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In this post, we are going to solve a math question which is very important for CHSL, CGL, SSC GD, SBI PO, IBPS etc. Q. Renu Spends 68% of her income. When her income increases by 40 and expenditure by 30%. Her saving is? Sol.  Step-by-Step  Step1: Assume that Renu's original income = 100 Her original expenditure = 68 of 100 = 68 Her original saving = 100 - 68 = 32 Step2: After 40% increase in income New income = 140 Step3: After 30% increase expenditure New expenditure = 30 of 68 = 20.4 Step4: Calculate new saving: New saving = New income + New expenditure = 140-88.4 =51.6 Step5: Increase in savings: Increase in saving = New saving - original saving = 51.6 - 32 = 19.6 Step6: Percentage increase in saving: Percentage increase = Increase in saving/Original saving * 100 Percentage increase = 19.6/32 * 100 = 61.25 Therefore, the Renu saves 61.25%. Below is the Youtube video where you can easily unde...

Theorem: The Union of two subgroups of a group G may not be a subgroup of G    Proof:  Let G = Z be the additive group of integers.    ∀ ∈  H 1 ∪H 2 Let H1 = {2n: n = Z} = {....-4, -2, 0, 2, 4, 6, ....} H2 = {3n: n = Z} = {....-6, -3, 0, 3, 6, ....} Since          0 =   2(0)  ∈  H 1 , so  H 1 is a non-empty subset of G Let a, b =  H 1 . Then a = 2n 1 and b = 2n 2 for some n 1 , n 2   ∈  Z.  Now, a-b = 2n 1 - 2n 2 = 2(n 1 -n 2 )  ∈  H 1   a-b  ∈  H 1 a, b  ∈  H1 and so H1 is a subgroup of G.  Similarly, H2 is a subgroup of G. Also,  H 1 ∪H 2  = {..... -6, -4, -3, -2, 0, 2, 3, 4, 6, ...}  Now, 2, 3  ∈   H 1 ∪H 2 but 2-3=-1  ∉   H 1 ∪H 2  Thus,  H 1 ∪H 2  is not a subgroup of G. Youtube Video on this Theorem

Solution of (x² + y² 2xy)dx +2ydy=0 is a non-exact differential EQUATION  In this page, now you have to see that the Question (x^2 + y^2 + 2x)dx + 2ydy = 0 is the non exact differential equation and here is the solution of this question. In our website you have to see that we are updated higher Maths question and videos on it, that why you can not confuse for understand the question. Here, we find the integrating factor of this equation and also we solve the full question. This is the differential equation equation and we will solve it. SOLUTION: (x^2 +y^2 +2x)dx + 2ydy + 0 Comparing this with Mdx + Ndy = 0 Here, M = x^2 + y^2 + 2x and N= 2y dM/dy = 2y and dN/dx = 0 (dm/dy-dn/dx)/N = (2y-0)/2y = 1 or x^0 or f(x) e^integral f(x)dx = e^integral x^0dx = e^integral dx = e^x Now, Multiplying throughout in above equation e^x(x^2 + y^2 + 2x)dx + e^x2ydy = 0 Which is Exact differential equation integral e^x x^2 dx + integral e^xy^2dx + integral e^x 2xdx = c x^2 e^x - integral 2xe^x dx...