# BHARDWAJ CLASSES

## Sunday, April 29, 2018

Here we are providing Important Questions for Matrix and Determinant prepared by Expert Maths teachers. You can download these questions free in PDF Form.

## Class 12 Maths - Matrix and Determinant Important Questions

1. Find the value of $|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}|$

2. Evaluate $\Delta =|\begin{array}{ccc}0& \mathrm{sin}\alpha & -\mathrm{cos}\alpha \\ -\mathrm{sin}\alpha & 0& \mathrm{sin}\beta \\ \mathrm{cos}\alpha & -\mathrm{sin}\beta & 0\end{array}|$

3. Let $|\begin{array}{cc}3& x\\ x& 1\end{array}|=|\begin{array}{cc}3& 2\\ 4& 1\end{array}|$ . Find the value of $x$

4. If $A=2B$ where $A$ and $B$ are square matrices of order 2, then $|A|$ will be?

5. Evaluate the determinant of $|\begin{array}{cc}{\mathrm{log}}_{a}b& 1\\ 1& {\mathrm{log}}_{b}a\end{array}|$

6. Evaluate the determinant of $|\begin{array}{cc}\mathrm{cos}{25}^{0}& \mathrm{sin}{25}^{0}\\ \mathrm{sin}{65}^{0}& \mathrm{cos}{65}^{0}\end{array}|$

7. Find the value of x if  | x+1 x1 x3 x+2 |=| 4 1 1 3 |

8. Without expanding, Prove that  | ( a x + a x ) 2 ( a x a x ) 2 1 ( a y + a y ) 2 ( a y a y ) 2 1 ( a z + a z ) 2 ( a z a z ) 2 1 |=0

9. Without expanding, Prove that  | a 2 2ab b 2 b 2 a 2 2ab 2ab b 2 a 2 |= ( a 3 + b 3 ) 2

10. Show that  | b+c ab a c+a bc b a+b ca c |=3abc a 3 b 3 c 3

11. Prove that  | 1+a 1 1 1 1+b 1 1 1 1+c |=abc( 1+ 1 a + 1 b + 1 c )=abc+bc+ca+ab

12. Evaluate $|\begin{array}{ccc}1& \mathrm{sin}\theta & 1\\ -\mathrm{sin}\theta & 1& \mathrm{sin}\theta \\ -1& -\mathrm{sin}\theta & 1\end{array}|$ , Also prove that $2\le \Delta \le 4$ .

13. If $A+B+C=0$ , then prove that  Δ=| 1 cosC cosB cosC 1 cosA cosB cosA 1 |=0

14. Prove that  | (b+c) 2 a 2 a 2 b 2 ( c+a ) 2 b 2 c 2 c 2 ( a+b ) 2 |=2abc (a+b+c) 3

15. Show that  | ( b+c ) 2 ba ca ba ( a+c ) 2 bc ac bc ( b+a ) 2 |=2abc ( a+b+c ) 3

16. If a, b ,c are in A.P. Show that following  | x+1 x+2 x+a x+2 x+3 x+b x+3 x+4 x+c |=0

17. If a, b, c are positive and unequal, show that the value of the determinant $|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}|$ is always negative.

18. Show that  | 3a a+b a+c b+a 3b b+c c+a c+b 3c |=3( a+b+c )( ab+bc+ca )

19. Find the value of determinant  | 13 + 3 2 5 5 15 + 26 5 10 3+ 65 15 5 |

20. Prove that $|\begin{array}{ccc}x& 1& 1\\ \alpha & x& 1\\ \alpha & \beta & 1\end{array}|=\left(x-\alpha \right)\left(x-\beta \right)$

21. Prove that  | b+c a a b c+a b c c a+b |=4abc

22. Solve the following system of homogeneous equations : $2x−4y+3z=0 x+y−2z=0 2x+3y+z=0$ .

23. For what value of t, will the system $tx+3y-z=1,\text{\hspace{0.17em}}x+2y+z=2,\text{\hspace{0.17em}}-tx+y+2z=-1$ fail to have a unique solution? Will it have any solutions for this value of t?

24. If are not all zero such that $\begin{array}{l}ax+y+z=0\\ x+by+z=0\\ x+y+cz=0\end{array}$ Then prove that $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=1$ .

25. If $A=\left[\begin{array}{ccc}1& 3& 3\\ 1& 4& 3\\ 1& 3& 4\end{array}\right]$ verify that $A.\text{\hspace{0.17em}}adj\text{\hspace{0.17em}}A=\text{\hspace{0.17em}}|A|\text{\hspace{0.17em}}.\text{\hspace{0.17em}}I$ and find ${A}^{-1}$ .

26. Show that the matrix $A=\left[\begin{array}{ccc}1& 0& -2\\ -2& -1& 2\\ 3& 4& 1\end{array}\right]$ satisfies the equation ${A}^{3}-{A}^{2}-3A-{I}_{3}=O$ Hence find ${A}^{-1}$ If $A=\left[\begin{array}{cc}4& 3\\ 2& 5\end{array}\right]$ ., find  x and y such that  A 2 xA+yI=O.  Hence evaluate A 1 .

27. $\begin{array}{}\end{array}$Using matrix method, solve the following system of equations : 2x3y=1;x+3z=11;x+2y+z=7.

28. Show that the system of equations $x+2y=1\text{\hspace{0.17em}};\text{\hspace{0.17em}}3x+y=4$ is consistent and has a unique solution.

29. Show that the following system of equations is consistent : $\begin{array}{l}x-y+z=3\\ 2x+y-z=2\\ -x-2y+2z=1.\end{array}$ Also , find the solution.

30. A mixture is to be made of three foods $A,B,C$ . The three foods $A,B,C$ contain nutrients $P,Q,R$ as shown below :

31. Solve the equation using matrix method $2x+3y+3z=5 x−2y+z=−4 3x−y−2z=3$ .

32. State the condition under which the following system of linear equations have a unique solutions.  x+2y2z+5=0 x+3y+4=0 2y+z4=0

33. Use Product $\left[\begin{array}{ccc}1& -1& 2\\ 0& 2& -3\\ 3& -2& 4\end{array}\right]\left[\begin{array}{ccc}-2& 0& 1\\ 9& 2& -3\\ 6& 1& -2\end{array}\right]$ to solve the system of equations.  xy+2z=1 2y3z=1 3x2y+4z=2

34. Suppose the demand curve for automobile over some time period can we written as ${x}_{1}=15000-0.2{x}_{2}\text{\hspace{0.17em}}where\text{\hspace{0.17em}}{x}_{1}$ is the price of an automobile and ${x}_{2}$ is the corresponding quantity. Suppose that the supply curve is ${x}_{1}=600+0.4{x}_{2}$ . Use matrix theory to obtain ${x}_{1\text{\hspace{0.17em}},}\text{\hspace{0.17em}}{x}_{2}$

35. Two schools A and B decided to award prizes to their students for their values : honesty (x), punctuality (y) and obedience (z). School A decided to award a total of ₹11,000 for these three values to 5,4 and 3 students, respectively , while school B decided to award ₹10,700 for these three values to 4, 3and 5 students, respectively. If all the three prizes together amount to ₹2,700 then

i) Represent the above situation by a matrix equation and form linear equations by using matrix multiplication.
ii) Is it possible to solve the system of equations so obtained using matrices?
iii) Which value you prefer to be rewarded most and why?

36. 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 B 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?

Matrices and Determinants Important Questions in PDF