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MathWe have to prove that: tan [i*log (aib/a+ib)] = 2ab/(a^2  b^2) Start with denoting a + ib in the form r*( cos x + i sin x) Equate the real and complex components to get r*cos x = a and r*sin x =...

MathThe lines can be written in the slope intercept form y = mx + c x 3y  2 = 0 => 3y = x  2 => y = x/3  2/3 The slope is 1/3 2x + y  5 = 0 => y = 2x + 5 The slope is 2 As the slope of...

MathFor an arithmetic progression, the nth term is given by: a + (n  1)*d , where a is the first term and d is the common difference. Here the first term is given as 2 and we have the fifth term as...

MathThe electric bill is twice higher than the selling price. This makes the electric bill three times the selling price. If the electric bill is E and the selling price is S, E = 3*S The selling price...

MathThe length of a rectangular field is 12m more than width.The value of its area= 4 times the value...Let the length be L and the width be W Given that the length is 12 more than the width. ==> L = 12+ W.............(1) Also, given that the area is 4 times the perimeter. ==> A = 4*P ==>...

MathWe have to express the given expression 5/(x^25x+6) as partial fractions. 5/(x^25x+6) => 5 / ( x^2  3x  2x + 6) => 5/ ( x(x  3)  2( x  3)) => 5/ (x  2)(x  3) => A / (x  2) +...

MathGiven that : ln a = 2 ln b = 3 We need to find the values of ln a^2/b^3 We will use the logarithm properties to solve. We know that ln a/b = ln a  ln b Then we will simplify: ln a^2/b^3 = ln a^2 ...

MathWe have the sides of the triangle given as 3, 5 and 6. We find the area using Heron's formula as sqrt [ s*(sa)(sb)(sc)] s = (3 + 5 +6)/2 = 7 sqrt [ s*(sa)(sb)(sc)] => sqtr [...

MathGiven that g(x) = (x2)/(x^24) We need to find the integral of g(x). First we will simplify. g(x) = (x2)/(x2)(x+2) = 1/(x+2) ==> g(x) = (x+2)^1 Now we will integrate. ==> Intg g(x) dx =...

Math2x+ 2y = 5...........(1) x  3y = 2 ..........(2) We will use the substitution method to solve. From (2) we will re write x= 3y2 Now we will substitute into (1). ==> 2x +2y = 5 ==> 2(3y2)...

MathWe have 2z  3 + i = (3i)/(2+4i). We have to find z and z. 2z  3 + i = (3i)/(2+4i) => 2z  3 + i = (3i)(2  4i)/(2+4i)(2  4i) => 2z  3 + i = (3i)(2  4i)/(4+16) => 2z  3 + i =...

Mathf'(x) = x^2 5x 3 We need to find f(x). We know that f(x) = integral of f'(x). ==> f(x) = Int ( x^2 5x 3) dx = Int (x^2) dx  Int (5x) dx  Int 3 dx = x^3/3 ...

MathWe have the line y= 3x+2 and the curve g(x) = 2x^2 3x2 We need to find the points of intersection between g(x) and the line y. Then, we know that the points of intersection must verify the...

MathGiven the equation: x^3+ x^2 + x +1 = 0 We need to find x values that satisfies the equation. First we will simplify by factoring. We will factor x^2 from the first two terms. ==> x^2 ( x+1) +...

MathGiven the volume of the cylinder is v= 234 The height = 12 We need to find the surface area. First we will need to find the radius. We know that the volume if given by : v= r^2*pi*h where r is the...

MathLet x be the number of people attends. Let the revenue be y. Let the initial revenue when 120 people attend is 30*120 = 3600 y= people attend * price y= (120+x)*(30 1.5*(x/10)] y= (120+x) * ( 30...

MathThe system is consisted of 4 variables and 4 equations as follow: Let the variables be x,y,z, and u ==> 6x +2y z 5u = 25 ==> x +7z +3u = 7 ==> x y 10z +6u = 23 ==> 8y + z 11u = 21

Mathf(x) = 8x +4 Given that f(a) = 1.6 We need to find the values of a. We will substitute with x= a into the function f(x). ==> f(a) = 8a +4 = 1.6 Now we will solve for a. ==> 8a = 1.6 4 =...

MathGiven the formula: V = C(1r)^t v is the values c is the original cost = 40,000 r is the rate = 8% = 0.08 t is the number of years Given the information above, we need to find t such that v=...

MathGiven f(x) = 6x+4 We need to find the values of a such that f(a) = 1.3 We will substitute with x=a into the equation. ==> f(a) = 6a +4 But given that f(a) = 1.3 ==> f(a) = 6a+4 = 1.3 Now...

MathA=P (1+r)^t A = Value of CD P = original amount= 1000 r = the rate = 3% = 0.03 t= time We need to find t such that A is doubled ==> A = 2*1000 = 2000 Now we will substitute. ==> 2000 =...

MathWe have to simplify (e^x  e^x)^7 (e^x  e^x)^7 => (e^x  1/(e^x))^7 => [(e^x * e^x  1)/(e^x)]^7 => (e^x * e^x  1)^7/(e^x)^7 => (e^2x  1)^7 / e^7x We could expand the numerator...

MathFirst, we'll move all the powers of 7 to the right side of the equal sign. 2^x + 2^(x+1) + 2^(x+2) = 7^x + 7^(x+1)+7^(x+2) Now, we'll apply the property of exponentials: a^(b+c) = a^b*a^c According...

MathWe'll expand the binomial and we'll get: (1+x)^6 = a0 + a1*x + a2*x^2 + ... + a6*x^6 a0,a1,a2,...,a6 are the coefficients of polynomial. The sum of even coefficients is: a0 + a2 + a4 + a6 We know...

MathSince a1,a2,...,an are the terms of an A.P., we'll apply the rule of finding each term of the arithmetical sequence: a2 = a1 + d => a2 a1 = d a3 = a1 + 2d => a3  a1 = 2d...

MathWell rewrite 4^(2x+1) = 4^2x*4 and 6^(x+2) = 6^x*6^2 We'll rewrite the entire expression: 3^x*4^2x*4 = 6^x*6^2 We'll write 6^x = 2^x*3^x 3^x*4^2x*4 = 2^x*3^x*36 We'll divide by 4*3^x: 4^2x =...

MathThe formula for the surface area of the cone is the following: SA = area of the base + area of the cone. Area of the base = r^2 * pi Area of the cone = r*s*pi ( r is the radius of the base and s...

MathOnly values of a which are greater than or equal to 0 will make the expression inside the absolute symbol (that is a) negative. Therefore, for the property a = a to be meaningful, a must be...

MathYou have to remember that the number of solutions of an equation is equal to the highest power of the variable. A linear variable has 1 solution; a quadratic equation has 2 solutions, etc. The...

MathWhen the tire rotates 400 r/ m , then the distance the tire travels in one minute = 400 * circumference of the tire for 1 minute ==> D = 400 * C...............(1) Now we will calculate the...

MathYou want the value of x for which sin x  cos x = 0. Why are you taking the cases where cos x = 1, sin x = 0, etc. Instead, solve for x. sin x  cos x = 0 => sin x = cos x => sin x / cos x =...

MathYou have to solve x = sin x If you have a look at the graph of the sine function you see that it starts from the point (0,0). At this point the value of the sin function is zero and the value of x...

MathWe have the functions: f(x)=x, g(x)=6x8 and h(x) = 4x. We have to find u= fo(gof)/h. u = fo(gof)/h => u = fo[(g(f(x))/ h(x)] => u = fo[(6x  8) / (4  x)] => u = (6x  8)/ (4 ...

MathThe equation 2*lg(x1) = lg (ax3) has two equal roots. 2*lg(x1) = lg (ax3) => lg ( x1)^2 = lg (ax  3) taking the antilog of both the sides => (x  1)^2 = ax  3 => x^2  2x + 1 = ax ...

MathThe equation of the line between the points (6, 3) and (3, 4) is y – 3 = (x – 6)[(4 – 3)/(3 – 6)] => y – 3 = (x – 6)*( 1/3) The slope of the line is (1/3). The equation of the line...

MathLaw 1: log x + log y = log (x*y) Example: log 2 + log 5 = log (2*5) = log 10 = 1 Law 2: log x  log y = log x/y Example: log 100  log 10 = log 100/10 = log 10 = 1 Law 3: log x^a = a* log x...

MathThe slope of the tangent to a curve f(x) is the value of the first derivative of the curve for the appropriate x. Here we have x^2 + y^2 = 34 Differentiating both the sides 2x + 2y (dy/dx) = 0...

MathWe'll substitute x by 2 and we'll get: lim (lnxln2)/(x2 x) = lim (ln 2  ln 2)/(2  4) lim (lnxln2)/(x2 x) = 0/2 =0 But this is the value of the limit of the function, if and only if the...

MathWe have to prove that (1+2(sin a)(cos a))/(12(sin^2 a)=(cos a+sin a)/(cos asin a) Starting from the left hand side (1+2(sin a)(cos a))/(12(sin a)^2) replace 1 with (sin a)^2 + (cos a)^2 =>...

MathLet's denote the number of students as S. 5/7 of them are boys. If we denote the number of boys as B, B = (5/7)*S The rest of the students are girls. This is equal to (1  5/7)*S = (2/7)*S As the...

Mathln t =  ln 2 We will preview the properties of the logarithm equations. We know that: ln a^b = b*ln a Then we will rewrite the equation.  ln 2 = 1 * ln 2 = ln 2^1 = ln (1/2) ==> ln t = ln...

Mathx^2 + 6x + 6 = 3 + (x+3)^2 To prove, we will start form the left side and prove the right sides. First we will complete the square. ==> (x^2 + 6x +6) We will add and subtract( x's...

MathWe have to prove that 3+1/(x+1) has a unique solution. 3+1/(x+1) => [3(x +1) + 1] / (x+1) => [3x + 3 + 1] / (x +1) => [3x + 4] / (x + 1) We see that the maximum power of x in the function...

MathWe have to solve the inequation x/2 >= 1 + 4/x x/2 >= 1 + 4/x => x >= 2 + 8/x If we assume that x >=0, we can multiply both the sides of the inequation with x without changing the...

MathTo calculate cosine, using adjacent dided by hypotenuse. Tangent is opposite over adjacent. opposite =10 adjacent = 24 So you are missing hypotenuse. In a right triangle, c is the hypotenuse....

MathThe answer to this question is that the Arctic Wolf is 40 kg. and the bobcat is 8 kg. You can set up the equation as 1/5 X + X =48kg. Next, multiply 1/5 X by 5 to make it a whole number, or X....

MathWe have the function f(x) = sqrt x + a. At x= 16, the tangent to to the function is y = mx + 2 The slope of the tangent to the function f(x) = sqrt x + a at x = 16 is m. f'(x) = (1/2)/ sqrt x =>...

MathWe have f(x) = (x + 8)^2. We have to find the equation of the tangent to the graph at the point where x = 2. The slope of the tangent drawn to a graph f(x) at the point (x, f(x)) is the value of...

MathWe'll start using the condition from enunciation that the binomial coefficients of the given expansion are in arithmetical progression. Since it is not indicated what are the terms, we'll suppose...

MathWe have to prove: sin^4(theta)  cos^4(theta) = sin^2(theta)  cos^2(theta) First let's write the terms in a standard form and use x instead of theta. So we have to prove (sin x)^4  (cos x)^4 =...