answer：Okay, so I have this function f(x) = 3x + x - 5, and I need to find the root x₀ which is in the interval [a, b]. The conditions are that b - a = 1, and both a and b are positive integers. Then, I have to find a + b. Hmm, let me think about how to approach this.First, I should probably simplify the function f(x). It says f(x) = 3x + x - 5. Combining like terms, that would be 4x - 5. So, f(x) = 4x - 5. That seems straightforward.Now, a root of the function is a value x₀ where f(x₀) = 0. So, I need to solve the equation 4x - 5 = 0. Let me do that:4x - 5 = 0 4x = 5 x = 5/4 x = 1.25Okay, so the root is at x = 1.25. Now, the problem states that this root x₀ is in the interval [a, b], where b - a = 1, and a and b are positive integers. So, I need to find integers a and b such that a ≤ 1.25 ≤ b, and b - a = 1.Since b - a = 1, the interval [a, b] is just two consecutive integers. So, if a is an integer, then b is a + 1. Therefore, the interval is [a, a + 1]. I need to find a such that 1.25 is between a and a + 1.Let me list some integers and see where 1.25 falls:- If a = 1, then b = 2. The interval is [1, 2]. 1.25 is between 1 and 2, so that works.- If a = 2, then b = 3. The interval is [2, 3]. But 1.25 is less than 2, so that doesn't work.- Similarly, a can't be less than 1 because a is a positive integer, so a = 1 is the only possibility.Therefore, a = 1 and b = 2. So, a + b = 1 + 2 = 3.Wait, let me double-check. If a = 1, then b = 2, and the interval is [1, 2]. The root is at 1.25, which is indeed within [1, 2]. So, that makes sense.Is there any other possible a? Let's see:If a = 0, then b = 1, but a has to be a positive integer, so a = 0 is not allowed. Therefore, a must be 1.So, adding a and b, 1 + 2 = 3. That should be the answer.Just to make sure I didn't make any mistakes, let me recap:1. Simplified f(x) correctly: 3x + x is 4x, so f(x) = 4x - 5.2. Solved for root: x = 5/4 = 1.25.3. Determined that the interval [a, b] must be of length 1, with a and b positive integers.4. Found that a = 1 and b = 2 is the only interval containing 1.25.5. Therefore, a + b = 3.Everything seems to check out. I don't think I missed anything here.**Final Answer**The value of (a + b) is boxed{3}.

answer：Okay, so I have this problem where a student has a probability of passing a certain math test of 3/4. He takes the test n times in a row, and we need to find the minimum value of n such that the probability of him passing at least once is greater than 0.99.Hmm, let me think. So, the probability of passing the test is 3/4 each time, and he takes it n times. We want the probability that he passes at least once to be more than 0.99. I remember that when dealing with probabilities of at least one success in multiple trials, it's often easier to calculate the complement probability, which is the probability of not passing at all in those n trials. Then, subtract that from 1 to get the desired probability.So, the probability of not passing a single test is 1 - 3/4, which is 1/4. If he takes the test n times, the probability of not passing any of them is (1/4)^n. Therefore, the probability of passing at least once is 1 - (1/4)^n.We need this probability to be greater than 0.99. So, we can set up the inequality:1 - (1/4)^n > 0.99Let me write that down:1 - (1/4)^n > 0.99To solve for n, I can rearrange this inequality. Subtract 1 from both sides:- (1/4)^n > 0.99 - 1Which simplifies to:- (1/4)^n > -0.01Now, multiplying both sides by -1 will reverse the inequality:(1/4)^n < 0.01So, we have:(1/4)^n < 0.01Now, to solve for n, I can take the natural logarithm of both sides. Remember that taking the logarithm of both sides of an inequality is okay as long as both sides are positive, which they are here.Taking ln on both sides:ln((1/4)^n) < ln(0.01)Using the logarithm power rule, ln(a^b) = b*ln(a), so:n * ln(1/4) < ln(0.01)Now, ln(1/4) is equal to ln(1) - ln(4) = 0 - ln(4) = -ln(4). So, we can write:n * (-ln(4)) < ln(0.01)Dividing both sides by -ln(4) will reverse the inequality again because we're dividing by a negative number:n > ln(0.01) / (-ln(4))Let me compute the values step by step.First, compute ln(0.01). I know that ln(0.01) is the natural logarithm of 1/100, which is equal to ln(1) - ln(100) = 0 - ln(100) = -ln(100). Since ln(100) is approximately 4.60517 (because e^4.60517 ≈ 100), so ln(0.01) ≈ -4.60517.Next, compute ln(4). ln(4) is approximately 1.386294.So, substituting these values into the inequality:n > (-4.60517) / (-1.386294)The negatives cancel out, so:n > 4.60517 / 1.386294Let me compute that division. 4.60517 divided by 1.386294.Hmm, 1.386294 times 3 is approximately 4.158882, which is less than 4.60517. The difference is 4.60517 - 4.158882 = 0.446288.So, 0.446288 divided by 1.386294 is approximately 0.322.So, adding that to 3, we get approximately 3.322.Therefore, n > approximately 3.322.Since n must be an integer (he can't take the test a fraction of a time), we round up to the next whole number, which is 4.Wait, but let me double-check my calculations because sometimes when dealing with logarithms, it's easy to make a mistake.Alternatively, I can use logarithms with base 10 or another base, but natural logarithm is fine.Alternatively, let me compute 4.60517 divided by 1.386294 more accurately.Let me write it as:4.60517 / 1.386294 ≈ ?Let me perform the division step by step.1.386294 * 3 = 4.158882Subtract that from 4.60517: 4.60517 - 4.158882 = 0.446288Now, 0.446288 / 1.386294 ≈ 0.322So, total is 3.322, as before.So, n must be greater than approximately 3.322, so the smallest integer n is 4.But wait, let me check: if n=4, what is the probability?Compute (1/4)^4 = 1/256 ≈ 0.00390625So, 1 - 0.00390625 ≈ 0.99609375, which is approximately 0.9961, which is greater than 0.99.What about n=3?(1/4)^3 = 1/64 ≈ 0.0156251 - 0.015625 = 0.984375, which is less than 0.99.So, n=3 gives a probability of approximately 98.44%, which is below 99%, so n=4 is indeed the minimum value needed.Therefore, the minimum value of n is 4.**Final Answer**The minimum value of ( n ) is boxed{4}.

answer：First, I need to determine the number of cookies in 8 boxes. Since each box contains 12 cookies, multiplying 8 by 12 gives the total number of cookies in the boxes.Next, I'll calculate the number of cookies in 9 bags. Each bag has 7 cookies, so multiplying 9 by 7 provides the total number of cookies in the bags.Finally, to find out how many more cookies are in the boxes compared to the bags, I'll subtract the total number of cookies in the bags from the total number of cookies in the boxes.

answer：Okay, so I have this problem here with vectors a and b. Vector a is (3, -1) and vector b is (1, λ). The equation given is the square of the magnitude of (a - b) equals the sum of the squares of the magnitudes of a and b. I need to find λ. Hmm, okay, let me break this down step by step.First, I remember that the magnitude squared of a vector is just the sum of the squares of its components. So, for vector a, which is (3, -1), the magnitude squared would be 3 squared plus (-1) squared. Let me calculate that: 3² is 9, and (-1)² is 1, so 9 + 1 is 10. So, |a|² is 10.Similarly, for vector b, which is (1, λ), the magnitude squared is 1 squared plus λ squared. That would be 1 + λ². So, |b|² is 1 + λ².Now, the equation given is |a - b|² = |a|² + |b|². Let me write that out:|a - b|² = |a|² + |b|²I already know |a|² and |b|², so maybe I can compute |a - b|² and set it equal to 10 + (1 + λ²) and solve for λ.To find |a - b|², I need to first find the vector a - b. Vector subtraction is done component-wise, so subtracting b from a would be (3 - 1, -1 - λ). Let me compute that:a - b = (3 - 1, -1 - λ) = (2, -1 - λ)Now, the magnitude squared of this vector is 2 squared plus (-1 - λ) squared. So, that's 4 + (-1 - λ)².Let me expand (-1 - λ)². That would be (-1)² + 2*(-1)*(-λ) + (-λ)², which is 1 + 2λ + λ². So, putting it all together:|a - b|² = 4 + 1 + 2λ + λ² = 5 + 2λ + λ²So, according to the given equation, this should equal |a|² + |b|², which is 10 + (1 + λ²). Let me compute that:10 + 1 + λ² = 11 + λ²So, now I have the equation:5 + 2λ + λ² = 11 + λ²Hmm, okay, let me subtract λ² from both sides to simplify:5 + 2λ = 11Now, subtract 5 from both sides:2λ = 6Divide both sides by 2:λ = 3Wait, that seems straightforward. Let me double-check my steps to make sure I didn't make any mistakes.1. Calculated |a|² correctly: 3² + (-1)² = 9 + 1 = 10. That's right.2. Calculated |b|² correctly: 1² + λ² = 1 + λ². That's also correct.3. Subtracted vectors a - b: (3 - 1, -1 - λ) = (2, -1 - λ). Yep, that's correct.4. Calculated |a - b|²: 2² + (-1 - λ)² = 4 + (1 + 2λ + λ²) = 5 + 2λ + λ². That looks good.5. Set |a - b|² equal to |a|² + |b|²: 5 + 2λ + λ² = 10 + 1 + λ² = 11 + λ². Correct.6. Subtracted λ² from both sides: 5 + 2λ = 11. Then subtracted 5: 2λ = 6, so λ = 3. That all checks out.I don't see any mistakes in my calculations. So, λ must be 3.Wait a second, just to make sure, let me plug λ = 3 back into the original vectors and verify the equation.Vector a is (3, -1), vector b is (1, 3). Then, a - b is (3 - 1, -1 - 3) = (2, -4). The magnitude squared of a - b is 2² + (-4)² = 4 + 16 = 20.Now, |a|² is 10, |b|² is 1² + 3² = 1 + 9 = 10. So, |a|² + |b|² is 10 + 10 = 20. Which equals |a - b|². So, yes, it works out.Therefore, λ is indeed 3.**Final Answer**The value of lambda is boxed{3}.

answer：First, I need to determine how many goals each team scored in the second half. Let’s denote the number of goals the visiting team scored in the second half as ( x ).According to the problem, the home team scored twice as many goals as the visitors in the second half, so the home team scored ( 2x ) goals.At halftime, the score was 9-14 in favor of the visiting team. Adding the second-half goals:- The home team’s total score is ( 9 + 2x ).- The visiting team’s total score is ( 14 + x ).The home team won by one goal, so:[ 9 + 2x = 14 + x + 1 ][ 9 + 2x = 15 + x ]Subtracting ( x ) from both sides:[ 9 + x = 15 ]Subtracting 9 from both sides:[ x = 6 ]Therefore:- The home team scored ( 2x = 12 ) goals in the second half, making their total ( 9 + 12 = 21 ).- The visiting team scored ( x = 6 ) goals in the second half, making their total ( 14 + 6 = 20 ).The full-time score was 21-20.

answer：Okay, so I have this problem here about covering a square floor with identical square tiles. It says that if 37 tiles are laid along the two diagonals of the floor, how many tiles are needed to cover the entire floor? The options are (A) 148, (B) 324, (C) 361, (D) 1296, and (E) 1369.Hmm, let me try to visualize this. We have a square floor, which means the number of tiles along each side is the same. Let's denote the number of tiles along one side as 'n'. So, the total number of tiles needed to cover the entire floor would be n squared, right? That is, n x n = n².Now, the problem mentions that 37 tiles are laid along the two diagonals. I need to figure out how this relates to 'n'. In a square, there are two main diagonals: the primary diagonal (from top-left to bottom-right) and the secondary diagonal (from top-right to bottom-left). Each of these diagonals has 'n' tiles. However, if the square has an odd length, the two diagonals intersect at the center tile, meaning that tile is counted twice if we just add the two diagonals together.So, the total number of tiles along both diagonals would be 2n - 1. Wait, why? Because if n is the number of tiles on each diagonal, and they overlap at one tile in the center when n is odd, then the total number of unique tiles along both diagonals is 2n - 1. If n were even, the diagonals wouldn't overlap, so it would just be 2n. But in this case, since 37 is an odd number, that suggests that n must also be such that 2n - 1 = 37.Let me write that down:2n - 1 = 37So, solving for n:2n = 37 + 1 = 38n = 38 / 2 = 19Okay, so each side of the square floor has 19 tiles. Therefore, the total number of tiles needed to cover the entire floor is 19 squared.Calculating 19²:19 x 19. Hmm, 20 x 20 is 400, so 19 x 19 is 400 - 20 - 20 + 1 = 361. Wait, let me check that again. Alternatively, 19 x 19 is 361 because 10 x 10 is 100, 9 x 9 is 81, and then the cross terms: 10 x 9 + 9 x 10 = 180, so total is 100 + 180 + 81 = 361. Yeah, that's correct.So, the total number of tiles needed is 361, which is option (C).Wait, let me double-check my reasoning. If n = 19, then the number of tiles along each diagonal is 19, but since they overlap at the center, the total number of tiles along both diagonals is 19 + 19 - 1 = 37. That matches the given information. So, yes, n is 19, and the total tiles are 19² = 361. So, I think that's correct.I don't see any mistakes in my reasoning. So, I feel confident that the answer is (C) 361.**Final Answer**boxed{C}