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### 40 Maths General Knowledge Questions And Answers

1. Who discovered Zero (0)? Answer: Aryabhatta, AD 458 Explanation: Aryabhatta invented zero but he didn’t give any symbol for zero, Brahmagupta was the first to give symbol for zero and rules to compute with zero.

### Find the value of k, for which ​ [ f(x)=begin{cases} frac{sqrt{1+kx}-sqrt{1-kx}}{x}, text{if} -1le x<0 \ frac{2x+1}{x-1}, text{if} 0le x<1 end{cases} ]...

Given that, ​$$f(x)=\begin{cases} \frac{\sqrt{1+kx}-\sqrt{1-kx}}{x}, \text{if} \ -1\le x<0 \\ \frac{2x+1}{x-1}, \text{if} \ 0\le x<1 \end{cases}$$​

### First and Second fundamental theorem of calculus

If ​$$F'(x)=f(x)$$​ then F is called primitive or antiderivative of f. e.g.$$F(x)=x^2sin(\frac{1}{x})$$ ​$$\therefore F'(x)=2xsin(\frac{1}{x})+x^2.cos(\frac{1}{x}).(\frac{-1}{x^2})$$​

### A metric space (X, d) is disconnected iff there exists a non-empty proper open & closed subset of X which is both open and...

Let (X, d) be a metric space and suppose that it is disconnected. Claim: ​$$\exists$$​ a non-empty proper subset of X which is both open & closed.

### Continuous image of connected set is connected

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### A woman starts shopping with Rs. X and Y paise, spends Rs. 3.50 and is left with Rs. 2Y and 2X paise. The amount...

If ​$$Y>50$$​ then according to the information given, ​$$Y-50=2X$$​ and ​$$X-3=2Y$$​

### Linear Functionals, Annihilator and Double Dual

If V is a vector space over the field F and S be a subset of V, the annihilator of S is ​$$S^0$$​ and ​$$S^0$$​ is the set of linear functionals f on V such that ​$$f(\alpha)=0$$​ for each ​$$\alpha \in S$$

### The Matrix of Linear Transformation and Linear Functionals

Let V be n-dimensional vector space over the field F and W be an m-dimensional vector space over the same field F.

### Linear Transformation Part II – Inverse Linear Transformation and Isomorphism

A function T from V into W is called invertible if there exists a function S from W to V such that TS is an identity on W and ST is an identity on V.  i.e. ​$$TS=I_w, ST=I_v$$

### Linear Transformation Part I – Algebra of Linear Transformation

Let ​$$\alpha, \beta \in V$$​ and c \in F and S & T are linear transformation from V to W.  ​$$(S+T)(c\alpha+\beta)$$ $$=S(c\alpha+\beta)+T(c\alpha+\beta)$$

### Cayley Hamilton Theorem

Let K be the commutative ring with identity consisting of all polynomials in T. Choose an ordered basis ​$$\{\alpha_1, \alpha_2, ... ,\alpha_n\}$$​ for V. Let A be the matrix of T in the basis ​$$\{\alpha_1, \alpha_2, ... ,\alpha_n\}$$

### The Matrix of Linear Transformation and Linear Functionals

Let V be n-dimensional vector space over the field F and W be an m-dimensional vector space over the same field F.

### Any two closed subset of metric space are connected iff they are disjoint

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