
Math
We are given that: (x^2 y^6)/z = 3 ...(1) (x^2 z^5)/y^2 = 27 ..(2) We need to find one value of x^2y^2z^2 Multiply (1)*(2) => [(x^2 y^6)/z]*(x^2 z^5)/y^2 = 3*27 Open the brackets and simplify...

Math
Three geometric means have to be identified between 2 and 162. If the required numbers are a, b and c a geometric series is formed by 2, a, b, c, 162 162 = 2*r^3 r^3 = 81 => r = 3 a = 2*3 = 6 b...

Math
Yes, roots means the values of x for which f(x) is equal to 0. The roots represent points where the graph of the function crosses the xaxis. To find the roots of f(x)= 2x^2 + 3x 20, we equate it...

Math
The square root property is a method used to solve for an equation in the second degree, usually of the form `ax^2 + c = 0` . Notice that there is no `x` in the equation. However, aside from this...

Math
What is the length of the side f?
We have a triangle with sides d,e,f and angles D,E,F
(D,d) = (36`^@` ,62)
(E,e) = (42`^@` ,71)
(F,f) = (102`^@` ,f)
Using the sine rule `sinF/(f) = sinD/d = sinE/e = sin(36)/62 = 0.0095` `therefore` `f = sinF/0.0095 = sin102/0.0095 = 103.0` The length of f is 103.0

Math
`y=sqrt(4x10)` Using the chain rule `y' = 1/(2sqrt(4x10))*4=(2)/(sqrt(4x10))` Using the product and chain rules `y'' = (2)/(sqrt((4x10)^3))*(1/2)*(4)=4/((4x10)^(3/2))` There are no x...

Math
We 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 substituting...

Math
You should notice that `x o y = xy  3x  3y + 12` , hence, you need to substitute `x*x  3x  3x + 12` for `x o x` in equation `x o x = x` such that: `x*x  3x  3x + 12 = x => x^2  6x + 12...

Math
The sides of the triangular park are of 80 m , 100 m and 120 m. The area of the park can be calculated using Heron’s formula which gives the area as sqrt [s*(s – a)(s – b)(s – c)] where a, b and c...

Math
If general equation of hyperbola is `x^2/a^2+y^2/b^2=1` then we can calculate linear eccentricity `e>0` by using formula `e^2=a^2+b^2`...

Math
Since the amplitude of a sinusoidal function is the absolute value of the coefficient of the function, in this case the amplitude is 21. Also, the average of the sinusoidal function is 61, so the...

Math
We are given `y=2x^2+4x+5` The slope of the tangent line is given by `y'=4x+4` . The system will have two solutions if the line intersects the parabola twice. It will do so between the values...

Math
`14b^211b^2` Here (b^2) term is common for both parts. So we can take it out. `14b^211b^2` `= b^2(1411)` `= b^2(25)` `= 25b^2` `3x+2y8x7y` we have take common x and y terms...

Math
We have to prove that : sin(t)*sin(t) + cos(t)*cos(t) = 1 sin(t)*sin(t) + cos(t)*cos(t) = (1/2)(cos (t  t)  cos (t + t)) + (1/2)(cos (t  t) + cos (t + t)) => (1/2)(cos 0  cos 2t + cos 0 +...

Math
The line `y = ax + b` is the oblique asymptote for the function `y = (x^26x+m)/(2x4)` if there exists `a = lim_(x>+oo) y/x` and `b = lim_(x>+oo)(y  a*x)` . You need first to evaluate...

Math
The formual for total cost of driving cars rented from companies A,B,and C must be determined. So let, x  be the number of miles the car travels. y  the number of days the caris rented and Z ...

Math
f(x) = m/(x+1) mx To find where the function is increasing, we need to calculate the first derivative, if f'(x) is positive, then the function is increasing for the interval. Let us calculate...

Math
Let the numbers be x and y. Given that the sum of the numbers is 62. ==> x + y = 62 ==> y= 62 x.............(1) Also, we know that the product is 880. ==> x*y = 880............(2) We will...

Math
Let us assume that the numbers are x and y. Given that the sum is 57. Then we will write: x + y= 57 ............(1) Also we know that the product is 800. ==> x*y = 800 .............(2) Now we...

Math
You need to use the following properties of consecutive terms of an arithmetic progression `a_(n1),a_n,a_(n+1)` , such that: `a_n = (a_(n1) + a_(n+1))/2` Reasoning by analogy, yields: `17 = (a +...

Math
Permutations are the number of different ways numbers or, in this case, letters, can be arranged.The word helicopter has 10 letters but one of them is repeated (e). We take 10! (that is, 10...

Math
The number of combinations possible when r elements are chosen from a set with n elements is C(n, r) = `(n!)/(r!*(n  r)!)` The set of {A,B,C,D,E,F,G,H,I,J,K,L} has 12 distinct elements. The...

Math
What number is 16 more than its opposite? Let "the number" be represented by the variable x. The opposite of `x` is `x` . Since the number is 16 more than its opposite, we can add 16 to the...

Math
The series given is 10 , 27, 52, 85,… Subtracting subsequent terms gives 27 – 10 = 17 52 – 27 = 25 85 – 52 = 33 The difference we get does not provide any information about the series. Again...

Math
The series given is 4 , 44 , 444 , 4444 ,… This series is none of the general ones that we encounter like arithmetic series, geometric series or harmonic series. To find the nth term look at each...

Math
If the first term of an arithmetic series is a and the common difference is d, the nth term is given as a + (n  1)*d We have the 5th term as 10.8 => a + 4*d = 10.8 ...(1) The 12th term is 22...

Math
The mathematical notation capital pi can be used to represent (1*2*3*4*...*(infinity1)) / (2*3*4*5*...*(infinity)) in a compact form. This is written as `prod_(n=2)^oo (n1)/n`

Math
Vectors are like arrows, but they have different lengths and point in certain directions. I can't provide a lesson plan for you but can provide a link that might help.

Math
The derivative `dy/dx` has to be determined given that y^2 + 3y = 3x + 4 y^2 + 3y = 3x + 4 use implicit differentiation on both the sides => `2*y*(dy/dx) + 3*(dy/dx) = 3` => `(dy/dx)(2y + 3)...

Math
The next term of the series 2, 6, 12, 20, 30... has to be determined. Subtracting the terms given from the previous term gives the following values: 62 = 4, 12  6 = 6, 20  12 = 8, 30  20 = 10....

Math
If two lines do not intersect each other, they have an equal slope. The slope of the line (3t^2 + 2t)x  y = 21 is 3t^2 + 2t. The slope of the line x + y = 1 is 1. As the two lines are parallel to...

Math
Th general term of the the series: 60/121, 30/11, 15 , ... 219615/16 has to be determined. The difference of consecutive terms of the series is not equal, this is not an arithmetic progression. But...

Math
`y = ln(5x + 1)` The gradient of the above curve at any point is given by the first derivative. `y=ln(5x+1)` `dy/dx=[1/(5x+1)] xx 5 = 5/(5x+1)` Gradient at x=4 is given by; `(dy/dx)_(x=4) = [5/(5...

Math
The expression `(3*c^4*d^5)^2*12*c*d^4` has to be simplified. `(3*c^4*d^5)^2*12*c*d^4` Use the following formulas `(x^b)^c = x^(b*c)` , `x^a*x^b = x^(a+b)` =...

Math
A cubic polynomial that gives a remainder of 4 when it is divided by x + 2 can be written as `P_3 = (x+2)*P_2+ 4` , where `P_2 ` is any quadratic polynomial (can be monomial, binomial or...

Math
2x^4+9x^387x^249x+45=0 To solve this polynomial equation, factor the left side of the equation. (x5)(x+1)(x+9)(2x1)=0 Now solve for each separate term in the product. x5=0 x=5 x+1=0 x=1...

Math
C(x, 2) = x!/ ( 2! *(x  2)!) P(x , 2) = x! / (x  2)! Now C(x,2) + P(x,2) = 30 => x!/ ( 2! *(x  2)!) + x! / (x  2)! = 30 => x*(x  1)/2! + x*(x  1) = 30 => x*(x  1)/2 + x*(x  1) =...

Math
The solution of x/y + y/x = 8 can be obtained by equating z = x/y This makes x/y + y/x = 8 => z + 1/z = 8 => z^2 – 8z + 1 = 0 z1 = 8/2 + sqrt (8^2 – 4) / 2 => 4 + sqrt 60 / 2 => 4 +...

Math
A line can be expressed in the standard form y = mx + c where m is the slope and c is the yintercept. For the two line to be parallel they should have the same slope. Now, y = 7x / 2  17/2 has...

Math
The rate at which the cost of college education increases is assumed to be 6% per year. The cost of education right now is $14000. The cost of education after 18 years would be 14000*(1+0.06)^18 =...

Math
The function `y(x) = 5(3x^2 + 4x +7)^9` . Use the chain rule to find the derivative. `y'(x) = 5*9*(3x^2 + 4x +7)^8*(6x + 4)` = `45*(6x + 4)(3x^2 + 4x +7)^8` The derivative of `y(x) = 5(3x^2 + 4x...

Math
The function we have is : f(x)=1/(x4)^3  4 f(x)=1/(x4)^3  4 => [1  4(x  4)^3)]/(x  4)^3 Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator. Here...

Math
The equation to be solved is: √2x  2 = 4 √2x  2 = 4 => √2x = 2 + 4 => √2x = 6 => x = 6/√2 => x = 6*√2/(√2*√2) => x = 6*√2/2 => x = 3√2 The solution of the equation √2x  2 = 4...

Math
The sides of the triangle are 12 m, 14 m and 15 m. The angles of the triangle have to be found. Use the law of cosines. 12^2 = 13^2 + 14^2  2*13*14*cos X => cos X = 221/(2*13*14) => cos X =...

Math
We have 10100101, which is a binary number, so there are only two digits 1 and 0. 10100101 = 1*2^0 + 0*2^1 + 1*2^2 + 0*2^3 + 0*2^4 + 1*2^5 + 0*2^6 + 1*2^7 => 1 + 4 + 32 + 128 => 165 Binary...

Math
To determine if 6y^4 + 15y^3 + 28y + 6 is divisible by y  3, we can either do an actual division. But there is an easier way to check for divisibility. If 6y^4 + 15y^3 + 28y + 6 is divisible by y...

Math
When 4 dice are rolled together each of the die can get a number ranging from 1 to 6. The total number of options is therefore 6^4 = 1296. The ways in which 6 can be obtained is 1113, 1131, 1311,...

Math
We have to verify that f(x) and g(x) are inverse functions given that `f(x) = 2  x^(1/3)` `g(x) = (2  x)^3` f(g(x)) = f((2  x)^3) => 2  ((2  x)^3)^(1/3) => 2  (2  x) => 2  2...

Math
The initial population in the year 1991 is given as 364552. The rate of growth is also given as 2.2%. The number of growth periods is not limited by any factor as the population is not decreasing...

Math
The initial value of the car is $25000. The depreciation in year 1 is 20%. The value of the car after the first year is 25000*(10.2) = 25000*0.8 = 20000 Later, it depreciates by 10% each year....