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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...

Proof of Consistency Condition for Linear System Proof of Consistency Condition for Linear System To prove that a necessary and sufficient condition for the system of linear equations \(AX = B\) to be consistent is that the matrices \(A\) and \([A \mid B]\) have the same rank, we will use the rank criterion for the consistency of linear systems. Necessary Condition First, we show that if the system \(AX = B\) is consistent, then \(\text{rank}(A) = \text{rank}([A \mid B])\). System is consistent: This means there exists a solution \(X\) such that \(AX = B\). Augmented Matrix Representation: The system \(AX = B\) can be written as an augmented matrix \([A \mid B]\). Implication for ranks: Since \(AX = B\) has a solution, the vector \(B\) lies in the column space of \(A\). Therefore, the addition of...

Intersection and Union of Subspaces Intersection of Two Subspaces is a Subspace Proof: Let \( S \) and \( T \) be two subspaces of a vector space \( V \) over a field \( F \). We shall prove that \( S \cap T \) is also a subspace of \( V \). Let \( \alpha, \beta \in F \) and \( x, y \in S \cap T \). Then \( x, y \in S \) and \( x, y \in T \). Since \( S \) and \( T \) are subspaces of \( V \), we have: \[ \alpha x + \beta y \in S \] \[ \alpha x + \beta y \in T \] Therefore, \[ \alpha x + \beta y \in S \cap T \] Thus, \( \alpha x + \beta y \in S \cap T \) for all \( \alpha, \beta \in F \) and \( x, y \in S \cap T \). Hence, \( S \cap T \) is a subspace of \( V \). Union of Two Subspaces Need Not Be a Subspace Now, by giving an example we shall prove that the union of two subspaces need not be a subspace. Example: Let \( V = \mathbb{R}^3 = \{(x, y, z) : x, y, z \in \mathbb{R}\} \) be a vector space over the field \( \mathbb{R} \) of all rea...

Liouville's Theorem Art-15. Liouville's Theorem This theorem is very important, if you read this theorem carefully then some of the theorems coming up are based on this on which you can apply it and there are many questions where you can apply this theorem and you can solve those questions with Liouville's Theorem . Statement: If a function \( f(z) \) is analytic for all finite values of \( z \) and is bounded, then \( f(z) \) is constant. Proof: Let \( z_1, z_2 \) be any two points of the \( z \)-plane. Draw a circle \( C \) with center at the origin and radius \( R \), enclosing the points \( z_1 \) and \( z_2 \) so that: \( |z_1| < R \) and \( |z_2| < R \) Since \( f(z) \) is bounded, there exists \( M > 0 \) such that \( |f(z)| \leq M \) on \( C \). By Cauchy's integral formula, we have \( f(z_1) = \frac{1}{2\pi i} \int_C \frac{f(z)}{z - z_1} \, dz \) and \( f(...

It is very important to understand Rule of Divisibility because it is a way that a student can do faster calculations in their exam.  Divisibility rule are the shortcut way to determine that if a number is divisible by another number or not. It can enhance our calculation speed and accuracy and we can find the answer with in a second. Rule of Divisibility by 2: If the unit digit of a number is any 0,2,4,6 and 8 then that number will be completely divisible by 2.  Let's take a numbers:  98973232 8878438 893830 78733456 8768762494 Above, all the number are completely divisible by 2.  Let's take a number: 98973232 ÷ 2 = 49486616. Now, let's take a number where one's place have different number which is not divisible by 2.  For Example: 897987 ÷ 2 = 448993.5. Let's Understand from YouTube Rule of Divisibility by 3: If the sum of all the digits of the given number is completely divisible by...

  In this page, we are going to analysis the important topic or theorem in Complex Analysis.    Fundamental Theorem of Algebra in Complex Analysis is a very important topic to understand that can enhance your Higher Mathematics knowledge. Fundamental Theorem of Algebra in Complex Analysis Statement: If f(z) =  a 0 z n + a 1 z n-1 + ... + a n , a 0 ≠ 0 be a polynomial in z, then f(z) = 0 has at least one root in z-plane. Proof: If possible, let f(z) = 0 has no root in z-plane. Therefore, f(z)  ≠ 0 for any value of z in z-plane. We define, because, f(z) is analytic and f(z)  ≠ 0 therefore, F(z) is analytic for all values of z. Also, |F(z)| --> 0 as |z| -->  ∞  therefore, F(z) is bounded therefore, By Liouville's Theorem, F(z) is constant. ⇒ f(z) is constant, which is contradiction [because, f(z) is not constant polynomial] therefore, f(z) = 0 has at least one root in z-plane. Explanation in YouTube: