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If a and b are positive numbers explain a b. , with abc=4, then the minimum value of b is.


If a and b are positive numbers explain a b Suppose a and b are two real numbers such that the roots of the cubic equation a^3 - x^2 + bx – 1 = 0 asked May 5, 2019 in Olympiad by Taniska ( 65. b^2a+6 is raising to a power, not multiplying. Hence, the only viable case is a = b + 1, which confirms that a and b are consecutive Answer: You expect the result to be greater than a. -25 So, this statement is Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site No, you've only disproved one very specific possible counterexample. Simplify this solution. So)i) positive x positive: add a bunch of positive numbers a positive number of times. Let mh. In the positive reals we also have the problem of $2$ roots, yet people define $\sqrt x$ without problems. The product of two nonzero numbers is positive if both are positive or both are negative, and is negative otherwise. 3k points) jee; jee mains +1 vote. Can you explain this answer? in English & in Hindi are available as part of our courses for Class 10. Since, a and b are equal, they don't play a role in the a) Show that if a and b are positive integers with a ≥ b, then gcd(a, b) = a if a = b, gcd(a, b) = 2 gcd(a/2, b/2) if a and b are even, gcd(a, b) = gcd(a/2, b) if a is even and b is odd, and gcd(a, b) = gcd(a − b, b) if both a and b are odd. D. For example, Can you explain this answer? theory, EduRev gives you an ample number of questions to practice If A and B are positive integers, which of the following expressions is not an integer for Both the numbers are equal. a and b are both positive integers greater than 1. Prove that $a^n\mid b^n$ if and only if $a \mid b$. हैं, तब (x, y) के संभावित युग्मों की संख्या Let the harmonic mean of two positive real numbers a and b be 4, If q is a positive real number such that a, 5, q, b is an arithmetic progression, then the value(s) of |q -a| is (are) A. ~$ Therefore, it is impossible for both $~u~$ and $~v~$ to be A V factorial needs to be 35 factorial, a B factorial needs to be 35 factorial, and a factory needs to be factory. Besides giving the explanation of How many different pairs (a, b} of positive integers are there such that a < b and Correct answer is '3'. . Therefore, if A < B Let $a$ and $b$ be positive integers such that $b^n + n$ is a multiple of $a^n + n$ for all natural numbers $n$. To get a negative number, you need one negative and one positive number. An even number of negative numbers will give a positive answer. Prove that $a = b$. Suppose P is an end point of the latus rectum of the parabola y 2 = 4λx, and suppose the ellipse passes through the point P. C. If I add two positive numbers, they are added in the same direction. Which number is greater: -3/4 or 5/8? d. Notice that b/2 is less This video is part of the “Real Analysis” series I am making. Of course, as soon as you start accepting numbers other than positive reals, things start getting less orderly. Super Direction : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). The produce of a list of numbers is positive if the number of negative numbers in the list is even, and is negative if the Two positive integers are expressed as follows: p and q are prime numbers. Physically this says that we can empty an ocean \(b\) with a teaspoon a, provided we are willing to use the teaspoon a large number of times \(n\). ) If this is true, it'll greatly save my time in my work $\begingroup$ From the link given in the comment of @MartinR, you know that $~\displaystyle ab + \frac{1}{ab} \geq 2. Question: If you know that a, and both a and b are positive numbers, then what must be true about the relationship between the opposites of these numbers? Explain. a = b 144 ← equation of proportion . A is inversely proportional to B. And we can put around the numbers to avoid confusion. Eliminated. The magnitude of a is smaller than that of b. Biology. When A=4 B=36 Find the value of A when B=A. (a+b)/2 ==> (a+b ) will be an even integer and Even/2 can be odd or even as explained in option B. Draw a number line, then plot the whole number 5 on the line as a shaded dot. 6,9 If d = gcd(a,b) a = dr b = ds for some r and s. If a + b + c > 9c/4 and equation ax^2 + 2bx − 5c = 0 has non-real complex roots, then. For the (---->) direction, I realise we are supposed to break the proof into two cases. positive even. Example: 5 is really +5. Can you explain this Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. It can be, but they are all positive real numbers except 1. 14. From the triangle inequality, we get that the hypothenuse is smaller than the sum of the side lengths, which proves \begin{align} \sqrt{a+b} \leq \sqrt{a}+\sqrt{b}. (Closure of R+)If a and b are positive real numbers, then so are a+b and ab. If A is less than B, it means that A is smaller than B. Deduce that both {an} and {bn} are convergent. Asked in United Kingdom. View More. 13. This means that rational numbers include natural numbers, whole numbers, integers, The term − 3 b-3b − 3 b will be negative (since b b b is positive and it is being subtracted) and odd (since 3 is odd and b b b is odd, and the product of two odd numbers is odd). b) x is a negative number. Pivots and eigenvalues are not convenient for A + B. , with abc=4, then the minimum value of b is. 10 is further from zero than 6 is. Let a1 be their arithmetic mean and b1 their geometric mean: a1=a+b/2 b1=√ab Repeat this process so that, in general, an+1=an+bn/2, bn+1=√anbn. positive odd. व म. Let a, b and c be real numbers such that a+ 2b + c = 4 then the maximum value of ab + bc + ca is. A bona fide proof by contra-diction disproves every possible counterexample. (a, b) = gcd(a/2, b) if a is even and b is odd, and gcd(a, b) = gcd(a − b, b) if both a and b are odd Given that a is inversely proportional to b then the equation relating them is . While it's true that d must be a divisor of both a and b, the thing that really makes it the gcd rather than any old divisor is that r and s must also have no common factors except 1 (in math-speak, we'd say that r and s are coprime). e 2/2 = 1 but 4/2 = 2(Even) . a^2−2ab+b^2=25 Taking statement one alone, (a-b)^2 = 25 ==> a-b = +-5 Still, the number at the place of 'b' should be greater for the final value to be negative, i. x > 0. Step 5/5 5. 93% (674 rated) Explain. Here is my start: $a^n + n | b^n Let $ a $ and $ b $ be positive numbers with $ a > b. (Trichotomy law) If a and b are real numbers, then one and only one of the following three statements is true: a < b, a = b, or a > b. b/2 ==> We know that b is an even number, so b/2 can be odd or even depending on the value of b i. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion Click here 👆 to get an answer to your question ️ If you know that a < b, and both a and b are positive numbers, then what must be true about the relationship b, and both a and b are positive numbers, then what must be true about the relationship between the opposites of these numbers? Explain. Given any two positive real numbers, \(a\) and \(b\), there is a positive integer, \(n\) such that \(na > b\). Any number below zero is a negative number. Total possible ways are (1, 6), (6, 1), (2, 3), (3, 2). Can you explain this answer? in English & in Hindi B. Get a detailed solution and exclusive access to our masterclass to ensure you never miss a concept. Now, c is greater than d clearly states that the number c must be greater than d. Problem 6. We prove that if a divides b and b divides a then a = b or a = -b. If b=y, this is true. Identify three cases for a function NOT to be differentiable at a. × . (a + 2)/2 ==> a/2 + 1. x 0. I hope you can help me on that. 7\). What is a rational number? Rational numbers are in the form of p/q, where p and q can be any integer and q ≠ 0. Let a, b, c are positive real numbers forming an A. if b is not equal to 0 how can we showthat a by b is a rational number A fraction 19 is a rational number written as a quotient, or ratio, of two integers a and b where \(b≠0\). With 'and' I figured |0 + 0| = 0 and |0 - 0| = 0. If it is b^(2a+6)=b^2, then it works in all cases. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT. So we can eliminate Option C as well. Let $\sqrt{a},\sqrt{b}$ be the sides of a right-angles triangle. When b = a , then . with custom compareFunction you can use whatever logic: negative number or zero a before b; positive Click here 👆 to get an answer to your question ️ If you know that a < b, and both a and b are positive numbers, then what must be true about the relationship then what must be true about the relationship between the opposites of these numbers? Explain. This will be the location of the opposite (negative) of the whole number 5, which we indicate by the symbol −5. )(b) Suppose that a, b, and c are real numbers and 0 ≤ b ≤ c. And -10 is alos further from zero than -6 is. GCfiIBPG7Aw. Then $p^{3r Positive numbers are written with no sign or a '\({+}\)' sign in front of them and they are counted up from zero to the right on a number line. Given that, |a+b|=|a|+|b|. 64. All four terms are positive. 1/4. sign The sign of any number other than 0 is either positive or Remember a rational exponent, such as $(-4)$$^{a/b}$, represents a radical, namely $\sqrt[b]{(-4)^a}$, and a negative number can only be evaluated for an odd root (using real numbers). OR, as f(a+b)= f(a)+f(b) must be true for all positive numbers a and b, then you can randomly pick particular values of a and b and check for them: For example: \(a=2\) and \(b=3\) Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site A. Note that the negation of a positive number yields the same result as subtracting a positive number from zero. In that case, we have no choice but to have 2 negatives multiply to a positive. • If a | bc and gcd(a,c) = 1, then a | b. • If a 1a 2 ···a n is a perfect kth power and the a Let A and B be two sets. The sum of any list of negative numbers is negative. If the tangents to the parabola and the ellipse at the point P are perpendicular to each From a − b = p, we notice that for positive whole numbers, this scenario would not give us consecutive integers since both expressions will yield a value that exceeds 1. In addition to the A and B antigens, there is a protein called the Rh factor, which can be either present Two numbers, a and b, are added together and the sum is negative (less than zero). If a a a is positive and b b b is negative then ∣ a + b ∣ ≠ ∣ a ∣ + ∣ b ∣ \rvert a+b \rvert \ne\rvert a \rvert+ \rvert b \rvert ∣ a + b ∣ = ∣ a ∣ + ∣ b ∣ is true. b) Explain how to use (a) to construct an algorithm for computing the greatest common divisor of two For positive numbers, the number closest to zero is smaller. The absurdity can be the negation of one of the assumptions, or it can be a statement that is known to be false, i. youtube. If we negate A (multiply by -1), it becomes a negative number. Therefore, the number that needs to be multiplied by $-b$ to give $-a$ is the positive number $\frac a b$. Well, if I did A over B, C, that would be negative over positive times negat Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Q. 5 of Velleman's How To Prove It. Could someone exp There are four major blood groups determined by the presence or absence of two antigens, A and B, on the surface of red blood cells. If a and b are positive integers, what is the value of a+b? 1. VIDEO ANSWER: The number of the row that has the largest length is returned to if remeid calls the largest road as 2 to the array of ants. then what must be true about the relationship between the opposites of these numbers? Explain. Solutions for If two positive integers a and b are expressible in the form a = pq2and b = p3q ; p and q being prime numbers , then LCM (a,b) isa)pqb)p3q3c)p3q2d)p2q2Correct answer is option 'C'. a) x is a positive number. Prove that b/c + c/a − b/a ≥ 1. Join BYJU'S Learning Program For a matrix to be positive definite, all its leading principal minors (the determinants of the upper left submatrices) should be positive. The equation is valid for negative a,b. Calculus. Similarly, if we negate B, it also becomes a negative number. When a number has no sign it usually means that it is positive. jee main 2023; Share It On Facebook Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Here is an intuitive proof. Positive numbers go to the right and negative numbers are pointing left. Think of an example: if $a=5$ and $b=20$, then $a$ and $b$ are positive integers and $a$ divides $b$, but it is not the case that $k=1$ (in this example, $k=4$). Login. 11. This handles the case of why negating a negative number yields a positive number. If p is a prime divisor of a, and $p^r$ is the highest power of p dividing a. To find its opposite, reflect the number 5 through the origin. Super Gauth AI. PKN hey there, can you please explain how can option C be wrong. How to Prove It (2nd Edition) Edit edition Solutions for Chapter 6. In $\mathbb C$, we can still define $\sqrt x$ (or any power) using the principal branch of the complex logarithm. Step 3/5 3. The cancel out positive numbers. Does that help? $\endgroup$ – Joe When a and b are both positive: #### Expression: a − (− b) - This expression can be rewritten as a + b, since subtracting a negative is the same as adding a positive. Can you explain this answer? defined & explained in the simplest way possible. Much better to prove xᵀ(A + B)x > 0. negative odd. ) For those numbers, all of these properties hold except the existence of opposites (since the opposite of a positive number is negative). P. Lemma: If m > 0, lcm (ma, mb) = m . Furthermore, either 1/a or 1/b must be no less than ½ of 1/c, because if both fractions are less than ½ of 1/c, the sum will be less than 1/c , which implies that either a or b must be less than Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site But when we multiply both a and b by a negative number, the inequality swaps over! Notice that a<b becomes b<a after multiplying by (-2) But the inequality stays the same when multiplying by +3. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Suppose $a, b$ and $n$ are positive integers. I cannot figure out if the negative component of c would also be squared, making it a positive number, or if the negative is not squared because it is not in parentheses. a) If x is a positive real number, then x^{2} is a positive real number. a)Both assertion (A) and reason (R) are true and reason (R) Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Let a, b and λ be positive real numbers. This means that d divides both a and b. • For positive integers a and b, if d is a positive integer such that d | a, d | b, and for any d0, d0 | a and d0 | b implies that d0 | d, then d = gcd(a,b). 2,5 C. Two equal ratios expressed using different numerators and denominators are called equivalent fractions 22. (Unfortunately, I cannot remember where I've discovered it. asked Jun 27, 2019 in Mathematics by Taniska (65. Proof: First a . Since a, b, and c are positive integers, 1/a, 1/b, and 1/c are each less than or equal to 1. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Explain how to use a number line to find the difference between two integers. 186. Suppose, however, b=2 and a=3 Then 2^(12)=y^2, and y^2=4096 and y=64. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Blood Group B Positive Advantages And Disadvantages: In Medical science, blood group is an extremely important subject and knowledge of blood group is essential in various treatments because it is connected to the Find step-by-step Linear algebra solutions and the answer to the textbook question If A and B are positive definite, then A + B is positive definite. If a was negative and b positive, than the problem would look like -3 - 3, which would equal -6. All real numbers is any positive or negative number, it just can't be a number with the i in it (imaginary number). In the negative form, the numbers closer to 0 are considered to be greater. The concept that 'a' divides 'b' and how it relates to positive If a and b are positive numbers, is a < b? (1) a < b/2 + 2. Then introduce the concept of addition (of positive numbers) as moving to the right, subtraction (of positive numbers) as moving to the left. $$ So, you have two positive numbers, $~u~$ and $~v~$ such that $~u \times v \geq 4. Which of the following must be an odd integer? what if a=12 and b=4? then a/b will not be even. Not sufficient. This means they are greater than zero and are whole numbers. Hence, the required probability a. Write as a continued inequality: x 1 or x > 5. thanks in advance Akshay171297 Intern. 225 and thus Consider again the rational numbers \(a/b\) and \(c/d\). $ Let $ a_1 $ be their arithmetic mean and $ b_1 $ their geometric mean: $ a_1 = \frac {a + b}{2} $ $ b_1 = \sqrt {ab} $ Repeat this process so that, in general, $ a^{n + 1} Why is the product of negative numbers positive? Answer: Upon multiplication of a negative number by another negative number, the resultant operation is positive in nature. Inverses in Modular arithmetic We have the following rules for modular arithmetic: Sum rule: IF a ≡ b(mod m) THEN a+c ≡ b+c(mod m). it is known to be false If a and b are positive integers such that a-b and a/b are both even integers. For negative numbers, the number closest to zero is larger. is divided by a b c , then the resulting sequence is also in A. If A is a negative number, B is a positive number, and C is a negative number, then A over B, C is positive. Use a fixed-point iteration method to determine a solution accurate to within 1 0 b) Graph that continued inequality. If b = 2 and a = 2, then answer is NO but if b = 4 and a = 1, then the answer is YES. They are anti-numbers. e) Explain why these steps show that this statement is true whenever n ≥ (a) If a and b are positive real numbers, then a/b + b/a ≥ 2, with equality if and only if a = b. 3,4 B. Since lcm(ma, mb) is a multiple of ma, which is a multiple of m, we have m | lcm (ma, mb). Reason : If each term of an A. This coprime property isn't really important to the proof though (if we Yes indeed, two negatives make a positive, and we will explain why, with examples Lets talk about signs. Clarie. e. 3k points) rmo $$a+b <= 2 \sqrt (ab) $$ is incorrect for positive a,b. 8 (c) > 3. Step-by-step explanation: For this example, use 3 for a and 3 for b 3 - (- 3) = 6. Prove that if n is a positive integer, then $$ a^n − b^n ≤ na^{n−1}(a − b). plz explain. I've found this while googling some properties of positive semidefinite matrices. Assume a and b are natural numbers and answer the following questions. Solutions for If two positive integers a and b are expressible in the form a =pq2 and b=p3q , p and q being prime numbers, then HCF (a,b) isa)pqb)p3q3c)p3q2d)p2q2Correct answer is option 'A'. Explain why each case fails the conditions of differentiability. To Prove: The product of two negative numbers If three positive real numbers a, b, c are in A. Write that continued inequality as a compound inequality. Definition: Negative Number. Problem 5. Suppose P is an end point of the latus rectum of the parabola y 2 = 4λx, and suppose the ellipse 1 passes through the point P. If A and B are counting numbers and A is a multiple of B, how are the multiples of A and B related? Explain your answer and give some examples to illustrate. ii) positive x negative: add a bunch of negative anti-numbers. Step 4 Combine the terms a b ab ab and − 3 b -3b − 3 b Theorem: lcm(a, b) × gcd(a, b) = ab for any positive integers a, b. (Addition law for inequalities)If a, b, and c are real numbers and a < b, then a+c < b+c. Adding two numbers is like overlaying these directions. I hope this helps someone who is learning to write proofs The equation |a+b|=|a|+|b| is not always true for all rational numbers a and b. Show more VIDEO ANSWER: true or false. The opposite of a is also closer to zero than the opposite of b, so the opposite of a must be larger than the opposite of b. Since you only want hints: Try to prove the following identity: $$\gcd(a,b\cdot c) = \gcd(a,b)\cdot\gcd(a,c)$$ This identity basically finishes your proof, given your induction hypothesis. 0 x 6. LCM. of positive numbers bounded above, and let a = sup A, b = sup B. $\begingroup$ Oh wow, did not think about that all. a = b k ← k is the comstant of proportion . 752 and b can 2. 12. Is this the value? I guess because it's 45, what equal to it, and let's try to make up, so it's 10 to 453. x > 0 and x 6. by default it's comparing as strings, so you can give a function to compare as a numbers or whatever subproperty. You cannot multiply a number by itself to get a negative number. com/playlist?list=PLDiddI Given: a and b are both positive numbers and a is less than b, which means a is a number closer to 0. Q. Glossary Entries. Solution For If a and b are positive real numbers such that a+b=6, then the minimum value of (a4 +b1 ) is equal to : World's only instant tutoring platform. - If both a In this exercise, the numbers 'a' and 'b' are both positive integers. Which number is greater: 1/4 or 5/4? b. lcm (a, b). Mia. (b) If a and b are positive real numbers, then √ab ≤ (a+b)/2, with equality if and only if a = b. Can you explain this answer?, a detailed solution for How many different pairs (a, b} of positive integers are there such that Find step-by-step Calculus solutions and the answer to the textbook question Let a and b be positive numbers with a > b. I kept trying to start from the fact that if 1 < a,b, that a < a^2 and b < b^2. $$. For example, sqrt(ab) ≠ sqrt(a)sqrt(b) when a and b are not positive. If a < b and a,b are both positive, that means a and b are both on the right-hand side of the number line, and a is closer to the origin than b. So ONE or BOTH NUMBERS must be negative. The reason for this is that while we do have two (positive and negative) choices for the square root it is simply more convenient to fix one unique output for the operation. For students between the ages of 11 and Let a and b are two positive real numbers such that a2 + b = 2, then the maximum value of term independent of x in the expansion of ax16+bx−139is _____ Moderate. $\begingroup$ I don't think this is fully satisfactory. Step 4/5 4. Here are the rules: If a < b, and c is Then, move on to multiplication of a negative number for a positive one: thus, $5 \times (-2)=-10$ becuse we add five time the quantity $-2$, and this is "obviously" a negative quantity (we move to the left with reference to $0$). negative even. That's very important with inequalities, because it means that we can multiply and divide Learn how to multiply and divide positive and negative numbers and use the sign rules to work out the sign of the answer with BBC Bitesize Maths. Solutions for If two positive integers a and b are written as a and x3y2 and b = xy3, where x, y are prime numbers, then HCF(a, b) isa)xyb)xy2c)x3y3d)x2y2Correct answer is option 'C'. Problem 4. Step 1. Search Instant Tutoring Private Courses Explore Tutors. Hi catty, We're looking for whether a/b < c/d. The sum of any list of positive numbers is positive. On a horizontal number line, negative numbers are usually If a, b, c are positive real numbers, then the number of real roots of the equation a x 2 + b | x | + c = 0 is Q. \end{align} Positive numbers are written with no sign or a '\({+}\)' sign in front of them and they are counted up from zero to the right on a number line. Reason: If each term of an A. Solutions for If three positive real numbers a, b and c (c > a) are in Harmonic Progression, then log (a + c) + log (a – 2b + c) is equal to:(2010)a)2 log (c – b)b)2 log (a – c)c)2 log (c – a)d)log a + log b + log cCorrect answer is option 'C'. A "perfect number" is defined as a positive integer that is equal to the sum of its distinct proper factors, which are the factors of the number other than Given: a and b are positive To Find: value of a+b Statement 1: (3^a)(3^b) = 81 => (3^a)(3^b) = 3^4 => a+b = 4 Sufficient. -----b(2a+6)=2y is a misunderstanding of raising to a power. 1) Negative numbers are numbers that are less than zero. (2) b, c and a are in A. Which, ofcourse is clearly visible 'a negative number is less than a Solutions for If two positive integers a and b are written as a = x3y2 and b = xy3; x, y are prime numbers, then LCM (a, b) isa)xyb)xy2c)x3y3d)x2y2Correct answer is option 'C'. $\begingroup$ It looks like the OP thinks that a proof by contradiction has to negate one of the assumptions or givens. There’s just one step to solve this. Figure \(\PageIndex{3}\) The integer above the fraction bar is called the numerator 20 and the integer below is called the denominator 21. Assertion :Let the positive numbers a , b , c be in A. High school teacher Your method is correct as well as all the 4 numbers are positive. So I use the Find step-by-step Discrete maths solutions and the answer to the textbook question Suppose that a and b are real numbers with 0 < b < a. Mark the correct choice as:Assertion : Let the positive numbers a,b,c be in A. Also, 1/a and 1/b must both be less than 1/c, implying that a and b must both be greater than c. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Suppose $A$ and $B$ are positive real numbers for which $\\log_AB=\\log_BA$. Now I know that the stages of a plant's life cycle include _____. c) x is a non-negative l i m x → 0 (a x + b x 2) 2 x = l i m x → 0 (1 + a x + b x − 2 2) 2 a x + b x − 2 L i m x → 0 (a x − 1 + b x − 1 x) = e l o g a b = a b = 6 Total number of possible ways in a,b can take values is 6 X 6 = 36. This is merely the assertion that any common divisor of a and b divides gcd(a,b). arrow_forward. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Square roots in real numbers: When we deal with the real number system, the square root operation $(\sqrt{})$, is usually defined to yield a positive number. Prove that the roots of the equation (a − b + c) x 2 + 2 (a − b) x + (a − b − c) = 0 are rational numbers for all real numbers a and b and for all rational c. If neither $A$ nor $B$ is $1$, and if $A\\neq B$, find the value of $AB$. University of technology in Ho Chi Minh City · Tutor for 5 For the two positive numbers a, b, if a, b and 1/18 are in a geometric progression, while 1/a , 10 and 1/b are in an arithmetic progression, then, 16a + 12b is equal to _____. and \(0. Take an example, Let a = 5 a=5 a = 5 and b = − 6 b=-6 b = − 6. It makes the most sense to define how negative numbers multiply in a way that respects the properties that hold for positive numbers. For example, $(-4)$$^{1/2}$ means $\sqrt{-4}$ which is a non-real answer. Solution. ~$ So, you can now prove that $$\left(a + \frac{1}{b}\right) \times \left(b + \frac{1}{a}\right) \geq 4. Complete the inductive step for k ≥ 21. 2<4<8 if A=2 B= 4 o correct 2<4 or a<b so what`s wring with it ? thanks and have a great day !:-) Again, a<b (You know we can multiply positive numbers on both sides of the inequality)(Given a and b are positive integers) Or, \(a*b<b^2\) (multiplying 'b' both sides) If two positive integers a and b are written as a = p3q2 and b = pq3; p, q are prime numbers, then HCF (a, b) is:a)pqb)pq2c)p3q3 d)p2q2Correct answer is option 'B'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The total number of possible ways is 4. asked Jan 23, 2020 in Mathematics by MukundJain Let a and b be nonzero integers. Result: A big positive gain. If x + y = 33, where x and y are LCM and HCF of two positive integers respectively, then number of possible pairs of (x, y) is यदि x + y = 33, जहाँ x तथा y दो धनात्मक पूर्णांकों के क्रमशः ल. although, for example, 3 ≡ 13 ≡ 23(mod 10), we would take the smallest positive such number which is 3. Find step-by-step Discrete maths solutions and the answer to the textbook question Show that if a and b are positive integers, then ab = gcd(a, b) · lcm(a, b). The rule works the same way when you have more than two numbers to multiply or divide. Hence the correct choice is (c). If a is negative, b can be either positive or negative. (2) a < b/2 - 2. 4 = 36 k ( multiply both sides by 36 ) 144 = k . VIDEO ANSWER: Determine whether these are valid arguments. -----b=2. Explain why only positive numbers (and 0) are needed to record temperature in \(\text{K}\). Show that if a and b are positive integers, and $a^3 | b^2$, then a | b. I figured I could use this, because in the event that a and b have magnitude less than 1, that you can re-arrange the Take any positive real number, take any (real) power of it, get a positive real number back. If the tangents to the parabola and the ellipse at the point P are perpendicular to each Click here 👆 to get an answer to your question ️ If you know that a , and both a and b are positive numbers, then what must be true about the relationship b I'm looking for an understandable proof of this theorem, and also a complex one involving beautiful math techniques such as analytic number theory, or something else. Explain how you know. In general, any positive number is greater than any negative number. 2 Problem 7E: (a) Prove that if a and b are positive real numbers, then a/b + b/a ≥ 2. Write in symbols. (Hint: Start with the fact that (a − b)2 ≥ 0. Statement 2: (3^a)(5^b) = 225 => Since it is not mentioned a and b are integers [if they were integers, then a=2 and b=2; as 225 = (3^2)(5^2)] a and b can take any values. Is the number that is farther from 0 always the greater number? Explain your reasoning. 3,6 D. We need to check whether |a+b|=|a|+|b| for all rational numbers a and b. The expression $ a^2 + b^2 - ab = c^2 $ is equivalent to $$ a^2 + b^2 - ab = c^2 \longleftrightarrow a^2 - ab = c^2 - b^2 \longleftrightarrow a(a - b) = (c - b)(c + b) $$ Now we multiply by $ b - c $ both sides (if $ b - c = 0 $ then there is nothing to prove): $$ a(a - b)(b - c) = (c - b)(c + b)(b - c) = -(c - b)^2 (c + b) $$ The right side is negative, because $ -(c - b)^2 \leq 0 \ Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Locate the whole number 5 and its opposite (negative) on the number line. Fortunately, we're told a useful bit of info in the question stem. Can you explain this answer? in English & in Hindi are Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site $\begingroup$ Do you have a theorem telling you that the product of two positive numbers is positive? $\endgroup$ – Robert Shore Commented Aug 22, 2021 at 20:31 Let a, b and λ be positive real numbers. , then 1/bc, 1/ac, 1/ab are also in A. a has to be the same on both sides of the equation, so does b and c. To find k use the condition when a = 4, b = 36, then . The result is a big amount of potential cancelling. a = a 144 ( multiply both sides by a ) We look at the contrapositive of this statement: $$\lnot (a^2 > b^2) \rightarrow \lnot (a < b < 0)\tag 2$$ We look at the left side and write it as: A and B are positive numbers. Glossary Terms. + is the positive sign, is the negative sign. QED . First, let's recall that 0 x Because of the way negative number multiplication is defined (and it is a made up, abstract definition, it doesn't "come" from anywhere), it is the case that multiplying a negative number $-b$ by the positive number $\frac a b$ gives you $-a$. Then, taking the highest powers of p and q in the values for a and b we get:. both sides are the same, just written Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site 1. If a,b,c are distinct positive real numbers such that b(a+c) = 2ac then the roots a x 2 + 2bx + c = 0 are : One way I explain to my students is that numbers have direction (a number line is the best depiction of this). स. Unlock the Full Solution and Master the Concept. Which number is farther from 0: -3/4 or 5/8? e. Thanks and enjoy the video!Real Analysis Playlist: https://www. Which number is farther from 0: 1/4 or 5/4? c. . A negative number is a number that is less than zero. I have: $$a^n\mid b^n$$ $$\implies b^n = a^n \cdot k$$ If A and B are counting numbers and A is a multiple of B, how are the multiples of A and B related? Explain your answer and give some examples to illustrate. and ax^2 + 2bx + 5c = 0 has real roots, if. is divided by abc, then the resulting sequence is also in A. This question can be tackled easily by the method below: You are given a/b < c/d ---> as all the numbers are positive you can cross multiply to get , a/c < b/d ---> (a/c) + 1 < (b/d) + 1 ---> (a+c)/c < (b+d)/d ---> (a+c) / (b+d) < c/ d which is statement I and is hence true. We conceded this is. An odd number of negative numbers will give a negative Okay, you say, multiplying by -1 flips the sign of a positive number to a negative number, but why should I think that multiplying -1 by a negative number flips the sign to be a positive number? Great question, let's prove that -1 x -1 = 1. If a is a factor of both b+2 and b-3 , then b could equal A) 21 B) 25 c ) 29 D) 33 Explain. Explain. (3) a. The magnitude is how far the number is from the origin. For any three positive real numbers a, b and c, `9(25a^2+b^2)+25(c^2-3ac)=15b(3a+c)` Then: (1) b, c and a are in G. (This is a simple case of the arithmetic-geometric mean inequality; √ab is the geometric mean of a and b, while (a+b)/2 is their arithmetic mean. For example, a can be 1. In fact, proof 'by contradiction' is translated from latin (reductio ad absurdam) as reducing to an absurdity. B. If a is less than b, and they are both positive, then a is closer to 0 than b. 1 answer. Expert Verified Solution Super Gauth AI. -----b=y only if 2a+6=2. Furthermore, a + b = 1 doesn't work because both a and b are positive whole numbers. 8>-2. Not possible! The conjunction must be and. , then 1 b c , 1 a c , 1 a b are also in A. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Solution for If a and b are positive numbers, find the maximum valueof f(x) = x a (1 - x)b, 0 ≤ x -1 Determine if the function is continuous [x-3, if x 2-1 everywhere. 4 (d). (3) Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site I put that the statement was true only when a = 0 and b = 0 but the correct answer was that it only held true for a = 0 OR b = 0. Therefore, if a^{2} is posi The problem is from exercise 12 of chapter 3. Say, 4. प. Download the App! If a and b are both positive numbers and a < b, what must be true about their absolute values? AI Recommended Answer: a < b A: a > b Therefore, a and b have different absolute values. Then, the hypothenuse is of length $\sqrt{a+b}$ by Pythagorean theorem. Joined: 16 Feb 2021 'b' can be only those numbers where the ratio remains even. 1 b) x gcd(a, b) = lcm(gc, gd) x g = g x lcm(c, d) × g = gcdg = (gc)(gd) = ab. Step 2/5 2. Since A is smaller than B, its negation (-A) will be greater than the negation of B (-B). faff pywh ozpqp qckjf ssopsb plmzq wtni mgoial sddwjr pkpxzzs