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MathTwo parallel do not intersect. Now, the lines we have are: y= 3(x2) and 2y = 6x 9 2y = 6x 9 => y = ( 6x  9)/2 => y = 3x  9/2 Now we see that y= 3(x2) => y  3x = 6 and y = 3x  9/2...

MathGiven the quadratic equation: x^2 + 7x + 9 = 0 We need to solve for the roots using the quadratic formula. Then we know that: a =1 b= 7 c = 9 The formula for the roots is: x = (b + sqrt(b^2...

MathGiven the points ( 2,3) and (5,8) . We need to find the equation of the line that passes through the given points. We will write the equation into the standard form. ==> yy1 = m (xx1) where...

MathGiven that sec(x) = 5/3 We need to find tan(x), cos(x), and (sin(x). First we know from trigonometric identities that: sec(x) = 1/cos(x) ==> 5/3 = 1/cosx ==> cosx = 3/5 Now we will calculate...

MathWe need to prove that: (tanx*sinx) / (sec^2 x 1) = cosx We will start from the left side and prove that it equals cosx. First we will rewrite using the trigonometric identities. We know that secx...

MathHere we have to find the integral of y=e^x/(3+e^x) Int [y] = Int [e^x/(3+e^x)] Take 3+e^x as t. = dt/dx = e^x So we have Int [e^x/(3+e^x) dx] => Int [dt/t} => ln t + C substitute t with...

MathWe'll rewrite the first term recalling the quotient property of exponentials: 5^(3x) = 5^3/5^x We'll rewrite the equation: 5^3/5^x + 5^x  20 = 0 We'll multiply by 5^x both sides: 5^3 + 5^2x ...

MathWe'll impose the constraint of existence of logarithms: x>0 lg x > 0 x > 10^0 x > 1 lg lg x > 0 lg x > 1 x > 10 The interval of admissible values of x is (10 ; +infinite)....

MathTo solve for x : log(x+21)^1/2+log(x21)^1/2=2(1+log2 We use the property, log a+log b = log ab. log(x+21)^(1/2) +log(x21)^(1/2) = 2(1+log2. log(x+21)^(1/2)*(x21)^(1/2) = 2+2log2 log...

MathWe consider the inverse function arcsin 3t > pi/6. We'll apply sine functin both sides: sin (arcsin 3t) > sin pi/6 Since the sine function is increasing, we'll keep the direction of the...

Math(1+i)^2008. To simplify, first we will rewrite the exponent. ==> ( 1+ i)^2008 = (1+ i)^(2*1004) Now we know from exponent properties that x^ab= (x^a)^b ==> (1+i)^(2*1004) =[ (1+i)^2]^1004 Now...

MathGiven y= x^3/ (x^4 +10^5 We need to find the integral of y. ==> intg y = intg ( x^3/(x^4 +1)^5 Let us assume that u = x^4 + 1 ==> du = 4x^3 dx Now we will substitute. ==> intg y = intg (...

MathGiven the equations: (x^3 1) and ( x^4 + x^2 +1) We need to find the greatest common factor. First we will factor each equation. x^3 1 = (x1) ( x^2 +x +1) x^4 + x^2 + 1 = ( x^2 +x +1) ( x^2 x...

MathThe limit `lim_(x>0) (1cosx)/x^2` has to be determined. If we substitute x = 0, the result `(1cosx)/x^2` is the indeterminate form `0/0` . In this case it is possible to use l'Hospital's rule...

MathWe'll have to show that for any positive number epsilon, there is N = N(epsilon), so that: an  1/5 < epsilon for any n > N(epsilon) We'll substitute an by it's given expression:...

MathWe'll calculate the definite integral of the function that describes the speed: v(t) = t^2  5t + 6, using the LeibnizNewton formula. Int v(t)dt = Int (t^2  5t + 6)dt Int (t^2  5t + 6)dt = Int...

MathGiven the system: a = 2+ b .............(1) b = 3c .............(2) a+b = c ...............(3) We will use the substitution method to solve. We will substitute (2) into (1). ==> a= 2+ b = 2 +...

MathGiven the line segment AB such that B(2, 12) and the midpoint m(2,5). We need to find the coordinates of the point A. We will use the midpoint formula to find A. We know that: xm = ( xA+xB)/2...

Math4^(x1) = 8*2^3x First we will simplify the equation. We will rewrite the numbers 4 and 8 as exponent of 2. ==> 4 = 2^2 ==> 8 = 2^3 ==> (2^2)^(x1) = 2^3 * 2^3x Now from exponent...

MathGiven the line passes through the point ( 3, 4). Then, we will write the equation of the line: yy1  m (xx1) where (x1,y1) is any point passes through the line and m is the slope. ==> y 4 =...

MathWe know that the volume of the cylinder is given by: V = r^2* pi * h such that: V is the volume, r is the radius, and h is the height of the cylinder. Given the height h= 12 ==> V = r^2 * pi *...

MathGiven the series: b, 3, 4b , 12 . Let r be the common difference. Then we know that: 3 = b*r................(1) 4b = b*r^2.............(2) 12 = b*r^3.................(3) First we will rewrite...

MathGiven the first derivative: f'(x) = (2x+3)/x We need to find f(x). First we will simplify f'(x). ==> f'(x) = 2x/x + 3/x ==> f'(x) = 2 + 3/x Now we know that f(x) = intg f'(x). ==> f(x) =...

MathTo solve the equation: log (2x+2) = 1  log 3x We rewrite equation as below by adding log3x to both sides: log(2x+2)+log3x = 1 log (2x+2)*3x = 1. => (2x+2)*3x = 10^1 , As log a = b => a =...

MathWe'll solve the system using elimination method. We'll multiply the 1st equation by 3: 6x  12y = 36 (3) We'll multiply the 2nd equation by 4: 4x + 12y = 12 (4) We'll add (3) + (4): 6x  12y + 4x +...

MathGiven the equation: 2+ 3 l 3x +1l = 11 First we need to isolate the absolute value. Let us subtract 2 from both sides: ==> 3 l 3x + 1l = 11  2 ==> 3 l 3x + 1l = 9 Now we will divide by 3....

MathWe have to find x if x^(1+log3 x)=9x^2 x^(1+log3 x)=9x^2 Take the log to the base 3 on both the sides. (1 + log3 x) log3 x = log3 (3x^2) => (1 + log3 x)* log3 x = 2 log3 3x => (1 + log3 x)*...

Mathsin3x = 2sin^3 x First we will rewrite: sin3x = sin(2x+x) But we know that : sin(A+B) = sinAcosB + cosAsinB ==> sin(2x+x) = sin2xcosx + sinxcos2x But sin2x = 2sinxcosx ==> sin3x =...

Matha/ax1 + b/bx1 = a+b => ( a ax) / ax + ( bbx)/ bx = (a+ b) We will multiply by abx^2 ==> (a ax)(bbx) = (a+b) abx^2 ==> (ab  2abx + abx^2 = (a+b) abx^2 ==> ab  2abx = (a+b)(...

MathWe have to solve the equation (cos x + cos 3x + cos 5x) / (sin x + sin 3x + sin 5x) = sqrt 3. Now we know that cos x + cos y= 2cos[x+y)/2]*cos [(xy)/2] =>cos x + cos (5x)= 2...

MathThe equation is corrected to d(t) = 16t^2+bt+20, as the velocity is upwards is taken as positive. So when t= 0, d(0) = 20 ft which is platform height. The height to which a person jumps is given...

MathWe have to find x and y given that (x1)/i + (y+1)/2 = (x+2)/3 + (y1)/i (x1)/i + (y+1)/2 = (x+2)/3 + (y1)/i => (x1)i/i + (y+1)i/2 = (x+2)i/3 + (y1)i/i => (x  1) + yi/2 + i/2 = xi/3 +...

MathWe have the line 2y + 4x  8 = 0. Now the equation of a line parallel to it is 2y + 4x +r=0. As the parallel line passes through (2, 1) => 2y + 4x +r=0 => 2*1 + 4*2 + r = 0 => 2 + 8 + r...

MathWe have to find x given 3*4^x + 2*9^x = 5*6^x. 3*4^x + 2*9^x = 5*6^x => 3*2^2x + 2*3^2x = 5*2^x*3^x Divide all the terms by 3^2x => 3*(2/3)^2x + 2 = 5*(2/3)^x => 3*(2/3)^2x  5*(2/3)^x + 2...

MathWe'll impose the constraints of existence of logarithm: 3x + 6 > 0 3x > 6 x > 2 2x + 1 > 0 x > 1/2 2x + 1 different from 1 x different from 0. Now, we'll solve taking...

MathWe have g' = 5x^4+16x^3. To determine g we need the integral of g' Int [ 5x^4+16x^3 ] => Int [ 5x^4 ] + Int [ 16x^3] => 5x^5 / 5 + 16x^4 / 4 => x^5 + 4x^4 + C Therefore g = x^5 + 4x^4 + C

MathThe area of a right triangle with legs x+ 1 and 2x + 4 is (1/2)*(x+1)*(2x +4). So we have the area given by (1/2)[ 2x^2 + 4x + 2x + 4] => (1/2) 2x^2 + 6x + 4 Now it is not possible to find the...

MathIf 2 vectors are perpendicular, then the dot product is zero. u*v = 0 The dot vector is: u*v = u*v*cos (u,v) Since u and v are perpendicular, then cos 90 = 0. u*v = 0 u*v = (3i+5j)(ai6j) We'll...

MathGiven the inequality: 16^(x1) > 2^2x+2 First we will write: 16 = 2^4. ==> 2^4^(x1) > 2^2x+2 We know that x^a^b = x^ab ==> 2^4(x1) > 2^(2x+2) ==> 2^(4x4) > 2^(2x+2) Now...

MathWe have to solve for x using the equation 2/e^x = 5/(1+e^x) 2/e^x = 5/(1+e^x) => 2* ( 1 + e^x) = 5* e^x => 2 + 2*e^x = 5*e^x => 2 = 3*e^x => (2/3) = e^x take the natural logarithm of...

MathGiven the hyperbola : x^2 / 9  y^2 /4 = 1 To find the points of the hyperbola that have x= 4, we will substitute with x= 4 and determine y values. ==> 4^2 / 9  y^2 /4 = 1 ==> 16/9  y^2...

MathGiven the equation: l 5x + 8 l = 17 We have the absolute value of (5x+8). Then, we have two cases: Case(1): (5x + 8) = 17 Now we will subtract 8 from both sides. ==> 5x = 17  8 ==> 5x = 9...

Mathlet f(x) = 3/ (x^1/2) We need to find the integral of f(x). ==> intg f(x) = intg ( 3/(x^1/2) dx We know that if 1/x^a = x^a ==> intg f(x) = intg 3x^1/2 dx = 3intg...

MathWe know that 2x  y  10 = 0 is tangential to x^2+y^24x+2y=0 if there is only one point of contact. Now 2x  y  10 = 0 => y = 2x  10 Substituting this in x^2+y^24x+2y=0, we get x^2+ ( 2x ...

Mathlet f(x) = cosx*e^2x We need to find the integral of f(x). ==> intg f(x) = intg cosx * e^2x dx We will use partial integration to solve. Let us assume that: u= e^2x ==> du = 2e^2x dx dv...

MathLet there be n terms in the set. The average value is 45; therefore the sum of the n terms is 45n. The addition of two terms a and b to the set, increases the total sum of the terms to 45n + a + b...

MathLet the numbers be x and y. The sum of the two numbers is x + y = a. The difference of the two numbers is x – y = b So we have two equations x + y = a … (1) x – y = b … (2) Add the two...

MathA soap bubble is spherical in shape. The volume of a sphere of radius r is V = (4/3)*pi*r^3. We are given dV/ dt as 10. As V = (4/3)*pi*r^3, => dV/dt = (4/3)*pi*(3r^2)*(dr/dt) => dr / dt =...

MathWe can prove this by induction. For n = 1, n (n+1) (2n+1)/6 = 1*2*3/6 = 1. Therefore the relation is true. Now, if we assume 1^2 + 2^2 + 3^2 …n^2 = n (n+1) (2n+1)/6 1^2 + 2^2 + 3^2 …n^2 + (n...

MathFirst we have to find dy/dx for the curve. dy/dx(y^2 – 6x^2 + 4y + 19) = 0 => 2y dy/dx – 12x + 4dy/dx = 0 => y dy/dx – 6x + 2dy/dx = 0 => dy/dx = 6x /(y+2) Now at the point (2, 1)...