A linear equation can be written as $ax+b=0$, where $a$ and $b$ are constants. Write the equation $3(x-1)=5(2x+1)$ in the form of $ax+b=0$.

Bala's teacher asked him what should be added to $\frac{-3}{7}$ so that the sum is the multiplicative inverse of, $\frac{-3}{7}+\frac{1}{3}$?

Which of these numbers is the correct answer to the problem Bala is trying to solve?

Rahul has a machine which does operations on rational numbers. He feeds four rational numbers, $a, b, c$, and $d$ into the machine. The machine did the operations as per the figure below and gave number $e$, which Rahul says is a rational number.$$

Which of these reasons could justifiy Rahul?

Hari is solving the sum his teacher gave him. The sum is, $-3\frac{1}{2}+1\frac{1}{2}$. Shown below the Hari's steps to find the sum. Check the steps and answer the question below.

Step $\left(1\right)$: $-3\frac{1}{2}+1\frac{1}{2}=\left(1\frac{1}{2}\right)+\left(-3\frac{1}{2}\right)$

Step $\left(2\right)$: $\left(1\frac{1}{2}\right)+\left(-3\frac{1}{2}\right)=\left(1\frac{1}{2}\right)-\left(3\frac{1}{2}\right)$

Step $\left(3\right)$: $\left(1\frac{1}{2}\right)-\left(3\frac{1}{2}\right)=\left(3\frac{1}{2}\right)-\left(1\frac{1}{2}\right)$

Step $\left(4\right)$: $\left(3\frac{1}{2}\right)-\left(1\frac{1}{2}\right)=2$

Hari's friend says there is a mistake in his steps. In which of these steps the error is introduced?

Mahesh's computer program accepts rational numbers along with the operations to be done on them. The program gives the final value. The schematic diagram of a set of variables representing rational numbers and operations loaded on to the machine is given below.

Which of these expressions is the output of the computer program?

Meena tries to solve the problem $-4\frac{2}{3}\times 3\frac{2}{3}$. She converted both into decimal form, $-4\frac{2}{3}=-4.\overline{6}$ and $3\frac{2}{3}=3.\overline{6}$ and multipled the decimal numbers, to get, $-4.\overline{6} \times 3.\overline{6} \approx -16.56$ 

To verify the sum, she wrote the steps.

Step $\left(\mathrm{i}\right)$ $-4\frac{2}{3}\times 3\frac{2}{3}=\left(-4+\frac{2}{3}\right)\times \left(3+\frac{2}{3}\right)$

Step $\left(\mathrm{ii}\right) =-4\times \left(3+\frac{2}{3}\right)+\frac{2}{3}\times \left(3+\frac{2}{3}\right)$

Step $\left(\mathrm{iii}\right) =-4\times 3+-4\times \frac{2}{3}+\frac{2}{3}\times 3+\frac{2}{3}\times \frac{2}{3}$

Step $\left(\mathrm{iv}\right) =-12+\frac{-8}{3}+2+\frac{4}{9}=-10+\frac{-24}{9}+\frac{4}{9}$$=-10+\frac{-20}{9}\approx -12.2$

The verification failed as the answers do not match. Diagnose the step where the error is introduced.