Ncert Solutions Maths class 12th

$=-sin\alpha \left(-si{n}^2\beta sin\alpha -sin\alpha co{s}^2\beta \right)-0\left(cos\alpha cos\beta sin\alpha sin\beta -cos\alpha sin\beta sin\alpha cos\beta \right)+cos\alpha \left(cos\alpha co{s}^2\beta +cos\alpha si{n}^2\beta \right)$

$=si{n}^2\alpha \left(si{n}^2\beta +co{s}^2\beta \right)+co{s}^2\alpha \left(co{s}^2\beta +si{n}^2\beta \right)$

$=si{n}^2\alpha +co{s}^2\alpha$

=1

L.H.S.= $\left|\begin{array}{ccc}\hfill a\hfill & \hfill a^2\hfill & \hfill bc\hfill \\ \hfill b\hfill & \hfill b^2\hfill & \hfill ca\hfill \\ \hfill c\hfill & \hfill c^2\hfill & \hfill ab\hfill \end{array}\right|$

Multiplying R₁, by a, R₂ by b and R₃ by c

$=\frac{1}{abc}\left|\begin{array}{ccc}\hfill a^2\hfill & \hfill a^3\hfill & \hfill abc\hfill \\ \hfill b^2\hfill & \hfill b^3\hfill & \hfill abc\hfill \\ \hfill c^2\hfill & \hfill c^3\hfill & \hfill abc\hfill \end{array}\right|$

Taking abc common from c_{3 = $=\frac{abc}{abc}\left|\begin{array}{ccc}\hfill a^2\hfill & \hfill a^3\hfill & \hfill 1\hfill \\ \hfill b^2\hfill & \hfill b^3\hfill & \hfill 1\hfill \\ \hfill c^2\hfill & \hfill c^3\hfill & \hfill 1\hfill \end{array}\right|$}

Inter changing c₁ and c_{3 = $=-\left|\begin{array}{ccc}\hfill 1\hfill & \hfill a^3\hfill & \hfill a^2\hfill \\ \hfill 1\hfill & \hfill b^3\hfill & \hfill b^2\hfill \\ \hfill 1\hfill & \hfill c^3\hfill & \hfill c^2\hfill \end{array}\right|$}

Inter changing c_{2 and} c_{3 = $=\left|\begin{array}{ccc}\hfill 1\hfill & \hfill a^2\hfill & \hfill a^3\hfill \\ \hfill 1\hfill & \hfill b^2\hfill & \hfill b^3\hfill \\ \hfill 1\hfill & \hfill c^2\hfill & \hfill c^3\hfill \end{array}\right|=R.H.S$}

Kindly go through the solution

Let k, y and z be the cost per kg of onion, wheat and rice. Then, we form a system of equations as follows

         4k + 3y + 2z = 60

         2k + 4y + 6z = 90

         6k + 2y + 3z = 70

In matrix form we can write,

AX = B

= 0 - 3 (-30) + 2 (-20)

= 90 - 40

= 50 ≠ 0.

∴ The system has a unique solution.

The cost of onion, wheat and rice per kg are? 5? 8 and? 8 respectively.

Kindly go through the solution

The given system of eqⁿ can be written in matrix form as

          AK = B

∴ x= 2, y = 1 and z = 3 .

The given system of equation can be represented in matrix form as AK = B

The given system of eqⁿ can be written in matrix form as AK = B.

The given system of solution in matrix form is AK = B.

The given system of equation in matrix form is

          AK = B