Determinants

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.

The exemplar is beyond the NCERT textbook as it is conceptually more deep and challenging than the NCERT textbook exercises. These MCQs, LA, and SA questions help students improve their logical reasoning and help them prepare for competitive exams. It tests students on higher-order and application-based questions and promotes analytical thinking.

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}True.\\ Let\Delta =\left|\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill \left(1+sin\theta \right)\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1+cos\theta \hfill \end{array}\right|\\ C_1\to C_1-C_2,C_2\to C_2-C_3\\ =\left|\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill -sin\theta \hfill & \hfill sin\theta \hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill -cos\theta \hfill & \hfill 1+cos\theta \hfill \end{array}\right|\\ Expandingalong{C}_3\\ =1\left|\begin{array}{cc}\hfill -sin\theta \hfill & \hfill sin\theta \hfill \\ \hfill 0\hfill & \hfill -cos\theta \hfill \end{array}\right|=sin\theta cos\theta -0=sin\theta cos\theta \\ =\frac{1}{2}.2sin\theta cos\theta =\frac{1}{2}sin2\theta \\ =\frac{1}{2}*1=\frac{1}{2}\left[Maximumvalueofsin2\theta =1\right]\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}True.\\ Giventhat\Delta =\left|\begin{array}{ccc}\hfill a\hfill & \hfill p\hfill & \hfill x\hfill \\ \hfill b\hfill & \hfill q\hfill & \hfill y\hfill \\ \hfill c\hfill & \hfill r\hfill & \hfill z\hfill \end{array}\right|=16\\ L.H.S.{\Delta }_1=\left|\begin{array}{ccc}\hfill p+x\hfill & \hfill a+x\hfill & \hfill a+p\hfill \\ \hfill q+y\hfill & \hfill b+y\hfill & \hfill b+q\hfill \\ \hfill r+z\hfill & \hfill c+z\hfill & \hfill c+r\hfill \end{array}\right|\\ C_1\to C_1+C_2+C_3\\ =\left|\begin{array}{ccc}\hfill 2p+2x+2a\hfill & \hfill a+x\hfill & \hfill a+p\hfill \\ \hfill 2q+2y+2b\hfill & \hfill b+y\hfill & \hfill b+q\hfill \\ \hfill 2r+2z+2c\hfill & \hfill c+z\hfill & \hfill c+r\hfill \end{array}\right|\\ =2\left|\begin{array}{ccc}\hfill p+x+a\hfill & \hfill a+x\hfill & \hfill a+p\hfill \\ \hfill q+y+b\hfill & \hfill b+y\hfill & \hfill b+q\hfill \\ \hfill r+z+c\hfill & \hfill c+z\hfill & \hfill c+r\hfill \end{array}\right|\left[Taking2commonfrom{C}_1\right]\\ C_1\to C_1-C_2\\ =2\left|\begin{array}{ccc}\hfill p\hfill & \hfill a+x\hfill & \hfill a+p\hfill \\ \hfill q\hfill & \hfill b+y\hfill & \hfill b+q\hfill \\ \hfill r\hfill & \hfill c+z\hfill & \hfill c+r\hfill \end{array}\right|\\ C_3\to C_3-C_1\\ =2\left|\begin{array}{ccc}\hfill p\hfill & \hfill a+x\hfill & \hfill a\hfill \\ \hfill q\hfill & \hfill b+y\hfill & \hfill b\hfill \\ \hfill r\hfill & \hfill c+z\hfill & \hfill c\hfill \end{array}\right|\\ Splittingup{C}_2\\ =2\left|\begin{array}{ccc}\hfill p\hfill & \hfill a\hfill & \hfill a\hfill \\ \hfill q\hfill & \hfill b\hfill & \hfill b\hfill \\ \hfill r\hfill & \hfill c\hfill & \hfill c\hfill \end{array}\right|+2\left|\begin{array}{ccc}\hfill p\hfill & \hfill x\hfill & \hfill a\hfill \\ \hfill q\hfill & \hfill y\hfill & \hfill b\hfill \\ \hfill r\hfill & \hfill z\hfill & \hfill c\hfill \end{array}\right|=2\left(0\right)+2\left|\begin{array}{ccc}\hfill p\hfill & \hfill x\hfill & \hfill a\hfill \\ \hfill q\hfill & \hfill y\hfill & \hfill b\hfill \\ \hfill r\hfill & \hfill z\hfill & \hfill c\hfill \end{array}\right|\\ =2\left|\begin{array}{ccc}\hfill p\hfill & \hfill x\hfill & \hfill a\hfill \\ \hfill q\hfill & \hfill y\hfill & \hfill b\hfill \\ \hfill r\hfill & \hfill z\hfill & \hfill c\hfill \end{array}\right|\Rightarrow 2\left|\begin{array}{ccc}\hfill a\hfill & \hfill p\hfill & \hfill x\hfill \\ \hfill b\hfill & \hfill q\hfill & \hfill y\hfill \\ \hfill c\hfill & \hfill r\hfill & \hfill z\hfill \end{array}\right|\left(C_1\leftrightarrow C_{3}and{C}_2\leftrightarrow C_3\right)\\ =2*16=32\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}True.\\ Let\Delta =\left|\begin{array}{ccc}\hfill x+a\hfill & \hfill p+u\hfill & \hfill l+f\hfill \\ \hfill y+b\hfill & \hfill q+v\hfill & \hfill m+g\hfill \\ \hfill z+c\hfill & \hfill r+w\hfill & \hfill n+h\hfill \end{array}\right|\\ Splittingup{C}_1\\ =\left|\begin{array}{ccc}\hfill x\hfill & \hfill p+u\hfill & \hfill l+f\hfill \\ \hfill y\hfill & \hfill q+v\hfill & \hfill m+g\hfill \\ \hfill z\hfill & \hfill r+w\hfill & \hfill n+h\hfill \end{array}\right|+\left|\begin{array}{ccc}\hfill a\hfill & \hfill p+u\hfill & \hfill l+f\hfill \\ \hfill b\hfill & \hfill q+v\hfill & \hfill m+g\hfill \\ \hfill c\hfill & \hfill r+w\hfill & \hfill n+h\hfill \end{array}\right|\\ Splittingup{C}_{2}inboth\text{\hspace{0.17em}determinants}\\ =\left|\begin{array}{ccc}\hfill x\hfill & \hfill p\hfill & \hfill l+f\hfill \\ \hfill y\hfill & \hfill q\hfill & \hfill m+g\hfill \\ \hfill z\hfill & \hfill r\hfill & \hfill n+h\hfill \end{array}\right|+\left|\begin{array}{ccc}\hfill x\hfill & \hfill u\hfill & \hfill l+f\hfill \\ \hfill y\hfill & \hfill v\hfill & \hfill m+g\hfill \\ \hfill z\hfill & \hfill w\hfill & \hfill n+h\hfill \end{array}\right|+\left|\begin{array}{ccc}\hfill a\hfill & \hfill p\hfill & \hfill l+f\hfill \\ \hfill b\hfill & \hfill q\hfill & \hfill m+g\hfill \\ \hfill c\hfill & \hfill r\hfill & \hfill n+h\hfill \end{array}\right|+\left|\begin{array}{ccc}\hfill a\hfill & \hfill u\hfill & \hfill l+f\hfill \\ \hfill b\hfill & \hfill v\hfill & \hfill m+g\hfill \\ \hfill c\hfill & \hfill w\hfill & \hfill n+h\hfill \end{array}\right|\\ Similarlybysplitting{C}_{3}ineach\text{\hspace{0.17em}determinants},wewillget\mathrm{8determinants}.\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}True.\\ Let\Delta =\left|\begin{array}{ccc}\hfill sinA\hfill & \hfill cosA\hfill & \hfill sinA+cosB\hfill \\ \hfill sinB\hfill & \hfill cosA\hfill & \hfill sinB+cosB\hfill \\ \hfill sinC\hfill & \hfill cosA\hfill & \hfill sinC+cosB\hfill \end{array}\right|\\ Splittingup{C}_3\\ =\left|\begin{array}{ccc}\hfill sinA\hfill & \hfill cosA\hfill & \hfill sinA\hfill \\ \hfill sinB\hfill & \hfill cosA\hfill & \hfill sinB\hfill \\ \hfill sinC\hfill & \hfill cosA\hfill & \hfill sinC\hfill \end{array}\right|+\left|\begin{array}{ccc}\hfill sinA\hfill & \hfill cosA\hfill & \hfill cosB\hfill \\ \hfill sinB\hfill & \hfill cosA\hfill & \hfill cosB\hfill \\ \hfill sinC\hfill & \hfill cosA\hfill & \hfill cosB\hfill \end{array}\right|\\ =0+\left|\begin{array}{ccc}\hfill sinA\hfill & \hfill cosA\hfill & \hfill cosB\hfill \\ \hfill sinB\hfill & \hfill cosA\hfill & \hfill cosB\hfill \\ \hfill sinC\hfill & \hfill cosA\hfill & \hfill cosB\hfill \end{array}\right|\left[?C_{1}and{C}_{3}areidentical\right]\\ =cosAcosB\left|\begin{array}{ccc}\hfill sinA\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill sinB\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill sinC\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right|\left[TakingcosAandcosBcommonfrom{C}_{2}and{C}_{3}respectively\right]\\ =cosAcosB\left(0\right)\left[?C_{2}and{C}_{3}areidentical\right]\\ =0\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}False.\\ Since?adjA?=?A{?}^{n-1}wherenistheorderofthesquarematrix.\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}True.\\ Let\Delta =\left|\begin{array}{ccc}\hfill x+1\hfill & \hfill x+2\hfill & \hfill x+a\hfill \\ \hfill x+2\hfill & \hfill x+3\hfill & \hfill x+b\hfill \\ \hfill x+3\hfill & \hfill x+4\hfill & \hfill x+c\hfill \end{array}\right|\\ R_2\to 2R_2-\left(R_1+R_3\right)\\ =\left|\begin{array}{ccc}\hfill x+1\hfill & \hfill x+2\hfill & \hfill x+a\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2b-\left(a+c\right)\hfill \\ \hfill x+3\hfill & \hfill x+4\hfill & \hfill x+c\hfill \end{array}\right|\\ a,b,careinA.P.\\ \therefore b-a=c-b\Rightarrow 2b=a+c\\ =\left|\begin{array}{ccc}\hfill x+1\hfill & \hfill x+2\hfill & \hfill x+a\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill x+3\hfill & \hfill x+4\hfill & \hfill x+c\hfill \end{array}\right|=0\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}True.\\ \text{\hspace{0.17em}Since}\left|A\right|=12\\ IfAisasquarematrixofordern\\ then\left|AdjA\right|={\left|A\right|}^{n-1}\\ \therefore \left|AdjA\right|={\left|A\right|}^{3-1}={\left|A\right|}^2={\left(12\right)}^2=144\left[n=3\right]\end{array}$

This is a  Fill in the blanks as classified in NCERT Exemplar

Sol:

$\begin{array}{l}True.\\ \left|3AB\right|=3^3\left|AB\right|=27\left|A\right|\left|B\right|=27*5*3\left[?\left|KA\right|=K^n\left|A\right|\right]\end{array}$