G Tewani Solutions for Chapter: Determinant, Exercise 3: DPP

If $u=ax+by+cz, v=ay+bz+cx, w=az+bx+cy$, then the value of  $\left|\begin{array}{lll}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|\times \left|\begin{array}{lll}x& y& z\\ y& z& x\\ z& x& y\end{array}\right|$ is

If the number of positive integral solutions of $u+v+w=n$, is denoted by $P_n$, then the absolute value of $\left|\begin{array}{ccc}{P}_n& P_{n+1}& P_{n+2}\\ P_{n+1}& P_{n+2}& P_{n+3}\\ P_{n+2}& P_{n+3}& P_{n+4}\end{array}\right|$ is

If $f\left(x\right),g\left(x\right)$ and $h\left(x\right)$ are polynomials of degree $4$ and $\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ a& b& c\\ p& q& r\end{array}\right|=m{x}^4+n{x}^3+r{x}^2+sx+t$ be an identity in $x,$then $$$\left|\begin{array}{ccc}{f}^{\text{'}\text{'}\text{'}}\left(0\right)-f^{\text{'}\text{'}}\left(0\right)& g^{\text{'}\text{'}\text{'}}\left(0\right)-g^{\text{'}\text{'}}\left(0\right)& h^{\text{'}\text{'}\text{'}}\left(0\right)-h^{\text{'}\text{'}}\left(0\right)\\ a& b& c\\ p& q& r\end{array}\right|$ is

If $\Delta \left(x\right)=\left|\begin{array}{ccc}x-2& {\left(x-1\right)}^2& x^3\\ x-1& x^2& {\left(x+1\right)}^3\\ x& {\left(x+1\right)}^2& {\left(x+2\right)}^3\end{array}\right|$, the coefficient of $x$ in $\Delta \left(x\right)$ is

If $f\left(x\right)=\left|\begin{array}{ccc}\sin x& \cos x& \tan x\\ x^3& x^2& x\\ 2x& 1& x\end{array}\right|$ , then $\underset{x\to 0}{\lim }\frac{f\left(x\right)}{x^2}=$?

If $x\in R; a_i, b_i, c_i\in R$ for $i=1, 2, 3$ and $\left|\begin{array}{lll}{a}_1+b_{1}x& a_{1}x+b_1& c_1\\ a_2+b_{2}x& a_{2}x+b_2& c_2\\ a_3+b_{3}x& a_{3}x+b_3& c_3\end{array}\right|=0$, then which of the following may be true?

If $a_i,i=1,2,\dots ,9$ are perfect odd squares, then $\left|\begin{array}{lll}{a}_1& a_2& a_3\\ a_4& a_5& a_6\\ a_7& a_8& a_9\end{array}\right|$ is always a multiple of

The value of the determinant $\left|\begin{array}{ccc}\cos (\mathrm{\theta }+\mathrm{\alpha })& -\sin (\mathrm{\theta }+\mathrm{\alpha })& \cos 2\mathrm{\alpha }\\ \mathrm{sin\theta }& \mathrm{cos\theta }& \mathrm{sin\alpha }\\ -\mathrm{cos\theta }& \mathrm{sin\theta }& \mathrm{\lambda cos\alpha }\end{array}\right|$ is