 # Ch04. Determinants Class 12 Maths (combo)

## Achievements

The following badges can be earned while learning.   ## Certificate

Certificate on successful completion of this course. THIS IS A COMBO OF TWO COURSES

COURSE – 1

In this online course, you will learn determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.. This course also has solutions of NCERT Exercise questions.

## Course Content

Please note, this course comprises of live session recordings of Ashish Kumar (Agam Sir). Please ignore, if you have heard Agam Sir discussing about live session or doubts session in the course. Live sessions or doubts sessions are no longer offered.

COURSE – 2

This online course is an extended part of Ch04. Determinants Class 12 Maths. The course is based on the assignments by Ashish Kumar (Agam Sir), which have questions from NCERT Exemplar, Board’s Question Bank, R. D. Sharma etc. The PDF of assignments can be downloaded within the course. Please note, the assignments do not have questions based on properties of determinants.

## Course Content

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The following list of questions are just meant for reference before purchasing membership. The list might or might not include NCERT Questions as it depends on the chapter/course. Some chapters have NCERT questions combined in the Assignments and some chapters have separate NCERT questions and Assignments. For complete details, please check the index of the course in the "About Course".

### Assignment – 1

1. If {\displaystyle x=-9} is a root of {\displaystyle \left| \,\begin{matrix} x & 3 & 7  \\    2 & x & 2  \\    7 & 6 & x  \\ \end{matrix}\, \right|=0} , then find the other two roots?
2. Examine whether the given matrix {\displaystyle \text{A}=\,\,\left( \begin{matrix} 1 & -2 & 3  \\    -2 & -1 & 0  \\    4 & -2 & 5  \\ \end{matrix} \right)} is singular.
3. If the points {\displaystyle (a,\,\,b),\,(a',\,\,b')} and {\displaystyle (a-a',\,\,b-b')} are collinear. Show that {\displaystyle ab'=a'b} .
4. Find the value of p, such that the matrix {\displaystyle \left[ \begin{matrix} -1 & 2  \\    4 & p  \\ \end{matrix} \right]} is singular.
5. Find the value of {\displaystyle \left| \begin{matrix} a+ib & c+id  \\    -c+id & a-ib  \\ \end{matrix} \right|} .
6. Show that the points (a + 5, a – 4), (a – 2, a + 3) and (a, a) do not lie on a straight line for any value of a.
7. Evaluate the following determinant: {\displaystyle \left| \,\begin{matrix} \cos 15{}^\circ  & \sin 75{}^\circ   \\    \sin 15{}^\circ  & \cos 75{}^\circ   \\ \end{matrix}\, \right|}
8. Show that if the determinant {\displaystyle \left| \,\begin{matrix} 3 & -2 & \sin 3\theta   \\    -7 & 8 & \cos 2\theta   \\    -11 & 14 & 2  \\ \end{matrix}\, \right|=0} , then {\displaystyle \sin \theta =0\text{ or }\frac{1}{2}} .
9. Find the value of {\displaystyle \theta} satisfying {\displaystyle \left| \,\begin{matrix} 1 & 1 & \sin 3\theta  \\    -4 & 3 & \cos 2\theta  \\    7 & -7 & -2  \\ \end{matrix}\, \right|=0} .

### Assignment – 2

1. If {\displaystyle A=\left[ \begin{matrix} 0 & 0 & 1  \\    0 & 1 & 0  \\    1 & 0 & 0  \\ \end{matrix} \right]} then find {\displaystyle {{A}^{2}}} . Hence find {\displaystyle {{A}^{6}}} .
2. If {\displaystyle {{A}^{-1}}=\left[ \begin{matrix} 3 & -1 & 1  \\   -15 & 6 & -5  \\    5 & -2 & 2  \\ \end{matrix} \right]} and {\displaystyle B=\left[ \begin{matrix}   1 & 2 & -2  \\    -1 & 3 & 0  \\    0 & -2 & 1  \\ \end{matrix} \right]} , find {\displaystyle {{(AB)}^{-1}}} .
3. If {\displaystyle \text{A}=\left[ \begin{matrix} 2 & -1  \\    3 & 4  \\ \end{matrix} \right]} , find {\displaystyle \left| \,{{\text{(}{{\text{A}}^{-\text{1}}}\text{)}}^{-\text{1}}}\, \right|} .
4. If the matrix {\displaystyle \text{A}=\left[ \begin{matrix} 1 & 1 & 2  \\    0 & 2 & -3  \\    3 & -2 & 4  \\ \end{matrix} \right]} and {\displaystyle {{\text{B}}^{-\text{1}}}=\left[  \begin{matrix}    1 & 2 & 0  \\    0 & 3 & -1  \\    1 & 0 & 2  \\ \end{matrix} \right]} , then compute {\displaystyle {{\text{(AB)}}^{-\text{1}}}} .
5. If A is the square matrix of order 3 and | A | = 2, find the value of {\displaystyle |-3A|} .
6. If A = 2B where A and B are of square matrix of order {\displaystyle 3\,\times \,3} and |B| = 5. What is |A|?
7. If A is non-singular matrix of order 3 and {\displaystyle |A|\,=\,-3} find {\displaystyle |adj(A)|} .
8. Given a square matrix of order {\displaystyle 3\,\times \,3} such that | A | = 12 find the value of  {\displaystyle |A.adj(A)|} .
9. If A is a square matrix of order 3 such that {\displaystyle |adj(A)|\,\,=\,\,8}   find | A |.
10. If A and B are matrices of order 3 and {\displaystyle |A|\,=\,\,5,\,\,|B|\,\,=\,\,3} , then find {\displaystyle |3AB|} .
11. Given square matrix A of order {\displaystyle 3\,\,\times \,\,3} , such that {\displaystyle |\text{A }|\,=\,-\text{5}} , find the value of {\displaystyle |\text{A}\text{.adj(A)}|} .
12. If A is invertible matrix of order 3, then find {\displaystyle |{{A}^{-1}}|} & {\displaystyle {{({{A}^{2}})}^{-1}}} .
13. If the value of a third order determinant is 12, then what is the value of the determinant formed by replacing each element by its cofactor?
14. There are two values of a which makes determinant {\displaystyle \left| \,\begin{matrix} 1 & -2 & 5  \\    2 & a & 1  \\    0 & 4 & 2a  \\ \end{matrix}\, \right|=86} . Find the sum of these numbers.
15. If {\displaystyle |A|\,=3} and {\displaystyle A={{[{{a}_{ij}}]}_{3\times 3}}} and {\displaystyle {{C}_{ij}}} the cofactors of {\displaystyle {{a}_{ij}}} then what is the value of {\displaystyle {{a}_{13}}{{C}_{13}}+{{a}_{23}}{{C}_{23}}+{{a}_{33}}{{C}_{33}}} .

### Assignment – 3

1. Find the product of matrices {\displaystyle A=\left( \begin{matrix} 1 & -1 & 0  \\    2 & 3 & 4  \\    0 & 1 & 2  \\ \end{matrix} \right)} , {\displaystyle \text{B}=\left( \begin{matrix}    2 & 2 & -4  \\    -4 & 2 & -4  \\    2 & -1 & 5  \\ \end{matrix} \right)} and use it for solving the equations:  {\displaystyle x-y=3,\,\,\,\,2x+3y+4z=17,\,\,\,\,\,y+2z=7} .
2. Three shopkeepers A, B, C are using polythene, handmade bags (prepared by prisoners), and newspaper’s envelope as carry bags. it is found that the shopkeepers A, B, C are using (20,30,40) , (30,40,20,) , (40,20,30) polythene , handmade bags and newspapers envelopes The shopkeepers A, B, C spent ₹250, ₹220 & ₹200 on these carry bags respectively. Find the cost of each carry bags using matrices. Keeping in mind the social & environmental conditions, which shopkeeper is better? & why?
3. In a Legislative assembly election, a political party hired a public relation firm to promote its candidate in three ways; telephone, house calls and letters. The numbers of contacts of each type in three cities A, B & C are (500, 1000, and 5000), (3000, 1000, 10000) and (2000, 1500, 4000), respectively. The party paid ₹3700, ₹7200, and ₹4300 in cities A, B & C Find the costs per contact using matrix method. Keeping in mind the economic condition of the country, which way of promotion is better in your view?
4. Using matrix method solve the following system of equations                                                              x + 2y + z = 7, xy + z =4,                 x + 3y +2z = 10                                                 If x represents the no. of persons who take food at home, y represents the no. of parsons who take junk food in market and z represent the no. of persons who take food at hotel. Which way of taking food you prefer and way?
5. A school has to reward the students participating in co-curricular activities (Category I) and with 100% attendance (Category II) brave students (Category III) in a function. The sum of the numbers of all the three category students is 6. If we multiply the number of category III by 2 and added to the number of category I to the result, we get 7. By adding second and third category would to three times the first category we get 12. Form the matrix equation and solve it.
6. For keeping Fit X people believes in morning walk, Y people believe in yoga and Z people join Gym. Total no. of people are 70. further 20% 30% and 40% people are suffering from any disease who believe in morning walk, yoga and GYM respectively. Total no. of such people is If morning walk cost ₹0 Yoga cost ₹500/month and GYM cost ₹400/ month and total expenditure is ₹23000. Calculate the no. of each type of people. Why exercise is important for health?
7. An amount of ₹600 crores is spent by the government in three schemes. Scheme A is for saving girl child from the cruel parents who don’t want girl child and get the abortion before her birth. Scheme B is for saving of newlywed girls from death due to dowry. Scheme C is planning for good health for senior citizen. Now twice the amount spent on Scheme C together with amount spent on Scheme A is ₹700 crores. And three times the amount spent on Scheme A together with amount spent on Scheme B and Scheme C is ₹1200 crores. Find the amount spent on each Scheme using matrices? What is the importance of saving girl child from the cruel parents who don’t want girl child and get the abortion before her birth?
8. If {\displaystyle A=\left( \begin{matrix} 1 & 2 & 0 \\   -2 & -1 & -2  \\    0 & -1 & 1  \\  \end{matrix} \right)} , find {\displaystyle {{A}^{-1}}} . Using {\displaystyle {{A}^{-1}}} , solve the system of equations {\displaystyle x-2y=102x-y-z=8-2y+z=7}
9. Given {\displaystyle A=\left[ \begin{matrix} 2 & 2 & -4  \\    -4 & 2 & -4  \\    2 & -1 & 5  \\ \end{matrix} \right],\,B=\left[ \begin{matrix}    1 & -1 & 0  \\    2 & 3 & 4  \\    0 & 1 & 2  \\ \end{matrix} \right]} , find {\displaystyle BA} and use this to solve the system of equations {\displaystyle y+2z=7x-y=32x+3y+4z=17}
10. A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received ₹2,800 as interest. However, if trust had interchanged money in bonds, they would have got ₹100 less as interest. Using matrix method, find the amount invested by the trust. Interest received on this amount will be given to Helpage India as donation. Which value is reflected in this question?
Syllabus medium English Hinglish (Hindi + English) 12 Online learning Self-learning from videos Mathematics