A company produces three products every day. Their production on certain day is 45 tons. It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product.

Using the concepts of matrices and determinants, answer the following questions.
(i) If x, y and z respectively denotes the quantity (in tons) of first, second and third product produced, then which of the following is true?

(ii) If \(\left(\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 0 & -2 \\ 1 & -1 & 1 \end{array}\right)^{-1}=\frac{1}{6}\left(\begin{array}{ccc} 2 & 2 & 2 \\ 3 & 0 & -3 \\ 1 & -2 & 1 \end{array}\right)\) , then the inverse of \(\left(\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 0 & -1 \\ 1 & -2 & 1 \end{array}\right)\) is

(iii) x :y : z is equal to

(iv) Which of the following is not true?

Each triangular face of the Pyramid of Peace in Kazakhstan is made up of 25 smaller equilateral triangles as shown in the figure.

Using the above information and concept of determinants, answer the following questions
(i) If the vertices of one of the smaller equilateral triangle are (0, 0), \(\begin{equation} (3, \sqrt{3}) \end{equation}\) and \(\begin{equation} (3,-\sqrt{3}) \end{equation}\) then the area of such triangle is

(ii) The area of a face of the Pyramid is

(iii) The length of a altitude of a smaller equilateral triangle is

(iv) If (2, 4), (2, 6) are two vertices of a smaller equilateral triangle, then the third vertex will lie on the line represented by

(v) Let A(a, 0), B(O, b) and C(1, 1) be three points. If \(\begin{equation} \frac{1}{a}+\frac{1}{b}=1 \end{equation}\) ,then the three points are

Two schools A and B want to award their selected students on the values of Honesty, Hard work and Punctuality. The school A wants to award Rs. x each, Rs. y each and Rs.z each for the three respective values to its 3, 2 and 1 students respectivefy with a total award money of Rs. 2200. School B wants to spend Rs. 3100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school A). The total amount of award for one prize on each value is Rs. 1200.

Using the concept of matrices and determinants, answer the following questions.
(i) What is the award money for Honesty?

(ii) What is the award money for Punctuality?

(iii) What is the award money for Hard work?

(iv) If a matrix P is both symmetric and skew-symmetric, then IPI is equal to

(v) If P and Q are two matrices such that PQ = Q and QP = P, then IQ²1 is equal to

Area of a triangle whose vertices are (x₁, y₁), (x_2' y₂) and (x_3, y₃) is given by thedeterminant
\(\begin{equation} \Delta=\frac{1}{2}\left|\begin{array}{lll} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array}\right| \end{equation}\) 
Since, area is a positive quantity, so we always take the absolute value of the determinant Δ. Also, the area of the triangle formed by three collinear points is zero.
Based on the above information, answer the following questions
(i) Find the area of the triangle whose vertices are (-2, 6), (3, -6) and (1, 5).

(ii) If the points (2, -3), (k, -1) and (0, 4) are collinear, then find the value of 4k

(iii) If the area of a triangle ABC, with vertices A(1, 3), B(O, 0) and C(k, 0) is 3 sq. units, then a value of k is

(iv) Using determinants, find the equation of the line joining the points A(1, 2) and B(3, 6).

(v) If A = (11, 7), B = (5, 5) and C = (-1, 3), then

Gaurav purchased 5 pens, 3 bags and 1 instrument box and pays Rs. 16. From the same shop, Dheeraj purchased 2 pens, 1 bag and 3 instrument boxes and pays Rs. 19, while Ankur purchased 1 pen, 2 bags and 4 instrument boxes and pays Rs. 25.
 
Using the concept of matrices and determinants, answer the following questions.
(i) The cost of one pen is

(ii) What is the cost of one pen and one bag?

(iii) What is the cost of one pen and one instrument box?

(iv) Which of the following is correct?

(v) From the matrix equation AB = AC, it can be concluded that B = C provided