Maths

This is a Long Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Let\text{?}\text{?}\text{?}\text{?}I={\displaystyle ?\underset{II}{e^{?3x}}}\underset{I}{co{s}^{3}x}dx\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}=co{s}^{3}x.{\displaystyle ?e^{?3x}\text{?}dx}?{\displaystyle ?\left(D\left(co{s}^{3}x\right).{\displaystyle ?e^{?3x}\text{?}dx}\right)dx}\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}=co{s}^{3}x.\frac{e^{?3x}}{?3}?{\displaystyle ?\left(3co{s}^{2}x\left(?sinx\right).\frac{e^{?3x}}{?3}\right)dx}\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}=?\frac{1}{3}{e}^{?3x}co{s}^{3}x?{\displaystyle ?co{s}^{2}xsinx.}{e}^{?3x}dx\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}=?\frac{1}{3}{e}^{?3x}co{s}^{3}x?{\displaystyle ?\left(1?si{n}^{2}x\right)sinx.}{e}^{?3x}dx\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}=?\frac{1}{3}{e}^{?3x}co{s}^{3}x?{\displaystyle ?sinx.e^{?3x}dx}+{\displaystyle ?\underset{I}{si{n}^{3}x}}.\underset{II}{e^{?3x}}dx\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}=?\frac{1}{3}{e}^{?3x}co{s}^{3}x?{\displaystyle ?sinx.e^{?3x}dx}+si{n}^{3}x{\displaystyle ?e^{?3x}dx?{\displaystyle ?\left(D\left(si{n}^{3}x\right).{\displaystyle ?e^{?3x}\text{?}dx}\right)dx}}\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}=?\frac{1}{3}{e}^{?3x}co{s}^{3}x?{\displaystyle ?sinx.e^{?3x}dx}+si{n}^{3}x.\frac{e^{?3x}}{?3}?{\displaystyle ?\left(3si{n}^{2}x.cosx.\frac{e^{?3x}}{?3}\right)dx}\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}=?\frac{1}{3}{e}^{?3x}co{s}^{3}x?{\displaystyle ?sinx.e^{?3x}dx}?\frac{1}{3}{e}^{?3x}si{n}^{3}x+{\displaystyle ?si{n}^{2}xcosx.}{e}^{?3x}dx\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}=?\frac{1}{3}{e}^{?3x}co{s}^{3}x?{\displaystyle ?sinx.e^{?3x}dx}?\frac{1}{3}{e}^{?3x}si{n}^{3}x+{\displaystyle ?\left(1?co{s}^{2}x\right)cosx.}{e}^{?3x}dx\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}I=?\frac{1}{3}{e}^{?3x}co{s}^{3}x?\left[sinx.\frac{e^{?3x}}{?3}?{\displaystyle ?cosx.\frac{e^{?3x}}{?3}dx}\right]?\frac{1}{3}{e}^{?3x}si{n}^{3}x+{\displaystyle ?cosx.e^{?3x}}dx?{\displaystyle ?co{s}^{3}x.e^{?3x}dx}\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}=?\frac{1}{3}{e}^{?3x}co{s}^{3}x+sinx.\frac{e^{?3x}}{3}?{\displaystyle ?cosx.\frac{e^{?3x}}{?3}dx}?\frac{1}{3}{e}^{?3x}si{n}^{3}x+{\displaystyle ?cosx.e^{?3x}}dx?I\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}2I=\frac{e^{?3x}}{?3}\left[co{s}^{3}x+si{n}^{3}x\right]?\left[sinx.\frac{e^{?3x}}{3}?{\displaystyle ?cosx.\frac{e^{?3x}}{?3}dx}\right]+{\displaystyle ?cosx.e^{?3x}}dx\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}=\frac{e^{?3x}}{?3}\left[co{s}^{3}x+si{n}^{3}x\right]+\frac{1}{3}sinx.e^{?3x}?\frac{1}{3}{\displaystyle ?cosx.e^{?3x}}dx+{\displaystyle ?cosx.e^{?3x}}dx\\ ?\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}2I=\frac{e^{?3x}}{?3}\left[co{s}^{3}x+si{n}^{3}x\right]+\frac{1}{3}sinx.e^{?3x}+\frac{2}{3}{\displaystyle ?cosx.e^{?3x}}dx\\ Now,\text{?}{I}_1=\frac{2}{3}{\displaystyle ?\underset{I}{cosx}}.\underset{II}{e^{?3x}}dx\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}=\frac{2}{3}\left[cosx.{\displaystyle ?e^{?3x}dx}?{\displaystyle ?\left(D\left(cosx\right).{\displaystyle ?e^{?3x}\text{?}dx}\right)dx}\right]\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}=\frac{2}{3}\left[cosx.\frac{e^{?3x}}{?3}?{\displaystyle ??sinx.\frac{e^{?3x}}{?3}dx}\right]\\ \text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}=\frac{2}{3}\left[cosx.\frac{e^{?3x}}{?3}?\frac{1}{3}{\displaystyle ?sinx.e^{?3x}dx}\right]\end{array}$

Some of the most frequently tested topics for high scores are finding the adjoint and inverse of a matrix, solving equations using Cramer's Rule, expansion using cofactors, and properties of determinants. If students practice questions based on these topics, they can get high scores in the CBSE Board exam and entrance exams.

In finding inverses of matrices, solving systems of linear equations, and understanding matrix transformations, the determinants play a crucial role. The Determinants concept is widely used in physics, engineering, computer Science and competitive exams like JEE. Understanding determinants is important for higher mathematical studies.