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MathWe have the functions f(x) = 2x7 and g(x) = sqrt (2x)  1. We have to find fog(x). fog(x) = f(g(x)) => f( sqrt 2x  1) => 2*(sqrt 2x  1)  7 => 2* sqrt 2x  2  7 => 2* sqrt 2x  9...

MathWe have a number that can be expressed in terms of factors as a*4^3*6^2*13^11. The number given, needs to have 11^2 and 3^3 as factors. Now, in a*4^3*6^2*13^11 we see there is no factor of 11. 6^2...

MathA practical example of an inverse function could be the following. A particle travels at a varying speed which is given by v(t) = 3t + 6, where t is the time travelled by the particle. Now if we...

MathIn the problem, the train is travelling at 140 km/h. It crosses the man which can be taken as a single point in 2 seconds and it requires 8 seconds to cross the platform. The extra 6 seconds are...

MathWe are given the word MATHEMATICS. The 4 letter words that we have to create have to start with M and end with S. Now, for the letters at the 2nd and the 3rd place, we have 2 As, 2 Ts, H, E, M I...

MathWe'll have to determine the intercepting point of the line AB and the line that represents the first bisectrix. For this reason, we'll have to solve the system formed from the equations of the line...

MathFirst, let's write f(n+1) and f(n), substituting x by n+1 and n: f(n+1) = (n+1)^3 + 3(n+1) + 2 f(n+1) = n^3 + 1 + 3n(n+1) + 3(n+1) + 2 f(n+1) = n^3 + 3(n+1)(n+1) + 3 f(n+1) = n^3 + 3(n+1)^2 + 3...

MathThe angle of elevation from a point 50 feet from the base of the tree to the top of the tree is 30 degrees. Now tan 30 = height of tree/ distance of the point from the base => tan 30 = H / 50...

MathWe have to prove that 1  2(sin a)^2 = (cos a)^4  (sin a)^4. We know that (sin a)^2 + (cos a)^2 = 1 (cos a)^4  (sin a)^4 => [(cos a)^2  (sin a)^2][(cos a)^2 + (sin a)^2] => [(cos a)^2...

Mathx + y = pi/4 if and only if tan (x+y) = tan pi/4 = 1 So, we'll have to prove that tan (x+y) = 1. We'll apply the formula of tangent of the sum of 2 angles: tan (x + y) = (tan x + tan y)/(1  tan...

MathThe area of a triangle is given as (1/2)*base*height. Now in the given triangle, the height is twice the base and the area is 576 => (1/2)* base*2*base = 576 => base^2 = 576 => base = 24...

MathA tin can's diameter is 5 inches and it's height is two more than three times the diameter . Find...The volume of a cylinder with a base of area A and height h is A*h. Here we have a can, the diameter of which is 5 inches. The area is equal to pi*(5/2)^2. Its height is 2 more than three times the...

MathBy definition, C(2n3,2) = (2n3)!/2!*(2n32)! C(2n3,2) = (2n3)!/2!*(2n5)! But (2n5)! = (2n  5)!(2n  4)(2n  3) 2! = 1*2 = 2 C(2n3,2) = (2n  5)!(2n  4)(2n  3)/2*(2n  5)! We'll simplify...

MathA square is inscribed in a circle, side of the square is 2*squareroot2. What is the circumference...Given that the side of the square is 2sqrt2. Then, we know that the diagonal pf the square is the diagonal of the circle. Let us calculate. The diagonal = sqrt(side^2 + side^2)...

MathGiven that: f(x) = 1/(x1)(x2) We need to find f'(0) First we need to find the first derivative f'(x). Let f(x) = 1/(x1) *(1/(x2) = u*v such that: u = 1/(x1) ==> u' = 1/(x1)^2 v= 1/(x2)...

MathGiven the polynomial f(x) = x^4 + x^2 + 1 Divided by g(x) = x^2 + 2x + 3. We need to find the remainder. First we will divide and determine the quotient. ==> f(x) = g(x)*P(x) + R where R is the...

MathWe have the equation x^23x+1=0 x^2  3x + 1 = 0 x1 = [b + sqrt ( b^2  4ac)]/2a => [ 3 + sqrt(9  4)]/2 => 3/2 + sqrt 5/2 x2 = 3/2  sqrt 5/2 The sum of the square of these roots is: (3/2 +...

MathWe have to solve lg(8x+9) + lgx = 1 + lg(x^2  1) for x. Now, we use the relation that lg a + lg b = lg(a*b) lg(8x+9) + lgx = 1 + lg(x^2  1) => lg [( 8x + 9)*x] = 1 + lg ( x^2  1) => lg [...

MathWe have the sum of the roots of the quadratic equation as 5 and the product of the roots as 6. Let the roots be A and B. Here we don't need to consider the quadratic equation. We can find A and B...

MathWe have to solve the equation x = 6[(x2)^1/2  1] x = 6[(x2)^1/2  1] => x = 6*( x  2)^1/2  6 => x + 6 = 6*( x  2)^1/2 take the square of both the sides => x^2 + 36 + 12x = 36 ( x ...

MathWe have to find the derivative of y= 1/(x^3+3x^2+2x) y = 1/(x^3+3x^2+2x) = (x^3+3x^2+2x)^1 Using the chain rule y' = (x^3+3x^2+2x)^2 * ( 3x^2 + 6x + 2) => y' =  ( 3x^2 + 6x + 2) /...

MathThis is a simple quadratic function. To solve x or y, we need to put the coefficient of x or y same for each equation so that x or y can be eliminated when we subtract or add the two equations....

MathWe have to find the sum of the extremes of f(x) = (x^2 + 3x  3) / (x1). We have to differentiate f(x). f'(x) = [(x^2 + 3x  3)'*(x  1)  (x^2 + 3x  3)*(x  1)']/(x  1)^2 => [(2x + 3)(x  1)...

MathIt is given that Xn = sum[ 1/k*(k+1)*(k+2)]. We can write 1/k*(k+1)*(k+2) as A / k + B/(k+1) + C/(k+2) => 1/k*(k+1)*(k+2) = A / k + B/(k+1) + C/(k+2) => 1 = A(k+1)(k+2) + B(k)(k+2) +...

MathTo determine the points that lie on both the curves y=x^2+x+1 and y=x^22x+6, we have to equate the two. Doing this gives x^2 + x + 1 = x^2  2x + 6 => x^2 + x^2 + x + 2x + 1  6 = 0 =>...

MathWe have to find the values of x for which the following inequation holds : 3x  6 < 2x + 4 3x  6 < 2x + 4 subtract 2x from both the sides => 3x  2x  6 < 2x  2x + 4 => x  6...

MathThe domain of a function f(x) is all the values of x for which f(x) is a determinate quantity. Here f(x) = 5 / (5x  7) f(x) is not defined for 5x  7 = 0 => 5x = 7 => x = 7/5 Therefore the...

MathJustin builds 15 snowballs in an hour and 2 snowballs melt every 15 minutes. So in an hour he makes 15 snowballs of which 2*4 = 8 melt leaving him with 7. As he needs to build 210 snowballs, it...

MathYou have not provided an equation but a mathematical expression as 54x^4+2x. To factorize this: 54x^4 + 2x separate the common factors which are 2 and x => 2x*27*x^3 + 2x => 2x( 27x^3 + 1)...

MathWe need to prove that: 1/(1+sinx) = sec^2 x  tanx*secx We will start from the right side an prove the right side. ==> sec^2 (x)  tanx*secx. We know that sec(x) = 1/cos(x) and tanx = sinx/cosx...

MathWe need to prove that: sin2A = 2tanA/ (1+tan^2 A) We will start from the right side and prove the left side. We know that tanA = sinA/cosA ==> 2tanA / (1+tan^2 A) = 2(sinA/cosA) / [1+...

MathWe have to prove that: sin 2x/ (1 + cos 2x) = tan x To do this we use the relations : sin 2x = 2 sin x*cos x and cos 2x = (cos x)^2  (sin x)^2. sin 2x/ (1 + cos 2x) => [2 sin x*cos x] / [1 +...

MathWe need to prove that: cos3A = 3cosA + 4cos^3 A We will start from the left sides and prove the right sides. We know that: cos3A = cos(2A + A) Now we will use the trigonometric identities to...

Mathsin^4 A + 2cos^2 A  cos^4 A = 1 First, we will rearrange terms. ==> sin^4 A  cos^4 A + 2cos^2 A = 1 Now we know that: (a^2  b^2) = (ab)(a+b) ==> sin^4 A cos^4 A = (sin^2 A  cos^2...

MathThe initial velocity ufeet/sec. The bell is at a height of 20 feet above. Whether the contestantwill be able to jump a 20 feet height from the spring board is the question. So the actual equationis...

MathIf someone has a velocity of 32 ft/sec, will they be able to ring the bell( more info below)?At a...The initial velocity is u feet/sec. The bell is at a height of 20 feet above the platform. Whether the contestant will be able to jump a 20 feet height from the spring board is the question. So the...

MathIf one root of the quadratic equation ax^2 + 8x + 12 = 0 is twice the other, what is the value of...We have the equation ax^2 + 8x + 12 = 0. The roots of a quadratic equation ax^2 + bx + c = 0 are x1 = [b + sqrt (b^2 – 4ac)]/2a and x2 = [b  sqrt (b^2 – 4ac)]/2a. Here one of the roots is...

MathHow can math be used in daily life?I am trying to write an essay, and its topic is 'math in daily...I think that many math topics have meaning and relevancy and are dependent on the path one takes in terms of finding real world application. For example, sports is largely dependent on sports....

MathWe have to solve 13^x  20 = 13^ (3  x) 13^x  20 = 13^ (3  x) => 13^x  20 = 13^3/ 13^x let 13^x = y => y  20 = 13^3 / y => y^2  20 y  13^3 = 0 y1 = [20 + sqrt (20^2 + 4*13^3)]/ 2 =...

MathWe have the equation 2/arc tanx  arc tanx = 1 to solve. let y = arc tan x. 2/arc tanx  arc tanx = 1 => 2/y  y = 1 mutiply all the terms with y => 2  y^2 = y move the terms to one side...

MathWe have to find the first derivative of [cos ( x^3 + 13)]^3. Here we use the chain rule to arrive at the solution. ([cos ( x^3 + 13)]^3 )' => 3* [cos ( x^3 + 13)]^2 * [sin ( x^3 + 13)] * 3x^2...

MathGiven the derivative of the function s(t) , we have to find s(t). For this we integrate the derivative of the function. s(t) = Int [ s'(t)] => s(t) = Int [ 36*t^5 + 4*t^3] => s(t) = Int [...

MathWe have to find the absolute value of the vector z = u + v if u = i  j and v = 2i + 4j. z = u + v => z = i  j + 2i + 4j add the terms with i and those with j => z = 3i + 3j The absolute...

MathThe required quadratic equation has roots 1 + i and 1  i. => [x  (1 + i)]*[x  (1  i)] = 0 => [x  1  i][x  1 + i] = 0 => x^2  x + xi  x + 1  i  xi + i + 1 =0 cancel the common...

MathWe have to find x given that 8^(4x6)1/64=0 8^(4x6)  1/64 = 0 => 8^(4x6) = 1/64 => 8^(4x6) = 8^2 as the base is the same on both the sides we can equate the exponent. => 4x  6 = 2...

MathWe have to find the indefinite integral of f(x)=x^4(x^5+5)^5. Int [f(x) dx] = Int [ x^4(x^5+5)^5 dx] let t = x^5 + 5 => dt/dx = 5x^4 => x^4 dx = dt /5 Int [ x^4(x^5+5)^5 dx] => Int [ (1/5)...

MathWe'll apply the rule of multiplying 2 complex number, put in polar form: [cos (a1) + i*sin (a1)]*[cos (a2) + i*sin (a2)] = [cos (a1+a2) + i*sin (a1+a2)] We'll also apply Moivre's rule: [cos (a1) +...

MathWe have two vertexes of the triangle as A(0,4) and B(3,0). Let the third vertex be C(m, n). As C lies on x + y = 0, m + n = 0 => m = n The line joining AB has the equation: y  0 = [( 4  0)/ (...

MathThe roots of a quadratic equation ax^2 + bx + c = 0 are given by x1 = [b + sqrt (b^2 – 4ac)]/2a and x2 = [b  sqrt (b^2 – 4ac)]/2a. If the two roots are to be equal, sqrt (b^2 – 4ac) should...

MathThe mode M of a frequency distribution with class interval c ,the modal frequency fm and frequencices fm1 and fm+1 of the preceding and succeeding the modal class interval is given by: M = L +...