💡Okay, so I have this problem here about deductive reasoning, and I need to figure out how many of the given statements are correct. There are four statements labeled ① to ④, and I need to evaluate each one to see if it's true or false. Then, based on that, I can determine the correct answer from the options A to D.First, let me recall what deductive reasoning is. Deductive reasoning is a logical process where conclusions are drawn from general statements or premises. It's often described as moving from the general to the specific. So, if I have a general rule or principle, I can apply it to specific cases to reach a conclusion. That seems to align with statement ①, which says, "Deductive reasoning is reasoning from general to specific." That sounds correct to me.Now, statement ② says, "The conclusion obtained from deductive reasoning is definitely correct." Hmm, this is interesting. I remember that for a conclusion in deductive reasoning to be correct, the premises must be true, and the reasoning must be logically valid. If either the premises are false or the reasoning is flawed, the conclusion might not be correct. So, just because you use deductive reasoning doesn't automatically make the conclusion correct. It depends on the truth of the premises and the structure of the argument. Therefore, statement ② might not be entirely accurate.Moving on to statement ③: "The general pattern of deductive reasoning is in the form of a 'syllogism'." A syllogism is a form of deductive reasoning consisting of two premises and a conclusion. For example, "All men are mortal. Socrates is a man. Therefore, Socrates is mortal." This is a classic syllogism. So, yes, deductive reasoning often follows this pattern. Therefore, statement ③ seems correct.Lastly, statement ④ states, "The correctness of the conclusion obtained from deductive reasoning depends on the major premise, minor premise, and the form of reasoning." This makes sense because, as I thought earlier, the conclusion's correctness hinges on the truth of the premises (major and minor) and the logical structure of the reasoning. If any of these are flawed, the conclusion might not hold. So, statement ④ is also correct.To summarize my thoughts:- Statement ①: Correct- Statement ②: Incorrect- Statement ③: Correct- Statement ④: CorrectSo, out of the four statements, three are correct. Therefore, the number of correct statements is 3, which corresponds to option C.But wait, let me double-check statement ②. It says the conclusion is "definitely correct." I think the key here is that deductive reasoning can lead to a necessarily correct conclusion only if the premises are true and the reasoning is valid. If the premises are false, the conclusion might still be incorrect even if the reasoning is logically sound. So, it's not that the conclusion is definitely correct in all cases, but rather that if the premises are true and the reasoning is valid, then the conclusion must be true. So, statement ② might be misleading because it doesn't specify the conditions under which the conclusion is correct. Therefore, I think my initial assessment was right; statement ② is incorrect.Yes, I'm confident now that statements ①, ③, and ④ are correct, making the total number of correct statements three.

💡Okay, so I have this problem here with three equations involving variables a, b, c, x, y, z. The equations are:14x + by + cz = 0,ax + 24y + cz = 0,ax + by + 43z = 0.And I'm told that a ≠ 14, b ≠ 24, c ≠ 43, and x ≠ 0. I need to find the value of the expression:(a)/(a - 14) + (b)/(b - 24) + (c)/(c - 43).Hmm, okay. So, first, let me try to understand what's going on here. We have a system of three linear equations with variables x, y, z, and coefficients a, b, c. The fact that x, y, z are real numbers and the equations equal zero suggests that this system has a non-trivial solution, meaning that there's a solution where x, y, z are not all zero. Since x ≠ 0, that tells me that the system is dependent, so the determinant of the coefficient matrix must be zero.Wait, is that the right approach? Let me think. If the system has a non-trivial solution, then the determinant of the coefficients matrix should be zero. So, maybe I can set up the determinant and solve for a, b, c? But the problem is asking for an expression involving a, b, c, not their specific values. Hmm.Alternatively, maybe I can manipulate the equations to express x, y, z in terms of each other or in terms of a, b, c. Let me try that.Looking at the first equation: 14x + by + cz = 0.The second equation: ax + 24y + cz = 0.The third equation: ax + by + 43z = 0.I notice that the first two equations both have the term cz. Maybe I can subtract the first equation from the second to eliminate cz.So, subtracting equation 1 from equation 2:(ax + 24y + cz) - (14x + by + cz) = 0 - 0.Simplify:(ax - 14x) + (24y - by) + (cz - cz) = 0.Which becomes:(a - 14)x + (24 - b)y = 0.So, (a - 14)x = (b - 24)y.Similarly, maybe I can subtract equation 2 from equation 3 to eliminate ax.Subtracting equation 2 from equation 3:(ax + by + 43z) - (ax + 24y + cz) = 0 - 0.Simplify:(ax - ax) + (by - 24y) + (43z - cz) = 0.Which becomes:(b - 24)y + (43 - c)z = 0.So, (b - 24)y = (c - 43)z.Similarly, maybe I can subtract equation 1 from equation 3 to eliminate by.Subtracting equation 1 from equation 3:(ax + by + 43z) - (14x + by + cz) = 0 - 0.Simplify:(ax - 14x) + (by - by) + (43z - cz) = 0.Which becomes:(a - 14)x + (43 - c)z = 0.So, (a - 14)x = (c - 43)z.Okay, so now I have three relationships:1. (a - 14)x = (b - 24)y.2. (b - 24)y = (c - 43)z.3. (a - 14)x = (c - 43)z.So, from these, I can express y and z in terms of x.From the first equation: y = [(a - 14)/(b - 24)]x.From the second equation: z = [(b - 24)/(c - 43)]y.But since y is expressed in terms of x, substitute that into z:z = [(b - 24)/(c - 43)] * [(a - 14)/(b - 24)]x.Simplify: The (b - 24) terms cancel out, so z = [(a - 14)/(c - 43)]x.Alternatively, from the third equation, z = [(a - 14)/(c - 43)]x, which matches.So, now I have y and z in terms of x.So, y = [(a - 14)/(b - 24)]x,z = [(a - 14)/(c - 43)]x.Now, let's plug these expressions for y and z back into one of the original equations to find a relationship between a, b, c.Let me choose the first equation: 14x + by + cz = 0.Substitute y and z:14x + b * [(a - 14)/(b - 24)]x + c * [(a - 14)/(c - 43)]x = 0.Factor out x:x [14 + b*(a - 14)/(b - 24) + c*(a - 14)/(c - 43)] = 0.Since x ≠ 0, the expression in the brackets must be zero:14 + [b(a - 14)]/(b - 24) + [c(a - 14)]/(c - 43) = 0.Let me factor out (a - 14):14 + (a - 14)[b/(b - 24) + c/(c - 43)] = 0.Hmm, interesting. Let me write this as:(a - 14)[b/(b - 24) + c/(c - 43)] = -14.Wait, I need to find the expression (a)/(a - 14) + (b)/(b - 24) + (c)/(c - 43). Maybe I can relate this to the equation I just got.Let me denote S = (a)/(a - 14) + (b)/(b - 24) + (c)/(c - 43).I need to find S.From the equation above, I have:(a - 14)[b/(b - 24) + c/(c - 43)] = -14.Let me denote T = b/(b - 24) + c/(c - 43).So, (a - 14)T = -14.Then, T = -14/(a - 14).But S = a/(a - 14) + T.So, S = a/(a - 14) + (-14)/(a - 14).Simplify:S = [a - 14]/(a - 14) = 1.Wait, that's nice. So, S = 1.Therefore, the value of the expression is 1.Wait, let me double-check my steps to make sure I didn't make a mistake.Starting from the first equation, substituted y and z in terms of x, ended up with an equation involving a, b, c. Then, expressed T as the sum of b/(b - 24) and c/(c - 43), found that (a - 14)T = -14, so T = -14/(a - 14). Then, S is a/(a -14) + T, which is a/(a -14) -14/(a -14) = (a -14)/(a -14) = 1. That seems correct.Alternatively, another approach is to consider that the system has a non-trivial solution, so the determinant of the coefficients matrix must be zero. Let me try that approach to confirm.The coefficient matrix is:[14, b, c;a, 24, c;a, b, 43].The determinant should be zero.Compute the determinant:14*(24*43 - c*b) - b*(a*43 - c*a) + c*(a*b - 24*a) = 0.Wait, that's a bit messy, but let's compute it step by step.First, expand along the first row:14 * det[[24, c], [b, 43]] - b * det[[a, c], [a, 43]] + c * det[[a, 24], [a, b]].Compute each minor:First minor: 24*43 - c*b = 1032 - bc.Second minor: a*43 - c*a = a(43 - c).Third minor: a*b - 24*a = a(b - 24).So, putting it all together:14*(1032 - bc) - b*(a(43 - c)) + c*(a(b - 24)) = 0.Simplify each term:14*1032 = 14448,14*(-bc) = -14bc,- b*(a(43 - c)) = -a b (43 - c),c*(a(b - 24)) = a c (b - 24).So, the equation becomes:14448 -14bc - a b (43 - c) + a c (b - 24) = 0.Let me expand the terms with a:- a b (43 - c) + a c (b - 24) = -43 a b + a b c + a b c -24 a c.Wait, that's:-43ab + abc + abc -24ac.Combine like terms:-43ab + 2abc -24ac.So, the equation is:14448 -14bc -43ab + 2abc -24ac = 0.Hmm, that seems complicated. Maybe I can factor out some terms.Let me factor out a from the terms involving a:a(-43b + 2bc -24c) + (-14bc + 14448) = 0.Hmm, not sure if that helps. Alternatively, maybe I can factor terms with bc:Looking at the equation:14448 -14bc -43ab + 2abc -24ac = 0.Let me rearrange terms:2abc -14bc -43ab -24ac +14448 = 0.Factor bc from the first two terms:bc(2a -14) -43ab -24ac +14448 = 0.Hmm, maybe factor a from the next two terms:bc(2a -14) -a(43b +24c) +14448 = 0.Hmm, still complicated. Maybe I can factor out (a -14) or something similar.Wait, perhaps another approach. Let me recall that in the first method, I found that S = 1. Maybe I can relate this determinant equation to S.Alternatively, perhaps I can express the determinant equation in terms of S.But this seems too involved. Maybe my first approach was better.Alternatively, let me think about the system of equations. Since x ≠ 0, the system has a non-trivial solution, so the determinant is zero, which gives me the equation above. But perhaps instead of computing the determinant, I can use the fact that the system is consistent and express ratios between x, y, z as I did earlier, leading to the expression S = 1.Since both methods lead me to the same conclusion, I think the answer is indeed 1.**Final Answer**boxed{1}

💡Okay, so I have this problem about a trapezoid. Let me first recall what a trapezoid is. A trapezoid is a quadrilateral with at least one pair of parallel sides, which are called the bases. The other two sides are called the legs. Now, the problem says that there's a segment parallel to the bases and equal to their geometric mean. I need to prove that this segment divides the non-parallel sides (the legs) into segments with lengths in the ratio of the square root of the ratio of the lengths of the bases.Alright, let me break this down. Let's denote the trapezoid as ABCD, where AB and CD are the bases with lengths a and c respectively. The segment that's parallel to these bases and equal to their geometric mean would be MN, where M is on side AD and N is on side BC. So, MN is parallel to AB and CD, and its length is sqrt(ac).I need to show that the ratio of the lengths of the segments into which MN divides the legs AD and BC is sqrt(a/c). That is, if MN intersects AD at M and BC at N, then AM/MD = BN/NC = sqrt(a/c).Hmm, okay. Maybe I can use similar triangles here. Since MN is parallel to AB and CD, the triangles formed by these segments should be similar. Let me visualize this. If I draw MN parallel to AB and CD, then triangles AMN and ABC should be similar, and similarly, triangles MND and CDA should be similar.Wait, actually, since MN is between AB and CD, maybe it's better to think of the trapezoid being divided into two smaller trapezoids by MN. But I'm not sure if that's the right approach. Maybe I should consider the triangles formed by the legs and the segment MN.Alternatively, perhaps using the properties of trapezoids and the concept of the geometric mean. The geometric mean of two numbers is the square root of their product, so MN = sqrt(ac). I need to relate this to the ratio of the segments on the legs.Let me denote the lengths of the segments on the legs. Let's say that AM = x and MD = y on side AD, so that x + y = AD. Similarly, BN = p and NC = q on side BC, so that p + q = BC. I need to show that x/y = sqrt(a/c) and p/q = sqrt(a/c).Since MN is parallel to AB and CD, the triangles AMN and ABC are similar. So, the ratio of their corresponding sides should be equal. Therefore, AM/AB = AN/AC. Wait, but I'm not sure about that. Maybe I should think about the ratio of the lengths of the bases.Alternatively, since MN is parallel to AB and CD, the ratio of the lengths of the segments on the legs should be equal to the ratio of the lengths of the bases. But in this case, MN is the geometric mean, so maybe the ratio is sqrt(a/c).Let me try to set up some equations. Let's denote the height of the trapezoid as h. Then, the area of the trapezoid is (a + c)/2 * h. But I'm not sure if the area is directly useful here.Wait, maybe using the concept of similar triangles. If I draw a line parallel to the bases, it creates similar triangles. So, the ratio of similarity would be the ratio of the lengths of the corresponding sides.Let me consider triangle ABD and triangle MND. Wait, no, that might not be the right approach. Maybe I should consider the triangles formed by the legs and the segment MN.Alternatively, perhaps using coordinate geometry. Let me place the trapezoid on a coordinate system. Let me set point A at (0, 0), point B at (a, 0), point C at (d, h), and point D at (e, h). But this might get complicated. Maybe there's a simpler way.Wait, I remember that in a trapezoid, if a line is drawn parallel to the bases, dividing the legs proportionally, then the length of that line is the weighted average of the lengths of the bases. Specifically, if the ratio of the segments on the legs is k, then the length of the segment is (k*c + a)/(k + 1). But in this case, the segment is the geometric mean, so sqrt(ac). Therefore, we can set up the equation sqrt(ac) = (k*c + a)/(k + 1) and solve for k.Let me write that down:sqrt(ac) = (k*c + a)/(k + 1)Multiply both sides by (k + 1):sqrt(ac)*(k + 1) = k*c + aExpand the left side:k*sqrt(ac) + sqrt(ac) = k*c + aBring all terms to one side:k*sqrt(ac) + sqrt(ac) - k*c - a = 0Factor out k:k*(sqrt(ac) - c) + (sqrt(ac) - a) = 0Solve for k:k = (a - sqrt(ac))/(sqrt(ac) - c)Factor out sqrt(a) from numerator and denominator:k = sqrt(a)*(sqrt(a) - sqrt(c)) / (sqrt(c)*(sqrt(a) - sqrt(c)))Simplify:k = sqrt(a)/sqrt(c) = sqrt(a/c)So, the ratio k is sqrt(a/c), which is the ratio of the segments on the legs. Therefore, the segment MN divides the legs AD and BC into segments with lengths in the ratio sqrt(a/c).Wait, let me check my steps. I set up the equation for the length of the segment MN as a weighted average, which is correct. Then I solved for k and ended up with k = sqrt(a/c). That seems right. So, the ratio of the segments on the legs is indeed sqrt(a/c).I think this makes sense. The geometric mean of the bases leads to the ratio of the segments on the legs being the square root of the ratio of the bases. So, the proof is complete.

💡First, determine the total number of days Bob takes the pills: 3 weeks × 7 days/week = 21 days.Next, calculate the total cost per day: 966 ÷ 21 days = 46/day.Let the cost of one red pill be ( r ) dollars. Since a blue pill costs 4 more than a red pill, the cost of one blue pill is ( r + 4 ) dollars.Bob takes one blue pill and one red pill each day, so the total daily cost is:[r + (r + 4) = 2r + 4]Set up the equation for the daily cost:[2r + 4 = 46]Solve for ( r ):[2r = 46 - 4 2r = 42 r = 21]Therefore, the cost of one blue pill is:[r + 4 = 21 + 4 = 25]

💡Alright, so I have this problem about a parallelepiped. Let me try to visualize it first. A parallelepiped is like a 3D figure formed by six parallelograms. It's kind of a slanted box, right? The problem says that one of its edges is equal to ( l ). Okay, so let's say this edge is ( DD_1 ) in the parallelepiped ( ABCD A_1 B_1 C_1 D_1 ). It's bounded by two adjacent faces, and these faces have areas ( m^2 ) and ( n^2 ). Also, the planes of these two faces form a 30-degree angle. I need to find the volume of this parallelepiped.Hmm, volume of a parallelepiped. I remember that the volume can be found using the scalar triple product of the vectors defining the edges. But maybe there's a simpler way since we have some areas and an angle between the faces.Let me think about the given information. One edge is ( l ), and two adjacent faces have areas ( m^2 ) and ( n^2 ). The angle between these two faces is 30 degrees. So, if I can find the lengths of the other edges, I can maybe compute the volume.Wait, the areas of the faces are given. So, if I consider one face, say ( AA_1 D_1 D ), its area is ( m^2 ). Since it's a parallelogram, the area is base times height. If ( DD_1 = l ), then the base could be ( l ), and the height would be something else. Similarly, the other face ( CC_1 D_1 D ) has an area of ( n^2 ).Maybe I can denote the other edges as ( a ) and ( b ), so that the areas of the faces are ( a times l = m^2 ) and ( b times l = n^2 ). So, ( a = frac{m^2}{l} ) and ( b = frac{n^2}{l} ).Now, the angle between the two faces is 30 degrees. Since the faces are adjacent, the angle between them is the dihedral angle, which is the angle between the two planes. This angle is related to the angle between the normals of the two planes, but I'm not sure how to translate that into the volume.Alternatively, maybe I can think of the volume as the product of the area of the base and the height. If I take one face as the base, then the height would be the distance from the opposite face perpendicular to the base. But since the faces are inclined at 30 degrees, the height might involve some sine or cosine of that angle.Let me try to formalize this. Suppose I take the face with area ( m^2 ) as the base. Then, the height corresponding to this base would be related to the other face's area and the angle between them. Since the angle between the faces is 30 degrees, the height might be ( frac{n^2}{l} times sin(30^circ) ).Wait, let me think. If the two faces meet at a dihedral angle of 30 degrees, then the height from the opposite edge would involve the sine of that angle. So, maybe the height ( h ) is ( frac{n^2}{l} times sin(30^circ) ).Since ( sin(30^circ) = 0.5 ), then ( h = frac{n^2}{2l} ).So, the volume would be the area of the base times the height, which is ( m^2 times frac{n^2}{2l} = frac{m^2 n^2}{2l} ).Does that make sense? Let me double-check. If I have two adjacent faces with areas ( m^2 ) and ( n^2 ), and the angle between them is 30 degrees, then the volume should indeed involve the product of these areas, adjusted by the sine of the angle between them, and scaled by the edge length.Alternatively, thinking in terms of vectors, if I have vectors ( vec{a} ) and ( vec{b} ) defining the edges, the volume is the magnitude of the scalar triple product ( |vec{a} cdot (vec{b} times vec{c})| ). But since I don't have all the vectors, maybe this approach is more complicated.I think my initial approach is correct. By considering the areas and the angle between the faces, I can relate the heights and compute the volume accordingly. So, I'll go with ( frac{m^2 n^2}{2l} ) as the volume.**Final Answer**The volume of the parallelepiped is boxed{dfrac{m^2 n^2}{2l}}.

💡Okay, so I have this problem where I need to find the range of values for ( a ) given some conditions about two propositions ( p ) and ( q ). Let me try to break it down step by step.First, let me understand what each proposition means.Proposition ( p ): The function ( y = a^x ) is monotonically increasing on ( mathbb{R} ).I remember that for exponential functions ( y = a^x ), if ( a > 1 ), the function is indeed monotonically increasing. If ( 0 < a < 1 ), the function is monotonically decreasing. So, if ( p ) is true, that means ( a > 1 ). Conversely, if ( p ) is false, ( a ) must be less than or equal to 1.Proposition ( q ): The inequality ( ax^2 - ax + 1 > 0 ) holds for all ( x in mathbb{R} ).This is a quadratic inequality. For a quadratic ( ax^2 + bx + c ), it is always positive if the leading coefficient ( a ) is positive and the discriminant ( Delta = b^2 - 4ac ) is negative. So, let's compute the discriminant for ( q ).Given ( ax^2 - ax + 1 ), the discriminant ( Delta ) is:[Delta = (-a)^2 - 4 cdot a cdot 1 = a^2 - 4a]For the quadratic to be always positive, we need ( Delta < 0 ):[a^2 - 4a < 0][a(a - 4) < 0]This inequality holds when ( a ) is between 0 and 4. Since ( a > 0 ) is given, the solution is ( 0 < a < 4 ). So, if ( q ) is true, ( a ) must be in ( (0, 4) ). If ( q ) is false, then ( a ) is either less than or equal to 0 or greater than or equal to 4. But since ( a > 0 ), the false condition for ( q ) simplifies to ( a geq 4 ).Now, the problem states that both ( p ) and ( q ) are false, and ( p ) or ( q ) is true. Hmm, that seems a bit confusing. Let me parse that again.Wait, actually, the problem says: "If ( p ) and ( q ) are false, and ( p ) or ( q ) is true, find the range of values for ( a )."Wait, that seems contradictory because if both ( p ) and ( q ) are false, then ( p ) or ( q ) should also be false. But the problem says that ( p ) or ( q ) is true. So, maybe I misread the problem.Let me check again: "If ( p ) and ( q ) are false, and ( p ) or ( q ) is true, find the range of values for ( a )."Hmm, that seems contradictory because if both ( p ) and ( q ) are false, then ( p ) or ( q ) must be false. So, perhaps the problem is saying that ( p ) and ( q ) cannot both be true or both be false? Or maybe it's saying that ( p ) and ( q ) are both false, but ( p ) or ( q ) is true, which is a contradiction. That can't happen.Wait, maybe the problem is phrased differently. Let me read it again carefully:"If ( p ) and ( q ) are false, and ( p ) or ( q ) is true, find the range of values for ( a )."Wait, that seems like a logical contradiction because if both ( p ) and ( q ) are false, then ( p ) or ( q ) must be false. So, perhaps the problem is misstated? Or maybe I'm misunderstanding the logical connectives.Alternatively, maybe it's saying that ( p ) and ( q ) are both false, but ( p ) or ( q ) is true, which is impossible. Therefore, perhaps the problem is actually saying that ( p ) and ( q ) cannot both be true or both be false, but that's not exactly what it says.Wait, perhaps the problem is saying that ( p ) and ( q ) are both false, and ( p ) or ( q ) is true. But that's impossible because if both are false, then ( p ) or ( q ) is false. So, maybe the problem is misstated.Alternatively, perhaps the problem is saying that ( p ) and ( q ) are both false, but ( p ) or ( q ) is true, which is a contradiction, so there is no solution. But that can't be, because the problem is asking for a range.Wait, perhaps the problem is actually saying that ( p ) and ( q ) cannot both be true or both be false, but that's not what it says. Let me read it again:"If ( p ) and ( q ) are false, and ( p ) or ( q ) is true, find the range of values for ( a )."Wait, perhaps the problem is saying that ( p ) and ( q ) are both false, but ( p ) or ( q ) is true, which is impossible, so there is no solution. But that can't be, because the problem is asking for a range.Alternatively, maybe the problem is saying that ( p ) and ( q ) are both false, but ( p ) or ( q ) is true, which is a contradiction, so there is no solution. But that can't be, because the problem is asking for a range.Wait, perhaps I misread the problem. Let me check again:"Given ( a > 0 ), let proposition ( p ): the function ( y = a^x ) is monotonically increasing on ( mathbb{R} ); proposition ( q ): the inequality ( ax^2 - ax + 1 > 0 ) holds for ( forall x in mathbb{R} ). If ( p ) and ( q ) are false, and ( p ) or ( q ) is true, find the range of values for ( a )."Wait, so the problem is saying that ( p ) and ( q ) are both false, but ( p ) or ( q ) is true. That is, both ( p ) and ( q ) are false, but at least one of them is true. But that's a contradiction because if both are false, then ( p ) or ( q ) is false. So, perhaps the problem is misstated.Alternatively, maybe the problem is saying that ( p ) and ( q ) are not both true, and ( p ) or ( q ) is true. That is, exactly one of ( p ) or ( q ) is true. That would make sense because if exactly one is true, then ( p ) and ( q ) are not both true, and ( p ) or ( q ) is true.Wait, but the problem says "If ( p ) and ( q ) are false, and ( p ) or ( q ) is true". So, maybe it's a translation issue or a misstatement.Alternatively, perhaps the problem is saying that ( p ) and ( q ) cannot both be true, and ( p ) or ( q ) is true. That would mean that exactly one of ( p ) or ( q ) is true.But in the original problem, it's stated as "If ( p ) and ( q ) are false, and ( p ) or ( q ) is true". So, perhaps it's a translation issue, and the intended meaning is that ( p ) and ( q ) are not both true, and ( p ) or ( q ) is true, meaning exactly one is true.Alternatively, perhaps the problem is saying that ( p ) and ( q ) are both false, but ( p ) or ( q ) is true, which is impossible, so there is no solution. But that can't be, because the problem is asking for a range.Wait, maybe the problem is saying that ( p ) and ( q ) are both false, but ( p ) or ( q ) is true, which is a contradiction, so there is no solution. But that can't be, because the problem is asking for a range.Alternatively, perhaps the problem is saying that ( p ) and ( q ) are both false, but ( p ) or ( q ) is true, which is impossible, so the range is empty. But that seems unlikely.Wait, perhaps I'm overcomplicating this. Let me try to approach it differently.Given that ( p ) and ( q ) are both false, but ( p ) or ( q ) is true. Wait, that's impossible because if both are false, then ( p ) or ( q ) is false. So, perhaps the problem is misstated.Alternatively, perhaps the problem is saying that ( p ) and ( q ) are not both true, and ( p ) or ( q ) is true. That would mean that exactly one of ( p ) or ( q ) is true.So, let's assume that the problem is saying that exactly one of ( p ) or ( q ) is true. That is, ( p ) is true and ( q ) is false, or ( p ) is false and ( q ) is true.So, let's consider both cases.Case 1: ( p ) is true and ( q ) is false.We know that ( p ) is true when ( a > 1 ).( q ) is false when ( a geq 4 ) (since for ( q ) to be true, ( a ) must be in ( (0,4) ), so false when ( a geq 4 )).So, in this case, ( a > 1 ) and ( a geq 4 ). The intersection of these is ( a geq 4 ).Case 2: ( p ) is false and ( q ) is true.( p ) is false when ( 0 < a leq 1 ).( q ) is true when ( 0 < a < 4 ).So, the intersection here is ( 0 < a leq 1 ).Therefore, combining both cases, the range of ( a ) is ( (0,1] cup [4, infty) ).Wait, that makes sense. So, the problem might have a translation issue, but the intended meaning is that exactly one of ( p ) or ( q ) is true, leading to the range ( (0,1] cup [4, infty) ).So, I think that's the answer.