Embibe Experts Solutions for Chapter: Solved Mathematics Paper-1, Exercise 1: Exercise

Show that the function $f:Z\to 2N_{0,}$ given by $f\left(x\right)=2\left|x\right|, x\in Z$ and $N_0=N\cup \left\{0\right\}$   is surjective but not injective.

Find $\int \frac{2x}{\sqrt[3]{x^2+1}}dx$

Solve $\left|\begin{array}{ccc}0& x-a& x-b\\ x+a& 0& x-c\\ x+b& x+c& 0\end{array}\right|=0$.

Solve the differential equation: $xdy-ydx=\sqrt{x^2+y^2}dx$, given that $y=0$  when $x=1$.

Solve the following differential equation :

$\left(x+2y^3\right)\frac{dy}{dx}=y$

An examinee has to answer a question on a multiple choice type test. The answer to this question is either known to him or else he simply guesses at it. The probability that he knows the answer is $\frac{4}{5}$. He is correct in $25\%$ cases when he guesses at the answer. Find the probability that he knows the answer given that he answered it correctly.

Using elementary row operation and elementary column operations separately, find the inverse of $B=\left(\begin{array}{lll}1& 2& 0\\ 0& 1& 2\\ 0& 0& 1\end{array}\right)$.

Using product rule, show that the following four points with given position vectors are coplanar: $-\hat{i}+4\hat{j}-3\hat{k}, 3\hat{i}+2\hat{j}-5\hat{k},-3\hat{i}+8\hat{j}-5\hat{k},-3\hat{i}+2\hat{j}+\hat{k}$.