💡Alright, so I have this trigonometric equation to solve: sin(4x) * sin(6x) = cos(4x) * cos(6x). Hmm, okay, let me think about how to approach this. I remember there are some trigonometric identities that might help simplify this equation. First, I notice that both sides of the equation involve products of sine and cosine functions with arguments 4x and 6x. Maybe I can use a product-to-sum formula or something similar. Wait, actually, there's an identity that relates the product of sines and cosines. Let me recall it.I think the identity is something like cos(A + B) = cos A cos B - sin A sin B. Oh, that's right! So if I rearrange the given equation, I can write it as cos(4x)cos(6x) - sin(4x)sin(6x) = 0. That looks exactly like the right-hand side of the cosine addition formula. So, applying that identity, the equation simplifies to cos(4x + 6x) = 0, which is cos(10x) = 0.Okay, so now the equation is cos(10x) = 0. I need to find the smallest positive angle x that satisfies this equation. I know that cosine equals zero at 90 degrees, 270 degrees, 450 degrees, and so on. In general, cosine is zero at (90 + 180k) degrees, where k is any integer. So, 10x must be equal to one of these angles.Let me write that down: 10x = 90 + 180k degrees, where k is an integer. To find x, I can divide both sides by 10: x = (90 + 180k)/10 degrees. Simplifying that, x = 9 + 18k degrees.Now, I need the smallest positive angle x. Let's consider k = 0 first: x = 9 + 0 = 9 degrees. That's positive. What if k = -1? Then x = 9 - 18 = -9 degrees, which is negative, so we don't want that. So, the smallest positive solution is x = 9 degrees.Wait, let me double-check. If I plug x = 9 degrees back into the original equation, does it hold? Let's compute both sides.First, sin(4x) = sin(36 degrees) and sin(6x) = sin(54 degrees). So, sin(36) * sin(54). Similarly, cos(4x) = cos(36 degrees) and cos(6x) = cos(54 degrees). So, cos(36) * cos(54).Calculating sin(36) * sin(54): sin(36) is approximately 0.5878, and sin(54) is approximately 0.8090. Multiplying them gives roughly 0.5878 * 0.8090 ≈ 0.4755.Now, cos(36) * cos(54): cos(36) is approximately 0.8090, and cos(54) is approximately 0.5878. Multiplying them gives roughly 0.8090 * 0.5878 ≈ 0.4755.So, both sides are approximately equal, which means x = 9 degrees is indeed a solution. Is there a smaller positive solution? Let's see. If I take k = 0, x = 9 degrees. If I take k = -1, x = -9 degrees, which is negative. So, 9 degrees is the smallest positive solution.Alternatively, I could think about the unit circle. The equation cos(10x) = 0 implies that 10x must be at an angle where cosine is zero, which occurs at 90 degrees, 270 degrees, etc. The smallest positive angle is 90 degrees, so 10x = 90 degrees, leading to x = 9 degrees. That confirms my earlier result.I think I'm confident that x = 9 degrees is the correct answer. It satisfies the original equation, and there's no smaller positive angle that does so.**Final Answer**The smallest positive angle ( x ) is boxed{9} degrees.

💡Okay, so I have this problem where I need to find the value of ( ac + bd ) given four equations with four variables: ( a ), ( b ), ( c ), and ( d ). The equations are:1. ( a + b + c = 6 )2. ( a + b + d = -3 )3. ( a + c + d = 0 )4. ( b + c + d = -9 )Hmm, since there are four variables and four equations, it seems like I can solve for each variable individually and then plug them into ( ac + bd ) to find the answer. Let me try to figure out how to do that step by step.First, I notice that each equation is missing one variable. For example, the first equation is missing ( d ), the second is missing ( c ), the third is missing ( b ), and the fourth is missing ( a ). Maybe if I add all these equations together, I can find the sum of all variables and then use that to find each variable's value.Let me add all four equations:( (a + b + c) + (a + b + d) + (a + c + d) + (b + c + d) = 6 + (-3) + 0 + (-9) )Simplifying the left side:Each variable is added three times because each variable is missing from one equation. So, ( a ) appears in three equations, ( b ) appears in three equations, and so on. Therefore, the left side becomes:( 3a + 3b + 3c + 3d )On the right side, adding the constants:( 6 - 3 + 0 - 9 = -6 )So, the equation becomes:( 3a + 3b + 3c + 3d = -6 )I can factor out a 3:( 3(a + b + c + d) = -6 )Divide both sides by 3:( a + b + c + d = -2 )Okay, so the sum of all four variables is -2. Now, I can use this to find each variable by subtracting each original equation from this sum.Let me write down the sum again:( a + b + c + d = -2 )Now, let's subtract each original equation one by one.1. Subtracting the first equation ( a + b + c = 6 ) from the sum:( (a + b + c + d) - (a + b + c) = -2 - 6 )Simplifying:( d = -8 )So, ( d = -8 ).2. Subtracting the second equation ( a + b + d = -3 ) from the sum:( (a + b + c + d) - (a + b + d) = -2 - (-3) )Simplifying:( c = -2 + 3 = 1 )So, ( c = 1 ).3. Subtracting the third equation ( a + c + d = 0 ) from the sum:( (a + b + c + d) - (a + c + d) = -2 - 0 )Simplifying:( b = -2 )So, ( b = -2 ).4. Subtracting the fourth equation ( b + c + d = -9 ) from the sum:( (a + b + c + d) - (b + c + d) = -2 - (-9) )Simplifying:( a = -2 + 9 = 7 )So, ( a = 7 ).Now, I have all the values:- ( a = 7 )- ( b = -2 )- ( c = 1 )- ( d = -8 )Now, I need to compute ( ac + bd ). Let's calculate each product separately.First, ( ac ):( a times c = 7 times 1 = 7 )Next, ( bd ):( b times d = (-2) times (-8) = 16 )Now, add these two results together:( ac + bd = 7 + 16 = 23 )So, the value of ( ac + bd ) is 23.Wait, let me double-check my calculations to make sure I didn't make any mistakes.Starting with the sum of all four equations:( 3a + 3b + 3c + 3d = -6 ) simplifies to ( a + b + c + d = -2 ). That seems correct.Subtracting each equation:1. ( d = -2 - 6 = -8 ) – correct.2. ( c = -2 - (-3) = 1 ) – correct.3. ( b = -2 - 0 = -2 ) – correct.4. ( a = -2 - (-9) = 7 ) – correct.Calculating ( ac ) and ( bd ):( 7 times 1 = 7 ) and ( (-2) times (-8) = 16 ). Adding them gives 23. That seems right.I think I did everything correctly. So, the answer should be 23.**Final Answer**boxed{23}

💡Okay, so I've got this problem here with six positive variables: p, q, r, u, v, and w. The problem gives me three equations:1. ( p^2 + q^2 + r^2 = 49 )2. ( u^2 + v^2 + w^2 = 64 )3. ( pu + qv + rw = 56 )And I need to find the value of ( frac{p + q + r}{u + v + w} ).Hmm, let me think about how to approach this. I remember that when dealing with sums of squares and products, the Cauchy-Schwarz inequality often comes into play. Maybe that's useful here.The Cauchy-Schwarz inequality states that for any real vectors, the square of their dot product is less than or equal to the product of their magnitudes squared. In mathematical terms, for vectors ( mathbf{a} ) and ( mathbf{b} ), it's:[ (mathbf{a} cdot mathbf{b})^2 leq (mathbf{a} cdot mathbf{a})(mathbf{b} cdot mathbf{b}) ]In this problem, if I let ( mathbf{a} = (p, q, r) ) and ( mathbf{b} = (u, v, w) ), then the dot product ( mathbf{a} cdot mathbf{b} = pu + qv + rw = 56 ). The magnitudes squared of these vectors are given by the first two equations: ( ||mathbf{a}||^2 = 49 ) and ( ||mathbf{b}||^2 = 64 ).Applying Cauchy-Schwarz, we get:[ (pu + qv + rw)^2 leq (p^2 + q^2 + r^2)(u^2 + v^2 + w^2) ]Plugging in the given values:[ 56^2 leq 49 times 64 ]Calculating both sides:Left side: ( 56^2 = 3136 )Right side: ( 49 times 64 = 3136 )So, both sides are equal. I remember that equality in Cauchy-Schwarz occurs if and only if the vectors are linearly dependent, which in this case means that each component of one vector is a scalar multiple of the corresponding component of the other vector. So, there exists some constant ( k ) such that:[ p = k u ][ q = k v ][ r = k w ]Alright, so that gives me a relationship between the variables. Now, let's use this to find the value of ( k ).Since ( p = k u ), ( q = k v ), and ( r = k w ), I can substitute these into the first equation:[ p^2 + q^2 + r^2 = (k u)^2 + (k v)^2 + (k w)^2 = k^2 (u^2 + v^2 + w^2) ]We know that ( p^2 + q^2 + r^2 = 49 ) and ( u^2 + v^2 + w^2 = 64 ), so substituting these in:[ 49 = k^2 times 64 ]Solving for ( k^2 ):[ k^2 = frac{49}{64} ]Taking the square root of both sides, since all variables are positive, ( k ) must be positive:[ k = frac{7}{8} ]Great, so now we know that ( p = frac{7}{8} u ), ( q = frac{7}{8} v ), and ( r = frac{7}{8} w ).Now, the question asks for ( frac{p + q + r}{u + v + w} ). Let's compute the numerator:[ p + q + r = frac{7}{8} u + frac{7}{8} v + frac{7}{8} w = frac{7}{8} (u + v + w) ]So, when we take the ratio:[ frac{p + q + r}{u + v + w} = frac{frac{7}{8} (u + v + w)}{u + v + w} ]The ( u + v + w ) terms cancel out, leaving:[ frac{7}{8} ]That seems straightforward. Let me just double-check my steps to make sure I didn't make any mistakes.1. Recognized that Cauchy-Schwarz applies here because we have sums of squares and a dot product.2. Applied the inequality and noticed that both sides are equal, which implies equality condition.3. Set up the proportional relationships between the variables.4. Substituted into the given equations to solve for the constant of proportionality ( k ).5. Expressed ( p + q + r ) in terms of ( u + v + w ) and found the ratio.Everything seems to check out. I don't see any errors in the logic or calculations. So, I'm confident that the answer is ( frac{7}{8} ).**Final Answer**boxed{dfrac{7}{8}}

💡First, I need to determine the cost price of one ball. Let's denote the cost price of one ball as ( x ).The cost price of 13 balls would then be ( 13x ).The selling price of 13 balls is given as Rs. 720.The loss incurred is equal to the cost price of 5 balls, which is ( 5x ).Using the formula for loss:[text{Loss} = text{Cost Price} - text{Selling Price}]Substituting the known values:[5x = 13x - 720]To find ( x ), I'll solve the equation:[5x = 13x - 720]Subtract ( 5x ) from both sides:[0 = 8x - 720]Add 720 to both sides:[720 = 8x]Divide both sides by 8:[x = frac{720}{8} = 90]Therefore, the cost price of one ball is Rs. 90.

💡Okay, so I have this problem with an ellipse and a hyperbola that share the same foci, F₁ and F₂. The ellipse is given by the equation x²/m + y² = 1, where m > 1, and the hyperbola is given by x²/n - 3y² = 1, where n > 0. I need to find the angle ∠F₁PF₂ where P is one of their intersection points. The options are 30°, 60°, 90°, or 120°. Hmm, okay, let's break this down step by step.First, I remember that for both ellipses and hyperbolas, the distance between the foci is related to the parameters of the curves. For an ellipse, the distance between the foci is 2c, where c = √(m - 1) because the standard form of an ellipse is x²/a² + y²/b² = 1, and here a² = m and b² = 1. So, c = √(a² - b²) = √(m - 1).For the hyperbola, the standard form is x²/a² - y²/b² = 1, so here a² = n and b² = 1/3. The distance between the foci for a hyperbola is also 2c, where c = √(a² + b²) = √(n + 1/3).Since both curves share the same foci, their c values must be equal. So, √(m - 1) = √(n + 1/3). Squaring both sides, we get m - 1 = n + 1/3. Therefore, m = n + 4/3. Okay, that's a useful relationship between m and n.Now, let's think about the point P where the ellipse and hyperbola intersect. At this point P, both equations are satisfied. So, P lies on both the ellipse and the hyperbola. Let me denote the distances from P to F₁ and F₂ as d₁ and d₂, respectively.For the ellipse, I remember that the sum of the distances from any point on the ellipse to the two foci is constant and equal to 2a, which in this case is 2√m. So, d₁ + d₂ = 2√m.For the hyperbola, the difference of the distances from any point on the hyperbola to the two foci is constant and equal to 2a, which here is 2√n. So, |d₁ - d₂| = 2√n.Since P is on the right branch of the hyperbola, I think d₁ - d₂ = 2√n. So, we have two equations:1. d₁ + d₂ = 2√m2. d₁ - d₂ = 2√nI can solve these two equations to find d₁ and d₂. Adding the two equations, I get 2d₁ = 2√m + 2√n, so d₁ = √m + √n. Subtracting the second equation from the first, I get 2d₂ = 2√m - 2√n, so d₂ = √m - √n.Now, I need to find the angle ∠F₁PF₂. To find this angle, I can use the Law of Cosines in triangle F₁PF₂. The Law of Cosines states that for any triangle with sides a, b, c opposite angles A, B, C respectively, c² = a² + b² - 2ab cos(C).In this case, the sides of the triangle are d₁, d₂, and the distance between F₁ and F₂, which is 2c, where c is the distance from the center to each focus. Earlier, we found that c = √(m - 1) = √(n + 1/3). So, the distance between F₁ and F₂ is 2√(m - 1) = 2√(n + 1/3).Let me denote the distance between F₁ and F₂ as D. So, D = 2√(m - 1). Then, according to the Law of Cosines:D² = d₁² + d₂² - 2d₁d₂ cos(∠F₁PF₂)I need to compute D², d₁², and d₂². Let's compute each term step by step.First, D² = [2√(m - 1)]² = 4(m - 1).Next, d₁² = (√m + √n)² = m + 2√(mn) + n.Similarly, d₂² = (√m - √n)² = m - 2√(mn) + n.So, d₁² + d₂² = [m + 2√(mn) + n] + [m - 2√(mn) + n] = 2m + 2n.Now, let's compute 2d₁d₂. d₁d₂ = (√m + √n)(√m - √n) = m - n. So, 2d₁d₂ = 2(m - n).Putting it all together into the Law of Cosines equation:4(m - 1) = (2m + 2n) - 2(m - n) cos(∠F₁PF₂)Simplify the right-hand side:4(m - 1) = 2m + 2n - 2(m - n) cos(∠F₁PF₂)Let me rearrange this equation to solve for cos(∠F₁PF₂):2(m - n) cos(∠F₁PF₂) = 2m + 2n - 4(m - 1)Simplify the right-hand side:2m + 2n - 4m + 4 = -2m + 2n + 4So,2(m - n) cos(∠F₁PF₂) = -2m + 2n + 4Divide both sides by 2:(m - n) cos(∠F₁PF₂) = -m + n + 2Now, recall from earlier that m = n + 4/3. Let me substitute m with n + 4/3 in the equation:(n + 4/3 - n) cos(∠F₁PF₂) = -(n + 4/3) + n + 2Simplify the left-hand side:(4/3) cos(∠F₁PF₂) = -n - 4/3 + n + 2Simplify the right-hand side:-n + n cancels out, so we have -4/3 + 2 = (-4/3 + 6/3) = 2/3So, we have:(4/3) cos(∠F₁PF₂) = 2/3Multiply both sides by 3:4 cos(∠F₁PF₂) = 2Divide both sides by 4:cos(∠F₁PF₂) = 2/4 = 1/2So, cos(∠F₁PF₂) = 1/2. The angle whose cosine is 1/2 is 60°, since cos(60°) = 0.5.Therefore, ∠F₁PF₂ is 60°, which corresponds to option B.Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from the Law of Cosines, I substituted all the terms correctly. I used the relationship m = n + 4/3, which came from equating the foci distances. Then, I substituted m in terms of n and simplified. The algebra seems correct, and the final result for the cosine is 1/2, leading to 60°. Yeah, I think that's right.I also considered whether P is on the right branch of the hyperbola, which affects the sign in the hyperbola's distance difference. I assumed it's on the right branch, so d₁ - d₂ = 2√n. If it were on the left branch, it would be d₂ - d₁ = 2√n, but since the problem doesn't specify, I think assuming the right branch is fine because the angle would be the same regardless of the branch due to symmetry.So, I feel confident that the angle is 60°, so the answer is B.**Final Answer**boxed{B}

💡First, I need to find the number ( x ) such that ( 1.003^x = 1.012 ).To solve for ( x ), I'll take the natural logarithm of both sides of the equation:[ln(1.003^x) = ln(1.012)]Using the logarithmic identity ( ln(a^b) = b cdot ln(a) ), I can simplify the left side:[x cdot ln(1.003) = ln(1.012)]Next, I'll solve for ( x ) by dividing both sides by ( ln(1.003) ):[x = frac{ln(1.012)}{ln(1.003)}]Now, I'll calculate the natural logarithms:[ln(1.012) approx 0.011918][ln(1.003) approx 0.002995]Finally, I'll divide these values to find ( x ):[x approx frac{0.011918}{0.002995} approx 3.978]Rounded to three decimal places, the number is:[x approx 3.978]