
Math
The 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...

Math
We 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...

Math
x + 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...

Math
The 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...

Math
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...

Math
By 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...

Math
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)...

Math
Given 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)...

Math
Given 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...

Math
We 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 +...

Math
We 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 [...

Math
We 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...

Math
We 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 ...

Math
We 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) /...

Math
This 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....

Math
We 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)...

Math
It 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) +...

Math
To 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 =>...

Math
We 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...

Math
The 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...

Math
Justin 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...

Math
You 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)...

Math
We 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...

Math
We 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+...

Math
We 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 +...

Math
We 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...

Math
sin^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...

Math
The 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...

Math
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...

Math
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...

Math
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....

Math
We 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 =...

Math
We 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...

Math
We 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...

Math
Given 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 [...

Math
We 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...

Math
The 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...

Math
We 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...

Math
We 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)...

Math
We'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) +...

Math
We 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)/ (...

Math
The 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...

Math
The 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 +...

Math
Let the A be the point on the ground, G be the base of the church and S be the church steeple. So AGH is right angled triangle with angle AGS = 90 degrees. So AG*tan GAS = GS. => AG tan 20 =...

Math
Given the function: f(x) = x^2  7x + 12.5 Let a =1 b= 7 We need to solve by completing the square. Then we will add and subtract the values of (b/2)^2 ==> b= 7 ==> b/2 = 7/2 ==? (b/2)^2...

Math
We know that the path of the water forms a parabola. Then the equation of the parabola is given by: f(x) = ax^2 We will assume that the highest point of the water path is the origin point (0,0) The...

Math
Let the speed be S = 6 kn/h Then, we need to calculate the distance we can jog in 1.5 hours. ==> Let the time be T = 1.5 ==> Let the distance be D km. ==> S = D/T ==> D = S*T = 6*1.5 =...

Math
According to the law motion of an object projected vertically above with an initial velocity u ft/sec is subject to a constant accelerationdue to gravitation g. So the velocity b is a vector in a...

Math
Given the function f(x) = x^2  4x 1 We need to find the inverse function of f(x). Let us assume that y= f(x). ==> y = x^2  4x 1 We will complete the square . ==> y = x^2  4x 1 +4 4...

Math
Given that the number is between 100 and 200, Then the number is a three digit number. Then, the hundreds digit is 1. ==> Let the number be x such that "a" is the tens and "b" is the units....