CBSE CLASS 12 MATHS IMPORTANT QUESTIONS - PROPERTIES OF DETERMINANTS

Here we are providing Important Practice Questions based on Properties of Determinants prepared by Experts for 2019-20 Board Examinations.

Properties Of Determinants - Extra Practice  Questions 

  $$\begin{array}{l}1.\text{Prove that }\left|\begin{array}{ccc}yz-x^2& zx-y^2& xy-z^2\\ zx-y^2& xy-z^2& yz-x^2\\ xy-z^2& yz-x^2& zx-y^2\end{array}\right|\text{ is divisible by }\left(x+y+z\right),\text{ andhence find the quotient}\text{.}\\ \text{2}\text{.}\left|\begin{array}{ccc}{\left(x+y\right)}^2& zx& zy\\ zx& {\left(z+y\right)}^2& xy\\ zy& xy& {\left(z+x\right)}^2\end{array}\right|=2xyz{\left(x+y+z\right)}^3\\ 3.\text{}\text{Show that the }\Delta \text{ABC is an isosceles triangle if the determinant}\\ \left|\begin{array}{ccc}1& 1& 1\\ 1+\mathrm{cosA}& 1+\cos B& 1+\cos C\\ {\cos }^{2}A+\cos A& {\cos }^{2}B+\cos B& {\cos }^{2}C+\cos C\end{array}\right|=0\\ 4.\text{ Write the value of }\left|\begin{array}{ccc}a-b& b-c& c-a\\ b-c& c-a& a-b\\ c-a& a-b& b-c\end{array}\right|\\ 5.\text{ Solve for}x:\left|\begin{array}{ccc}a+x& a-x& a-x\\ a-x& a+x& a-x\\ a-x& a-x& a+x\end{array}\right|=0\\ 6.\left|\begin{array}{ccc}{\left(b+c\right)}^2& a^2& bc\\ {\left(c+a\right)}^2& b^2& ac\\ {\left(a+b\right)}^2& c^2& ab\end{array}\right|=\left(a-b\right)\left(b-c\right)\left(a+b+c\right)\left(a^2+b^2+c^2\right)\\ 7.\text{ If a, b and c all non -zero and }\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=0,\text{ then prove that}\\ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1=0\\ 8.\text{ if f}\left(x\right)=\left|\begin{array}{ccc}a& -1& 0\\ ax& a& -1\\ a{x}^2& ax& a\end{array}\right|,\text{ using properties of determinant, find the value f}\left(2x\right)-f\left(x\right).\\ 9.\left|\begin{array}{ccc}{a}^3& 2& a\\ b^3& 2& b\\ c^3& 2& c\end{array}\right|=2\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a+b+c\right).\end{array}$$ $$\begin{array}{l}10.\text{ Write the value of }\left|\begin{array}{ccc}x+y& y+z& z+x\\ z& x& y\\ -3& -3& -3\end{array}\right|.\\ 11.\left|\begin{array}{ccc}{a}^2& bc& ac+c^2\\ a^2+ab& b^2& ac\\ ab& b^2+bc& c^2\end{array}\right|=4a^2{b}^2{c}^2.\\ 12.\text{If }a+b+c\ne 0and\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|,\text{ then prove that with properties of determinant a = b = c}\text{.}\\ \text{13}\text{.}\left|\begin{array}{ccc}1+a^2-b^2& 2ab& -2b\\ 2ab& 1-a^2+b^2& 2a\\ 2b& -2a& 1-a^2-b^2\end{array}\right|={\left(1+a^2+b^2\right)}^3\\ 14.\left|\begin{array}{ccc}x+2& x+6& x-1\\ x+6& x-1& x+2\\ x-1& x+2& x+6\end{array}\right|=0,\text{ find the value of }x.\\ 15.\left|\begin{array}{ccc}1& x& x+1\\ 2x& x\left(x-1\right)& x(x+1)\\ 3x\left(1-x\right)& x\left(x-1\right)\left(x-2\right)& x\left(x+1\right)\left(x-1\right)\end{array}\right|=6x^2\left(1-x^2\right).\\ 16.\left|\begin{array}{ccc}1& a& a^2\\ a^2& 1& a\\ a& a^2& 1\end{array}\right|={\left(1-a^3\right)}^2\\ 17.\left|\begin{array}{ccc}2y& y-z-x& 2y\\ 2z& 2z& x-y-z\\ x-y-z& 2x& 2x\end{array}\right|={\left(x+y+z\right)}^3\\ 18.\left|\begin{array}{ccc}a+b+2c& a& b\\ c& b+c+2a& b\\ c& a& c+a+2b\end{array}\right|=2{\left(a+b+c\right)}^2\\ 19.\left|\begin{array}{ccc}{x}^2+1& xy& xz\\ xy& y^2+1& yz\\ xz& yz& z^2+1\end{array}\right|=1+x^2+y^2+z^2\\ 20.\left|\begin{array}{ccc}x+y& x& x\\ 5x+4y& 4x& 2x\\ 10x+8y& 8x& 3x\end{array}\right|=x^3\end{array}$$ $\begin{array}{l}21.\left|\begin{array}{ccc}b+c& c+a& a+b\\ q+r& r+p& p+q\\ y+z& z+x& x+y\end{array}\right|=2\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ x& y& z\end{array}\right|.\\ 22.\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=abc+bc+ca+ab\\ 23.\left|\begin{array}{ccc}a+x& y& z\\ x& a+y& z\\ x& y& a+z\end{array}\right|=a^2\left(a+x+y+z\right).\\ 24.\left|\begin{array}{ccc}x+\lambda & 2x& 2x\\ 2x& x+\lambda & 2x\\ 2x& 2x& x+\lambda \end{array}\right|=\left(5x+\lambda \right){\left(\lambda -x\right)}^2.\\ 25.\left|\begin{array}{ccc}{a}^2& bc& ac+c^2\\ a^2+ab& b^2& ac\\ ab& b^2+bc& c^2\end{array}\right|=4a^2{b}^2{c}^2\\ 26.\left|\begin{array}{ccc}1& x& x^2\\ x^2& 1& x\\ x& x^2& 1\end{array}\right|={\left(1-x^3\right)}^2.\\ 27.\left|\begin{array}{ccc}x& x+y& x+2y\\ x+2y& x& x+y\\ x+y& x+2y& x\end{array}\right|=9y^2\left(x+y\right).\\ 28.\left|\begin{array}{ccc}3x& -x+y& -x+z\\ x-y& 3y& z-y\\ x-z& y-z& 3z\end{array}\right|=3\left(x+y+z\right)\left(xy+yz+zx\right).\\ 29.\left|\begin{array}{ccc}{a}^2+1& ab& ac\\ ab& b^2+1& bc\\ ca& cb& c^2+1\end{array}\right|=1+a^2+b^2+c^2.\\ 30.\left|\begin{array}{ccc}1+a^2-b^2& 2ab& -2b\\ 2ab& 1-a^2+b^2& 2a\\ 2b& -2a& 1-a^2-b^2\end{array}\right|={\left(1+a^2+b^2\right)}^3\\ 31.\left|\begin{array}{ccc}b+c& q+r& y+z\\ c+a& r+p& z+x\\ a+b& p+q& x+y\end{array}\right|=2\left|\begin{array}{ccc}a& p& x\\ b& q& y\\ c& r& z\end{array}\right|\end{array}$ $\begin{array}{l}32.\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^3& b^3& c^3\end{array}\right|=\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a+b+c\right).\\ 33.{\text{If a, b, c are the p}}^{th},\text{}\text{}\text{}{q}^{th}and{r}^{th}\text{terms respectively of a G}\text{.P}\text{., prove that}\\ \left|\begin{array}{ccc}\log a& p& 1\\ \log b& q& 1\\ \log c& r& 1\end{array}\right|\text{ =0}\end{array}$

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