NCERT Solutions Class 12 Maths Chapter-9 (Determinants) Exercise 9.1

NCERT Solutions Class 12 Maths from class 12th Students will get the answers of Chapter-9 (Determinants)Exercise 9.1 This chapter will help you to learn the basics and you should expect at least one question in your exam from this chapter.
We have given the answers of all the questions of NCERT Board Mathematics Textbook in very easy language, which will be very easy for the students to understand and remember so that you can pass with good marks in your examination.

NCERT Question-Answer

Class 12 Mathematics

Chapter-9 (Determinants)

Questions and answers given in practice

Chapter-9 (Determinants)

Exercise 9.1

Q1. Determine order and degree(if defined) of differential equation $\frac{d^{4}y}{d{x}^4}+\sin \left(y^{\prime \prime \prime }\right)=0$

Answer. $\frac{d^{4}y}{d{x}^4}+\sin \left(y^{\prime \prime \prime }\right)=0$ $\Rightarrow y^{\prime \prime \prime \prime }+\sin \left(y^m\right)=0$ The highest order derivative present in the differential equation is $y^{\prime \prime \prime \prime }$. Therefore, its order is four. The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.

Q2. Determine order and degree(if defined) of differential equation $y^{\prime }+5y=0$

Answer. $y^{\prime }+5y=0$ $\begin{array}{c}\text{The highest order derivative present in the differential}\\ \text{equation is}{y}^{\prime }.\text{Therefore, its order is one.}\\ \text{It is a polynomial equation in}{y}^{\prime }\text{. The highest power raised to}\\ y^{\prime }\text{is}1.\text{Hence, its degree is one.}\end{array}$

Q3. Determine order and degree(if defined) of differential equation $\left(\begin{array}{c}(\frac{ds}{dt}{)}^4+3s\frac{d^{2}s}{d{t}^2}=0\end{array}\right).$

Answer. $\left(\begin{array}{c}(\frac{ds}{dt}{)}^4+3s\frac{d^{2}s}{d{t}^2}=0\end{array}\right).$ $\begin{array}{c}\text{The highest order derivative present in the given differential equation is}\frac{d^{2}s}{d{t}^2}\text{. Therefore,}\\ \text{its order is two.}\\ \text{It is a polynomial equation in}\frac{d^{2}s}{d{t}^2}\text{and}\frac{ds}{dt}\text{. The power raised to}\frac{d^{2}s}{d{t}^2}\text{is}1\\ \text{Hence, its degree is one.}\end{array}$

 $\begin{array}{c}\text{Q4. Determine order and degree(if defined) of differential}\\ {\left(\frac{d^{2}y}{d{x}^2}\right)}^2+\cos \left(\frac{dy}{dx}\right)=0\end{array}$

Answer. $\begin{array}{c}{\left(\frac{d^{2}y}{d{x}^2}\right)}^2+\cos \left(\frac{dy}{dx}\right)=0\\ \text{The highest order derivative present in the given differential}\\ \text{equation is}\frac{d^{2}y}{d{x}^2}\text{. Therefore, its order is}2\text{.}\\ \text{The given differential equation is not a polynomial equation}\\ \text{in its derivatives. Hence, its degree is not defined.}\end{array}$

Q5. $\begin{array}{c}\text{Determine order and degree(if defined) of differential equation}\\ \frac{d^{2}y}{d{x}^2}=\cos 3x+\sin 3x\end{array}$

Answer. $\begin{array}{c}\frac{d^{2}y}{d{x}^2}=\cos 3x+\sin 3x\\ \Rightarrow \frac{d^{2}y}{d{x}^2}-\cos 3x-\sin 3x=0\\ \text{The highest order derivative present in the differential.}\\ \text{equation is}\frac{d^{2}y}{d{x}^2}\text{. Therefore, its order is two.}\\ \text{It is a polynomial equation in}d{x}^2\text{and the power raised to}\frac{d^{2}y}{d{x}^2}\\ \text{Hence, its degree is one.}\end{array}$

 $\begin{array}{c}\text{Q6. Determine order and degree(if defined) of differential}\\ \text{equation}{\left(y^{\prime \prime }\right)}^2+{\left(y^{\prime \prime }\right)}^3+{\left(y^{\prime }\right)}^4+y^5=0\end{array}$

Answer. $\begin{array}{c}{\left(y^{\prime \prime }\right)}^2+{\left(y^{\prime }\right)}^3+\left(y^{\prime }\right)+y^5=0\\ \text{The highest order derivative present in the differential equation is}{y}^{\prime \prime }.\text{Therefore, its}\\ \text{order is three.}\\ \text{The given differential equation is a polynomial equation in}{y}^{\prime \prime },y^{\prime },\text{and}{y}^{\prime }\end{array}$ $\begin{array}{c}\text{The highest power raised to}{y}^{\prime \prime \prime }\text{is}2\text{. Hence, its degree is}2\text{.}\end{array}$

Q7. $\begin{array}{c}\text{Determine order and degree(if defined) of differential}\\ \text{equation}{y}^{\prime \prime \prime }+2y^{\prime \prime }+y^{\prime }=0\end{array}$

Answer. $\begin{array}{c}{y}^{\prime \prime }+2y^{\prime \prime }+y^{\prime }=0\\ \text{The highest order derivative present in the differential}\\ \text{equation is}{y}^{\prime \prime \prime }.\text{Therefore, its order is three.}\end{array}$ $\begin{array}{c}\text{It is a polynomial equation in}{y}^{\prime \prime \prime },y^{\prime \prime }\text{and}{y}^{\prime }\text{. The highest power}\\ \text{raised to}{y}^{\prime \prime \prime }\text{is}1.\text{Hence, its degree is}1\text{.}\end{array}$

Q8. $\begin{array}{c}\text{Determine order and degree(if defined) of differential}\\ \text{equation}{y}^{\prime }+y=e^x\end{array}$

Answer. $\begin{array}{c}{y}^{\prime }+y=e^x\\ \Rightarrow y^{\prime }+y-e^x=0\\ \text{The highest order derivative present in the differential}\\ \text{equation is}{y}^{\prime }\text{. Therefore, its order is one.}\\ \text{The given differential equation is a polynomial equation in}{y}^{\prime }\\ \text{and the highest power raised to}{y}^{\prime }\text{is one. Hence, its degree is}\\ \text{one.}\end{array}$

Q9. $\begin{array}{c}\text{Determine order and degree(if defined) of differential}\\ \text{equation}{y}^{\prime \prime }+{\left(y^{\prime }\right)}^2+2y=0\end{array}$

Answer. $\begin{array}{c}{y}^{\prime \prime }+{\left(y^{\prime }\right)}^2+2y=0\\ \text{The highest order derivative present in the differential}\\ \text{equation is}{y}^{\prime \prime }.\text{Therefore, its order is two.}\\ \text{The given differential equation is a polynomial equation in}{y}^{\prime \prime }\\ \text{Hence, its degree is one.}\end{array}$

Q10. $\begin{array}{c}\text{Determine order and degree(if defined) of differential}\\ \text{equation}{y}^{\prime \prime }+2y^{\prime }+\sin y=0\end{array}$

Answer. $\begin{array}{c}{y}^{\prime \prime }+2y^{\prime }+\sin y=0\\ \text{The highest order derivative present in the differential}\\ \text{equation is}{y}^{\prime \prime }.\text{Therefore, its order is two.}\\ \text{This is a polynomial equation in}{y}^{\prime \prime }\text{and}{y}^{\prime }\text{and the highest}\\ \text{power raised to}{y}^{\prime \prime }\text{is one. Hence, its degree is one.}\end{array}$

Q11. $\begin{array}{c}\text{The degree of the differential equation}\\ {\left(\frac{d^{2}y}{d{x}^2}\right)}^3+{\left(\frac{dy}{dx}\right)}^2+\sin \left(\frac{dy}{dx}\right)+1=0\\ \left(\text{A)}3\right(B\left)2\right(C\left)1\right(D)\text{not defined}\end{array}$

Answer. $\begin{array}{c}{\left(\frac{d^{2}y}{d{x}^2}\right)}^3+{\left(\frac{dy}{dx}\right)}^2+\sin \left(\frac{dy}{dx}\right)+1=0\\ \text{The given differential equation is not a polynomial equation}\\ \text{in its derivatives. Therefore, its degree is not defined.}\\ \text{Hence, the correct answer is D.}\end{array}$

Q12. $\begin{array}{c}\text{The order of the differential equation}\\ 2x^2\frac{d^{2}y}{d{x}^2}-3\frac{dy}{dx}+y=0\\ \left(A\right)2\left(B\right)1\left(C\right)0\left(D\right)\text{not defined}\end{array}$

Answer. $\begin{array}{c}2x^2\frac{d^{2}y}{d{x}^2}-3\frac{dy}{dx}+y=0\\ \text{The highest order derivative present in the given differential}\\ \text{equation is}\frac{d^{2}y}{d{x}^2}\text{. Therefore, its order is two.}\\ \text{Hence, the correct answer is A.}\end{array}$

Chapter-9 (Determinants)

• Exercise 9.2
• Exercise 9.3
• Exercise 9.4
• Exercise 9.5
• Exercise 9.6