
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...

Math
Given the geometric series : 1/4 + x + 1/36 + 1/108+ .... We need to find the value of x. Let (r) be the common difference between terms. Then we know that: x = 1/4 * r...........(1) 1/36 = x *...

Math
First we will recall the formula of the volume of the cone. We know that the volume if given by the equation: V = (1/3)*r^2*pi * h where r is the radius and h is the height. Now, given that V = 142...

Math
Given the equations: x*y = 144.............(1) x+ y= 30 .............(2) We will use the substitution method to find the values of x and y. from (2) we know that y= 30x Now we will substitute into...

Math
Given the numbers 42, 126, and 210 We need to find the greatest common factor ( GCF) First we will factor each numbers. ==> 42 = 2*3*7 ==> 126 = 2*3*3*7 ==> 210 = 2*3*5*7 Now we will...

Math
Given the equations: 3x^2  2y 2x^2  3y Also we know that x= 3 and y= 5 We need to know the difference in values between both equations. First we will substitute x and y values in each equation....

Math
The car travels 27 mile / gallon. We need to calculate the cost for the car to travel 2727 miles. First we need to determine how many gallons does the car needs to travel 2727 miles. We know that:...

Math
The complex number is written in the form z = a+ bi. We need to determine the result of the product of two complex numbers. Let z1= a+ bi and z2 = c+di Now we need to calculate the product of...

Math
The area of the rectangle is 2.9 square meters. We know that the perimeter is 2l + 2w if l is length and w is width. We know that l = w + .8 So 2l + 2w = 7 2 (w + .8) + 2w = 7 2w + 1.6 + 2w = 7 4w...

Math
If (2n+1)/n is a natural number, then 2n+1 is divisible by n. Then we will rewrite as follow. 2n+1)/n = k where k is a natural number. ==> (2n/n + 1/n ) = k ==> 2 + 1/n = k Now since k is a...

Math
1/100 + 2/100 + 3/100 + ....+ 99/100 Since the denominators are the same, then we will add the numerators. ==> (1+2+3+...+99)/100 Now we notice that the numerator is a sum of a series Then we...

Math
ctgx + cosx = 1+ ctg*cosx First we will rewrite the identities. We know that: ctg(x) = cosx/sinx. ==> cosx/sinx + cosx = 1 + cosx/sinx * cosx Now we will simplify. ==> (cosx + cosx*sinx)/sinx...

Math
4sinx*cosx 1 = 2(sinxcosx) We will expand the brackets. ==> 4sinxcosx 1 = 2sinx 2cosx Now we will move all terms to the left side. ==> 2cosx 2sinx +4sinxcosx 1 = 0 ==> 2cosx...

Math
We can calculate the monthly payment for a mortgage using the formula: M = P [ i*(1 + i)^n] / [ ((1 + i)^n)1] where M is the monthly payment, i is the annual rate of interest divided by 12, n...

Math
The function for the profit is given as: P(x)=(5x400)/(x+600) Note: Here, the domain is x > 0. I have taken this as the domain under the assumption that you cannot sell less than 0 kilograms of...

Math
The function f(x) = (3x^0)  x^(1/2) Now any number to the power 0 is 1 So f(x) = 3  x^(1/2) f(9) = 3  9^(1/2) => 3  3 => 0 Therefore f(9) = 0

Math
The picture below represents the graph of the function `y = 4^x` . b) You need to reflect the graph over the x axis such that: Notice that the red curve is the graph of original function `y =...

Math
We have to solve 3^(x^2+4x)=(1/9)^2 for x. 3^(x^2+4x)=(1/9)^2 => 3^(x^2+4x)=(1/3^2)^2 => 3^(x^2+4x)=(3^2)^2 => 3^(x^2+4x)= 3^4 As the base 3 is the same, we can equate the exponent....

Math
The monthly payment for a mortgage can be calculated using the formula: M = P [ i*(1 + i)^n] / [ ((1 + i)^n)1], where M is the monthly payment, i is the annual rate of interest divided by 12, n...

Math
Starting with y = 2^x, it is not possible to get the result y = 3*log(3) (125*x^3) We can achieve the same by the following: y = 3*log(3) (125*x^3) Use log b^a = a*log b => y = 3*log (3)...