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MathWe need to find the angle that satisfies: (tan x)^2 = 6*sec x  10 (tan x)^2 = 6*sec x  10 =>(sin x)^2 / (cos x)^2 = 6 / cos x  10 => [(1  (cos x)^2)/(cos x)^2] = 6 / cos x  10 let y =...

MathWe have to determine the real roots of x^2  9 = 6/(x^24) x^2  9 = 6/(x^24) => (x^2  9)(x^2  4) = 6 => x^4  9x^2  4x^2 + 36 = 6 => x^4  9x^2  4x^2 + 42 = 0 => x^4  13x^3 +...

MathThe equation to be solved is : 7^5x7^x*7^36=0 7^5x  7^x*7^36 = 0 => 7^5x  7^(x + 36) = 0 => 7^5x = 7^(x + 36) As the base is the same we equate the exponent 5x = x + 36 => 4x = 36 =>...

MathWe have y = 54 + 11x^3  11x^5 y' = 33*x^2  55*x^4 y' = 0 => 33*x^2  55*x^4 = 0 => 11x^2( 3  5x^2) = 0 => x = 0 and 3  5x^2 = 0 => 5x^2 = 3 => x^2 = 3/5 => x = sqrt (3/5) and...

MathWe have to calculate the value of lim x>1 [(x^2+12x13)/(x1)]. We cannot substitute x = 1 directly as that yields an indeterminate form. Instead we do the following. lim x>1...

MathYou cannot ask more than one question at a time. So I am only providing one basic method. We need to find the derivative of the function y=2x^210x+13 +2(x^25x+6). y= 2x^210x+13 +2(x^25x+6)...

MathWe have to solve : 5^x^2 = 15625 / 5^x 5^x^2 = 15625 / 5^x => 5^x^2 = 5^6 / 5^x => 5^x^2 = 5^(6  x) As the base 5 is the same we can equate the exponent => x^2 = 6  x => x^2 + x  6 =...

MathA Cartesian plane can be divided into 4 parts known as quadrants based on the value of the x and y coordinates. Starting from the top right quadrant which is the first quadrant in a clockwise...

MathWe have to find the value of lim n>+inf. [ a^(1/n)] We know that as n tends to infinity 1/n tends to 0. lim n>+inf. [ a^(1/n)] => lim (1/n)>0 [ a^(1/n)] Any number raised to the...

MathWe have to verify that (1+ (tan x)^2) *(cos x)^2 = 1 Starting with the left hand side (1+ (tan x)^2) *(cos x)^2 tan x = sin x / cos x => (1 + (sin x)^2/(cos x)^2) * (cos x)^2 => [(cos x)^2 +...

MathWe are given that m+4n=5 and 2m+4n=6. We need to find the value of 3m + 4n We solve the given equations for m and n m + 4n = 5 ...(1) 2m + 4n = 6 ...(2) (2)  (1) => m = 1 substitute in (1) 1 +...

MathFirst, we'll substitute x by the value of accumulation point. lim ln(x2)/(x3) = ln(32)/(33) = ln1/0 = 0/0 We've get an indetermination, so, we'll apply L'Hospital rule. We'll differentiate both...

MathThe linear function through the given points is a straight line. The equation of the line passing through (x1, y1) and (x2, y2) is: (y  y1)/(x  x1) = (y2  y1)/(x2  x1) Here we have the points...

MathFirst, we'll substitute x by 5 and we'll verify if it is an indetermination: lim (x^26x+5)/(x^225) = (5^26*5+5)/(5^225) = (3030)/(2525) = 0/0 Since we've get an indetermination, that means...

MathWe have to find the first derivative of y=(4x^2+2x)/x^2. Here it is not necessary to use the quotient rule. y=(4x^2+2x)/x^2 => y = 4x^2 / x^2 + 2x / x^2 => y = 4 + 2/x => y = 4 + 2*x^1 y'...

MathWe have F(x) = direct integral from 1 to x^2 of ((t^3)+(t^(1/2))) dt and we have to find F'(x). F(x) = Int[ (t^3 + t^0.5)dt], t = 1 to x^2 => F(x) = [t^4/4 + t^1.5 / 1.5 + C], t = 1 to x^2 =>...

MathWe have to solve (x+18)(x14)(x+5) > 0. Now (x+18)(x14)(x+5) > 0, if either none of the factors are negative or two of the factors are negative. None of the factors are negative (x+18) >...

MathWe have f(x) = 4x^2 + 4 and g(x) = 3x  2. (fg)(x) = f(x) * g(x) => (4x^2 + 4)(3x  2) open the brackets and multiply => 4x^2*3x + 4*3x  2*4x^2  8 => 12x^3 + 12x  8x^2  8 => 12x^3 ...

MathWe have to simplify (sec x  cosec x) / (tan x  cot x) further. (sec x  cosec x) / (tan x  cot x) use sec x = 1/ cos x, cosec x = 1/ sin x , tan x = sin x / cos x and cot x = cos x / sin x =>...

MathWe have to solve cos 2x  2*(sin x)^2 + 2 = 0 cos 2x  2*(sin x)^2 + 2 = 0 Use the identity cos 2x = 1 2^(sin x)^2 => 1 2*(sin x)^2  2^(sin x)^2 +2 = 0 => 4*(sin x)^2 = 3 => (sin x)^2...

MathWe have to determine if the equation 5/(x+5)  5=0 has a unique root. 5/(x+5)  5=0 => 5/(x+5) = 5 => 5 = 5x + 25 => 5x = 20 => x = 20/5 => x = 4 Actually we don't need to solve...

MathAt the point of intersection of the lines the x and y coordinates are the same We have y = x +14 and y = 4x  11. Equating the ycoordinates ,we get x + 14 = 4x  11 => 5x = 25 => x = 5 y...

MathWe have to prove the identity 3+(4+x)^3=x^3+12x^2+115 Starting with the left hand side 3+(4+x)^3 => 3 + 4^3 + x^3 + 3*16*x + 3*4*x^2 => 3 + 64 + x^3 + 48x + 12x^2 => x^3 + 12x^2 + 48x +...

MathThe equation of a line in the form x/a + y/b = 1 gives the x and y intercepts as a and b The line we have is y = 12x + 8 y = 12x + 8 => y  12x = 8 divide all the terms by 8 => 12x/8 + y/8 =...

MathWe have to factor x^18  y^18. x^18  y^18 => (x^9)^2  (y^9)^2 => (x^9  y^9)(x^9 + y^9) => (x^3^3  y^3^3)(x^3^3 + y^3^3) => (x^3  y^3)(x^6 + y^6 + x^3*y^3)(x^3 + y^3)(x^6 + y^6 ...

MathWe have z  2z' = 2  4i As z = x + i*y and z' = x  i*y z  2z' = 2  4i => x + i*y  2*(x  i*y) = 2  4*i => x + i*y  2*x + 2*i*y = 2  4*i =>  x +3*i*y = 2  4*i equate the real and...

MathAn example of two imaginary numbers which when multiplied give a real number is of the form a+ bi and a  bi. They are called complex conjugates. When two complex conjugates are multiplied, the...

MathIf the derivative of a function is 0, it indicates an extreme point. y = x^2  6x + 3 y' = 2x  6 If y' = 0 => 2x  6 = 0 => x = 3 f(3) = 3^2  6*3 +3 => 9  18 + 3 =>  6 At x = 3, we...

MathThe equation will become: 3sin x/cos x + cos x/cos x = 4sin x/cos x We'll substitute the fraction sin x/cos x = tan x 3*tan x + 1 = 4*tan x We'll subtract 4*tan x both sides: 3*tan x + 1  4*tan x...

MathGiven the equation of the circle: x^2  6x + y^2  2y = 14 We need to find the radius and the center of the circle. Then, we need to rewrite into the standard form as follows: (xa)^2 + (yb)^2 =...

MathTo prove that the polynomial is divisible by (x1)^2, that means that x = 1 is a root of polynomial and it's first derivative. For this reason, we'll substitute x by 1 in the expresison of...

MathLet the number we need to find be N. As half of the number added to its third part is eight less than the number, we get N/2 + N/3 = N  8 => (3N + 2N)/6 = (N  8) => 5N = 6N  48 => N =...

MathWe have (x3)/(x1)=(x+1)/(x+2), and we have to solve for x. (x3)/(x1)=(x+1)/(x+2) => (x  3)(x + 2) = (x + 1)(x  1) => x^2  x  6 = x^2  1 => x = 5 => x = 5 The required value...

MathWe have to prove that (4  i*sqrt 6)/(2  i*sqrt 6)=(sqrt 3 + 2i*sqrt 2)/(sqrt 3 + i*sqrt 2) The left hand side: (4  i*sqrt 6)/(2  i*sqrt 6) multiply the numerator and denominator by (2 + i*sqrt...

MathThe problem provides the equations that relates the binomial coefficients of the `(r1),r ` and `(r+1)` terms, such that: `(C_n^r)/(C_n^(r+1)) = 1/3 => (C_n^(r+1)) = 3(C_n^r)` Using the...

MathWe have to multiply (x*y^2*z^3v^4) and (x*y^2*z^3+v^4) We can see that the terms are of the form a  b and a + b with a=x*y^2*z^3 and b = v^4 We use the relation (a  b)(a + b) = a^2  b^2...

MathThe 4 basic arithmetical operations are: addition, subtraction, multiplication and division. If we'll solve any linear equation, we'll fall into one from the 4 forms. For instance: 1) If x + a = b,...

MathWe are given that the company makes profits of 50,000 in the first year and they are predicted to increase every year in such a manner that a geometric series is formed. As the common ratio is r,...

MathWe have to find the extremes of f(x) = ln(2x^220x+53). At the extremes the value of the first derivative is 0. f(x) = ln(2x^220x+53) f'(x) = 4x  20 / (2x^2  20x + 53) 4x  20 / (2x^2  20x +...

MathWe'll multiply the 1st equation by 1+x both sides: y*(1+x) = 5x We'll remove the brackets: y + x*y = 5x We'll move all terms to one side: y+x + x*y  5 = 0 (3) We'll factorize th second equation...

MathGiven the equation: 3x^2 + 5x 12 = 0 We need to solve for x. We will use the formula to find the roots of the quadratic equations. We know that: x = [ b + sqrt(b^24ac)/ 2a ==> x1= ( 5 +...

MathLet the dimensions of the rectangle be L and W. We know that the area is A = L*W ==> L*W = 84 ............(1) Also, we know that the perimeter is given by P = 2L + 2W ==> 2L + 2W = 38 ==>...

MathGiven that g(x)= (2x^2+3)*ln x We need to find the derivative g'(x). We will use the product rule to determine g'(x). We know that: if f(x) = u*v ==> f'(x) = u'v + uv' We will assume that g(x) =...

MathGiven that: f(x) = (2x5) / (3x+2) We need to find f'(x). We will use the qoutient rule to determine the derivative. ==> We know that : if f(x) = u/v , then f'(x) = ( u'v  uv')/v^2 Then we will...

MathGiven the curve: f(x) = 4x^2  4x + 5 We need to find the minimum value of the curve f(x). First we know that the coefficient of x^2 is positive. Then, f(x) has a minimum value. Now we will...

MathGiven the point (2,3) passes through the line. We know that the equation of the line is given by : yy1 = m(xx1) where m is the slope and (x1,y1) is any point on the line. We will substitute with...

MathGiven the line segment AB such that B(3,3). Also, given that M(2,4) is the point of AB. We need find the coordinates of the point A. We will use the midpoint formula to determine A. We know...

MathWe are given the points A and B and we need to find the distance between them. We will use the formula of the distance between two points. ==> D = sqrt(x1x2)^2 + (y1y)^2 Now we will subsitute...

MathGiven that 2z +3i = 3z +2 We need to find the absolute value of z. First we need to rewrite the number into the form z= a+ bi. Then the absolute values is lzl = sqrt(a^2+b^2). ==> 2z + 3i =...

MathWe have the inequality : 2 l 2x4 l  5 =< 3 We need to find the values of x that verifies the equation. First we need to isolate the absolute values on one side. We will add 5 to both sides....