
Math
We have to find the roots of : x^3 – 2x^2 – 23x + 60 x^3 – 2x^2 – 23x + 60 = 0 As the equation has a highest power of x, we will have 3 roots. From the numeric term equal to 60 we know that...

Math
We have to prove the sum of the squares of the first n numbers is given by n(n+1)(n+2)/6 We know that (x+1)^3 = x^3 + 3x^2 + 3x + 1, now if we write the cubes of the numbers 1 to n+1, we get 1^3 =...

Math
We have f(x) = 5x  2 and g(x) = x^(1/3) [ I have assumed by cubed sqrt you mean raised to the power (1/3)] We have to find fog(  8) fog(x) = f(g(x)) = f( x^(1/3)) =5 * x^(1/3)  2 For x = 8...

Math
We have to solve the system of equations: y=x^2+4 y=2x+7 Equate the value of y => x^2+4 = 2x+7 => x^2  2x  3 = 0 solving the quadratic equation we get two values of x. => x^2  3x + x ...

Math
The rate of depreciation is given as 30%. This means that the value of the car reduces by 30% every year. Or the value of the car after a year is 1  30% = 70% of its value in the previous year....

Math
The man can make the trip from city A t city B at a rate inversely proportional to the time he takes. Let the distance between the two cities be D. When he travels at 60 kilometers per hour he...

Math
Given f(x) = ax+bx Since wee need to differentiate with respect to a, then we will assume that b is a constant. ==> f'(x)= (ax)' +(bx)' = a'*x + a*x' + bx' = a'x + a...

Math
Let the amount that the manufacturer gets per chocolate be C. The supermarket marks up the price by 3% and it is sold at $3. So, we have C *( 1 + 3%) = 3 => C* (1 + 0.03) = 3 => C*( 1.03) =...

Math
Use cross multiple: y(2x+1) = 6x8 2yx +y = 6x  8 2yx  6x = y8 2x(y3) = (y+8) x = (y+8)/2(y3) If y = 2, x = 5 This is the only one meeting the requirement.

Math
We have to solve: 6(7^(2x1) + 7^(x1)  1 = 0. Note : I am not sure where the initial brackets that you have started ends, so I'm solving the following equation 7^2(x1) + 7^(x1)  1 = 0 Take...

Math
We have to simplify : 6 (log c/log a) + 5 log(a) d  7 log(a) b Now we use the relation that log(a)b = log b / log a 6 (log c/log a) + 5 log(a) d  7 log(a) b => 6(log c/ log a) + 5( log d/ log...

Math
We have to solve :  x  4 =  2x 7 Now the absolute value x of x as well as x is equal to x.  x  4 =  2x 7 gives us x  4 = 2x  7 and ( x  4) = 2x  7 x  4 = 2x  7 => x  2x =...

Math
We have to solve x^2  19x = 90 x^2  19x = 90 we use the relation that the roots of a quadratic equation ax^2+ bx + c = 0 are given by [b + sqrt (b^2  4ac)]/2a and [b + sqrt (b^2  4ac)]/2a...

Math
We are given that f(x) = x^2  3x + 8. f(x2) can be found by replacing x by (x – 2) in f(x) = x^2  3x + 8 => (x – 2)^2 – 3*(x – 2) + 8 Similarly for f(x + 2) replace x with (x + 2) in...

Math
The average age of 10 men initially is 20. So the total age of the men is 10*20 = 200. When the two men join the group later, the average age of the group increases by 3. So the total age of all...

Math
We have to express [5x^2 + 10 x + 3] / [x^3 + 2x^2 – x – 2] as partial fractions. First, we factorize the denominator [5x^2 + 10 x + 3] / [x^3 + 2x^2 – x – 2] => [5x^2 + 10 x + 3] /...

Math
We have to solve 7^(3x) = 8^(2x) Now the base of the two terms are not the same, so we use logarithms. Take the logarithm of both the sides. log( 7^(3x)) = log(8^(2x)) we use the relation log (a^b)...

Math
We have to find y such that sin y = cos 3y. We know that sin(y) = cos(pi/2y),so we can convert the given equation to: sin(y) = cos(pi/2y) = cos(3y) We now equate pi/2  y and 3y pi/2  y = 3y...

Math
Integers are only whole numbers and include both positive as well as negative whole numbers and 0. An integer cannot not have a decimal or a fractional part. The set of integers is {... −2, −1,...

Math
We have to determine x given that: sin x * sin 2x  cos x * cos 2x > sin 6x We know that cos (a  b) = cos a * cos b  sin a * sin b =>  cos ( 2x + x) > sin 6x => cos 3x > sin 6x...

Math
x^532. We have to factorise this. Solution: Let f(x) = x^5 32. Put x= 2 in f(x) = x^5 32. f(2) = 2^5  32 = 3232 = 0. So by remainder theorem, (x2) is a factor of x^532. We know that...

Math
When we need to do an operation on two rational numbers, first we need to have a common denominator. If the ddenominators are not the same, then we will have to multiply the denominators by each...

Math
The zeroes of a polynomial are simply those values of the variables where the polynomial as a whole equals zero. So, in other words, to find the zeroes of a polynomial, you have to set the whole...

Math
Given that z is a complex number, then we will rewrite into the complex form z= a + bi We know that the absolute values of z is given by: lzl = sqrt(a^2 + b^2) Then we will write the log of a...

Math
A matrix is consisting from rows and columns. These rows and columns contain numbers that are called elements of the matrix. The size of a matrix is mbyn, where m represents the number of rows...

Math
We know that (x + y)^2 = x^2 + y^2 + 2xy 9x^2 + 12xy + 4y^2 => (3x)^2 + 2*(3x)*(2y) + (2y)^2 = 49 = (3x + 4y)^2 = 49 => (3x + 4y)^2 = 7^2 => (3x + 4y) = 7 For x = 1 and y = 1 we have 3x +...

Math
To answer this question, split up the line integral into two pieces: intc (x + 2y)dx and intc (x  y)dy. Our parameter is t, 0<=t<=pi/4 (I assume, because your problem statement gives...

Math
Notice that the force P acts along x axis, the force Q acts along y axis and the force R acts on a support that expresses the hypotenuse of right triangle that has as legs the supports of P and Q...

Math
We can write the given polynomial 6*x^4+17*x^32*x^2+x6 as: 6*x^4+17*x^32*x^2+x6 => 6x^4 + 18x^3  x^3  3x^2 + x^2 + 3x  2x  6 => 6x^3( x + 3)  x^2(x + 3) + x(x + 3)  2(x + 3) =>...

Math
We'll put the qudratic equal to zero: 3x^2 + 18x  7 = 0 We'll multiply by 1 both sides: 3x^2  18x + 7 = 0 We'll apply the quadratic formula: x1 = [b + sqrt(b^2  4ac)]/2a a,b,c are the...

Math
A line is represented mathematically by a "linear" equation, which takes the following standard form: y = mx + b This equation defines the point y given the input x, and parameters m and b. m is...

Math
We have the two functions : f(x)=x^2+3x and g(x)=x+3. To determine for what values of x is f(x) = g(x) , equate the two f(x) = g(x) => x^2+3x = x+3 => x^2 + 2x  3 = 0 => x^2 + 3x  x  3...

Math
Your answer can be derived from an understanding of the unit of current. In your problem, 0.2 A of current is flowing. (We'll assume that this value remains unchanged for the duration of the...

Math
The bicycle's wheel has a diameter of d cm. Th radius of the wheel is d/2 cm. When the wheel rotates once the bicycle moves forward by 2*pi*d/2 = pi*d cm. If the bicycle is travelling at x km/h, x...

Math
We need to find the area of the concrete borders. The swimming pool is of dimensions 25m by 10m. The concrete border at the sides which are 25 m each is 2.5 m wide and is 5 m wide at the ends of 10...

Math
We have to determine the coordinates of points that lie in the locus given the constraint: modulus(z  2) ≤ modulus(z  2i) The modulus of an expression x + yi = sqrt ( x^2 + y^2) We have...

Math
Let one of the numbers be x. Then the next consecutive number is (x+1) Given that the sum of the squares is 145. ==> x^2 + (x+1)^2 = 145 ==> x^2 + x^2 + 2x + 1 = 145 ==> 2x^2 + 2x  144 =...

Math
We have to solve 2a5 < 5(a1) for a. 2a5 < 5(a1) open the brackets => 2a  5 < 5a  5 => 2a  5a < 5 + 5 => 3a < 0 => a < 0 => a > 0 Therefore a > 0

Math
We are given that f'(x) = 8x^3 + 3x^2 + 2. To find f(x) we need to find the integral of f'(x). Int[f'(x)] = Int [ 8x^3 + 3x^2 + 2] => 8x^4 / 4 + 3x^3 / 3 + 2x + C => 2x^4 + x^3 + 2x + C Also,...

Math
We have cosec x = 1/sin x. sin (x) = sin x. Now sin x * cosec(x) => sin x * (1 / sin (x)) => sin x * ( 1 / sin x) => 1 * (sin x / sin x) = 1 Therefore we prove that sin x *...

Math
We have the two functions h(x) = x^3+ x and g(x) = 2x + 3. Now we need to find g(h(2)). h(2) = 2^3 + 2 = 8 + 2 = 10 g(h(2)) = g(10) = 2*10 + 3 = 20 + 3 = 23 Therefore g(h(2)) = 23.

Math
So simplify 5/k + (k+3)/(k+5), we make the denominator the same for all the terms 5/k+(k+3)/(k+5) => [5(k + 5) + k(k+3)]/ k(k+5) => [5k + 25 + k^2 + 3k]/k(k+5) => (k^2 + 8k + 25)/k(k+5)...

Math
Given the set of numbers 14 , 8, 16, 14 , and x We know that the mean = the median. Let us arrange the numbers from smallest to largest. ==> 8, 14, 14, 16, x Since we so not know that value of x...

Math
At the xintercept the value of the ycoordinate is 0. So to find the xintercept of y = x^2 – 4x + 4 we can substitute y = 0 and solve. x^2 – 4x + 4 = 0 => (x  2)^2 = 0 we get x = 2....

Math
Given that the center of the circle is the point (0,0). We know that the equation of the circle is given by : (xa)^2 + (yb)^2 = r^2 where (a,b) is the center and r is the radius. ==> x^2 + y^2...

Math
The rectangle has a perimeter of 30 cm. Its length is twice its width. Let us denote length by L and width by W. So we have L = 2W Perimeter = 2L + 2W = 30 => 2* 2W + 2W = 30 => 4W + 2W =...

Math
We have the set of numbers {2, 3, 7, 12, 15, 22, 72, 108}. Here we see there are 8 terms in total out of which 12 , 72 and 108 or three terms are divisible by both 2 and 3. So the probability of...

Math
When looking at the clock, we notice that the circumference is divided into 12 equal sections. These sections represents the hours. When the clock is exactly 1:00, the hands will be on the numbers...

Math
Given that the original price of the DVD player is $210. First we need to calculate the cost of the DVD player after the 30% off. ==> The sale price = original price  discount ==> Sale price...

Math
Given the line: 9x + 4y = 7 We need to determine the slope of any line that is parallel to the line 9x+4y=7. We know that the slopes of two parallel lines are equal. Then, we will determine the...