4.3 EXERCISE

Short Answer (S.A.)

Using the properties of determinants in Exercises 1 to 6, evaluate:

1. $ \begin{vmatrix} x^{2}-x+1 & x-1 \\ x+1 & x+1\end{vmatrix} $

2. $ \begin{vmatrix} a+x & y & z \\ x & a+y & z \\ x & y & a+z\end{vmatrix} $

3. $ \begin{vmatrix} 0 & x y^{2} & x z^{2} \\ x^{2} y & 0 & y z^{2} \\ x^{2} z & z y^{2} & 0\end{vmatrix} $

4. $ \begin{vmatrix} 3 x & -x+y & -x+z \\ x-y & 3 y & z-y \\ x-z & y-z & 3 z\end{vmatrix} $

5. $ \begin{vmatrix} x+4 & x & x \\ x & x+4 & x \\ x & x & x+4\end{vmatrix} $

6. $ \begin{vmatrix} a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b\end{vmatrix} $

Using the proprties of determinants in Exercises 7 to 9, prove that:

7. $ \begin{vmatrix} y^{2} z^{2} & y z & y+z \\ z^{2} x^{2} & z x & z+x \\ x^{2} y^{2} & x y & x+y\end{vmatrix} =0$

8. $ \begin{vmatrix} y+z & z & y \\ z & z+x & x \\ y & x & x+y\end{vmatrix} =4 x y z$

9. $ \begin{vmatrix} a^{2}+2 a & 2 a+1 & 1 \\ 2 a+1 & a+2 & 1 \\ 3 & 3 & 1\end{vmatrix} =(a-1)^{3}$

10. If $A+B+C=0$, then prove that $ \begin{vmatrix} 1 & \cos C & \cos B \\ \cos C & 1 & \cos A \\ \cos B & \cos A & 1\end{vmatrix} =0$

11. If the co-ordinates of the vertices of an equilateral triangle with sides of length ’ $a$ ’ are $\left(x_1, y_1\right),\left(x_2, y_2\right),\left(x_3, y_3\right)$, then $\left|\begin{array}{lll}x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1\end{array}\right|^2=\frac{3 a^4}{4}$.

12. Find the value of $\theta$ satisfying $\left[\begin{array}{ccc}1 & 1 & \sin 3 \theta \\ -4 & 3 & \cos 2 \theta \\ 7 & -7 & -2\end{array}\right]=0$.

13. If $\left[\begin{array}{ccc}4-x & 4+x & 4+x \\ 4+x & 4-x & 4+x \\ 4+x & 4+x & 4-x\end{array}\right]=0$, then find values of $x$.

14. If $a_1, a_2, a_3, \ldots, a_r$ are in G.P., then prove that the determinant $ \begin{vmatrix} a _{r+1} & a _{r+5} & a _{r+9} \\ a _{r+7} & a _{r+11} & a _{r+15} \\ a _{r+11} & a _{r+17} & a _{r+21}\end{vmatrix} $ is independent of $r$.

15. 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 $a$.

16. Show that the $\triangle ABC$ is an isosceles triangle if the determinant

$ \Delta=\begin{bmatrix} 1 & 1 & 1 \\ 1+\cos A & 1+\cos B & 1+\cos C \\ \cos ^{2} A+\cos A & \cos ^{2} B+\cos B & \cos ^{2} C+\cos C \end{bmatrix} =0 . $

17. Find $A^{-1}$ if $A=\begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{bmatrix} $ and show that $A^{-1}=\frac{A^{2}-3 I}{2}$.

Long Answer (L.A.)

18. If $A=\left[\begin{array}{ccc}1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1\end{array}\right]$, find $A^{-1}$.

Using $\mathrm{A}^{-1}$, solve the system of linear equations $x-2 y=10,2 x-y-z=8,-2 y+z=7$.

19. Using matrix method, solve the system of equations $3 x+2 y-2 z=3, x+2 y+3 z=6,2 x-y+z=2$.

20. Given $\mathrm{A}=\left[\begin{array}{ccc}2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5\end{array}\right], \mathrm{B}=\left[\begin{array}{ccc}1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{array}\right]$, find $\mathrm{BA}$ and use this to solve the system of equations $y+2 z=7, x-y=3,2 x+3 y+4 z=17$.

21. If $a+b+c \neq 0$ and $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=0$, then prove that $a=b=c$.

22. Prove that $\left|\begin{array}{lll}b c-a^2 & c a-b^2 & a b-c^2 \\ c a-b^2 & a b-c^2 & b c-a^2 \\ a b-c^2 & b c-a^2 & c a-b^2\end{array}\right|$ is divisible by $a+b+c$ and find the quotient.

23. If $x+y+z=0$, prove that $ \begin{vmatrix} x a & y b & z c \\ y c & z a & x b \\ z b & x c & y a\end{vmatrix} =x y z \begin{vmatrix} a & b & c \\ c & a & b \\ b & c & a\end{vmatrix} $

Objective Type Questions (M.C.Q.)

Choose the correct answer from given four options in each of the Exercises from 24 to 37.

24. If $ \begin{vmatrix} 2 x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix} 6 & -2 \\ 7 & 3\end{vmatrix} $, then value of $x$ is

(A) 3

(B) $\pm 3$

(C) $\pm 6$

(D) 6

25. The value of determinant $ \begin{vmatrix} a-b & b+c & a \\ b-a & c+a & b \\ c-a & a+b & c\end{vmatrix} $

(A) $\quad a^{3}+b^{3}+c^{3}$

(B) $3 b c$

(C) $a^{3}+b^{3}+c^{3}-3 a b c$

(D) none of these

26. The area of a triangle with vertices $(-3,0),(3,0)$ and $(0, k)$ is 9 sq. units. The value of $k$ will be

(A) 9

(B) 3

(C) -9

(D) 6

27. The determinant $ \begin{vmatrix} b^{2}-a b & b-c & b c-a c \\ a b-a^{2} & a-b & b^{2}-a b \\ b c-a c & c-a & a b-a^{2}\end{vmatrix} $ equals

(A) $ a b c(b-c)(c-a)(a-b)$

(B) $(b-c)(c-a)(a-b)$

(C) $(a+b+c)(b-c)(c-a)(a-b)$

(D) None of these

28. The number of distinct real roots of $ \begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{vmatrix} =0$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is

(A) 0

(B) 2

(C) 1

(D) 3

29. If A, B and C are angles of a triangle, then the determinant

$ \begin{vmatrix} -1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1 \end{vmatrix} \text{ is equal to } $

(A) 0

(B) -1

(C) 1

(D) None of these

30. Let $f(t)= \begin{vmatrix} \cos t & t & 1 \\ 2 \sin t & t & 2 t \\ \sin t & t & t\end{vmatrix} $, then $\lim _{t \to 0} \frac{f(t)}{t^{2}}$ is equal to

(A) 0

(B) $ -1$

(C) 2

(D) 3

31. The maximum value of $\Delta= \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+\sin \theta & 1 \\ 1+\cos \theta & 1 & 1\end{vmatrix} $ is $(\theta$ is real number)

(A) $\frac{1}{2}$

(B) $\frac{\sqrt{3}}{2}$

(C) $\sqrt{2}$

(D) $\frac{2 \sqrt{3}}{4}$

32. If $f(x)= \begin{vmatrix} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0\end{vmatrix} $, then

(A) $\quad f(a)=0$

(B) $\quad f(b)=0$

(C) $\quad f(0)=0$

(D) $\quad f(1)=0$

33. If A=$\left[\begin{array}{ccc}2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3\end{array}\right]$ , then $A^{-1}$ exists if

(A) $\quad \lambda=2$

(B) $\quad \lambda \neq 2$

(C) $\quad \lambda \neq-2$

(D) None of these

34. If $A$ and $B$ are invertible matrices, then which of the following is not correct?

(A) $ adj A=|A| \cdot A^{-1}$

(B) $ det(A)^{-1}=[det(A)]^{-1}$

(C) $(AB)^{-1}=B^{-1} A^{-1}$

(D) $ (A+B)^{-1}=B^{-1}+A^{-1}$

35. If $x, y, z$ are all different from zero and $ \begin{vmatrix} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z\end{vmatrix} =0$, then value of $x^{-1}+y^{-1}+z^{-1}$ is

(A) $x y z$

(B) $x^{-1} y^{-1} z^{-1}$

(C) $-x-y-z$

(D) -1

36. The value of the determinant $ \begin{vmatrix} x & x+y & x+2 y \\ x+2 y & x & x+y \\ x+y & x+2 y & x\end{vmatrix} $ is

(A) $9 x^{2}(x+y)$

(B) $9 y^{2}(x+y)$

(C) $3 y^{2}(x+y)$

(D) $7 x^{2}(x+y)$

37. There are two values of $a$ which makes determinant, $\Delta= \begin{vmatrix} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a\end{vmatrix} =86$, then sum of these number is

(A) 4

(B) 5

(C) -4

(D) 9

Fill in the blanks

38. If $A$ is a matrix of order $3 \times 3$, then $|3 A|=$____________

39. If $A$ is invertible matrix of order $3 \times 3$, then $|A^{-1}|$____________

40. If $x, y, z \in R$, then the value of determinant $ \begin{vmatrix} (2^{x}+2^{-x})^{2} & (2^{x}-2^{-x})^{2} & 1 \\ (3^{x}+3^{-x})^{2} & (3^{x}-3^{-x})^{2} & 1 \\ (4^{x}+4^{-x})^{2} & (4^{x}-4^{-x})^{2} & 1\end{vmatrix} $ is equal to____________

41. If $\cos 2 \theta=0$, then $ \begin{vmatrix} 0 & \cos \theta & \sin \theta \\ \cos \theta & \sin \theta & 0 \\ \sin \theta & 0 & \cos \theta\end{vmatrix} ^{2}=$____________

42. If $A$ is a matrix of order $3 \times 3$, then $(A^{2})^{-1}=$____________

43. If $A$ is a matrix of order $3 \times 3$, then number of minors in determinant of $A$ are____________

44. The sum of the products of elements of any row with the co-factors of corresponding elements is equal to____________

45. If $x=-9$ is a root of $ \begin{vmatrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x\end{vmatrix} =0$, then other two roots are____________

46. $ \begin{vmatrix} 0 & x y z & x-z \\ y-x & 0 & y-z \\ z-x & z-y & 0\end{vmatrix} =$____________

47. If $f(x)= \begin{vmatrix} (1+x)^{17} & (1+x)^{19} & (1+x)^{23} \\ (1+x)^{23} & (1+x)^{29} & (1+x)^{34} \\ (1+x)^{41} & (1+x)^{43} & (1+x)^{47}\end{vmatrix} =A+B x+C x^{2}+\ldots$, then $A=$____________

State True or False for the statements of the following Exercises:

48. $(A^{3})^{-1}=(A^{-1})^{3}$, where $A$ is a square matrix and $|A| \neq 0$.

49. $(a A)^{-1}=\frac{1}{a} A^{-1}$, where $a$ is any real number and $A$ is a square matrix.

50. $|A^{-1}| \neq|A|^{-1}$, where $A$ is non-singular matrix.

51. If $A$ and $B$ are matrices of order 3 and $|A|=5,|B|=3$, then $|3 AB|=27 \times 5 \times 3=405$.

52. If the value of a third order determinant is 12 , then the value of the determinant formed by replacing each element by its co-factor will be 144 .

53. $ \begin{vmatrix} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c\end{vmatrix} =0$, where $a, b, c$ are in A.P.

54. $|adj . A|=|A|^{2}$, where $A$ is a square matrix of order two.

55. The determinant $ \begin{vmatrix} \sin A & \cos A & \sin A+\cos B \\ \sin B & \cos A & \sin B+\cos B \\ \sin C & \cos A & \sin C+\cos B\end{vmatrix} $ is equal to zero.

56. If the determinant $ \begin{vmatrix} x+a & p+u & l+f \\ y+b & q+v & m+g \\ z+c & r+w & n+h\end{vmatrix} $ splits into exactly K determinants of order 3, each element of which contains only one term, then the value of $K$ is 8 .

57. Let $\Delta= \begin{vmatrix} a & p & x \\ b & q & y \\ c & r & z\end{vmatrix} =16$, then $\Delta_1= \begin{vmatrix} p+x & a+x & a+p \\ q+y & b+y & b+q \\ r+z & c+z & c+r\end{vmatrix} =32$.

58. The maximum value of $ \begin{vmatrix} 1 & 1 & 1 \\ 1 & (1+\sin \theta) & 1 \\ 1 & 1 & 1+\cos \theta\end{vmatrix} $ is $\frac{1}{2}$.

SOLUTIONS

1. $x^{3}-x^{2}+2$

2. $a^{2}(a+x+y+z)$

3. $2 x^{3} y^{3} z^{3}$

4. $3(x+y+z)(x y+y z+z x)$

5. $16(3 x+4)$

6. $(a+b+c)^{3}$

12. $\theta=n \pi \quad$ or $n \pi+(-1)^{n} \ \big(\frac{\pi}{6}\big)$

13. $x=0,-12$

18. $x=0, y=-5, z=-3$

19. $x=1, y=1, z=1$

20. $x=2, y=-1, z=4$

24. C

25. C

26. B

27. D

28. C

29. A

30. A

31. A

32. C

33. D

34. D

35. D

36. B

37. C

38. $27|A|$

39. $\frac{1}{|A|}$

40. Zero

41. $\frac{1}{2}$

42. $(A^{-1})^{2}$

43. 9

44. Value of the determinant

45. $x=2 y=7$

46. $(y-z)(z-x)(y-x+x y z)$

47. Zero

48. True

49. False

50. False

51. True

52. True

53. True

54. False

55. True

56. True

57. True

58. True