Determinants Assignment PDF

1 4. Determinants 1 MARK 1. Evaluate a  ib c  id. c  id a  ib 2 3 5 2. Find the cofactor of a12 in the following: 6 0 4....

1 4. Determinants 1 MARK 1. Evaluate a  ib c  id. c  id a  ib 2 3 5 2. Find the cofactor of a12 in the following: 6 0 4. 1 5 7 3. Evaluate sin 300 cos 300  sin 600 cos 600. a11 a12 a13 4. Given determinant a 21 a 22 a 23 , Find the value of a11A 21  a12 A 22  a13A 23 Where a 31 c32 a 33 Aij is cofactor of element a ij.  1 2  5. Find the value of p, such that the matrix   is singular.  4 p  2 3   8 3a  6. If A=   ,KA=   find a ,b, c and K. ans=a=-4,b=-10,c=0,K=4 5 0   2b c  7. Given I2. Find |I2|. Also find |3I2|. 8. Find the value of x, such that the points (0, 2), (x, 1) and (3, 1) are collinear. 9. If for matrix A, |A| = 3, find |5A|, Where matrix A is of order 2x2 x y 10. If points (2,0), (0, 5) and (x, y) are collinear, then show that   1. 2 5 11. If A=diag.[3, −5,7] B=diag.[−1,2,4] find 3A+4B. ans.= diag.[5, −7,37] 12. Give an example of two nonzero matrices A and B of order 2×2 such that AB=O. 13. Write the equation on using elementary row operation 𝑅 → 𝑅 + 2𝑅 in the given  5 7   1 2  1 1   19 25   7 6  1 1  matrix equation     ans=      7 9   3 2  2 3   7 9   3 2  2 3  14. A is a non-singular matrix of order 3 and |A|= - 4. Find |adj A|. 15. Given a square matix A of order 3 × 3, such that |A| = 12, find the value of |A. adj A|. 2x  5 3 16. If =0. Find x 5x  2 9 k 2 17. For what value of k, the matrix.  has no inverse ? 3 4 Manju Bala 8383001236 Pawan Gupta 9999102886  2 2 3 4 18. Write the value of the determinant 5 6 8. 6x 9x 12x 19. Find the cofactor of the element of first row and second column (a12 ) in the following 2 3 5 determinant 6 0 4 1 5 7 20. If A is a square matrix of order 3 and | 3A | k | A |, then write the value of k.  1 2 21. If A=   ,then find K if |2𝐴|=K|𝐴| ans=4  4 2 1 2 3 2 3 4 3 5 7 22. If 4 6 8  4 6 8  x y z.find x and u. ans= x=4 ,u=5 5 1 3 5 1 3 u v w 23. What positive value of x makes the following pair of determinant equal? 2x 3 16 3 , 5 x 5 2  2  1 24. Write the adjoint of the following matrix  . 4 3 4 8 6 2 4 3 25. Evaluate 1 1 2  1 1 2 ans:0 5 1 2 10 2 4 26. What is the value of the following determinant ? 4 a bc 4 b ca 4 c ab 5  x x  1 27. For what value of x, the matrix  is singular?  2 4  28. If A and B are square matrices of same order 3,such that |𝐴|=2 and AB=2I.write the value of |𝐵|. Ans:1  2 5 29. Write A-1 for A =  . 1 3 cos150 sin150 30. Evaluate. sin 750 cos 750 Manju Bala 8383001236 Pawan Gupta 9999102886  3  sin   cos   31. If A=   be a singular matrix, find the value of  when  𝜖[0,2𝜋]  cos   sin    3 5 7 ans: , , , 4 4 4 4 1 2 32. If A=   find |𝐴 |,without actually find 𝐴 ans 0 3 2 3  33. If A   -1  , write A in terms of A.  5  2  x x 3 4 34. If  , write the positive value of x. 1 x 1 2 35. Let A be a square matrix of order 3 × 3. Write the value of |2A|, where |A|=4. 36. Let A be a square matrix of order 3 × 3 and |𝑎𝑑𝑗 𝐴|=361,find |𝐴|. Ans:±19 102 18 36 37. Write the value of the determinant 1 3 4. 17 3 6  2 1 3   38. Determine the invertibility of the matrix  3 4 1  ans. Not invertible.  3 2 5   39. Using determinants,for what value of k,the following equations have a unique solution? 2x-7y=21 5x+ky=7 ans. k≠ − x 1 x 1 4 1 40. If  then write the value of x. x 3 x 2 1 3 1 2 2 x 41. Find x, if . 4 8 x 4 4 1 0 42. Give 2 1 4 , find (i) minor of an element a23 (ii) cofactor of an element a23. 1 0 3 43. Area of a triangle with vertices (k,0), (1, 1) and (0,3) is 5 square units, Find the values (s) of k. a11 a12 a13 44. Given determinant a 21 a 22 a 23 , Find the value of a11A 21  a12 A 22  a13A 23 Where a 31 c32 a 33 Aij is cofactor of element a ij.  1 2  45. Find the value of p, such that the matrix   is singular.  4 p 46. Given I2. Find |I2|. Also find |3I2|. Manju Bala 8383001236 Pawan Gupta 9999102886  4  3 3  47. If the matrix A=   and 𝐴 =𝜆A,then evaluate 𝜆. Ans: 𝜆=6  3 3  48. Find the value of x, such that the points (0, 2), (1, x) and (3, 1) are collinear. 49. If for matrix A, |A| = 3, find |5A|, Where matrix A is of order 2x2. x y 50. If points (2, 0), (0, 5) and (x, y) are collinear, then show that   1. 2 5 51. A is a non-singular matrix of order 3 and |A|= - 4. Find |adj A|. 52. Let A= 𝑎 is a matrix of order 2x2 such that |A| =-15 and 𝑐 represents the cofactor of 𝑎 ,then find 𝑎 𝑐 + 𝑎 𝑐. Ans :-15 53. Given a square matrix A of order 3 × 3, such that |A| = 12, find the value of |A. adj A|. 1 a bc 54. Using properties of determinants, prove that 1 b c  a  0. 1 c ab cosec2  cot 2  1 55. Without expanding. show that cot  cosec  1  0 2 2 42 40 2 2 3 x 3 56. Find value of x, if . 4 5 2x 5 57. If |𝐴𝑑𝑗 𝐴|=64 where A is a 3× 3 matrix find |𝐴|𝑎𝑛𝑑 |2𝐴| ANS. |𝐴| =±8,. |2𝐴| =±64 58. If A and are the matrices of order 3 and |𝐴| =5,and |𝐵| = 3,then find |3𝐴𝐵| ans.405 59. Find equation of line joining (1, 2) and (3, 6) using determinants. 3 y 3 2 60. Let  , find the possible values of x, y  N. Also find the values, if x = y. x 1 4 1 61. Show that the point A (a, b + c), B (b, c + a) and C (c, a + b) are collinear. 62. If the value of a third order determinant is 12,then find the value of the determinant formed by replacing each element by its cofactor. Ans 144 x2  x  1 x  1 63. Evaluate the determinant ans x 3  x 2  2 x 1 x 1 x y yz zx 64. Write the value of the determinant z x y ans = 0 3 3 3  4  3k 3  65. Show that the matrix   is never singular matrix,for any k.  1  2k 2   cos  sin   66. If A=   ,then for any natural number n,find the value of Det(𝐴 ). Ans=1for n∈   sin  cos   𝑁 Manju Bala 8383001236 Pawan Gupta 9999102886  5 2x 5 6 5 67. If  , then find x. 8 x 8 3 1 x x2 1 1 1 68. If   1 y y 2 , 1  yz zx xy , then prove that   1  0. 1 z z2 x y z x p q 69. Show that   p x q  ( x  p) ( x 2  px  2q2 ) q q x 0 ba ca 70. If   a  b 0 c  b , then show that  is equal to zero. ac bc 0 71. Prove that (A 1 ) /  (A / )1 , where A is an invertible matrix. Choose the correct answer from the given four options in each of the Examples 10 and 11. Ax x2 1 A B C 2 72. Let   By y 1 and 1  x y z , then Cz z 2 1 zy zx xy (A) 1   (B)   1 (C)   1  0 (D) None of these cos x  sin x 1 73. If x , y  R , then the determinant   sin x cos x 1 lies in the interval cos ( x  y )  sin ( x  y ) 0 (A)   2 , 2  (B)  1, 1 (C)   2 , 1  (D)  1,  2  Fill in the blanks in each of the Examples 12 to 14. sin 2 A cot A 1 2 74. If A, B, C are the angles of a triangle, then   sin B cot B 1  …………….. 2 sin C cot C 1 23  3 5 5 75. The determinant   15  46 5 10 is equal to ……………………. 3  115 15 5 sin 2 23 sin 2 67 cos180 76. The value of the determinant    sin 67  sin 23 cos 2 180  ………………….. 2 2 cos180 sin 2 23 sin 2 67 2x 5 6 2 77. If  , then value of x is 8 x 7 3 (A) 3 (B)  3 (C)  6 (D) 6 Manju Bala 8383001236 Pawan Gupta 9999102886  6  2  3  78. If A   0 2 5  , then A 1 exists if  1 1 3  (A)   2 (B)   2 (C)   2 (D) None of these 79. If A and B are invertible matrices, then which of the following is not correct? (B) d et (A)1   d et (A)  1 (A) adj A | A|  A 1 (C) (AB)1  B1A 1 (D) (A  B)1  B1  A 1 1 2 5 80. There are two values of a which makes determinant,   2 a 1  86, then sum of 0 4 2a these number is (A) 4 (B) 5 (C) –4 (D) 9 Fill in the blanks in each of the Exercises 71 —81. 81. If A is a matrix of order 3 × 3, then |3 A|= ___________. 82. If A is invertible matrix of order 3 × 3, then | A 1 | ___________  2x  2 x   2 x  2 x  2 2 1  3x  3 x   3 x  3 x  2 2 83. If x, y, z  R, then the value of determinant 1 is equal to  4 x  4 x   4 x  4 x  2 2 1 ___________ 2 0 cos  sin  84. If cos 2  0, then cos  sin  0  ___________ sin  0 cos  85. If A is a matrix of order 3 × 3, then (A 2 )1  ___________. 86. If A is a matrix of order 3 × 3, then number of minors in determinant of A are ___________. 87. The sum of the products of elements of any row with the co-factors of corresponding elements is equal to ___________. (1  x )17 (1  x )19 (1  x )23 88. If f ( x )  (1  x )23 (1  x )29 (1  x )34  A  B x  Cx 2  then A = ____________. (1  x )41 (1  x )43 (1  x )47 State True or False for the statements of the following Exercises:  A3    1 3 89.  A 1 , where A is a square matrix and | A|  0. 90. | A 1 | | A| 1 , where A is non-singular matrix. 91. If A and B are matrices of order 3 and |A| = 5, |B| = 3, then | 3 AB|  27  5  3  405. Manju Bala 8383001236 Pawan Gupta 9999102886  7 92. 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. x 1 x  2 xa 93. x 2 x3 x  b  0, , where a, b, c are in A.P. x 3 x 4 x c 94. |adj A| = |𝐴| , where A is a square matrix of order 2 sin A cos A sin A  cos B 95. The determinant sin B cos A sin B  cos B is equal to zero. sin C cos A sin C  cos B x a pu l f 96. If the determinant y  b q  v m  g splits into exactly K determinants of order 3, each zc rw nh element of which contains only one term, then the value of K is 8. a p x px a x a p 97. Let   b q y  16, then 1  q  y by b  q  32. c r z rz cz cr 1 1 1 1 98. The maximum value of 1 (1  sin ) 1 is. 2 1 1 1  cos  1. 𝑎 + 𝑏 − 𝑐 − 𝑑 2. 46 3. 1 4. 0 5. 8 6. a=-4,b=-10,c=0,K=4 7. 9 8. x=3 9. 75 10. 11. diag. 12.  19 25   7 6  1 1  [5, −7,37] 13.      7 9   3 2  2 3  14. 16 15. 1728 16. X=-13 17. K= 18. 0 19. 𝐴 =46 20. K=27 21. K=4 22. X=4,u=5 23. X=±4  3 1 25. 0 24.    4 2  26. 0 27. X=3 28. 1  3 5  29. 𝐴 =    1 2  Manju Bala 8383001236 Pawan Gupta 9999102886  8 30. 0  3 5 7 32. 33. 𝐴 = A 31. , , , 4 4 4 4 34. X=2 35. 32 36. ±19 37. 0 38. Not invertible. 39. k≠ − 40. x=2 41. x=±2√2 42. 43. K= or - 44. 0 45. P=-8 46. 9 47. 𝜆=6 48. x= 49. 75 50. 51. 16 52. -15 53. 1728 54. 55. 56. X=2 57. ±64 58. 59. 60. 61. 62. 144 63. x 3  x 2  2 64. 0 65. 66. 1for n∈ 𝑁 67. X=±3 68. 69. 70. 71. 72. C 73. A 74. 0 75. 0 76. 0 77. C 78. D 79. D 80. C 81. 27|𝐴| 82. 1 83. 0 84. 85. A  1 2 86. 9 87. VALUE OF THE 88. 0 89. True,since DETERMINANT. A  n 1   A 1  n Where n𝜖𝑁 90. FALSE SINCE 91. TRUE 92. TRUE 93. TRUE 1 A1  A 94. false 95. 0 96. true 97. true 98. true 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. Manju Bala 8383001236 Pawan Gupta 9999102886