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?

If there is a statement involving the natural number n such that
(i) The statement is true for n = 1
(ii) When the statement is true for n = k (where k is some positive integer), then the statement is also true for n = k + 1.
Then, the statement is true for all natural numbers n.
Also, if A is a square matrix of order n, then A ² is defined as AA. In general, A^m = AA .... A (m times), where m is any positive integer.
Based on the above information, answer the following questions.
(i) If \(A=\left[\begin{array}{ll} 3 & -4 \\ 1 & -1 \end{array}\right]\),then for any positive integer n,

(ii) If \(A=\left[\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right]\),then |Aⁿ| where \(n \in N\), is equal to

(iii) If \(A=\left[\begin{array}{ll} 1 & 0 \\ 1 & 1 \end{array}\right]\), and \(I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\) then which of the following holds for all natural numbers \(n \geq 1\) ?

(iv) Let \(A=\left[\begin{array}{lll} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{array}\right]\) , and \(A^{n}=\left[a_{i j}\right]_{3 \times 3}\) for some positive integer n, then the cofactor of a₁₃ is

(v) If A is a square matrix such that IAI = 2, then for any positive integer n, IAⁿl is equal to

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

Let \(\begin{equation} A=\left[\begin{array}{ll} 1 & 0 \\ 2 & 1 \end{array}\right] \end{equation}\) , and U₁ U₂ are first and second columns respectively of a 2 x 2 matrix U.
Also, let the column matrices U_I and U₂ satisfying \(\begin{equation} A U_{1}=\left[\begin{array}{l} 1 \\ 0 \end{array}\right] \end{equation}\) and \(\begin{equation} A U_{2}=\left[\begin{array}{l} 2 \\ 3 \end{array}\right] \end{equation}\) 
Based on the above information, answer the following questions
(i) The matrix U₁ + U₂ is equal to

(ii) The value of IUI is

(iii) If \(\begin{equation} X=\left[\begin{array}{ll} 3 & 2 \end{array}\right] U\left[\begin{array}{l} 3 \\ 2 \end{array}\right] \end{equation}\),then the value of IXI =

(iv) The minor of element at the position a₂₂ in U is

(v) If \(\begin{equation} U=\left[a_{i j}\right]_{2 \times 2} \end{equation}\) , then the value of a₁₁A₁₁ + a₁₂A_l2 where A_ij denotes the cofactor of a_ij is

The upward speed v(t) of a rocket at time t is approximated by \(\begin{equation} v(t)=a t^{2}+b t+c, 0 \leq t \leq 100 \end{equation}\) ,where a, band c are constants. It has been found that the speed at times t = 3, t = 6 and t = 9 seconds are respectively 64, 133 and 208 miles per second.

If \(\begin{equation} \left(\begin{array}{ccc} 9 & 3 & 1 \\ 36 & 6 & 1 \\ 81 & 9 & 1 \end{array}\right)^{-1}=\frac{1}{18}\left(\begin{array}{ccc} 1 & -2 & 1 \\ -15 & 24 & -9 \\ 54 & -54 & 18 \end{array}\right) \end{equation}\) ,then answer the following questions
(i) The value of b + c is

(ii) The value of a + c is

(iii) v(t) is given by

(iv) The speed at time t = 15 seconds is

(v) The time at which the speed of rocket is 784 miles/see is

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

Three shopkeepers Salim, Vijay and Venket are using polythene bags, handmade bags (prepared by prisoners) and newspaper's envelope as carry bags. It is found that the shopkeepers Salim, Vijay and Venket are using (20, 30, 40), (30, 40, 20) and (40, 20, 30) polythene bags, handmade bags and newspaper's envelopes respectively. The shopkeepers Salim, Vijay and Venket spent Rs. 250, Rs. 270 and Rs. 200 on these carry bags respectively.

Using the concept of matrices and determinants, answer the following questions.
(i) What is the cost of one polythene bag?

(ii) What is the cost of one handmade bag?

(iii) What is the cost of one newspaper envelope

(iv) Keeping in mind the social conditions, which shopkeeper is better?

(v) Keeping in mind the environmental conditions, which shopkeeper is better?

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
(1) 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?

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

Minor of an element aij of a determinant is the determinant obtained by deleting its ith row and /h column in a_ij lies and is denoted by M_ij.
Cofactor of an element a_ij'denoted by A_ij,is defined by \(\begin{equation} A_{i j}=(-1)^{i+j} M_{i j} \end{equation}\), where M_ij is minor of a_ij.
Also, the determinant of a square matrix A is the sum of the products of the elements of any row (or column with their corresponding cofactors.
For example if \(\begin{equation} A=\left[a_{i j}\right]_{3 \times 3}, \text { then }|A|=a_{11} A_{11}+a_{12} A_{12}+a_{13} A_{13} \end{equation}\) .
Based on the above information, answer the following questions
(i) Find the sum of the cofactors of all the elements of \(\begin{equation} \left|\begin{array}{cc} 1 & -2 \\ 4 & 3 \end{array}\right| \end{equation}\) 

(ii) Find the minor of a₂₁ of \(\begin{equation} \left|\begin{array}{ccc} 5 & 6 & -3 \\ -4 & 3 & 2 \\ -4 & -7 & 3 \end{array}\right| \end{equation}\) 

(iii) In the determinant \(\begin{equation} \left|\begin{array}{ccc} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{array}\right| \end{equation}\) find the value of a₃₂·A₃₂

(iv) If \(\begin{equation} \Delta=\left|\begin{array}{lll} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{array}\right| \end{equation}\), find the value of a₃₂·A₃₂ .

(v) If \(\begin{equation} \Delta=\left|\begin{array}{ccc} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{array}\right| \end{equation}\), then find the value of \(\begin{equation} |\Delta| \end{equation}\).

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