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MathWe have the equation 2x5=9 Now if 2x  5 > 0 => x > 5/2 Also (2x  5) = 9 => 2x = 14 => x = 7 As 7 > 5/2 , this is valid. If 2x  5 < 0 => x < 5/2 Also , ( 5  2x ) =...

MathWe have to solve for x given that x=(x+6)^1/2 x = (x+6)^1/2 square both the sides => x^2 = x + 6 => x^2  x  6 = 0 => x^2  3x + 2x  6 =0 => x (x  3) + 2(x  3) = 0 => (x + 2)(x ...

MathThe power of M with respect to circle C is: p(M) = d^2  r^2 d  is the distance from M to the center of the circle C. We'll calculate d^2 = 8^2 + 3^2 d^2 = 64 + 9 d^2 = 73 p(M) = 73  25 p(M) = 48...

MathLet the equation of the circle be (x  h)^2 + (y  k)^2 = r^2 Now the center lies on the line 3x  y  2 = 0 => 3h  k  2 = 0 Also, as the circle passes through the points (3,1) and (1,3) (3 ...

MathWe have to find the definite integral of y = 4x^7 + 5x^1 + 6sinx if x=1 to x=2. Int [ y] = Int [4x^7 + 5x^1 + 6 sin x] => Int [4x^7] + Int [5x^1] + Int [6 sin x] => (4/8)x^8 + 5 ln x  6...

MathWe have to find the integral of f(x)=7x^1+6/(1+x^2)+7x^2 Now Int [ f(x)] = Int [ 7x^1+6/(1+x^2)+7x^2] = Int [7x^1] + Int [6/(1+x^2)] + Int [7x^2] = 7 ln x + 6 arc tan x + (7/3)x^3 + C Therefore...

MathWe'll use the integral test to determine the convergence of the given series. We know that if: Int f(x)dx, x = 1 to x = infinite, converges, then the series Sum1/(n^2+17) converges. We'll determine...

MathWe have to find given that 9^3a = 57. Now 9^3a = 57 take the logarithm to the base 3 on both the sides => log(3) [9^3a] = log (3) 57 => 3a* log (3) 9 = log (3) [ 19*3] => 3a * log (3) 3^2...

MathWe have to solve for x given that 7^(3x5) = 1/49. Now 7^(3x5) = 1/49 => 7^(3x5) = 1/ 7^2 => 7^(3x5) = 7^2 Now the base 7 is the same for terms on both the sides, we can equate the...

MathWe are given that f(x) = 13x11 and g(x)=14x^2+13x+11. Now, we have to find g(f(6)) f(6) = 13x11 = 13*6  11 = 89 g(f(6)) = g( 89) = 14*(89)^2+13*(89)+11 = 110894  1157 + 11 = 109748...

MathWe have to solve u + (9) >= 10 u – 9 >= 10 add 9 to both the sides u – 9 + 9 >= 10 + 9 => u > = 1 Therefore u is greater than or equal to 1.

MathLet’s take a quadratic equation ax^2 + bx + c = 0 The roots of the equation are given by x1 = [b + sqrt (b^2 + 4ac)]/ 2a and x2 = [b  sqrt (b^2 + 4ac)]/ 2a Now if all the terms of the equation...

MathA quadratic equation ax^2 + bx + c = 0 has roots that differ by the number 3. Let the roots be r and r + 3 We now have (x – r) (x – r – 3) = 0 => x^2 – rx – x(r + 3) + r(r + 3) = 0...

MathWe have to find the derivative of the function f(x) = [(2x 1)/ (x +7)] ^3. Start by using the Power rule. This gives f’(x) = 3*[(2x 1)/ (x +7)] ^2 * d/dx [(2x 1)/ (x +7)]. Now use the quotient...

MathWe have the equations of three lines: x + 6y = 7, 8x + y = 1 and 4x + 5y = 0. Now, we need to find their points of intersection. x + 6y = 7 and 8x + y = 1 x + 6y = 7 => x = 7 – 6y...

MathWhen we multiply a 2X3 matrix with a 3X2 matrix we get a 2X2 matrix. Now the term T11 of the resulting matrix is equal to the sum of the products of each term in the first row of first matrix with...

MathThe sum of two numbers is given as 38 and the sum of their squares is 730. Let the numbers be denoted by A and B. So we have A + B = 38 and A^2 + B^2 = 730. A + B = 38 => A = 38 – B Substitute...

MathThe instantaneous rate of change of y = x^2 is dy/dx = 2x and y = x^3 separately is dy/dx = 3x^2. For the first function y = x^2 dy/dx at x= 1, is 2, at x=2 is dy/dx = 2x =2*2 = 4, and at x=...

MathGiven the functions: f(x) = 4x + 3 g(x) = x^2 + 6x  8 We need to find (fog)(x). ==> fog(x) = f(g(x)) = f ( x^2 + 6x  8) Now we will substitute between the brackets with x into f(x). ==>...

MathGiven the system: 5x + 3y = 18 ...........(1) 2y  2x = 4 We will divide by 2: ==> y  x = 2 .............(2) Now we will use the substitution method to solve for x and y. First, we will...

MathGiven the equation x^2 + 3x 10 = 0 Let us assume that f(x) is a quadratic function such that: f(x) = x^2 + 3x  10 We know that : a =1 b= 3 c = 10 Then, we will use the formula to find the...

MathGiven the angle a and angle b. Angle a is 35 degrees more than 3 times b. Then we will write: a = 3b + 35 .............(1) Also, given that angle a and angle b are supplementary. We know that two...

MathOnce you have drawn the graph, look for the points where the line meets the xaxis and the yaxis. The xcoordinate of the point where the line meets the xaxis is the x intercept, and the...

MathGiven ab+7b  3a  21 We need to factor the equation. ==> We will write: ==> ( ab + 7b ) + ( 3a  21) Now we will factor b from the first two terms. ==> b*(a + 7) + ( 3a21) Now we will...

MathWe can also use ac method here. a = 2, b = 9 and c = 5. So, a*c = 2*5 = 10. We need to find two numbers which has a product of 10, and has a sum of 9. Factors of 10 and their sums are the...

MathIf d is deaths and c is the cholesterol level, then here's how to do this: d + 525000 = 5000c d + 525000 = 5000(210) d + 525000 = 1050000 d = 525000 (because you subtract 525000 from 1050000 and...

MathOne way to do this is to find a way to get these fractions to have the same denominator. In this case, that you could be 15. So 1/3 (for rent) is the same as 5/15. 1/5 (electricity) is the same...

MathThe car depreciates by 30% every year. So if its initial value is V, the next year it will be V*(1  30%) = V*(1  0.3) = V*0.7. The year after that it will be V*(0.7)^2 and the value similarly...

MathThe rate of growth of Alberta's population is 3.1% per year. Therefore if the population in the base year is P, after n number of years the population will increase at a compounded rate and will be...

MathWe know that if: Int f(x)dx, x = 2 to x = infinite, converges, then the series Sum 1/n*ln n converges. We'll determine the integral: Int f(x)dx = Int dx/x*ln x, x = 2 > x = infinite lim Int...

MathWe'll factorize by 2 the expression under the square root: 2x^2+4x+6 = 2(x^2  2x  3) We'll rewrite x^2  2x  3 = x^2  2x + 1  4 x^2  2x + 1  4 = (x  1)^2  4 Int f(x)dx = sqrt2 Int...

MathWe have to solve for x and y given that: y=6/x and 2^(x+y)=32. Now 2^(x+y)=32 => 2^(x+ y) = 2^5 As the base is the same we can equate the exponents => x+ y = 5 => x = 5  y Substitute this...

MathWe have to solve for x given that 5*6^x  2*3^2x  3*2^2x = 0 Now 5*6^x  2*3^2x  3*2^2x = 0 => 5*2^x*3^x  2*3^2x  3*2^2x = 0 Factor out 2^2x => 5*3^x/2^x  2*3^2x/2^2x  3 = 0 =>...

MathGiven the radius r= 8/9. We need to find the area of the circle. We know that the formula of the area of a circle is given by: A = r^2 * pi where r is the radius. Then, we will substitute with r=...

MathIf a and b are the legs and c is the hypotenuse, we can find this using the Pythagorean Theorem. This theorem says that a^2 +b^2 = c^2 So then we have 5^2 + 17^2 = c^2 25 + 289 = c^2 c^2 = 314 c =...

MathWe'll determine the general term bn, and then, we'll utter any other term of the progression. From enunciation: Sn=b1+b2+b3+...+bn (4^n)1=b1+b2+b3+...+bn bn=(4^n)1[b1+b2+b3+...+b(n1)] But...

MathLet us assume that ABC is a right angle triangle where AC is the hypotenuse, BC is the base, and AB is the height. Given the height of the triangle is 7 m and the area is 15 3/4. ==> AB = 7 We...

MathWe'll write the vector b as: v = x*i + y*j We'll apply the definition of dot product. u*v = (5i+2j)(xi + yj) u*v = 5xi^2 + 5yij + 2xij + 2yj^2 since the product of vectors ij = 0 and i^2 = j^2 = 1...

MathLet us assume that ABC is a right angle triangle such that the base is AB = 2/3 m and the height is BC = 3 1/3. We need to find the area of the triangle. We know that the area of the triangle is...

MathGiven the radius of a circle is r= 65.72 ft. We need to find the circumference (C). We know that the formula of the circumference of a circle is given by: C = 2*r*pi (where C is the circumference...

MathWe'll write the fraction 1/(3n+1)(3n+4) as an algebraic sum of simple ratios: 1/(3n+1)(3n+4) = (1/3)[1/(3n+1)  1/(3n + 4)] We'll substitute n by 1 and we'll get: s1 = 1/(3+1)  1/(3 + 4) =...

MathThe triangle is made up of the line ax + by + 4 = 0, the x – axis and the yaxis. Now the x and y axes are perpendicular. The equation of the third line ax + by + 4 = 0 can be written as ax/4 +...

MathWe are given that 12 times the sum of the digits of a three digit number is equal to the number. Let the number be abc where the digits of the number are a, b and c. Now (a + b + c)*12 = 100*a +...

MathWe need the equation of a line parallel to the x axis and passing through (3, 6). Now a line parallel to the xaxis is one that has the same value of y for all values of x. As the line required...

MathThe line x = 2 is a line that has a value of x equal to 2 for all values of y. This is a vertical line. The line perpendicular to x = 2, would have to be a horizontal line. We also know that the...

Mathfind the equation of a line prependicular to x= 2, passing through the point(1, 3)equation of a...Given the line x= 2 , Then we conclude that the line is parallel to the yaxis. Then the line will be perpendicular to the yaxis. Then the line is parallel to the xaxis. Then the slope is m= 0....

Mathfind the equation of a line prependicular to y=2 passing through (3, 1)equation of a line given...We will write the equation of the line into the standard form. ==> yy1 = m (xx1) where (x1,y1) is any point passes through the line and m is the slope. ==> Given the points ( 3,1) passes...

MathAt least 2 clocks chiming at true time implies one of the following exclusive 4 possibilities: (i) Only particular 2 clocks are chiming at the true time or (ii) only particular 3 clocks are...

MathWe'll use logarithmic differentiation: y = (2x)^sqrt x We'll take logarithms both sides: ln y = ln [(2x)^sqrt x] ln y = sqrt x * ln (2x) We'll differentiate both sides, using product rule to the...

MathFind the equation of the tangent line to the inverse derivative of f(x)=x^3+7x+2 in the point (10;1)To determine the inverse of the derivative, we'll have to determine the derivative. f'(x) = (x^3+7x+2)' f'(x) = 3x^2 + 7 We know that: f'(x)*[f^1(x)]' = 1 We'll divide by f'(x): [f^1(x)]' =...