Download App

DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS4 Marks
Question
Show that the following system of linear equations is consistent and also find solution:
2x + 2y − 2z = 1
4x + 4y − z = 2
6x + 6y + 2z = 3
✓

Answer

This system can be written as:

$\begin{bmatrix}2&2&-2\\ 4&4&-1\\ 6&6&2\end{bmatrix}\begin{bmatrix}\text{x}\\ \text{y}\\ \text{z}\end{bmatrix}=\begin{bmatrix}1\\ 2\\ 3\end{bmatrix}$

or $\text{AX = B}$

$\text{|A|}=2{(14)}-2(14)-2{(0)}=0$

So, A is singular and the system has either no solution or infinite solutions according as

$\text{(Adj A)}\times\text{(B)}\neq0$ or $\text{(Adj A)}\times\text{(B)}=0$

Let Cij be the co-factors of aij in A

$\text{C}_{11}=14\\ \text{C}_{21}=-16\\ \text{C}_{31}=6$

$\text{C}_{12}=-14\\ \text{C}_{22}=16\\ \text{C}_{32}=-6$

$\text{C}_{1}=0\\ \text{C}_{23}=0\\ \text{C}_{33}=0$

$\text{adj A}=\begin{bmatrix}14&-14&0\\ -16&16&0\\ 0&0&0\end{bmatrix}^\text{T}=\begin{bmatrix}14&-16&6\\ -14&16&-6\\ 0&0&0\end{bmatrix}$

$(\text{adj A})\times\text{B}=\begin{bmatrix}14&-16&0\\ -14&16&-6\\ 0&0&0\end{bmatrix}\begin{bmatrix}1\\ 2\\ 3\end{bmatrix}=\begin{bmatrix}14-32+18\\ -14+32-18\\ 0+0+0\end{bmatrix}=\begin{bmatrix}0\\ 0\\ 0\end{bmatrix}$

So, $\text{AX}=\text{B}$ has infinite solutions.

Now, let z = k

So, 2x + 2y = 1 + 2k

4x + 4y = 2 + k

which can be written as:

$\begin{bmatrix}2&2\\ 4&4\end{bmatrix}\begin{bmatrix}\text{x}\\ \text{y}\end{bmatrix}=\begin{bmatrix}1+2\text{k}\\ 2+\text{k}\end{bmatrix}$

or $\text{AX = B}$

|A| = 0, z = 0

Again, 

$2\text{x}+2\text{y}=1$

$4\text{x}+4\text{y}=2$

Let $\text{y = k}$

$2\text{x}=1-2\text{k}$

$\text{x}=\frac{1}{2}-\text{k}$

Hence, $\text{x}=\frac{1}{2}-\text{k}$

$\text{y}=\text{k}$

$\text{z}=0$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from DETERMINANTS→DETERMINANTS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A manufacturer has three machines installed in his factory. machines I and II are capable of being operated for at most 12 hours whereas Machine III must operate at least for 5 hours a day. He produces only two items, each requiring the use of three machines. The number of hours required for producing one unit each of the items on the three machines is given in the following table:
Item Number of hours required by the machine
  I II III
A 1 2 1
B 2 1 $\frac{5}{4}$
He makes a profit of Rs. 6.00 on item A and Rs. 4.00 on item B. Assuming that he can sell all that he produces, how many of each item should he produces so as to maximize his profit? Determine his maximum profit. Formulate this LPP mathematically and then solve it.
→
If $\text{A}=\begin{bmatrix}\cos2\theta&\sin2\theta\\-\sin2\theta&\cos2\theta\end{bmatrix},$ find A2.
→
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$\text{f}(\text{x})=\tan^{-1}\text{x}\text{ on }[0,1]$
→
Find the distance of the point (1, -2, 4) from plane passing throuhg the point (1, 2, 2) and perpendicular of the planes x - y + 2z = 3 and 2x - 2y + z + 12 = 0
→
Compute the adjoint of the following matrices:

$\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$

Verify that (adjoint A)A = |A|I = A (adjoint A) for the above matrices.

→
Evaluate the following integrals:
$\int\frac{1}{\sin\text{x}+\sin2\text{x}}\ \text{dx}$
→
If f(x) = x2 - 2x, find f(A), where $\text{A}=\begin{bmatrix}0&1&2\\4&5&0\\0&2&3\end{bmatrix}$
→
Find the vector and Cartesian equations of the line through the point (1, 2, – 4) and perpendicular to the two lines.
$\overrightarrow{\text{r}} = (8\hat{\text{i}} - 19\hat{\text{j}} + 10\hat{\text{k}})+\lambda(3\hat{\text{i}} - 16\hat{\text{j}} + 7\hat{\text{k})}$ and $\overrightarrow{\text{r}} = (15\hat{\text{i}} - 29\hat{\text{j}} + 5\hat{\text{k}})+\mu(3\hat{\text{i}} - 8\hat{\text{j}} + 5\hat{\text{k})}.$
→

Solve the following LPP graphically:

Maximize Z = 20x + 10y

Subject to the following constraints

$\text{x}+2\text{y}\leq28$

$3\text{x}+\text{y}\leq24$

$\text{x}\geq2$

$\text{x},\text{y}\geq0$

→
A hospital dietician wishes to find the cheapest combination of two foods, A and B, that contains at least 0.5 milligram of thiamin and at least 600 calories. Each unit of A contains 0.12 milligram of thiamin and 100 calories, while each unit of B contains 0.10 milligram of thiamin and 150 calories. If each food costs 10 paise per unit, how many units of each should be combined at a minimum cost?
→