11th Standard Syllabus & Materials
11th Standard
TN 11th Tamil இயற்கை வேளாண்மை,சுற்றுச்சூழல் -செய்யுள் - மனோன்மணீயம் Important Questions And Answers Study Material - QB365 Set A
NEW11th Standard
TN 11th Tamil என்னுயிர் என்பேன் -துணைப்பாடம் - இசைத்தமிழர் இருவர் Important Questions And Answers Study Material - QB365 Set A
NEW11th Standard
TN 11th Tamil மொழி கலை -செய்யுள் - ஒவ்வொரு புல்லையும் Important Questions And Answers Study Material - QB365 Set A
NEW11th Standard
TN 11th Tamil பீடு பெற நில் - இலக்கணம் - பகுபத உறுப்புகள் Important Questions And Answers Study Material - QB365 Set A
NEW11th Standard
TN 11th Tamil பீடு பெற நில் - துணைப்பாடம் - வாடிவாசல் Important Questions And Answers Study Material - QB365 Set A
NEW11th Standard
TN 11th Tamil பீடு பெற நில் - செய்யுள் - குறுந்தொகை Important Questions And Answers Study Material - QB365 Set A
Published on: 30/08/2019
Matrices And Determinants
Download Tamil Nadu 11th Standard Business Maths and Statistics question papers, model tests, one-mark questions, important questions, and public exam papers in PDF format. Free study materials and answer keys for TN State Board students.
Questions + Answers key
Take MCQ Business Maths and Statistics Test
1.
If any three rows or columns of a determinant are identical then the value of the determinant is ________.
0
2
1
3
2.
If \(\begin{vmatrix} 4 & 3 \\ 3 & 1 \end{vmatrix}=-5\) then value of \(\begin{vmatrix} 20 & 15 \\ 15 & 5 \end{vmatrix}\) is ________.
-5
-125
-25
0
3.
If \(\begin{vmatrix} x & 2 \\ 8 &5 \end{vmatrix}=0\) then the value of x is ________.
\({{-5}\over{6}}\)
\({{5}\over{6}}\)
\({{-16}\over{5}}\)
\({{16}\over{5}}\)
4.
If \(\triangle=\begin{vmatrix} {a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & {a}_{23} \\ {a}_{31} & {a}_{32} & {a}_{33} \end{vmatrix}\) and Aij is cofactor of aij, then value of \(\triangle\) is given by ________.
a11 A31 + a12 A32 + a13 A33
a11 A11 + a12 A21 + a13 A31
a21 A11 + a22 A12 + a23 A13
a11 A11 + a21 A21 + a31 A31
5.
If A = \(\begin{vmatrix}cos \theta & sin \theta \\ -sin \theta&cons\theta \end{vmatrix},\) then |2A| is equal to ________.
4 cos 2 \(\theta\)
4
2
1
6.
The technology matrix of an economic system of two industries is \(\left[ \begin{matrix} 0.8 & 0.2 \\ 0.9 & 0.7 \end{matrix} \right] \) Test whether the system is viable as per Hawkins – Simon conditions.
7.
If \(A=\left[ \begin{matrix} 2 & 4 \\ -3 & 2 \end{matrix} \right] \)then, find A -1.
8.
If \(A=\left| \begin{matrix} -2 & 6 \\ 3 & -9 \end{matrix} \right| \)then, find A-1
9.
Find adj A for \(A=\left[ \begin{matrix} 2 & 3 \\ 1 & 4 \end{matrix} \right] \)
10.
Evaluate\(\left| \begin{matrix} 1 & a & { a }^{ 2 } \\ 1 & b & { b }^{ 2 } \\ 1 & c & { c }^{ 2 } \end{matrix} \right| \)= (a–b) (b–c) (c–a)
11.
Show that \(\left[ \begin{matrix} 8 & 2 \\ 4 & 3 \end{matrix} \right] \)is non – singular.
12.
An economy produces only coal and steel. These two commodities serve as intermediate inputs in each other’s production. 0.4 tonne of steel and 0.7 tonne of coal are needed to produce a tonne of steel. Similarly 0.1 tonne of steel and 0.6 tonne of coal are required to produce a tonne of coal. No capital inputs are needed. Do you think that the system is viable? 2 and 5 labour days are required to produce a tonnes of coal and steel respectively. If economy needs 100 tonnes of coal and 50 tonnes of steel, calculate the gross output of the two commodities and the total labour days required.
13.
The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs.320. The cost of 2kg onion, 4 kg wheat and 6 kg rice is Rs.560. The cost of 6 kg onion, 2 kg wheat and 3 kg rice is Rs. 380. Find the cost of each item per kg by matrix inversion method.
1.
(a)
0
2.
\(\left|\begin{array}{cc} 20 & 15 \\ 15 & 5 \end{array}\right|=5 \times 5\left|\begin{array}{ll} 4 & 3 \\ 3 & 1 \end{array}\right|\)
\(=5 \times 5 \times-5\)
\(=-125\)
3.
5x - 16 = 0
5x = 16
\(x=\frac{16}{5}\)
4.
(Corresponding co-factor)
5.
\(|A|=\cos ^2 \theta+\sin ^2 \theta=1\)
\(|2 A|=2^2(1)=4\)
6.
B = \(\left[ \begin{matrix} 0.8 & 0.2 \\ 0.9 & 0.7 \end{matrix} \right] \)
I - B=\(\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right] \)-\(\left[ \begin{matrix} 0.8 & 0.2 \\ 0.9 & 0.7 \end{matrix} \right] \)
=\(\left[ \begin{matrix} 0.2 & -0.2 \\ - 0.9 & 0.3 \end{matrix} \right] \)
|I - B|= \(\left[ \begin{matrix} 0.2 & -0.2 \\ - 0.9 & 0.3 \end{matrix} \right] \)
= (0.2)(0.3) - (-0.2)(-0.9)
= 0.06 - 0.18
= 0.12 < 0
Since |I - B| is negative, Hawkins – Simon conditions are not satisfied.
Therefore, the given system is not viable.
7.
\(A=\left[ \begin{matrix} 2 & 4 \\ -3 & 2 \end{matrix} \right] \)
\(|A|=\left[ \begin{matrix} 2 & 4 \\ -3 & 2 \end{matrix} \right] \)
=16 ≠ 0
Since A is a nonsingular matrix, A -1 exists
Now adj \(A=\left[ \begin{matrix} 2 & -4 \\3 & 2 \end{matrix} \right] \)
\({ A }^{ -1 }=\frac { 1 }{ \left| A \right| } adjA\)
\(=\frac { 1 }{ 16 } \left[ \begin{matrix} 2 & -4 \\ 3 & 2 \end{matrix} \right] \)
8.
\(|A|=\left| \begin{matrix} -2 & 6 \\ 3 & -9 \end{matrix} \right| =0\)
Since A is a singular matrix, A -1 does not exist.
9.
\(A=\left[ \begin{matrix} 2 & 3 \\ 1 & 4 \end{matrix} \right] \)
adj A = [Aij]T
\(=\left[ \begin{matrix} 4 & -3 \\ -1 & 2 \end{matrix} \right] \)
10.
\(\left| \begin{matrix} 1 & a & { a }^{ 2 } \\ 1 & b & { b }^{ 2 } \\ 1 & c & { c }^{ 2 } \end{matrix} \right| =\left| \begin{matrix} 0 & a-b & { a }^{ 2 }-{ b }^{ 2 } \\ 0 & b-c & { b }^{ 2 }-{ c }^{ 2 } \\ 1 & c & { c }^{ 2 } \end{matrix} \right| \begin{matrix} { R }_{ 1 }\rightarrow { R }_{ 1 }-{ R }_{ 2 } \\ { R }_{ 2 }\rightarrow { R }_{ 2 }-{ R }_{ 3 } \end{matrix}\)
\(=\left| \begin{matrix} 0 & a-b & (a-b)(a+b) \\ 0 & b-c & (b-c)(b+c) \\ 1 & c & { c }^{ 2 } \end{matrix} \right| \)
\(=(a-b)(b-c)\left| \begin{matrix} 0 & 1 & a+b \\ 0 & 1 & b+c \\ 1 & c & { c }^{ 2 } \end{matrix} \right| \)
= (a - b)(b - c)[0 - 0 + {b + c - (a + b)}]
= (a - b)(b - c)(c - a)
11.
Let A = \(\left[ \begin{matrix} 8 & 2 \\ 4 & 3 \end{matrix} \right] \)
|A| = \(\left| \begin{matrix} 8 & 2 \\ 4 & 3 \end{matrix} \right| \)
= 24 – 8 = 16 ≠ 0
\(\therefore\) A is a non-singular matrix
12.
Here the technology matrix is given under
| Steel | Coal | Final demand | |
| Steel | 0.4 | 0.1 | 50 |
| Coal | 0.7 | 0.6 | 100 |
| Labour days | 5 | 2 | - |
The technology matrix is B = \(\left[ \begin{matrix} 0.4 & 0.1 \\ 0.7 & 0.6 \end{matrix} \right] \)
I - B = \(\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right] \)- \(\left[ \begin{matrix} 0.4 & 0.1 \\ 0.7 & 0.6 \end{matrix} \right] \) = \(\left[ \begin{matrix} 0.6 & -0.1 \\ -0.7 & 0.4 \end{matrix} \right] \)
|I - B| = \(\left| \begin{matrix} 0.6 & -0.1 \\ -0.7 & 0.4 \end{matrix} \right|\)
= (0.6)(0.4) – (–0.7)(–0.1)
= 0.24 – 0.07 = 0.17
Since the diagonal elements of I – B are positive and value of |I-B| is positive, the system is viable.
adj(I - B) = \(\left[ \begin{matrix} 0.4 & 0.1 \\ 0.7 & 0.6 \end{matrix} \right] \)
(I - B)-1 = \(\frac{1}{|I -B |} \)adj(I - B)
= \(\frac{1}{0.17} \) \(\left[ \begin{matrix} 0.4 & 0.1 \\ 0.7 & 0.6 \end{matrix} \right] \)
X = (I – B)–1D, where D =\(\left[ \begin{matrix} 50 \\ 100 \end{matrix} \right] \)
= \(\frac{1}{0.17} \)\(\left[ \begin{matrix} 0.4 & 0.1 \\ 0.7 & 0.6 \end{matrix} \right] \)\(\left[ \begin{matrix} 50 \\ 100 \end{matrix} \right] \)
= \(\frac{1}{0.17} \)\(\left[ \begin{matrix} 30 \\ 95 \end{matrix} \right] \)
= \(\left[ \begin{matrix} 176.5 \\ 558.8 \end{matrix} \right] \)
Steel output = 176.5 tonnes
Coal output = 558.8 tonnes
Total labour days required
= 5(steel output) + 2( coal output)
= 5(176.5) + 2(558.8)
= 882.5 + 1117.6 = 2000.1
\(\simeq\) 2000 labour days.
13.
Let x, y and z be the cost of onion, wheat and rice per kg respectively.
Then, 4x + 3y + 2z = 320
2x + 4y + 6z = 560
6x + 2y + 3z = 380.
It can be written as
\(\left[ \begin{matrix} 4 & 3 & 2 \\ 2 & 4 & 6 \\ 6 & 2 & 3 \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 320 \\ 560 \\ 380 \end{matrix} \right] \)
AX = B
where \(A=\left[ \begin{matrix} 4 & 3 & 2 \\ 2 & 4 & 6 \\ 6 & 2 & 3 \end{matrix} \right] ,X= \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] \)and \(B =\left[ \begin{matrix} 320 \\ 560 \\ 380 \end{matrix} \right] \)
\(|A|=\left[ \begin{matrix} 4 & 3 & 2 \\ 2 & 4 & 6 \\ 6 & 2 & 3 \end{matrix} \right]\)= 50 ≠ 0
A–1 exist
A11 = 0 A12 = 30 A13 = –20
A21 = –5 A22 = 0 A23 = 10
A31 = 10 A32 = –20 A33 = 10
[Aij] =\(\left[ \begin{matrix} 0 & 30 & -20 \\ -5 & 0 & 10 \\ 10 & -20 & 10 \end{matrix} \right]\)
adj A = [Aij]T
=\(\left[ \begin{matrix} 0 & 5 & -10 \\ 30 & 0 & -20 \\ -20 & 10 & 10 \end{matrix} \right]\)
\(A^{-1}=\frac{1}{|A|}adjA\)
=\(\frac{1}{50}\left[ \begin{matrix} 0 & -5 & 10 \\ 30 & 0 & -20 \\ -20 & 10 & 10 \end{matrix} \right]\)
\(X=A^{-1} B\)
= \(\frac{1}{50}\left[ \begin{matrix} 0 & -5 & 10 \\ 30 & 0 & -20 \\ -20 & 10 & 10 \end{matrix} \right]\)\(\left[ \begin{matrix} 320 \\ 560 \\ 380 \end{matrix} \right] \)
= \(\frac{1}{5}\left[ \begin{matrix} 0 & -5 & 10 \\ 30 & 0 & -20 \\ -20 & 10 & 10 \end{matrix} \right]\)\(\left[ \begin{matrix} 32 \\ 56 \\ 38 \end{matrix} \right] \)
= \(\frac{1}{5}\left[ \begin{matrix} 0 & -280 & +380 \\ 960 & 0 & -760 \\ -640 & +560 & +380 \end{matrix} \right]\)
=\(\frac{1}{5}\left[ \begin{matrix} 100 \\ 200 \\ 300 \end{matrix} \right] \)
\(\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] \)=\(\left[ \begin{matrix} 20 \\ 40 \\ 60 \end{matrix} \right] \)
x = 20, y = 40, z = 60
Cost of 1 kg onion is Rs.20, cost of 1 kg wheat is Rs.40 and cost of 1 kg rice is Rs.60.
11th Standard Syllabus & Materials
11th Standard
TN 11th Tamil பீடு பெற நில் - செய்யுள் - காவடிச்சிந்து Important Questions And Answers Study Material - QB365 Set A
NEW11th Standard
TN 11th Tamil பீடு பெற நில் - உரைநடை - மலை இடப்பெயர்கள் : ஓர் ஆய்வு Important Questions And Answers Study Material - QB365 Set A
NEW11th Standard
TN 11th Tamil மாமழை போற்றுதும் - துணைப்பாடம் - யானை டாக்டர் Important Questions And Answers Study Material - QB365 Set A
NEW11th Standard
TN 11th Tamil மாமழை போற்றுதும் - செய்யுள் - ஐங்குறுநூறு Important Questions And Answers Study Material - QB365 Set A
Tamilnadu Stateboard 11th Standard Subjects
Maths
Commerce
Economics
Biology
Business Maths and Statistics
Accountancy
Computer Science
Physics
Chemistry
Maths
Biology
Economics
Physics
Chemistry
History
Business Maths and Statistics
Computer Science
Accountancy
Computer Applications
History
Computer Technology
Commerce
Computer Applications
Computer Technology
Tamil
English
French
Tamilnadu Stateboard Standards
