List of top Mathematics Questions on Determinants asked in CBSE CLASS XII

(b) Given \[ A = \begin{bmatrix} 2 & 3 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2 \end{bmatrix}, \text{ find } A^{-1}. \text{ Use it to solve the following system of equations:} \] \[ x + y + z = 1 \] \[ 2x + 3y + 2z = 2 \] \[ x + y + 2z = 4. \]

If \( \begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix} = kabc \), then the value of \( k \) is:

If \( \begin{vmatrix} -a & b & c
a & -b & c
a & b & -c \end{vmatrix} = kabc \), then the value of \( k \) is:

The cost of 4 kg onion,3 kg wheat and 2kg rice is Rs 60.The cost of 2 kg onion,4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.

Evaluate\(\begin{vmatrix} 1 & x & y\\ 1 & x+y & y\\1&x&x+y \end{vmatrix}\)

Let \(A=\begin{bmatrix}1&-2&1\\ -2&3&1\\ 1&1&5\end{bmatrix}\)verify that -
\((i)[adjA]^{-1}=adj(A^{-1})\)
\((ii)(A^{-1})^{-1}=A\)

Without expanding the determinant, prove that
\(\begin{vmatrix} a &a^2 &bc \\ b&b^2 &ca \\ c&c^2 &ab \end{vmatrix}=\begin{vmatrix} 1 &a^2 &a^3 \\ 1&b^2 &b^3 \\ 1&c^2 &c^3 \end{vmatrix}\)

Using the property of determinants and without expanding, prove that: \(\begin{vmatrix}a-b&b-c&c-a\\b-c&c-a&a-b\\c-a&a-b&b-c\end{vmatrix}\)=0

If A=\(\begin{bmatrix}3&1\\-1&2\end{bmatrix}\),show that A²-5A+7I=0.Hence find A^-1.

If A=\(\begin{bmatrix}1&1&-2\\2&1&-3\\5&4&-9\end{bmatrix}\),find IAI.

By using properties of determinants,show that:
(i)\(\begin{vmatrix} a-b-c & 2a & 2a\\ 2b & b-c-a & 2b \\ 2c&2c&c-a-b\end{vmatrix}\)=(a+b+c)³
(ii)\(\begin{vmatrix} x+y+2z & x & y\\ z & y+z+2x & y\\ z &x&z+x+2y \end{vmatrix}\)=2(x+y+z)³

By using properties of determinants, show that:
(i)\(\begin{vmatrix} x+4 & 2x & 2x\\ 2x & x+4 & 2x \\2x &2x&x+4\end{vmatrix}\)=(5x+4)(4-x)²
(II)\(\begin{vmatrix} y+k & y & y\\ y & y+k & y \\y &y&y+k\end{vmatrix}\)=k²(3y+k)

By using properties of determinants,show that:I\(\begin{vmatrix} x & x^2 & yz\\ y & y^2 & zx\\z&z^2&xy \end{vmatrix}\)=(x-y)(y-z)(z-x)(xy+yz+zx)

Using the property of determinants and without expanding,prove that:\(\begin{vmatrix} b+c & q+r & y+z\\ c+a & r+p & z+x\\ a+b&p+q&x+y \end{vmatrix}\)=2\(\begin{vmatrix} a & p & x\\ b & q & y\\c&r&z \end{vmatrix}\)

Choose the correct answer.
If \(a,b,c\),are in A.P.,then the determinant
\(\begin{vmatrix}x+2& x+3& x+2a\\ x+3& x+4& x+2b\\ x+4& x+5& x+2c\end{vmatrix}\) is:

Write Minors and Cofactors of the elements of following determinants:
I. \(\begin{vmatrix}1&0&0\\0&1&0\\0&0&1\end{vmatrix}\)II. \(\begin{vmatrix}1&0&4\\3&5&-1\\0&1&2\end{vmatrix}\)

Find equation of line joining (1,2) and (3,6) using determinants
Find equation of line joining (3,1) and (9,3) using determinants

Solve the system of the following equations
\(\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4\)
\(\frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1\)
\(\frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2\)

Using properties of determinants, prove that:
\(\begin{vmatrix} \alpha &\alpha^2 &\beta+\gamma \\ \beta&\beta^2 &\gamma+\alpha \\ \gamma&\gamma^2 &\alpha+\beta \end{vmatrix}\)=(\(\beta-\gamma\))( \(\gamma-\alpha\))(\(\alpha-\beta\))(\(\alpha+\beta+\gamma\))

Evaluate\(\begin{vmatrix} x &y &x+y \\ y&x+y &x \\ x+y&x &y \end{vmatrix}\)

Using properties of determinants,prove that:
\(\begin{vmatrix}sinα& cosα& cos(α+δ)\\ sinβ& cosβ& cos(β+δ)\\ sinγ& cosγ& cos(γ+δ)\end{vmatrix}=0\)

Using properties of determinants,prove that:
\(\begin{vmatrix} 3a& -a+b & -a+c\\ -b+a & 3b & -b+c \\-c+a&-c+b&3c\end{vmatrix}\)\(=3(a+b+c)(ab+bc+ca)\)

Using properties of determinants,prove that:
\(\begin{vmatrix}1& 1+p& 1+p+q\\ 2& 3+2p& 4+3p+2q\\ 3& 6+3p& 10+6p+3q\end{vmatrix}=1\)

Using properties of determinants,prove that:
\(\begin{vmatrix} x & x^2 & 1+px^3\\ y & y^2 & 1+py^3\\z&z^2&1+pz^3 \end{vmatrix}\)\(=(1+pxyz)(x-y)(y-z)(z-x)\)

Choose the correct answer.
If x,y,z are nonzero real numbers,then the inverse of matrix
A=\(\begin{bmatrix}x& 0& 0\\ 0& y& 0\\0&0& z\end{bmatrix}\)is