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MathThe condition for the given 3 lengths form a triangle is that the greatest of the 3 sides should be less than the other two sides. Or if a is the greatest of the 3 sides and the other two sides...

MathWE know that the three towns form a triangle. For the formation of the three towns, the length of the 3 sides a, b and c shold be such that the grest side a < b+c, where b and c are the other...

MathThe question given could be analysed in general. But this attempt is only to the limitted sense of solving problems related with rational fractions and rational algebraic expressions. If a...

MathWe are given two points P(7,11), Q(2,4). Now for the distance between P and Q we use sqrt[(27)^2+(411)^2] = sqrt [ 81 + 49] = sqrt 130 = 11.40 approximately. The slope of the line PQ is ( 4 ...

MathWe have 5^x  3^x = 2^x. Now to prove that x = 1 is a solution for the equation we substitute x = 1 and we should get the same value on both the sides 5^x  3^x = 2^x substituting x = 1 in the...

MathWe have to solve 3^(x2)  9 = 0. Now we see that 9 = 3^2. 3^(x2)  9 = 0 => 3^(x 2) = 9 => 3^(x 2) = 3^2 As the base 3 is same on both the sides, the power should also be the same. =>...

MathGiven the function y=3x^3 + 3^x. We need to determine whether y is an increasing function. To verify, we need to determine the first derivative ( y'). If y' is positive, then the function is...

MathWe are given that lines P and Q are parallel. The slope of line P is 5 and the slope of line Q is 10/v. Now we have to find v. Here we use the property that the slope of parallel lines is the...

MathWe know that tan x = adjacent side/ opposite side. Now consider x to be angle in a right angled triangle such that tan x = 4/5. The hypotenuse of the right angled triangle is equal to sqrt( 4^2 +...

MathWe have to find the exact value of sec 2x given that sin x = 3/5 and cos x = 4/5 Now sec x = 1/cos x. To find the value of sec 2x, we need to find cos 2x and then take its inverse. The relation for...

MathHere we have to use set theory. Now the total number of investors surveyed is 80. All the investors have either one mutual fund or two stocks. The total number of investors is equal to those who...

MathSince the center of the circle is located in the 3rd quadrant, the values of it's coordinates are negative. From enunciation we conclude that x = 3 <=>h =3, because the radius of the...

MathThe first step is to isolate (sin y)^2 to the left side. For this reason, we'll add (sin y)^2 both sides: 11 (sin y)^2 + (sin y)^2 = 13 12 (sin y)^2 =13 We'll divide by 12: (sin y)^2 = 13/12 sin y...

MathWe have to solve : x^2  y^2 = 9 and x  y = 1 using the method of substitution. Now x  y = 1 => x = 1 + y Substitute x = 1+ y in x^2  y^2 = 9 => ( 1 + y)^2  y^2 = 9 => 1 + y^2 + 2y ...

MathGiven f(2) = 9 , we have to find k in the function f(x) = x^2  kx  3. Now f(x) = x^2  kx  3, for x =2 => f(2) = 2^2  k*2  3 = 9 => 4  2k  3 = 9 => 2k = 4  3  9 => 2k = 8...

MathWe have to find a given that 3 log 4(a2) = 9/2. Now 3 log 4(a2) = 9/2 => log 4 (a  2) = (9/2)/3 => log 4 (a  2) = (3/2) take the anti log of both the sides => (a  2) = 4^(3/2) => a...

MathWe have to find the multiplicative inverse of 4  2i. Now the multiplicative inverse of any number X is defined as X^1, so that X*X^1 = 1. For 4  2i, let the multiplicative inverse be M. Now M*...

MathWe have to find the tangent to the curve y = x^2 + 8x + 12, at the point where x= 4. Now the slope of the tangent to a curve is given by the derivative of the equation representing the curve. In...

MathWe'll write the numerator and denominator in polar form: z = r(cos t + i*sin t) The reason of putting the numerator and denominator in polar form is to use Moivre's rule. We'll put the numerator in...

MathWe have to prove that x+y=<sqrt 2z, if z=x+i*y. Now assume that the given relation is true. So x+y=<sqrt 2z. Taking the square of both the sides => (x+y)^2 =< [(sqrt...

MathWe have to solve 4x^2 + 3x  5 > 2 4x^2 + 3x  5 > 2 => 4x^2 + 3x  7 > 0 => 4x^2 + 7x – 4x – 7 > 0 => x (4x + 7) – 1(4x + 7) >0 => (x  1) (4x + 7) > 0 This is...

MathWe have to simplify (4+3i)^2. Now, let's open the brackets. (4+3i)^2 => 4^2 + (3i)^2 + 2*4*3i => 16 + 9*i^2 + 24i => 16  9 + 24i => 7 + 24i Therefore (4+3i)^2 = 7 + 24i

MathThe zeroes of a polynomial are the roots of that polynomial. We also know that the complex solutions come in complex conjugate pairs. So, if 2i is a solution of the equation, that means that 2i is...

MathWe have a 3M solution and a 4M solution and we are supposed to mix them to create a 3.6M solution. Now if 2 liters of the 4M solution are used, let the volume of the 3M solution used be equal to...

MathWe have to solve for x given the following equation: 1/2 +7/10x = 13/20. 1/2 + 7/10x = 13/20 multiply all terms with 20x => 1*20x/2 + 7*20x/10x = 13*20x/20 => 10x + 7*2 = 13x Subtract 10x...

MathWE have to simplify [(4t^216) / 8] / [(t2) / 6] [(4t^216) / 8] / [(t2) / 6] => [(4t^216) / 8] * [6 / (t2)] => [(4t^216)*6 / 8*(t2)] => [(2t  4)(2t + 4)*6 / 8*(t2)] =>[4(t ...

MathWe have to find the integral of y=1/(x1)(x+4) We know that the integral of 1/(1+x) ln (1+x). We need to convert the given expression to a sum of two terms with a similar form. Let's write...

MathWe'll form a right angle triangle, whose hypotenuse is the distance from origin to the point and one cathetus is its abscisa and the other cathetus is the ordinate. We'll note the distance as r: r...

MathWe'll establish the cost of an apple as x and the cost of an orange as y. We'll write mathematically the phrase "5 apples and 4 oranges cost $ 3.40": 5x + 4y = 3.4 (1) We'll write mathematically...

MathWe'll write the rate of the 1st investment: Interest = Price*rate*time We'll substitute the known data: 945 = 6000*r*3.5 We'll divide by 6000*3.5: r = 945/(6000*3.5) r = 0.045 or 4.5% per year...

MathFirst, we'll impose the constraints of existence of logarithms: x  24/5>0 x > 24/5 2x>0 x>0 The common interval of admissible values for x is (24/5 ,+inf.). Now,we'll could solve the...

MathWe'll solve the equation considering 4 cases of study. The expressions 3x4 and 34x can be either positive or negative. Case 1: 3x4 = 34x Case 2: (3x4) = 34x Case 3: 3x4 = (34x) Case 4:...

MathTo convert a complex number a + bi into the polar form we need to find r which is equal to sqrt (a^2 + b^2) and the angle theta given as arc tan (b/a) For the complex number 7  5i we have a = 7...

MathLet u be the initial velocity of the object thrown up verically. So the ovject goes on loosing its velocity till the final velocity of the object becomes zero. Therefore v = ugt = 0, where v is...

MathThe stone thrown upwards will slow down under the influence of gravity till it reaches 0 velocity/speed and will again start falling down with increasing velocity in downward direction till it hits...

MathWe know that the primes are integers divisible by 1 and themselves only: The set of the primes is: 2, 3, 5, 7, 11, 13, 17, .... Then the first three primes are 2 , 3, and 5 Let the other prime be...

MathWe have the equation of the circle given as x^2 + y^2 + 8x + 6y = 0. Now, the equation of a circle with center (h, k) and radius r is (x − h)^2 + (y − k)^2 = r^2 So we convert x^2 + y^2 + 8x +...

MathWe have to find the perpendicular distance of the point (5, 6) from the line 2x + 3y + 4 = 0. Now, we know that the perpendicular distance of the point (m, n) from the line Ax + By + C = 0 is...

MathWe need to find equivalent fractions for 0.60 , 1.7 , 0.05 , 2.3 , 0.15 and 0.625. An easy way to do this is write them as a number divided by a power of 10. Then for equivalent fractions multiply...

Math[2][x] + [y] = [4]....(1) [2] [x] +[4](y)] = [5]....(2) Where all values of the form [n] belongs to the z sub 7 system of numbers. Subtract (1) from (2) and we get: [4][y]  [y] = = [5][4]....

MathZ5 multiplication table. The Z5 congruence class consists of the numbers 0,1,2,3,4. So Multiplication any 2 numbers in z system is given below: In Z5 system we represent numbers by the rule: a*b =...

MathWe have the equation of the curve as y = x^3  2x^2 + 5 To find the slope or the gradient of the tangent drawn at any point on this curve, we need to find the differential. y’ = 3x^2 – 4x For...

Math358x = 17(mod313). To find the solution x, where x is in Z. We proceed like this: Let us find the gcd in the usual procedure: 358 = 313*1 +45. 313 = 45*6+43 45 = 43*1+2. 1= 2*21 +1. Therefore , we...

MathHow many TVs will give maximum profit and what is the maximum profit?The daily profit, P, made by...The function for the profit earned per TV manufactured is given as P = 100x − 0.02x^2. Now we need to find the number of TV's that should be manufactured to earn the maximum profits. To solve...

MathLet the length of the rectangle be L and the width be W Given the the Length is twice the width: Then, we will write: L = 2W.............(1) Also, given that the area is 98. We know that the area...

MathLet the equation of the line be : y y1 = m ( xx1) where ( x1, y1) any point passes through the line and m is the slope. Given that the line passes through the origin point Then the line passes...

MathLet mark's age be M and his sister's age is S Given that mark is 4 years older than his sister: Then M = S + 4 ............(1) Now after 3 years: Mark's age = M + 3 His sister's age = S + 3 But...

MathLet the numbers be x and y Given that the sum of both numbers is 23: Then we will write: x+ y = 23..............(1) Also, given that the difference between the numbers is 1: x  y =...

MathLet the vertices of the triangle be : A(1,1) B( 2,0) and C ( 4,2) First we will calculate the length of the sides: AB = sqrt( 2+1)^2+ ( 01)^2 = sqrt(3^2 + 1) = sqrt( 10) ==> AB...

Math2 < 2x4 < 6 To solve the inequality, first we need to solve the same ways we solve the equality. We need to isolate x by itself : Let us add 4 to all sides: ==> 2 + 4 < 2x4 + 4 <...