of 3 and the number of 1s is even. Which of the following statements is most accurate regarding the nature of $L$?
In the context of formal language theory, consider a language $L$ defined over the alphabet $ \Sigma = {0, 1} $ that consists of all strings in which the number of 0s is a multiple of 3 and the number of 1s is even. Which of the following statements is most accurate regarding the nature of $L$?
Consider a universal Turing machine tasked with simulating another Turing machine $M$ on input $w$. If $M$ has $k$ states and its tape alphabet contains $m$ symbols, and the input $w$ has length $n$, what is the minimal time complexity lower bound for the universal Turing machine to simulate $t$ steps of $M$?
Consider a universal Turing machine tasked with simulating another Turing machine $M$ on input $w$. If $M$ has $k$ states and its tape alphabet contains $m$ symbols, and the input $w$ has length $n$, what is the minimal time complexity lower bound for the universal Turing machine to simulate $t$ steps of $M$?
Given a set of $n$ distinct elements, what is the number of ways to arrange $k$ of these elements in a circle, where two arrangements are considered the same if one can be obtained from the other by rotation?
Given a set of $n$ distinct elements, what is the number of ways to arrange $k$ of these elements in a circle, where two arrangements are considered the same if one can be obtained from the other by rotation?
Consider a geometric progression (GP) with a first term $a_1 = 2$ and a common ratio $r = 3$. Determine the $n$th term, $a_n$, where $n$ approaches infinity, and concomitantly evaluate the limit of the series if it exists; if the limit does not exist, provide a rigorous justification.
Consider a geometric progression (GP) with a first term $a_1 = 2$ and a common ratio $r = 3$. Determine the $n$th term, $a_n$, where $n$ approaches infinity, and concomitantly evaluate the limit of the series if it exists; if the limit does not exist, provide a rigorous justification.
Given an infinite geometric progression (GP) with a first term $a = 4$ and a common ratio $r = \frac{1}{2}$, analyze the convergence of the series. Furthermore, derive the closed-form expression for the sum to infinity, $S_\infty$, and contrast this result with Riemann's criteria for convergence.
Given an infinite geometric progression (GP) with a first term $a = 4$ and a common ratio $r = \frac{1}{2}$, analyze the convergence of the series. Furthermore, derive the closed-form expression for the sum to infinity, $S_\infty$, and contrast this result with Riemann's criteria for convergence.
Consider three terms in a GP with a sum of 21 and a product of 216. Construct a system of equations to represent these conditions. Subsequently, employ advanced algebraic techniques to solve for the individual terms, thoroughly justifying each step in the solution process and accounting for any potential extraneous solutions.
Consider three terms in a GP with a sum of 21 and a product of 216. Construct a system of equations to represent these conditions. Subsequently, employ advanced algebraic techniques to solve for the individual terms, thoroughly justifying each step in the solution process and accounting for any potential extraneous solutions.
Given the formula for the sum of a finite geometric progression, $S_n = a \frac{(r^n - 1)}{r - 1}$, where $r > 1$, derive $S_4$ when $a = 2$ and $r = 3$. Further, analyze the computational complexity of this calculation within the context of large-scale numerical computation, accounting for potential overflow errors.
Given the formula for the sum of a finite geometric progression, $S_n = a \frac{(r^n - 1)}{r - 1}$, where $r > 1$, derive $S_4$ when $a = 2$ and $r = 3$. Further, analyze the computational complexity of this calculation within the context of large-scale numerical computation, accounting for potential overflow errors.
If the sum of the initial three terms of a GP is $\frac{13}{12}$, and their product totals $\frac{27}{8}$, deduce the exact values of these terms. Investigate whether an explicit algebraic solution always exists for such problems, and outline the constraints necessary for guaranteed solutions.
If the sum of the initial three terms of a GP is $\frac{13}{12}$, and their product totals $\frac{27}{8}$, deduce the exact values of these terms. Investigate whether an explicit algebraic solution always exists for such problems, and outline the constraints necessary for guaranteed solutions.
Given the quadratic equation $x^2 - 5x + 6 = 0$, employ modern algebraic techniques, such as Galois theory or resultant methods, to solve for $x$. Further, explore the philosophical implications of solution existence within various algebraic closures.
Given the quadratic equation $x^2 - 5x + 6 = 0$, employ modern algebraic techniques, such as Galois theory or resultant methods, to solve for $x$. Further, explore the philosophical implications of solution existence within various algebraic closures.
For the quadratic equation $x^2 - 4x + 5 = 0$, determine not only the nature of its roots (real, distinct, complex conjugates) but also rigorously prove their nature using discriminant analysis and complex analysis principles, considering potential numerical instability in computation.
For the quadratic equation $x^2 - 4x + 5 = 0$, determine not only the nature of its roots (real, distinct, complex conjugates) but also rigorously prove their nature using discriminant analysis and complex analysis principles, considering potential numerical instability in computation.
Given the equation $x^2 - 3x + 4 = 0$, use Vieta's formulas to determine relationships involving the coefficients and roots. Subsequently, express the sum of the roots as a function of the equation's parameters, and analyze the computational efficiency of this approach relative to directly solving for the roots.
Given the equation $x^2 - 3x + 4 = 0$, use Vieta's formulas to determine relationships involving the coefficients and roots. Subsequently, express the sum of the roots as a function of the equation's parameters, and analyze the computational efficiency of this approach relative to directly solving for the roots.
Given an arithmetic progression where the first term is 4 and the tenth term is 31, determine the sum of the first 10 terms, rigorously justifying each step based on the established theorems of arithmetic series.
Given an arithmetic progression where the first term is 4 and the tenth term is 31, determine the sum of the first 10 terms, rigorously justifying each step based on the established theorems of arithmetic series.
If the common difference of an arithmetic progression is 6 and the 20th term is 122, employing meticulous algebraic manipulation, ascertain the value of the first term, showing all intermediate steps.
If the common difference of an arithmetic progression is 6 and the 20th term is 122, employing meticulous algebraic manipulation, ascertain the value of the first term, showing all intermediate steps.
In a geometric progression with the first term being 3 and the common ratio being $\frac{1}{3}$, precisely calculate the sum of the first 4 terms, expressing the result as an exact fraction.
In a geometric progression with the first term being 3 and the common ratio being $\frac{1}{3}$, precisely calculate the sum of the first 4 terms, expressing the result as an exact fraction.
Given that $a_2 = 2$ and $a_3 = 3$ in a geometric progression, meticulously determine the 6th term ($a_6$), clearly stating all assumptions and logical deductions.
Given that $a_2 = 2$ and $a_3 = 3$ in a geometric progression, meticulously determine the 6th term ($a_6$), clearly stating all assumptions and logical deductions.
For an infinite geometric progression with the first term $a_1 = 5$ and common ratio $r = \frac{1}{5}$, meticulously compute the sum to infinity, providing a rigorous justification for convergence.
For an infinite geometric progression with the first term $a_1 = 5$ and common ratio $r = \frac{1}{5}$, meticulously compute the sum to infinity, providing a rigorous justification for convergence.
Given that the sum of a geometric progression is represented by $S_n = \frac{a_1(1 - r^n)}{1 - r}$, where $S_n = 5$, $a_1 = 2$, and $n = 3$, solve for the common ratio $r$ with an accuracy of two decimal places, detailing the algebraic steps involved.
Given that the sum of a geometric progression is represented by $S_n = \frac{a_1(1 - r^n)}{1 - r}$, where $S_n = 5$, $a_1 = 2$, and $n = 3$, solve for the common ratio $r$ with an accuracy of two decimal places, detailing the algebraic steps involved.
Given that in a geometric progression, the first term $a_1 = 7$ and the sum to infinity $S_\infty = 21$, rigorously derive the common ratio $r$, elaborating on the conditions required for the existence of such a sum.
Given that in a geometric progression, the first term $a_1 = 7$ and the sum to infinity $S_\infty = 21$, rigorously derive the common ratio $r$, elaborating on the conditions required for the existence of such a sum.
Solve the quadratic equation $x^2 + 6x + 9 = 0$ using a complete and mathematically rigorous method (e.g., completing the square, quadratic formula), and specify the nature of the roots.
Solve the quadratic equation $x^2 + 6x + 9 = 0$ using a complete and mathematically rigorous method (e.g., completing the square, quadratic formula), and specify the nature of the roots.
Flashcards
What is a Set?
What is a Set?
An ordered collection of objects.
What is a Power Set?
What is a Power Set?
The set containing all the subsets of a given set (including the empty set and the set itself).
What is a Finite Set?
What is a Finite Set?
A set with a finite number of elements.
What is an Infinite Set?
What is an Infinite Set?
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What is Cartesian Product?
What is Cartesian Product?
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What is Cardinality?
What is Cardinality?
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sin(90° – A) = cos A?
sin(90° – A) = cos A?
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What is the fundamental trigonometric identity?
What is the fundamental trigonometric identity?
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What does 'n' represent in GP Sum formula?
What does 'n' represent in GP Sum formula?
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What is the sum of 5, 10, 15,..., 100?
What is the sum of 5, 10, 15,..., 100?
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What is the 5th term of a GP?
What is the 5th term of a GP?
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What is the sum of an infinite GP?
What is the sum of an infinite GP?
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What are the solutions to a quadratic equation?
What are the solutions to a quadratic equation?
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What does the discriminant of a quadratic show?
What does the discriminant of a quadratic show?
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How do you find the sum of the roots of a quadratic equation?
How do you find the sum of the roots of a quadratic equation?
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What is the modulus of a complex number?
What is the modulus of a complex number?
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Mathematical Induction
Mathematical Induction
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Base Case
Base Case
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Terminating Decimal
Terminating Decimal
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Non-Terminating Recurring Decimal
Non-Terminating Recurring Decimal
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Irrational Number
Irrational Number
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Arithmetic Progression (AP)
Arithmetic Progression (AP)
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Common Difference (in AP)
Common Difference (in AP)
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Sum of n Terms
Sum of n Terms
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Order of (5A - 2B) if A and B are 3 x n
Order of (5A - 2B) if A and B are 3 x n
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Identity Matrix
Identity Matrix
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Identity (vs. Equation)
Identity (vs. Equation)
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Proper Fraction (in Algebra)
Proper Fraction (in Algebra)
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Partial Fractions
Partial Fractions
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Induction Step
Induction Step
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Arithmetic Sequence
Arithmetic Sequence
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Arithmetic Series Sum
Arithmetic Series Sum
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Common Difference (AP)
Common Difference (AP)
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Common Ratio (GP)
Common Ratio (GP)
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Sum to Infinity (GP)
Sum to Infinity (GP)
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Discriminant
Discriminant
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Product of Roots
Product of Roots
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Complex Number
Complex Number
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Study Notes
-
Base case for induction proof of 1 + 2 + 3 +...+ n = n(n+1)/2 is when n = 1.
-
If P(k) is true for k = 1 and P(k+1) is true when P(k) is true, conclude P(n) is true for all n ≥ 1.
Decimal Expansion
- The decimal expansion of 22/7 is non-terminating and recurring
Number Types
- (3 + √7)(3 - √7) is not irrational
- The sum of a rational and an irrational number is always irrational
- The product of two irrational numbers is either rational or irrational
Divisibility
- The largest number that divides 72 and 128, leaving remainders 7 and 11, is 13.
Mathematical Induction
- A valid use of mathematical induction is proving identities
- For 2ⁿ > n² where n ≥ 5, the induction hypothesis is 2ᵏ > k².
- Given n³ + 2n is divisible by 3 for all n ≥ 1, the inductive step shows P(k + 1) – P(k) is divisible by 3
Arithmetic Progression (AP)
- The sum of the first n terms of the AP 2, 5, 8,... is n/2 * (2a + (n - 1)d)
- If the 10th term of an AP is 25 and the first term is 5, the common difference is 2
- In an AP, if a = 3, d = 2, and S_n = 55, find n = 5
- For an AP: The 4th and 8th terms are 10 and 22, respectively, the common difference is 3.
- The sum of all integers from 1 to 100 divisible by 5 is 1050
- To calculate the 15th term of the AP where a=5 and d = 3, you get 47
- To find the sum of first 20 terms of an AP is 210, a = 5, find d = 1
- 6 terms of the AP 7, 10, 13... are needed to make 72
- In an AP, the first term is 4, and the 10th term is 31. Find the sum of the first 10 terms is 175
- The common difference of an AP is 6, and the 20th term is 122. The first term is 8
Geometric Progression (GP)
- If the first term of a GP is 2 and the common ratio is 3, find the 5th term is 162
- The sum of the infinite GP with a = 4 and r = 1/2 is 8
- The sum of three terms in GP is 21 and their product is 216 the terms is 3,6,12
- The sum of a finite GP is given by: Sn = a(r^n - 1) / (r - 1), when r > 1; if a = 2, r = 3; S₄ = 80
- The first term of a GP is 3, and the common ratio is 1/3. The sum of the first 4 terms is 40/9
- If a = 2 and r = 3 for a GP, the 6th term is 486
- In an infinite GP, if a = 5 and r = 1/5, the sum is 6.25
- With a sum of a GP, S_n = a(1 - r^n)/(1 - r), where a = 2 and r = 0, S₅ is 2.0
- If three first terms of a GP is 13/2 and the product is 27/8, the terms are ¾, 3/2, 9/4
- In a GP, if a = 7 and S∞ = 21. The value for r is 2/3
Solving Equations
Basic Equation:
- For x² - 5x + 6 = 0; Solutions are x=2, and x=3
- For x² + 7x + 10 = 0; Solutions are x = -2 and x = -5.
- For z² + 1 = 0; x = ±i.
- For x² + 6x + 9 = 0; Solution is: x = -3, -3
Roots of Equations:
- The roots of x² - 4x + 5 = 0 are complex conjugates
- Sum of the roots of x² - 3x + 4 is 3
- If one root of x² + px + q = 0 is 3, and p = -5, q equals -12
- Product of the roots of x² – 12x + 9 = 0, the product equals 9
Nature of Quadratic Equations:
- Given α and β are roots of x² – 4x + 5 = 0. Find the value of α + β + αβ = 9, which is the sum of the roots plus their product
- the discriminant of x² – 2x + 7 = 0 is negative
Complex Numbers:
- (2 + 3i)(1 − i) simplifies to 5 + i
- The argument of -1 + i is 3π/4
- If z = 1 + 2i, then |z|² = 5
Simplifying Complex Numbers:
- i³⁵ simplifies to -i
- (1 + i)² in the form a + bi is 0 + 2i
- The modulus of 3 - 4i is 5
- If z = 2 + 3i, then z̅ = 2 - 3i
- The solution of z2 = −4 is ± 2i
- 1 / (1 + i) in the form of a + bi is ½ - ½i
Miscellaneous
- 1² + 2² + 3² +...+ n² = (n(n+1)(2n+1))/6, the base case for n = 1 is 1.
- SOH CAH TOA: The value of tan 60°/cot 30° is 3
- 1−cos²A is equal to sin²A
- sin (90° –A) and cos A are equal
- If cos X = 1/2, then tan X is equal to √(5/2)
- If cos X = a/b, then sin X is equal to √(b² - a²)/b
- The value of sin 60° cos 30° + sin 30° cos 60° is 1
- 2 tan 30°/(1 + tan²30°) = sin 60°
- If A and B are two matrices of the order 3 × m and 3 × n, respectively, and m = n, then the order of the matrix (5A – 2B) is 3 × n
- An ordered collection of objects is called is a Set
- The set {1, 3, 5, 7, 9} defines O as a set of odd positive intergers less than 10
- Power set of an empty set has exactly one subset
- Given A = {1, 2} and B = {a, b}, The Cartesian product of 𝐴 × 𝐵 = {(1, a), (1, b), (2, a), (2, b)}
- The Cartesian Product B x A is not equal to the Cartesian product A x B
- The cardinality of the set of odd positive integers less than 10 is 5
- Given A = {1, 2} and B = {1, 2, 3}; A ≠ B
- The set of positive integers is infinite
- The Cardinality of the Power set of the set {0, 1, 2} is 8
- For 2p+q, p-2q4 −3 = 4, −3 ; 5r - s, 3sr1 = 11, 24, then the value of p + q + 2s is 8
- 1, 0, 0; 0, 1, 0; 0, 0, 1 is called idenity matrix
- (2x+1)/((x+1)(x-1)) is an an equation proper function
- (5x + 4)² = 25x² + 40x+16 is true for all values of x
- Partial fractions of x − 2 / (x − 1)(x + 2) are of the form A /x+1 + B /x-1
- (x + 3)² = x² + 6x + 9 is an equation for An identity linear equation
- Given 2ⁿ > n² for n ≥ 4, the induction step in the equation needs to be 2{k+1} ≥ (k + 1)² – To prove P(n) is: n! > 2ⁿ for n ≥ 4, P(k+1): k! (k + 1) > 2{k+1}, what is based on: (k + 1)> (2^(k+1))
- To prove 3^n > n^3 for n >= 4, the first inductive step is (3^{k+1}> (k+1)^3
- log (m + n) = log m + log n is an incorrect statement
- log ba x logcb x logac = 1
Logarithims
- If log 2 = 0.30103, log 200 = 2.30103
- Log 144 = 2log4 + 2log2
- loge x + loge (1 + x) = 0 is equal to x² + x - 1 = 0
- If log₁₀2 = 0.3010, the value of log₁₀80 is 1.9030
- log₃(27)³ = 9
- If log₃(log₂ x) = 1, Find the value of x = 8
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