💡First, I need to determine the total cost of the pens. There are 30 pens, and each pen costs 18. So, the total cost for the pens is 30 multiplied by 18, which equals 540.Next, I'll calculate the total cost of the pencils. Let's denote the number of pencils as P. Each pencil costs 2, so the total cost for the pencils is 2 times P, or 2P.The sum of the costs of pens and pencils is given as 690. Therefore, I can set up the equation: 540 (cost of pens) plus 2P (cost of pencils) equals 690.To find the value of P, I'll subtract 540 from both sides of the equation: 2P equals 690 minus 540, which is 150.Finally, I'll solve for P by dividing both sides of the equation by 2. This gives P equals 150 divided by 2, which is 75.Therefore, the number of pencils purchased is 75.

💡Alright, so I have this equation to solve: 3z minus 4 times the conjugate of z equals -1 minus 40i. Hmm, complex numbers can be tricky, but I think I remember that to solve equations involving complex numbers, I can represent z as a + bi, where a and b are real numbers. Then, the conjugate of z, which is denoted as (bar{z}), would be a - bi. Okay, let me write that down. Let z = a + bi. Then, (bar{z}) = a - bi. Now, I need to substitute these into the equation 3z - 4(bar{z}) = -1 - 40i. So, substituting, I get:3(a + bi) - 4(a - bi) = -1 - 40i.Alright, let me expand this out. First, distribute the 3 into (a + bi):3 * a = 3a,3 * bi = 3bi.So, that gives me 3a + 3bi.Next, distribute the -4 into (a - bi):-4 * a = -4a,-4 * (-bi) = +4bi.So, that gives me -4a + 4bi.Now, combine these two results:3a + 3bi - 4a + 4bi.Let me combine like terms. The real parts are 3a and -4a, and the imaginary parts are 3bi and 4bi.So, combining the real parts: 3a - 4a = -a.Combining the imaginary parts: 3bi + 4bi = 7bi.So, putting it all together, the left side simplifies to -a + 7bi.Now, the equation is:-a + 7bi = -1 - 40i.Hmm, okay. So, for two complex numbers to be equal, their real parts must be equal, and their imaginary parts must be equal. That means:Real part: -a = -1,Imaginary part: 7b = -40.Alright, let's solve for a and b.Starting with the real part equation: -a = -1. To solve for a, I can multiply both sides by -1:a = 1.Okay, that was straightforward. Now, the imaginary part equation: 7b = -40. To solve for b, I'll divide both sides by 7:b = -40 / 7.Hmm, that's a fraction. I wonder if I can simplify that, but 40 divided by 7 is approximately 5.714, but since it's exact, I should keep it as a fraction. So, b = -40/7.Now, with a and b found, I can write the complex number z as:z = a + bi = 1 + (-40/7)i.Which simplifies to:z = 1 - (40/7)i.Let me double-check my work to make sure I didn't make any mistakes. Starting from the substitution:3z - 4(bar{z}) = -1 - 40i,z = 1 - (40/7)i,(bar{z}) = 1 + (40/7)i.Plugging back into the equation:3*(1 - (40/7)i) - 4*(1 + (40/7)i).Calculate each term:3*1 = 3,3*(-40/7)i = -120/7 i.So, the first part is 3 - (120/7)i.Now, the second term:-4*1 = -4,-4*(40/7)i = -160/7 i.Wait, hold on, that's not right. Because it's -4 times (1 + (40/7)i), so:-4*1 = -4,-4*(40/7)i = -160/7 i.So, the second part is -4 - (160/7)i.Now, combine the two parts:First part: 3 - (120/7)i,Second part: -4 - (160/7)i.Adding them together:3 - 4 = -1,-120/7 i - 160/7 i = (-120 - 160)/7 i = (-280)/7 i = -40i.So, combining, we get -1 - 40i, which matches the right side of the original equation. That means my solution is correct.Therefore, the complex number z that satisfies the equation is 1 - (40/7)i.

💡First, I need to determine the total distance the train travels while crossing the bridge. This includes both the length of the train and the length of the bridge.Next, I'll calculate the speed of the train in meters per second by dividing the total distance by the time taken to cross the bridge.Finally, I'll convert the speed from meters per second to kilometers per hour using the appropriate conversion factor.

💡Okay, so I have this problem where I need to find the range of x for which the three sides of a triangle are 3, 1+2x, and 8. The options are given as A: 2 < x < 5, B: 3 < x < 8, C: 4 < x < 7, and D: 5 < x < 9. I need to figure out which one is correct.First, I remember that for any triangle, the sum of any two sides must be greater than the third side. This is called the triangle inequality theorem. So, I need to apply this theorem to the given sides.Let me list the sides again: 3, 1+2x, and 8. I need to make sure that the sum of any two sides is greater than the third side. That means I have to set up three inequalities.1. The first inequality would be 3 + (1 + 2x) > 8. Let me simplify that: 3 + 1 + 2x > 8 4 + 2x > 8 Subtract 4 from both sides: 2x > 4 Divide both sides by 2: x > 22. The second inequality is 3 + 8 > 1 + 2x. Let's work that out: 11 > 1 + 2x Subtract 1 from both sides: 10 > 2x Divide both sides by 2: 5 > x or x < 53. The third inequality is (1 + 2x) + 8 > 3. Let's see: 1 + 2x + 8 > 3 9 + 2x > 3 Subtract 9 from both sides: 2x > -6 Divide both sides by 2: x > -3Wait, so from the first inequality, I get x > 2, from the second, x < 5, and from the third, x > -3. So, combining these, the most restrictive conditions are x > 2 and x < 5. The third inequality, x > -3, is less restrictive because x has to be greater than 2 anyway.So, putting it all together, x must be greater than 2 and less than 5. That means the range of x is 2 < x < 5, which corresponds to option A.But just to make sure I didn't make a mistake, let me double-check each inequality.First inequality: 3 + (1 + 2x) > 8Simplify: 4 + 2x > 82x > 4x > 2That seems correct.Second inequality: 3 + 8 > 1 + 2xSimplify: 11 > 1 + 2x10 > 2x5 > xYes, that's right.Third inequality: (1 + 2x) + 8 > 3Simplify: 9 + 2x > 32x > -6x > -3That's also correct, but since x has to be greater than 2, this doesn't affect the range.So, combining the first two inequalities, x must be between 2 and 5. Therefore, the correct answer is A: 2 < x < 5.

💡I need to calculate the value of the expression ((12! / 3!) times (5^3 - 3 times 7^2)).First, I'll compute the factorial part, (12! / 3!). Since (12! = 12 times 11 times 10 times 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1) and (3! = 3 times 2 times 1), dividing these simplifies to (12 times 11 times 10 times 9 times 8 times 7 times 6 times 5 times 4).Next, I'll evaluate the exponential part, (5^3 - 3 times 7^2). Calculating each term:- (5^3 = 125)- (7^2 = 49)- (3 times 49 = 147)Subtracting these gives (125 - 147 = -22).Now, I'll multiply the two results together: ((12 times 11 times 10 times 9 times 8 times 7 times 6 times 5 times 4) times (-22)). Simplifying by canceling out common factors, I find that (11) and (-22) can be reduced, leading to (-2).Finally, multiplying all the remaining numbers step by step:- (12 times 10 = 120)- (120 times 9 = 1080)- (1080 times 8 = 8640)- (8640 times 7 = 60480)- (60480 times 6 = 362880)- (362880 times 5 = 1814400)- (1814400 times 4 = 7257600)- (7257600 times (-2) = -14515200)So, the final result is (-14,515,200).

💡Alright, let me try to figure this problem out. It's about parametric equations and polar coordinates, which I remember are ways to describe curves and lines in the coordinate system. I need to find the general equation of a curve and the rectangular equation of a line, and then find the distance between two points. Okay, let's take it step by step.First, part (1) asks for the general equation of curve C and the rectangular coordinate equation of line l. Starting with curve C, the parametric equations are given as:x = -1 + 2 cos αy = 2 + 2 sin αHmm, these look familiar. They seem like the parametric equations of a circle. I remember that the general parametric equations for a circle are x = h + r cos θ and y = k + r sin θ, where (h, k) is the center and r is the radius. So in this case, comparing, the center of the circle should be at (-1, 2) and the radius is 2, since the coefficients of cos α and sin α are both 2.So, to find the general equation, I can use the standard form of a circle: (x - h)^2 + (y - k)^2 = r^2. Plugging in the values, it should be (x + 1)^2 + (y - 2)^2 = 4. Let me double-check that. If I expand this, I get x^2 + 2x + 1 + y^2 - 4y + 4 = 4, which simplifies to x^2 + y^2 + 2x - 4y + 1 = 0. That seems right.Now, for the polar coordinate equation of line l: ρ cos θ - 2ρ sin θ + 4 = 0. I need to convert this into rectangular coordinates. I remember that in polar coordinates, ρ cos θ is x and ρ sin θ is y. So substituting these in, the equation becomes x - 2y + 4 = 0. That looks straightforward. So the rectangular equation is x - 2y + 4 = 0.Okay, part (1) seems done. Now, part (2) is a bit more involved. Given point P(-4, 0), line l intersects curve C at points A and B. The midpoint of AB is Q, and I need to find the distance |PQ|.First, I need to find the points of intersection between line l and curve C. So, I have the equation of the line x - 2y + 4 = 0 and the equation of the circle (x + 1)^2 + (y - 2)^2 = 4.I can solve these two equations simultaneously. Let me express x from the line equation: x = 2y - 4. Then substitute this into the circle equation:(2y - 4 + 1)^2 + (y - 2)^2 = 4Simplify inside the first bracket: 2y - 3So, (2y - 3)^2 + (y - 2)^2 = 4Expanding both squares:(4y^2 - 12y + 9) + (y^2 - 4y + 4) = 4Combine like terms:4y^2 + y^2 = 5y^2-12y - 4y = -16y9 + 4 = 13So, 5y^2 - 16y + 13 = 4Subtract 4 from both sides: 5y^2 - 16y + 9 = 0Now, solve this quadratic equation for y. Using the quadratic formula: y = [16 ± sqrt(256 - 180)] / 10Calculate discriminant: 256 - 180 = 76So, y = [16 ± sqrt(76)] / 10Simplify sqrt(76): sqrt(4*19) = 2 sqrt(19)Thus, y = [16 ± 2 sqrt(19)] / 10 = [8 ± sqrt(19)] / 5So, the y-coordinates of A and B are (8 + sqrt(19))/5 and (8 - sqrt(19))/5.Now, find the corresponding x-coordinates using x = 2y - 4.For y = (8 + sqrt(19))/5:x = 2*(8 + sqrt(19))/5 - 4 = (16 + 2 sqrt(19))/5 - 20/5 = (-4 + 2 sqrt(19))/5Similarly, for y = (8 - sqrt(19))/5:x = 2*(8 - sqrt(19))/5 - 4 = (16 - 2 sqrt(19))/5 - 20/5 = (-4 - 2 sqrt(19))/5So, points A and B are:A: [(-4 + 2 sqrt(19))/5, (8 + sqrt(19))/5]B: [(-4 - 2 sqrt(19))/5, (8 - sqrt(19))/5]Now, find the midpoint Q of AB. The midpoint formula is average of x-coordinates and average of y-coordinates.x-coordinate of Q:[ (-4 + 2 sqrt(19))/5 + (-4 - 2 sqrt(19))/5 ] / 2 = [ (-8)/5 ] / 2 = (-8)/10 = -4/5y-coordinate of Q:[ (8 + sqrt(19))/5 + (8 - sqrt(19))/5 ] / 2 = [16/5] / 2 = 8/5So, Q is at (-4/5, 8/5).Now, point P is at (-4, 0). I need to find the distance between P(-4, 0) and Q(-4/5, 8/5).Using the distance formula: |PQ| = sqrt[(x2 - x1)^2 + (y2 - y1)^2]Compute x2 - x1: (-4/5) - (-4) = (-4/5) + 4 = (-4/5 + 20/5) = 16/5Compute y2 - y1: 8/5 - 0 = 8/5So, |PQ| = sqrt[(16/5)^2 + (8/5)^2] = sqrt[(256/25) + (64/25)] = sqrt[320/25] = sqrt[12.8] = sqrt[64*5/25] = (8 sqrt(5))/5Wait, let me check the calculations again.Wait, 16/5 squared is 256/25, and 8/5 squared is 64/25. Adding them gives 320/25, which simplifies to 64*5/25, which is 64/5. So sqrt(64/5) is (8)/sqrt(5), which rationalized is (8 sqrt(5))/5. Yeah, that's correct.So, |PQ| is 8 sqrt(5)/5.I think that's the answer. Let me just recap to make sure I didn't make any mistakes.1. Converted parametric equations to circle equation: correct.2. Converted polar equation to rectangular: correct.3. Solved for intersection points: substituted x = 2y -4 into the circle equation, expanded, solved quadratic: seems correct.4. Found midpoint Q: averaged x and y coordinates: correct.5. Calculated distance PQ: used distance formula, simplified correctly.Looks good to me. So, the value of |PQ| is 8 sqrt(5)/5.**Final Answer**The value of (|PQ|) is boxed{dfrac{8sqrt{5}}{5}}.