Question

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

medium

$\left| {\,\begin{array}{*{20}{c}}1&1&1\\{bc}&{ca}&{ab}\\{b + c}&{c + a}&{a + b}\end{array}\,} \right|$

$1$

$0$

$(a - b)(b - c)(c - a)$

$(a + b)(b + c)(c + a)$

Solution

(c) $\left| {\,\begin{array}{*{20}{c}}1&1&1\\{bc}&{ca}&{ab}\\{b + c}&{c + a}&{a + b}\end{array}\,} \right|$=$\left| {\,\begin{array}{*{20}{c}}0&0&1\\{c(b – a)}&{a(c – b)}&{ab}\\{b – a}&{c + a}&{a + b}\end{array}\,} \right|$

$\{ {C_1} \to {C_1} – {C_2},\,{C_2} \to {C_2} – {C_3}\} $

= $(b – a)\,\,(c – b)\,\,\left| {\,\begin{array}{*{20}{c}}0&0&1\\c&a&{ab}\\1&1&{a + b}\end{array}\,} \right|$ = $(b – a)\,(c – a)\,\,(c – a)$

$ = (a – b)\,\,(b – c)\,\,(c – a)$.

Std 12

Mathematics

यदि $ab + bc + ca = 0$ और $\left| {\,\begin{array}{*{20}{c}}{a – x}&c&b\\c&{b – x}&a\\b&a&{c – x}\end{array}\,} \right| = 0$, तो $x$ का एक मान होगा

hard

यदि ${U_n} = \left| {\,\begin{array}{*{20}{c}}n&1&5\\{{n^2}}&{2N + 1}&{2N + 1}\\{{n^3}}&{3{N^2}}&{3N}\end{array}\,} \right|$ , तब $\sum\limits_{n = 1}^N {{U_n}} $ का मान है

hard

$\left|\begin{array}{ccc}1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y\end{array}\right|$ का मान ज्ञात कीजिए।

medium

$\left| {\,\begin{array}{*{20}{c}}{441}&{442}&{443}\\{445}&{446}&{447}\\{449}&{450}&{451}\end{array}\,} \right|$ का मान है

easy

यदि $\omega $ इकाई का एक घनमूल हो, तो $\left| {\,\begin{array}{*{20}{c}}{x + 1}&\omega &{{\omega ^2}}\\\omega &{x + {\omega ^2}}&1\\{{\omega ^2}}&1&{x + \omega }\end{array}\,} \right| = $